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abstract: 'We discuss the implications for gravitational wave detectors of a class of modified gravity theories which dispense with the need for dark matter. These models, which are known as [*Dark Matter Emulators*]{}, have the property that weak gravitational waves couple to the metric that would follow from general relativity without dark matter whereas ordinary particles couple to a combination of the metric and other fields which reproduces the result of general relativity with dark matter. We show that there is an appreciable difference in the Shapiro delays of gravitational waves and photons or neutrinos from the same source, with the gravitational waves always arriving first. We compute the expected time lags for GRB 070201, for SN 1987a, and for Sco-X1. We estimate the probable error by taking account of the uncertainty in position, and by using three different dark matter profiles.'
author:
- 'S. Desai'
- 'E. O. Kahya'
- 'R. P. Woodard'
title: Gravity Gets There First with Dark Matter Emulators
---
Introduction {#sec:intro}
============
The direct detection of gravitational waves from astrophysical sources would enable us to open a new window into the universe and get insights which are complementary to electromagnetic astronomy [@Cutler02]. Many ground-based interferometric detectors such as LIGO, VIRGO, GEO600 and TAMA have been online for several years. In October 2007, LIGO completed a long science run to collect one year of coincident data at design sensitivity [@LIGO] and the VIRGO detector also joined this science run in the last five months. During the latest LIGO science run, the sensitivity of the 4 km Hanford and Livingston LIGO detectors to detect binary neutron-star coalescence with mass 1.4 $M_{\odot}$ with signal to noise ratio greater than 8 (after averaging over all orientations and sky positions) was about 15 Mpc [@LIGO]. Analysis of the latest LIGO and VIRGO data for gravitational wave (GW) searches from a wide variety of sources is in progress [@Papa].
An important science goal pursued is the search for impulsive transient GW signals from sources with electromagnetic and/or neutrino counterparts. Some examples of such sources include core-collapse supernovae, gamma-ray bursts (GRBs), soft gamma-ray repeaters (SGRs), pulsar glitches, low mass X-ray binaries, blazar flares, optical transients, etc [@extrig08]. These “triggered” searches allow us to get better sensitivity for a given false alarm rate as compared to an all-sky search at all times and to design custom-made analysis algorithms taking into account our knowledge of the source astrophysics. Conversely, there has been a proposal to look for optical and infrared counterparts at the time of coincident GW burst candidates [@Kanner08]. An overview and benefits of such triggered searches carried out by the current interferometric gravitational wave detectors are reviewed in Ref. [@extrig08]. There have been proposals to determine neutrino mass using simultaneous neutrino and GW observations from core-collapse supernova [@Arnaud01]. Similar triggered GW searches will also be important for the future LISA experiment [@Kocsis].
In all present and past triggered searches for gravitational waves, the analysis is done by looking at the data from GW detectors within a narrow time window (of about hundreds of seconds) around the time of the electromagnetic trigger. With this assumption, one can detect gravitational waves only if the propagation time of photons/neutrinos is the same as that of gravitational waves. In general relativity, photons, neutrinos and gravitational waves propagate on the same null geodesics. Hence the total time of propagation is the light travel time delay plus the Shapiro time-delay due to intervening matter [@Shapiro64]. For electromagnetic waves, Shapiro delay has been detected in a wide variety of systems such as radar ranging to Venus, Doppler tracking of Cassini spacecraft and in binary pulsars [@Will06]. From the relative arrival times of photons and neutrinos from SN 1987a, we also know that the Shapiro time-delay for neutrinos is the same as that for photons to within 0.5% [@Longo88; @Krauss88].
The conventional view is that general relativity describes gravity on cosmic scales. If this is so, the gravitation of stars and gas is not sufficient to account for the velocity dispersions in clusters [@Zwi], or for the rotation curves of spiral galaxies [@RTF; @RFT1; @RFT2], or for the weak lensing in galactic clusters [@TVW; @FKSW; @SKFBW; @CLKHG; @Mellier; @WTKAB]. Big Bang Nucleosynthesis severely limits the extent to which the deficit can be made up of unseen but ordinary matter [@Olive]; the remainder must consist of an exotic, nonrelativistic substance which has never been detected except gravitationally. This [*dark matter*]{} must vastly predominate over ordinary matter. For example, only about one fifth of our galaxy’s mass is made up of normal matter, with the rest being composed of dark matter [@Trimble87]. Thus the dominant contribution to the Shapiro delay for photons from GRBs and other sources is due to the gravitational potential of the intervening dark matter.
None of the proposed dark matter candidates have been detected, either directly in a laboratory experiment or indirectly through their annihilation products [@Bertone04; @ADMX; @Hooper08; @CDMS]. This prompts the suspicion that perhaps it is gravity which must be modified, rather than the universe’s inventory of nonrelativistic matter. Of course that would invalidate the assumption which is the basis for all current and proposed GW searches from sources seen in photons and neutrinos [@extrig08; @Arnaud01]. A previous study [@Kahya07; @Kahya08] has considered the consequences for gravitational wave detection of a certain class of modified gravity theories known as [*Dark Matter Emulators*]{}. In this paper we correct a mistake in the original work [@Kahya07] that led to the wrong sign for the effect, and we work out explicit results for three interesting sources.
Section II defines and motivates Dark Matter Emulators. In Section III we review three popular dark matter profiles which these models are designed to obviate. Section IV computes the expected time lag between the arrival of the pulse of gravitational waves from some cosmic event and its optical or neutrino counterpart. In Section V we give explicit results for three sources of interest. Section VI gives a very brief discussion of other alternate gravity models, and our conclusions comprise Section VII.
Dark Matter Emulators
=====================
Certain regularities in cosmic structures suggest modified gravity. One of these is the Tully-Fisher relation, which states that the luminosity of a spiral galaxy is proportional to the fourth power of the peak velocity in its rotation curve [@TF]. If luminous matter is insignificant compared to dark matter, why should such a relation exist? Another regularity is Milgrom’s Law, which states that the need for dark matter occurs at gravitational accelerations of $a_0 \simeq 10^{-10}~{\rm m/s}^2$ [@KT]. A third regularity is that $a_0$ also seems to give the internal accelerations of pressure-supported objects ranging over six orders of magnitude in size — from massive molecular clouds within our own galaxy to X-ray clusters of galaxies [@SM].
A modification of Newtonian gravity which explains these regularities was proposed by Milgrom in 1983 [@Milg]. His model, Modified Newtonian Dynamics (MOND), was soon given a Lagrangian formulation in which conservation of energy, 3-momentum and angular momentum are manifest [@BM]. However, there was for years no successful relativistic generalization which could be employed to study cosmological evolution. Even in the context of static, spherically symmetric geometries, $$ds^2 \equiv -B(r) c^2 dt^2 + A(r) dr^2 + r^2 d\Omega^2 \; , \label{ds2}$$ the early formulation of MOND fixed only $B(r)$, not $A(r)$. It was therefore incapable of making definitive predictions about gravitational lensing.
A relativistic extension of MOND has recently been proposed by Bekenstein [@Bek]. This model is known as TeVeS for “Tensor-Vector-Scalar.” In addition to reproducing the MOND force law at low accelerations, TeVeS has acceptable post Newtonian parameters, and it gives a plausible amount of gravitational lensing [@Bek]. When TeVeS is used in place of general relativity + dark matter to study cosmological evolution, the results are in better agreement with data than many thought possible [@Skordis; @Skordis2; @Dodelson; @Bourliot; @ZFS]. The model does have problems with stability [@Clayton; @CWW]. The Bullet Cluster is sometimes cited as a fatal blow for the model [@SMC] but opinion on this differs [@AFZ; @Stacy; @FFB], and this system in any case poses problems for dark matter [@BrM2; @Stacy].
What concerns us here is the curious property of TeVeS that small amplitude gravitational waves are governed, as in general relativity, by the metric $g_{\mu\nu}$, whereas matter couples to a “disformally transformed” metric which involves the vector and scalar fields, $$\widetilde{g}_{\mu\nu} = e^{-2\phi} (g_{\mu\nu} + A_{\mu} A_{\nu})
- e^{2\phi} A_{\mu} A_{\nu} \; .$$ The Scalar-Vector-Tensor gravity (SVTG) theory proposed by Moffat also has different metrics for matter and small amplitude gravitational waves [@JWM; @BrM]. The appearance of this feature in two very different models is the result of trying to reconcile solar system tests with modified gravity at ultra-low accelerations. Solar system tests strongly predispose the Lagrangian to possess an Einstein-Hilbert term [@Will; @BEF]. On the other hand, failed attempts to generalize MOND [@SW1] have led to a theorem that one cannot get sufficient weak lensing from a stable, covariant and purely metric theory which reproduces the Tully-Fisher relation without dark matter [@SW2]. Hence the MOND force must be carried by some other field, and it is a combination of this other field and the metric which determines the geodesics for ordinary matter. However, the dynamics of small amplitude gravitational waves are still set by the linearized Einstein equation. This simple observation makes for a sensitive and generic test.
We define a [*Dark Matter Emulator*]{} as any modified gravity theory for which:
1. [Ordinary matter couples to the metric $\widetilde{g}_{\mu\nu}$ that would be produced by general relativity + dark matter; and]{}
2. [Small amplitude gravitational waves couple to the metric $g_{\mu\nu}$ produced by general relativity without dark matter.]{}
Now consider a cosmic event such as a supernova which emits simultaneous pulses of gravitational waves and either neutrinos or photons. If physics is described by a dark matter emulator then the pulse of gravitational waves will reach us on a lightlike geodesic of $g_{\mu\nu}$, whereas neutrinos and photons travel along a lightlike geodesic of $\widetilde{g}_{\mu\nu}$. If significant propagation occurs over regions that would be dark matter dominated in general relativity then there will be a measurable lag between arrival times.
Currently the only observational constraint on the speed $v_{\rm g}$ of gravity relative to that of ordinary matter $v_{\rm m}$ derives from the consequences of gravitational Cherenkov radiation from particles moving faster than gravity [@Caves; @Moore01]. From observations of the highest energy cosmic rays Moore and Nelson infer the bound, $v_{\rm m}
- v_{\rm g} < 2 \times 10^{-15} c$ [@Moore01]. Although the original study of Dark Matter Emulators [@Kahya07] in fact violated this bound, that was the result of incorrectly choosing the dimensional constant in a certain logarithm. In the next section we show that the speed of gravity is always greater than that of light for Dark Matter Emulators. A discussion of the Shapiro delay calculation in some other alternate gravity theories can be found in Refs. [@Carlip04; @Asada07].
Three Dark Matter Profiles
==========================
We shall specialize to static, spherically symmetric distributions of dark matter, consistent with the invariant element (\[ds2\]). It is well to bear in mind that hierarchical structure formation will not necessarily result in spherically symmetric distributions [@WNEF]. There is even evidence that the dark matter halo conjectured to surround our own galaxy is not spherical [@LJM].
For a pressureless, static, spherically symmetric system the Einstein equations take the form, $$\begin{aligned}
\frac{B}{A} \Biggl[ \frac{A'}{r A} + \Bigl(\frac{A-1}{r^2}\Bigr)\Biggr]
& = & \frac{8\pi G}{c^2} \, \rho \; , \label{Eqn1} \\
\frac{B'}{r B} - \Bigl(\frac{A-1}{r^2}\Bigr) & = & 0 \; . \label{Eqn2}\end{aligned}$$ If the potential $B(r)$ goes to a constant at infinity we can choose the time units so that equation (\[Eqn2\]) has the exact solution, $$B(r) = \exp\Biggl[-\int_r^{\infty} dr' \, \Bigl(\frac{A(r') - 1}{r'}\Bigr)
\Biggr] \; .$$ However, our study requires only small corrections to $A(r)$ and $B(r)$, $$A(r) \equiv 1 + \Delta A(r) \qquad , \qquad B(r) \equiv 1 + \Delta B(r) \; .$$ This not only simplifies (\[Eqn1\]-\[Eqn2\]), it also means we can dispense with the contribution of ordinary matter to the mass density $\rho(r)$. The reason is that we are computing the difference in propagation times along null geodesics between the same points in two different geometries. The geometry felt by gravitational waves is sourced only by ordinary matter, while the geometry felt by photons and neutrinos is sourced by the sum of ordinary matter and dark matter. At first order in the mass density, the effect of ordinary matter cancels out when computing the difference in propagation times between the two geometries! We will henceforth consider $\rho(r)$ to be the density of dark matter.
The potentials $\Delta A(r)$ and $\Delta B(r)$ can be given simple expressions in terms of the mass function, $$M(r) \equiv 4 \pi \int_0^r dr' \, \rho(r') \; .$$ The linearized solution of (\[Eqn1\]) is, $$\Delta A(r) = \frac{8 \pi G}{c^2 r} \int_0^r dr' \, r^{\prime 2} \rho(r') =
\frac{2 G}{c^2} \, \frac{M(r)}{r} \; . \label{DA}$$ Note that $\Delta A(r)$ is positive semi-definite. From (\[DA\]) we find $\Delta B(r)$, $$\Delta B(r) = -\!\!\int_r^{\infty} \!\!\!\!\! dr' \, \frac{\Delta A(r')}{r'} =
-\Delta A(r) - \frac{2 G}{c^2} \!\!\!\int_r^{\infty} \!\!\!\!\! dr' \,
\frac{M(r')}{r'} \; . \label{DBfromDA}$$ Note that $\Delta B(r)$ is negative semi-definite and in fact less than or equal to $-\Delta A(r)$. This guarantees that gravitational waves travel faster than photons or neutrinos, so there is no problem with the bound of Moore and Nelson [@Moore01].
Our study requires the dark matter density functions for our own galaxy and (for the most distant source) for the Andromeda galaxy. We took these in the form of fits to three popular density profiles whose analytic forms are presented at the end of this section. Given the current rough quality of the observational data, a Dark Matter Emulator that reproduced the potentials $\Delta A(r)$ and $\Delta B(r)$ for any of these profiles would be judged successful. One can therefore regard the slightly different time delays that result as one measure of the theoretical uncertainty. The fits for our own galaxy appear in Table \[MWPs\] and were taken from the study by Ascasibar, Jean, Boehm and Knödlseder of the positron annihilation line from the galactic center [@Ascasibar]. The fits for Andromeda appear in Table \[M31Ps\] and were done by Tempel, Tam and Tenjes [@Tempel].
Isothermal Halo Profile
-----------------------
We shall use a variant of the Isothermal Halo Profile in which the density vanishes after a cutoff radius $r_c$ [@King; @Einasto], $$\rho(r) = \Biggl[ \frac{\rho_0}{1 + (\frac{r}{r_0})^2} -
\frac{\rho_0}{1 + (\frac{r_c}{r_0})^2} \Biggr] \theta(r_c - r) \; .$$ Such a cutoff is inevitable, even in MOND, owing to the presence of other galaxies. Of course it is also necessary to make the potential $\Delta B(r)$ vanish at infinity.
For $r < r_c$ the mass function and potentials are, $$\begin{aligned}
M(r) & \!\!\!=\!\!\! & 4\pi \rho_0 r_0^3 \Biggl\{\frac{r}{r_0}
\!-\! \tan^{-1}\Bigl(\frac{r}{r_0}\Bigr) \!-\!\frac{r^3}{3 r_0 (r_0^2 + r_c^2)}
\Biggr\} , \qquad \\
\Delta A(r) & \!\!\! =\!\!\! & \frac{8\pi G \rho_0 r_0^2}{c^2}
\Biggl\{ \!\! 1 \!-\! \frac{r_0}{r} \tan^{-1}\!\Bigl(\frac{r}{r_0}\Bigr) \!-\!
\frac{r^2}{3 (r_0^2 + r_c^2)} \!\Biggr\} , \qquad \\
\Delta B(r) & \!\!\!= \!\!\!& \frac{8\pi G \rho_0 r_0^2}{c^2}
\Biggl\{ -1 \!+\! \frac{r_0}{r} \tan^{-1}\Bigl(\frac{r}{r_0}\Bigr) \nonumber \\
& & \hspace{1.7cm} + \frac{3 r_c^2 - r^2}{6 (r_0^2 + r_c^2)} - \frac12
\ln\Bigl[\frac{r_c^2 + r_0^2}{r^2 + r_0^2}\Bigr] \Biggr\} . \qquad\end{aligned}$$ For $r > r_c$ the mass is constant and the (equal and opposite) potentials fall off like $1/r$, $$\begin{aligned}
M(r) & \!\!\!=\!\!\! & 4\pi \rho_0 r_0^3 \Biggl\{\frac{r_c}{r_0}
\!-\! \tan^{-1}\Bigl(\frac{r_c}{r_0}\Bigr) \!-\!\frac{r_c^3}{3 r_0 (r_0^2
+ r_c^2)} \Biggr\} , \qquad \\
\Delta A(r) & \!\!\! =\!\!\! & \frac{8\pi G \rho_0 r_0^2}{c^2} \Biggl\{\!\!
\frac{r_c (2 r_c^2 + 3 r_0^2)}{3 r (r_c^2 + r_0^2)} - \frac{r_0}{r}
\tan^{-1}\!\Bigl(\frac{r_c}{r_0}\Bigr) \!\!\Biggr\} , \qquad \\
\Delta B(r) & \!\!\! =\!\!\! & \frac{8\pi G \rho_0 r_0^2}{c^2} \Biggl\{\! -
\frac{r_c (2 r_c^2 \!+\! 3 r_0^2)}{3 r (r_c^2 \!+\! r_0^2)} \!+\! \frac{r_0}{r}
\tan^{-1}\!\Bigl(\frac{r_c}{r_0}\Bigr) \!\!\Biggr\} . \qquad\end{aligned}$$
It should be noted that Ascasibar, Jean, Boehm and Knödlseder used a simpler version of the isothermal profile without a cutoff radius $r_c$. This would cause the potential $\Delta B(r)$ to eventually become positive, which violates the bound of Moore and Nelson [@Moore01]. It also doesn’t make any sense when one considers the effect of other galaxies. Because the isothermal profile is the most closely related to MOND we considered it important to include results for this profile, so we used the values of $\rho_0$ and $r_0$ given by Ascasibar, Jean, Boehm and Knödlseder [@Ascasibar], along with $r_c = 219~{\rm kpc}$. This choice for $r_c$ causes the ratio of the total masses of the Milky Way and Andromeda galaxies for the isothermal profile to agree with that of the NFW profile considered in the next subsection.
NFW Profile
-----------
The NFW profile was the result of studying the equilibrium density profiles of dark matter halos in numerical simulations of structure formation [@NFW], $$\rho(r) = \frac{\rho_0}{\frac{r}{r_0} [1 + \frac{r}{r_0}]^2} \; .$$ The associated mass function and potentials are, $$\begin{aligned}
M(r) & = & 4\pi \rho_0 r_0^3 \times \Biggl\{ \ln\Bigl[1 + \frac{r}{r_0}\Bigr]
- \frac{r}{r_0 + r}\Biggr\} \; , \\
\Delta A(r) & = & \frac{8\pi G \rho_0 r_0^2}{c^2} \times \Biggl\{ \frac{r_0}{r}
\ln\Bigl[1 + \frac{r}{r_0}\Bigr] - \frac{r_0}{r_0 + r} \Biggr\} \; , \qquad \\
\Delta B(r) & = & \frac{8\pi G \rho_0 r_0^2}{c^2} \times -\frac{r_0}{r}
\ln\Bigl[1 + \frac{r}{r_0}\Bigr] \; .\end{aligned}$$
Moore Profile
-------------
A later effort along the same lines showed a better fit to a density function which is more sharply peaked at the center [@Moore], $$\rho(r) = \frac{\rho_0}{(\frac{r}{r_0})^{\frac32} [1 +
\frac{r}{r_0}]^{\frac32}} \; .$$ The associated mass function and potentials are, $$\begin{aligned}
M(r) & = & 4\pi \rho_0 r_0^3 \times \frac23 \ln\Biggl[1 +
\Bigl(\frac{r}{r_0}\Bigr)^{\frac32}\Biggr] \; , \\
\Delta A(r) & = & \frac{8\pi G \rho_0 r_0^2}{c^2} \times \frac23 \frac{r_0}{r}
\ln\Biggl[1 + \Bigl(\frac{r}{r_0}\Bigr)^{\frac32}\Biggr] \; , \\
\Delta B(r) & = & \frac{8\pi G \rho_0 r_0^2}{c^2} \times \Biggl\{
-\frac23 \frac{r_0}{r} \ln\Biggl[1 + \Bigl(\frac{r}{r_0}\Bigr)^{\frac32}
\Biggr] \nonumber \\
& & + \ln\Biggl[1 + \Bigl(\frac{r_0}{r}\Bigr)^{\frac12}\Biggr]
-\frac13 \ln\Biggl[1 + \Bigl(\frac{r_0}{r}\Bigr)^{\frac32}\Biggr] \nonumber \\
& & \hspace{1.8cm} - \frac2{\sqrt{3}} \tan^{-1}\Biggl[\frac{\sqrt{3 r_0}}{2
\sqrt{r} - \sqrt{r_0}} \Biggr] \Biggr\} \; . \qquad\end{aligned}$$
The Shapiro Delay for Given $M(r)$
==================================
It is more convenient to convert the spatial coordinates from spherical $(r,\theta,\phi)$ to Cartesian $x^i$, $$\vec{x} \equiv r \Bigl( \sin\theta \cos\phi, \sin\theta \sin\phi,
\cos\theta \Bigr) \equiv r \hat{r} \; .$$ The invariant element (\[ds2\]) has a simple expression in terms of these coordinates, $$ds^2 = -c^2 dt^2 + d\vec{x} \cdot d\vec{x} -\Delta B \, c^2 dt^2 +
\Delta A \, (\hat{r} \cdot d\vec{x})^2 \; . \label{line1}$$ Before moving on we digress to note that specializing to $ds^2 = 0$ and identifying the velocity of photons and neutrinos as $\vec{v}_{\rm m} =
d\vec{x}/dt$ results in an equation for the speed of effectively massless, ordinary matter, $$0 = -(1 + \Delta B) c^2 + \vec{v}_{\rm m} \cdot \vec{v}_{\rm m} +
\Delta A (\hat{r} \cdot \vec{v}_{\rm m})^2 \; . \label{veq}$$ Now recall that we are ignoring the role of ordinary matter in both the metrics of gravity (where it is the only part of the mass density) and of ordinary matter. It follows that the speed of gravity is $v_{\rm g}$ is $c$. Treating to first order in the potentials and assuming $\Delta A(r)
\geq 0$ and $\Delta B(r) \leq 0$, we see that $v_{\rm m} - v_{\rm g} \leq 0$.
We can express the invariant element (\[line1\]) as the flat space contribution plus a perturbation, $$ds^2 \equiv (\eta_{\mu\nu} + h_{\mu\nu}) dx^{\mu} dx^{\nu} \; . \label{line2}$$ Comparing (\[line1\]) and (\[line2\]) allows us to read off the $3+1$ decomposition of the graviton field $h_{\mu\nu}$, $$h_{00} = -\Delta B \quad , \quad h_{0i} = 0 \quad {\rm and} \quad h_{ij} =
\Delta A \, \hat{r}^i \hat{r}^j \; .$$ One advantage of Cartesian coordinates is that the affine connection vanishes for the flat background. It is easy to give the first correction, $$\begin{aligned}
\Gamma^{\mu}_{~\rho\sigma} & = & \Delta \Gamma^{\mu}_{~\rho\sigma}
+ O(h^2) \; , \\
\Delta \Gamma^{\mu}_{~\rho\sigma} & = & \frac12 \eta^{\mu\nu} \Bigl(
h_{\nu \rho , \sigma} + h_{\sigma \nu , \rho} - h_{\rho \sigma , \nu}
\Bigr) \; .\end{aligned}$$
We need the null geodesic $\chi^{\mu}(\tau)$ that connects the spacetime points $x_1^{\mu} = (0,\vec{x}_1)$ and $x_2^{\mu} = (ct,\vec{x}_2)$. It obeys the geodesic equation, $$\ddot{\chi}^{\mu} + \Gamma^{\mu}_{~\rho\sigma}\Bigl(\chi(\tau)\Bigr)
\dot{\chi}^{\rho} \dot{\chi}^{\sigma} = 0 \; ,$$ subject to the conditions, $$\begin{aligned}
\chi^{\mu}(0) & = & x_1^{\mu} \; , \\
\chi^i(1) & = & x_2^i \; , \\
g_{\mu\nu}(x_1) \dot{\chi}^{\mu}(0) \dot{\chi}^{\nu}(0) & = & 0 \; .\end{aligned}$$ Of course we solve this perturbatively in the potentials. The zeroth order solution is of course the flat space result, $$\chi_0^{\mu}(\tau) = x_1^{\mu} + \Delta x^{\mu} \tau \; .$$ Here the temporal and spatial components of the interval $\Delta x^{\mu}$ are, $$\Delta x^0 \equiv \Vert \vec{x}_2 - \vec{x}_1\Vert \qquad {\rm and} \qquad
\Delta x^i \equiv x_2^i - x_1^i \; .$$
The first order corrections to the spatial components of the geodesic are, $$\begin{aligned}
\lefteqn{\chi_1^i(\tau) = \tau \int_0^1 \! d\tau' \, (1-\tau')
\Delta \Gamma^i_{~\rho\sigma}\Bigl(x + \Delta x \tau\Bigr) \Delta x^{\rho}
\Delta x^{\sigma} } \nonumber \\
& & \hspace{.5cm} - \int_0^{\tau} \! d\tau' \, (\tau - \tau')
\Delta \Gamma^i_{~\rho\sigma}\Bigl(x_1 + \Delta x \tau\Bigr) \Delta x^{\rho}
\Delta x^{\sigma} \; . \qquad\end{aligned}$$ Of course it is from the first order temporal correction that we infer the time lag. This correction is more complicated, $$\begin{aligned}
\lefteqn{\chi_1^0(\tau) = \frac{\tau}{2 \Delta x} h_{\rho\sigma}(x_1)
\Delta x^{\rho} \Delta x^{\sigma} } \nonumber \\
& & + \frac{\tau}{\Delta x} \int_0^1 \! d\tau' \, (1 - \tau') \Delta x^i
\Delta \Gamma^i_{~\rho\sigma}\Bigl(x + \Delta x \tau\Bigr) \Delta x^{\rho}
\Delta x^{\sigma} \nonumber \\
& & \hspace{.5cm} - \int_0^{\tau} \! d\tau' \, (\tau - \tau')
\Delta \Gamma^0_{~\rho\sigma}\Bigl(x_1 + \Delta x \tau\Bigr) \Delta x^{\rho}
\Delta x^{\sigma} \; . \qquad \label{DX0}\end{aligned}$$ Ignoring ordinary matter makes the graviton geodesics identical to $\chi_0^{\mu}(\tau)$. Hence the time lag between the arrival of gravitational waves and the arrival of photons or neutrinos is (to first order in the potentials), $$c \Delta t \equiv \chi_1^0(1) = \frac1{2 \Delta x} \int_0^1 \! d\tau
\, h_{\mu\nu}\Bigl(x_1 + \Delta x \tau\Bigr) \Delta x^{\mu}
\Delta x^{\nu} \; . \label{thelag}$$
Expression (\[thelag\]) can be simplified a great deal further. First expand out the potentials, $$h_{\mu\nu} \Delta x^{\mu} \Delta x^{\nu} = - \Delta B \, \Delta x^2
+ \Delta A \, (\hat{r} \cdot \Delta \vec{x})^2 \; .$$ (Note that the time lag is positive semi-definite because $\Delta B \leq
0$ and $\Delta A \geq 0$.) Now use relation (\[DBfromDA\]) for $\Delta
B(r)$ in terms of $\Delta A(r)$ and partially integrate to reach the form, $$\begin{aligned}
\lefteqn{c\Delta t = -\frac{\Delta x}{2} \Delta B(r_2) } \nonumber \\
& & \hspace{.5cm} + \frac{\Delta x}{2} \int_0^1 \! d\tau \, \Delta A(r)
\Biggl[ \frac{\tau}{r} \frac{\partial r}{\partial \tau} + \Bigl( \frac{\hat{r}
\cdot \Delta \vec{x}}{\Delta x}\Bigr)^2 \Biggr] \; , \qquad \\
& & = \frac{\Delta \vec{x} \cdot \vec{x}_1}{2 \Delta x} \, \Delta B(r_1)
- \frac{\Delta \vec{x} \cdot \vec{x}_2}{2 \Delta x} \, \Delta B(r_2)
\nonumber \\
& & \hspace{.5cm} + \Delta x \int_0^1 \!d\tau \, \Delta A(r) \Biggl[1 +
\frac{ (\vec{x}_1 \cdot \Delta \vec{x})^2 - r_1^2 \Delta x^2}{\Delta x^2 r^2}
\Biggr] \; . \qquad\end{aligned}$$ It is useful to define the constant $C$, $$C \equiv \frac1{\Delta x^2} \sqrt{r_1^2 \Delta x^2 - (\vec{x}_1 \cdot
\Delta \vec{x})^2} \; .$$ Finally, we change variables from $\tau$ to $r$, $$r(\tau) = \Delta x \sqrt{\Bigl(\tau + \Bigl(\frac{\vec{x}_1 \cdot \Delta
\vec{x}}{\Delta x^2}\Bigr)^2 + C^2} \; .$$ Assuming $r_2 < r_1$ the result is, $$\begin{aligned}
\lefteqn{c \Delta t = \frac{\Delta \vec{x} \cdot \vec{x}_1}{2 \Delta x} \,
\Delta B(r_1) - \frac{\Delta \vec{x} \cdot \vec{x}_2}{2 \Delta x} \,
\Delta B(r_2) } \nonumber \\
& & \hspace{2cm} + \int_{r_2}^{r_1} \!\!\! dr \, \frac{2 G M(r)}{c^2 r}
\sqrt{1 - \Bigl(\frac{C \Delta x}{r}\Bigr)^2 } \; . \qquad \label{cdt}\end{aligned}$$ For $r_1 < r_2$ we take the other root of the solution for $\tau(r)$, which reverses the upper and lower limits in (\[cdt\]).
Results
=======
We have worked out explicit results for three typical sources at vastly different distances: GRB 070201, SN 1987a and Sco-X1. Their celestial coordinates are given in Table \[Coords\]. Table \[Coords\] also gives the centers of the Milky Way and Andromeda dark matter halos.
GRB 070201 was a short hard gamma ray burst whose angular error box corresponded to a $0.124^{\circ}$ quadrilateral which overlapped with the Andromeda galaxy [@Mazets07]. Short hard gamma-ray bursts are believed to be caused by the mergers of two neutron stars or a neutron star and a black hole [@Nakar07]. If GRB 070201 derived from such a merger, with masses close to 1.4 $M_{\odot}$ and a reasonable orientation, the GW signal should have been seen if its distance was $780~{\rm kpc}$ [@Anderson07]. It is however possible that the GRB did not originate in the Andromeda galaxy, and in that case the signal from a compact object merger may be inaccessible to LIGO.
No gravitational waves were found from a search done within $\pm$ 180 s time-window with the LIGO Hanford detectors around the time of this GRB [@070201; @Dietz08]. One interpretation of this null result is that GRB 070201 was a SGR flare [@070201; @Ofek07]. However, it is also possible that physics is described by a Dark Matter Emulator, in which case the pulse of gravitational waves would have arrived long before the electromagnetic signal. Table \[Results\] gives our results for the time lag one would expect at the central position using each of the three dark matter density profiles. Although the time lags differ by as much as 69 days, none of the lags is less than two years.
Table \[Errors\] considers another measure of the likely error by specializing to the isothermal profile and varying the angular position (at the fixed distance of $780~{\rm kpc}$) over the four vertices of the angular error box. In this case distinct results are reported for the contributions from the Milky Way and Andromeda halos, which are of course independent at linearized order. As expected, varying the position has no effect on the contribution from the Milky Way halo but it can change the contribution from the Andromeda halo by as much as 15 days. We should however stress that if this delay calculation was done by assuming that this GRB is at a distance of $780~{\rm kpc}$. If this GRB did not originate in Andromeda, then the calculated delay would be much larger.
SN 1987a was a core collapse supernova in the Large Magellanic Cloud at a distance of 51.4 kpc [@SN1987a]. Neutrinos were observed by the Kamiokande-II [@Kam1; @Kam2] and Irvine-Michigan-Brookhaven [@Bion; @Brat] detectors. The optical signal arrived several hours later because photons must traverse the optically dense stellar environment [@SN1987a]. The total Shapiro delay for SN 1987A from the visible and dark matter distribution has also been independently estimated [@Longo88; @Mena07] to be between 0.29 to 0.36 years. If the oblateness of SN 1987a was in relation to that of the Sun, the current gravitational wave detectors would probably not have seen anything had they been operating at the time [@Dimmel]. However, advanced LIGO would detect such a supernova out to $0.8~{\rm Mpc}$ [@Dimmel]. This includes the Andromeda galaxy, which doubles the expected rate and also ensures that the signal passes through dark matter dominated regions. Of course the effective coverage from neutrino detectors will remain limited to our galaxy and its satellites [@BFV; @Vogel].
Table \[Results\] gives our results for the expected time lag from a Dark Matter Emulator which reproduces each of the three dark matter profiles. These results include only the effect of the Milky Way halo. In contrast to the much more distant GRB 070201, the scatter between the various models for SN 1987a is much smaller — a mere 2.7 days.
Sco-X1 (located at a distance of 2.8 kpc) is one of the brightest Low Mass X-ray Binaries (LMXBs). LMXBs are potential sources of gravitational waves from r-modes getting excited due to accretion, or from a deformed crust [@Bildsten98; @Andersson99; @Heyl02]. One proposed search is to look for coincidences between the data from LIGO and Rossi X-Ray Timing satellite [@extrig08]. This search also assumes that gravitational waves and X-ray photons arrive at the same time.
Table \[Results\] reports the expected time lag for each of the three dark matter profiles, again from the Milky Way halo. Although the time lag is still easily observable at $\sim 4.9~{\rm days}$, the agreement between the three models for this source is excellent. The largest discrepancy is just $.1~{\rm day}$.
Other Modified Gravity Theories
===============================
Since the 1970s, there have been various proposed tests of general relativity through gravitational wave observations (See Ref. [@Alexander07] for a recent review). Most of these tests are in the strong field regime. In this section, we list some other non-GR gravity theories which also predict a non-zero time-delay between photons and GWs and are not yet ruled out through other observations.
These are massive graviton theories and brane-world models. In massive graviton theories [@Will97], the gravitational waves would arrive [ *after*]{} the photons, with the delay being dependent on the graviton mass. However, the Moore-Nelson lower bound on the speed of gravitational waves imposes stringent constraints on the validity of massive graviton models. It should be noted that there is no consistent interacting theory for massive spin two particles which is limited to a finite number of fields [@Deser].
In various brane-world models, gravitational waves propagate faster than photons or neutrinos, depending on the curvature of the bulk [@Csaki01; @Chung02; @Ahmadi07]. Therefore, it is not possible to calculate model-independent time-delays for the three sources we considered in this paper.
Conclusions
===========
The power and generality of our analysis derives from ignoring the details of how a Dark Matter Emulator dispenses with the need for galactic dark matter. We merely assume that it does, which implies that ordinary matter must couple to the metric that general relativity would predict with dark matter. The special characteristic of Dark Matter Emulators is that weak gravitational waves couple to the metric that general relativity would predict without dark matter. [*Both of these metrics can be inferred from observation, and all geometrical quantities worked out, without regard to the details of specific models.*]{} Although a Dark Matter Emulator is not the only conceivable way of evading the no-go theorem [@SW2] while preserving solar system tests [@Will], it is the only way that has so far been given a concrete realization.
If dark matter does not exist and the observed cosmic motions and lensing instead derive from a Dark Matter Emulator then the assumption upon which all triggered gravitational wave searches are based breaks down. In this case the optical or neutrino identification of a plausible gravitational wave source would not imply a simultaneous pulse of gravitational waves but rather that such a pulse occurred [*earlier*]{}. Even for nearby sources such as Sco-X1 (at about $2.8~{\rm kpc}$) the gravitational waves would arrive almost five days earlier. For a source in the Andromeda galaxy the time difference would be over two years.
It is obviously premature to proclaim that the failure of triggered searches to reveal any coincident gravitational wave pulse implies that physics is described by a Dark Matter Emulator. But if plausible sources continue to produce null results this possibility has to be considered. In that case the key question becomes the accuracy with which one can estimate the expected time lag. Some measure of this is given by the spread in Table \[Results\] for different reasonable dark matter profiles. Table \[Errors\] considers variations in the angular position, and there will be comparable results for varying the much less well-determined distances. Based on these analyses it seems unlikely that the uncertainty can be reduced below the level of a few percent. This has important implications for the way data needs to be kept and for the types of searches that should be contemplated.
Of course a gravitational wave signal might be loud enough to show up in all-sky untriggered searches [@LIGOs4]. In that case the gravitational wave signal would trigger electromagnetic and neutrino searches. A single gold-plated event in which the counterpart signal arrived after a plausible delay would be powerful evidence in support of Dark Matter Emulators. Conversely, a single detection of coincident signals would rule out the entire class of Dark Matter Emulators. This is a novel way of using gravitational wave detectors to test alternate gravity theories in the ultra-weak field regime. Indeed, this is an ideal test because Dark Matter Emulators do not change aspects of the tensor component of a gravitational wave signal such as the number of polarizations or, to any reasonable accuracy, the travel time between Earth-bound detectors. So there need be no change in the data analysis algorithms that would be used in any case.
We would like to thank H. Asada, P. Gondolo, J. Kanner, B. Kocsis, G. Lake, S. Marka, G. Moore, E. Tempel and B. Whiting. This work was partially supported by NSF grants NSF-428-51 29NV0 and PHY-0653085, by the Institute for Fundamental Theory at the University of Florida and by the Center for Gravitational Wave Physics at Penn State. The Center for Gravitational Wave Physics is funded by the National Science Foundation under Cooperative Agreement PHY 01-14375.
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|
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|
---
abstract: 'We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a [*skew-symmetric vanishing lattice*]{} in the sense of Janssen. Using the classification of vanishing lattices over $\mathbb{Z}$, we completely determine the structure of the monodromy groups of the $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$ Hitchin fibrations. As an application we determine the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration.'
address: 'School of Mathematical Sciences, The University of Adelaide, Adelaide SA 5005, Australia'
author:
- David Baraglia
title: 'Monodromy of the $SL(n)$ and $GL(n)$ Hitchin fibrations'
---
Introduction
============
Monodromy of the Hitchin fibration {#secmonohit}
----------------------------------
In this paper we determine the monodromy of the $SL(n,\mathbb{C})$-Hitchin fibration $h : \mathcal{M}(n,L) \to \mathcal{A}$ for all $n$. Here $\mathcal{M}(n,L)$ is the moduli space of $L$-twisted $SL(n,\mathbb{C})$-Higgs bundles on a compact Riemann surface $\Sigma$ of genus $g > 1$ and $L$ is a line bundle which is either the canonical bundle $K$, or satisfies $l = deg(L) > 2g-2$. Our proof also gives the monodromy of the corresponding $GL(n,\mathbb{C})$-Hitchin fibration. For $SL(2,\mathbb{C})$, the monodromy was first determined in [@cop] using a combinatorial approach and later revisited in [@bs1] from a more geometric point of view. It does not seem possible to extend the arguments used in [@cop], [@bs1] to rank $n > 2$. In this paper we introduce new techniques that apply for all $n$. When $n > 2$, our results are completely new.\
Recall that the Hitchin fibration is a proper, surjective holomorphic map $h : \mathcal{M}(n,L) \to \mathcal{A}$ from $\mathcal{M}(n,L)$ to the affine space $\mathcal{A} = \bigoplus_{j=2}^n H^0(\Sigma , L^j)$ [@hit1; @hit2; @nit]. As shown by Hitchin [@hit2], when $L = K$ the Hitchin map gives $\mathcal{M}(n,L)$ the structure of an algebraically completely integrable system with respect to a natural holomorphic symplectic structure on $\mathcal{M}(n,L)$. For $L \neq K$, we do not have a holomorphic symplectic structure, but it remains the case that the non-singular fibres of the Hitchin map are abelian varieties. Finding the monodromy of this fibration is a natural problem which has a number of important applications, some of which are described in \[secapp\].\
To explain our results we will mainly focus on the $SL(n,\mathbb{C})$ case. The $GL(n,\mathbb{C})$ case is covered by Theorem \[thmglmon\]. Let $\mathcal{D} \subset \mathcal{A}$ denote the locus of singular fibres of the $SL(n,\mathbb{C})$-Hitchin fibration and let $\mathcal{A}^{\rm reg} = \mathcal{A} \setminus \mathcal{D}$ be the regular locus. Let $\mathcal{M}^{\rm reg}$ be the points of $\mathcal{M}$ lying over $\mathcal{A}^{\rm reg}$, so that $h : \mathcal{M}^{\rm reg} \to \mathcal{A}^{\rm reg}$ is a non-singular bundle of abelian varieties. The monodromy of the $SL(n,\mathbb{C})$-Hitchin system is the local system $\underline{\Lambda}$ on $\mathcal{A}^{\rm reg}$ whose fibre over a point $a \in \mathcal{A}^{\rm reg}$ is the underlying lattice $\underline{\Lambda}_a = H_1( h^{-1}(a) , \mathbb{Z} )$ of the abelian variety $h^{-1}(a)$. Equivalently $\underline{\Lambda}$ is the dual of the Gauss-Manin local system $R^1 h_* \mathbb{Z}$. Choose a basepoint $a_0 \in \mathcal{A}^{\rm reg}$ and let $\Lambda_P$ be the fibre of $\underline{\Lambda}$ over $a_0$ (we use subscript $P$ because $\Lambda_P$ is the lattice of a Prym variety, see \[sec2\]). The local system $\underline{\Lambda}$ is then equivalent to a representation $\rho_{SL} : \pi_1( \mathcal{A}^{\rm reg} , a_0) \to Aut( \Lambda_P)$. We call $\rho_{SL}$ the [*monodromy representation*]{} of the Hitchin fibration. The image $\Gamma_{SL} \subseteq Aut( \Lambda_P )$ of $\rho_{SL}$ will be called the [*monodromy group*]{} of the $SL(n,\mathbb{C})$-Hitchin fibration. The smooth fibres of the Hitchin fibration are Prym varieties associated to certain branched covers of $\Sigma$ called [*spectral curves*]{}, recalled in \[sec2\]. This construction shows that the smooth fibres are equipped with a natural polarization, which defines a non-degenerate skew-symmetric bilinear pairing $\langle \; , \; \rangle : \Lambda_P \times \Lambda_P \to \mathbb{Z}$ invariant under the monodromy representation. Thus the monodromy of the $SL(n,\mathbb{C})$-Hitchin fibration is given by a triple $( \Lambda_P , \langle \; , \; \rangle , \Gamma_{SL})$, consisting of:
- [a lattice $\Lambda_P$,]{}
- [a skew-symmetric $\mathbb{Z}$-valued bilinear form $\langle \; , \; \rangle$ on $\Lambda_P$,]{}
- [a subgroup $\Gamma_{SL} \subset Aut(\Lambda_P)$ preserving $\langle \; , \; \rangle$.]{}
Vanishing cycles
----------------
The monodromy group is generated by a certain collection of vanishing cycles. To describe these vanishing cycles, we choose a basepoint $a_0 \in \mathcal{A}^{\rm reg}$ of the form $a_0 = (0,0, \dots , 0 , a_n)$, where $a_n \in H^0(\Sigma , L^n)$ has only simple zeros. Let $tot(L)$ denote the total space of $L$ and $\pi : tot(L) \to \Sigma$ the projection. The spectral curve $S$ associated to $a_0$ is given by: $$S = \{ \lambda \in tot(L) \; | \; \lambda^n + a_n(\pi(\lambda)) = 0 \} \subset tot(L).$$ Spectral curves of this form carry a Galois action of the cyclic group $\mathbb{Z}_n$ by deck transformations, and will thus be referred to as [*cyclic spectral curves*]{}. One of the key insights of this paper is that the monodromy of the Hitchin fibration can be computed by studying cyclic spectral curves and small perturbations of them.\
Given a cyclic spectral curve $S$, we construct vanishing cycles as follows. Let $\pi : S \to \Sigma$ denote the restriction of $\pi$ to $S$. Then $\pi : S \to \Sigma$ is a degree $n$ branched cover of $\Sigma$, branched over the zeros $b_1,b_2, \dots , b_k$ of $a_n$, where $k = nl$. Let $u_1, \dots , u_k \in S$ be the corresponding ramification points in $S$. The spectral curve construction identifies the lattice $\Lambda_P$ with the kernel of the Gysin homomorphism $\pi_* : H^1(S , \mathbb{Z}) \to H^1(\Sigma , \mathbb{Z})$. Suppose that $\gamma : [0,1] \to \Sigma$ is an embedded path in $\Sigma$ joining $b_i$ to $b_j$, where $i \neq j$ and assume that $\gamma$ does not meet any other branch point. Then $\pi^{-1}( \gamma( [0,1] ) ) \subset S$ consists of $n$ paths $\gamma^1 , \gamma^2 , \dots , \gamma^n : [0,1] \to S$ joining $u_i$ to $u_j$. Let $t : S \to S$ be the generator of the cyclic Galois action given by $\lambda \mapsto e^{2\pi i/n}\lambda$ and order the paths so that $t \gamma^i = \gamma^{i+1}$, $i = 1 , \dots , n-1$. Then $(\gamma^1 - \gamma^2)$ is a $1$-cycle in $S$. Let $l_\gamma \in H_1(S , \mathbb{Z})$ be its homology class and $c_\gamma \in H^1(S , \mathbb{Z})$ the Poincaré dual class. Similarly the cycles $(\gamma^2 -\gamma^3) , \dots , (\gamma^{n-1} - \gamma^n) , (\gamma^n - \gamma^1)$ correspond to the cohomology classes $tc_\gamma , \dots , t^{n-2}c_\gamma , t^{n-1}c_\gamma \in H^1(S,\mathbb{Z})$. Clearly these cycles are in the kernel of $\pi_* : H^1(S , \mathbb{Z}) \to H^1(\Sigma , \mathbb{Z})$, so they are elements of $\Lambda_P$. As the paths $\gamma^1 , \dots , \gamma^n$ are only determined by $\gamma$ up to cyclic permutation, the cycles $c_\gamma , tc_\gamma , \dots , t^{n-1}c_\gamma$ are likewise only determined by $\gamma$ up to cyclic permutation. We call $c_\gamma , tc_\gamma , \dots , t^{n-1}c_\gamma$ the [*vanishing cycles associated to $\gamma$*]{}. We then have:
Let $a = t^ic_\gamma \in \Lambda_P$ be a vanishing cycle associated to $\gamma$. Then the $SL(n,\mathbb{C})$ monodromy group $\Gamma_{SL}$ contains the Picard-Lefschetz transformation $T_a : \Lambda_P \to \Lambda_P$ associated to $a$, given by: $$\label{equtransvection}
T_a(x) = x + \langle a , x \rangle a.$$
Vanishing lattices
------------------
To state our main results, we need to recall the notion of a [*skew-symmetric vanishing lattice*]{} [@jan1]. Let $R$ denote the ring $\mathbb{Z}$ or $\mathbb{Z}_2$ and let $V$ be a free $R$-module of rank $\mu$ equipped with a bilinear form $\langle \; , \; \rangle : V \times V \to R$ which is alternating, i.e. $\langle x , x \rangle = 0$ for all $x \in V$. For any $a \in V$ we have an endomorphism $T_a : V \to V$, called a [*symplectic transvection*]{}, given by Equation (\[equtransvection\]). Clearly $T_a$ acts as an automorphism of $V$ which preserves $\langle \; , \; \rangle$. Let $j : V \to V^*$ be the map $j(x) = \langle x , \; \rangle$, let $V_0$ be the kernel of $j$ and $V'$ the image of $j$. Let $Sp^{\#}V$ denote the automorphisms of $(V , \langle \; , \; \rangle)$ acting trivially on $V^*/V'$ and note that $T_a$ belongs to $Sp^{\#}V$ for any $a \in V$.
Let $\Delta \subseteq V$ and let $\Gamma_\Delta$ be the subgroup of $Sp^{\#}V$ generated by $\{ T_\alpha \}_{\alpha \in \Delta}$. We say that $(V , \langle \; , \; \rangle , \Delta)$ is a [*(skew-symmetric)-vanishing lattice*]{} over $R$ if:
1. [$\Delta$ is a $\Gamma_\Delta$-orbit.]{}
2. [$\Delta$ spans $V$.]{}
3. [If $\mu >1$, then there exists $\delta_1,\delta_2 \in \Delta$ for which $\langle \delta_1 , \delta_2 \rangle = 1$.]{}
We also call $\Gamma_{\Delta}$ the [*monodromy group*]{} of the vanishing lattice.
Main Results {#secmainres}
------------
Let $\mathcal{VC} \subset \Lambda_P$ be the collection of all vanishing cycles $c_\gamma , \dots , t^{n-1}c_\gamma$ associated to paths $\gamma$ joining pairs of branch points. Let $\Gamma_{\mathcal{VC}}$ be the subgroup of $Sp^{\#}( \Lambda_P )$ generated by transvections $T_a$, where $a \in \mathcal{VC}$ and set $\Delta_P = \Gamma_{\mathcal{VC}} \cdot \mathcal{VC}$.
We have that $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ is a vanishing lattice. Let $\Gamma_{\Delta_P}$ be the subgroup of $Sp^{\#}(\Lambda_P)$ generated by transvections in $\Delta_P$. Then $\Gamma_{\mathcal{VC}} = \Gamma_{\Delta_P} = \Gamma_{SL}$. That is, the monodromy group of the $SL(n,\mathbb{C})$-Hitchin fibration is the monodromy group $\Gamma_{\Delta_P}$ of the vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$.
Thus to describe the monodromy group $\Gamma_{SL}$ and its action on $\Lambda_P$, it suffices to classify the vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$. The classification is as follows (here we use the notation for vanishing lattices introduced in [@jan1; @jan2], which is recalled in \[secthevanlat\]):
The vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ is isomorphic to:
1. [$A'(1,1, \dots , 1 , 2 , 2 , \dots , 2; 0)$, if $n=2$,]{}
2. [$O_a^{\#}(1,1, \dots , 1 , n , n , \dots , n ; 0)$, where $a = (m(m-1)/2)l$, if $n =2m+1$ is odd,]{}
3. [$O_a^{\#}(1,1, \dots , 1 , n , n , \dots , n ; 0)$, where $a = m(l/2)$, if $n=2m$ and $l$ are even and $n>2$,]{}
4. [$Sp^{\#}(1,1, \dots , 1 , n , n , \dots , n ; 0)$, if $n$ is even, $l$ is odd and $n > 2$.]{}
In this classification, the number of $1$’s is $(n-2)(g-1) + n(n-1)l/2 - 1$ and the number of $n$’s is $g$.
As mentioned previously, our results also yield the monodromy of the $GL(n,\mathbb{C})$-Hitchin fibration. Fix a basepoint $a_0 \in \mathcal{A}^{\rm reg}$ with spectral curve $S$ and let $\Lambda_S = H^1(S, \mathbb{Z})$. Notice that $\Lambda_P$ is a sublattice of $\Lambda_S$. The monodromy of the $GL(n,\mathbb{C})$-Hitchin fibration is given by a representation $\rho_{GL} : \pi_1( \mathcal{A}^{\rm reg} , a_0) \to Aut( \Lambda_S)$. Let $\Gamma_{GL} \subseteq Aut( \Lambda_S)$ be the image of $\rho_{GL}$. We have:
\[thmglmon\] The monodromy group $\Gamma_{GL}$ is the subgroup of $Aut(\Lambda_S)$ generated by transvections $T_{\alpha} : \Lambda_S \to \Lambda_S$, where $\alpha \in \Delta_P$.
In \[secappl\] we give an application of our monodromy computations. Let $\mathcal{M}(n,d,L)$ denote the moduli space of rank $n$ $L$-twisted Higgs bundles with trace-free Higgs field and determinant equal to a fixed line bundle of degree $d$. The Hitchin fibration $h : \mathcal{M}(n,d,L) \to \mathcal{A}$ may be defined for any value of $d$. Let $F_a = h^{-1}(a)$ be a non-singular fibre of the Hitchin fibration lying over $a \in \mathcal{A}^{\rm reg}$. Then:
\[thmrestriction0\] Let $\omega \in H^2( F_a , \mathbb{Q} )$ be the cohomology class of the polarization on $F_a$. The image $$Im( H^*( \mathcal{M}(n,d,L) , \mathbb{Q} )) \to H^*( F_a , \mathbb{Q} ) )$$ of the restriction map in cohomology is the subspace spanned by $1 , \omega , \omega^2 , \dots , \omega^u$, where $u = dim_{\mathbb{C}}(F_a)$ is the dimension of the fibre.
This result generalises to $SL(n,\mathbb{C})$ a result proved by Thaddeus for $SL(2,\mathbb{C})$ in [@tha] (see also the appendix of [@chm]). When $n$ and $d$ are coprime, the moduli space $\mathcal{M}(n,d,L)$ is smooth and Theorem \[thmrestriction0\] can be proven by a straightforward generalisation of the argument given in [@chm]. The argument breaks down when $n$ and $d$ are not coprime, in which case Theorem \[thmrestriction0\] is a new result.
Applications {#secapp}
------------
Knowledge of the monodromy is important for applications of the moduli space of Higgs bundles in which the Hitchin fibration plays a prominent role. We describe some of these applications here.\
[**Cohomology of the moduli space and Ngô’s support theorem**]{}. Let $\mathcal{M}_{GL}(n,d,L)$ denote the moduli space of $L$-twisted Higgs bundles of rank $n$, degree $d$. Assume that $n$ and $d$ are coprime, in which case $\mathcal{M}_{GL}(n,d,L)$ is non-singular. We have the Hitchin fibration $h : \mathcal{M}_{GL}(n,d,L) \to \mathcal{A}_{GL}$ whose monodromy is isomorphic to $\underline{\Lambda}$ for any value of $d$. Let $\mathcal{A}_{GL}^{\rm ell} \subset \mathcal{A}_{GL}$ be the elliptic locus of $\mathcal{A}_{GL}$, the locus of points for which the corresponding spectral curve is reduced and irreducible. Ngô’s support theorem [@ngo] (see also [@cm]) applied to the restriction $h^{\rm ell} : \mathcal{M}_{GL}^{\rm ell}(n,d,L) \to \mathcal{A}_{GL}^{\rm ell}$ of $h$ over $\mathcal{A}_{GL}^{\rm ell}$ implies that the perverse sheaves on $\mathcal{A}^{\rm ell}_{GL}$ occuring in the decomposition theorem for $h^{\rm ell}$ are supported on the whole of $\mathcal{A}_{GL}^{\rm ell}$. The decomposition theorem then reduces to: $$Rh^{\rm ell}_* \mathbb{Q}[ dim(\mathcal{M}_{GL}(n,d,L)) ] = \bigoplus_i IC_{\mathcal{A}_{GL}^{\rm ell}}( \wedge^i \underline{\Lambda}^* \otimes_{\mathbb{Z}} \mathbb{Q} )[f-i],$$ where $f$ is the dimension of the fibres of the Hitchin fibration. This raises the possibility of computing the cohomology of $\mathcal{M}_{GL}^{\rm ell}(n,d,L)$ through a knowledge of the local system $\underline{\Lambda}$. In turn, this gives us partial information about the cohomology of the full moduli space $\mathcal{M}_{GL}(n,d,L)$. In fact for $d > 2g-2$, it was shown by Chaudouard and Laumon that there are no new supports when $\mathcal{A}_{GL}^{\rm ell}$ is replaced by $\mathcal{A}_{GL}$ [@chla]. A similar result holds in the $SL(n,\mathbb{C})$ case, except there are additional supports related to the endoscopy theory of $SL(n,\mathbb{C})$ [@cat]. In a related direction, we note that local monodromy calculations were used extensively in [@chm] in the proof of the $P = W$ conjecture for $GL(2,\mathbb{C}), SL(2,\mathbb{C}), PSL(2,\mathbb{C})$. We expect our monodromy calculations to be similarly useful in tackling this conjecture for higher rank groups.\
[**Wall crossing and the hyperkähler metric**]{}. In the work of Gaiotto, Moore and Neitzke on the Kontsevich-Soibelman wall-crossing formula for $\mathcal{N} = 2$ supersymmetric quantum field theories, a relation is established between hyperkähler metrics and counts of BPS states [@gmn1]. In particular, this can be applied to the hyperkähler metric on the moduli space of Higgs bundles (in the untwisted case $L = K$). Through a twistorial construction, the hyperkähler metric on the moduli space of Higgs bundles is encoded in a family of complex symplectic forms $\omega(\zeta)$ parametrised by $\zeta \in \mathbb{CP}^1$. In [@gmn1], the authors consider local Darboux coordinates for the symplectic forms $\omega(\zeta)$. The Darboux coordinates $\chi_\gamma$, are locally defined $\mathbb{C}^*$-valued functions on $\mathcal{M}^{\rm reg}$ depending on a choice of point $\zeta \in \mathbb{C}^* \subset \mathbb{CP}^1$ and local section $\gamma$ of the monodromy local system $\underline{\Lambda}$. The functions $\chi_\gamma$ depend multiplicatively on $\gamma$ in the sense that $\chi_{\gamma + \gamma'} = \chi_\gamma \chi_{\gamma'}$ and are Darboux coordinates in the sense that their Poisson brackets are given by the formula: $$\{ \chi_\gamma , \chi_{\gamma'} \} = \langle \gamma , \gamma' \rangle \chi_{\gamma} \chi_{\gamma'}.$$ The locally defined coordinates $\chi_\gamma$ do not patch together globally, instead they satisfy a wall-crossing formula as one crosses a real codimension $1$ subspace of $\mathcal{M}^{\rm reg} \times \mathbb{C}^*$. The wall-crossing formula is given by a holomorphic symplectomorphism, which may be expressed in terms of symplectomorphisms $\mathcal{K}_\gamma$ of the form $$\mathcal{K}_\gamma : \chi_{\gamma'} \mapsto \chi_{\gamma'}( 1 - (-1)^{q(\gamma)}\chi_{\gamma})^{\langle \gamma' , \gamma \rangle },$$ where $q : \underline{\Lambda} \to \mathbb{Z}_2 = \{ 0, 1 \}$ is a quadratic function on $\underline{\Lambda}$ (see \[secquadratics\]). Interestingly, the existence of a monodromy invariant quadratic function on $\underline{\Lambda}$ is a key ingredient in the classification of $\underline{\Lambda}$ as a vanishing lattice. The proposal of [@gmn1] is that the existence of coordinates $\chi_\gamma$ satisfying the wall-crossing formula and some further technical conditions completely determines the hyperkähler metric on the moduli space, through twistor theory. Clearly it is important for this proposal to be able to describe the local system $\underline{\Lambda}$ and its quadratic function $q$.\
[**Connected components of real character varieties**]{}. Let $G$ be a real or complex reductive Lie group. A representation $\theta : \pi_1(\Sigma) \to G$ is called [*reductive*]{} if the action of $\pi_1(\Sigma)$ on the Lie algebra of $G$ obtained by composing $\theta$ with the adjoint representation is a direct sum of irreducible representations. Let $Hom^{\rm red}(\pi_1(\Sigma) , G)$ be the space of reductive representations given the compact-open topology. The group $G$ acts on $Hom^{\rm red}(\pi_1(\Sigma) , G)$ by conjugation and the quotient $$Rep(G) = Hom^{\rm red}(\pi_1(\Sigma) , G)/G$$ is a Hausdorff space [@ric], called the [*character variety*]{} of representations of $\pi_1(\Sigma)$ in $G$. The non-abelian Hodge correspondence gives a homeomorphism between $Rep(G)$ and the moduli space $\mathcal{M}_G$ of polystable $G$-Higgs bundles, with $L = K$. If $G_\mathbb{C}$ is a complex reductive group and $G_{\mathbb{R}}$ the split real form, then as shown by Schaposnik [@sch0 Theorem 4.12], the image of the natural map $\mathcal{M}_{G_{\mathbb{R}}} \to \mathcal{M}_{G_{\mathbb{C}}}$ meets the smooth fibres of the Hitchin fibration in points of order $2$. The points of order $2$ in smooth fibres are described by the monodromy with $\mathbb{Z}_2$-coefficients $\underline{\Lambda}[2] = \underline{\Lambda} \otimes_{\mathbb{Z}} \mathbb{Z}_2$. Thus one may hope to determine the number of connected components of $\mathcal{M}_{G_{\mathbb{R}}}$ and hence of $Rep(G_{\mathbb{R}})$ by counting the number of orbits of the local system $\underline{\Lambda}[2]$. This strategy has been successfully carried out for $SL(2,\mathbb{R})$ in [@sch1] and extended to $GL(2,\mathbb{R})$, $PGL(2,\mathbb{R})$ and $Sp(4,\mathbb{R})$ with maximal Toledo invariant in [@bs1]. With the calculation of the monodromy $\underline{\Lambda}$ in this paper one can similarly obtain the number of components of the character varieties for $SL(n,\mathbb{R}), GL(n,\mathbb{R}), PSL(n,\mathbb{R})$ and $Sp(2n,\mathbb{R})$ with maximal Toledo invariant for any value of $n$. There is however a caveat that so far for $n>2$, we are only able to count the number of connected components which intersect the regular locus. To turn these counts into a complete proof it will be necessary to show that there are no components of the corresponding character varieties lying entirely within the singular locus. We hope to address this problem in future work.\
The strategy of counting components of $Rep(G_\mathbb{R})$ through monodromy may also be applied to non-split real forms. One may define spectral data or cameral data for arbitrary real forms and at least in some cases, this gives rise to families of smooth spectral curves or cameral curves. The generic fibre $F$ of the Hitchin fibration restricted to $\mathcal{M}_{G_\mathbb{R}}$ is generally not a discrete space anymore, but one can consider the monodromy action on $\pi_0(F)$ and by counting the number of orbits obtain a count of the components of $Rep(G_\mathbb{R})$.
Structure of the paper
----------------------
This paper is organised as follows. In \[sec2\] we recall the spectral curve construction of the regular locus of the Hitchin fibration. In \[secthelattices\], we study in detail the structure of the lattices $\Lambda_S , \Lambda_P$, where $\Lambda_S = H^1(S , \mathbb{Z})$ is the integral cohomology of the spectral curve over the basepoint $a_0$. In particular, we work out the intersection form $\langle \; , \; \rangle$ in Section \[secintf\]. In Section \[secquadratics\] we find that under conditions on $n$ and $l$, there is a natural quadratic function on $\Lambda_P$ and we compute its Arf invariant. This is an important input for the classification of the vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$. In \[secvc\] we construct vanishing cycles associated to paths $\gamma$ in $\Sigma$ joining pairs of branch points. In \[secthevanlat\], we use the vanishing cycles of \[secvc\], to construct the vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$. We then recall the classification of vanishing lattices over $\mathbb{Z}_2$ in Section \[secz2class\], over $\mathbb{Z}$ in Section \[seczclass\] and use this to completely determine the structure of $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$. In \[secmain\], we prove that the monodromy group $\Gamma_{SL}$ of the $SL(n,\mathbb{C})$-Hitchin fibration coincides with the monodromy group $\Gamma_{\Delta_P}$ of the vanishing lattice. In \[secappl\], we apply our monodromy computations to give a proof of Theorem \[thmrestriction\], describing the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration.
Acknowledgements
----------------
We would like to thank Laura Schaposnik, Tamás Hausel and Nigel Hitchin helpful discussions. The author is supported by the Australian Research Council grant DE160100024.
Spectral curves and the regular locus {#sec2}
=====================================
We recall some basic facts about the moduli space of $SL(n,\mathbb{C})$-Higgs bundles, in particular the spectral curve construction of the regular locus. An [*$SL(n,\mathbb{C})$ $L$-twisted Higgs bundle*]{} is a pair $(E,\Phi)$ where $E$ is a rank $n$ holomorphic vector bundle on $\Sigma$ with trivial determinant and $\Phi$ is a trace-free holomorphic endomorphism $\Phi : E \to E \otimes L$. As shown by Nitsure [@nit], one may define notions of semistability and $S$-equivalence for twisted Higgs bundles and construct a moduli space $\mathcal{M}(n,L)$ of $S$-equivalence classes of semistable $SL(n,\mathbb{C})$ $L$-twisted Higgs bundles. The moduli space $\mathcal{M}(n,L)$ is a quasi-projective complex algebraic variety [@nit]. As the rank $n$ and line bundle $L$ will be fixed throughout, we will omit them from the notation and simply write $\mathcal{M}$ for the moduli space.\
Recall the [*Hitchin fibration*]{}, also known as the [*Hitchin map*]{} or [*Hitchin system*]{} is a surjective holomorphic map $h : \mathcal{M} \to \mathcal{A}$ from $\mathcal{M}$ to the affine space $\mathcal{A} = \bigoplus_{j=2}^n H^0(\Sigma , L^j)$ [@hit1; @hit2; @nit]. Using the notion of spectral curves recalled below, it can be shown that the non-singular fibres of the Hitchin map are abelian varieties [@hit2; @bnr]. Let $\mathcal{D} \subset \mathcal{A}$ denote the locus of singular fibres of the Hitchin map and let $\mathcal{A}^{\rm reg} = \mathcal{A} \setminus \mathcal{D}$ be the regular locus. We let $\mathcal{M}^{\rm reg}$ denote the points of $\mathcal{M}$ lying over the regular locus, so that $h : \mathcal{M}^{\rm reg} \to \mathcal{A}^{\rm reg}$ is a locally trivial torus bundle. Our goal in this paper is to determine the monodromy of this torus bundle.\
We recall the spectral curve construction of $\mathcal{M}^{\rm reg}$ from $\mathcal{A}^{\rm reg}$. For this, let $tot(L)$ denote the total space of $L$ and $\pi : tot(L) \to \Sigma$ the projection. We let $\lambda \in H^0( tot(L) , \pi^*L)$ denote the tautological section of $\pi^*(L)$ on $tot(L)$. Given $a = (a_2,a_3, \dots , a_n) \in \mathcal{A}$, let $s_a$ be the section of $\pi^*L^n$ on $tot(L)$ given by $s_a = \lambda^n + a_2 \lambda^{n-2} + \dots + a_n$. The vanishing locus of $s_a$ defines a curve $S_a \subset tot(L)$ called the [*spectral curve*]{} associated to $a$. We use $\pi : S_a \to \Sigma$ to denote the restriction of $\pi : tot(L) \to \Sigma$. As in [@hit2], Bertini’s theorem implies that $S_a$ is smooth for generic $a \in \mathcal{A}$. Moreover it can be shown that $S_a$ is smooth if and only if the corresponding fibre of the Hitchin system $h : \mathcal{M} \to \mathcal{A}$ is non-singular [@kp]. When this is the case the corresponding fibre of the Hitchin system is: $$h^{-1}(a) = \{ M \in Jac_{ln(n-1)/2}(S_a) \; | \; Nm(M) = L^{n(n-1)/2} \},$$ where $Jac_{d}(S_a)$ denotes the degree $d$ component of $Pic(S_a)$ and $Nm : Pic(S) \to Pic(\Sigma)$ is the norm map associated to $\pi : S_a \to \Sigma$ [@bnr]. Note that $h^{-1}(a)$ is naturally a torsor over the abelian variety $Prym(S_a, \Sigma)$, which is defined as: $$Prym(S_a , \Sigma) = \{ M \in Jac(S_a) \; | \; Nm(M) = \mathcal{O} \}.$$ The abelian variety $Prym(S_a , \Sigma)$ is called the [*Prym variety*]{} associated to $\pi : S_a \to \Sigma$.\
Let $q : \mathcal{S} \to \mathcal{A}^{\rm reg}$ denote the family of smooth spectral curves parametrised by $\mathcal{A}^{\rm reg}$. This may be defined as the set of pairs $(a,x) \in \mathcal{A}^{\rm reg} \times tot(L)$ such that $s_a(x) = 0$. This is a family of non-singular curves parametrised by $\mathcal{A}^{\rm reg}$ and so we may construct the relative Jacobian $j : Jac(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg}$. Let $Nm : Jac(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg} \times Jac(\Sigma)$ be the fibrewise norm map and let $p : Prym(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg}$ be the fibrewise kernel of $Nm$. Then $Prym(\mathcal{S}/\mathcal{A}^{\rm reg})$ is a family of Prym varieties parametrised by $\mathcal{A}^{\rm reg}$.
If $n$ is odd or $l$ is even, then there exists on $\Sigma$ a square root $L^{(n-1)/2}$ of $L^{n-1}$. The map sending $a \in \mathcal{A}^{\rm reg}$ to $\pi^*( L^{(n-1)/2} ) \in Jac_{ln(n-1)/2}(S_a)$ is a section of $\mathcal{M}^{\rm reg} \to \mathcal{A}^{\rm reg}$ and this gives an isomorphism $\mathcal{M}^{\rm reg} \simeq Prym(\mathcal{S}/\mathcal{A}^{\rm reg})$.
We define the following local systems on $\mathcal{A}^{\rm reg}$:
- [$\underline{\Lambda} = Hom( R^1 h_* \mathbb{Z} , \mathbb{Z})$, the monodromy of the Hitchin fibration.]{}
- [$\underline{\Lambda}_J = Hom(R^1 j_* \mathbb{Z},\mathbb{Z})$, the monodromy of the family of Jacobians.]{}
- [$\underline{\Lambda}_P = Hom(R^1 p_* \mathbb{Z},\mathbb{Z})$, the monodromy of the family of Prym varieties.]{}
- [$\underline{\Lambda}_S = Hom(R^1q_*\mathbb{Z},\mathbb{Z})$, the monodromy of the family of spectral curves.]{}
- [$\underline{\Lambda}_\Sigma$ will denote the trivial local system with coefficient group $H^1(\Sigma , \mathbb{Z})$.]{}
We have the following relations between local systems:
- [$\underline{\Lambda}_S \simeq \underline{\Lambda}_J$]{}
- [$\underline{\Lambda} \simeq \underline{\Lambda_P}$]{}
- [The fibrewise norm map $Nm : Jac(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg} \times Jac(\Sigma)$ induces a short exact sequence: $$\label{equseslocsys}
0 \longrightarrow \underline{\Lambda}_P \longrightarrow \underline{\Lambda}_J \buildrel Nm \over \longrightarrow \underline{\Lambda}_\Sigma \longrightarrow 0.$$ ]{}
(1). For any smooth spectral curve $S$ there is a canonical isomorphism $H^1(S,\mathbb{Z}) \simeq H^1( Jac(S) , \mathbb{Z})$. Clearly this isomorphism extends to a canonical isomorphism of local systems $R^1 q_* \mathbb{Z} \simeq R^1 j_* \mathbb{Z}$ on $\mathcal{A}^{\rm reg}$ associated to the families $q : \mathcal{S} \to \mathcal{A}^{\rm reg}$ and $j : Jac(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg}$. Thus $\underline{\Lambda}_S \simeq \underline{\Lambda}_J$.\
(2). Recall that $Prym(\mathcal{S}/\mathcal{A}^{\rm reg})$ is the pre-image under the fibrewise norm map $Nm : Jac(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg} \times Jac(\Sigma)$ of $\mathcal{A}^{\rm reg} \times \{ \mathcal{O} \}$. Similarly $\mathcal{M}^{\rm reg}$ is the pre-image under $Nm$ of $\mathcal{A}^{\rm reg} \times\{ L^{n(n-1)/2} \}$. Thus $p : Prym(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg}$ is a bundle of abelian varieties and $h : \mathcal{M}^{\rm reg} \to \mathcal{A}^{\rm reg}$ is a bundle of torsors for $Prym(\mathcal{S}/\mathcal{A}^{\rm reg})$. In particular this gives a canonical isomorphism $R^1h_* \mathbb{Z} \simeq R^1p_* \mathbb{Z}$ of local systems. Hence $\underline{\Lambda} \simeq \underline{\Lambda_P}$.\
(3). We have a short exact sequence of bundles of abelian varieties over $\mathcal{A}^{\rm reg}$: $$Prym(\mathcal{S}/\mathcal{A}^{\rm reg}) \to Jac(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg} \times Jac(\Sigma).$$ Taking homology of the fibres gives the short exact sequence (\[equseslocsys\]).
Instead of working with local systems directly we will fix a basepoint $a_0 \in \mathcal{A}^{\rm reg}$ and work with representations of $\pi_1( \mathcal{A}^{\rm reg} , a_0)$. Let $\pi : S \to \Sigma$ denote the spectral curve corresponding to the basepoint $a_0$. We let $\Lambda_S$ denote $H^1(S,\mathbb{Z})$, let $\Lambda_\Sigma$ denote $H^1(\Sigma , \mathbb{Z})$ and let $\Lambda_P$ denote the kernel of the Gysin map $\pi_* : \Lambda_S \to \Lambda_\Sigma$. Then $\underline{\Lambda} \simeq \underline{\Lambda}_P$ corresponds to a representation $\rho_{SL} : \pi_1( \mathcal{A}^{\rm reg} , a_0) \to Aut( \Lambda_P)$ which we call the [*monodromy representation of the $SL(n,\mathbb{C})$ Hitchin fibration*]{}. Similarly $\underline{\Lambda}_S \simeq \underline{\Lambda}_J$ corresponds to a representation $\rho_{GL} : \pi_1( \mathcal{A}^{\rm reg} , a_0) \to Aut( \Lambda_S )$, which we refer to as the [*monodromy representation of the $GL(n,\mathbb{C})$ Hitchin fibration*]{}. We note that (\[equseslocsys\]) corresponds to a short exact sequence of representations: $$0 \longrightarrow \Lambda_P \longrightarrow \Lambda_S \buildrel \pi_* \over \longrightarrow \Lambda_\Sigma \longrightarrow 0.$$
Although it will play no part in subsequent calculations, we note that the dual local system $\underline{\Lambda}^*$ and corresponding representation $\rho_{SL}^* : \pi_1(\mathcal{A}^{\rm reg} , a_0) \to Aut( \Lambda_P^*)$ give the monodromy of the Hitchin fibration for the group $PGL(n,\mathbb{C})$. This is an instance of Langlands duality for Hitchin systems, which in general implies that Langlands dual groups give rise to dual monodromy representations.
The lattices $\Lambda_S,\Lambda_P$ {#secthelattices}
==================================
Decomposition of $\Lambda_S$ {#secdecomps}
----------------------------
In this section we fix a basepoint $a_0 \in \mathcal{A}^{\rm reg}$ and examine the structure of the lattices $\Lambda_S,\Lambda_P$ in detail. For this we choose $a_0$ to be of the form $a_0 = (0,0, \dots , 0 , a_n )$, for some $a_n \in H^0(\Sigma , L^n)$. The corresponding spectral curve $S$ is given by the equation $\lambda^n + a_n = 0$ and it is clear that $S$ is smooth if and only if $a_n$ has only first order zeros. Let $b_1 , b_2 , \dots , b_k \in \Sigma$ be the zeros of $a_n$, where $k = nl$ and let $u_1, \dots , u_k \in S$ be the corresponding ramification points.
Let $D$ be the open unit disc in $\mathbb{C}$ of radius $1$ centred at $0$ and $\overline{D}$ the corresponding closed unit disc. We say that an embedding $i : \overline{D} \to \Sigma$ is a [*trivialising disc*]{} if $i(D)$ contains the branch points of $\pi : S \to \Sigma$ and the restriction of $S$ to $\Sigma \setminus D$ is the trivial covering space ($n$ disjoint copies of $\Sigma \setminus D$).
\[proptd\] For any $a_n \in H^0(\Sigma , L^n)$ with simple zeros, the corresponding spectral curve $\pi : S \to \Sigma$ admits a trivialising disc.
Let $\Sigma' = \Sigma \setminus \{b_1 , \dots , b_k \}$. Let $C \subset H_1( \Sigma' , \mathbb{Z})$ be the subgroup generated by cycles $l_1,l_2, \dots l_k$ around the points $b_1, \dots , b_k$, so that we have an exact sequence: $$\xymatrix{
0 \ar[r] & C \ar[r] & H_1( \Sigma' , \mathbb{Z}) \ar[r] & H_1(\Sigma , \mathbb{Z}) \ar[r] & 0.
}$$ The monodromy of the branched cover $S \to \Sigma$ defines a homomorphism $\phi_S : H_1(\Sigma' , \mathbb{Z}) \to \mathbb{Z}_n$ such that $\phi_S(l_i) = 1$ for each $i$. Let $i : \overline{D} \to \Sigma$ be an embedding of $\overline{D}$ in $\Sigma$ for which $i(D)$ contains the branch points. Then $i$ determines a splitting $s_i : H_1(\Sigma , \mathbb{Z}) \to H_1(\Sigma' , \mathbb{Z})$ given by the composition of the isomorphism $H_1(\Sigma, \mathbb{Z}) \simeq H_1(\Sigma \setminus i(D) , \mathbb{Z})$ with the inclusion induced map $H_1(\Sigma \setminus i(D) , \mathbb{Z}) \to H_1(\Sigma' , \mathbb{Z})$. We have that $i$ gives a trivialising disc if and only if the composition $\phi_S \circ s_i : H_1(\Sigma , \mathbb{Z}) \to \mathbb{Z}_n$ is the trivial homomorphism.\
Choose a branch point $b_i$ and let $l$ be an embedded loop in $\Sigma'$ with underlying homology class of the form $[l] = l_i + s_i(m)$, where $m \in H_1(\Sigma , \mathbb{Z})$ is the homology class of $l$ as a loop in $\Sigma$. Let $\tau_l : \Sigma \to \Sigma$ be a Dehn twist around $l$, supported in a neighbourhood of $l$ containing no branch points. So $\tau_l$ preserves the branch points and acts a homeomorphism $\tau_l : \Sigma' \to \Sigma'$. The composition $i' = \tau_l \circ i : \overline{D} \to \Sigma$ gives a new embedding and corresponding splitting $s_{i'} : H_1(\Sigma , \mathbb{Z}) \to H_1(\Sigma' , \mathbb{Z})$. From the commutative diagram: $$\xymatrix{
& & \\
H_1(\Sigma , \mathbb{Z}) \ar[d]^-{{\tau_l}_*} \ar@/^2pc/[rr]^-{s_i} & \ar[l]_-{\cong} H_1(\Sigma \setminus i(D) , \mathbb{Z}) \ar[r] \ar[d]^-{{\tau_l}_*} & H_1(\Sigma' , \mathbb{Z}) \ar[d]^-{{\tau_l}_*} \\
H_1(\Sigma , \mathbb{Z}) \ar@/_2pc/[rr]_-{s_{i'}} & \ar[l]_-{\cong} H_1(\Sigma \setminus i'(D), \mathbb{Z}) \ar[r] & H_1(\Sigma' , \mathbb{Z}) \\
& &
}$$ we see that $s_{i'} = {\tau_l}_* \circ s_i \circ {\tau_l}_*^{-1}$. One finds that $({\tau_l}_* s_i {\tau_l}_*^{-1})(a) = s_i(a) + \langle a , m \rangle l_i$, hence $(\phi_S \circ s_{i'})(a) = (\phi_S \circ s_i)(a) + \langle a , m \rangle$. From this we see that by applying to $i$ a series of Dehn twists around suitably chosen loops, we can obtain an embedding $\tilde{i}$ of $\overline{D}$ with splitting $s_{\tilde{i}}$ such that $\phi_S \circ s_{\tilde{i}}$ is trivial, as required.
By Proposition \[proptd\], there exists a trivialising disc $i : \overline{D} \to \Sigma$. Let $D'$ be an open disc obtained from $D$ by shrinking the radius slightly, but for which $i(D')$ still contains all branch points. Let $U_0 = i(D)$ and $U_1 = \Sigma \setminus \overline{i(D')}$, where $\overline{i(D')}$ is the closure of $i(D')$ in $\Sigma$. The Mayer-Vietoris sequence applied to the cover $V_0 = \pi^{-1}(U_0), V_1 = \pi^{-1}(U_1)$ of $S$ gives: $$\xymatrix{
0 \ar[r] & H_2(S , \mathbb{Z} ) \ar[r]^-{\partial} & H_1(V_0 \cap V_1 , \mathbb{Z}) \ar[r]^-{({i_0}_*, {i_1}_*)} & H_1(V_0 , \mathbb{Z}) \oplus H_1( V_1 , \mathbb{Z}) \ar[r]^-{{j_0}_*-{j_1}_*} & H_1(S,\mathbb{Z}) \ar[r] & 0,
}$$ where $i_a$ is the inclusion $V_0 \cap V_1 \to V_a$ and $j_a$ is the inclusion $V_a \to S$. Since $i : D \to \Sigma$ is a trivialising disc we have that $V_1$ consists of $n$ disjoint copies of $U_1$ and that the inclusion $\coprod_{i=1}^n U_1 \to \coprod_{i=1}^n \Sigma$ induces an isomorphism $H_1( V_1 , \mathbb{Z}) \to \oplus_{i=1}^n H^1(\Sigma , \mathbb{Z})$. Similarly, $V_0 \cap V_1$ is homotopy equivalent to $n$ copies of the circle. These circles correspond to the $n$ lifts to $S$ of the boundary of the disc $i(D)$, hence the map ${i_1}_*$ is trivial. Let $\Lambda_{S,0}$ be the cokernel of ${i_0}_* : H_1( V_0 \cap V_1 , \mathbb{Z}) \to H_1( V_0 , \mathbb{Z})$ and $\Lambda_{S,1} = H_1( V_1 , \mathbb{Z}) = \oplus_{i=1}^n H^1(\Sigma , \mathbb{Z})$. Then the above sequence gives an isomorphism $H_1(S,\mathbb{Z}) \simeq \Lambda_{S,0} \oplus \Lambda_{S,1}$. By Poincaré duality we have $\Lambda_S = H^1(S,\mathbb{Z}) \simeq H_1(S,\mathbb{Z})$, so $$\label{equdecomps}
\Lambda_S \simeq \Lambda_{S,0} \oplus \Lambda_{S,1}.$$ We will make extensive use of this decomposition of $\Lambda_S$ in the following sections.
$\mathbb{Z}[t]$-module structure {#secztmod}
--------------------------------
Let $t : S \to S$ be the map sending $\lambda$ to $\xi \lambda$, where $\xi = e^{2\pi i/n}$. Then $t$ generates a $\mathbb{Z}_n$-action on $S$. We can thus view $\Lambda_S$ as a $\mathbb{Z}[t]$-module, where the action of $t$ satisfies $t^n = 1$. Clearly (\[equdecomps\]) is a direct sum of $\mathbb{Z}[t]$-modules. The lattice $\Lambda_{S,1}$ was seen to consist of a direct sum of $n$ copies of $\Lambda_\Sigma = H^1(\Sigma , \mathbb{Z}) \simeq H_1(\Sigma , \mathbb{Z})$. Further, $t$ acts on $\Lambda_{S,1}$ by cyclic permutation of these $n$ copies so that as $\mathbb{Z}[t]$-modules we have: $$\Lambda_{S,1} = \frac{\mathbb{Z}[t]}{\langle t^n - 1 \rangle} \otimes_{\mathbb{Z}} \Lambda_\Sigma.$$
Next consider $\Lambda_{S,0}$. From its definition $\Lambda_{S,0}$ fits into an exact sequence: $$\xymatrix{
0 \ar[r] & H_2(S , \mathbb{Z} ) \ar[r]^-{\partial} & H_1(V_0 \cap V_1 , \mathbb{Z}) \ar[r]^-{ {i_0}_*} & H_1( V_0 , \mathbb{Z}) \ar[r] & \Lambda_{S,0} \ar[r] & 0.
}$$ Up to homotopy $V_0 \cap V_1$ can be identified with the boundary of $V_0$ and $i_0$ with the inclusion map. We have that $V_0$ is a degree $n$ branched cover of the unit disc $D \subset \mathbb{C}$. Applying a suitable homeomorphism to $D$, we can assume that the branch points $b_1,b_2, \dots , b_k$ lie on the real axis with $-1/2 = b_1 < b_2 < \dots < b_k = 1/2$. A deformation retraction of $D$ onto the interval $[-1/2 , 1/2]$ lifts to a deformation retraction of $V_0$ onto $\pi^{-1}( [-1/2,1/2])$.\
The branched cover $\pi^{-1}( [-1/2 , 1/2]) \to [-1/2,1/2]$ is depicted in Figure \[figbranch\]. For $1 \le i \le k-1$, let $\gamma_i$ be the path joining $b_i$ to $b_{i+1}$ along the interval $[b_i , b_{i+1}]$. Let $\gamma_i^1 , \gamma_i^2 , \dots , \gamma_i^n$ be the lifts of $\gamma_i$ to paths in $\pi^{-1}( [-1/2 , 1/2])$ joining $u_i$ to $u_{i+1}$, as shown in Figure \[figbranch\]. We may order the lifts in such a way that $\gamma_i^j = t^{j-1}\gamma_i^1$. Let $c_i$ be the homology class of the $1$-cycle $\gamma_i^1 - \gamma_i^2$. Then $t^{j-1}c_i$ is the homology class of $\gamma_i^j - \gamma_i^{j+1}$. Note that $t^{n-1}c_i = \gamma_i^n - \gamma_i^1 = -(c_i + tc_i + \dots + t^{n-2}c_i)$, but that the cycles $c_i , tc_i , \dots , t^{n-2}c_i$ are independent. In fact, we have:
The homology group $H_1( V_0 , \mathbb{Z})$ is free as a $\mathbb{Z}$-module, with basis given by the cycles $t^j c_i $ for $1 \le i \le k-1$, $0 \le j \le n-2$.
We have shown that $V_0$ admits a deformation retraction to $\pi^{-1}( [-1/2,1/2])$. By considering Figure \[figbranch\], the proposition is easily seen to follow.
(0,0) circle(0.1); (4,0) circle(0.1); (8,0) circle(0.1);
(0,0) to \[out=80, in=180\] (2,1.7) ; (2,1.7) to \[out=0, in=100\] (4,0); (0,0) to \[out=60, in=180\] (2,1.2) ; (2,1.2) to \[out=0, in=120\] (4,0); (0,0) to \[out=300, in=180\] (2,-1.2) ; (2,-1.2) to \[out=0, in=240\] (4,0); (0,0) to \[out=280, in=180\] (2,-1.7) ; (2,-1.7) to \[out=0, in=260\] (4,0); (2,-0.6) – (2,0.6) ;
(4,0) to \[out=80, in=180\] (6,1.7) ; (6,1.7) to \[out=0, in=100\] (8,0); (4,0) to \[out=60, in=180\] (6,1.2) ; (6,1.2) to \[out=0, in=120\] (8,0); (4,0) to \[out=300, in=180\] (6,-1.2) ; (6,-1.2) to \[out=0, in=240\] (8,0); (4,0) to \[out=280, in=180\] (6,-1.7) ; (6,-1.7) to \[out=0, in=260\] (8,0); (6,-0.6) – (6,0.6) ;
at (0.4,0) [$u_1$]{}; at (4.4,0) [$u_2$]{}; at (8.4,0) [$u_3$]{};
at (2,2) [$\gamma_1^1$]{}; at (2,0.9) [$\gamma_1^2$]{}; at (2.2,-0.9) [$\gamma_1^{n-1}$]{}; at (2,-2) [$\gamma_1^n$]{};
at (6,2) [$\gamma_2^1$]{}; at (6,0.9) [$\gamma_2^2$]{}; at (6.2,-0.9) [$\gamma_2^{n-1}$]{}; at (6,-2) [$\gamma_2^n$]{};
(8.8,0) – (10,0);
(6,-2.3) –(6,-3.3) ; at (6.2,-2.8) [$\pi$]{};
(0,-4) circle(0.1); (4,-4) circle(0.1); (8,-4) circle(0.1);
at (0,-3.6) [$b_1$]{}; at (4,-3.6) [$b_2$]{}; at (8,-3.6) [$b_3$]{};
(0,-4) –(2,-4); (2,-4) –(4,-4); (4,-4) –(6,-4); (6,-4) –(8,-4);
at (2,-3.6) [$\gamma_1$]{}; at (6,-3.6) [$\gamma_2$]{};
(8.8,-4) – (10,-4);
Next, we will determine the image of ${i_0}_* : H_1( V_0 \cap V_1 , \mathbb{Z} ) \to H_1( V_0 , \mathbb{Z})$. For this we introduce a convention on the ordering of the paths $\gamma_i^j$ as follows. For each $i$, let $D_i$ be a small disc in $D$ centered around the point $b_i$. We choose these discs small enough so that they are mutually disjoint. Let $\tilde{D}_i = \pi^{-1}(D_i)$. Then $\gamma_i^1 , \gamma_i^2 , \dots , \gamma_i^{n}$ divide $\tilde{D}_i$ into $n$ segments. We will assume that the lifts $\gamma_i^j$ have been ordered in such a way that for $i < i < k$ and $1 \le j \le n-1$, we have that $\gamma_{i-1}^j \cap \pi^{-1}(D_i)$ lies in the segment between $\gamma_i^j$ and $\gamma_i^{j+1}$. By induction on $i$, such an ordering exists. Henceforth we will assume that such an ordering has been chosen. Figure \[figorder\] shows the order of paths entering and leaving the point $u_i$ under our convention.
(0,0) coordinate (P); (60:3) coordinate (Q); (120:3) coordinate (R);
\(P) – (3,0); (P) – (Q); (P) – (R) ;
\(P) – (30:3); (P) – (90:3); (P) – (150:3);
at (0,-0.4) [$u_i$]{}; at (3.4,0) [$\gamma_i^1$]{}; at (60:3.4) [$\gamma_i^2$]{}; at (120:3.4) [$\gamma_i^3$]{}; at (30:3.4) [$\gamma_{i-1}^1$]{}; at (90:3.4) [$\gamma_{i-1}^2$]{}; at (150:3.4) [$\gamma_{i-1}^3$]{};
(-2,0) arc (180:340:2);
\[propboundary\] The image of ${i_0}_* : H_1( V_0 \cap V_1 , \mathbb{Z} ) \to H_1( V_0 , \mathbb{Z})$ is the subgroup spanned by $\partial , t\partial , \dots , t^{n-1}\partial$, where
$$\partial = c_1 + (1+t)c_2 + (1+t+t^2)c_3 + \dots + (1 + t + \dots + t^{n-2})c_{k-1}.$$
Recall that $V_0 \cap V_1$ may be identified with the boundary of $V_0$, which a disjoint union of $n$ circles. Moreover $t : S \to S$ cyclically permutes the circles. Let $\partial_D$ be a clockwise loop in $D$ enclosing the branch points $b_1 , \dots , b_k$. Let $\partial$ be any lift of $\partial_D$ to a loop in $S$. It follows that the image of ${i_0}_*$ is spanned by the cycles $\partial , t\partial , \dots , t^{n-1}\partial$. Thus it remains to determine the cycle $\partial$. We think of $\partial$ as consisting of an upper segment $\partial^+$ and a lower segment $\partial^-$.
(0,0) circle(0.05); (1,0) circle(0.05); (5,0) circle(0.05);
(1.4, 0)–(4.6, 0);
(0,0) to \[out=30, in=180\] (2.5,1) ; (2.5,1) to \[out=0, in=150\] (5,0) ;
(0,0) to \[out=330, in=180\] (2.5,-1) ; (2.5,-1) to \[out=0, in=210\] (5,0) ;
at (2.5,1.4) [$\partial^+$]{}; at (2.5,-1.4) [$\partial^-$]{};
at (-0.2,0) [$b_1$]{}; at (0.8,0) [$b_2$]{}; at (5.3,0) [$b_k$]{};
Due to our ordering conventions, it is easy to see that $\partial^+$ can be lifted to $\gamma_1^1 + \gamma_2^1 + \dots + \gamma_{k-1}^1$ and $\partial^-$ to $-\gamma_1^2 - \gamma_2^3 - \gamma_3^4 - \dots - \gamma_{k-1}^n$. We then have: $$\begin{aligned}
\partial &= (\gamma_1^1 - \gamma_1^2) + (\gamma_2^1 - \gamma_2^3) + \dots + (\gamma_{k-1}^1 - \gamma_{k-1}^n) \\
& = c_1 + (1+t)c_2 + (1+t+t^2)c_3 + \dots + (1+t + \dots + t^{n-2})c_{k-1}
\end{aligned}$$ as required.
As a $\mathbb{Z}[t]$-module, $\Lambda_{S,0}$ is isomorphic to $k-2$ copies of $\mathbb{Z}[t]/ \langle 1 + t + t^2 + \dots + t^{n-1} \rangle$.
We shall use the same notation $t^j c_i$ for the cycle in $H_1( V_0 , \mathbb{Z})$ and for its image in $\Lambda_{S,0}$. This should not cause confusion, since from this point onward we will only be concerned with $\Lambda_{S,0}$ as opposed to $H_1( V_0 , \mathbb{Z})$. We have that the $t^j c_i$ span $\Lambda_{S,0}$, but are not linearly independent, since by Proposition \[propboundary\] we have the relation $$c_1 + (1+t)c_2 + (1+t+t^2)c_3 + \dots + (1+t+ \dots + t^{n-2})c_{k-1} = 0.$$
Intersection form {#secintf}
-----------------
Consider the intersection pairing $\langle \; , \; \rangle : \Lambda_S \times \Lambda_S \to \mathbb{Z}$ defined by the cup product. Clearly $\langle \; , \; \rangle$ is $t$-invariant and the decomposition $\Lambda_S = \Lambda_{S,0} \oplus \Lambda_{S,1}$ is orthogonal. Under the identification $\Lambda_{S,1} = \oplus_{i=1}^n H^1(\Sigma , \mathbb{Z})$, we have that the restriction $\langle \; , \; \rangle|_{\Lambda_{S,1}}$ is given by $n$ copies of the usual intersection form on $H^1(\Sigma , \mathbb{Z})$. The intersection form on $\Lambda_{S,0}$ is given by the following:
\[propints\] We have the following intersection pairings: $$\begin{aligned}
\langle c_i , tc_i \rangle &= 1, &
\langle c_i , t^j c_i \rangle &= 0, \; \text{ for } 2 \le j \le n-2, \\
\langle c_i , c_{i+1} \rangle &= 1, &
\langle c_i , t^j c_{i+1} \rangle &= 0, \; \text{ for } 2 \le j \le n-2, \\
\langle c_i , tc_{i+1} \rangle &= -1, &
\langle c_i , t^j c_{i'} \rangle & = 0, \; \text{ whenever } |i-i'| > 1.
\end{aligned}$$
We choose representatives of the cycles $t^j c_i$ meeting transversally and directly compute their intersection. Figure \[figint1\] shows the computation of $\langle c_1 , tc_1 \rangle = 1$ and Figure \[figint2\] shows the computations of $\langle c_1 , c_2 \rangle = 1$ and $\langle c_1 , tc_2 \rangle = -1$. The remaining intersections are computed similarly.
The intersection relations described in Proposition \[propints\] may be visualised as a graph, shown in Figure \[figintgr\]. Here the vertices correspond to the elements $\{ t^j c_i\}$, for $1 \le i \le k-1$ and $0 \le j \le n-2$. We draw an oriented edge from from $u$ to $v$ whenever $\langle u , v \rangle = 1$.
(0,0) coordinate (P); (60:3) coordinate (Q); (120:3) coordinate (R);
\(P) – (3,0); (P) – (Q); (P) – (R) ;
at (0,-0.4) [$u_1$]{}; at (3.4,0) [$\gamma_1^1$]{}; at (60:3.4) [$\gamma_1^2$]{}; at (120:3.4) [$\gamma_1^3$]{};
(3,0.4) coordinate (A); (1.85,2.40) coordinate (B); (0,0.4) coordinate (C); (0.35,-0.2) coordinate (D);
\(A) .. controls (C) and (D) .. (B);
(1.15,2.80) coordinate (E); (-1.15,2.80) coordinate (F); (-0.35,0.2) coordinate (G); (0.35,0.2) coordinate (H);
\(E) .. controls (G) and (H) .. (F);
at (3,0.7) [$c_1$]{}; at (1,3) [$tc_1$]{};
(6,0) coordinate (S); (6,0)+(60:3) coordinate (T); (6,0)+(120:3) coordinate (U);
\(S) – (9,0); (S) – (T); (S) – (U) ;
at (6,-0.4) [$u_2$]{}; at (9.4,0) [$\gamma_1^1$]{}; at (7.7, 2.94) [$\gamma_1^2$]{}; at (4.3, 2.94) [$\gamma_1^3$]{};
(9,-0.4) coordinate (A1); (7.15, 2.80) coordinate (B1); (6,-0.4) coordinate (C1); (5.65,0.2) coordinate (D1);
(A1) .. controls (C1) and (D1) .. (B1);
(7.85,2.40) coordinate (E1); (4.15,2.40) coordinate (F1); (6.35,-0.2) coordinate (G1); (5.65, -0.2) coordinate (H1);
(E1) .. controls (G1) and (H1) .. (F1);
at (9,-0.7) [$c_1$]{}; at (8.3,2.4) [$tc_1$]{};
(0,0) coordinate (P); (60:3) coordinate (Q); (120:3) coordinate (R);
(30:3) coordinate (S);
\(P) – (3,0); (P) – (Q); (P) – (R) ; (P) – (S); (P) – (0,3);
at (0,-0.4) [$u_2$]{}; at (3.4,0) [$\gamma_2^1$]{}; at (60:3.4) [$\gamma_2^2$]{}; at (120:3.4) [$\gamma_2^3$]{};
(3,0.4) coordinate (A); (1.85,2.40) coordinate (B); (0,0.4) coordinate (C); (0.35,-0.2) coordinate (D);
\(A) .. controls (C) and (D) .. (B);
(2.8,1.15) coordinate (E); (-0.4,3) coordinate (F); (0.2,-0.35) coordinate (G); (-0.4,0) coordinate (H);
\(E) .. controls (G) and (H) .. (F);
at (3.2,0.6) [$c_2$]{}; at (-0.7,3.2) [$c_1$]{};
at (30:3.4) [$\gamma_1^1$]{}; at (0,3.4) [$\gamma_1^2$]{};
(1.15,2.80) coordinate (E); (-1.15,2.80) coordinate (F); (-0.35,0.2) coordinate (G); (0.35,0.2) coordinate (H);
\(E) .. controls (G) and (H) .. (F);
at (1,3) [$tc_2$]{};
(0,0) circle(0.05); (1,0) circle(0.05); (2,0) circle(0.05); (8,0) circle(0.05); (0,-1) circle(0.05); (1,-1) circle(0.05); (2,-1) circle(0.05); (8,-1) circle(0.05); (0,-2) circle(0.05); (1,-2) circle(0.05); (2,-2) circle(0.05); (0,-4) circle(0.05); (0,-5) circle(0.05); (1,-5) circle(0.05); (8,-5) circle(0.05);
(8,-4) circle(0.05); (7,-5) circle(0.05); (7,-4) circle(0.05); (7,0) circle(0.05);
(0,0)–(1,0); (1,0)–(2,0); (0,-1)–(1,-1); (1,-1)–(2,-1); (0,-2)–(1,-2); (1,-2)–(2,-2); (0,0)–(0,-1); (1,0)–(1,-1); (2,0)–(2,-1); (0,-1)–(0,-2); (1,-1)–(1,-2); (2,-1)–(2,-2);
(8,0)–(8,-1); (0,-4)–(0,-5); (0,-5)–(1,-5);
(1,-1)–(0,0); (2,-1)–(1,0); (1,-2)–(0,-1); (2,-2)–(1,-1); (1,-5)–(0,-4);
(0,-2)–(0,-2.3); (1,-2)–(1,-2.3); (2,-2)–(2,-2.3);
(2,0)–(2.3,0); (2,-1)–(2.3,-1); (2,-2)–(2.3,-2);
(8,-1)–(8,-1.3);
(0,-4)–(0,-3.7); (0,-4)–(0.3,-4);
(1,-5)–(1,-4.7); (1,-5)–(1.3,-5);
(2,0)–(2.3,-0.3); (2,-1)–(2.3,-1.3); (2,-2)–(2.3,-2.3); (1,-2)–(1.3,-2.3); (0,-2)–(0.3,-2.3);
(7,0)–(8,0); (8,-1)–(7,0);
(7,-4)–(8,-4); (7,-4)–(7,-5); (8,-4)–(8,-5); (7,-5)–(8,-5); (8,-5)–(7,-4);
(8,-4)–(8,-3.7); (8,-4)–(7.7,-3.7);
(7,-4)–(7,-3.7); (7,-4)–(6.7,-3.7); (7,-4)–(6.7,-4);
(7,-5)–(6.7,-4.7); (7,-5)–(6.7,-5);
(7,0)–(7,-0.3); (7,0)–(6.7,0); (8,-1)–(7.7,-1);
(0,-2.4)–(0,-3.6); (2.4,0)–(6.6,0); (8,-1.4)–(8,-3.6); (1.4,-5)–(6.6,-5);
at ( 0 , 0.2 ) [$c_1$]{}; at (-0.3 , -1 ) [$tc_1$]{}; at (-0.4 , -2 ) [$t^2c_1$]{};
at (-0.6 , -4 ) [$t^{n-3}c_1$]{}; at (-0.6 , -5 ) [$t^{n-2}c_1$]{};
at (1 , 0.2) [$c_2$]{}; at (2 , 0.2) [$c_3$]{};
at (7,0.2) [$c_{k-2}$]{}; at (8 , 0.2) [$c_{k-1}$]{}; at (8.6 , -1) [$tc_{k-1}$]{}; at (8.9 , -4) [$t^{n-3}c_{k-1}$]{}; at (8.9 , -5) [$t^{n-2}c_{k-1}$]{};
Decomposition of $\Lambda_P$ {#secdp}
----------------------------
Recall the decomposition $\Lambda_S = \Lambda_{S,0} \oplus \Lambda_{S,1}$. Let $i_u : \Lambda_{S,u} \to \Lambda_S$ for $u=0,1$ be the inclusions and $j_u : \Lambda_S \to \Lambda_{S,u}$ the projections. Recall also the identification $\Lambda_{S,1} = \frac{\mathbb{Z}[t]}{\langle t^n - 1 \rangle} \otimes_{\mathbb{Z}} \Lambda_\Sigma$.
\[proppi\] The map $\pi^* : \Lambda_\Sigma \to \Lambda_S$ factors as $\pi^* = i_1 \circ \pi_1^*$, where $\pi_1^* : \Lambda_\Sigma \to \Lambda_{S,1}$ is given by $\pi_1^*(a) = (1+t+ \dots + t^{n-1})a$. The map $\pi_* : \Lambda_S \to \Lambda_\Sigma$ factors as $\pi_* = (\pi_1)_* \circ j_1$, where $(\pi_1)_* : \Lambda_{S,1} \to \Lambda_\Sigma$ is given by $(\pi_1)_*( t^j a) = a$, for $a \in \Lambda_\Sigma$.
Let $i : D \to \Sigma$ be a trivialising disc and $\Lambda_S = \Lambda_{S,0} \oplus \Lambda_{S,1}$ the corresponding decomposition. Any class in $\Lambda_\Sigma$ can be represented by a cycle lying outside of $D$ and the factorsation $\pi^* = i_1 \circ \pi_1^*$ follows. Similarly, any class in $\Lambda_{S,0}$ is represented by a cycle whose image under $\pi$ lies in $D$. So $\pi_*$ is trivial on $\Lambda_{S,0}$ and the factorisation $\pi_* = (\pi_1)_* \circ j_1$ follows.
The decomposition $\Lambda_S = \Lambda_{S,0} \oplus \Lambda_{S,1}$ induces a similar decomposition $\Lambda_P = \Lambda_{P,0} \oplus \Lambda_{P,1}$, where $\Lambda_{P,0} = \Lambda_{S,0}$ and $\Lambda_{P,1}$ is the kernel of the map $(\pi_1)_* : \Lambda_{S,1} \to \Lambda_\Sigma$ defined in Proposition \[proppi\]. Moreover, we have an isomorphism of $\mathbb{Z}[t]$-modules $\Lambda_{P,1} \simeq \mathbb{Z}[t]/\langle 1 + t + \dots + t^{n-1} \rangle \otimes_{\mathbb{Z}} \Lambda_\Sigma$.
\[corpolarization\] The polarization type of $\Lambda_P$ is $(1,1, \dots , 1 , n , n , \dots , n )$, where $1$ occurs $(n-2)(g-1) + n(n-1)l/2 - 1$ times and $n$ occurs $g$ times.
Since the decomposition $\Lambda_S = \Lambda_{S,0} \oplus \Lambda_{S,1}$ is orthogonal and $\langle \; , \; \rangle$ is unimodular on $\Lambda_S$, it follows that the restriction of $\langle \; , \; \rangle$ to $\Lambda_{S,0}$ or $\Lambda_{S,1}$ is unimodular. So the type of $\Lambda_{P,0} = \Lambda_{S,0}$ is $(1,1, \dots , 1)$. Choose a symplectic basis $a_1,b_1 , \dots , a_g , b_g$ of $\Lambda_\Sigma$. There is an induced decomposition of $\Lambda_{P,1}$ into $g$ orthogonal $\mathbb{Z}[t]$-submodules $M_1,M_2, \dots , M_g$, where $M_u$ is spanned by $\{ (1-t)a_u , t(1-t)a_u , \dots , t^{n-2}(1-t)a_u , (1-t)b_u , t(1-t)b_u , \dots , t^{n-2}(1-t)b_u \}$. The intersection matrix on $M_u$ can be obtained as the tensor product of the Cartan matrix for $A_{n-1}$ with the standard $2 \times 2$ symplectic matrix $\left[\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right]$. It follows that the type of $M_u$ is given by the invariants of the $A_{n-1}$ Cartan matrix, which are $(1,1, \dots , 1 , n)$, where there are $n-1$ entries equal to $1$. Thus the type of $\Lambda_{P,1}$ is of the form $(1,1, \dots , n , \dots , n)$, where there are $g$ copies of $n$.
In order to apply the classification of vanishing lattices in Section \[secthevanlat\], it will be necessary to consider cohomology with $\mathbb{Z}_2$-coefficients. Let $\Lambda_S[2] = \Lambda_S \otimes_\mathbb{Z} \mathbb{Z}_2$ be the mod $2$ reduction of $\Lambda_S$ and similarly define $\Lambda_P[2],\Lambda_\Sigma[2]$.
\[proporthogsplit\] Suppose $n$ is odd. Then $\pi^* : \Lambda_\Sigma[2] \to \Lambda_S[2]$ gives a splitting of the sequence $$\xymatrix{
0 \ar[r] & \Lambda_P[2] \ar[r] & \Lambda_S[2] \ar[r]^-{\pi_*} & \Lambda_\Sigma[2] \ar[r] & 0.
}$$ The decomposition $$\Lambda_S[2] = \Lambda_P[2] \oplus \Lambda_\Sigma[2]$$ induced by this splitting is orthogonal.\
Suppose $n$ is even. Then the image of $\pi^* : \Lambda_\Sigma[2] \to \Lambda_S[2]$ is contained in $\Lambda_P[2]$. We have that $\pi^*\Lambda_\Sigma[2]$ is the null space of $\langle \; , \; \rangle|_{\Lambda_P[2]}$ and hence the quotient $H = \Lambda_P[2]/\pi^*\Lambda_\Sigma[2]$ has an induced non-degenerate alternating bilinear form $\langle \; , \; \rangle_H$. The filtration $$\pi^* \Lambda_\Sigma[2] \subset \Lambda_P[2] \subset \Lambda_S[2]$$ can be split in such a way that under the induced isomorphism $$\begin{aligned}
\Lambda_S[2] &\simeq \Lambda_\Sigma[2] \oplus (\Lambda_P[2]/\Lambda_\Sigma[2]) \oplus (\Lambda_S[2]/\Lambda_P[2]) \\
&\simeq \Lambda_\Sigma[2] \oplus H \oplus \Lambda_\Sigma[2],
\end{aligned}$$ we have that $\pi^*(a) = (a,0,0)$, $\pi_*(a,b,c) = c$ and $$\langle (a,b,c) , (a',b',c') \rangle = \langle a, c' \rangle + \langle b , b' \rangle_H + \langle a' , c \rangle,$$ for all $a,a',c,c' \in \Lambda_\Sigma[2]$ and $b,b' \in H$.
For $a \in \Lambda_\Sigma$ we have $\pi_* \pi^* (a) = na$. Thus when $n$ is odd we have that $\pi^* : \Lambda_\Sigma[2] \to \Lambda_S[2]$ is a splitting of $\pi_* : \Lambda_S[2] \to \Lambda_\Sigma[2]$. For $a \in \Lambda_\Sigma[2]$, $b \in \Lambda_P[2]$, we have $\langle \pi^*(a) , b \rangle = \langle a , \pi_*(b) \rangle = 0$. So for $n$ odd, the splitting provided by $\pi^*$ is orthogonal. Now assume that $n$ is even. We have shown that $\pi^* \Lambda_\Sigma[2] \subset \Lambda_P[2]$. By non-degeneracy of $\langle \; , \; \rangle$ on $\Lambda_S[2]$ we have that $\Lambda_P^\perp[2] = ker(\pi_*)^\perp = im(\pi^*)$, so that $\pi^* \Lambda_\Sigma[2]$ is the nullspace of $\langle \; , \; \rangle|_{\Lambda_P[2]}$ as claimed. The quotient $H = \Lambda_P[2]/\pi^*\Lambda_\Sigma[2]$ then has an induced non-degenerate alternating bilinear form $\langle \; , \; \rangle_H$. The last claim concerning the splitting of the filtration is straightforward.
Quadratic functions on $\Lambda_S[2],\Lambda_P[2]$ {#secquadratics}
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Under conditions on $n$ and $l$ we will construct monodromy invariant quadratic functions on $\Lambda_S[2], \Lambda_P[2]$ and compute their Arf invariants.
Let $\overline{V}$ be a finite dimensional $\mathbb{Z}_2$-vector space and let $\langle \; , \; \rangle$ be a bilinear form on $\overline{V}$ which is alternating, i.e. $\langle x , x \rangle = 0$ for all $x \in \overline{V}$. A [*quadratic function*]{} associated to $(\overline{V} , \langle \; , \; \rangle)$ is a $\mathbb{Z}_2$-valued function $q$ on $\overline{V}$ satisfying: $$q(x+y) = q(x) + q(y) + \langle x , y \rangle$$ for all $x,y \in \overline{V}$.
Let $\overline{V}_0 = \{ x \in \overline{V} \; | \; \langle x , \; \rangle = 0 \}$ and let $p$ be the dimension of $\overline{V}_0$. Then $\overline{V}/\overline{V}_0$ is even-dimensional as it carries a non-degenerate alternating bilinear form. Let $2r$ be the dimension of $\overline{V}/\overline{V}_0$, so that $\overline{V}$ has dimension $\mu = 2r + p$. If $q$ is a quadratic function, observe that $q|_{\overline{V}_0}$ is linear. If $q$ vanishes on $\overline{V}_0$, then $q$ descends to a quadratic function on $\overline{V}/\overline{V}_0$ and we define the [*Arf invariant*]{} ${\rm Arf}(q) \in \mathbb{Z}_2$ of $q$ to be $\sum_{i=1}^r q(e_i)q(f_i)$, where $e_1 , \dots , e_r , f_1 , \dots , f_r \in \overline{V}$ project to a symplectic basis of $\overline{V}/\overline{V}_0$. This can be shown to be independent of the choice of $e_1 , \dots , f_r$. If $q|_{\overline{V}_0}$ is not identically zero, then ${\rm Arf}(q)$ is left undefined. For fixed $r,p$ there are at most $3$ isomorphism classes of quadratic functions, distinguished by whether the Arf invariant is $0,1$ or undefined.\
Let $X$ be a compact Riemann surface with canonical bundle $K_X$. Spin structures on $X$ may be identified with square roots of $K_X$. A spin structure gives a $KO$-orientation and hence a Gysin homomorphism $\varphi_X : KO(X) \to KO^{-2}(pt) = \mathbb{Z}_2$. If $E$ is a holomorphic vector bundle on $X$ with orthogonal structure, then $\varphi_X([E])$ is the mod $2$ index [@ati]: $$\varphi_X([E]) = dim \left( H^0( X , E \otimes K_X^{1/2} ) \right) \; ( \text{mod } 2).$$ Note that if $A$ is any line bundle on $X$, then $A\oplus A^{*}$ has an orthogonal structure given by pairing $A$ and $A^*$. By Riemann-Roch, we have $\varphi_X(A \oplus A^*) = deg(A) \; ( \text{mod } 2)$. The restriction of $\varphi_X$ to $H^1(X,\mathbb{Z}_2)$, the space of $\mathbb{Z}_2$-line bundles, is a quadratic function with constant term, in the sense that: $$\varphi_X( a + b ) = \varphi_X(a) + \varphi_X(b) + \langle a , b \rangle + \varphi_X(0).$$ In particular, if we let $\tilde{\varphi}_X = \varphi_X + \varphi_X(0)$, then $\tilde{\varphi}_X|_{H^1(X,\mathbb{Z}_2)}$ is a quadratic function. The spin structure $K_X^{1/2}$ is called [*even*]{} or [*odd*]{} according to whether $\varphi_X(0)$ is $0$ or $1$. If $X$ has genus $g_X$ then $\varphi_X$ has $2^{g_X-1}(2^{g_X}+1)$ zeros [@ati]. It follow that the Arf invariant of $\tilde{\varphi}|_{H^1(X,\mathbb{Z}_2)}$ equals $\varphi_X(0)$.\
Now consider the case of a spectral curve $\pi : S \to \Sigma$ and let $K_S$ denote the canonical bundle of $S$. We will use the mod $2$ index to construct a monodromy invariant quadratic function on $\Lambda_S[2]$ provided that either $n$ is odd or $n$ and $l$ are both even. We will later see that when $n$ is even and $l$ is odd, there are no monodromy invariant quadratic functions on $\Lambda_S[2]$. By the adjunction formula we have $K_S \pi^*(K^{-1}) = \pi^*(L^{n-1})$. Under the assumption that either $n$ is odd or $n$ and $l$ are both even, there exists a square root $L^{(n-1)/2}$ of $L^{n-1}$ on $\Sigma$. This defines a relative $KO$-orientation for $\pi : S \to \Sigma$ and hence a Gysin homomorphism $\pi_{!} : KO(S) \to KO(\Sigma)$. If $E$ is a holomorphic vector bundle on $S$ with orthogonal structure, then $\pi_{!}$ is related to taking the direct image by: $$\pi_! [E] = [\pi_* (E \otimes \pi^*( L^{(n-1)/2}) )].$$ Note that by relative duality, $\pi_* (E \otimes \pi^*( L^{(n-1)/2}) )$ inherits an orthogonal structure from the orthogonal structure on $E$. Now choose a square root $K^{1/2}$ of $K$ and set $K_S^{1/2} = \pi^*( L^{(n-1)/2} K^{1/2})$. As we have chosen our spin structures to be compatible with the relative spin structure, we have that $\varphi_S = \varphi_\Sigma \circ \pi_!$. Moreover, since $K^{1/2}$ and $L^{(n-1)/2}$ are defined on $\Sigma$, it is clear that $\tilde{\varphi}_S |_{\Lambda_S[2]}$ is a monodromy invariant quadratic function on $\Lambda_S[2]$.
\[proparfinvq\] Suppose that either $n$ is odd or $n$ and $l$ are even. Choose square roots of $K$ and $L^{(n-1)}$ on $\Sigma$ so that the function $\varphi_S : KO(S) \to \mathbb{Z}_2$ is defined. Then:
1. [The restriction of $\tilde{\varphi}_S$ to $\Lambda_P[2]$ is independent of the choice of square roots and defines a monodromy invariant quadratic function $q : \Lambda_P[2] \to \mathbb{Z}_2$.]{}
2. [If $n = 2m+1$ is odd, then $Arf(q) = (m(m-1)/2)l \; (\text{mod } 2)$.]{}
3. [If $n = 2m$ is even, then $Arf(q) = m(l/2) \; (\text{mod } 2)$.]{}
To show independence of choices, suppose we replace $K^{1/2}$ and $L^{(n-1)/2}$ by $K^{1/2} \otimes A_1$ and $L^{(n-1)/2} \otimes A_2$, where $A_1,A_2 \in \Lambda_\Sigma[2]$. Then $K_S^{1/2}$ is replaced with $K_S^{1/2} \otimes \pi^*(A)$, where $A = A_1 \otimes A_2$. This has the effect of replacing the function $x \mapsto \varphi_S(x)$ with $x \mapsto \varphi_S(x+\pi^*[A])$. But if $x \in \Lambda_P[2]$ then $\langle x , \pi^*[A] \rangle = 0$, so $$\begin{aligned}
\varphi_S(x+\pi^*[A]) + \varphi_S(\pi^*[A]) &= \varphi_S(x) + \varphi_S(\pi^*[A]) + \langle x , \pi^*[A] \rangle + \varphi_S(0) + \varphi_S(\pi^*[A])\\
& = \varphi_S(x) + \varphi_S(0) \\
& = \tilde{\varphi}_S(x),
\end{aligned}$$ which shows independence of $\tilde{\varphi}_S$ on the choice of square roots.\
Now suppose that $n = 2m+1$ is odd. If $A \in \Lambda_\Sigma[2]$, we find that $$\begin{aligned}
\varphi_S( \pi^*[A]) &= \varphi_\Sigma( \pi_! \pi^*[A]) \\
&= \varphi_\Sigma( [A \otimes ( L^m \oplus L^{m-1} \oplus \dots \oplus L^{-m} ) ]) \\
&= ml + (m-1)l + \dots + l + \varphi_\Sigma([A]) \\
& = (m(m-1)/2 )l + \varphi_\Sigma([A]).
\end{aligned}$$ In this calculation we have used that $\pi_* \mathcal{O}_S = \mathcal{O}_\Sigma \oplus L^{-1} \oplus L^{-2} \oplus \dots \oplus L^{-(n-1)}$ [@bnr]. It follows that $\varphi_S(0) = (m(m-1)/2)l + \varphi_\Sigma(0)$ and that $\tilde{\varphi}_S(\pi^*[A]) = \tilde{\varphi}_\Sigma([A])$. Thus the restriction of $\tilde{\varphi}_S$ to $\Lambda_\Sigma[2]$ has Arf invariant equal to $\varphi_\Sigma(0)$. Now since $n$ is odd, we have an orthogonal decomposition $\Lambda_S[2] = \Lambda_P[2] \oplus \Lambda_\Sigma[2]$ and from the additivity of the Arf invariant we have $\varphi_S(0) = Arf(q) + \varphi_\Sigma(0)$. Hence $Arf(q) = \varphi_S(0) + \varphi_\Sigma(0) = (m(m-1)/2)l$.\
Lastly, suppose that $n$ and $l$ are even and set $n = 2m$. If $A \in \Lambda_\Sigma[2]$, we find that $$\begin{aligned}
\varphi_S( \pi^*[A]) &= \varphi_\Sigma( \pi_! \pi^*[A]) \\
&= \varphi_\Sigma( [A \otimes ( L^{(2m-1)/2} \oplus L^{(2m-3)/2} \oplus \dots \oplus L^{-(2m-1)/2} ) ]) \\
&= (2m-1)l/2 + (2m-3)l/2 + \dots + l/2. \\
&= m(l/2) \; ( \text{mod } 2).
\end{aligned}$$ In particular, $\varphi_S(0) = m(l/2)$ and $\tilde{\varphi}_S( \pi^*[A]) = 0$ for all $A \in \Lambda_\Sigma[2]$. By Proposition \[proporthogsplit\], we have $\Lambda_S[2] = \Lambda_\Sigma[2] \oplus H \oplus \Lambda_\Sigma[2]$, where $H$ is orthogonal to the two factors of $\Lambda_\Sigma[2]$. Let $J$ be the subspace of $\Lambda_S[2]$ given by $J = \Lambda_\Sigma[2] \oplus 0 \oplus \Lambda_\Sigma[2]$. Then we have an orthogonal decomposition $\Lambda_S[2] = H \oplus J$. By the above computation, $\tilde{\varphi}_S$ has at least $2^{2g}$ zeros on the subspace $J$, hence the Arf invariant of $\tilde{\varphi}_S |_J$ is $0$. This implies that $Arf( \tilde{\varphi}_S |_H ) = Arf( \tilde{\varphi}_S|_{\Lambda_S[2]} ) = \varphi_S(0) = m(l/2)$. Now recall that $\Lambda_P[2] = \pi^*\Lambda_\Sigma[2] \oplus H$, and that $\pi^*\Lambda_\Sigma[2]$ is the null space of $\langle \; , \; \rangle|_{\Lambda_P[2]}$. We have just shown that $\tilde{\varphi}_S$ vanishes on $\pi^* \Lambda_\Sigma[2]$, so the Arf invariant of $\tilde{\varphi}_S |_{\Lambda_P[2]}$ is defined and equals the Arf invariant of $\tilde{\varphi}_S |_H$, which is $m(l/2)$.
Constructing vanishing cycles {#secvc}
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General construction
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Our goal is to construct vanishing cycles associated to singular spectral curves and to show that the corresponding transvections occur in the monodromy of the Hitchin fibration. These vanishing cycles will occur as a special case of the general construction given in this section.\
Let $X$ be a Riemann surface which may be non-compact and let $f : Y \to X$ be a degree $n$ branched cover satisfying the following conditions:
- [All branch points have ramification index $n$.]{}
- [There is an action of the cyclic group $\mathbb{Z}_n$ on $Y$ by deck transformations.]{}
We let $t : Y \to Y$ be a generator of the $\mathbb{Z}_n$-action. Let $b_1, b_2 , \dots , b_k$ denote the branch points of $f : Y \to X$. By assumption each branch point $b_i \in X$ has a unique ramification point $u_i \in Y$ lying over it and $f^{-1}(b_i) = u_i$.\
Let $i \neq j$ and suppose that $\gamma : [0,1] \to X$ is an embedded path in $X$ joining $b_i$ to $b_j$ for which $\gamma(t)$ is not a branch point for any $t \in (0,1)$. Then $f^{-1}( \gamma([0,1]) )$ is the union of $n$ paths $\gamma^1 , \gamma^2 , \dots , \gamma^n : [0,1] \to Y$, each of which goes from $u_i$ to $u_j$. We may order the paths such that $t\gamma^i = \gamma^{i+1}$ for $i=1 , \dots , n-1$. We think of $\gamma^i$ as $1$-chains in $Y$ with boundary $u_j - u_i$. Thus $\gamma^1 - \gamma^2$ is a $1$-cycle in $Y$. Let $l_\gamma \in H_1(Y , \mathbb{Z})$ be the underlying homology class. Similarly $(\gamma^2 - \gamma^3) , \dots , (\gamma^{n-1} - \gamma^n) , (\gamma^n - \gamma^1)$ define cycles $tl_\gamma , \dots , t^{n-2}l_\gamma , t^{n-1}l_\gamma \in H_1(Y , \mathbb{Z})$. Note that while $l_\gamma$ depends on the choice of lift $\gamma^1$, the collection $\{ l_\gamma , tl_\gamma , t^2l_\gamma , \dots , t^{n-1}l_\gamma \}$ depends only on $\gamma$.
We call $l_\gamma , tl_\gamma , \dots , t^{n-1}l_\gamma $ the [*vanishing cycles associated to $\gamma$*]{}.
Let $\gamma , \gamma'$ be two embedded paths from $b_i$ to $b_j$ which avoid all other branch points. We say that $\gamma,\gamma'$ are [*isotopic*]{}, if they are homotopic through a path of embedded paths from $b_i$ to $b_j$ which avoid the other branch points. If $\gamma,\gamma'$ are isotopic then clearly $\gamma$ and $\gamma'$ define the same set of vanishing cycles.
Vanishing cycles of the $A_{n-1}$ singularity {#secan1}
---------------------------------------------
We now consider a local calculation of vanishing cycles around $A_{n-1}$ singularities. This will subsequently be converted into a global calculation for spectral curves.\
Let $\mathbb{C}^2$ have coordinates $(\lambda,z)$ and consider the function $f : \mathbb{C}^2 \to \mathbb{C}$ given by $f(\lambda , z) = \lambda^n +z^2$. The zero locus of $f$ is a hypersurface in $\mathbb{C}^2$ with an isolated singularity at $(0,0)$. The germ of this hypersurface around $(0,0)$ is the $A_{n-1}$ plane curve singularity. Let $\mathbb{C}^n = \mathbb{C}^2 \times \mathbb{C}^{n-2}$ have coordinates $(\lambda , z , u_1 , u_2 , \dots , u_{n-2})$ and consider the map $\tilde{f} : \mathbb{C}^n \to \mathbb{C}^{n-1}$ given by $$\tilde{f}(\lambda , z , u_1 , u_2 , \dots , u_{n-2} ) = ( \lambda^n + u_{n-2}\lambda^{n-2} + \dots + u_2 \lambda^2 + u_1 \lambda + z^2 , u_1 , u_2, \dots , u_{n-2}).$$ This is a versal deformation of the $A_{n-1}$ singularity. Let $B$ be a sufficiently small open ball around the origin in $\mathbb{C}^n$ with closure $\overline{B}$ and boundary $\partial B$. For any such $B$, there exists a sufficiently small open ball $B'$ around the origin in $\mathbb{C}^{n-1}$ such that $\tilde{f}$ is a submersion along $\partial B \cap \tilde{f}^{-1}(B')$. Let $D' \subset B'$ be the set of critical values of $\tilde{f}$ restricted to $\overline{B} \cap \tilde{f}^{-1}(B')$. The restriction of $\tilde{f}$ to $\overline{B} \cap \tilde{f}^{-1}( B' \setminus D')$ is a smooth fibre bundle over $B' \setminus D'$, the [*Milnor fibration*]{} associated to the germ of $\tilde{f}$ around $0$. Let $b \in B' \setminus D'$ be a regular value of $\tilde{f}$ in $B'$ and let $\overline{X}_b = \tilde{f}^{-1}(b) \cap \overline{B}$ be the [*(compact) Milnor fibre*]{}. The compact Milnor fibre is a compact manifold with boundary $\partial \overline{X}_b = \tilde{f}^{-1}(b) \cap \partial B$. The [*geometric monodromy*]{} [@loo] of the Milnor fibration is given by a representation $\rho_{geom} : \pi_1( B' \setminus D' , b ) \to Iso^0( \overline{X}_b , \partial \overline{X}_b )$, where $Iso^0( \overline{X}_b , \partial \overline{X}_b )$ is the group of relative isotopy classes of homeomorphisms of $\overline{X}_b$ which are the identity on $\partial \overline{X}_b$.\
Let $w_0 \in \mathbb{C} \setminus \{0\}$ be small enough that $(w_0 , 0 , 0, \dots , 0) \in B'$. Then we take $b = (w_0 , 0 , \dots , 0) \in B'$ as a basepoint. The fibre of $\tilde{f}$ over $b$ is given by the equation $\lambda^n + z^2 - w_0 = 0$, which is smooth for any $w_0 \neq 0$, so $b \in B' \setminus D'$. We have: $$\overline{X}_b = \tilde{f}^{-1}(b) \cap \overline{B} = \{ (\lambda , z) \; | \; \lambda^n + z^2 - w_0 = 0, \; \; (\lambda,z ,0 , \dots , 0) \in \overline{B} \}.$$ Observe that $\overline{X}_b$ is a branched cover of a closed ball in $\mathbb{C}$ via the map $(\lambda , z) \mapsto z$. This is a degree $n$ cyclic branched cover with branch points $\pm\sqrt{w_0}$. As with cyclic spectral curves, we let $t : \overline{X}_b \to \overline{X}_b$ be the generator of the cyclic action given by $t(\lambda , z) = (\xi \lambda ,z)$, where $\xi = e^{2\pi i/n}$. Let $\gamma : [0,1] \to \mathbb{C}$ be the straight line in $\mathbb{C}$ joining $-\sqrt{w_0}$ to $\sqrt{w_0}$ (the choice of which square root of $w_0$ is taken to be $\sqrt{w_0}$ will be unimportant). Associated to $\gamma$ we have the vanishing cycles $l_\gamma , tl_\gamma , \dots , t^{n-1}l_\gamma$.
\[propgeommono\] The geometric monodromy representation $\rho_{geom} : \pi_1( B' \setminus D' , b ) \to Iso^0( \overline{X}_b , \partial \overline{X}_b )$ is generated by Dehn twists of $\overline{X}_b$ around the loops $l_\gamma , tl_\gamma , \dots , t^{n-2}l_\gamma$.
Of course the Dehn twist around $t^{n-1}\gamma$ is also in the image of the geometric monodromy representation, but can be expressed in terms of $l_\gamma , \dots , t^{n-2}l_\gamma$.
Recall that $D'$ is the set of critical values of $\tilde{f}$ restricted to $\overline{B} \cap \tilde{f}^{-1}(B')$. It is easy to see that $D'$ is given by: $$D' = \{ (w , u_1 , u_2 , \dots, u_{n-2} ) \in B' \; | \; \lambda^n + u_{n-2} \lambda^{n-2} + \dots + u_1 \lambda - w \text{ has multiple roots } \}.$$ Then $\pi_1( B' \setminus D' , b)$ is the $n$-th Artin braid group with generators $\sigma_1 , \sigma_2 , \dots , \sigma_{n-1}$ exchanging pairs of roots of $\lambda^n - w_0$. More precisely, let $q \in \mathbb{C}$ be an $n$-th root of $w_0$, so $q^n = w_0$. Then $$\lambda^n - q^n = (\lambda - q)(\lambda - \xi q) \dots (\lambda- \xi^{n-1}q).$$ We construct loops in $B' \setminus D'$ representing the generators $\sigma_1 , \dots , \sigma_{n-1}$ as follows. For $i = 1, \dots , n-1$, we can find a loop in $B' \setminus D'$ exchanging $\xi^{i-1} q , \xi^i q$ along paths joining $\xi^{i-1} q$ and $\xi^{i}q$ while keeping all other roots fixed. To be specific, let $\tau_i^+,\tau_i^- : [0,1] \to \mathbb{C}$ be the paths $$\begin{aligned}
\tau_i^+(t) &= \xi^{i-1}\left( 1 + \frac{1-\xi}{2}(e^{i\pi t} - 1) \right)q, \\
\tau_i^-(t) &= \xi^{i-1}\left( \xi + \frac{\xi-1}{2}(e^{i\pi t} - 1) \right)q.
\end{aligned}$$ Then we let $\sigma_i \in \pi_1(B' \setminus D' , b)$ be the loop in which $\xi^{i-1}q$ moves on $\tau_i^+$, $\xi^i q$ moves on $\tau_i^-$ and all other roots fixed. It follows that the geometric monodromy corresponding to $\sigma_i$ is given by a Dehn twist around a cycle $c_i$, which we now define. Let $\tau_i(t) = \xi^{i-1}\left( 1 + (\xi-1)t \right)q$ be the straight line path joining $\xi^{i-1}q$ to $\xi^{i}q$. The cycle $c_i$ may be taken to be the pre-image of $\tau_i$ in $\overline{X}_b$ under the branched double cover $(\lambda , z) \mapsto \lambda$. We have shown that $\rho_{geom}(\sigma_i)$ is a Dehn twist around the loop $c_i$. To conclude we note that it is easy to see that the cycles $c_1,c_2, \dots , c_{n-1}$ are homologous to $l_\gamma , tl_\gamma , t^{n-2}l_\gamma$ (possibly after a cyclic re-ordering of $l_\gamma , \dots , t^{n-1}l_\gamma$).
Vanishing cycles for spectral curves
------------------------------------
In this section we will construct a collection of vanishing cycles $\alpha \in H^1(S , \mathbb{Z})$ such that the corresponding transvections $T_\alpha$ generate the monodromy action of the Hitchin system. We continue to assume that we have chosen a basepoint $a_0 \in \mathcal{A}^{\rm reg}$ of the form $a_0 = (0, \dots , 0 , a_n)$, where $a_n$ has only simple zeros.\
Let $i \neq j$ and suppose that $\gamma : [0,1] \to \Sigma$ is an embedded path in $\Sigma$ joining $b_i$ to $b_j$ for which $\gamma(t)$ is not a branch point for any $t \in (0,1)$. Let $l_\gamma , tl_\gamma , \dots , t^{n-1} l_\gamma \in H_1(S , \mathbb{Z})$ be the corresponding vanishing cycles. We let $c_\gamma , tc_\gamma , \dots , t^{n-1}c_\gamma \in H^1(S , \mathbb{Z})$ be the Poincaré dual cohomology classes. We note that $\pi_*( t^j c_\gamma) = 0$ for all $j$, so $t^j c_\gamma \in \Lambda_P$. In particular the transvection $T_{t^j c_\gamma} : \Lambda_S \to \Lambda_S$ preserves $\Lambda_P$. The main result of this section is the following:
\[thmallowedmonodromy\] The transvections $T_{c_\gamma} , T_{tc_\gamma} , \dots , T_{t^{n-1}c_\gamma }$ belong to the $SL(n,\mathbb{C})$ monodromy group.
Let $D$ denote the unit disc in $\mathbb{C}$ and choose an embedding $e : D \to \Sigma$ such that $b_i = e( -1/2 )$, $b_j = e(1/2)$ and $\gamma(t) = e(t-1/2)$. We can also choose $D$ so that the image $e(D)$ contains no other branch points. For each $w \in D$, let $D_w$ be the degree $2$ divisor given by $D_w = e(\sqrt{w}) + e(-\sqrt{w})$. Then $D_w$ consists of two distinct points for $w \neq 0$ and $D_0 = 2 e(0)$. We also have $D_{1/4} = b_i + b_j$. Let $D^*$ be the divisor $D^* = \sum_{k \neq i,j} b_j$. Denote by $S^m \Sigma$ the $m$-th symmetric product and $\alpha : S^m \Sigma \to Jac_m(\Sigma)$ the Abel-Jacobi map, where $Jac_m(\Sigma)$ is the space of degree $m$ line bundles on $\Sigma$. We also let $\widetilde{S}^m \Sigma$ denote the subspace of $S^m \Sigma$ consisting of $m$-tuples of distinct points on $\Sigma$. For $m > 2g-2$, the restriction $\widetilde{S}^m \Sigma \to Jac_m(\Sigma)$ of the Abel-Jacobi map to $\widetilde{S}^m \Sigma$ is known to be a Serre fibration [@doli]. Now consider the constant map $f : D \to \widetilde{S}^{nl-2}\Sigma$ given by $f(w) = D^*$. Then $\alpha \circ f$ is the constant map $D \to Jac_{nl-2}(\Sigma)$ taking the value $\alpha(D^*) = L^n \otimes [D_{1/4}]^{-1}$. By the lifting property of Serre fibrations, $f$ is homotopic to a map $f' : D \to \widetilde{S}^{nl-2}\Sigma$ for which $\alpha(f'(w)) = L^n \otimes [D_w]^{-1}$. Here we take homotopies relative to the point $1/4 \in D$. Thus we can assume that $f'(1/4) = D^*$.\
Consider the family of divisors $\{ D_{w} + f'(w) \}_{w \in D}$. We would like it to be true that for all $w \in D$, the divisors $D_{w}$ and $f'(w)$ have no points in common. Unfortunately this may not be the case. To avoid this, we will need to change $f'$ by a homotopy and also shrink the disc $D$ as we now explain. Define spaces $X$ and $Y$ as follows: let $X = \{ (N , w) \in \widetilde{S}^{nl-2}\Sigma \times D \; | \; [N] = L^n \otimes [D_w]^{-1} \}$. Let $Y$ be the subspace of $X$ consistsing of pairs $(N,w)$ for which some point of $N$ belongs to $D_w$. Then $X$ and $Y$ naturally fiber over $D$ giving a commutative diagram: $$\xymatrix{
Y \ar[r]^-{i} \ar[dr] & X \ar[d]^-{p} \\
& D
}$$ Clearly $X$ is a complex manifold and $Y$ a subvariety of codimension $1$. We also find that the projection $p : X \to D$ is a submersion. The map $f' : D \to X$ is a section of $p$. We may replace $f'$ by a homotopic map $f'' : D \to X$, which is a section of $p$ such that $f''(D)$ meets $Y$ transversally in smooth points of $Y$ and away from the fibre of $Y$ over $0 \in D$. We again choose our homotopy relative to the basepoint $1/4 \in D$, so that $f''(1/4) = D^* \notin Y$. So $Y$ meets $f''(D)$ in a discrete set of points different from $f''(1/4)$ and $f''(0)$. Choose an embedded path $p(t) : [0,1] \to D$ in $D$ from $1/4$ to $0$ with $f''( p(t) ) \notin Y$ for all $t$. Let $D'$ be a tubular neighborhood of the path $p([0,1])$ not meeting any points of $Y$. We can assume $D'$ is homeomorphic to a disc. Replacing $D$ with $D'$, we have obtained a map $f'' : D' \to X$ such that $f''(1/4) = D'$ and such that the divisor $f''(w)$ does not intersect with $D_w$ for any value of $w \in D'$.\
Let $\sqrt{p(t)}$ denote the square root of $p(t)$ that goes from $-1/2$ to $0$. We obtain a path $\hat{p} : [0,1] \to D$ from $-1/2$ to $1/2$ by taking $\sqrt{p(t)}$ from $-1/2$ to $0$ and then taking $-\sqrt{p(t)}$ in reverse from $0$ to $1/2$. Let $\gamma' : [0,1] \to \Sigma$ be $e \circ \hat{p}$. Then $\gamma'$ is isotopic to $\gamma$ in the space of embedded paths from $b_i$ to $b_j$ avoiding all other branch points. In particular, the vanishing cycles $c_\gamma, tc_\gamma , \dots , t^{n-1}c_\gamma$ coincide with the cycles $c_{\gamma'} , tc_{\gamma'} , \dots , t^{n-1}c_{\gamma'}$.\
Consider now the family of effective divisors $\{ D_w + f''(w) \}_{w \in D'}$ parametrised by $w \in D'$. By construction, this family has the following properties:
- [For every $w$, the line bundle associate to $D_w + f''(w)$ is $L^n$.]{}
- [When $w = 1/4$, the divisor $D_{1/4} + f''(1/4)$ is the divisor of zeros of $a_n$.]{}
- [For $w \neq 0$, the divisor $D_w + f''(w)$ has no multiple points.]{}
- [For $w=0$, the divisor $D_0 + f''(0)$ has one point of multiplicity $2$ and no other multiple points.]{}
- [If $w$ follows a loop based at $1/4$ going once around $0$, this has the effect of swapping $b_i, b_j$ along paths isotopic to $\gamma$ while all other branch points move on contractible loops.]{}
By (i), we have constructed a map $d : D' \to \mathbb{P}( H^0(\Sigma , L^n) )$. Since $D'$ is contractible this can be lifted to a map $a : D' \to H^0( \Sigma , L^n)$, where the divisor of $a(w)$ is $d(w)$. We may further choose $a$ so that $a(1/4) = a_n$. Let $a'_n = a(0) \in H^0(\Sigma , L^n)$. Then $a'_n$ has a single zero of order $2$ and no other multiple zeros. Let $x = e(0) \in \Sigma$ be the double zero of $a'_n$. Let $u \in tot(L)$ be the origin of the fibre of $L$ lying over $x$. The spectral curve $S_{a'_n}$ given by $\lambda^n + a'_n = 0$ has exactly one singularity, located at the point $u$. In a suitable local coordinate $z$ centered at $x$ we have that $a'_n(z) = z^2 (dz)^n$ and $S'_{a'_n}$ is locally given by $\lambda^n + z^2 = 0$, a plane curve singularity of type $A_{n-1}$. In what follows, we aim to show that the monodromy of this $A_{n-1}$ singularity occurs as monodromy of the Hitchin fibration and that this monodromy is generated by Picard-Lefschetz transformations in the vanishing cycles $c_\gamma , tc_\gamma , \dots , t^{n-1}c_\gamma$.\
Let $\mathcal{A}'$ be the affine subspace of $\mathcal{A}$ consisting of points $a = (a_2 , a_3 , \dots , a_{n-1} , a'_n )$ whose $H^0(\Sigma , L^n)$-term is given by $a'_n$. For each $a \in \mathcal{A}'$, the spectral curve $S_a$ passes through the point $u \in tot(L)$ and we may consider the germ of the hypersurface $S_a \subset tot(L)$ around the point $u$. Thus $\mathcal{A}'$ gives a family of deformations of the $A_{n-1}$ singularity of $S_{a'_n}$ located at $u$. We claim this is a versal family of deformations. Under the family of deformations of the $A_{n-1}$ singularity defined by the space $\mathcal{A}'$, we have that the germ of $\lambda^n + z^2$ is deformed to $\lambda^n + a_{n-2}(z) \lambda^{n-2} + \dots + a_2(z) \lambda^2 + a_1(z) \lambda + z^2$. The Kodaira-Spencer map for this family (see [@loo]) is to the map $\mathcal{A}' \to \mathbb{C}^{n-2}$ sending $(a_2 , a_3 , \dots , a_{n-1} , a'_n)$ to $(a_2(x) , a_3(x) , \dots , a_{n-1}(x))$. It is clear that for any point $x \in \Sigma$, this map surjects to $\mathbb{C}^{n-2}$, which shows that $\mathcal{A}'$ provides a versal deformation of the singularity. It follows that the geometric monodromy of Section \[secan1\] is realised as the monodromy of the Hitchin fibration around certain loops in $\mathcal{A}^{\rm reg}$, defined in the vicinity of $a'_n$. These loops act on $S$ by Dehn twists around the vanishing cycles of this $A_{n-1}$ singularity. Clearly a Dehn twist around a vanishing cycle $v \in H^1(S,\mathbb{Z})$ acts on cohomology by the corresponding Picard-Lesfschetz transformation $T_v$. To complete the proof of the theorem, it remains to identify the vanishing cycles associated to the $A_{n-1}$ singularity of $S_{a'_n}$ with specific cohomology classes in $H^1(S,\mathbb{Z})$.\
Let $B$ be a small open ball around $u \in tot(L)$. Let $U \subset \mathcal{A}$ be an open neighbourhood of $(0,0, \dots , 0 , a'_n)$ in $\mathcal{A}$ sufficiently small so that for any $a \in U$, the spectral curve $S_a$ is smooth outside of $B$. Then after possibly further shrinking $U$, we know that the monodromy of the Hitchin fibration associated to loops in $U \setminus \mathcal{D} \cap U$ is generated by transvections of the vanishing cycles associated to the $A_{n-1}$ singularity of $S_{a'_n}$. Let $t_0 \in [0,1)$ be such that $a'_0 = a( p(t_0)) \in U$. Let $\gamma'_0$ be an embedded path in $e(D)$ joining $e(\sqrt{p(t_0)})$ to $e(-\sqrt{p(t_0)})$, in fact we may take $\gamma'_0(t)$ to be the restriction of $\gamma'$ to $[t_0/2 , 1 - t_0/2]$. Then by Proposition \[propgeommono\], the monodromy of the Hitchin fibration over $U \setminus \mathcal{D} \cap U$ is generated by transvections in the vanishing cycles $c_{\gamma'_0} , \dots , t^{n-1} c_{\gamma'_0} \in H^1( S_{a'_0} , \mathbb{Z})$ associated to $\gamma'_0$. Consider the path in $\mathcal{A}^{\rm reg}$ joining $a_n$ to $a'_0$ given by restricting $a( p(t) )$ to $[0,t_0]$. The Gauss-Manin connection over this path defines an isomorphism $H^1( S_{a'_0} , \mathbb{Z}) \simeq H^1( S , \mathbb{Z})$ and it is easy to see that the vanishing cycles $c_{\gamma'_0} , tc_{\gamma'_0} , \dots , t^{n-1}c_{\gamma'_0} \in H^1(S_{a'_0} , \mathbb{Z})$ associated to $\gamma'_0$ are mapped to the vanishing cycles $c_{\gamma'} , tc_{\gamma'} , \dots , t^{n-1}c_{\gamma'} \in H^1(S , \mathbb{Z})$ associated to $\gamma'$. This proves the theorem.
In the process of proving Theorem \[thmallowedmonodromy\] we constructed a family $a : D' \to H^0(\Sigma , L^n)$ of sections of $L^n$ parametrised by a space $D'$ homeomorphic to a disc. Moreover, $D'$ contained exactly one point $0 \in D'$ for which the corresponding spectral curve was singular. If we take a loop in $D'$ that winds once anti-clockwise around $0$, the corresponding monodromy transformation is seen to be $T_{c_\gamma} T_{tc_\gamma} T_{t^2 c_\gamma} \dots T_{t^{n-2}c_\gamma}$.
Next, we will apply Theorem \[thmallowedmonodromy\] to construct some specific examples of vanishing cycles. In fact we will later see that the vanishing cycles constructed below already are sufficient to generate the monodromy group. Recall from Section \[secdecomps\] that on choosing a trivialising disc $i : \overline{D} \to \Sigma$, we obtain a decomposition $\Lambda_S = \Lambda_{S,0} \oplus \Lambda_{S,1}$, where $\Lambda_{S,0}$ is spanned by the cycles $\{ t^j c_i \}_{i,j}$ and $\Lambda_{S,1}$ can be identified with $\mathbb{Z}[t]/\langle t^n-1 \rangle \otimes_{\mathbb{Z}} \Lambda_\Sigma$. We note that once a choice of trivialising disc is given, the identification $$\label{identification}
\Lambda_{S,1} \simeq \mathbb{Z}[t]/\langle t^n-1 \rangle \otimes_{\mathbb{Z}} \Lambda_\Sigma$$ is determined only up to composition with a power of $t$.
\[propallowedcycles\] For $i = 1,2, \dots , k-1$, there exists a path from $b_i$ to $b_{i+1}$ whose associated vanishing cycles are $\{ c_i , tc_i , \dots , t^{n-1}c_i\}$. Let $a_1, b_1 , \dots , a_g , b_g$ be a symplectic basis for $\Lambda_\Sigma$. For $u = 1,2, \dots , g$, there exist paths $\gamma_{a_u}$ and $\gamma_{b_u}$ from $b_1$ to $b_2$ such that under a suitable choice of identification (\[identification\]), the associated vanishing cycles are $\{ c_1 + (1-t)a_u , t(c_1 + (1-t)a_u) , \dots , t^{n-1}(c_1 + (1-t)a_u) \}$ and $\{ c_1 + (1-t)b_u , t(c_1 + (1-t)b_u) , \dots , t^{n-1}(c_1 + (1-t)b_u) \}$.
Let $\gamma_i$ be the path from $b_i$ to $b_{i+1}$ constructed in Section \[secztmod\] ($\gamma_i$ is depicted in Figure \[figbranch\]). Then by the definition of $c_i , tc_i , \dots , t^{n-1}c_i$, it is clear that these are the vanishing cycles associated to $\gamma_i$. Consider a path $\gamma_{a_1}$ in $\Sigma$ starting at $b_1$ and moving left to the boundary of the trivialising disc, going around the loop $a_1$, then returning back to the trivialising disc and terminating at $b_2$. The corresponding vanishing cycles will clearly have the form $t^j( c_1 + (1-t)t^w a_1)$, for some value of $w$. Similarly define paths $\gamma_{b_1} , \dots , \gamma_{a_g} , \gamma_{b_g}$. The corresponding vanishing cycles will have the form $t^j( c_1 + (1-t)t^w a_u)$ or $t^j( c_1 + (1-t)t^w b_u)$ for the same value of $w$. Choosing a different identification in (\[identification\]) if necessary, we may assume $w=0$.
The vanishing lattice {#secthevanlat}
=====================
Construction of vanishing lattice on $\Lambda_P$ {#secvanlat}
------------------------------------------------
In this section we give $\Lambda_P$ the structure of a vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$, made out of vanishing cycles constructed in Section \[secvc\]. We then recall the classification of vanishing lattices in [@jan1; @jan2] and use these results to classify the vanishing lattice on $\Lambda_P$. In Section \[secmain\] we will show that the group generated by transvections in $\Delta_P$ is precisely the monodromy group $\Gamma_{SL}$ of the Hitchin fibration.\
Vanishing lattices were classified for $R = \mathbb{Z}_2$ in [@jan1] and $R = \mathbb{Z}$ in [@jan2]. If $(V , \langle \; , \; \rangle , \Delta)$ is a vanishing lattice over $\mathbb{Z}$ then we obtain a vanishing lattice $(\overline{V} , \langle \; , \; \rangle , \overline{\Delta})$ over $\mathbb{Z}_2$, where $\overline{V} = V \otimes_{\mathbb{Z}} \mathbb{Z}_2$ and $\overline{\Delta}$ is the image of $\Delta$ under mod $2$ reduction.
\[propsgen\] Let $V$ be a free $R$-module of finite rank with alternating bilinear form $\langle \; , \; \rangle$. Let $S \subseteq V$ be a subset of $V$ satisfying the following conditions:
1. [$S$ spans $V$.]{}
2. [For any two distinct elements $u,v \in S$, there exists a sequence $u = s_1 , s_2 , \dots , s_m = v$ of elements of $S$ such that $\langle s_i , s_{i+1} \rangle = \pm 1$ for, $1 \le i \le m-1$.]{}
Let $\Gamma_S$ be the subgroup of $Sp^{\#}V$ generated by $\{ T_s \}_{s \in S}$ and set $\Delta = \Gamma_S \cdot S$. Then $(V , \langle \; , \; \rangle , \Delta)$ is a vanishing lattice and $\Gamma_{\Delta} = \Gamma_S$.
Clearly $\Delta$ spans $V$, as $S$ spans $V$. If $V$ has rank $\mu > 1$ then by (i), $S$ has more than one element. Hence by (ii) there exists $\delta_1,\delta_2 \in S \subseteq \Delta$ with $\langle \delta_1 , \delta_2 \rangle = 1$. Next, we claim that for any two elements $u,v \in S$, we have $u = g(v)$ for some $g \in \Gamma_S$. By (ii) it suffices to show this in the case that $\langle u , v \rangle = \pm 1$. The claim follows, as $u = T_v T_u(v)$, if $\langle u , v \rangle = 1$ and $u = T_v^{-1} T_u^{-1}(v)$, if $\langle u , v \rangle = -1$. Our claim shows that $\Delta$ is a $\Gamma_S$-orbit, hence also a $\Gamma_{\Delta}$-orbit. Thus $(V,\langle \; , \; \rangle , \Delta)$ is a vanishing lattice. Moreover, since $\Delta$ is a $\Gamma_S$-orbit, we have that for any $\alpha \in \Delta$ there exists $u \in S$ and $g \in \Gamma_S$ with $\alpha = g(u)$. Then $T_\alpha = g \circ T_u \circ g^{-1} \in \Gamma_S$, hence $\Gamma_\Delta = \Gamma_S$.
Consider $\Lambda_P$ equipped with the intersection form $\langle \; , \; \rangle$. We will give $(\Lambda_P , \langle \; , \; \rangle)$ the structure of a vanishing lattice. For this let $\{ a_u , b_u \}_{u=1}^g$ be a symplectic basis for $\Lambda_\Sigma$. Let $S_P \subset \Lambda_P$ be the subset: $$S_P = \{ t^j c_i \}_{\substack{ 0 \le j \le n-2 \\ 1 \le i \le nl-1 }} \cup \{ t^j( c_1 + (1-t)a_u) \}_{\substack{ 0 \le j \le n-2 \\ 1 \le u \le g }} \cup \{ t^j( c_1 + (1-t)b_u) \}_{\substack{ 0 \le j \le n-2 \\ 1 \le u \le g }}$$ By Proposition \[propallowedcycles\], we see that elements of $S_P$ are vanishing cycles associated to certain paths in $\Sigma$. By Theorem \[thmallowedmonodromy\], it follows that the transvections $\{ T_v \}_{v \in S_P}$ belong to the $SL(n,\mathbb{C})$-monodromy group $\Gamma_{SL}$. Clearly $S_P$ satisfies the conditions of Proposition \[propsgen\], so we obtain a vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P )$, where $\Gamma_{\Delta_P}$ is the group generated by transvections in $S_P$ and $\Delta_P = \Gamma_{\Delta_P} \cdot S_P$. So $\Gamma_{\Delta_P}$ is a subgroup of $\Gamma_{SL}$. We will eventually show that $\Gamma_{\Delta_P} = \Gamma_{SL}$.
Let $\mathcal{VC} \subset \Lambda_P$ be the set of all vanishing cycles associated to paths $\gamma$ joining pairs of branch points in $\Sigma$ and let $\Gamma_{\mathcal{VC}}$ be the group generated by transvections in $\mathcal{VC}$. We will eventually be able to show that $\Delta_P = \Gamma_{\mathcal{VC}} \cdot \mathcal{VC}$ and $\Gamma_{\mathcal{VC}} = \Gamma_{\Delta_P}$, so that $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ is the same as the vanishing lattice described in the introduction.
Classification over $\mathbb{Z}_2$ {#secz2class}
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We recall the classification of vanishing lattices in [@jan1; @jan2]. Since the classification over $\mathbb{Z}$ depends on the classification over $\mathbb{Z}_2$, we begin with the $\mathbb{Z}_2$ case. In this section $\overline{V}$ will be a finite dimensional $\mathbb{Z}_2$ vector space of dimension $\mu$, equipped with an alternating bilinear form $\langle \; , \; \rangle$. Let $\overline{V}_0$ be the null space of $\langle \; , \; \rangle$ and let $p$ be the dimension of $\overline{V}_0$. Then $\mu = 2r+p$ for some integer $r$.\
If $q$ is a quadratic function on $\overline{V}$, we see that the transvection $T_v$ associated to $v \in \overline{V}$ preserves $q$ if and only if $q(v) = 1$. We observe that to any basis $B$ of $\overline{V}$, we can associate a unique quadratic function $q_B$ with the property that $q_B(v) = 1$ for all $v \in B$. Clearly the group generated by transvections by elements of $B$ preserves $q_B$.
Let $(\overline{V} , \langle \; , \; \rangle , \overline{\Delta} )$ be a vanishing lattice. A basis $B$ of $\overline{V}$ is called [*weakly distinguished*]{} if $\Gamma_{\overline{\Delta}}$ is generated by $\{ T_v \}_{v \in B}$. In this case, $\Gamma_{\overline{\Delta}}$ preserves $q_B$, hence $q_B(\alpha)=1$ for all $\alpha \in \overline{\Delta}$.
Note that a vanishing lattice does not necessarily admit a weakly distinguished basis.\
To any basis $B$ of $\overline{V}$, we construct a graph $Gr(B)$ as follows. The vertices of $Gr(B)$ are elements of $B$ and for every distinct pair of element $u,v \in B$, there is a single edge joining $u$ and $v$ if and only if $\langle u , v \rangle = 1$.
\[remwdb\] If $(\overline{V} , \langle \; , \; \rangle , \overline{\Delta} )$ is a vanishing lattice and $B$ a weakly distinguished basis consisting of elements of $\overline{\Delta}$, then $(\overline{V} , \langle \; , \; \rangle , \overline{\Delta})$ can be completely recovered from $Gr(B)$. Indeed, $\Gamma_{\overline{\Delta}}$ is the group generated by $\{ T_v \}_{v \in B}$ and $\overline{\Delta} = \Gamma_{\overline{\Delta}} \cdot B$. Note however that different graphs can give rise to the same underlying vanishing lattice.
We now state the classification in [@jan1] of vanishing lattices over $\mathbb{Z}_2$. If $(\overline{V}, \langle \; , \; \rangle , \overline{\Delta})$ is a vanishing lattice which admits a weakly distinguished basis $B$ whose elements belong to $\overline{\Delta}$, then by Remark \[remwdb\] it is enough to simply give the graph $Gr(B)$. The remaining cases will be described individually. Vanishing lattices over $\mathbb{Z}_2$ can be broadly classified into three main types: [*symplectic, orthogonal*]{} and [*special*]{}. In both the orthogonal and special cases, the are three sub-cases to consider.\
[**Case 1: Symplectic.**]{} In this case $\overline{\Delta} = \overline{V} \setminus \overline{V}_0$ and $\Gamma_{\overline{\Delta}} = Sp^{\#}V$. Symplectic vanishing lattices do not admit weakly distinguished bases and the group $Sp^{\#}V$ does not preserve any quadratic functions. For fixed values of $(r,p)$ there is only one symplectic vanishing lattice, which is denoted by $Sp^{\#}(2r,p)$.\
[**Case 2: Orthogonal.**]{} In this case, a weakly distinguished basis $B$ exists, $\overline{\Delta} = \{ v \in \overline{V} \setminus \overline{V}_0 \; | \; q_B(v) = 1 \}$ and $\Gamma_{\overline{\Delta}} = O^{\#}(q_B)$ is the subgroup of $Sp^{\#}V$ preserving $q_B$. There are three sub-cases which are distinguished according to whether the Arf invariant of $q_B$ is $0,1$ or undefined. In all cases, a weakly distinguished basis can be chosen so that $Gr(B)$ is one of the following:
(0,0) – (3.4,0) ; (4.6,0) – (7,0) ; (2,0) – (2,-1) ; (3.6,0) – (4.4,0); (0,0) circle(0.1); (1,0) circle(0.1); (2,0) circle(0.1); (3,0) circle(0.1); (2,-1) circle(0.1); (5,0) circle(0.1); (6,0) circle(0.1); (7,0) circle(0.1); (7,-0.5) circle(0.1); (7,-1.5) circle(0.1); at (0,0.4) [$2$]{}; at (1,0.4) [$3$]{}; at (2,0.4) [$4$]{}; at (3,0.4) [$5$]{}; at (2.4,-1) [$1$]{}; at (7.4,0) [$2r$]{}; at (7.7,-0.5) [$2r+1$]{}; at (7.7,-1.5) [$2r+p$]{}; (6,0) – (7,-0.5) ; (7,-0.7) – (7,-1.3) ; (6,0) – (7,-1.5) ;
at (0 , -2) [for which ${\rm Arf}(q_B) = \begin{cases} 1 \; \; \text{ if } r = 2,3 \;( \text{mod }4), \\ 0 \;\; \text{ if } r = 0,1 \;( \text{mod }4). \end{cases}$]{} ;
(-1,0) – (3.4,0) ; (4.6,0) – (7,0) ; (2,0) – (2,-2) ; (3.6,0) – (4.4,0); (2,-2) circle(0.1); (-1,0) circle(0.1); (0,0) circle(0.1); (1,0) circle(0.1); (2,0) circle(0.1); (3,0) circle(0.1); (2,-1) circle(0.1); (5,0) circle(0.1); (6,0) circle(0.1); (7,0) circle(0.1); (7,-0.5) circle(0.1); (7,-1.5) circle(0.1); at (-1,0.4) [$3$]{}; at (0,0.4) [$4$]{}; at (1,0.4) [$5$]{}; at (2,0.4) [$6$]{}; at (3,0.4) [$7$]{}; at (2.4,-2) [$1$]{}; at (2.4,-1) [$2$]{}; at (7.4,0) [$2r$]{}; at (7.7,-0.5) [$2r+1$]{}; at (7.7,-1.5) [$2r+p$]{}; (6,0) – (7,-0.5) ; (7,-0.7) – (7,-1.3) ; (6,0) – (7,-1.5) ;
at (0,-3) [for which ${\rm Arf}(q_B) = \begin{cases} 1 \; \; \text{ if } r = 0,1 \;( \text{mod }4), \\ 0 \;\; \text{ if } r = 2,3 \;( \text{mod }4). \end{cases}$]{};
(-1,0) – (3.4,0) ; (4.6,0) – (7,0) ; (2,0) – (2,-1) ; (3.6,0) – (4.4,0); (-1,0) circle(0.1); (0,0) circle(0.1); (1,0) circle(0.1); (2,0) circle(0.1); (3,0) circle(0.1); (2,-1) circle(0.1); (5,0) circle(0.1); (6,0) circle(0.1); (7,0) circle(0.1); (7,-0.5) circle(0.1); (7,-1.5) circle(0.1); at (-1,0.4) [$2$]{}; at (0,0.4) [$3$]{}; at (1,0.4) [$4$]{}; at (2,0.4) [$5$]{}; at (3,0.4) [$6$]{}; at (2.4,-1) [$1$]{}; at (7.7,0) [$2r+1$]{}; at (7.7,-0.5) [$2r+2$]{}; at (7.7,-1.5) [$2r+p$]{}; (6,0) – (7,-0.5) ; (7,-0.7) – (7,-1.3) ; (6,0) – (7,-1.5) ;
at (0,-2) [for which ${\rm Arf}(q_B)$ is undefined.]{};
For fixed $(r,p)$ there are at most three orthogonal vanishing lattices, according to whether the Arf invariant is $0$, $1$ or undefined. We denote these by $O^{\#}_0(2r,p), O^{\#}_1(2r,p)$ and $O^{\#}(2r,p)$. Note that $O^{\#}(2r,p)$ only exists for $p > 0$.\
[**Case 3: Special.**]{} There are three sub-cases, of which two admit weakly distinguished bases. We describe these cases first. In the first subcase $B$ can be chosen so that $Gr(B)$ is:
(2,0) – (3.4,0) ; (4.6,0) – (7,0) ; (3.6,0) – (4.4,0); (2,0) circle(0.1); (3,0) circle(0.1); (5,0) circle(0.1); (6,0) circle(0.1); (7,0) circle(0.1); (7,-0.5) circle(0.1); (7,-1.5) circle(0.1); at (2,0.4) [$1$]{}; at (3,0.4) [$2$]{}; at (6,0.4) [$2r-1$]{}; at (7.4,0) [$2r$]{}; at (7.7,-0.5) [$2r+1$]{}; at (7.7,-1.5) [$2r+p$]{}; (6,0) – (7,-0.5) ; (7,-0.7) – (7,-1.3) ; (6,0) – (7,-1.5) ;
We have ${\rm Arf}(q_B) = \begin{cases} 1 \; \; \text{ if } r = 1,2 \;( \text{mod }4), \\ 0 \;\; \text{ if } r = 0,3 \;( \text{mod }4). \end{cases}$ This vanishing lattice is denoted $A^{ev}(2r,p)$.\
In the second subcase, $Gr(B)$ is:
(2,0) – (3.4,0) ; (4.6,0) – (7,0) ; (3.6,0) – (4.4,0); (2,0) circle(0.1); (3,0) circle(0.1); (5,0) circle(0.1); (6,0) circle(0.1); (7,0) circle(0.1); (7,-0.5) circle(0.1); (7,-1.5) circle(0.1); at (2,0.4) [$1$]{}; at (3,0.4) [$2$]{}; at (6,0.4) [$2r$]{}; at (7.7,0) [$2r+1$]{}; at (7.7,-0.5) [$2r+2$]{}; at (7.7,-1.5) [$2r+p$]{}; (6,0) – (7,-0.5) ; (7,-0.7) – (7,-1.3) ; (6,0) – (7,-1.5) ;
We have ${\rm Arf}(q_B) = \begin{cases} 1 \; \; \text{ if } r = 1 \;( \text{mod }4), \\ 0 \;\; \text{ if } r = 3 \;( \text{mod }4), \\ \text{undefined} \; \; \text{ if } r = 2,4 \;( \text{mod }4). \end{cases}$ This vanishing lattice is denoted $A^{odd}(2r,p)$.\
The third subcase is obtained by taking the vanishing lattice $A^{odd}(2r,p+1)$ and taking the quotient by the subspace spanned by $e_1 + e_3 + e_5 + \dots + e_{2r+1}$, where $e_i$ denotes the basis vector corresponding to the $i$-th vertex. The resulting vanishing lattice is denoted by $A'(2r,p)$. Note that the quadratic form of the $A^{odd}(2r,p+1)$ vanishing lattice descends to the quotient if and only if $r$ is odd.\
The following is a very useful criterion for determining whether or not a vanishing lattice is special:
\[propnotspecial\] Let $(\overline{V} , \langle \; , \; \rangle , \overline{\Delta})$ be a vanishing lattice over $\mathbb{Z}_2$. The following are equivalent:
- [$(\overline{V} , \langle \; , \; \rangle , \overline{\Delta})$ is not of special type.]{}
- [There exists $\overline{V}_1 \subseteq \overline{V}$, $\overline{\Delta}_1 \subseteq \overline{\Delta}$ such that $(\overline{V}_1 , \langle \; , \; \rangle , \overline{\Delta}_1)$ is a vanishing lattice of type $O^{\#}_1(6,0)$.]{}
The condition that $(\overline{V} , \langle \; , \; \rangle , \overline{\Delta})$ contains $O^{\#}_1(6,0)$ can be re-stated more simply as the condition that there exists $ e_1,e_2,e_3,e_4,e_5,e_6 \in \overline{\Delta}$ such that the graph of $\{ e_1,e_2,e_3,e_4,e_5,e_6\}$ is the $E_6$ Dynkin diagram:
(0,0) – (4,0) ; (2,0) – (2,-1) ;
(0,0) circle(0.1); (1,0) circle(0.1); (2,0) circle(0.1); (3,0) circle(0.1); (2,-1) circle(0.1); (4,0) circle(0.1);
at (0,0.4) [$2$]{}; at (1,0.4) [$3$]{}; at (2,0.4) [$4$]{}; at (3,0.4) [$5$]{}; at (2.4,-1) [$1$]{}; at (4,0.4) [$6$]{};
Let $(\Lambda_P , \langle \; , \rangle , \Delta_P)$ be the vanishing lattice constructed in Section \[secvanlat\] and let $(\Lambda_P[2] , \langle \; , \; \rangle , \overline{\Delta}_P)$ be its mod $2$ reduction.
\[thmz2class\] Let $\mu = (n-1)(nl + 2g-2)$, let $p = 2g$ if $n$ is even and $p=0$ if $n$ is odd. Further define $r$ such that $\mu = 2r + p$. The vanishing lattice $(\Lambda_P[2] , \langle \; , \; \rangle , \overline{\Delta}_P)$ is isomorphic to:
1. [$A'(2l-2,2g)$, if $n=2$,]{}
2. [$O_a^{\#}(2r,0)$, where $a = (m(m-1)/2)l$, if $n =2m+1$ is odd,]{}
3. [$O_a^{\#}(2r,2g)$, where $a = m(l/2)$, if $n=2m$ and $l$ are even and $n>2$,]{}
4. [$Sp^{\#}(2r,2g)$, if $n$ is even, $l$ is odd and $n > 2$.]{}
Consider first the case that $n = 2$. Then $$S_P = \{ c_1 , c_2, \dots , c_{2l-1} , c_1 + (1-t)a_1 , c_1 + (1-t)b_1 , \dots , c_1+(1-t)a_g , c_1 + (1-t)b_g \}.$$ The intersection graph of $S_P$ is given by:
(2,0) – (3.4,0) ; (4.6,0) – (7,0) ; (3.6,0) – (4.4,0); (2,0) circle(0.1); (3,0) circle(0.1); (5,0) circle(0.1); (6,0) circle(0.1); (7,0) circle(0.1); (7,-0.5) circle(0.1); (7,-1.5) circle(0.1); at (2,0.4) [$c_{2l-1}$]{}; at (3,0.4) [$c_{2l-2}$]{}; at (6,0.4) [$c_2$]{}; at (7.5,0) [$c_1$]{}; at (8.4,-0.5) [$c_1 + (1-t)a_1$]{}; at (8.4,-1.5) [$c_1 + (1-t)b_g$]{}; (6,0) – (7,-0.5) ; (7,-0.7) – (7,-1.3) ; (6,0) – (7,-1.5) ;
So $(\Lambda_P[2] , \langle \; , \; \rangle , \overline{\Delta}_P)$ must either be of type $A^{odd}(2l-2,2g+1)$ or $A'(2l-2,2g)$. However, we have the relation $c_1 + c_3 + c_5 + \dots + c_{2l-1} = 0$, so the vanishing lattice is of type $A'(2l-2,2g)$.\
Consider the case $n > 2$. We will show that $(\Lambda_P[2] , \langle \; , \; \rangle , \overline{\Delta}_P)$ contains a copy of $O^{\#}_1(6,0)$. In fact, consider the elements $tc_3 , tc_1 , c_2+tc_2 , c_3 , c_4 , c_5 \in \overline{\Delta}_P$. Note that $c_2+tc_2 \in \overline{\Delta}_P$ because $c_2 + tc_2 = T_{c_2}(tc_2)$. The intersection graph of these elements is the $E_6$ Dynkin diagram:
(0,0) – (4,0) ; (2,0) – (2,-1) ;
(0,0) circle(0.1); (1,0) circle(0.1); (2,0) circle(0.1); (3,0) circle(0.1); (2,-1) circle(0.1); (4,0) circle(0.1);
at (0,0.4) [$tc_1$]{}; at (1,0.4) [$c_2+tc_2$]{}; at (2,0.4) [$c_3$]{}; at (3,0.4) [$c_4$]{}; at (2.4,-1) [$tc_3$]{}; at (4,0.4) [$c_5$]{};
Hence by Proposition \[propnotspecial\], the vanishing lattice $(\Lambda_P[2] , \langle \; , \; \rangle , \overline{\Delta}_P)$ is not special for $n > 2$. If $n$ is odd or $n$ and $l$ are even, we saw in Section \[secquadratics\], that $\Lambda_P[2]$ has a monodromy invariant quadratic function $q$ and we calculated the Arf invariant of $q$ in Proposition \[proparfinvq\]. Thus if $n$ is odd or $n$ and $l$ are both even, then $(\Lambda_P[2] , \langle \; , \; \rangle , \overline{\Delta}_P)$ is of orthogonal type with the specified Arf invariant.\
Lastly, in the case that $n > 2$ is even and $l$ is odd we will show that $(\Lambda_P[2] , \langle \; , \; \rangle , \overline{\Delta}_P)$ is not of orthogonal type, hence it must be symplectic. Suppose on the contrary that there is a quadratic function $q : \Lambda_P[2] \to \mathbb{Z}_2$ invariant under $\Gamma_{\overline{\Delta}_P}$. Then we must have $q( t^j c_i ) = 1$ for all $i$ and $j$. From this it follows that $q( c_1 + (1+t)c_2 + \dots + (1+t+ \dots+ t^{n-2})c_{k-1}) = 1$. But this is impossible, since $c_1 + (1+t)c_2 + \dots + (1+t+\dots+t^{n-2})c_{k-1} = 0$.
Classification over $\mathbb{Z}$ {#seczclass}
--------------------------------
Let $(V , \langle \; , \; \rangle , \Delta)$ be a vanishing lattice over $\mathbb{Z}$. Recall [@bou] that for an alternating bilinear form $\langle \; , \; \rangle$ on $V$ there exists a basis $\{ e_1 , f_1 , \dots , e_r , f_r , g_1 , \dots , g_p \}$ of $V$ for which the matrix of $\langle \; , \; \rangle$ has the form $$\left[\begin{matrix} 0 & d_1 & & & & & & & & \\
-d_1 & 0 & & & & & & & & \\
& & 0 & d_2 & & & & & & \\
& & -d_2 & 0 & & & & & & \\
& & & & \ddots & & & & & \\
& & & & & 0 & d_r & & & \\
& & & & & -d_r & 0 & & & \\
& & & & & & & 0 & & \\
& & & & & & & & \ddots & \\
& & & & & & & & & 0
\end{matrix}\right]$$ where the $d_i$ are positive integers and $d_i$ divides $d_{i+1}$ ($i=1, \dots , r-1$). The $d_i$ are called the elementary divisors of $\langle \; , \; \rangle$ and are uniquely determined.\
Let $(\overline{V} , \langle \; , \; \rangle , \overline{\Delta})$ be the mod $2$ reduction, let $\overline{V}_0$ be the null space of $\langle \; , \; \rangle$ on $\overline{V}$. Choose an element $\delta \in \overline{\Delta}$ and let $\overline{V}_{00} = \{ v \in \overline{V}_0 \; | \; v + \delta \in \overline{\Delta} \}$. Then $\overline{V}_{00}$ is a subspace of $\overline{V}_0$ and does not depend on the choice of $\delta$ [@jan1 Lemma 2.11]. Let $j : V \to V^*$ be the homomorphism $j(x) = \langle x , \; \rangle$ and consider the homomorphism $$j^{-1}(2V^*) \to j^{-1}(2V^*)/2V \simeq \overline{V}_0 \to \overline{V}_0/\overline{V}_{00}.$$ In [@jan1] it is shown that $\overline{V}_0/\overline{V}_{00}$ is either $0$ or $\mathbb{Z}_2$. Hence we obtain a homomorphism $\phi : j^{-1}(2V^*) \to \mathbb{Z}_2$, where in the case $\overline{V}_0/\overline{V}_{00} = 0$, we take $\phi = 0$. Define $$k_0(V) = \max \{ k \; | \; \phi( j^{-1}(2^k V^*)) \neq 0 \},$$ with the conventions that $k_0(V) = \infty$ if no such maximum value of $k$ exists and $k_0(V) = 0$ if $\phi = 0$. Then we have:
Let $(V_1 , \langle \; , \; \rangle , \Delta_1)$ and $(V_2 , \langle \; , \; \rangle , \Delta_2)$ be vanishing lattices over $\mathbb{Z}$. They are isomorphic if and only if (i) they are isomorphic as lattices with bilinear form, (ii) their mod $2$ reductions are isomorphic as vanishing lattices and (iii) $k_0(V_1) = k_0(V_2)$.
Thus integral vanishing lattices are classified by their mod $2$ reduction, the invariants $d_1,d_2, \dots , d_r , p$ of the bilinear form and the invariant $k_0$. Moreover, we have that $\overline{V}_{00} = \overline{V}_0$ except in the $O^{\#}$ and $A^{odd}$ cases. Following Janssen, we use notation like $O^{\#}(d_1,\dots , d_r ; p ; k_0)$ to denote a vanishing lattice with mod $2$ reduction of type $O^{\#}$ and invariants $d_1, \dots , d_r , p , k_0$. In all cases other than $O^{\#}$ and $A^{odd}$ we may omit $k_0$ from the notation. We will denote by $u$ the number of $d_i$’s which are odd.
Up to isomorphism integral vanishing lattices are given by the following list: $$\begin{aligned}
O^{\#}_1(d_1, \dots , d_r ; p) \\
O^{\#}_0(d_1, \dots , d_r ; p) && (u \ge 3) \\
O^{\#}(d_1 , \dots , d_r ; p ; k_0) && (u \ge 2, r > u \text{ or } p>0, k_0 > 0) \\
Sp^{\#}(d_1 , \dots , d_r ; p)\\
A^{ev}(d_1 , \dots , d_r ; p) && (\text{if } u=1, \text{ then } r=1 \text{ and } p=0) \\
A^{odd}(d_1, \dots, d_r ; p ; k_0) && (r>u \text{ or } p>0; \; k_0 = 0 \text{ iff } u=1) \\
A'(d_1 , \dots , d_r ; p) && (u \ge 2).
\end{aligned}$$
Now we can determine the isomorphism class of the integral vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P )$:
The vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ is isomorphic to:
1. [$A'(1,1, \dots , 1 , 2 , 2 , \dots , 2; 0)$, if $n=2$,]{}
2. [$O_a^{\#}(1,1, \dots , 1 , n , n , \dots , n ; 0)$, where $a = (m(m-1)/2)l$, if $n =2m+1$ is odd,]{}
3. [$O_a^{\#}(1,1, \dots , 1 , n , n , \dots , n ; 0)$, where $a = m(l/2)$, if $n=2m$ and $l$ are even and $n>2$,]{}
4. [$Sp^{\#}(1,1, \dots , 1 , n , n , \dots , n ; 0)$, if $n$ is even, $l$ is odd and $n > 2$.]{}
In this classification, the number of $1$’s is $(n-2)(g-1) + n(n-1)l/2 - 1$ and the number of $n$’s is $g$.
The mod $2$ classification of $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ was given in Theorem \[thmz2class\]. In particular we saw that $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ does not have type $O^{\#}$ or $A^{odd}$, so we do not need to consider the invariant $k_0$. The intersection form $\langle \; , \; \rangle$ is non-degenerate on $\Lambda_P$, so $p=0$ and the invariants $(d_1 , \dots , d_r)$ are precisely the polarization type of the Prym variety $Prym(S,\Sigma)$. This was calculated in Corollary \[corpolarization\].
Proof of the main theorem {#secmain}
=========================
Let $\delta$ be a smooth point of the discriminant locus $\mathcal{D}$. Let $D_\delta$ be a copy of the unit disc in $\mathbb{C}$, embedded in $\mathcal{A}$ such that $D_\delta$ intersects $\mathcal{D}$ transversally at the point $\delta$ and meets no other point of $\mathcal{D}$. Consider a loop $\gamma_\delta$ in $\mathcal{A} \setminus \mathcal{D}$, which starts at the basepoint $a_0$, follows a path $p$ from $a_0$ to the boundary of $D_\delta$, goes once around the boundary of $D_\delta$ and goes back to $a_0$ along $p^{-1}$. Such a loop will be called a [*meridian*]{}.\
Kouvidakis and Pantev showed in the case $L = K$, that the discriminant locus $\mathcal{D} \subseteq \mathcal{A}$ is an irreducible hypersurface [@kp]. Their proof easily extends to the case where $L \neq K$ and $deg(L) > deg(K)$. It follows that $\pi_1( \mathcal{A} \setminus{D} , a_0)$ is generated by meridians and since $\mathcal{D}$ is irreducible, any two meridians are conjugate in $\pi_1(\mathcal{A}\setminus{D} , a_0)$. Let $\mathcal{D}^0 \subset \mathcal{D}$ be the locus of points $a \in \mathcal{D}$ for which the corresponding spectral curve $S_a$ is irreducible and has an ordinary double point as its only singularity. Kouvidakis and Pantev also showed that $\mathcal{D}^0$ is a non-empty Zariski open subset of $\mathcal{D}$, in the case $L=K$ [@kp]. This is clearly also true in the case $L \neq K$ and $deg(L) > deg(K)$. Since $\mathcal{D}^0$ is Zariski dense in $\mathcal{D}$, we see that $\pi_1(\mathcal{A} \setminus{D} , a_0)$ is generated by meridians around points in $\mathcal{D}^0$.\
Let $\delta \in \mathcal{D}^0$ and suppose that $D_\delta$ is an embedded disc meeting $\mathcal{D}$ transversally in $\delta$, as above. Consider the family $X \to D_\delta$ of spectral curves over $D_\delta$ defined by pullback under the incusion $D_\delta \subset \mathcal{A}$. The fibre $X_\delta$ over $\delta$ of this family has an isolated non-degenerate singularity. It follows that the monodromy $\rho( \gamma_\delta) \in Aut( \Lambda_S )$ of a meridian $\gamma_\delta$ around $\delta$ is a Picard-Lefschetz transformation: $$\rho(\gamma_\delta)(x) = T_\alpha(x) = x + \langle \alpha , x \rangle \alpha,$$ where $\alpha \in \Lambda_S$ is the vanishing cycle associated to $\gamma_\delta$. In fact, since $\pi_* \circ \rho(\gamma_\delta) = \pi_*$, we see that $\alpha \in \Lambda_P$. We then have:
\[lemgenvan\] Let $\Delta \subseteq \Lambda_P$ be the set of vanishing cycles associated to meridians around points in $\mathcal{D}^0$. The monodromy group $\Gamma_{SL}$ is generated by the transvections $\{ T_\alpha \}_{\alpha \in \Delta}$. For any two vanishing cycles $\alpha , \beta \in \Delta$, we have $\alpha = g \beta$ or $\alpha = -g \beta$ for some $g \in \Gamma_{SL}$.
Only the last statement of the lemma requires proof. Let $\alpha , \beta \in \Delta$. Then $T_\alpha = \rho(\gamma_1) , T_\beta = \rho(\gamma_2)$ for some meridians $\gamma_1,\gamma_2$. But we have seen that all meridians are conjugate, so there exists $x \in \pi_1( \mathcal{A} \setminus {D} , a_0)$ for which $\gamma_1 = x \gamma_2 x^{-1}$. Applying $\rho$, we get $T_\alpha = g T_\beta g^{-1} = T_{g \beta }$, where $g = \rho(x) \in \Gamma_{SL}$. As $\langle \; , \; \rangle$ is non-degenerate it is easy to see that $T_\alpha = T_{g \beta}$ implies that $\alpha = g\beta$ or $\alpha = -g\beta$.
\[lemdeltaz\] Let $(V , \langle \; , \; \rangle , \Delta)$ be an integral vanishing lattice and $x \in V$. Then $x \in \Delta$ if and only of there exists $y \in V$ and $\delta \in \Delta$ such that $\langle x , y \rangle = 1$ and $x - \delta \in 2V$.
\[thmmain1\] Let $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ be the integral vanishing lattice constructed in Section \[secvanlat\] and $\Gamma_{\Delta_P} \subseteq Aut(\Lambda_P)$ the group generated by transvections by elements of $\Delta_P$. We have an equality $\Gamma_{SL} = \Gamma_{\Delta_P}$.
We have already established the inclusion $\Gamma_{\Delta_P} \subseteq \Gamma_{SL}$, which follows by Theorem \[thmallowedmonodromy\] and Proposition \[propallowedcycles\]. It remains to prove the reverse inclusion.\
Consider first the case $n > 2$. By Lemma \[lemgenvan\], it is sufficient to show that $T_\alpha \in \Gamma_{\Delta_P}$, where $\alpha$ is the vanishing cycle of a meridian around a point in $\mathcal{D}^0$. It is easy to see that the vanishing cycles $\{ t^j c_i \}$ constructed in Section \[secvc\] are vanishing cycles associated to meridians. Applying Lemma \[lemgenvan\], we have that there exists a $g \in \Gamma_{SL}$ for which $\alpha = gc_1$ or $\alpha = -gc_1$. As $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ is a vanishing lattice, it can be shown that there exists $h \in \Gamma_{\Delta_P}$ such that $hc_1 = -c_1$ [@jan1]. So replacing $g$ by $gh$ if necessary, we can assume $\alpha = gc_1$. If $n$ is odd or $n$ and $l$ are both even, then $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ is of orthogonal type. Let $\overline{\Delta}_P \subset \Lambda_P[2]$ be the mod $2$ reduction of $\Delta_P$. Then as explained in Section \[secz2class\], $\overline{\Delta}_P = \{ v \in \overline{V} \setminus \overline{V}_0 \; | \; q(v) = 1 \}$, where $\overline{V} = \Lambda_P[2]$ and $q$ is the monodromy invariant quadratic function. It follows that the mod $2$ reduction of $\alpha$ belongs to $\overline{\Delta}_P$, since $q(gc_1) = q(c_1) = 1$. Moreover $\langle \alpha , y \rangle = 1$, where $y = g(tc_1)$, so by Lemma \[lemdeltaz\] we have $\alpha \in \Delta_P$. If $n$ is even and $l$ is odd, then $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ is of symplectic type. So $\overline{\Delta}_P = \overline{V} \setminus \overline{V}_0$ and by a similar argument we have $\alpha \in \Delta_P$.\
Lastly, consider the case $n=2$. It was shown in [@bs1] that $\Gamma_{SL}$ is generated by transvections $T_{c_\gamma}$, where $c_\gamma$ is the cycle associated to a path $\gamma$ joining two branch points, as in Theorem \[thmallowedmonodromy\]. It is also clear that $c_\gamma$ is the vanishing cycle of a meridian. Suppose that $\gamma$ joins branch points $b_i,b_j$ and assume $i < j$ (the case $i>j$ is similar). Let $\overline{c} \in \Lambda_P[2]$ be the mod $2$ reduction of $c_\gamma$. Then it is easy to see that $\overline{c}$ must have the form $\overline{c} = c_i + c_{i+1} + \dots + c_{j-1} + (1+t)a$ for some $a \in \Lambda_\Sigma[2]$. In the case $n=2$, we have that $\overline{\Delta}_P \subset \Lambda_P[2]$ is of special type. Then using [@jan1 Lemma 3.11], we see that $\overline{c} \in \overline{\Delta}_P$. By the same argument as in the $n > 2$ case, we have $c_\gamma = g(c_1)$ for some $g \in \Gamma_{SL}$. So $\langle c_\gamma , y \rangle = 1$ for $y = g(c_2)$. Then by Lemma \[lemdeltaz\], we have $c_\gamma \in \Delta_P$.
The next theorem shows that the monodromy of the $GL(n,\mathbb{C})$ Hitchin fibration is determined by the vanishing lattice $(\Lambda_P , \langle \; , \; \rangle , \Delta_P)$ together with the extension $\Lambda_P \to \Lambda_S \to \Lambda_\Sigma$.
The monodromy group $\Gamma_{GL}$ is the subgroup of $Aut(\Lambda_S)$ generated by transvections $T_{\alpha} : \Lambda_S \to \Lambda_S$, where $\alpha \in \Delta_P$.
By the same argument used for $\Gamma_{SL}$, we have that $\Gamma_{GL}$ is the subgroup of $Aut(\Lambda_S)$ generated by transvections in vanishing cycles associated to meridians around points in $\mathcal{D}^0$. Let $\Gamma_{GL,\Delta_P}$ be the subgroup of $Aut(\Lambda_S)$ generated by transvections in $\Delta_P$. Recall from Section \[secvanlat\] the subset $S_P \subset \Delta_P$. We let $\Gamma_{GL,S_P}$ be the subgroup of $Aut(\Lambda_S)$ generated by transvections in $S_P$.\
In the proof of Theorem \[thmmain1\], we established that if $\alpha$ is such a vanishing cycle, then $\alpha \in \Delta_P$. Hence $\Gamma_{GL} \subseteq \Gamma_{GL,\Delta_P}$. On the other hand we have that the elements of $S_P$ are vanishing cycles associated to meridians, so $\Gamma_{GL,S_P} \subseteq \Gamma_{GL}$. Arguing as in the proof of Proposition \[propsgen\], we see that the subgroup of $Aut(\Lambda_S)$ generated by transvections in $\Delta_P$ is also generated by transvections in $S_P$. Hence $\Gamma_{GL,S_P} = \Gamma_{GL} = \Gamma_{GL,\Delta_P}$. In particular, $\Gamma_{GL}$ is the subgroup of $Aut(\Lambda_S)$ generated by transvections in $\Delta_P$.
Let $\mathcal{VC} \subset \Lambda_P$ be the set of vanishing cycles associated to paths in $\Sigma$ joining pairs of branch points and let $\Gamma_{\mathcal{VC}}$ be the group generated by transvections by cycles in $\mathcal{VC}$. Then $\Delta_P = \Gamma_{\mathcal{VC}} \cdot \mathcal{VC}$ and $\Gamma_{\mathcal{VC}} = \Gamma_{\Delta_P} = \Gamma_{SL}$.
The elements of $S_P$ are all vanishing cycles associated to certain paths joining branch points, so $S_P \subseteq \mathcal{VC}$ and $\Gamma_{S_P} \subseteq \Gamma_{\mathcal{VC}}$, where $\Gamma_{S_P}$ is the group generated by transvections in $S_P$. By Proposition \[propsgen\], we have $\Gamma_{S_P} = \Gamma_{\Delta_P}$ and $\Gamma_{\Delta_P} = \Gamma_{SL}$ by Theorem \[thmmain1\], so $\Gamma_{SL} \subseteq \Gamma_{\mathcal{VC}}$. On the other hand $\Gamma_{\mathcal{VC}} \subseteq \Gamma_{SL}$, by Theorem \[thmallowedmonodromy\]. So $\Gamma_{\mathcal{VC}} = \Gamma_{SL}$. It remains to show that $\mathcal{VC} \subseteq \Delta_P$. Let $\alpha \in \mathcal{VC}$. Then as $T_\alpha$ is the monodromy around a meridian, arguing along the same lines as in the proofs of Lemma \[lemgenvan\] and Theorem \[thmmain1\] gives $\alpha = g c_1$ for some $g \in \Gamma_{SL}$. But $g c_1 \in \Delta_P$, so we have shown that $\mathcal{VC} \subseteq \Delta_P$ and hence $\Gamma_{\mathcal{VC}} \cdot \mathcal{VC} \subseteq \Gamma_{SL} \cdot \Delta_P = \Delta_P$. However, $\Delta_P$ is an orbit of $\Gamma_{SL} = \Gamma_{\mathcal{VC}}$, so we must have $\Gamma_{\mathcal{VC}} \cdot \mathcal{VC} = \Delta_P$.
Application to the topology of Higgs bundle moduli spaces {#secappl}
=========================================================
Let $\mathcal{M}(n,d,L)$ denote the moduli space of rank $n$ $L$-twisted Higgs bundles with trace-free Higgs field and determinant equal to a fixed line bundle $D$ of degree $d$. Up to isomorphism, $\mathcal{M}(n,d,L)$ depends on $D$ only through the degree $d$. We can define the Hitchin fibration $h : \mathcal{M}(n,d,L) \to \mathcal{A}$ and one finds that for any value of $d$, the moduli space $\mathcal{M}(n,d,L)$ is a torsor for the family of Prym varieties $p : Prym(\mathcal{S}/\mathcal{A}^{\rm reg}) \to \mathcal{A}^{\rm reg}$ as defined in Section \[sec2\]. In particular, this implies that the monodromy local system of $\mathcal{M}(n,d,L)$ is $\underline{\Lambda}$, independent of $d$. Let $a \in \mathcal{A}^{\rm reg}$ and let $F_a = h^{-1}(a)$ be the corresponding non-singular fibre of the Hitchin fibration. Our goal this section will be to prove the following:
\[thmrestriction\] Let $\omega \in H^2( F_a , \mathbb{Q} )$ be the cohomology class of the polarization on $F_a$. The image $$Im( H^*( \mathcal{M}(n,d,L) , \mathbb{Q} )) \to H^*( F_a , \mathbb{Q} ) )$$ of the restriction map in cohomology is the subspace spanned by $1 , \omega , \omega^2 , \dots , \omega^u$, where $u = dim_{\mathbb{C}}(F_a)$ is the dimension of the fibre.
To prove this theorem we need to use a result concerning symplectic vector spaces over finite fields of prime order. Let $p$ be an odd prime, $V$ a vector space over $\mathbb{Z}_p$ of dimension $2v$ and $\langle \; , \; \rangle$ a non-degenerate alternating bilinear form over $V$. Given a symplectic basis $\{ e_1 , f_1, e_2 , f_2 , \dots , e_v , f_v \}$ and $m \in \{0,1,2, \dots , v\}$, let $\alpha_{2m} \in \wedge^{2m} V$ be given by $$\label{equalpha}
\alpha_{2m} = \sum_{ i_1 < i_2 < \dots < i_m } (e_{i_1} \wedge f_{i_1} ) \wedge \dots \wedge (e_{i_m} \wedge f_{i_m} ).$$ It can be shown that $\alpha_{2m}$ is independent of the choice of symplectic basis and is invariant under the group $Sp( V , \mathbb{Z}_p)$ of symplectic transformations of $V$ [@de].
\[leminvs\] The subspace of $\wedge^* V$ invariant under $Sp(V , \mathbb{Z}_p)$ is spanned by $1 , \alpha_2 , \alpha_4 , \dots , \alpha_{2v}$.
We use induction on the dimension $2v$ of $V$. Assume the result holds in dimension $2v-2$. Choose a symplectic basis $B = \{e_1,f_1 , \dots , e_v , f_v\}$ for $V$. For $i = 1,2, \dots , v$, let $V_i$ be the subspace spanned by $B \setminus \{ e_i , f_i \}$, let $\iota_i : V_i \to V$ be the inclusion and $\pi_i : V \to V_i$ the projection with kernel spanned by $e_i,f_i$.\
Let $\lambda \in \wedge^k V$ be invariant. Consider first the case where $k$ is odd. For any $i$ we have that $\pi_i(\lambda)$ is invariant under $Sp( V_i , \mathbb{Z}_p)$, so $\pi_i(\lambda) = 0$ by induction. It follows that $\lambda = e_i \wedge \alpha + f_i \wedge \beta + e_i \wedge f_i \wedge \gamma$, where $\alpha,\beta,\gamma$ are in the image of $\iota_i : \wedge^* V_i \to \wedge^* V$. But $\lambda$ is invariant under the transvections $T_{e_i},T_{f_i}$ and it follows easily that $\alpha = \beta = 0$, so that $\lambda$ is a multiple of $e_i \wedge f_i$. Since $i$ was arbitrary, we have that $\lambda$ is a multiple of $e_1\wedge f_1 \wedge \dots \wedge e_v \wedge f_v$. But $\lambda$ has odd degree so this can only happen if $\lambda = 0$.\
Consider the case where $k = 2m$ is even. Then for each $i$, we have that $\pi_i( \lambda)$ is invariant under $Sp(V_i , \mathbb{Z}_p)$, so by induction we have $\pi_i(\lambda) = c_i \pi_i(\alpha_{2m})$ for some $c_i \in \mathbb{Z}_p$. Clearly this can only happen if $c_1 = c_2 = \dots = c_m$ and thus $\pi_i( \lambda -c_1 \alpha_{2m}) = 0$ for all $i$. By the same argument as used in the case where $k$ is odd, we see that either $\lambda - c_1 \alpha_{2m} = 0$, or $m = v$ and $\lambda-c_1 \alpha_{2v}$ is a multiple of $e_1 \wedge f_1 \wedge \dots \wedge e_v \wedge f_v = \alpha_{2v}$. In either case $\lambda$ is a multiple of $\alpha_{2m}$.
\[leminvcohom\] The space $H^*( F_a , \mathbb{Q} )^{\rho_{SL}}$ of monodromy invariant rational cohomology classes on $F_a$ is spanned by $1,\omega , \omega^2 , \dots , \omega^u$.
Suppose that $\mu \in H^k( F_a , \mathbb{Q} )^{\rho_{SL}}$ is a monodromy invariant cohomology class on $F_a$. Multiplying $\mu$ by a sufficiently large positive integer, it suffices to assume that $\mu \in H^k( F_a , \mathbb{Z} )^{\rho_{SL}}$. Let $p$ be an odd prime not dividing $n$ and let $V = \Lambda_P^* \otimes_{\mathbb{Z}} \mathbb{Z}_p$. Then since $H^k( F_a , \mathbb{Z}) = \wedge^k \Lambda_P^*$, we have $H^k( F_a ,\mathbb{Z}_p) = \wedge^k V$. By Corollary \[corpolarization\], the polarization type of $\langle \; , \; \rangle$ on $\Lambda_P$ is $(1,1 , \dots , 1 , n , n , \dots , n)$, where there are $g$ copies of $n$. It follows that the mod $p$ reduction of $\langle \; , \; \rangle$ is a non-degenerate alternating bilinear form on $\Lambda_P \otimes_{\mathbb{Z}} \mathbb{Z}_p$ and so by duality induces a non-degenerate alternating bilinear form $\langle \; , \; \rangle$ on $V$ preserved by the monodromy action. Reduction mod $p$ thus induces a homomorphism $\phi : \Gamma_{SL} \to Sp( V , \mathbb{Z}_p)$ and the reduction of $\mu$ mod $p$ gives an element $\mu_p \in \wedge^k V$ invariant under $\phi(\Gamma_{SL})$. Since $p$ is odd, it follows from [@bh1 Theorem 2.7 and Theorem 6.5] and the fact that $\Gamma_{SL}$ is generated by transvections in the set $S_P$ given in Section \[secvanlat\], that $\phi$ is actually surjective. Then by Lemma \[leminvs\], we have that $\mu_p = 0$ if $k$ is odd and $\mu_p$ is a multiple of $\alpha_{2m}$ if $k = 2m$ is even.\
Suppose first that $k$ is odd. Then $\mu$ is divisible by infinitely many primes, hence $\mu = 0$. Now suppose that $k = 2m$ is even. Then for every odd prime $p$ not dividing $n$ we have that the mod $p$ reduction of $\mu$ is a multiple of $\alpha_{2m}^{(p)}$ (here we use a superscript $p$ to remind us that $\alpha_{2m} = \alpha_{2m}^{(p)}$ depends on $p$). Suppose also that $p > m$. Then from Equation (\[equalpha\]), we have that $\alpha_{2m}^{(p)}$ is the mod $p$ reduction of $\omega^m/m!$, where $\omega \in \wedge^2 \Lambda_P^*$ is the alternating form $\langle \; , \; \rangle$ thought of as a $2$-form. Let $\tau \in \wedge^{2m} \Lambda_P^*$ be a primitive vector such that $\omega^m/m!$ is an integral multiple of $\tau$. The mod $p$ reduction of $\mu$ is a multiple of the mod $p$ reduction of $\tau$ for infinitely many primes $p$. It follows that $\mu$ is in the $\mathbb{Z}$-span of $\tau$ and thus $\mu$ is a rational multiple of $\omega^m$ in $\wedge^{2m} \Lambda_P^* \otimes_{\mathbb{Z}} \mathbb{Q}$.
As the restriction map $r : H^*(\mathcal{M}(n,d,L) , \mathbb{Q} ) \to H^*( F_a , \mathbb{Q} )$ factors through $\mathcal{M}^{\rm reg}(n,d,L)$, it is clear that the image of $r$ is contained in $H^*( F_a , \mathbb{Q} )^{\rho_{SL}}$, the subgroup of monodromy invariants. By Lemma \[leminvcohom\], $H^*( F_a , \mathbb{Q} )^{\rho_{SL}}$ is spanned by $1,\omega , \dots , \omega^u$. On the other hand since $\mathcal{M}(n,d,L)$ is quasi-projective, there exists a class $\alpha \in H^2( \mathcal{M}(n,d,L) , \mathbb{Q})$ whose restriction $r(\alpha)$ to $F_a$ is a Kähler class, hence non-zero. Thus $r(\alpha)$ must be some non-zero rational multiple of $\omega$. It follows that the image of $r$ is precisely the span of $1,\omega , \omega^2 , \dots , \omega^u$.
When $n$ and $d$ are coprime and $L=K$, Theorem \[thmrestriction\] may also be proved as follows. By Theorem 7 of [@mar], the cohomology ring of $\mathcal{M}(n,d,K)$ is generated by the Künneth factors of the Chern classes of the universal $PGL(n,\mathbb{C})$-Higgs bundle on $\mathcal{M}(n,d,K) \times \Sigma$. Then by a generalisation of the proof of Proposition 5.1.2 in [@chm], one can determine the image of the generators on restriction to a non-singular fibre. The advantage of our proof is that it applies without restriction on the values of $n$ and $d$ and does not require knowing a set of generators for the cohomology of $\mathcal{M}(n,d,L)$.
[99]{} M. F. Atiyah, Riemann surfaces and spin structures. [*Ann. Sci. École Norm. Sup.*]{} Vol. [**4**]{}, no. 4 (1971) 47-62. D. Baraglia, L. Schaposnik, Monodromy of rank $2$ twisted Hitchin systems and real character varieties, to appear in [*Trans. Amer. Math. Soc.*]{}, arXiv:1506.00372 (2015). A. Beauville, M. S. Narasimhan, S. Ramanan, Spectral curves and the generalised theta divisor. [*J. Reine Angew. Math.*]{} [**398**]{} (1989), 169-179. N. Bourbaki, Éléments de mathématique. Algèbre. Chapitre 9. Reprint of the 1959 original. Springer-Verlag, Berlin, 2007. 211 pp. R. Brown, S. P. Humphries, Orbits under symplectic transvections. I. [*Proc. London Math. Soc.*]{} (3) [**52**]{} (1986), no. 3, 517-531. M. A. A. de Cataldo, L. Migliorini, The decomposition theorem, perverse sheaves and the topology of algebraic maps. [*Bull. Amer. Math. Soc. (N.S.)*]{} [**46**]{} (2009), no. 4, 535-633. M. A. A. de Cataldo, T. Hausel, L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case $A_1$. [*Ann. of Math.*]{} (2) [**175**]{} (2012), no. 3, 1329-1407. M. A. A. de Cataldo, A support theorem for the Hitchin fibration: the case of $SL_n$, arXiv:1601.02589 (2016). P.-H. Chaudouard, G. Laumon, Un théorème du support pour la fibration de Hitchin. [*Ann. Inst. Fourier (Grenoble)*]{} [**66**]{} (2016), no. 2, 711-727. D. J. Copeland, Monodromy of the Hitchin map over hyperelliptic curves. [*Int. Math. Res. Not.*]{} (2005), no. 29, 1743-1785. B. De Bruyn, On the Grassmann modules for the symplectic groups. [*J. Algebra*]{} [**324**]{} (2010), no. 2, 218-230. I. Dolgachev, A. Libgober, On the fundamental group of the complement to a discriminant variety. In: Algebraic geometry, Lecture Notes in Math. 862, 1-25, Springer, Berlin-New York, 1981. D. Gaiotto, G. W. Moore, A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory. [*Comm. Math. Phys.*]{} [**299**]{} (2010), no. 1, 163-224. N. J. Hitchin, The self-duality equations on a Riemann surface. [*Proc. London Math. Soc.*]{} (3) [**55**]{} (1987), no. 1, 59-126. N. J. Hitchin, Stable bundles and integrable systems. [*Duke Math. J.*]{} [**54**]{} (1987), no. 1, 91-114. W. A. M. Janssen, Skew-symmetric vanishing lattices and their monodromy groups. [*Math. Ann.*]{} [**266**]{} (1983), no. 1, 115-133. W. A. M. Janssen, Skew-symmetric vanishing lattices and their monodromy groups. II. [*Math. Ann.*]{} [**272**]{} (1985), no. 1, 17-22. A. Kouvidakis, T. Pantev, The automorphism group of the moduli space of semistable vector bundles. [*Math. Ann.*]{} [**302**]{} (1995), no. 2, 225-268. E. J. N. Looijenga, Isolated singular points on complete intersections. London Mathematical Society Lecture Note Series, [**77**]{}. Cambridge University Press, Cambridge, 1984. xi+200 pp. E. Markman, Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces. [*J. Reine Angew. Math.*]{} [**544**]{} (2002), 61-82. B. C. Ngô Le lemme fondamental pour les algébres de Lie. [*Publ. Math. Inst. Hautes Études Sci.*]{} No. [**111**]{} (2010), 1-169. N. Nitsure, Moduli space of semistable pairs on a curve. [*Proc. London Math. Soc.*]{} (3) [**62**]{} (1991), no. 2, 275-300. R. W. Richardson, Conjugacy classes of $n$-tuples in Lie algebras and algebraic groups. [*Duke Math. J.*]{} [**57**]{} (1988), no. 1, 1-35. L. P. Schaposnik, Spectral data for $G$-Higgs bundles. DPhil thesis. arXiv:1301.1981 (2013). L. P. Schaposnik, Monodromy of the $SL2$ Hitchin fibration. [*Internat. J. Math.*]{} [**24**]{} (2013), no. 2, 1350013, 21 pp. M. Thaddeus, Topology of the moduli space of stable vector bundles over a compact Riemann surface, Masters thesis, University of Oxford (1989).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The analysis of the multicolour photometric observations of MW Lyr, a large modulation amplitude Blazhko variable, shows for the first time how the mean global physical parameters vary during the Blazhko cycle. About $1-2\%$ changes in the mean radius, luminosity and surface effective temperature are detected. The mean radius and temperature changes are in good accordance with pulsation model results, which show that these parameters do indeed vary within this order of magnitude if the amplitude of the pulsation changes significantly. We interpret the phase modulation of the pulsation to be a consequence of period changes. Its magnitude corresponds exactly what one expects from the detected changes of the mean radius assuming that the pulsation constant remains the same during the modulation. Our results indicate that during the modulation the pulsation remains purely radial, and the underlying mechanism is most probably a periodic perturbation of the stellar luminosity with the modulation period.'
author:
- 'J. Jurcsik$^{1}$[^1], Á. Sódor$^{1}$, B. Szeidl$^{1}$, Z. Kolláth$^{1}$, H. A. Smith$^{4}$, Zs. Hurta$^{1,2}$,'
- 'M. Váradi$^{1,3}$, A. Henden$^{5}$, I. Dékány$^{1}$, I. Nagy$^{2}$, K. Posztobányi$^{6}$, A. Szing$^{7}$,'
- |
K. Vida$^{1,2}$, and N. Vityi$^{2}$\
\
$^{1}$Konkoly Observatory of the Hungarian Academy of Sciences, H–1525 Budapest PO Box 67, Hungary\
$^{2}$Dept. of Astronomy, Eötvös University, H–1518 Budapest PO Box 49, Hungary\
$^{3}$Observatoire de Genéve, Universite de Genève, CH–1290, Sauverny, Switzerland\
$^{4}$Dept. of Physics and Astronomy, Michigan State Univ., East Lansing, MI 48824, USA\
$^{5}$American Association of Variable Star Observers, 49 Bay State Road, Cambridge, MA 02138, USA\
$^{6}$AEKI, KFKI Atomic Energy Research Institute, Thermohydraulic Department, H–1525 Budapest 114, PO Box 49, Hungary\
$^{7}$University of Szeged, Dept. of Exp. Physics and Astron. Obs., H–6720 Szeged, Dóm tér 9, Hungary
date: 'Accepted 2008 ..... Received 2008 ...; in original form 2008 Jul 24'
title: 'An extensive photometric study of the Blazhko RR Lyrae star MW Lyr: II. Changes in the physical parameters[^2]'
---
\[firstpage\]
stars: horizontal branch – stars: variables: other – stars: individual: MW Lyr – stars: oscillations (including pulsations) – methods: data analysis – techniques: photometric
Introduction
============
Neither multicolour photometric nor spectroscopic observations have been previously obtained with good enough time coverage of both the pulsation and modulation cycles of a Blazhko RR Lyrae star that they could be used to derive any phenomenological conclusion about the changes in the global physical properties of the star during the modulation cycle. To cover the pulsation variations of different shape in each phase of the Blazhko modulation with observations several hundreds of hours of observing time were needed. This can be achieved only with telescopes ’dedicated’ to the study of the phenomenon. As $80-90\%$ of the telescope time of the automated 60cm telescope of the Konkoly Observatory is allocated to study RR Lyrae stars, during recent years we could first obtain $BVR_CI_C$ photometric time series of Blazhko variables that are extended enough for such an investigation.
We have detected $0.010-0.005$ mag systematic changes in the intensity weighted mean $\langle{V}\rangle, \langle{B}\rangle-\langle{V}\rangle$, and $\langle{V}\rangle-\langle{I_C}\rangle$ brightness and colours of RR Gem and SS Cnc in different phases of the Blazhko modulation (see Fig. 12 and Fig. 9 in @rrgI and [@sscnc], respectively). The intensity weighted quantities indicated slight brightness and temperature increases in RR Gem at the time of the largest amplitude of the pulsation, while in SS Cnc the brightest mean magnitude and bluest mean $\langle{V}\rangle-\langle{I_C}\rangle$ colour occurred during the decreasing pulsation amplitude phase of the modulation. The small amplitude of the modulation of these stars, and also the ambiguity of the equivalent static colours of RR Lyrae stars [@bono] make, however, these results somewhat uncertain.
The light curve analysis, utilizing mostly the $V$ data of our $\sim 1000$ hours extended multicolour CCD observations of MW Lyr, an RR Lyrae star showing large amplitude Blazhko modulation, was presented in @mw1 [hereafter Paper I]. The large amplitude of the modulation of MW Lyr makes it possible to detect changes in the mean magnitudes and colours more definitely if there are any indeed.
In order to exploit the most information possible from the multicolour light curves of RR Lyrae stars we have recently developed the IP method [Inverse Photometric method, @cikk]. This method gives good estimates of the mean physical parameters and their variations with pulsation phase exclusively from photometric data without any spectroscopic observations. Using the extended $BVI_C$ time series of MW Lyr the IP method shows the differences between the pulsation in different phases of the modulation, and whether or not there are any changes in the mean global physical parameters of the star during the Blazhko cycle.
Data and method
===============
{width="17.2cm"}
The photometric data utilized in this paper were published in Paper I. The observed colours and magnitudes are dereddened using $E(B-V)=0.10$ mag interstellar extinction given by the [@schlegel] catalogue and $A_V=3.1E(B-V)$ and $E(V-I_C)=1.27E(B-V)$ relations.
In the present paper the mean light and colour curves of MW Lyr and the actual light and colour curves in 20 different phase bins of the 16.546 d modulation cycle are investigated. The colour-magnitude and two-colour loops of the pulsation in different phases of the modulation are shown in Fig. \[lck\]. This is the first multicolour photometric observation of a Blazhko variable that makes a detailed study of the colour behaviour changes during the Blazhko cycle possible.
The analysis is performed using the IP method [@cikk], that finds which $T_{eff}$ and $V_{rad}$ variations during the pulsation results in luminosity, radius and effective gravity changes that best match the light curves using synthetic magnitudes and bolometric corrections of static atmosphere models [@kurucz]. The input data of the IP method are the Fourier fits of the time series data. Solutions are obtained using the Fourier representations of either the $V$, $B-V$, and $V-I_C$ or the $V, B,$ and $I_C$ time series.
Those parameters that definitely must not change with the Blazhko period, namely the metallicity, mass and the distance of the star are determined from the mean light curves of the entire data set and are fixed when studying the changes during the Blazhko cycle. Two mean data sets are investigated to determine these parameters, one corresponding to the original observations, and another where data are corrected for the phase modulation with appropriate time transformation of the observations. In Paper I it was demonstrated that if phase modulation is due to pulsation period changes during the Blazhko cycle, this treatment is valid and the mean light curve of the time transformed data represents the mean pulsation variation better than the mean light curve of the original data.
As no spectroscopic observation of MW Lyrae has ever been obtained, the metallicity is calculated from the [@jk96] formula which derives the \[Fe/H\] from the period and the $\Phi_{31}$ phase difference of the $V$ light curve. This formula gives $-0.6$ and $-0.4$ \[Fe/H\] values for the mean light curves of the original and the time transformed data, respectively. Therefore, it seems to be a reasonable choice to use the IP method with \[M/H\]=$-0.5 $ model atmosphere grid for MW Lyr assuming solar scaled chemical composition distribution as significant enhancement of the $\alpha$ elements are observed typically only in more metal poor variables [see e.g., Fig 9. in @alpha].
The mass and distance of MW Lyr are derived by applying the IP method using the \[Fe/H\]=$-0.5$ model atmosphere grid on the mean light and colour curves of the entire original and the time transformed data sets. One basic input of the IP method is an initial $V_{rad}(\varphi)$ curve ($\varphi$ means the pulsation phase) which is varied in the minimization process. [@cikk] defined this initial $V_{rad}(\varphi)$ curve twofold, first using the template $V_{rad}(\varphi)$ curve scaled according to the $A_{V_{rad}} - V_{Amp}$ relation given by [@liu], secondly utilizing the $V_{rad}(\varphi)-I_C(\varphi)$ relation that was shown to be valid for RRab stars in [@cikk]. For variables showing no or only small amplitude light curve modulation the two types of initial $V_{rad}(\varphi)$ functions led to similar final results. The $A_{V_{rad}} - V_{Amp}$ relation of individual Blazhko variables during the modulation cycle is, however, steeper than that was defined by [@liu], and from the $V_{rad}(\varphi)-I_C(\varphi)$ relation valid for nonmodulated RRab stars [see Fig. 2 in @dist]. Blazhko variables obey the same amplitude relation as normal RRab stars at around their maximum amplitude phase [@dist]. As a consequence, the amplitude of the mean radial velocity variation of a Blazhko star is supposed to be smaller than that of a nonmodulated star with photometric amplitude similar to the amplitude of its mean light curve. The slope of the $\Delta A_{V_{rad}}/\Delta A(V)$ amplitude ratio is 30 for regular RRab stars while it is about 41 for Blazhko variables. The amplitudes of the initial radial velocity curves used in the analysis of MW Lyr, calculated either from the $I_C$ light curves or Liu’s template, are therefore scaled differently from normal RRab stars. Both mean light curve sets are modelled assuming three values for the $\Delta A_{V_{rad}}/\Delta V_{Amp}$ ratio, 37, 41, and 45, respectively, and with zero points of the $A_{V_{rad}} - V_{Amp}$ relation determined from matching the maximum amplitude value of the $V_{rad}$ curve to the $A_{V_{rad}} - V_{Amp}$ relation of nonmdulated RR Lyrae stars.
[rrrrrrr]{} d \[pc\] & $\overline{M_V}$ & $\overline{L/L_{\odot}}$& $\overline{T_{eff}}$ & $\overline{R/R_{\odot}}$ & $\overline {\log g_{stat}}$ &$\overline{{\mathfrak{M/M}}_{\odot}}$\
&&&&\
\
&\
3595 & 0.74 & 44.7 & 6892.5 & 4.64 &2.982 &0.76\
147 & 0.09 & 3.6 & 2.5 & 0.19 & 0.010 & 0.08\
&\
3460 & 0.83 & 41.4 & 6891.9 & 4.47 & 2.973& 0.69\
145 & 0.09 & 3.4 &2.5 & 0.18 & 0.010 & 0.07\
&\
3343 & 0.90 & 38.6& 6890.3 & 4.32 & 2.965& 0.63\
138 & 0.09 & 3.2 &2.4 & 0.18 & 0.010 & 0.06\
\
&\
3367 & 0.89 & 39.0 & 6889.4 & 4.35 & 2.966 & 0.64\
106 & 0.07 & 2.4 & 2.7 & 0.14 & 0.009 & 0.05\
&\
3223 & 0.98 &35.7 & 6888.0 & 4.16 &2.956 & 0.57\
111 & 0.08 & 2.4 & 2.3 & 0.14 & 0.008 & 0.05\
&\
3104 & 1.06& 33.2 & 6887.0 & 4.01 &2.947 & 0.52\
109 & 0.08 & 2.3 & 2.3 & 0.14 & 0.009 & 0.05\
As there is no information on the changes of the actual shapes of the $V_{rad}(\varphi)$ curves during the Blazhko cycle we do not know whether any of the two approximations of the radial velocity variation that the IP method uses is valid for Blazhko variables. In the first method when using Liu’s template, the shape of the initial $V_{rad}$ curve is always the same only its amplitude is scaled, while in the second one when calculating the $V_{rad}(\varphi)$ curve from the actual $I_C$ light curve, it varies in accordance with the variations of the light curve’s shape. Having no a priori knowledge on the $V_{rad}(\varphi)$ of a Blazhko star, in order to make even less restrictions on its shape, in the IP process an even smaller weight of the initial $V_{rad}(\varphi)$ curve is given than that has been used for unmodulated RRab stars in [@cikk] to allow a larger variance of its Fourier parameters.
Table \[tab1\] summarizes the mean global parameters of MW Lyr using the original and the time transformed data and 3 possible values for the $A_{V_{rad}} - V_{Amp}$ relation of Blazhko variables. The estimated uncertainties are the rms scatter of the results running the IP method using 8 settings:
1. [Initial $V_{rad}(\varphi)$ calculated from Liu’s $V_{rad}$ template curve or from the $V_{rad}(\varphi)-I_C(\varphi)$ relation.]{}
2. [Large and small weights are given to the initial $V_{rad}(\varphi)$ function, the first keeps it close to its initial shape, the second allows the amplitudes and phases of the lower Fourier components of the $V_{rad}(\varphi)$ curve to vary substantially. In [@cikk] it was shown that giving small weights to the initial $V_{rad}(\varphi)$ curve leads to unreliable shape of the solution $V_{rad}(\varphi)$ curve. The results of the IP method are the most reliable for unmodulated RRab stars if large and medium weights of the initial $V_{rad}(\varphi)$ curve are given. However, as we do not know how the radial velocity curve of a Blazhko variable vary, we decided to use small weights of the initial $V_{rad}(\varphi)$ curves instead of medium weights. Any result that is independent from this choice is practically independent of the exact shape of the radial velocity variations, and as so, is a robust solution for the detected changes during the Blazhko cycle.]{}
3. [Input data are the Fourier fits of the $V$, $B-V$, and $V-I_C$ or the $V, B,$ and $I_C$ time series. These input data are not identical representations of the observations because the differences between the Fourier fits of the magnitudes do not reproduce exactly the Fourier fits of the colour indices.]{}
As mentioned earlier and discussed in Paper I, we regard the time transformed data that corrects the phases of the pulsation to eliminate the phase modulation/pulsation period changes to be a more reliable representation of the data than the original time series. Therefore, in the detailed analysis of the light curves in different phases of the modulation, the distance and mass of MW Lyrae are fixed to the mean values of the results obtained for the time transformed data assuming $\Delta A_{V_{rad}} / \Delta V_{Amp} = 41 $, namely to 3460 pc and 0.69 $\mathfrak{M_{\odot}}$ values, respectively. This mass is somewhat larger than the mass of higher metallicity horizontal branch stars in the instability strip according to stellar evolutionary models (e.g., ACS Survey, @dotter; Padova evolutionary database, @salasnich; Y2 evolutionary tracks, @demarque). The IP method calculates the mass of the star from the pulsation equation of fundamental mode variables. The masses of some of the unmodulated RRab stars derived by the IP method are too large on evolutionary grounds, but the combination of the results of direct Baade-Wesselink analysis with the pulsation mass formula may also lead to too large mass values compared to the evolutionary values in some cases. E.g., 0.87 and 0.85 $\mathfrak{M_{\odot}}$ masses were derived in [@cikk] for WY Ant and UU Cet, respectively, from the pulsation equation and the results of the Baade-Wesselink analysis of [@kbw]. To resolve this discrepancy is, however, beyond the scope of the present paper.
It is important to note, however, that when running the IP code using any of the physical parameter combinations listed in Table 1, similar solutions for the variations of the global physical parameters with the Blazhko phase are obtained. It is found that only the mean values of these parameters depend on which mean light curve solution is accepted. All the conclusions of the next sections remain unchanged if other values of the fixed parameters, \[Fe/H\], distance and mass, were selected, and analyses are performed using the original data with other possible values of the $\Delta A_{V_{rad}} / \Delta V_{Amp}$ amplitude ratio.
![Residual $V$, $B-V$, and $V-I_{C}$ data (crosses) are plotted for the largest and smallest amplitude phases of the modulation, for Blazhko phases 20 and 13, respectively. The mean pulsation curves in both phases are subtracted. The lines show the deviations of 8 variant fits of the IP method from the mean pulsation curves.[]{data-label="res"}](lc.eps){width="9.3cm"}
Changes in the physical parameters during the Blazhko cycle
===========================================================
The detailed analysis of the light curves in different phases of the modulation is carried out by running the IP code using atmosphere models with \[Fe/H\]=$-0.5$ on 20 segments of the time transformed photometric data with the 8 settings specified in (i), (ii), and (iii) points. The distance and mass of MW Lyr are fixed to the 3460 pc and 0.69 $\mathfrak{M_{\odot}}$ values, respectively.
The IP method fits the light and colour curves by finding the appropriate changes in the surface radius and temperature of the star. The fitting accuracy is about 0.02 mag. Just for comparison, in Paper I we have shown that due to some stochastic/chaotic behaviour of the modulation the best Fourier fit to the complete $V$ data set reproduces the observations with only 0.02 mag accuracy. The Fourier fits of the light curves of the time transformed data in the 20 phase bins of the modulation give, on average, also 0.02 mag rms residual, which is significantly larger than the observational inaccuracy. This is partially due to modulations detected in Paper I with periods that are non-integer multiplets of the main modulation period, and to small but nonperiodic perturbations of the modulation. Fig. \[res\] plots the residual $V, B-V$, and $V-I_{C}$ data in the largest and smallest amplitude phases of the modulation, in Blazhko phase 20 and 13, respectively. The deviations of the fitted curves from the mean light curves of the data are also drawn. The fits seem to match the light curves within the required accuracy, with the largest deviations at around the rising branch of the light curves (phase 0.85-1.00) where the observed colours of the dynamic atmosphere of RR Lyrae stars definitely cannot be fitted accurately with static atmosphere model results. Moreover, in this phase of the pulsation the modulation is not strictly regular (as it was shown in paper I), which may also explain the larger residuals of the solutions of the IP method here.
In Figs. \[t\]-\[lg\] the results of the IP method for the ${T_{eff}}, L/L_{\odot}, R/R_{\odot}, V_{rad},$ and $ \log g_{eff}$ changes are shown in two phases of the Blazhko modulation, one at the largest and another at the smallest pulsation amplitude. The upper panels show when the initial $V_{rad}$ curve is calculated according to the $V_{rad}-I_C$ relation while bottom panels show results using the template $V_{rad}$ curve defined by [@liu]. Solid lines show the results when the $V_{rad}$ curve is kept close to its initial shape, while the dashed lines indicate how the results change if the lower Fourier parameters of the initial $V_{rad}$ curves are allowed to vary. Each setting is applied using either the fits to the $B, V,$ and $I_C$ or the $V, B-V $and $V-I_C$ data, however, these differences have only very marginal effect on the results.
Fig. \[mindenv\] displays the light curve variation of MW Lyr in 20 phase bins of the modulation according to the time transformed data. For comparison, in Figs. \[mindenl\]-\[mindeng\] the variations of the physical parameters derived form the IP method are shown. The left and right panels show if the initial $V_{rad}$ curves are calculated from the $I_C$ light curves or from Liu’s template, while top and bottom panels are for large and small weights of the initial $V_{rad}$ curves, respectively.
There are only minor changes in the luminosity and surface effective temperature curves depending on the choice and weights of the initial $V_{rad}$ curves. The amplitude of the temperature and luminosity variations during the pulsation of the star are significantly different in the different phases of the modulation. The temperature and luminosity at maximum amplitude vary between 6300 and 8800 K and between 26 and 95 L$_\odot$, while at minimum amplitude only between 6500 and 7700 K, and between 31 and 58 L$_\odot$, respectively. The changes in the amplitude of the temperature and luminosity variations are about $50 \%$! These results are very robust, the ranges of the detected changes in the variations of these parameters in different phases of the modulation are independent of the choice and alterations of the initial radial velocity curve.
Substantial changes in the radius, radial velocity and surface effective gravity curves can be also seen. It is important to note that in those solutions when the shape of the initial radial velocity curve is allowed to vary substantially the results are very similar if Liu’s template is used or the initial $V_{rad}$ curve is calculated from the $I_C$ light curve. The radial velocity and, as a consequence, the radius variations are somewhat different if the $V_{rad}$ curve is kept close to its initial template, or it is allowed to vary significantly. In the first case significant changes in the amplitude of the radius variation during the Blazhko cycle can be detected, while in the second case not the amplitude but the shape of the radius curves vary. The only radial velocity observations of a Blazhko variable with good enough phase coverage of both the pulsation and the modulation cycles were recently obtained for RR Lyr [@chadid]. Fig. 9 in [@chadid] shows that the radial velocity curves of RR Lyr remain smooth during the modulation and there is no sign of a double wave shape similar to what our results indicate if the shapes of the radial velocity curves are allowed to differ significantly from their initial templates during the large amplitude phase of the modulation. Most probably the real variations of the radial velocity curves are close to those shown in the upper panels of Fig. \[mindenvr\] and not those in the bottom panels, i.e, the radial velocity curves of Blazhko variables differ from the $V_{rad}$ templates of unmodulated RRab stars mostly in their amplitudes and not in their shapes. If this is indeed the case, then there is also about $50\%$ change in the amplitude of the radius variation of the pulsation during the Blazhko cycle.
![$T_{eff}$ variations in the large amplitude (left panels) and small amplitude (right panels) phases of the modulation. Top and bottom panels show results if the initial $V_{rad}$ curve is defined from the $V_{rad}-I_C$ relation and from the template curve given by @liu, respectively. Four curves are plotted in each panel, giving large (solid lines) and small weights (dashed lines) to the initial $V_{rad}$ curves and using the $V, B-V$, and $V-I_C$ or the $B,V,$ and $I_C $ curves as input data. These latter curves, however, highly overlap in each of the plots shown in Figs. \[t\]-\[lg\].[]{data-label="t"}](t.eps){width="9.3cm"}
![The same as Fig. \[t\] but for the luminosity variations. []{data-label="l"}](l.eps){width="9.3cm"}
![The same as Fig. \[t\] but for the radius variations.[]{data-label="r"}](r.eps){width="9.3cm"}
![The same as Fig. \[t\] but for the radial velocity curves.[]{data-label="vr"}](vr.eps){width="9.3cm"}
![The same as Fig. \[t\] but for the effective surface gravity variations.[]{data-label="lg"}](lg.eps){width="9.3cm"}
![The time transformed $V$ light curves of MW Lyrae in 20 phase bins of the modulation.[]{data-label="mindenv"}](20m.eps){width="6.cm"}
![Luminosity variations in 20 phase bins of the modulation. Left and right panels show results using the transformation of the $I_C$ light curves and Liu’s template as initial radial velocity curves, respectively. In the top panels the $V_{rad}$ curves are kept close to their initial curve, while in the bottom panels the initial $V_{rad}$ curves ar allowed to vary significantly in the fitting process.[]{data-label="mindenl"}](20l.eps){width="9cm"}
![The same as in Fig \[mindenl\] but for the temperature variations in 20 phase bins of the modulation.[]{data-label="mindent"}](20t.eps){width="9cm"}
![The same as in Fig \[mindenl\] but for the radius variations in 20 phase bins of the modulation.[]{data-label="mindenr"}](20r.eps){width="9cm"}
![The same as in Fig \[mindenl\] but for the radial velocity curves in 20 phase bins of the modulation.[]{data-label="mindenvr"}](20vr.eps){width="9cm"}
![The same as in Fig \[mindenl\] but for the effective surface gravity variations in 20 phase bins of the modulation.[]{data-label="mindeng"}](20g.eps){width="9cm"}
Changes in the mean physical parameters during the Blazhko cycle
================================================================
From photometric data different types of mean magnitudes and colours can be derived: average values of the intensities or the magnitudes or mean colours calculated from the difference of the mean magnitudes or from colour data. The intensity averaged magnitudes are brighter than the magnitude averaged ones, while for the colours the $(m_1-m_2)> \langle m_1-m_2 \rangle > \langle m_1 \rangle - \langle m_2 \rangle$ relation holds. It was shown, however, by [@bono] that none of these mean magnitudes and colours match the equivalent static values of RR Lyrae stars in the full ranges of their possible parameter domain. Therefore, to draw any conclusion for the changes of the mean physical parameters during the Blazhko cycle from the changes of the mean colours and magnitudes is somewhat ambiguous.
In the left panels in Fig. \[minden\] the observed mean data derived from the photometry of MW Lyr are shown for 20 Blazhko phase bins. The period changes correspond to the pulsation period variation determined in Paper I from the phase of the $f_0$ pulsation frequency. These plots show definitely that each of the mean colours and magnitudes vary with 0.01-0.02 mag amplitude, but these changes are of the opposite sign for the different averages, making any firm conclusion of the mean temperature and luminosity changes impossible.
{width="16.6cm"}
However, after finding the appropriate pulsation curves of the physical parameters in different phases of the modulation with the IP method we can derive their mean values directly. In the middle and right panels in Fig. \[minden\] the mean physical parameter changes are shown if the $V_{rad}-I_C$ relation and Liu’s template are used to define the initial $V_{rad}$ curves, respectively. For each phase bin four data of the mean parameters are plotted corresponding to the results with different settings of the IP method as shown in the panels in Figs. \[t\]-\[lg\].
Fig. \[minden\] shows unambiguously that during the Blazhko cycle the mean physical parameters do indeed vary. The amplitudes and signs of these variations are independent of the choice of the initial $V_{rad}$ curve and of which setting of the IP code is used. We have also tested whether there is any change in these variations if any other possible combinations of the mass and distance of MW Lyr as listed in Table \[tab1\] are adopted, or if the original data are used instead of the time transformed ones, or if different $\Delta A_{V_{rad}}/\Delta A(V)$ ratio is applied. Any of these possibilities have only minor influence on the results shown in Fig. \[minden\], only the parameters’ ranges are shifted corresponding to the data given in Table \[tab1\].
We can therefore sate that, for the first time, we succeeded in reliably detecting changes in the mean global physical parameters of a Blazhko variable during the Blazhko cycle. $1-2\%$ changes in each of the parameters are evident. MW Lyr is larger by about $0.04 R_\odot$, more luminous by about $1.00 L_\odot$ and cooler by about 50 K in the large pulsation amplitude phases of the modulation as compared to its small pulsation amplitude phases. Though we cannot exclude the possibility that the detected changes in the mean global parameters arise from a complicated conjunction of different unknown processes, the most straightforward conclusion is that they simply reflect real changes in the mean global physical parameters.
The relative radius variation $\Delta R / R$ is 0.009 which leads to $\Delta P / P =0.014$ relative period change as a consequence of the $\Delta P / P \approx 3/2 \times \Delta R / R$ pulsation relation. The detected changes in the pulsation period is $\Delta P / P= 0.016$. This good agreement between the period changes measured directly from the phase modulation and calculated from the radius variation proves that both the measured radius changes and our treatment to regard the phase modulation of the pulsation as pulsation period changes should be real. The phase relation between the radius and period changes are not synchronized, the period is about the longest when the mean stellar radius and the pulsation amplitude is the smallest. [@stothers] found a similar phase connection between the pulsation amplitude and pulsation period changes in RR Lyrae.
Our results also reveal that the intensity mean $V$ magnitude reflects the luminosity changes and the $(B-V)$ and $\langle B-V \rangle$, or $(V-I_C)$ and $\langle V-I_C \rangle$ mean colours reproduce the temperature changes correctly, notwithstanding the large amplitude variation of the pulsation. Consequently, these values can be regarded as good representatives of the equivalent static values of RR Lyrae stars. The temperatures derived from $\langle B \rangle - \langle V \rangle$ or $\langle V \rangle - \langle I_C \rangle$ colours may differ from these temperatures by 100-250 K, the difference is the largest when the amplitude of the pulsation is large.
Comparison with pulsation model results
=======================================
Assuming that the variation of MW Lyr can be fully described by radial pulsation, some hints on the character of the variation of the mean physical parameters can be drawn from numerical pulsation calculations. Since there are no nonlinear pulsation models that can reproduce the Blazhko phenomenon, such calculations can reveal only partly the relations found in this paper. For our calculations we used the Florida-Budapest pulsation code with standard RR Lyrae parameters [see @model]. Even the state of the art pulsation codes contain only a relatively simple model of convection. In these models only one dynamical equation is used for the turbulent energy, with additional source functions of turbulent energy and convective flux. According to our experience, the interaction of this simple turbulent equation with the hydrodynamics and radiation transfer does not give rise to any time-scale or modulation that can be connected to Blazhko cycles. If turbulence plays an important role in Blazhko phenomenon, as suggested by [@stothers], then to model the effect, additional physics, e.g. with a more complicated turbulent-convection formalism, or with the inclusion of interaction with magnetic field, should be included in the codes. In the lack of any more sophisticated model only some aspects of the nonlinear dynamics of pulsation can be tested in comparison with the observations.
The natural timescales that come from numerical calculations are the pulsation period and the e-fold time of amplitude variation during the onset of pulsation (the last one is related to the linear growth rate of pulsation). Since the growth rate normalized with the pulsation frequency, is of the order of a few percent, the e-fold time of pulsation growth or decay is of the order of the Blazhko period. It means that any change in the stellar structure can be realized in pulsational changes on the period of the Blazhko cycle.
As a first test, nonlinear pulsation calculations are initiated with a small velocity perturbation with the corresponding eigenvector of the linear stability analysis. Then the onset of pulsation occurs through a transient, with the growth of pulsation amplitude. We used this phase of the numerical calculations to estimate the dependence of the mean physical quantities on the amplitude.
A more physical test is to check the effect of the variation of turbulent parameters on the mean physical parameters. From the 6 turbulent ($\alpha$) parameters of the Florida-Budapest code we selected the efficiency of eddy viscosity (the dissipation due to turbulence), since this parameter ($\alpha_\nu$) does not affect the static structure and the linear growth-rates of the pulsation modes. With this test the amplitude of the relaxed pulsation can be changed and it again gives the dependence of mean values of some of the global parameters on the amplitude. From a finite amplitude (limit cycle) model the hydrocode can be restarted with a slightly modified eddy viscosity parameter, which gives an efficient way to get the pulsation model for different values of $\alpha_\nu$, and finally the averaged physical quantities as the function of amplitude.
The two tests gave basically identical results, proving that the amplitude dependence of some of the averaged quantities is not a transient event, but a real dynamic effect. We calculated several models of both fundamental and overtone pulsations. As expected, the mean radius of the star increases with the pulsation amplitude according to the results of both tests. We found that approximately $\Delta R/R_0 \approx 0.015 \Delta A_{Bol}$ holds. The mean effective temperature decreases with increasing amplitude: $\Delta T_{eff} \approx 100 \cdot \Delta A_{Bol}$. These results agree well with the observed dependencies as shown in Fig \[zoli\]. The figure compares the observed relative changes in mean temperature and radius as a function of the observed bolometric amplitude (calculated from the luminosity curve solution of the IP method) with model predictions. The measured amplitude dependencies are $\Delta T_{eff} \approx 80 \cdot \Delta A_{Bol}$ and $\Delta R/R_0 \approx 0.012 \Delta A_{Bol}$.
![The observed changes of the mean temperature (left panel) and relative mean radius (right panel) variations as a function of the bolometric amplitude variation of MW Lyr in different phases of the modulation (dots) are compared with pulsation model predictions (lines).[]{data-label="zoli"}](zoli.eps){width="8.9cm"}
We have to note, however, that the calculations reveal only one of the mechanisms that should be taken into consideration to understand the Blazhko effect, namely the standard dynamics of radial pulsation. For example, if one considers the standard relation between $\Delta P/P$ and $\Delta R/R_0$, the order of the observed variations of these quantities agrees with each other as shown in Sect. 4. But since the whole mechanism is more complicated, it is not astonishing that the period and radius variations are not in phase as expected from the simple relation. Since in these tests the static structure of the star has been fixed, it is not expected that all the observed variations during the Blazhko cycle could be estimated. First of all, this simplified dynamic system has only a very limited effect on the pulsation period (of the order of $10^{-4}$ ). It indicates that structural changes are necessary to reproduce all the observational data, e.g. by varying the other turbulent parameters in the code.
Another deficiency of our simple model is that since there is no built in mechanism besides the kinetic energy of pulsation in the code, which results in time varying storage of radiative energy on a longer time scale than the pulsation, the mean luminosity changes cannot be compared to the observations. Moreover, dynamical effects can also play an important role in the complete picture of the phenomenon. This effect, however, cannot be investigated without a self-consistent model which naturally provides the periodic or quasi periodic modulation of turbulent structure. To get a consistent model of Blazhko phenomenon, based on the dynamics of turbulent convection, more sophisticated models and lot more computational efforts are needed. The goal of this paper cannot be to find the solution of the long standing problem of Blazhko effect. However, it turns out, that some of the observational results can be well estimated even with our simple treatment. Whether it is a fortunate coincidence or the indication that variations on the turbulent convective structure of RR Lyrae stars are an important ingredient of Blazhko phenomenon, should be answered in further studies.
Colour dependent characteristics of the modulation and pulsation frequencies
============================================================================
The detected changes in the mean global physical parameters during the Blazhko cycle prove that the physically meaningful frequency of the modulation is $f_m$, the frequency of the modulation itself, and not any of the side lobe frequencies appearing close to the pulsation frequency or its harmonics in the Fourier spectrum of the light curve. A similar conclusion was already drawn in [@sscnc] based on the colour behaviour of the amplitudes and the phases of the pulsation, modulation side lobe, and the modulation frequencies of RR Gem and SS Cnc. It was found that while the modulation side lobe frequencies have similar amplitude ratios and phase differences in the different colours as the pulsation frequencies have, the modulation frequency has discrepant values.
Table \[f0\] summarizes the amplitude ratios of the pulsation and modulation frequencies of MW Lyr calculated from the Fourier parameters given in Table 5 in Paper I. Only frequency components with $V$ amplitudes larger than the $V$ amplitude of the $f_\mathrm{m}$ frequency are considered. For MW Lyr the phase differences of the $f_\mathrm{m}$ frequency component do not show discrepant behaviour but its $A(B)/A(V)$ amplitude ratio is about 0.1 larger, and its $A(V)/A(I_C)$ amplitude ratio is 0.1 smaller than the amplitude ratios of the pulsation and modulation side lobe frequency components.
Based on, on the one hand, the similarity between the colour dependence of the amplitudes of the modulation side lobe frequencies and that of the pulsation frequencies and, on the other hand, on the discrepant amplitudes of the modulation frequency itself, we conclude that the side lobe frequencies are most probably combination frequencies of the pulsation frequencies ($kf_0$) and the modulation frequency ($f_\mathrm{m}$). As being combination frequencies they inherit the properties of their larger amplitude component, namely that of a pulsation frequency component. In this case, the independent frequency of the modulation is, in fact, the modulation frequency, $f_\mathrm{m}$.
[lc@c@c@c]{} frequencies& $A(B)/A(V)$& range&$A(V)/A(I_C)$& range\
$f_\mathrm{m}$&1.379&&1.400&\
$kf_0\,{^a}$ &$1.263$&$1.185 / 1.336$& $1.522$&$1.438 / 1.625$\
$kf_0\pm f_\mathrm{m}\,{^b}$ &$1.302$&$1.251 / 1.352$& $1.577$&$1.505 / 1.623$\
\
\
Conclusions
===========
As a conclusion, based on the results shown in Paper I and in this paper we can interpret the Blazhko modulation of MW Lyr as follows.
- [The frequency that characterizes the Blazhko modulation of RRab stars is the modulation frequency, $f_\mathrm{m}$, as we have found that the observed modulations are governed by global physical changes that also occur with the Blazhko period. If this proves to be indeed the case, then theories that tie the Blazhko modulation to a side frequency of a pulsation frequency component ($kf_0\pm f_\mathrm{m}$) misinterpret the Blazhko phenomenon [e.g. @dz].]{}
- [The global mean physical parameters of the star change about $1-2\%$ during the Blazhko cycle. Most probably the primary changes occur in the mean stellar luminosity and the detected changes in mean stellar temperature, radius, and pulsation period are the consequences of the luminosity changes. ]{}
- [The phase modulation during the Blazhko cycle reflects simply the oscillations of the pulsation period, while the amplitude changes are due to the changes in the mean global physical parameters of the star (L, T, R) reflecting periodic alterations in the atmosphere structure of the star.]{}
Numerical models of the onset of radial nonlinear pulsation display very similar dependence of mean radius and effective temperature on amplitude to the observed ones. It also suggests that, at least in part, the variation of mean quantities can be understood by the simple dynamics of radial pulsation. However, to unfold the whole picture, a presently unknown mechanism of the variation of the internal structure of the star (e.g., turbulent/convective properties) needs to be revealed.
Similar conclusions has been recently drawn about the nature of the Blazhko phenomenon by [@stothers]. He supposed that the underlying mechanism disturbing periodically the amplitude and period of the purely radial mode pulsation is cyclic weakening and strengthening of turbulent convection in the stellar envelope. The suggested triggering mechanism, namely a turbulent convective dynamo, would, however, most probably result in much less regular modulation behaviour than observed in MW Lyr. Another difficulty with Stother’s idea is how the multiperiodic nature of the modulation observed in some Blazhko stars e.g., CZ Lac [@sodor and Sódor, Jurcsik et al. in preparation] and XZ Cyg [@xzc] can be explained. Consequently, while we think that though our results support the idea that during the Blazhko cycle the pulsation remains purely radial, further theoretical efforts are needed in order to find the reason for the drastic changes observed in its main properties.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank the comments of the anonymous referee which helped to improve the lucidity of the paper significantly. The financial support of OTKA grants K-68626 and T-048961 is acknowledged. HAS thanks the US National Science Foundation for support under grants AST 0440061 and AST 0607249.
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[^1]: E-mail: [email protected]
[^2]: Based on observations collected mainly with the automatic 60 cm telescope of Konkoly Observatory, Budapest, Svábhegy
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Identity alignment models assume precisely annotated images manually. Human labelling is unrealistic on large sized imagery data. Detection models introduce varying amount of noise and hamper identity alignment performance. In this work, we propose to refine images by removing the undesired pixels. This is achieved by learning to eliminate less informative pixels in identity alignment. To this end, we formulate a method of automatically detecting and removing identity class irrelevant pixels in auto-detected bounding boxes. Experiments validate the benefits of our model in improving identity alignment.'
bibliography:
- 'id\_align.bib'
title: Identity Alignment by Noisy Pixel Removal
---
Introduction {#sec:intro}
============
Identity alignment aims at matching people across non-overlapping camera views distributed at different locations by comparing person bounding box images [@gong2014person]. In real-world scenarios, [*automatic detection*]{} [@felzenszwalb2010object] is [ *essential*]{} for identity alignment to scale up to large size data, e.g. more recent identity alignment benchmarks CUHK03 [@Li_DeepReID_2014b] and Market-1501 [@zheng2015scalable]. Most existing identity alignment test datasets (Table \[tab:datasets\]) are [*manually cropped*]{}, as in VIPeR [@VIPeR] and iLIDS [@RankSVMReId_BMVC10], thus they do not fully address the identity alignment challenge in real world. However, auto-detected bounding boxes are not accurate for identity alignment tasks due to potentially more pixels of background noise, occlusion, and inaccurate bounding box alignment (Fig. \[Fig:1\]). This is evident from that the rank-1 identity alignment rate on CUHK03 drops significantly from 61.6% on manually-cropped to 53.4% on auto-detected bounding boxes by best hand-crafted models [@wang2016highly], that is, a 8.2% rank-1 drop; and from 75.3% on manually-cropped [@xiao2016learning] to 68.1% on auto-detected [@Gated_SCNN] by strong deep learning models, that is, a 7.2% rank-1 drop. Moreover, currently reported “auto-detected” identity alignment performances on both CUHK03 and Market-1501 have further benefited from artifical [*human-in-the-loop cleaning process*]{}, which discarded “bad” detections with $<50\%$ IOU (intersection over union) overlap with corresponding manually cropped bounding boxes. Poorer detection bounding boxes are considered as “distractors” in Market-1501 and not given identity alignment labelled data for model learning. In this context, there is a need for [*noisy pixel removal*]{} within auto-detected bounding boxes as an integral part of learning to optimise identity alignment accuracy in a fully automated process.
-0.3cm \[tab:datasets\]
![ Comparisons of person bounding boxes by manually cropping (MC), automatically detecting (AD), and identity irrelevant pixel removal learning (IIPRL). Often AD contains more background clutter (a,d,e). Both AD and MC may suffer from occlusion (c), or a lack of identity discriminative pixel selection (b). []{data-label="Fig:1"}](images/boxes_new.pdf "fig:"){width="1\linewidth"} -0.15cm
There is very little attempt in the literature for solving this problem of removing noisy pixels within auto-detected bounding boxes for optimising identity alignment, except a related recent study on joint learning of object detection and identity alignment [@xiao2016end]. Our approach however differs from that by operating on any detectors [*independently*]{} so to benefit continuously from a wide range of detectors being rapidly developed by the wider community. Other related possible strategies include local patch calibration for mitigating misalignment in pairwise image matching [@SalienceReId_CVPR13; @shen2015person; @zheng2015partial; @Gated_SCNN] and local saliency learning for region soft-selective matching [@zhao2013person; @hanxiao2014GTS; @SalienceReId_CVPR13; @liu2016end]. These methods have shown to reduce the effects from viewpoint and pose change on identity alignment accuracy. However, [*all*]{} of them assume that person images are reasonably accurate.
In this work, we consider the problem of optimising pixels within any auto-detected person bounding boxes for maximising identity alignment tasks. The [**contributions**]{} of this study are: [**(1)**]{} We formulate a novel [**I**]{}dentity [**I**]{}rrelavent [**P**]{}ixel Removal [**L**]{}earning (IIPRL) model for useful pixels selection given identity matching discriminative constraints. Specifically, IIPRL is designed to discover identity-useful pixels within auto-detected bounding boxes by optimising recursively selecting regions subject to satisfying identity alignment pairwise label constraints (Fig. \[Fig:architecture\]). In contrast to existing selection methods, this global pixel region selection approach is more scalable in practice. This is because that most saliency models are local-region based and assume accurate inter-image consistency, or it requires expensive manipulation of local patch correspondence [*independently*]{}, difficult to scale. The IIPRL model is directly estimated under a pre-defined identity alignment matching criterion to jointly maximise an identity alignment model. Moreover, the IIPRL pixel removing strategy has the flexibility to be readily integrated with different deep learning models and detectors therefore can benefit directly from models rapidly developed elsewhere. [**(2)**]{} We introduce a simple yet powerful deep identity alignment model based on the GoogLeNet-V3 architecture [@szegedy2016rethinking]. This model is learned directly by the identity class loss rather than the more common pairwise [@Li_DeepReID_2014b; @Ahmed2015CVPR] or triplet loss function [@ding2015deep]. This loss selection not only significantly simplifies training data batch construction (e.g. random sampling with no notorious tricks required [@krizhevsky2012imagenet]), but also makes our model more scalable in practice given a large size training population or imbalanced training data from different camera views. We conducted extensive experiments on two large datasets to demonstrate the advantages of the proposed IIPRL model over a wide range of contemporary and state-of-the-art person identity alignment methods.
Related Work {#sec:related}
============
Most existing identity alignment methods [@PCCA_CVPR12; @KISSME_CVPR12; @PRD_PAMI13; @CVPR13LFDA; @wang2014person; @wang2016pami; @Li_DeepReID_2014b; @ding2015deep; @wang2016towards; @Liao_2015_ICCV; @wang2016human; @ma2017person; @chen2017person] focus on supervised learning of person identity-discriminative information. Representative learning algorithms include ranking by pairwise constraints [@Anton_2015_CoRR; @wang2016pami; @loy2013person; @wang2016person], discriminative subspace/distance metric learning [@KISSME_CVPR12; @PRD_PAMI13; @CVPR13LFDA; @xiong2014person; @liao2015person; @zhang2016learning; @chen2017person; @peng2017joint], and deep learning [@ShiZLLYL15; @ding2015deep; @Li_DeepReID_2014b; @Ahmed2015CVPR; @xiao2016learning; @ding2015deep; @wangjoint; @li2017person]. They typically require a large quantity of person bounding boxes and inter-camera pairwise identity labels, which is prohibitively expensive to collect manually.
[**Automatic Detection**]{} Recent works [@Li_DeepReID_2014b; @zheng2015scalable; @zheng2016PRW; @zheng2016PRW] have started to use automatic person detection for identity alignment benchmark training and test. Auto-detected person bounding boxes contain more noisy background and occlusions with misaligned person cropping (Fig. \[Fig:1\]), impeding discriminative identity alignment model learning. A joint learning of person detection and identity alignment was also investigated [@xiao2016end]. However, the problem of [*post-detection*]{} noisy pixel removal for identity alignment studied in this work has not been addressed in the literature. Noisy pixel removal can benefit independently from detectors rapidly developed by the wider community.
[**Saliency Selection** ]{} Most related identity alignment techniques are localised patch matching [@shen2015person; @zheng2015partial; @Gated_SCNN] and saliency detection [@zhao2013person; @hanxiao2014GTS; @SalienceReId_CVPR13; @liu2016end]. They are inherently unsuitable by design to cope with poorly detected person images, due to their stringent requirement of tight bounding boxes around the whole person.
![ The IIPRL noisy pixel removal model. (a) An identity sensitive learning branch based on the deep GoogLeNet-V3 network optimised by a multi-class cross entropy loss (orange arrows). (b) A irrelevant pixel removal learning branch designed as a deep Q-network optimised by identity alignment class label constraints in the deep feature space from branch (a) (blue arrows). For model deployment, the trained noisy pixel removal branch (b) computes the optimal regions for each probe and all the gallery images, extract the deep features from these optimal regions in the multi-class identity alignment branch (a) and perform L2 distance matching (green arrows). []{data-label="Fig:architecture"}](images/pipeline_new.pdf){width="0.99\linewidth"}
Identity Irrelevant Pixel Removal Modelling {#sec:method}
===========================================
The [I]{}dentity Irrelevant Pixel Removal Learning ([IIPRL]{}) model has two sub-networks: [**(I)**]{} A multi-class [*discrimination network*]{} $\mathcal{D}$ by deep learning from a training set of auto-detected person bounding boxes (Fig. \[Fig:architecture\](a)). This part is flexible with many options from existing deep identity alignment networks and beyond [@ding2015deep; @wangjoint; @xiao2016learning; @wang2016highly]. [**(II)**]{} An identity alignment [*refinement network*]{} $\mathcal{A}$ by learning recursively a better sub-region with its deep feature representation from $\mathcal{D}$ that can maximise identity-matching given identity alignment label constraints (Fig. \[Fig:architecture\](b)).
Noisy Pixel Removal Formulation {#sec:DQN}
-------------------------------
We formulate the identity alignment noisy pixel removal as a discriminative learning problem. This allows to correlate directly the identity alignment noisy pixel removal process with the learning objective of an “agent” by recursively [ *rewarding*]{} or punishing the learning process. In essence, the aim of model learning is to achieve an optimal identity discriminative attending action policy $a =
\pi(\mathbf{s})$ of an agent, i.e. a mapping function, that projects a state observation $\mathbf{s}$ (model input) to an action prediction $a$. In this work, we exploit the Q-learning technique for learning the proposed IIPRL agent, due to its sample efficiency advantage for a small set of actions [@watkins1989learning; @gu2016q]. Formally, we aim to learn an optimal state-value function which measures the maximum sum of the current reward ($R_t$) and all the future rewards ($R_{t+1}, R_{t+2}, \cdots$) discounted by a factor $\gamma$ at each time step $t$: $$\label{eq:Q}
Q^*(\mathbf{s}, a) = \max_\pi \mathbb{E}\big[R_t + \gamma R_{t+1} + \gamma^2R_{t+2} + \cdots \;\; | \;\; \mathbf{s}_t = \mathbf{s},
a_t = a, \pi \big]$$ Once $Q^*(\mathbf{s}, a)$ is learned, the optimal policy $\pi^*(\mathbf{s})$ can be directly inferred by selecting the action with the maximum $Q^*(\mathbf{s}, a)$ value in model deployment. More specifically, the agent interacts with each data sample in a sequential episode, which can be considered as a Markov decision process (MDP) [@Puterman1994MDP]. For our purpose, we need to design a specific MDP for identity alignment discriminative noisy pixel removal, as described below.
Markov Decision Process for Noisy Pixel Detection {#sec:ideal}
-------------------------------------------------
We design a MDP for identity alignment noisy pixel removal in auto-detected bounding boxes. In particular, we consider each input person bounding box image as a dynamic environment. An IIPRL agent interacts with this dynamic environment to locate the optimal identity alignment sensitive window. To guide this discriminative learning process, we further consider a reward that can encourage those attending actions to improve identity alignment performance and maximise the cumulative future reward in Eqn. . As such, we define actions, states, and rewards as follows.
![ Illustration of noisy pixel removal. []{data-label="Fig:actions"}](images/actions_new.pdf "fig:"){width="1\linewidth"} -0.4cm
**Actions**: An action set $\mathbf{A}$ is defined to facilitate the IIPRL agent to determine the location and size of an “salient window” (Fig. \[Fig:actions\]). Specifically, an attending action $a$ is defined by the location shift direction ($a_d \in \{\text{left, right, top, bottom}\}$) and shift scale ($a_e \in \mathbf{E}$). We also introduce a termination action as a search process stopping signal. $\mathbf{A}$ consists of a total of ($4\times |\mathbf{E}|+1$) actions. Formally, let the upper-left and bottom-right corner coordinates of the current salient window and an updated window be $\left[x_1, y_1, x_2, y_2\right]$ and $\left[ x_1', y_1', x_2', y_2' \right]$ respectively, the action set $\mathbf{A}$ can then be defined as: $$\begin{aligned}
&\mathbf{A} = \{ x_1'= x_1 + \alpha \Delta x, \;\; x_2'=x_2 - \alpha \Delta x, \;\; y_1'=y_1 + \alpha \Delta y, \;\; y_2'= y_2 - \alpha \Delta y, \;\; \text{T}\} , \\
&\text{where} \;\;\; \alpha \in \mathbf{E}, \;\;\;
\Delta x = x_2 - x_1, \;\;\;
\Delta y = y_2 - y_1, \;\;
\text{T} = \text{termination} . \nonumber\end{aligned}$$ Computationally, each action except termination in $\mathbf{A}$ modifies the environment by cutting off a horizontal or vertical stripe. We set $\mathbf{E} = \{5\%, 10\%, 20\%\}$ by cross-validation in our experiments, resulting in total 13 actions. Such a small action space with multi-scale changes has three merits: (1) Only a small number of simple actions are contained, which allows more efficient and stable agent training; (2) Fine-grained actions with small changes allow the IIPRL agent sufficient freedoms to utilise small localised regions in auto-detected bounding boxes for subtle identity matching. This enables more effective elimination of undesired background clutter whilst retaining identity discriminative information; (3) The termination action enables the agent to be aware of the satisfactory condition met for noisy pixel removal and stops further actions when optimised.
**States**: The state $\mathbf{s}_t$ of our MDP at time $t$ is defined as the concatenation of the feature vector $\mathbf{x}_t \in \mathbb{R}^d$ (with $d$ identity alignment feature dimension) of current attending window and an action history vector $\mathbf{h}_t \in \mathbb{R}^{|\mathbf{E}| \times n_\text{step}}$ (with $n_\text{step}$ a pre-defined maximal action number per bounding box), i.e. $\mathbf{s}_t = \left[ \mathbf{x}_t, \mathbf{h}_t \right]$. Specifically, at each time step, we extract the feature vector $\mathbf{x}_t$ of current window by the trained [identity alignment network]{} $\mathcal{D}$. The action history vector $\mathbf{h}_t$ is a binary vector for keeping a track of all past actions, represented by a $|\bm{A}|$-dimensional (13 actions) one-hot vector where the corresponding action bit is encoded as one, all others as zeros.
**Rewards**: The reward function $R$ (Eqn. ) defines the agent task objective. In our context, we therefore correlate directly the reward function of the IIPRL agent’s behaviour with the identity alignment matching criterion. Formally, at time step $t$, suppose the IIPRL agent observes a person image $\mathbf{I}_t$ and then takes an action $a_t = a \in \mathbf{A}$ to attend the image region $\mathbf{I}_t^a$. Given this region shift from $\mathbf{I}_t$ to $\mathbf{I}_t^a$, its state $\mathbf{s}_t$ changes to $\mathbf{s}_{t+1}$. We need to assess such a state change and signify the agent if this action is encouraged or discouraged by an award or a punishment. To this end, we propose three reward function designs, inspired by pairwise constraint learning principles established in generic information search and person identity alignment.
[*Notations* ]{} From the labelled training data, we sample two other [*reference*]{} images w.r.t. $\mathbf{I}_t$: (1) A [*cross-view positive*]{} sample $\mathbf{I}_t^+$ sharing the same identity as $\mathbf{I}_t$ but not the camera view; (2) A [*same-view negative*]{} sample $\mathbf{I}_t^-$ sharing the camera view as $\mathbf{I}_t$ but not the identity. We compute the features of all these images by $\mathcal{D}$, denoted respectively as $\mathbf{x}_t, \mathbf{x}_t^a, \mathbf{x}_t^+$, and $\mathbf{x}_t^-$.
[***(I) Reward by Relative Comparison*** ]{} Our first reward function $R_{t}$ is based on relative comparison, in spirit of the triplet loss for learning to rank [@liu2009learning]. It is formulated as: $$\label{eq:_Hierachical_2}
R_t=R_{rc}(\mathbf{s}_t,a)
= \Big(f_\text{match}(\mathbf{x}_t^a, \mathbf{x}_t^-) -
f_\text{match}(\mathbf{x}_t^a, \mathbf{x}_t^+)\Big) -
\Big(f_\text{match}(\mathbf{x}_t, \mathbf{x}_t^-) -
f_\text{match}(\mathbf{x}_t, \mathbf{x}_t^+)\Big)$$ where $f_\text{match}$ defines the identity alignment matching function. We use the Euclidean distance metric given the GoogLeNet-V3 deep features. Intuitively, this reward function commits (i) a positive reward if the attended region becomes more-matched to the [*cross-view positive*]{} sample whilst less-matched to the [*same-view negative*]{} sample, or (ii) a negative reward otherwise. When $a$ is the termination action, i.e. $\mathbf{x}^a_t = \mathbf{x}_t$, the reward value $R_{rc}$ is set to zero. In this way, the IIPRL agent is supervised to attend the regions subject to optimising jointly two tasks: (1) being more discriminative and/or more salient for the target identity in an inter-view sense (cross-view identity alignment), whilst (2) pulling the target identity further away from other identities in an intra-view sense (discarding likely shared view-specific background clutter and occlusion therefore focusing more on genuine person appearance). Importantly, this multi-task objective design favourably allows appearance saliency learning to intelligently select the most informative parts of certain appearance styles for enabling holistic clothing patten detection and ultimately more discriminative identity alignment matching (e.g. Fig. \[Fig:1\](b) and Fig. \[Visuliaztion\_DQN\_V2\_training\](b)).
[***(II) Reward by Absolute Comparison*** ]{} Our second reward function considers only the compatibility of a true matching pair, in the spirit of positive verification constraint learning [@chopra2005learning]. Formally, this reward is defined as: $$\label{eq:_Hierachical_3}
R_t=R_{ac}(\mathbf{s}_t,a)
= \Big( f_\text{match}(\mathbf{x}_t, \mathbf{x}_t^+) \Big) -
\Big( f_\text{match}(\mathbf{x}_t^a, \mathbf{x}_t^+)\Big)$$ The intuition is that, the cross-view matching score of two same-identity images depends on how well irrelevant background clutter/occlusion is removed by the current action. That is, a good attending action will increase a cross-view matching score, and vice verse.
[***(III) Reward by Ranking*** ]{} Our third reward function concerns the true match ranking change brought by the agent action, therefore simulating directly the identity alignment deployment rational [@REIDchallenge]. Specifically, we design a binary reward function according to whether the rank of true match $\mathbf{x}_{t}^{+}$ is improved when $\mathbf{x}_{t}$ and $\mathbf{x}_{t}^{a}$ are used as the probe separately, as: $$R_t=R_{r}(\mathbf{s}_t,a)=\left\{
\begin{array}{lr}
+1, \ \ \text{if} \;\;
\text{Rank}(\mathbf{x}_{t}^{+}|\mathbf{x}_{t}) >
\text{Rank}(\mathbf{x}_{t}^{+}|\mathbf{x}_{t}^a)\\
-1, \ \ \text{otherwise}\\
\end{array}
\right.
\label{eqn:ranking}$$ where $\text{Rank}(\mathbf{x}_{t}^{+}|\mathbf{x}_{t})$ ($\text{Rank}(\mathbf{x}_{t}^{+}|\mathbf{x}_{t}^a)$) represents the rank of $\mathbf{x}_{t}^{+}$ in a gallery against the probe $\mathbf{x}_{t}$ ($\mathbf{x}_{t}^a$). Therefore, Eqn. gives support to those actions of leading to a higher rank for the true match, which is precisely the identity alignment objective. In our implementation, the gallery was constructed by randomly sampling $n_g$ (e.g. 600) cross-view training samples. We evaluate and discuss the above three reward function choices in the experiments (Sec. \[sec:exp\]).
Model Implementation, Training, and Deployment
----------------------------------------------
[**Implementation and Training** ]{} For the multi-class discrimination network $\mathcal{D}$ in the IIPRL model, we deploy the GoogLeNet-V3 network [@szegedy2016rethinking] (Fig. \[Fig:architecture\](a)), a generic image classification CNN model [@szegedy2016rethinking]. It is trained from scratch by a softmax classification loss using person identity labels of the training data. For the identity alignment noisy pixel removal network $\mathcal{A}$ in the IIPRL model, we design a neural network of 3 fully-connected layers (each with 1024 neurons) and a prediction layer (Fig. \[Fig:architecture\](b)). This implements the state-value function Eqn. . For optimising the sequential actions for identity alignment noisy pixel removal, we utilise the $\epsilon$-greedy learning algorithm [@mnih2015human] during model training: The agent takes (1) a random action from the action set $\mathbf{A}$ with the probability $\epsilon$, and (2) the best action predicted by the agent with the probability $1-\epsilon$. We begin with $\epsilon=1$ and gradually decrease it by $0.15$ every 1 training epoch until reaching $0.1$. The purpose is to balance model exploration and exploitation in the training stage so that local minimum can be avoided. To further reduce the correlations between sequential observations, we employ the experience replay strategy [@mnih2015human]. In particular, a fixed-sized memory pool $\mathbf{M}$ is created to store the agent’s $N$ past training sample (experiences) $e_t = (\mathbf{s}_t, a_t, R_t,
\mathbf{s}_{t+1})$ at each time step $t$, i.e. $\{ e_{t-N+1}, \cdots, e_t \}$. At iteration $i$, a mini-batch of training samples is selected randomly from $\mathbf{M}$ to update the agent parameters $\mathbf{\theta}$ by the loss function: $$L_i(\mathbf{\theta}_i) = \mathbb{E}_{(\mathbf{s}_t, a_t, R_{t}, \mathbf{s}_{t+1})\sim \text{Uniform}(\mathbf{M})}
\Big( R_{t} + \gamma \max_{a_{t+1}}Q(\mathbf{s}_{t+1}, a_{t+1}; \tilde{\mathbf{\theta}}_i) - Q(\mathbf{s}_t, a_t; \mathbf{\theta}_i) \Big)^2,$$ where $\tilde{\mathbf{\theta}}_i$ are the parameters of an intermediate model for predicting training-time target values, which are updated as $\mathbf{\theta}_i$ at every $\varsigma$ iterations, but frozen at other times.
[**Deployment** ]{} During model deployment, we apply the learned noisy pixel removal network $\mathcal{A}$ to all test probe and gallery bounding boxes for extracting their useful window images. The deep features of these window images are used for person identity alignment matching by extracting the 2,048-D output from the last fully-connected layer of the discrimination network $\mathcal{D}$. We employ the L2 distance as the identity alignment matching metric.
Experiments {#sec:exp}
===========
**Datasets** For evaluation, we used two large benchmarking identity alignment datasets generated by automatic person detection: \[key\]CUHK03 [@Li_DeepReID_2014b], and Market-1501 [@zheng2015scalable] (details in Table \[tab:datasets\]). CUHK03 also provides an extra version of bounding boxes by human labelling therefore offers a like-to-like comparison between the IIPRL noisy pixel removal and human manually cropped images. Example images are shown in (a),(b) and (c) of Fig. \[Fig:1\].
**Evaluation Protocol** We adopted the standard CUHK03 1260/100 [@Li_DeepReID_2014b] and Market-1501 750/751 [@zheng2015scalable] training/test person split. We used the single-shot setting on CUHK03, both single- and multi-query setting on Market-1501. We utilised the cumulative matching characteristic (CMC) to measure identity alignment accuracy. For Market-1501, we also used the recall measure of multiple truth matches by mean Average Precision (mAP).
**Implementation Details** We implemented the proposed IIPRL method in the TensorFlow framework [@abadi2016tensorflow]. We trained an GoogLeNet-V3 [@szegedy2016rethinking] multi-class identity discrimination network $\mathcal{D}$ from scratch for each identity alignment dataset at a learning rate of 0.0002 by using the Adam optimiser [@kingma2014adam]. The final FC layer output feature vector (2,048-D) together with the L2 distance metric is used as our identity alignment matching model. All person bounding boxes were resized to $299 \times 299$ in pixel. We trained the $\mathcal{D}$ by 100,000 iterations. We optimised the IIPRL noisy pixel removal network $\mathcal{A}$ by the Stochastic Gradient Descent algorithm [@bottou2012stochastic] with the learning rate set to 0.00025. We used the relative comparison based reward function (Eqn. ) by default. The experience replay memory ($\mathbf{M}$) size was 100,000. We fixed the discount factor $\gamma$ to 0.8 (Eqn. ). We allowed a maximum of $n_\text{step}=5$ action rounds for each episode in training $\mathcal{A}$. The intermediate regard prediction network was updated every $\varsigma=100$ iterations. We trained the $\mathcal{A}$ by 10 epochs.
-0.2cm \[tab:CMC\_state\_of\_art\]
![ Qualitative evaluations of the IIPRL model: [**(a)**]{} Two examples of action sequence for noisy pixel removal given by action 1 (Blue), action 2 (Green), action 3 (Yellow), action 4 (Purple), action 5 (Red); [**(b)**]{} Two examples of cross-view IIPRL selection for identity alignment; [**(c)**]{} Seven examples of IIPRL selection given by 5, 3, 5, 5, 4, 2, and 2 action steps respectively; [**(d)**]{} A failure case when the original auto-detected (AD) bounding box contains two people, manually cropped (MC) gives a more accurate box whilst IIPRL noisy pixel removal fails to reduce the distraction; [**(e)**]{} Four examples of IIPRL selection on the Market-1501 “distractors” with significantly poorer auto-detected bounding boxes when IIPRL shows greater effects. []{data-label="Visuliaztion_DQN_V2_training"}](images/Figure4_v2.pdf "fig:"){width="1\linewidth"} -0.4cm
[**Comparisons to the State-of-the-Arts**]{} We compared the IIPRL model against 24 different contemporary and the state-of-the-art identity alignment methods (Table \[tab:CMC\_state\_of\_art\]). It is evident that IIPRL achieves the best identity alignment performance, outperforming the strongest competitor GS-CNN [@Gated_SCNN] by $2.9\%$ (71.0-68.1) and $20.9\%$ (86.7-65.8) in Rank-1 on CUHK03 and Market-1501 respectively. This demonstrates a clear positive effect of IIPRL’s noisy pixel removal on person identity alignment performance by filtering out bounding box misalignment and random background clutter in auto-detected person images. To give more insight and visualise both the effect of IIPRL and also failure cases, qualitative examples are shown in Fig. \[Visuliaztion\_DQN\_V2\_training\].
-0.3cm \[tab:Attention Strategy\]
[**Evaluations on Noisy Pixel Removal** ]{} We further compared in more details the IIPRL model against three state-of-the-art identity alignment models (eSDC [@SalienceReId_CVPR13], CAN [@liu2016end], GS-CNN [@Gated_SCNN]), and two baseline noisy pixel removal methods (Random, Centre) using the GoogLeNet-V3 identity alignment model (Table \[tab:Attention Strategy\]). For [*Random*]{}, we attended randomly person bounding boxes by a ratio (%) randomly selected from $\{95,90,80,70,50\}$. We repeated 10 times and reported the mean results. For [*Centre*]{}, we attended all person bounding boxes at centre by one of the same 5 ratios above. It is evident that the IIPRL (Relative Comparison) model is the best. The inferior identity alignment performance of eSDC, CAN and GS-CNN is due to their strong assumption on accurate bounding boxes. Both Random and Centre removal methods do not work either with even poorer identity alignment accuracy than that with “No Removal” selection. This demonstrates that optimal noisy pixel removal given by IIPRL is non-trivial. Among the three noisy pixel removal functions, [*Absolute Comparison*]{} is the weakest, likely due to the lack of reference comparison against false matches, i.e. no population-wise matching context in noisy pixel removal learning. [*Ranking*]{} fares better, as it considers reference comparisons. The extra advantage of [*Relative Comparison*]{} is due to the [*same-view negative*]{} comparison in Eqn.. This provides a more reliable background clutter detection since same-view images are more likely to share similar background patterns.
[**Auto-Detection+IIPRL vs. Manually Cropped** ]{} Table \[tab:compare\_labelling\] shows that auto-detection+IIPRL can perform similarly to that of [*manually cropped*]{} images in CUHK03 test[^1], e.g. $71.0\%$ vs. $71.9\%$ for Rank-1 score. This shows the potential of IIPRL in eliminating expensive manual labelling of bounding boxes and for scaling up identity alignment to large data deployment.
-0.3cm \[tab:compare\_labelling\]
[**Effect of Action Design** ]{} We examined three designs with distinct noisy pixel removal scales. Table \[tab:action\_scale\] shows that the most fine-grained design $\{5\%, 10\%, 20\%\}$ is the best. This suggests that the identity alignment by appearance is subtle and small regions make a difference in discriminative matching.
-0.3cm \[tab:action\_scale\]
Conclusion {#sec:conclusion}
==========
We presented an Identity Irrelevant Pixel Removal Learning (IIPRL) model for optimising identity alignment noisy pixel removal in auto-detected bounding boxes. This improves notably person identity alignment accuracy in a fully automated process required in practical deployments. The IIPRL model is formulated as a unified framework of discriminative identity learning by a deep multi-class discrimination network and noisy pixel removal learning by a deep Q-network. This achieves jointly optimal identity sensitive noisy pixel removal and identity alignment matching performance by a reward function subject to identity label pairwise constraints. Extensive comparative evaluations on two auto-detected identity alignment benchmarks show clearly the advantages and superiority of this IIPRL model in coping with bounding box misalignment and background clutter removal when compared to the state-of-the-art identity alignment models. Moreover, this IIPRL automatic noisy pixel removal mechanism comes near to be equal to human manual labelling of person bounding boxes on identity alignment accuracy, therefore showing a great potential for scaling up automatic identity alignment to large data deployment.
[^1]: The Market-1501 dataset provides no manually cropped person bounding boxes.
|
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"pile_set_name": "ArXiv"
}
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\[section\]
biblabel\[1\][\#1.]{}
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{
"pile_set_name": "ArXiv"
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---
abstract: |
Abstract
Every diagonal matrix [**D**]{} yields an endomorphism on the $n$-dimensional complex vector space. If one provides the $\mathbbm{C}^{n}$ with Hölder norms, we can compute the operator norm of [**D**]{}. We define homogeneous weighted spaces as a generalization of normed spaces. We generalize the Hölder norms for negative values, this leads to a proof of an extended version of the Hölder inequality. Finally, we formulate this version also for measurable functions.
author:
- |
VOLKER THÜREY\
Rheinstr. 91\
28199 Bremen, Germany [^1]
title: Another View on the Hölder Inequality
---
Subj-class: FA MSC-class: 46B, 46G Keywords: Hölder inequality, Operator norm
Introduction
=============
In this paper we generalize the well-known Hölder inequality (see, for instance, [@Meise/Vogt] or [@Elstrodt], or other books on functional analysis). So far nobody discussed the case of negative exponents in all details (for some discussions see e.g. [@Mitrinovic/Vasic],p.51). The main reason for this might be the fact that for $ p < 0$ the map $ (x_1, x_2, \ldots , x_n ) \longmapsto
\sqrt[p]{|x_1|^{p}+|x_2|^{p}+ \ldots + |x_n|^{p}} $ does not yield a norm for $ \mathbbm{C}^{n}$, because it is neither positive definit, nor the triangle inequality holds. Although it is worth to consider this map, since this leads to a natural extension of the often used Hölder inequality. To get this result, we first introduce [ *homogeneous weighted spaces* ]{} generalizing normed spaces. Then we define [ *Hölder weights* ]{} as a generalization of the Hölder norms, and the [ *operator weight* ]{} as a generalization of the operator norm. In our first rather inconvenient theorem we compute the operator weight of a diagonal matrix. The main result of this paper is then an extension of the Hölder inequality. Finally, we prove an analogic result for measurable functions. But here the proofs rely on the standard Hölder inequality.\
\
Let $ X $ be a complex vector space. Let $\|..\|$ denote a positive functional on $ X$, that means: $\|..\|$: $ X \longrightarrow$ $\mathbbm{R}^{+} \cup \{ 0 , \infty\}$. We consider three conditions,\
(1) $\| \vec{0}\|$ = 0 and for all *z $\in \ \mathbbm{C}$ and all *$\vec{x} \in X$ we have: *$\|z \cdot \vec{x}\|$ = $|z|$ $ \cdot \|\vec{x}\|$ (“homogenity”),\
(2) *$ \infty \notin \rm image(\|..\|) $ and $\| \vec{x}\| = 0$ [ if and only if]{} $ \vec{x} = \vec{0}$ (“positive definiteness”),\
(3) For all *$\vec{x} , \vec{y} \in X$ one has *$\|\vec{x}+\vec{y}\| \leq \|\vec{x}\| + \|\vec{y}\|$ (“triangle inequality”).******
\
$ \begin{array}[ ]{lll} \rm
\ \ \rm If \ \ \ \|..\| \ fullfils \ (1) & \rm then & \ \rm we \ call \ \|..\| \ \ a \
{ \it homogeneous \ weight } \ \rm on \ \it X, \\
\ \ \rm if \ \rm \ \ \|..\| \ fullfils \ (1) , (2) & \rm then & \ \rm we \ call \ \|..\| \ \ a \
{ \it pseudonorm } \ \rm on \ \it X, and \\
\ \ \rm if \ \ \ \|..\| \ fullfils \ (1), (2) \ and \ (3) & \rm then & \ \|..\| \ \ \rm is \ called
\ a \ { \it norm } \rm \ on \ \it X.
\end{array} $
Acording to this three cases we call the pair ( $ X , \|..\|$ ) a homogeneously weighted vector space (or [**hw space**]{}), a pseudonormed vector space, or a normed vector space, respectively.
For a linear map $F : ( X , \|..\|_X ) \ \longrightarrow \ ( Y , \|..\|_Y ) $ between complex homogeneously weighted vector spaces we denote by $ \| F \| := \inf \ \{ {\cal C} > 0 \ | \ \forall \vec{x} \ \in \ X :
\| F(\vec{x})\|_{Y} \leq {\cal C} \cdot \|\vec{x}\|_{X} \} $ the [ *operator weight* ]{} of $ F $ with respect to $ \|..\|_{X}, \ \|..\|_{Y} $.
Let [**A**]{} be a complex valued $m\times n$ matrix, $ m,n$ $\in$ $ \mathbbm{N}$. Then [**A**]{} defines a linear map, [**A**]{}: $\mathbbm{C}^{n} \rightarrow \mathbbm{C}^{m}$. Let $\|..\|_{X} , \: \|..\|_{Y}$ be homogeneous weights on $ X := \mathbbm{C}^{n}$ and $ Y := \mathbbm{C}^{m}$, respectively. Then the operator weight is $\|{\bf A}\| \ = \ \inf \{ {\cal C} > 0 \ | \ \forall \vec{x} \in \mathbbm{C}^{n} :
\|{\bf A}\vec{x}\|_{Y} \leq {\cal C} \cdot \|\vec{x}\|_{X} \} $.\
This definition turns {[**A**]{} $|$ [**A**]{}: ($\mathbbm{C}^{n},\|..\|_{X}) \rightarrow (\mathbbm{C}^{m},\|..\|_{Y}) $ and [**A**]{} is linear} into a [**hw space**]{}, which is a pseudonormed space, or a normed space, respectively, depending on the properties of the homogeneous weights $ \|..\|_{X}$ and $\|..\|_{Y}$.\
Now for every $n$ $\in$ $ \mathbbm{N}$ and for every $p$ $\in$ $ \{\infty, -\infty \} \cup \: \mathbbm{R}\backslash \{0\} $ we construct a homogeneous weight on $ \mathbbm{C}^{n}$.
\
For $\vec{x} = (x_1 , \ \ldots \ ,x_n ) \in \mathbbm{C}^{n} $ and for $p$ $\in$ $\left( 0, \infty \right)$ set $\|\vec{x}\|_{p} := \sqrt[p]{|x_1|^{p}+|x_2|^{p}+ \ \ldots \ + |x_n|^{p}}$,\
and for $p$ $\in$ $\left( -\infty , 0 \right)$ we set $$\|\vec{x}\|_{p} :=
\begin{cases}
\sqrt[p]{|x_1|^{p}+|x_2|^{p}+ \ \ldots \ + |x_n|^{p}} \quad &
\quad \mbox{for} \quad \prod_{i=1}^{n} x_i \neq 0 \\
0 & \quad \mbox{for} \quad \prod_{i=1}^{n} x_i = 0 \\
\end{cases}$$ and $\|\vec{x}\|_{\infty} := \max \{ \: |x_i|\: \ | \ i \in \{1,2, \ldots , n\} \ \} , $ and $\|\vec{x}\|_{-\infty} := \min \{ \: |x_i|\: \ | \ i \in \{1,2, \ldots , n\} \ \} $. These homogeneous weights will be called the [ *Hölder weights* ]{} on $\mathbbm{C}^{n}$.
Note that for $p$ $<$ 0 we have $\|\vec{x}\|_{p} = 0 \ \Longleftrightarrow \
\exists \ j \in \ \{ 1, \ldots , n \} $ and $x_j = 0 $. Furthermore, for all $n$ $>$ 1, these Hölder weights are pseudonorms if and only if $p$ $>$ 0, and they are norms if and only if $p$ $\geq$ 1.
In the case of a diagonal matrix [**D**]{}, $ {\bf D} : \ ( \mathbbm{C}^{n},\|..\|_{s})$ $\rightarrow$ ($\mathbbm{C}^{n},\|..\|_{t})$ and $\|..\|_{s} , \|..\|_{t}$ are Hölder weights, one easily verifies that $$\|{\bf D}\|
= \sup \ \{ \|{\bf D}\vec{x}\|_{t} \: | \: \vec{x} \in \mathbbm{C}^{n} \
\begin{rm} and \end{rm} \ \|\vec{x}\|_{s} = 1 \} .$$ This equality does not hold in general for arbitrary linear maps F : (X,$\|..\|_X$ ) $\longrightarrow$ ( Y,$\|..\|_Y$) due to the fact that there need not to exist an $\vec{x}$ with $ \|\vec{x}\|_{X} = 1 $.
Let us now restrict our attention to diagonal matrices to state our first theorem.
For $ n \geq $ 2 and $\vec{v} := ( v_1, \ldots , v_n ) \in \mathbbm{C}^{n} $ let $ \begin{array}[ ]{ccc}
\quad \ {\bf D} := \ &
\left( \begin{array}[ ]{ccc}
v_1 & & { \bf 0 } \\
& \ddots & \\
{ \bf 0 } & & v_n
\end{array} \right) &
\end{array} $ be the associated n-dimensional diagonal matrix, and let $ s, t \in \ \mathbbm{R} \backslash \{0\} \cup \{+\infty , \: -\infty \}$. Thus [**D**]{} is a linear endomorphism on $\mathbbm{C}^{n}$. Then we have for the operator weight $ \|{\bf D}\| $ with respect to $ \|..\|_{s} \ and \ \|..\|_{t}$ $$\|{\bf D}\|_{s,t} := \| {\bf D}\| =
\begin{cases}
\infty \ &
{ \rm if } \quad
( s < 0 < t \ \wedge \ \vec{v} \neq \vec{0} ) \hfill \qquad (\mathbb{A}), \\
\|\vec{v}\|_t \ &
{ \rm if } \quad ( \vec{v} = \vec{0} ) \ \ \ \vee \ \ \ ( t < 0 \wedge \prod_{i=1}^{n} v_i = 0 )
\ \ \ \vee \ \ \ ( s = \infty ) \hfill \qquad (\mathbb{B}), \\
\|\vec{v}\|_{\frac{s \cdot t}{s-t}} \ &
{ \rm if } \quad
( -\infty < t < s < 0 \ \wedge \ \prod_{i=1}^{n} v_i \neq 0 ) \ \quad \vee \\ & \quad \quad \quad ( 0 < t < s < \infty ) \ \ \vee \ \ \ ( -\infty < t < 0 < s < \infty ) \hfill \qquad (\mathbb{C}), \\
\|\vec{v}\|_{\infty} \ &
{ \rm if } \quad
( \ s \leq t < 0 \ \wedge \ \prod_{i=1}^{n} v_i \neq 0 ) \ \ \vee \ \
( 0 < s \leq t ) \hfill \qquad (\mathbb{D}), \\
\|\vec{v}\|_{-s} \ &
{ \rm if } \quad ( \ t = -\infty \ \wedge \ \prod_{i=1}^{n} v_i \neq 0 ) \hfill \qquad (\mathbb{E}). \end{cases}$$
Note that all possible cases are covered by $(\mathbb{A}) - (\mathbb{E})$. The above theorem allows us to deduce a theorem and two corollaries.
Let $ s, t \in \ \mathbbm{R} $ such that $ 0 \neq s \cdot t $, and for $ {\bf D} := \operatorname{diag}(v_1, \ldots, v_n)$ with $ \prod_{i=1}^{n} v_i \ \neq \ 0 $ we have $$\|{\bf D}\|_{s,t} \ = \ \|{\bf D}\|_{-t,-s} \ .$$
[ **\[Generalized Hölder Inequality\]** ]{}\
Let r, s, t $\in$ $\mathbbm{R}$ and $ 0 \neq$ $r \cdot s \cdot t$ and $\frac{1}{t}$ = $\frac{1}{r} \ + \ \frac{1}{s}$ . Then we have for every n $\in$ $ \mathbbm{N}$\
for all vectors $ \vec{v} := (v_1, \ \ldots , \ v_n) \: $ and $ \vec{x} := (x_1, \ \ldots , \ x_n) \ \in \mathbbm{C}^{n} $ $ ( with \ \ \vec{v} \cdot \vec{x} $ denotes multiplication by components $) $ $$\ t < r , s \quad \Longrightarrow \quad
\| \vec{v} \cdot \vec{x} \|_t \leq
\| \vec{v} \|_r \cdot \| \vec{x} \|_s \ ,$$ $$\ t > r , s \quad \Longrightarrow \quad
\| \vec{v} \cdot \vec{x} \|_t \geq
\| \vec{v} \|_r \cdot \| \vec{x} \|_s \ .$$
More explicitely we have the following corollary.
[ **\[Generalized Hölder Inequality\]** ]{}\
Let r, s, t $\in$ $\mathbbm{R}$ such that $ 0 \neq$ $r \cdot s \cdot t$ and $\frac{1}{t}$ = $\frac{1}{r} \ + \ \frac{1}{s}$ . Then for every n $\in$ $ \mathbbm{N}$ and for all numbers $ v_1, \ \ldots , \ v_n , \
x_1, \ \ldots , \ x_n \ \in \mathbbm{C} $ with $\prod_{i=1}^{n} \ v_i \cdot x_i$ $\neq 0 $ we have $$\ t < r , s \quad \Longrightarrow \quad
\sqrt[t]{\sum_{i=1}^{n}|v_i\cdot x_i|^{t}} \leq
\sqrt[r]{\sum_{i=1}^{n}|v_i|^{r}} \cdot \sqrt[s]{\sum_{i=1}^{n}|x_i|^{s}} \ ,$$ $$\ t > r , s \quad \Longrightarrow \quad
\sqrt[t]{\sum_{i=1}^{n}|v_i\cdot x_i|^{t}} \geq
\sqrt[r]{\sum_{i=1}^{n}|v_i|^{r}} \cdot \sqrt[s]{\sum_{i=1}^{n}|x_i|^{s}} \ .$$
If $\prod_{i=1}^{n} \ v_i \cdot x_i$ = 0 the inequality remains true provided the roots for negative exponents are defined.
Proof of Theorem 1
====================
First we handle the two easy cases.\
CASE $(\mathbb{A})$. Let $s < 0 < t$ and $ \vec{v} \ \neq \ \vec{0}$.\
Because ${\bf D}$ is not the 0-matrix, there is a $ j \in \{ 1, \ldots , n \} $ with $ \ v_j \neq 0$. Take for every $k$ $\in \mathbbm{N}\backslash\{1\}$ the vector $\vec{a}_k$ := ($ a_{k,1}, \: \ldots , \: a_{k,n} $) with $a_{k,j}$ := $k$ and for all $ i \in$ $\{1,2, \ldots , n \}\backslash \{j\}$ let $a_{k,i}$ := $ \sqrt[s]{\frac{1-k^{s}}{n-1}}$ . We have for every $k$ $\in \mathbbm{N}\backslash\{1\}$: $\|\vec{a}_k\|_s$ = 1 and $\|{\bf D}(\vec{a}_k)\|_t$ $\geq$ $|k \cdot v_j|$, and because of $ k \rightarrow \infty$ the right hand side goes to infinity, hence $\|{\bf D}\|_{s,t}$ = $\infty$.\
CASE $(\mathbb{B})$. Let $ \vec{v} = \vec{0}$, or $ t < 0 $ and $ \prod_{i=1}^{n} v_i = 0 $, or $ s = \infty $.\
If [**D**]{} is the 0-matrix we have for all $s,t$ : $\|{\bf D}\|_{s,t}$ = 0 . If $( t < 0 \wedge \prod_{i=1}^{n} v_i = 0 )$ one has at least one $ j \in \{1, \ldots , n \}$ with $ v_j = 0$. Then for $ \ \ \vec{x} \in$ $\mathbbm{C}^{n}$ we have $v_j \: x_j = 0$, hence $\|{\bf D}(\vec{x})\|_t$ = 0, hence $\|{\bf D}\|_{s,t}$ = 0 = $\|\vec{v}\|_{t}$ .\
In the case of $s = \infty $ take $\vec{e} := (1,1, \ldots , 1 )$, then we have $ \|\vec{e} \: \|_{\infty}$ = 1. If $ t \in \mathbbm{R} $ we get $ \|{\bf D} \vec{e} \: \|_{t} = \left[ \sum_{i=1}^{n} |v_i|^{t} \right]^{\frac{1}{t}}$. If $ t = -\infty $ we get $ \|{\bf D} \vec{e} \: \|_{-\infty} = \min \{ \: |v_i|\: \ | \ i \in \{1,2, \ldots , n\} \ \}$. Hence in CASE $(\mathbb{B})$ we always have $\|{\bf D}\|_{s,t}$ = $\|\vec{v}\|_{t}$ .\
\
The following two cases are more complicated and they need more attention. They will be treated together, because the proofs are similar.\
CASE $(\mathbb{C})$ and CASE $(\mathbb{D})$. Let either $ ( -\infty < t < s < 0 \ \wedge \ \prod_{i=1}^{n} v_i \neq 0 ), \ \ {\rm or } \ \
{ ( 0 < t < s < \infty ) },
\\ {\rm or} \ \ ( -\infty < t < 0 < s < \infty ), \ \ {\rm or} \ \
\ ( \ s \leq t < 0 \ \wedge \ \prod_{i=1}^{n} v_i \neq 0 ), \ \ {\rm or} \ \ ( 0 < s \leq t )$.\
The theorem is trivial if [**D**]{} is the 0-matrix, because then it clearly follows that 0 = $ \|\vec{v}\|_{\frac{s \cdot t}{s-t}} $ = $ \|\vec{v}\|_{\infty} $ = $ \|{\bf D}\|$. Hence, we assume $ \vec{v} \neq \vec{0} $. Let $\sf M$ $\in$ { $1,2, \ \ldots , \ n$ } such that $ |v_{\sf M}| $ = [ $ \max \{|v_1|, \ldots , |v_n| $}, ]{} hence $ |v_{\sf M}| $ $>$ 0. Now for the proof we will distinguish four different cases.\
[**Case a)**]{} $ 0 < s, t < \infty $.\
[**Case b)**]{} $-\infty < s , t < 0 \ { \rm and } \ \prod_{i=1}^{n} v_i \neq 0 $.\
[**Case c)**]{} $-\infty < t < 0 < s < \infty $.\
[**Case d)**]{} $ ( \ -\infty = s \leq t < 0 \ \ {\rm and} \ \ \prod_{i=1}^{n} v_i \neq 0 \ ) \ \
{\rm or} \ \ ( \ 0 < s \leq t = \infty \ ) $.\
We will prove the cases [**a,b,c** ]{} for $n = 2$ and then inductively for all $ n \in$ $ \mathbbm{N}\backslash\{1\}$.\
\
[**Case a)**]{} Let 0 $< s, t < \infty $.\
Let $ n = 2$. We have the $ 2\times2$ matrix := $ \operatorname{diag}(v_1,v_2)$. Without loss of generality let $v_{\sf M}$ = $ v_2$ ( $\neq$ 0 ). With $ b := {v_1}/{v_2}$ we have $|b|$ $\leq$ 1, and [**D**]{} = $ v_2$ $\cdot$ $ \left( \begin{array}[ ]{cc}
b & 0 \\
0 & 1
\end{array} \right) $ =: $ v_2$ $\cdot$ $\widetilde{{\bf D}}$.\
We have $\left\|{\bf D}\right\|_{s,t}$ = $|v_2|$ $\cdot$ $\|\widetilde{{\bf D}}\|_{s,t}$ = $|v_2|$ $\cdot \sup \ \{ \: \|\widetilde{{\bf D}}(\vec{x})\|_t \ | \ \vec{x} \in \mathbbm{C}^{2} \ { \rm and } \ \|\vec{x}\|_s = 1$ }. With $ \vec{x} := (x_1,x_2) $ we define a map $G$ : \[0,1\] $\rightarrow$ $\mathbbm{R}^{+} \cup \{ 0 \}$, but at first we will consider $G^{\:t}$ because it is easier ( $G$ and $G^{\:t}$ have extremums at the same values ). Define\
$G\:^{t}(y)$ := $ (\|\widetilde{{\bf D}}(\vec{x})\|_t)^{t}$ = $ y^{t}\cdot |b|^{t} + \left[\sqrt[s]{1-y^{s}} \right]^{t}$ for $ y := |x_1|, \ \|(x_1,x_2)\|_s = 1 $, hence $ y \ \in$ \[0,1\].\
First assume that $ s \neq t $. Elemantary analysis shows that $$(G\:^{t})'(y_E) = 0 \ \Leftrightarrow \
y_E = \sqrt[s]{\frac{1}{1+|b|^{\frac{s \cdot t}{t-s}}}} \ .$$ Instead of computing $(G\:^{t})''(y_E)$ we check the boundaries of the domain of $ G $, hence the maximum M$_{s,t} := \max\:\{\|\widetilde{{\bf D}}(\vec{x})\|_t \ \: | \ \: \vec{x} \in \mathbbm{C}^{2} \ \wedge \ \|\vec{x}\|_s = 1 \} \ = \
\max\:\{ G(y) \ | \ y \in [0,1] $ } is contained in the set $\{ \: G(y_E)\: ,\: G(0)\: , \: G(1) \: \} \ =
\ \{\: [1+|b|^{\frac{s \cdot t}{s-t}}]^{\frac{s-t}{s \cdot t}} \: , \: 1 \: , \: |b| \: \}$. To determine M$_{s,t}$ let us now consider the following three subcases.\
Subcase 1: $ s < t $ $\Longrightarrow$ M$_{s < t}$ = 1 and $\left\|{\bf D}\right\|_{s,t}$ = $|v_2|$ $\cdot$ M$_{s < t}$ = $ |v_2| $.\
Subcase 2: $ s > t $ $\Longrightarrow$ M$_{s > t}$ = $ [1+|b|^{\frac{s \cdot t}{s-t}}]^{\frac{s-t}{s \cdot t}}$ and $\left\|{\bf D}\right\|_{s,t}$ = $|v_2|$ $\cdot$ M$_{s > t}$ = $ [\: |v_2|^{\frac{s \cdot t}{s-t}} + |v_1|^{\frac{s \cdot t}{s-t}} \: ]^{\frac{s-t}{s \cdot t}}$.\
Subcase 3: $ s = t $ $\Longrightarrow$ By doing similar calculations as just now (in the case $ s \neq t $), we get M$_{s=t}$ = $G(0)$ = 1, hence $\left\|{\bf D}\right\|_{s,s}$ = $ |v_2|$, and the theorem has been proved for $n = 2$.
We have a continuous behaviour of $\left\|{\bf D}\right\|_{s,t}$ if $ s = t $, that means $$\lim_{\breve{t} \nearrow s} ( \left\|{\bf D}\right\|_{s,\breve{t}} ) \ = \
\lim_{ \breve{t} \nearrow s} ( \: [ \: |v_2|^{\frac{s \cdot \breve{t}}{s-\breve{t}}} +
|v_1|^{\frac{s \cdot \breve{t}}{s-\breve{t}}} \: ]^{\frac{s-\breve{t}}{s \cdot \breve{t}}} ) \ = \
\|\vec{v}\|_{\infty} \ = \
|v_2| \ = \ \left\|{\bf D}\right\|_{s,s} \ = \
\lim_{ \breve{s} \searrow t} ( \left\|{\bf D}\right\|_{\breve{s},t} ) \ .$$
Proof for $ n \geq $ 3.\
Assume that the theorem holds for $n - 1 $. Let $ {\sf m} \in $ $\{1 , \ldots , n-1\}$ with $|v_{\sf m}| := \max \{ |v_1|, \: \ldots , \: |v_{n-1}| $}, let $\vec{x} := ( x_1, \: \ldots , \: x_{n-1}, x_n ) \in \mathbbm{C}^{n}$. We distinguish two subcases.\
Subcase 1: $ s < t $ or $ s = t$.\
We have just proved the theorem for $n = 2$, that means that for arbitrary $ y_1, y_2, \ w_1, w_2 \ \in \: \mathbbm{C}$ we have $ \sqrt[t]{ |w_1 \: y_1|^{t} + |w_2 \: y_2|^{t} } \
\leq \ \max\{|w_1|, |w_2|\} \cdot \sqrt[s]{|y_1|^{s}+|y_2|^{s}} $ . By the assumption, we have $\sqrt[t]{\sum_{i=1}^{n-1}|v_ix_i|^{t} }$ $\leq$ $|v_{\sf m}| \cdot \sqrt[s]{\sum_{i=1}^{n-1} |x_i|^{s} }$ .\
By using the assumption and the theorem for $ n = 2 $, it follows that $$\begin{aligned}
\|{\bf D}(\vec{x})\|_t & = & \sqrt[t]{\sum_{i=1}^{n-1}|v_ix_i|^{t} + |v_nx_n|^{t}} \\
& \leq \ &
\sqrt[t]{ |v_{\sf m}|^{t} \cdot \left[ \sqrt[s]{\sum_{i=1}^{n-1}|x_i|^s} \right]^{t} + |v_nx_n|^{t} } \\
& \leq &
\max \{ |v_{\sf m}| , |v_n| \} \cdot \sqrt[s]{\sum_{i=1}^{n-1}|x_i|^{s} + |x_n|^{s } }
\ \ = \ \ |v_{\sf M}| \cdot \|\vec{x}\|_s \: .
\end{aligned}$$ Hence $\|{\bf D}\|_{s,t}$ $\leq$ $|v_{\sf M}|$.\
The vector $\vec{e_{\sf M}} := ( 0, \ldots , 0 , 1, 0, \ldots , 0 )$ shows that $\|{\bf D}(\vec{e_{\sf M}})\|_t$ = $|v_{\sf M}|$ $\cdot$ 1, hence $\|{\bf D}\|_{s,t}$ = $|v_{\sf M}|$.\
Subcase 2: $ s > t $.\
Let $\widetilde{w}$ := $ [\: \sum_{i=1}^{n-1}|v_i|^{\frac{s \cdot t}{s-t}} ]^{\frac{s-t}{s \cdot t}}.$ Because the theorem holds for $ n = 2$, we have for arbitrary $ y_1, y_2, \ w_1, w_2 \in \: \mathbbm{C}$: $ \sqrt[t]{ |w_1 \: y_1|^{t} + |w_2 \: y_2|^{t} }$ $\leq$ \[$ |w_1|^{ \: \frac{s t}{s-t}} + |w_2|^{\frac{s t}{s-t}} ]^{\frac{s-t}{st}}$ $\cdot$ $ \sqrt[s]{|y_1|^{s}+|y_2|^{s}} $.\
Because we assume the theorem for $ n-1 $, we have : $\sqrt[t]{\sum_{i=1}^{n-1}|v_ix_i|^{t} }$ $\leq$ $ \widetilde{w} \cdot \sqrt[s]{\sum_{i=1}^{n-1} |x_i|^{s} }$ .\
By using this and the theorem for $ n = 2$, we have $$\begin{aligned}
\|{\bf D}(\vec{x})\|_t & \ = \ &
\sqrt[t]{ \sum_{i=1}^{n-1}|v_i x_i|^{t} + |v_nx_n|^{t} } \\
& \ \leq \ &
\sqrt[t]{ \widetilde{w}^{ \: t} \cdot \left[ \sqrt[s]{\sum_{i=1}^{n-1}|x_i|^s} \right]^{t} + |v_nx_n|^{t} } \\ & \ \leq \ &
[ \widetilde{w}^{ \: \frac{s t}{s-t}} + |v_n|^{\frac{s t}{s-t}} ]^{\frac{s-t}{st}} \cdot \|\vec{x}\|_s
\ \ = \ \ \|\vec{v}\|_{\frac{s \cdot t}{s-t}} \cdot \|\vec{x}\|_s \ .
\end{aligned}$$ Hence $\|{\bf D}\|_{s,t}$ $\leq$ $\|\vec{v}\|_{\frac{s \cdot t}{s-t}}$ .\
Define for all $ i = 1,2, \ldots n \quad r_i := \sqrt[s-t]{|v_i|^{t}}$, and take the vector $\vec{z}$ := $ \frac{1}{\sqrt[s]{\sum_{i=1}^{n} |v_i|^{\frac{s \cdot t}{s-t}}}}\cdot ( r_1 , \ldots , r_n )$.\
One has $\|\vec{z}\|_s$ = 1 and $\|{\bf D}(\vec{z})\|_t$ = $\|\vec{v}\|_{\frac{s \cdot t}{s-t}}$ , that means the theorem is satisfied both in subcase 1 and in subcase 2, and the proof is finished if 0 $< s, t < \infty $.\
\
[**Case b)**]{} Let $-\infty \ < \ s, t \ <$ 0 and $ \prod_{i=1}^{n} v_i \neq 0 $.\
Let [**D**]{} = $ \left( \begin{array}[ ]{cc}
v_1 & 0 \\
0 & v_2
\end{array} \right) $ = $ v_2$ $\cdot$ $ \left( \begin{array}[ ]{cc}
b & 0 \\
0 & 1
\end{array} \right) $ =: $ v_2$ $\cdot$ $\widetilde{{\bf D}}$ , with $b$ := $v_1/v_2$, as above, and we have $|v_2|$ $\geq$ $|v_1|$ $>$ 0 and 1 $\geq$ $|b|$ $>$ 0. One has $\left\|{\bf D}\right\|_{s,t}$ = $|v_2|$ $\cdot \sup \{\|\widetilde{{\bf D}}(\vec{x})\|_t \ \ | \ \ \vec{x} \in \mathbbm{C}^{2} \ \ \wedge \ \|\vec{x}\|_s = 1$}, as above.\
But the domain of the map $ G\:^{t}(y) := (\|\widetilde{{\bf D}}(\vec{x})\|_t)^{t}$ = $ y^{t}\cdot |b|^{t} + \left[\sqrt[s]{1-y^{s}} \right]^{t}$ has changed.\
With $ \vec{x} =: (x_1,x_2) $ and $ \|\vec{x}\|_s = 1 $, $ y := |x_1| $, it has to be $ y >$ 1, ( because $ s $ is negative ).\
As above, we have $(G\:^{t})'(y_E) = 0 \ \Leftrightarrow \
y_E = \sqrt[s]{\frac{1}{1+|b|^{\frac{s \cdot t}{t-s}}}}$ = $[{1+|b|^{\frac{s \cdot t}{t-s}}}]^{-\frac{1}{s}}$ ,\
and the maximum M$_{s,t}$ := $ \sup\:\{\|\widetilde{{\bf D}}(\vec{x})\|_t \ \: | \ \: \vec{x} \in \mathbbm{C}^{2} \ \wedge \ \|\vec{x}\|_s = 1$ } = $ \sup\:\{ G(y) \ | \ y > 1 \} $\
is contained in the set $\{ \: G(y_E) , \: \lim_{ y \to 1} G(y) \: ,
\lim_{ y \to \infty} G(y) $ } = $\{\: [1+|b|^{\frac{s \cdot t}{s-t}}]^{\frac{s-t}{s \cdot t}} \: , \: |b| \: , \: 1 \: \}$.\
Again we consider three subcases.\
Subcase 1: $ s < t $ $\Rightarrow$ M$_{s < t}$ = 1 and $\left\|{\bf D}\right\|(s,t)$ = $|v_2|$ $\cdot$ M$_{s < t}$ = $ |v_2| $.\
Subcase 2: $ s > t $ $\Rightarrow$ M$_{s > t}$ = $ [1+|b|^{\frac{s \cdot t}{s-t}}]^{\frac{s-t}{s \cdot t}}$ and $\left\|{\bf D}\right\|(s,t)$ = $|v_2|$ $\cdot$ M$_{s > t}$ = $ [\: |v_2|^{\frac{s \cdot t}{s-t}} + |v_1|^{\frac{s \cdot t}{s-t}} \: ]^{\frac{s-t}{s \cdot t}}$.\
Subcase 3: $ s = t $ $\Rightarrow$ We get M$_{s=t}$ = $ \lim_{ y \to \infty} G(y)$ = 1, hence $\left\|{\bf D}\right\|(s,t)$ = $ |v_2|$, and the theorem has been proved for $ n = 2$. Now we finish [**Case b**]{} in a similar way to [**Case a**]{}.\
Subcase 1: $ s < t $ or $ s = t $.\
We have proved the theorem for $ n = 2 $. Because of $ t < 0 $, we have for arbitrary\
$ y_1, y_2, \ w_1, w_2 \in \mathbbm{C} $: $|w_1 \: y_1|^{t} + |w_2 \: y_2|^{t}
\geq [max\{|w_1|, |w_2|\}]^{t} \cdot \left[ \sqrt[s]{|y_1|^{s}+|y_2|^{s}} \right]^{t}$. Let\
[m]{} $\in$ $\{1 , \ldots , n-1\}$ with $|v_{\sf m}|$ := max {$ |v_1|, \: \ldots , \: |v_{n-1}| $ }, let $\vec{x} := ( x_1, \: \ldots , \: x_{n-1}, x_n ) \in \mathbbm{C}^{n}$.\
We assume the theorem for $ n - 1 $, hence we have : $\sqrt[t]{\sum_{i=1}^{n-1}|v_ix_i|^{t} }$ $\leq$ $|v_{\sf m}| \cdot \sqrt[s]{\sum_{i=1}^{n-1} |x_i|^{s} }$ .\
Because of $t <$ 0 , this is equivalent to $\sum_{i=1}^{n-1}|v_ix_i|^{t}$ $\geq$ $|v_{\sf m}|^{t} \cdot
\left[ \sqrt[s]{\sum_{i=1}^{n-1} |x_i|^{s}} \right]^{t}$. $$\begin{aligned}
{ \rm Thus \ it \ follows } \ \quad
\left[\|{\bf D}(\vec{x})\|_t\right]^{t} & = & \sum_{i=1}^{n-1}|v_ix_i|^{t} + |v_nx_n|^{t} \\
& \geq &
|v_{\sf m}|^{t} \cdot \left[ \sqrt[s]{\sum_{i=1}^{n-1}|x_i|^s} \right]^{t} + |v_nx_n|^{t} \\
& \geq &
\left[ \: \max\{ |v_{\sf m}| , |v_n| \} \: \right]^{t} \cdot
\left[\sqrt[s]{\sum_{i=1}^{n-1}|x_i|^{s} + |x_n|^{s }}\right]^{t}
= |v_{\sf M}|^{t} \cdot \|\vec{x}\|_s ^{t} \ .
\end{aligned}$$ Because of $ t <$ 0, this is equivalent to $\|{\bf D}(\vec{x})\|_t$ $\leq$ $|v_{\sf M}| \cdot \|\vec{x}\|_s$ . Hence $\|{\bf D}\|_{s,t}$ $\leq$ $|v_{\sf M}|$.\
To check equality, take for all sufficient large $ k \in$ $ \mathbbm{N}$ ( i.e. such that $ 2 - (1-\frac{1}{k})^{s} \ > \ 0$ )\
the vector $\vec{a_k}$ := ($ a_{k,1}, \: \ldots , \: a_{k,n} $) with $a_{k, {\sf M}}$ := $ \sqrt[s]{2 - (1-\frac{1}{k})^{s}}$ , and for every\
$ i \in$ $ \{1,2, \ldots , n \}\backslash \{\sf M\} $ take $a_{k,i}$ := $q_k$ := $ \sqrt[s]{\frac{( 1-\frac{1}{k} )^{s} \ -1}{n-1}}$ . We have for all such $ k $ : $\|\vec{a_k}\|_s$ = 1, and because of $ s, t <$ 0, we get $ \lim_{ k \to \infty}$ $(q_k)$ = $\sqrt[s]{0}$ = +$\infty$, hence $ \lim_{ k \to \infty}$ $((q_k)^{t})$ = 0, $$\begin{aligned}
{\rm and } \ \quad
\|{\bf D}(\vec{a_k})\|_t & = & \sqrt[t]{ |v_{\sf M}|^{t}
\cdot \left[ \sqrt[s]{2 - (1-\frac{1}{k})^{s}} \: \right] ^{t} +
\sum_{i=1, \ldots , n \wedge i \neq {\sf M}} |v_i|^t \cdot (q_k)^{t} } \\
& = & \ \ |v_{\sf M}| \cdot
\sqrt[t]{ \left[ \sqrt[s]{2 - (1-\frac{1}{k})^{s}} \: \right] ^{t} + \ \ (q_k)^{t} \ \cdot
\sum_{i=1, \ldots , n \wedge i \neq {\sf M}} |\frac{v_i}{v_{\sf M}}| ^{t} } \quad ,
\end{aligned}$$ hence $\ \lim_{ k \to \infty}$ $\|{\bf D}(\vec{a_k})\|_t$ = $ |v_{\sf M}|$. Thus $\|{\bf D}\|_{s,t}$ = $|v_{\sf M}|$.\
Subcase 2: $ s > t $.\
Let $\vec{x} := ( x_1, \: \ldots , \: x_{n-1}, x_n ) \in \mathbbm{C}^{n}$. We have proved the theorem for $n = 2$, that means\
$|y_1 w_1|^{t} + |y_2 w_2|^{t}$ $\geq$ $ [\: |y_1|^{\frac{s \cdot t}{s-t}} + |y_2|^{\frac{s \cdot t}{s-t}} \: ]^{\frac{s-t}{s}}$ $\cdot$ $\left[ \sqrt[s]{|w_1|^{s}+|w_2|^{s}} \right]^{t}$ for $ y_1, y_2, \ w_1, w_2 \in \mathbbm{C}$.\
Assume the theorem for $n - 1$, hence (because of $ t < 0 $)\
$\sum_{i=1}^{n-1}|v_ix_i|^{t}$ $\geq$ $ \left[ \sum_{i=1}^{n-1} |v_i|^{\frac{s \cdot t}{s-t} } \right]^{\frac{s-t}{s}} $ $\cdot$ $\left[ \sqrt[s]{\sum_{i=1}^{n-1} |x_i|^{s}} \right]^{t}$. By doing similar estimations as three times before, we get $\left[\|{\bf D}(\vec{x})\|_t\right]^{t}$ = $\sum_{i=1}^{n-1}|v_ix_i|^{t} + |v_nx_n|^{t}$ $\geq $ $ \left[ \sum_{i=1}^{n-1} |v_i|^{\frac{s \cdot t}{s-t} } \right ]^{\frac{s-t}{s}} $ $ \cdot \left[ \sqrt[s]{\sum_{i=1}^{n-1}|x_i|^s} \right]^{t} + |v_nx_n|^{t}$ $ \geq $ $ \left[ \sum_{i=1}^{n-1} |v_i|^{\frac{s \cdot t}{s-t} } + |v_n|^{\frac{s \cdot t}{s-t}} \right ]^{\frac{s-t}{s}} \cdot
\left[\sqrt[s]{\sum_{i=1}^{n-1}|x_i|^{s} + |x_n|^{s }}\right]^{t} $ = $\left[ \|\vec{v}\|_{\frac{s \cdot t}{s-t}} \right]^{t} \cdot \|\vec{x}\|_s ^{t}$ .\
Because of t $<$ 0, this is equivalent to $\|{\bf D}(\vec{x})\|_t$ $\leq$ $ \ \|\vec{v}\|_{\frac{s \cdot t}{s-t}} \cdot \ \|\vec{x}\|_s$ .\
Hence $\|{\bf D}\|_{s,t}$ $\leq$ $\|\vec{v}\|_{\frac{s \cdot t}{s-t}}$ . To check equality , one can use the same vector as above, i.e. , define for $i = 1,2, \ldots n : \ \ r_i := \sqrt[s-t]{|v_i|^{t}}$ , and $\vec{z}$ := $ \frac{1}{\sqrt[s]{\sum_{i=1}^{n} |v_i|^{\frac{s \cdot t}{s-t}}}}\cdot ( r_1 , \ \ldots , \ r_n )$.\
\
[**Case c)**]{} Let $ -\infty < t < 0 < s < \infty. $\
The proof is similar as the proofs before and we will not explain it in all details.\
In the case of $\prod_{i=1}^{n} v_i = 0$, in CASE $(\mathbb{B})$ we already have proved that $ \|{\bf D}\|_{s,t} $ = 0. Note that $\frac{s \cdot t}{s-t}$ $<$ 0, hence follows. Now assume $\prod_{i=1}^{n} v_i \neq 0$.\
Proof for $ n = 2 $. As in [**Case a**]{}, we consider the $ 2\times2$ matrix := $ \left( \begin{array}[ ]{cc}
v_1 & 0 \\
0 & v_2
\end{array} \right) $.\
With $v_{\sf M}$ = $ v_2$ and $ b := {v_1}/{v_2}$ we have [**D**]{} = $ v_2$ $\cdot$ $ \left( \begin{array}[ ]{cc}
b & 0 \\
0 & 1
\end{array} \right) $ =: $ v_2$ $\cdot$ $\widetilde{{\bf D}}$. One has $\left\|{\bf D}\right\|(s,t)$ = $|v_2|$ $\cdot$ $\|\widetilde{{\bf D}}\|(s,t)$ = $|v_2| \cdot \sup \{\|\widetilde{{\bf D}}(\vec{x})\|_t \ \ | \ \ \vec{x} \in \mathbbm{C}^{2} \ \wedge \ \|\vec{x}\|_s = 1 $ }. Again we consider the map $ G\:^{t}(y) := (\|\widetilde{{\bf D}}(\vec{x})\|_t)^{t}$ = $ y^{t}\cdot |b|^{t} + \left[\sqrt[s]{1-y^{s}} \right]^{t}$, ( here for all $ y $ in the open interval $ (0,1) $ ). As in [**Case a**]{}, we have: $(G\:^{t})'(y_E) = 0 \ \Leftrightarrow \
y_E = \sqrt[s]{\frac{1}{1+|b|^{\frac{st}{t-s}}}}$ , which yields a minimum for the map $G\:^{t}$, but a maximum for the map $ G $, and we get the maximum $\max\:\{\|\widetilde{{\bf D}}(\vec{x})\|_t \ \: | \ \: \vec{x} \in \mathbbm{C}^{2} \ \ { \rm and } \ \ \|\vec{x}\|_s = 1 \} $ = $\max\:\{ G(y) \ | \ y \in [0,1] \} \ = \ G(y_E) $.\
As above, we have $ G(y_E) = \: [1+|b|^{\frac{s \cdot t}{s-t}}]^{\frac{s-t}{s \cdot t}} $, and $\left\|{\bf D}\right\|_{s,t}$ = $|v_2|$ $\cdot$ $G(y_E)$ = $ [\: |v_2|^{\frac{s \cdot t}{s-t}} + |v_1|^{\frac{s \cdot t}{s-t}} \: ]^{\frac{s-t}{s \cdot t}}$, and the theorem is proved for $ n = 2 $.\
Because of $ t < 0 $, we have to continue as in [**Case b**]{} , subcase 2.\
Let $\vec{x} := ( x_1, \: \ldots , \: x_{n-1}, x_n ) \in \mathbbm{C}^{n}$, and let $ y_1, y_2, \ w_1, w_2 \in \mathbbm{C}$.\
We just have proved that $|y_1 w_1|^{t} + |y_2 w_2|^{t}$ $\geq$ $ [\: |y_1|^{\frac{st}{s-t}} + |y_2|^{\frac{st}{s-t}} \: ]^{\frac{s-t}{s}}$ $\cdot$ $\left[ \sqrt[s]{|w_1|^{s}+|w_2|^{s}} \right]^{t}$ holds.\
Assuming the theorem for $ n-1 $, we get $\sum_{i=1}^{n-1}|v_ix_i|^{t}$ $\geq$ $ \left[ \sum_{i=1}^{n-1} |v_i|^{\frac{st}{s-t} } \right]^{\frac{s-t}{s}} $ $\cdot$ $\left[ \sqrt[s]{\sum_{i=1}^{n-1} |x_i|^{s}} \right]^{t}$.\
Hence we compute as four times before\
$\left[\|{\bf D}(\vec{x})\|_t\right]^{t}$ = $\sum_{i=1}^{n-1}|v_ix_i|^{t} + |v_nx_n|^{t}$ $ \geq $ $ \left[ \sum_{i=1}^{n-1} |v_i|^{\frac{st}{s-t} } \right ]^{\frac{s-t}{s}} $ $ \cdot \left[ \sqrt[s]{\sum_{i=1}^{n-1}|x_i|^s} \right]^{t} + |v_nx_n|^{t}$\
$ \geq $ $ \left[ \sum_{i=1}^{n-1} |v_i|^{\frac{st}{s-t} } + |v_n|^{\frac{st}{s-t} } \right ]^{\frac{s-t}{s}}
\cdot \left[\sqrt[s]{\sum_{i=1}^{n-1}|x_i|^{s} + |x_n|^{s }}\right]^{t} $ = $\left[ \|\vec{v}\|_{\frac{s \cdot t}{s-t}} \right]^{t} \cdot \|\vec{x}\|_s ^{t}$ .\
Because of $ t <$ 0, this is equivalent to $\|{\bf D}(\vec{x})\|_t$ $\leq$ $ \ \|\vec{v}\|_{\frac{s \cdot t}{s-t}} \cdot \ \|\vec{x}\|_s$ , hence $\|{\bf D}\|(s,t)$ $\leq$ $\|\vec{v}\|_{\frac{s \cdot t}{s-t}}$. To check equality, one can use the same vector as two times before, i.e. define for $ i = 1,2, \ldots n : \ \ r_i := \sqrt[s-t]{|v_i|^{t}}$ , and $\vec{z}$ := $ \frac{1}{\sqrt[s]{\sum_{i=1}^{n} |v_i|^{\frac{s \cdot t}{s-t}}}}\cdot ( r_1 , \ \ldots , \ r_n )$.\
\
[**Case d)**]{} Let $ \ -\infty = s \leq t < 0 \: \ {\rm and } \: \
\prod_{i=1}^{n} v_i \neq 0 , \ \ \ { \rm or \ \ let } \ \ \ 0 < s \leq t = \infty. \ $\
If $ 0 < s $ $\leq t = \infty$, take $\vec{e_{\sf M}} := (0, \ldots , 0,1,0, \ldots , 0 ) , \ \ $ hence $ \ \
\|\vec{e_{\sf M}} \|_{s} = 1 $, and $ \|{\bf D} (\vec{e_{\sf M}} )\|_{\infty} = |v_{\sf M}| $, and $\|{\bf D}\|$ = $|v_{\sf M}| = \max \: \{ |v_1| , \ldots , |v_n| \}$ follows.\
If $-\infty = s \leq t < 0 $ and $ \prod_{i=1}^{n} v_i \neq 0 $ one can use the vector $\vec{e_k}$ ( for all $ \: k \in \mathbbm{N}$ ) with $ e_{k, \sf M}$ := 1, and for all $ i \in \{1, \ldots , n \}\backslash \{ \sf M \} $ $ e_{k,i} := k $, hence $\|\vec{e}_k\|_{-\infty} $ = 1, and\
$ \lim_{ k \to \infty}$ ($\|{\bf D}(\vec{e}_k)\|_{t})$ = $ |v_{\sf M}| = \|\vec{v}\|_{\infty} $ , and all four cases [**Case a**]{} $-$ [**Case d**]{} are proved, hence CASE $(\mathbb{C})$ and CASE $(\mathbb{D})$ are confirmed.\
\
It remains to prove one case of the theorem.\
CASE $(\mathbb{E})$. Let $ \ t = -\infty \ { \rm and } \ \prod_{i=1}^{n} v_i \neq 0. $ As it has been shown before, the statement is true if ( $ t = -\infty = s $ ) or ( $t = -\infty $ and $ s = \infty$ ). So assume $ t = -\infty < s \in \ \mathbbm{R}\backslash\{0\} $. Take a $\widetilde{t} \neq 0 \ \ $ with $ \ \ -\infty < \widetilde{t} < s $, it is already proved that $ \|{\bf D}\|_{s,\widetilde{t}} = \|\vec{v}\|_{\frac{s \cdot \widetilde{t}}{s-\widetilde{t}}} $ . Thus\
$ \|{\bf D}\|_{s,-\infty}$ = $\lim_{ \widetilde{t} \rightarrow -\infty}$ \[ $ \|{\bf D}\|_{s,\widetilde{t}}$ \] = $\lim_{ \widetilde{t} \rightarrow -\infty}$ \[ $ \|\vec{v}\|_{\frac{s \cdot \widetilde{t}}{s-\widetilde{t}}} $ \] = $\|\vec{v}\|_{-s} $ .\
For equality one takes the vector $\vec{z}$ := $ \|\vec{v}\|_{-s} \cdot ( \frac{1}{v_1} , \ldots , \frac{1}{v_n} ) $ = $ \sqrt[-s] {\sum_{i=1}^{n} \frac{1}{|v_i|^{s}}} \cdot ( \frac{1}{v_1} , \ldots , \frac{1}{v_n} ) $,\
hence $\|\vec{z}\|_{s} = 1 $ and $\|{\bf D}(\vec{z})\|_{-\infty} = \|\vec{v}\|_{-s}$ , and the proof of [**Theorem 1**]{} is finished.\
Proofs of Theorem 2 and the Corollaries
=======================================
The [ **Corollary 1**]{} follows immediately by observing that\
$\frac{s \cdot t}{s-t}$ = $\frac{(-t) \cdot (-s)}{(-t)-(-s)}$ , and $ s \leq t \ \Longleftrightarrow \ -t \leq -s $.\
\
Before we can prove [ [**Theorem 2**]{} ]{} we mention a fact, which is easy to confirm.
Let $ r, s, t \in \mathbbm{R}$, such that $ 0 \neq$ $r \cdot s \cdot t$ and $\frac{1}{t}$ = $\frac{1}{r} \ + \ \frac{1}{s}$ .\
Then either t $<$ r, s or t $>$ r, s.\
If furthermore $ t < 0, \ r, \ s $ or $ t > 0, \ r, \ s $ , then $ r \cdot s < 0 $ .
Now we are able to prove [[**Theorem 2**]{}]{}.
This theorem is trivial if $ n = 1$. So let $ n > 1 $. Let $ t < r , s $. Now take the [**Theorem 1**]{}, CASE $(\mathbb{C})$, and note that $ r = \frac{s \cdot t}{s-t}$ .\
Let $ t > r , s $ . In the case of $ \| \vec{v} \|_r $ $\cdot$ $ \| \vec{x} \|_s $ = 0, the inequality holds. Hence assume\
$ \| \vec{v} \|_r $ $\cdot$ $ \| \vec{x} \|_s \neq 0 $. Because of [**Fact 1**]{} and $\frac{1}{t}$ = $\frac{1}{r} \ + \ \frac{1}{s}$ , three cases are possible, namely $ 0 > t > r , s , \ \ $ or $ \ \ t > r > 0 > s , \ \ $ or $ \ \ t > s > 0 > r $.\
In the first two cases $s$ is negative, and because of $ \| \vec{x} \|_s \neq 0 $, $ x_i \neq 0 $ holds for every $i$. One has $\frac{1}{r}$ = $\frac{1}{t} \ + \ \frac{1}{-s}$ and (with [**Fact 1**]{} ) $ r < t , -s $. Let for all $ i \in \{1, \ldots , n \}$: $\widetilde{x_i}$ := $v_i \cdot x_i$\
and $z_i$ := $\frac{1}{x_i}$ . Because of $ r < t , -s $ we get\
$ \sqrt[r]{\sum_{i=1}^{n}|\widetilde{x_i} \cdot z_i|^{r}}$ $\leq$ $\sqrt[t]{\sum_{i=1}^{n}|\widetilde{x_i}|^{t}}$ $\cdot$ $\sqrt[-s]{\sum_{i=1}^{n}|z_i|^{-s}}$ $\Longleftrightarrow$ $\sqrt[r]{\sum_{i=1}^{n}|\widetilde{x_i} \cdot z_i|^{r}}$ $\cdot$ $\sqrt[+s]{\sum_{i=1}^{n}|z_i|^{-s}}$ $\leq$ $\sqrt[t]{\sum_{i=1}^{n}|\widetilde{x_i}|^{t}}$ $\Longleftrightarrow$ $\sqrt[r]{\sum_{i=1}^{n}|v_i|^{r}}$ $\cdot$ $\sqrt[s]{\sum_{i=1}^{n}|x_i|^{s}}$ $\leq$ $\sqrt[t]{\sum_{i=1}^{n}|v_i \cdot x_i|^{t}}$ $\Longleftrightarrow$ $ \| \vec{v} \|_r $ $\cdot$ $ \| \vec{x} \|_s $ $\leq$ $ \| \vec{v} \cdot \vec{x} \|_t $ .\
The remaining last case $ t > s > 0 > r $ is treated in the same way: because of $ 0 > r $ and $ \| \vec{v} \|_r \neq 0 $, $ v_i \neq 0 $ holds for every $i$. Hence define for all $ i \in \{1, \ldots , n \}$: $\widetilde{x_i}$ := $v_i \cdot x_i$ and $z_i$ := $\frac{1}{v_i}$ , and then one can go the same way as only just. This finishes the proof.
The [ **Corollary 2**]{} follows directly from [ **Theorem 2**]{}.
However, this version of the Hölder-inequality is not realy an extension, but equivalent with the usual one ( $1 = \frac{1}{r} \ + \ \frac{1}{s} \ $ and $ \ 1 < r,s \ \Longrightarrow $ $ \| \vec{v} \cdot \vec{x} \|_1 $ $\leq$ $ \| \vec{v} \|_r $ $\cdot$ $ \| \vec{x} \|_s $ ).\
For positive values of $ r, s, t $ one can find a short proof in [@Meise/Vogt],p.103. The general case which includes negative values is treated in the next section.
Measurable Functions
====================
In this last section we demonstrate that the generalized Hölder inequality also holds in the $ {\cal L}^{\: p} $ function spaces. The proofs rely mainly on the standard Hölder inequality. At first we have to define the $ {\cal L}^{\: p} $ spaces also for negative $ p$.\
Let $ (\Omega, {\cal A} , \mu ) $ be a measure space with $ \ \mu(\Omega) > 0 $. We use the conventions $ \infty \cdot 0 := 0 $ and $ \frac{1}{0} := \infty $. Let $ {\cal M}_{\Omega} := \{ \: f : (\Omega, {\cal A} , \mu ) \rightarrow \mathbbm{R}
\cup \{ -\infty, \infty \} \ | \ f { \rm \: is \: measurable \: } \}$. Define for every $p < 0$: $ {\cal L}^{\: p} := {\cal L}^{\infty} :=
\{ f \in {\cal M}_{\Omega} \ |
\ \ { \rm ess \ sup} \ \{ |f(\omega)| \ | \: \omega \in \Omega \} < \infty \ \} $.\
Then we define for all $ f \ \in {\cal M}_{\Omega} $ $$\| f \|_{p} :=
\begin{cases}
\sqrt[p]{\int_\Omega {|f|^{p}} \: d\mu } \: \quad &
\quad \Longleftrightarrow \quad 0 < \int_\Omega |f|^{p} \: d\mu < \infty \\
0 & \quad \Longleftrightarrow \quad \int_\Omega |f|^{p} \: d\mu = \infty \\
\infty & \quad \Longleftrightarrow \quad \int_\Omega |f|^{p} \: d\mu = 0 \\
\end{cases}$$ Note that for $ f \in {\cal L}^{\infty} , \ \| f \|_{p} \ < \ \infty $ holds. And for every $p > 0$ we take the usual definition, $ {\cal L}^{\: p} := \{ f: (\Omega, {\cal A} , \mu ) \rightarrow \mathbbm{R} \cup \{ -\infty, \infty \} \
| \ f \: \in {\cal M}_{\Omega} \ \ { \rm and } \ \ \int_\Omega {|f|^{p}} \: d\mu < \infty \}, $ and for all $ f \in {\cal M}_{\Omega}$ take $ \ \ \| f \|_{p} := \sqrt[p]{ \int_\Omega {|f|^{p}} \: d\mu }$ .\
By making an equivalence relation $N$ ( $ f \approx_N g $ $\Leftrightarrow$ $ f, g $ distinguish only on a zero set), and by defining $ \bf M_{\Omega} := {\cal M}_{\Omega}/_{\approx_N} $, and for all $p \ \in \mathbbm{R}\backslash\{0\}: \quad \ {\bf L}^{p} := {\cal L}^{\: p}/_{\approx_N}$, this definition makes that the pairs ( ${\cal M}_{\Omega} , \|..\|_{p} $ ), ( ${\bf M_{\Omega}}, \|..\|_{p} $ ), $( {\cal L}^{p} , \|..\|_{p} ) $ and $({ \bf L}^{p} , \|..\|_{p} ) $ are [**hw spaces**]{} for all $ p \in \mathbbm{R}\backslash\{0\}$. These homogeneous weights $ \|..\|_{p} $ we call the Hölder weights on ${\cal M}_{\Omega}$, ${\cal L}^{p}$, ${\bf M_{\Omega}}$ or $\bf L^{p} $, respectively. It is known that $( {\bf L}^{p} , \|..\|_{p} ) $ is a pseudonormed space if and only if $p > 0$, and it is a normed space if and only if $ p \geq$ 1.\
Now let us recall the well-known Hölder inequality and the reverse Hölder inequality for measurable functions. For two real numbers $ r, s$ such that 1 $ < $ $r , s $ and $ 1 \ = \ \frac{1}{r} \ + \ \frac{1}{s}$ , we have for all measurable functions $ f, g $ ( that means $ f , g $ $ \in {\cal M}_{\Omega}$ ): $ \| f \cdot g \|_1 $ $\leq$ $ \| f \|_r $ $\cdot$ $ \| g \|_s $ .\
For the next inequality see e.g. [@Elstrodt],p.226, or [@Mitrinovic/Vasic],p.51, or [@Hewitt/Stromberg],p.191.
Let $ r, s \in \mathbbm{R} \backslash \{0\} $ such that $ 1 > r , s $ and $ 1 \ = \ \frac{1}{r} \ + \ \frac{1}{s}$ .\
$ ($ Hence either $ r < 0 < s \quad or \quad s < 0 < r \ )$.\
Then one has for all measurable functions $ f , g , $ that a reverse Hölder inequality holds, i.e. $$\| f \cdot g \|_1 \geq
\| f \|_r \cdot \| g \|_s \ .$$
Assume $ r < 0 < s < 1 $. Now we have to distinguish three cases.\
1) $ \| f \|_r $ = 0. The inequality holds. ( Note that $ \infty \cdot 0 = 0 ) $.\
2) $ \| f \|_r = \infty . $ We have\
$ \| f \|_r = \infty \ \ \Longleftrightarrow \ \ {\int_\Omega |f|^{r} \ d\mu} = 0
\ \ \Longleftrightarrow \ \ |f|(\omega) = \infty \ $ (for almost all $ \omega \in \Omega) . $\
In the case of $ \| g \|_s $ = 0, the inequality holds. In the case of $ \| g \|_s > 0 $, there is a measurable set A with A $ \subset \Omega $, and $ \mu(A) > 0 \ $ and $ \
|g|(\omega) > 0 \ (\forall \: \omega \in A), $ hence it follows $ \ |f \cdot g|(\omega) = \infty \ $ (for almost all $ \omega \in A) $, hence $ \| f \cdot g \|_1 $ = $\infty $.\
3) $ 0 < \| f \|_r < \infty . $\
We have $ \frac{1}{s} \ = \ 1 \: + \: \frac{1}{-r} $ , hence $ 1 = \frac{1}{1/s} + \frac{1}{-r/s} \: \ $ and $ \ $ (with $ {\bf Fact \: 1} ) \
\ 1 < \frac{1}{s} , \: \frac{-r}{s} \: . $ Define $ v := |f|^{s} \cdot |g|^{s} $, $ w := |f|^{-s} , $ hence $ v , w \ \ \in {\cal M}_{\Omega}$, and we have by the Hölder inequality ( note that $ 0 < {\int_\Omega |w|^{\frac{-r}{s}} \ d\mu} < \infty \: ) $\
$ \| v \cdot w \|_1 $ $\leq$ $ \| v \|_{\frac{1}{s}} $ $\cdot$ $ \| w \|_{\frac{-r}{s}} $ $ \Longleftrightarrow $ $ \sqrt[s]{\int_\Omega |v \cdot w| \ d\mu} \leq
{\int_\Omega |v|^{\frac{1}{s} } \ d\mu} \ \cdot \ \sqrt[-r]{\int_\Omega |w|^{\frac{-r}{s}} \ d\mu} $\
$ \Longleftrightarrow $ $ \sqrt[s]{\int_\Omega |v \cdot w| \ d\mu} \cdot \sqrt[+r]{\int_\Omega |w|^{\frac{-r}{s}} \ d\mu} \leq
{\int_\Omega |v|^{\frac{1}{s} } \ d\mu} \quad \Longleftrightarrow \quad
\| g \|_s \cdot \| f \|_r \leq \| f \cdot g \|_1 $\
and all three cases of [ **Corollary 3** ]{} has been proved.
Now we are able to formulate the generalized Hölder inequality for measurable functions.
Let $ r, s, t \in \mathbbm{R}$ such that $ 0 \neq r \cdot s \cdot t$ and $\frac{1}{t}$ = $\frac{1}{r} \ + \ \frac{1}{s}$ .\
Then we have for all $ \ f , g \in {\cal M}_{\Omega} \ $ $$t < r , s \quad \Longrightarrow \quad
\| f \cdot g \|_t \leq
\| f \|_r \cdot \| g \|_s \ .$$ $$t > r , s \quad \Longrightarrow \quad
\| f \cdot g \|_t \geq
\| f \|_r \cdot \| g \|_s \ .$$
The proof is inspired by [@Meise/Vogt],p.103. We distinguish four cases.\
1) $t < r , s \quad { \rm and } \quad t > 0 $ 2) $t < r , s \quad { \rm and } \quad t < 0 $\
3) $t > r , s \quad { \rm and } \quad t > 0 $ 4) $t > r , s \quad { \rm and } \quad t < 0$\
We only show case 2. All the other cases follow along the same lines.\
Let $t < r , s \quad { \rm and } \quad t < 0$.\
Let $ f , g$ $ \in {\cal M}_{\Omega}$ . Then define $v, w $ $ \in {\cal M}_{\Omega}$ , by taking $v := |f|^{t}$ , $ w := |g|^{t}$ .\
Because of $ 1 = \frac{1}{r/t} + \frac{1}{s/t}$ , and $ 1 > \frac{r}{t} , \frac{s}{t} $ , and because of the previous [**Corollary 3**]{}, we have $ \| v \cdot w \|_1 $ $\geq$ $ \| v \|_\frac{r}{t} $ $\cdot$ $ \| w \|_\frac{s}{t} $ $\Longleftrightarrow$ $ \| f \cdot g \|_t $ $\leq$ $ \| f \|_r $ $\cdot$ $ \| g \|_s $ .
Acknowledgements:\
The author thanks Prof. Dr. Marc Keßeböhmer, Dr. Björn Rüffer and Dr. Gencho Skordev for support and help.
[99]{} Reinhold Meise, Dietmar Vogt, “ Introduction to Functional Analysis ”, Oxford University Press 1997 Jürgen Elstrodt, “ Maß- und Integrationstheorie ”, Springer 1996 $ \rm Mitrinovi\acute{c}, Vasi\acute{c}, $ “ Analytic Inequalities ”, Springer 1970 $ \rm Hewitt , Stromberg $, “ Real and Abstract Analysis ”, Springer 1969
[^1]: T: 49 (0)421/591777 E-Mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ has been investigated in its equilibrium liquid state with incoherent, inelastic neutron scattering. As compared to simple liquids, liquid PdNiCuP is characterized by a dense packing with a packing fraction above 0.5. The intermediate scattering function exhibits a fast relaxation process that precedes structural relaxation. Structural relaxation obeys a time-temperature superposition that extends over a temperature range of 540K. The mode-coupling theory of the liquid to glass transition (MCT) gives a consistent description of the dynamics which governs the mass transport in liquid PdNiCuP alloys. MCT scaling laws extrapolate to a critical temperature $T_c$ at about 20% below the liquidus temperature. Diffusivities derived from the mean relaxation times compare well with Co diffusivities from recent tracer diffusion measurements and diffsuivities calculated from viscosity via the Stokes-Einstein relation. In contrast to simple metallic liquids, the atomic transport in dense, liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ is characterized by a drastical slowing down of dynamics on cooling, a $q^{-2}$ dependence of the mean relaxation times at intermediate $q$ and a vanishing isotope effect as a result of a highly collective transport mechanism. At temperatures as high as $2\!\times\!T_c$ diffusion in liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ is as fast as in simple liquids at the melting point. However, the difference in the underlying atomic transport mechanism indicates that the diffusion mechanism in liquids is not controlled by the value of the diffusivity but rather by that of the packing fraction.'
author:
- 'A. Meyer'
date: ': Physical Review E (submitted) and cond-mat/0206364'
title: 'Atomic Transport in Dense, Multi-Component Metallic Liquids'
---
\[intro\]Introduction
=====================
We investigate microscopic dynamics in liquid PdNiCuP melts with inelastic neutron scattering. Our results show that PdNiCuP can in good approximation be regarded as multicomponent hard-sphere like system. As compared to liquid alkali-metals the packing in liquid PdNiCuP is much more dense. As a consequence, dynamics in liquid PdNiCuP can not be described by concepts developed for simple liquids. Instead, atomic transport in PdNiCuP melts is in excellent accordance with concepts developed in the context of glass formation.
Because of the short range nature of the interatomic potential with a strong repulsive core, the potential in simple metals can in good approximation be condensed to an effective hard sphere radius $R$. Therefore, the packing fraction $\varphi\!=\!\frac{4}{3}\pi\,n\, R^3$ ($n$ is the number density in unit volume) is an important parameter for the discussion of the transport mechanism [@PrAP73]. Due to the low compressibility of liquid metals the hard sphere radius is fairly temperature independent [@Pas88]. Liquid alkali metals are a paradigm of hard-sphere like fluids and their microscopic dynamics have been investigated in great detail [@BaZo94; @TyHa84]. At low packing fraction, e.g. at temperatures well above the melting temperature, atomic transport is dominated by binary collisions. This reflects itself in a Gaussian line shape of the quasielastic neutron scattering signal in the free particle limit towards large wavenumbers $q$. With increasing packing fraction, i.e. by approaching the melting temperature $T_m$, fluid dynamics play a more important role and towards small but finite $q$ the quasielastic line is well approximated by a Lorentzian function [@MoGl86; @BaZo94]. Alkali melts exhibit a packing fraction of about 0.4 at the melting point.
Colloidal suspensions [@MeUn93] and molecular dynamics simulated binary metallic melts [@KoAn94; @Tei96], that have a significantly higher packing fraction above 0.5, exhibit microscopic dynamics that is not governed by binary collisions and is typical for glass forming, molecular liquids [@WoAn76; @CuLH97; @Goe99]. Their dynamics has been described by predictions of the mode coupling theory (MCT) of the liquid to glass transition [@GoSj92]: Atomic transport slows down by orders of magnitude on a small change in temperature or pressure. The slowing down of the dynamics goes along with a spread of the quasielastic line as compared to the Lorentzian found in simple liquids corresponding to a stretching in time of correlation functions over a wider time range than expected for an exponential decay.
Mode coupling theory gives a microscopic explanation for this behaviour in terms of a non linear coupling of density fluctuations caused by feedback effects in the dense liquid. Atomic transport is envisioned as a highly collective process. At a critical packing fraction $\varphi_c$, or a critical temperature $T_c$ respectively, MCT predicts a change in the transport mechanism from liquid-like flow to glass like activated hopping processes. MCT calculations for a hard sphere system exhibit a critical packing fraction of 0.525 [@FuHL92]. In a colloidal suspension $\varphi_c \simeq 0.58$ has been found [@MeUn93]. Whereas in alkali melts the packing fractions are well below 0.5, new multicomponent alloys on Zr- [@PeJo93] and Pd-basis [@InNK97; @LuWG99; @NiIn02] at quasieutectic compositions exhibit packing fractions at the their liquidus temperature $T_{liq}$ of some 0.52 [@OhCR97; @LuGF02]. These alloys known for their bulk glass forming ability, exhibit viscosities at their melting temperatures that are 2-3 orders of magnitude larger than in simple metals and most alloys [@MaWB99; @HaBD02].
In previous neutron scattering experiments fast dynamics in viscous Zr$_{46.8}$Ti$_{8.2}$Cu$_{7.5}$Ni$_{10}$Be$_{27.5}$ have been investigated: Data analysis in the framework of MCT is in full agreement with the predicted scaling functions and gives a $T_c$ some 20% below $T_{liq}$ [@MeWP98]. In an experiment on viscous Pd$_{40}$Ni$_{10}$Cu$_{30}$P$_{20}$ (Pd40) the focus was on slow dynamics [@MeBS99]: The long–time decay of correlations exhibits the common features of glass forming liquids, i.e. structural relaxation obeys a stretching in time and a universal time-temperature superposition.
Here, inelastic neutron scattering results on liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ (Pd43) are reported. The small change in composition compared to the Pd$_{40}$Ni$_{10}$Cu$_{30}$P$_{20}$ results in an improved stability with respect to crystallization [@LuWG99] — a cooling rate as low as 0.09K/s is sufficient to avoid crystallization and to form bulk metallic glass [@ScJB00]. This allows measurements of transport coefficients at temperatures around the mode coupling $T_c$ with experimental techniques that only require heat treatment for some minutes, e.g. rheometry [@MaWB99] or radio tracer diffusion measurements [@Zoe02].
Compared to previous neutron scattering measurements [@MeWP98; @MeBS99] a different experimental setup has been used that covers an extended range of momentum transfers $q$. This allows a detailed investigation of the fast MCT $\beta$ relaxation and the correlation between structural relaxation and long range atomic transport. In addition, the temperature range has been extended by several 100K up to temperatures at which the diffusion of the atoms is of the order of that in simple monoatomic metals. The results are set in context to macroscopic transport coefficients, the dynamics in simple liquids and the predictions by the mode coupling theory of the liquid to glass transition.
\[exp\]Experimental
===================
Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ was prepared from a mixture of pure elements by induction melting in a silica tube. The melt was subject to a B$_2$O$_3$ flux treatment in order to remove oxide impurities. Differential scanning calorimetry with a heating rate of 40K/min resulted in a $T_g$ at 578K and a $T_{liq}$ at 863K in accordance with literature values [@LuWG99; @ScJB00]. For the neutron time-of-flight experiment a thin-walled Al$_2$O$_3$ container has been used giving a hollow cylinder sample geometry of 22mm in diameter and a thickness of 1mm. During the measurement the liquid was covered by a thin layer of $^{11}$B$_2$O$_3$ flux material. For the chosen geometry and neutron wavelength the sample scatters less than 2%. Therefore, multiple scattering, which would alter the data especially towards low $q$ [@Wut00], only has a negligible effect.
Microscopic dynamics in liquid PdNiCuP has been investigated on the neutron time-of-flight spectrometer IN6 at the Institut Laue-Langevin in Grenoble. An incident neutron wavelength of $\lambda\!=\!5.1\,\mbox{\AA}^{-1}$ yielded an accessible wavenumber range at zero energy transfer of $q\!=\!0.75-1.95\,\mbox{\AA}^{-1}$ at an energy resolution of $92\,\mu\mbox{eV}$ (FWHM). Regarding the scattering cross sections of the individual elements PdNiCuP is a 90% coherent scatterer. However, with the first structure factor maximum at $q_{0}\!\simeq\!2.9$Å$^{-1}$ our spectra are dominated by incoherent scattering, that is dominated by the contributions from Ni with $\simeq$73% and Cu with $\simeq$20%.
The alloy was measured at room temperature to obtain the instrumental energy resolution function. Data were collected in the liquid between 833K and 913K in steps of 40K and between 973K and 1373K in steps of 100K with a duration between 2 and 5 hours each. At each temperature empty cell runs were performed. The data at 833K, 30K below the liquidus, do not show signs of crystallization, which is in accordance with the time-temperature transformation of the undercooled liquid [@ScJB00]. In order to obtain the scattering law $S(q,\omega)$ (Fig. \[sqw\]), raw data were normalized to a Vanadium standard, corrected for self-absorption and container scattering, interpolated to constant $q$, and symmetrized with respect to energy with the detailed balance factor. Fourier deconvolution of $S(q,\omega)$ and normalization to 1 for $t=0$ gives the self correlation function $\Phi(q,t)$.
\[mct\]Mode-Coupling Theory Asymptotics
=======================================
The mode-coupling theory of the liquid to glass transition (MCT) [@GoSj92; @Goe99] is developed in the well-defined frame of molecular hydrodynamics [@BoYi80]. Starting with the particle density, $\vec{\rho_r}(t)= \sum_{i}
\delta(\vec{r} - \vec{R}_i(t))$, i.e. the positions, $\vec{R}_i$, of each particle, $i$, at time $t$, the Zwanzig-Mori formalism provides an exact equation of motion for the density correlation function, $\Phi_q(t) =
\langle\rho_q (t)^* \rho_q\rangle \,/\,\langle | \rho_q | ^2 \rangle$: $$\label{fullmct}
\ddot{\Phi}_q (t) + \Omega_q^2\, \Phi_q (t) +
\int_{0}^{t} M_q (t-t') \,\dot{\Phi}_q (t') \mbox{d}t' = 0$$ where $\Omega_q$ represents a phonon frequency at wavenumber $q$ and $M_q(t)$ a correlation function of force fluctuations which in turn is a functional of density correlations. The static part of the fluctuating force is a linear combination of density fluctuation pairs and depends therefore only on the interaction potential.
Since MCT does not aim to describe the detailed microscopic phonon distribution, but rather to give a universal picture of relaxational dynamics at longer times, the integral kernel is split into $$\label{split}
M_q(t) = \nu_q\delta(t) + \Omega_q^2 m_q(t),$$ where $\nu_q$ models the damping by “fast” modes and $m_q(t)$ accounts for memory effects through the coupling of “slow” modes. The basic idea of the mode-coupling theory of the liquid to glass transition is to consider as “slow” all products of density fluctuations. By derivation, $m_q(t)$ contains no terms linear in $\Phi_q(t)$. Therefore, in lowest order, it is a quadratic functional $$\label{kernel}
m_q (t) = \sum_{q_1+q_2=q} V_q(q_1,q_2) \,\Phi_{q_1}(t) \,\Phi_{q_2}(t).$$ In this approximation, the coupling coefficients $V_q$ are specified in terms of the static structure of the liquid.
Eqs. 1, 2 and 3 lead to the following scenario: a fast $\beta$-relaxation process, which can be visualized as a rattling of the atoms in the cages formed by their neighbouring atoms, prepares structural $\alpha$ relaxation, responsible for viscous flow. At an ideal glass transition temperature $T_c$ the transport mechanism crosses over from glass-like activated hopping processes to liquid-like collective motion. In other words, at $T_c$, the cages are no longer stable on the time scale of a diffusive jump.
A common feature of structural $\alpha$ relaxation in glass-forming liquids is a stretching in correlation functions over a wider time range than expected for exponential relaxation [@WoAn76; @CuLH97]. Experimental data in the $\alpha$ relaxation regime can usually be well described by a stretched exponential function $$\label{kww}
F(q,t) = f_q^c\,\exp{[-(t/\tau_q)}^{\beta_q}]$$ with an exponent $\beta_q\!<\!1$. $\tau_q$ is the relaxation time and $f_q^c\!<\!1$ accounts for the initial decay of correlations due to phonons and the fast relaxation process. For temperatures above $T_c$ mode coupling theory predicts a universal time-temperature superpostion of structural relaxation resulting in a temperature independent stretching exponent $\beta_q$. Many glass-forming liquids in contrast exhibit a temperature dependence of $\beta$ – in some $\beta$ is decreasing, in others increasing on temperature increase.
![\[sqw\] Scattering law $S(q,\omega)$ (logarithmic scale) of liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ obtained on the neutron time-of-flight spectrometer IN6. The signal is dominated by the incoherent scattering from Ni and Cu. The data at 300K represent the instrumental energy resolution. Measurements in the liquid ($T_{liq}\!=\!863\,$K) reveals a broad quasielastic signal. ](pdfig1-2.eps)
Solutions of the mode coupling equations for a hard sphere system confirm, that the MCT $\alpha$ relaxation is well described by a stretched exponential function and Eq. (\[kww\]) becomes a special, long time solution of mode coupling theory [@FuHL92]. For the mean relaxation times $$\label{mtau}
\left\langle\tau_q\right\rangle = \int\limits_0^\infty\! {\rm d}t\,
F(q,t) / f_q^c = \tau_q \,
\beta^{-1}\Gamma ({\beta^{-1}})$$ mode coupling theory predicts that $\left\langle\tau_q(T)\right\rangle$ is inversely proportional to the diffusivity $D(T)$ and obeys an asymptotic scaling law for temperatures above but close to $T_c$: $$\label{tauscal}
\langle\tau_q\rangle \propto 1 / D \propto [(T\!-\!T_c)/T_c]^{-\gamma}.$$
Asymptotic expansions of Eq. (\[fullmct\],\[kernel\]) show that in the intermediate $\beta$-relaxation regime around a crossover time, $t_{\sigma}$, and for temperatures close to $T_c$, the asymptotic form of the correlation function is independent of the detailed structure of the coupling coefficients and exhibits a universal factorization property: $$\label{betascal}
\Phi(q,t)= f_q^c + h_q\,g_{\rm \lambda}(t/t_{\sigma}),$$ where $f_q$ represents the Debye-Waller factor and $h_q$ an amplitude. The scaling function $g_{\rm \lambda}(\tilde{t})$ is defined by just one shape parameter $\lambda$. Close to $T_c$ mode coupling theory predicts the temperature dependence of $t_{\sigma}$ and $h_q$ with the asymptotic scaling functions: $$\label{tchq}
t_\sigma \propto (T-T_c)^{-1/2a} \quad{\rm and}\quad
h_q \propto (T-T_c)^{1/2}.$$ There are tables providing $a$, $\gamma$ and $g_{\lambda}$ as a function of $\lambda$ [@Goe90].
\[res\]Results
==============
Figure \[sqw\] shows the scattering law $S(q,\omega)$ of liquid PdNiCuP at $q\!=\!1.05$Å$^{-1}$. The signal is dominated by the incoherent scattering from Ni and Cu. $S(q,\omega)$ displays a quasielastic line with an increasing width on temperature increase and with wings extending up to several meV. Above some 3meV the quasielastic signal merges into a constant as expected for an ideal Debye solid and found in liquid GeO$_2$ [@MeSN01]. The quasielastic signal of $S(q,\omega)$ is fairly well separated from vibrations.
\[vib\]Vibrations
-----------------
There is no general understanding of the influence of phonons on the atomic transport in liquids. Whereas in monoatomic alkali melts one finds low-lying collective phonon modes that mediate mass transport [@WaSj82; @MoGG87; @BaZo94], the mode coupling theory of the liquid to glass transition does not consider the detailed phonon dynamics for its description of atomic transport in viscous liquids (Eq. \[split\]). In particular, dynamics in hard sphere-like colloidal suspensions, that do not exhibit vibrations, are in excellent agreement with MCT predictions [@MeUn93].
![\[gw\] Density of states $g(\omega)$ representing mainly the Ni and Cu vibrations in liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$. With a maximum at some 17meV (corresponding to some 0.04ps) vibrations are fairly well separated from the fast relaxational dynamics around 1ps. The small expansion coefficient reflects itself in a weak temperature dependence of $g(\omega)$.](pdfig10.eps)
In most molecular liquids broad phonon distributions overlap with the quasielastic signal. The phonon density of states exhibits a first maximum usually between 4 and 8meV [@WuPC95; @Tol01]. This aggravates the theoretical description of the relaxational dynamics. Figure \[gw\] displays the vibrational density of states $g(\omega)$ representing mainly the incoherent contributions of the Ni and Cu atoms of liquid PdNiCuP. $g(\omega)$ has been derived from the scattering law $S(q,\omega)$ using a procedure that assumes pure incoherent scattering and uses the multiphonon correction as described in [@WuKB93].
$g(\omega)$ displays a weak temperature dependence indicating that anharmonic contributions to the interatomic potential are minor. This is in accordance with an expansion coefficient of only $4 \times 10^{-5}$K$^{-1}$ [@LuGF02], that is an order of magnitude smaller than that of most molecular glass forming liquids. With a maximum in $g(\omega)$ at some 17meV, vibrations are fairly well separated from the quasielastic signal that extends up to several meV. In addition, PdNiCuP does not exhibit a “boson peak” – a maximum in $S(q,\omega)$ that is found in other glasses usually at a few meV – in the $q$ range investigated. This allows a detailed investigation of the atomic transport in liquid PdNiCuP.
\[alpha\] Structural Relaxation
-------------------------------
![\[sqt\] Normalized density correlation function $\Phi(q,t)$ of liquid PdNiCuP at 1.05Å$^{-1}$ as obtained by Fourier deconvolution of measured $S(q,\omega)$. Structural relaxation causes the final decay of $\Phi(q,t) \to 0$. The lines are fits a stretched exponential function (Eq. \[kww\]).](pdfig2.eps)
The quasielastic signal is best analyzed in the time domain with a removal of the instrumental resolution function. The density correlation function $\Phi(q,t)$ has been obtained by Fourier transformation of measured $S(q,\omega)$, division of the instrumental resolution function, and normalization with the value at $t=0$. Between 0 and $\simeq\!1$ps phonons and a fast process lead to a decrease in $\Phi(q,t)$ from 1 towards a plateau. Figure \[sqt\] displays the long-time decay of $\Phi(q,t)$ at $q\!=\!1.05$Å$^{-1}$ from this plateau towards zero in a semilogarithmic representation.
![\[tts\] Rescaling of the density correlation function in the $\alpha$ relaxation regime (for $t > 1$ps) using results from fits with a stretched exponential function: a time-temperature superposition of structural relaxation holds from $T_{liq} - 30$K to $T_{liq} + 510$K. The line is a fit with Eq. \[kww\] resulting in a stretching exponent $\beta\!=\!0.75$.](pdfig3.eps)
![\[tqs\] Rescaling of the density correlation function $\Phi(q,t)$ at 1073K. Within the accessible $q$ range the stretching of the structural relaxation is fairly $q$ independent. The line represents a stretched exponential function (Eq. \[kww\]) with a stretching exponent $\beta\!=\!0.75$.](pdfig4.eps)
In contrast to simple metallic liquids, liquid PdNiCuP exhibits a structural relaxation that shows stretching in time. The lines are fits with the stretched exponential function (Eq. \[kww\]). In a first fitting process the stretching exponent $\beta_q(T)$ was treated as a parameter: $\beta_q(T)$ varies weakly around a mean $\beta\!=\!0.75$. For the further data analysis a $\beta\!=\!0.75$ was used.
Structural relaxation described by mode coupling theory displays stretching and a time-temperature superposition. Using results from the fitting procedure with Eq. \[kww\] and a stretching exponent $\beta\!=\!0.75$ master curves $\Phi(q,(t/\tau_q))/f_q$ have been constructed for the data in the structural relaxation regime above some 1ps. Figure \[tts\] shows rescaled $\Phi(q,(t/\tau_q))/f_q$ at $q\!=\!1.05\,$Å$^{-1}$. Over the entire temperature range, that spans 540K, the $\Phi(q,(t/\tau_q))/f_q$ fall on a master curve: a time-temperature superposition of structural relaxation holds. We note, that the validity of the time-temperature superposition up to temperatures at which the transport coefficients approach that of simple liquids (\[diffsec\]) is in marked contrast to the idea, that viscous liquids exhibit a transition to a liquid with the stretching exponent $\beta$ approaching 1 on temperature increase.
Stretching of self correlation functions is generally found to be more pronounced in fragile glass-forming liquids, characterized by a curved temperature dependence of viscosity in an Arrhenius plot [@BoNA93]; e.g. the van der Waals liquid orthoterphenyl with a $\beta\!\simeq\!0.5$ [@Tol01]. In an intermediate system like hydrogen-bond forming glycerol a $\beta\!\simeq\!0.6$ has been reported [@WuCR96]. The stretching exponent in liquid PdNiCuP metals compares well to the value $\beta\!\simeq\!0.75$ found in covalent-network forming sodium disilicate melts [@MeSD02] and indicates that in this context the PdNiCuP alloy might classify as a fairly strong glass forming liquid.
Figure \[tqs\] shows rescaled $\Phi(q,(t/\tau_q))/f_q$ at 1073K for $q$ values in the range between 0.75Å$^{-1}$ and 1.95Å$^{-1}$. Structural relaxation in liquid PdNiCuP can be described with a stretched exponential function and a $q$ and temperature independent stretching exponent $\beta = 0.75 \pm 0.02$. $\beta_q$ shows a small but systematic decrease with increasing $q$ in agreement with MCT calculations for a hard-sphere system [@FuHL92]. However, a small variation of $\beta_q$ in the fitting procedure has no significant effect on the resulting mean relaxation times (Eq. \[mtau\]).
Diffusion {#diffsec}
---------
![\[tauq\] Mean relaxation times $\langle\tau_q\rangle$ of tagged particle motion in liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$. $1\,/\,\langle\tau_q\rangle$ shows a $q^2$ dependence as expected for long range atomic transport for $q \to 0$. The slope corresponds to the self diffusion coefficient $D(T)$.](pdfig5.eps)
In the hydrodynamic limit for $q \rightarrow 0$ one expects that the mean relaxation times $\langle\tau_q\rangle$ are indirect proportional to the square of the momentum transfer $q$ [@BoYi80]. In liquid PdNiCuP the $1\,/\,q^2$ dependence holds even up to 1.9Å$^{-1}$ (Fig. \[tauq\]) while the first structure factor maximum is at $q_0\!\simeq\!2.9$Å$^{-1}$. In liquid alkali-metals one observes a systematic deviation from a $q^2$ dependence of the quasielastic line width already at $q$ values that correspond to 1/10 of their structure factor maximum $q_0$. This deviation is explained by low-lying phonon modes that mediate atomic transport [@MoGG87; @BaZo94]. Mean relaxation times for self motion in the MCT hard sphere system, in contrast, vary rather well proportionally to $1\,/\,q^2$ for $q$ values extending even above the first structure factor maximum [@FuHL92]. We note that mixing of hard spheres with different radii should enforce this behaviour even more. It appears that the validity of $1\,/\,\langle\tau_q\rangle \propto q^2$ also for intermediate $q$ values is a signature of the atomic transport mechanism in dense liquids.
The $1\,/\,q^2$ dependence of the mean relaxation times, also demonstrates that structural relaxation leads to long range atomic transport with a diffusivity [@BoYi80] $$D= (\langle\tau_q\rangle q^2)^{-1}\, .$$ Figure \[diff\] displays the diffusivities $D$ in liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ as a function of $1\,/\,T$. Values range from $3\pm1\!\times\!10^{-11}$m$^2$s$^{-1}$ at 833K to $1.4\pm0.3\!\times\!10^{-9}$m$^2$s$^{-1}$ at 1373K. At the liquidus diffusion is about 2 orders of magnitude slower as compared to simple metallic liquids and most alloys. Diffusivities in liquid Pd40 show a similar temperature dependence but are larger by some 20%. The smaller mobility of the atoms in Pd43 appears to come along with the slightly better glass forming ability of the Pd43 alloy.
![\[diff\] Self diffusion coefficient $D$ of Ni and Cu in liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ (closed circles) and Pd$_{40}$Ni$_{10}$Cu$_{30}$P$_{20}$ (open circles from Ref. [@MeBS99]) derived from the mean relaxation times. The lines are fits with MCT $\tau$ scaling (Eq. \[tauscal\]). The data are well described with $T_c=700\pm30$ and $\gamma=2.7\pm0.2$.\
For comparison $^{57}$Co tracer diffusion in Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ (Ref. [@ZoRF02]) and diffusivities calculated from viscosity $\eta$ of liquid Pd$_{40}$Ni$_{10}$Cu$_{30}$P$_{20}$ (Ref. [@HaBD02]) via the Stokes-Einstein relation (Eq. \[stoke\]) are shown.](pdfig6-2.eps)
The lines in Figure \[diff\] represent fits with the MCT $\tau$ scaling law (Eq. \[tauscal\]) to the diffusivities $D(T)$. Although Eq. \[tauscal\] is only valid close to $T_c$, Eq. \[tauscal\] allows a rough estimate of the crossover temperature $T_c$ and the exponent $\gamma$. A fit to $D(T)$ of liquid Pd43 at temperatures up to 1073K yields a $T_c\!=\!700\!\pm\!30$K and a $\gamma\!=\!2.7\!\pm\!0.2$. The exponent $\gamma$ compares well to the $\gamma\!=\!2.62$ found in the MCT hard sphere system [@FuHL92]. The temperature dependence of the diffusivity in liquid PdNiCuP alloys is different from the $D \propto T^n$ behaviour (with $n\simeq2$) expected for uncorrelated binary collisions of hard spheres [@PrAP73; @KiYo87] and found with inelastic neutron scattering in expanded liquid alkali-metals at high temperature [@WiPH93].
Convection effects are a severe problem in macroscopic diffusion measurements in ordinary liquids under gravity conditions with respect to the absolute value of the self diffusivity $D$ and its temperature dependence. A $\mu$g experiment on liquid Sn revealed that well above the melting point convection even dominates the mass transport [@FrKW87]. The $\mu$g data in liquid Sn obey $D \propto T^2$ for temperatures between $T_m$ and $2 \times T_m$. Liquid Sn exhibits a packing fraction of $\simeq$0.4 as in liquid alkali-metals. Inelastic neutron scattering data are not affected by convection because it probes dynamics on significantly shorter times.
Diffusivities from recent $^{57}$Co tracer diffusion measurements in liquid Pd43 [@Zoe02] are in excellent agreement with the diffusivities obtained from inelastic neutron scattering (Fig. \[diff\]). This indicates that the mobility of Co is very similar to the Ni and Cu mobility observed in the incoherent neutron scattering signal and that convection effects play no significant role in the tracer diffusion experiment. The latter is in accordance with a viscosity $\eta$ that is two orders of magnitude larger as compared to simple metallic melts [@HaBD02].
In simple liquids shear viscosity $\eta$ and the self diffusivity $D$ generally obey the Stokes-Einstein relation [@BaZo94; @TyHa84; @KiYo87]: $$\label{stoke}
D = k_b T / (6\pi a \eta),$$ with reasonable values for the hydrodynamic radius $a$. Eq. \[stoke\] even holds in most molecular liquids [@ChSi97]. Figure \[diff\] shows the diffusivity $D_{\eta}$ calculated from the viscosity data of liquid Pd40 [@HaBD02] via Eq. \[stoke\] using $a=1.15$Å that represents the Ni hard sphere radius (Cu: 1.17Å) from Ref. [@Pas88]. The Stokes-Einstein relation holds with $D_{\eta} = D$. We note, that $a$ is similar to the mean next nearest neighbor distance displayed in the static structure factor $d = 2\pi / q_0 \simeq 1.1$Å. In multicomponent liquids, that exhibit a dense packing of hard spheres with comparable hard sphere radii one expects a similar mobility of the different components. In PdNiCuP the hard sphere radii of the different atoms have values within 20%. Because viscous flow represents the dynamics of all components, the validity of Eq. \[stoke\] in liquid PdNiCuP demonstrates that above $T_{liq}$ the mobility of the large and numerous Pd atoms is quite similar to the Ni and Cu atoms [@V1].
In colloidal suspensions the mean relaxation times $\langle\tau\rangle(\varphi)$ and the inverse of the diffusivity $1/D(\varphi)$ show the same dependence as a function of the packing fraction. The slope for $\langle\tau(\varphi)\rangle$ and $1/D(\varphi)$ results in a $\gamma=2.7$ [@MaMM98]. In molecular dynamics simulations on viscous NiP [@KoAn94], in contrast, the asymptotic MCT $\tau$ scaling prediction (Eq. \[tauscal\]) is violated: $\langle\tau(T)\rangle$ and $1/D(T)$ do not exhibit the same temperature dependence. Figures \[diff\] and \[transall\] demonstrate that in liquid PdNiCuP Eq. \[tauscal\] holds quite well.
Fast $\beta$ relaxation
-----------------------
![\[beta\] Normalized time correlation function $\Phi(q,t)$ of liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ in the $\beta$ relaxation regime around one picosecond. The solid lines represent fits with the MCT $\beta$ scaling function (Eq. \[betascal\]) giving a temperature and $q$ independent shape parameter $\lambda = 0.78\!\pm\!0.04$. Above $\simeq\,1$ps data are well described by a stretched exponential function (Eq. \[kww\], dashed lines).](pdfig7.eps)
Within the mode coupling theory of the liquid to glass transition, structural relaxation and therefore long range mass transport is preceded by a fast $\beta$ relaxation process whose time scale is typically in the order of a picosecond. For $q$ values well below the structure factor maximum and incoherent scattering the amplitude of the $\beta$ relaxation is increasing with increasing $q$. Figure \[beta\] displays the density correlation function of liquid PdNiCuP at $q\!=\!1.95\,\AA^{-1}$. Between 0 and $\simeq\!0.2$ps phonons lead to a rapid decay of atomic correlations (represented by the first data point in $\Phi(q,t)$ in Fig. \[beta\]), so that $\Phi(q,t)$ decreases from 1 towards a plateau. On approaching this plateau a fast relaxation is clearly seen in $\Phi(q,t)$ through an additional intensity below some 1ps. The lines are fits with the MCT $\beta$ scaling law (Eq. \[betascal\]). For $q<1.4$Å$^{-1}$ the limited dynamic range of the instrument prevents data analysis in the fast $\beta$ relaxation regime. Above 973K $\Phi(q,t)$ can not consistently be described with the asymptotic scaling laws (Eq. \[betascal\],\[tchq\]). The dynamic range in which Eq. \[betascal\] holds is increasing with decreasing temperature.
![\[tcbeta\] Mean amplitude $\langle h_q \rangle$ (triangles) and time scale $t_{\sigma}$ (circles) of fast $\beta$ relaxation in liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ rectified according the MCT predictions (Eq. \[tchq\]), with an exponent $a=0.29$ defined by $\lambda=0.78$. The inset shows the $q$ dependence of the amplitude $h_q$ between 833K ($\bullet$) and 953K ($\circ$). Both $h_q$ and $t_{\sigma}$ extrapolate to $T_c=720\!\pm\!20$K. $\lambda$ and $T_c$ obtained from the analysis of the $\beta$ relaxation regime are consistent with $T_c = 700\!\pm\!30$K and $\gamma=2.7\!\pm\!0.2$ from the analysis of the mean $\alpha$ relaxation times (Eq. \[tauscal\]). ](pdfig8-2.eps)
The data were fitted in a two-step procedure: starting with an arbitrary line shape parameter $\lambda$, fits to individual curves were used to estimate the scaling factors $f_q$, $h_q$, and $t_\sigma$ which physically represent the Debye–Waller factor, the amplitude of $\beta$-relaxation, and a characteristic time of $\beta$-relaxation. Using these values, the $\Phi(q,t)$ measured at different temperatures were superimposed on to master curves $(\Phi(q,t/t_\sigma)-f_q)/h_q$. After fixing a $q$-independent mean $t_\sigma$, the fit yielded a temperature and $q$ independent $\lambda\!=\!0.78\pm0.04$.
This result compares well to the hard spheres value $\lambda\!=\!0.766$ of the numerical MCT solution [@FuHL92], to the $\lambda\!=\!0.77\pm0.04$ found in liquid Zr$_{46.8}$Ti$_{8.2}$Cu$_{7.5}$Ni$_{10}$Be$_{27.5}$ [@MeWP98] and it is similar to the values in other viscous liquids [@Goe99; @Tol01]. $\lambda\!=\!0.78$ defines an exponent of the $\tau$ scaling law (Eq. \[tauscal\]) of $\gamma\!=\!2.7$ [@Goe90]. This is consistent with the temperature dependence of the mean relaxation times that give $\gamma\!=\!2.7\pm0.2$ (Fig. \[diff\]).
Figure \[tcbeta\] shows amplitude $h_q$ and time scale $t_{\sigma}$ of the fast $\beta$ relaxation in liquid PdNiCuP rectified according to Eq. \[tchq\]. The temperature dependence of $h_q$ and $t_{\sigma}$ is in accordance with the MCT predictions. Both $h_q$ and $t_{\sigma}$ extrapolate to $T_c=720\!\pm\!20$K that is close to the $T_c = 700\!\pm\!30$K obtained via the $\tau$ scaling law (Eq. \[tauscal\]). The relaxational dynamics in liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ can consistently be described with the universal scaling functions of the mode coupling theory of the liquid to glass transition at temperatures up to $\simeq 1.3 \times T_c$ with $T_c\!=\!710$K and $\lambda\!=\!0.78$.
\[matra\]Atomic transport mechanism
-----------------------------------
![\[transall\] Mass transport in PdNiCuP melts: At high temperatures diffusivities are similar to diffusivities in liquid Sn and Na close to $T_m$ (marked by the arrows). Time scales for viscous flow and Co tracer diffusion start to decouple and Co tracer diffusivities merge into an Arrhenius-type temperature dependence that extends down to the glass transition. Both occurs in the vicinity of mode coupling $T_c$. The line represents the MCT $\tau$ scaling law with $T_c\!=\!710$K and $\gamma\!=\!2.7$. ](pdfig9.eps)
Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ exhibits an excellent glass forming ability: a cooling rate of only 0.09K/s is sufficient to avoid crystallization and to form a glass [@ScJB00]. Consequently transport coefficients can continuously be measured from the equilibrium liquid down to the conventional glass transition temperature $T_g$. Figure \[transall\] displays diffusivities in viscous PdNiCuP alloys from Co tracer diffusion measurements, that cover more than 13 decades [@Zoe02; @ZoRF02], derived from the mean relaxation times and from viscosity [@HaBD02; @LuGF02] calculated via Stokes-Einstein (Eq. \[stoke\]).
At 1373K the diffusivity in liquid PdNiCuP is $1.4\,\times\,10^{-9}$m$^2$s$^{-1}$ which is similar to diffusivities known from monoatomic liquid metals at their respective melting temperature. E.g. close to the melting point inelastic neutron scattering reveals $D=4.2\,\times\,10^{-9}$m$^2$s$^{-1}$ in liquid sodium [@MoGl86] and tracer diffusion under $\mu$g conditions $D=2.0\,\times\,10^{-9}$m$^2$s$^{-1}$ in liquid tin [@FrKW87]. At high temperatures the Stokes-Einstein relation holds in liquid PdNiCuP (Fig. \[diff\]). On lowering the temperature diffusion slows down drastically. Time scales for viscous flow and Co tracer diffusion start to decouple in the vicinity of the mode coupling $T_c\simeq710\,$K and differ already by more than 3 orders of magnitude at 600K. Also around $T_c$ Co tracer diffusivities merge into an Arrhenius-type temperature dependence ($D(T) = D_0 \exp{(-H/k_BT)}$, where $D_0$ is a prefactor and $H$ the activation enthalpy) that extends down to the glass transition [@Zoe02; @ZoRF02].
The temperature dependence of the transport coefficients in Figure \[transall\] strongly supports the MCT prediction of a change in the atomic transport mechanism from viscous flow to glass like hopping at $T_c$. There are extensive tracer diffusion measurements in supercooled Zr$_{46.8}$Ti$_{8.2}$Cu$_{7.5}$Ni$_{10}$Be$_{27.5}$ at temperatures between the glass transition and about 200K below $T_c$ [@diffrev]. The diffusivities of various tracers exhibit a size dependence: the smaller the atoms the faster they diffuse and the smaller the activation enthalpy $H$. In viscous Zr$_{46.8}$Ti$_{8.2}$Cu$_{7.5}$Ni$_{10}$Be$_{27.5}$ the diffusivities of the various atoms approach each other with increasing temperature. Because in this alloy crystallization prevents access to the temperature range around $T_c$, one can not conclude from the present data whether the diffusivities of the various tracers merge at the MCT $T_c$. The isotope effect $E=(D_a/D_b-1)/(\sqrt{m_b/m_a}-1)$ (diffusivity $D$ and mass $m$ of two isotopes $a,b$) is a measure of the degree of collectivity of the atomic transport. For diffusion via single jumps in dense packed lattices $E$ is generally in the order of unity [@Meh90]. For uncorrelated binary collisions one also expects $E \to 1$. A vanishing isotope effect indicates a collective transport mechanism involving a large number of atoms. In liquid Sn an isotope effect of about 0.4 has been reported, that is increasing with increasing temperature [@FrKW87]. MD simulations on a binary Lennard-Jones liquid demonstrate that changes in the density in the order of 20% only result in a continuous increase in $E$ from 0 to $\simeq$0.3 [@Sch01]. In metallic glasses [@FaHR90] as well as in supercooled Zr$_{46.8}$Ti$_{8.2}$Cu$_{7.5}$Ni$_{10}$Be$_{27.5}$ [@EhHR98] and Pd$_{40}$Ni$_{10}$Cu$_{30}$P$_{20}$ [@ZoRF02] the isotope effect is close to zero as a result of a highly collective, thermally activated hopping process. Measurements of the isotope effect in Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ [@Zoe02] give $E\simeq0.05$ over the entire temperature range from the glass transition to the equilibrium liquid.
The atomic transport mechanism in dense PdNiCuP remains highly collective even in the equilibrium liquid in contrast to the findings in simple metallic liquids. Viscous PdNiCuP exhibits an expansion coefficient of $4\times10^{-5}$K$^{-1}$ [@LuGF02]. Between 1373K and 833K, the density and therefore, assuming a temperature independent hard sphere radius, the packing fraction changes only by about 2%. An increase in the radius of the attributed hard sphere radii [@Pas88] with increasing temperature results in an even smaller change in the packing fraction. This small increase in the packing fraction causes the drastical slowing down of dynamics on temperature decrease in accordance with the mode coupling scenario. Dense packing and the resulting collective transport mechanism extends also to high temperatures, where diffusivities are similar to that in simple metallic liquids. This indicates that the atomic transport mechanism in liquids is not controled by the value of the transport coefficients but rather by that of the packing fraction.
Conclusions
===========
Atomic transport in liquid Pd$_{43}$Ni$_{10}$Cu$_{27}$P$_{20}$ has been investigated with inelastic, incoherent neutron scattering. The PdNiCuP melt is characterized by a packing fraction of about 0.52 that is some 20% larger as compared to monoatomic alkali-melts. The self correlation function shows a two-step decay as known from other non-metallic glass-forming liquids: a fast relaxation process precedes structural relaxation. The structural relaxation exhibits stretching with a stretching exponent $\beta\simeq0.75$ and a time-temperature superpostion that holds for temperatures as high as $1.75 \times T_c$. The relaxational dynamics can consistently be described within the framework of the mode coupling theory of the liquid to glass transition with a temperature and $q$ independent line shape parameter $\lambda\simeq0.78$. Universal MCT scaling laws extrapolate to a critical temperature $T_c\simeq710$K some 20% below the liquidus.
Diffusivities derived from the mean relaxation times compare well with Co diffusivities from tracer diffusion measurements. Above $T_c$ diffusivities calculated from viscosity [@HaBD02] via the Stokes-Einstein relation are in excellent agreement with the diffusivities measred by tracer diffusion [@Zoe02] and neutron scattering. In contrast to simple metallic liquids the atomic transport in dense liquid PdNiCuP is characterized by a drastical slowing down of dynamics on approaching $T_c$, a $q^{-2}$ dependence of the mean relaxation times at intermediate $q$ and a vanishing isotope effect as a result of a highly collective transport mechanism in the dense packed liquid. At temperatures as high as $2\!\times\!T_c$ diffusion in liquid PdNiCuP is as fast as in simple monoatomic liquids at their melting points. However, the difference in the underlying atomic transport mechanism indicates that the diffusion mechanism in liquids is not controlled by the value of the diffusivity but rather by that of the packing fraction.
It is a pleasure to thank Winfried Petry and Helmut Schober for their support, Wolfgang Götze, Walter Schirmacher and Walter Kob for a critical reading of the manuscript, Joachim Wuttke for his ingenious data analysis program, the Institut Laue–Langevin for the beamtime on IN6, Stephan Roth for his help during the measurement and Helga Harlandt for her help with the sample preparation. This work is funded by the German DFG (SPP Phasenumwandlungen in mehrkomponentigen Schmelzen) under project Me1958/2-1.
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|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/order-theoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints, via the existential-pullback adjunction between fibres.'
author:
- Chris Heunen
- Bart Jacobs
bibliography:
- 'dagkercat-arxiv.bib'
title: Quantum Logic in Dagger Kernel Categories
---
Introduction {#IntroSec}
============
Dagger categories [$\mathbf{D}$]{} come equipped with a special functor $\dag\colon {\ensuremath{\mathbf{D}}}{\ensuremath{^{\mathrm{op}}}}\rightarrow {\ensuremath{\mathbf{D}}}$ with $X^\dag=X$ on objects and $f^{\dag\dag}=f$ on morphisms. A simple example is the category [[$\mathbf{Rel}$]{}]{}of sets and relations, where $\dag$ is reversal of relations. A less trivial example is the category [[$\mathbf{Hilb}$]{}]{}of Hilbert spaces and continuous linear transformations, where $\dag$ is induced by the inner product. The use of daggers, mostly with additional assumptions, dates back to [@MacLane61; @Puppe62]. Daggers are currently of interest in the context of quantum computation [@AbramskyC04; @Selinger07; @CoeckeP06]. The dagger abstractly captures the reversal of a computation.
Mostly, dagger categories are used with fairly strong additional assumptions, like compact closure in [@AbramskyC04]. Here we wish to follow a different approach and start from minimal assumptions. This paper is a first step to understand quantum logic, from the perspective of categorical logic (see *e.g.* [@MakkaiR77; @KockR77; @Taylor99; @Jacobs99a]). It grew from the work of one of the authors [@Heunen08b]. Although that paper enjoys a satisfactory relation to traditional quantum logic [@Heunen09b], this one generalises it, by taking the notion of dagger category as starting point, and adding kernels, to be used as predicates. The interesting thing is that in the presence of a dagger $\dag$ much else can be derived. As usual, it is quite subtle what precisely to take as primitive. A referee identified the reference [@Crown75] as an earlier precursor to this work. It contains some crucial ingredients, like orthomodular posets of dagger kernels, but without the general perspective given by categorical logic.
Upon this structure of “dagger kernel categories” the paper constructs pullbacks of kernels and factorisation (both similar to [@Freyd64]). It thus turns out that the kernels form a “bifibration” (both a fibration and an opfibration, see [@Jacobs99a]). This structure can be used as a basis for categorical logic, which captures substitution in predicates by reindexing (pullback) $f^{-1}$ and existential quantification by op-reindexing $\exists_f$, in such a way that $\exists_{f} \dashv
f^{-1}$. From time to time we use fibred terminology in this paper, but familiarity with this fibred setting is not essential. We find that the posets of kernels (fibres) are automatically orthomodular lattices [@Kalmbach83], and that the Sasaki hook and and-then connectives appear naturally from the existential-pullback adjunction. Additionally, a notion of Booleanness is identified for these dagger kernel categories. It gives rise to a generic construction that generalises how the category of partial injections can be obtained from the category of relations.
Apart from this general theory, the paper brings several important examples within the same setting—of dagger kernel categories. Examples are the categories [[$\mathbf{Rel}$]{}]{}and [[$\mathbf{PInj}$]{}]{}of relations and partial injections. Additionally, the category [[$\mathbf{Hilb}$]{}]{}is an example—and, interestingly—also the category [[$\mathbf{PHilb}$]{}]{}of Hilbert spaces modulo phase. The latter category provides the framework in which physicists typically work [@Lahti04]. It has much weaker categorical structure than [[$\mathbf{Hilb}$]{}]{}. We also present a construction to turn an arbitrary Boolean algebra into a dagger kernel category.
The authors are acutely aware of the fact that several of the example categories have much richer structure, involving for instance a tensor sum $\oplus$ and a tensor product $\otimes$ with associated scalars and traced monoidal structure. This paper deliberately concentrates solely on (the logic of) kernels. There are interesting differences between our main examples: for instance, [[$\mathbf{Rel}$]{}]{}and [[$\mathbf{PInj}$]{}]{}are Boolean, but [[$\mathbf{Hilb}$]{}]{}is not; in [[$\mathbf{PInj}$]{}]{}and [[$\mathbf{Hilb}$]{}]{}“zero-epis” are epis, but not in [[$\mathbf{Rel}$]{}]{}; [[$\mathbf{Rel}$]{}]{}and [[$\mathbf{Hilb}$]{}]{}have biproducts, but [[$\mathbf{PInj}$]{}]{}does not.
The paper is organised as follows. After introducing the notion of dagger kernel category in Section \[KernelSec\], the main examples are described in Section \[ExamplesSec\]. Factorisation and (co)images occur in Sections \[FactorisationSec\] and \[ImageCoimageSec\]. Section \[LogicSec\] introduces the Sasaki hook and and-then connectives via adjunctions, and investigates Booleanness. Finally, Sections \[HomsetOrderSec\] and \[ComplAtomSec\] investigate some order-theoretic aspects of homsets and of kernel posets (atomicity and completeness).
A follow-up paper [@Jacobs09a] introduces a new category [$\mathbf{OMLatGal}$]{} of orthomodular lattices with Galois connections between them, shows that it is a dagger kernel category, and that every dagger kernel category [$\mathbf{D}$]{} maps into it via the kernel functor ${\ensuremath{\mathrm{KSub}}}\colon{\ensuremath{\mathbf{D}}} \rightarrow {\ensuremath{\mathbf{OMLatGal}}}$, preserving the dagger kernel structure. This gives a wider context.
Daggers and kernels {#KernelSec}
===================
Let us start by introducing the object of study of this paper.
\[DagCatKerDef\] A *dagger kernel category* consists of:
1. a dagger category [$\mathbf{D}$]{}, with dagger $\dag\colon {\ensuremath{\mathbf{D}}}{\ensuremath{^{\mathrm{op}}}}\rightarrow {\ensuremath{\mathbf{D}}}$;
2. a zero object 0 in [$\mathbf{D}$]{};
3. kernels $\ker(f)$ of arbitrary maps $f$ in [$\mathbf{D}$]{}, which are dagger monos.
A *morphism of dagger kernel categories* is a functor $F$ preserving the relevant structure:
1. $F(f^\dag) = F(f)^\dag$;
2. $F(0)$ is again a zero object;
3. $F(k)$ is a kernel of $F(f)$ if $k$ is a kernel of $f$.
Dagger kernel categories and their morphisms form a category ${{\ensuremath{\mathbf{DagKerCat}}}\xspace}$.
A dagger kernel category is called Boolean if $m{\mathrel{\wedge}}n
= 0$ implies $m^{\dag} {\mathrel{\circ}}n = 0$, for all kernels $m,n$.
The name Boolean will be explained in Theorem \[BooleanLem\]. We shall later rephrase the Booleanness condition as: kernels are disjoint if and only if they are orthogonal, see Lemma \[KerLem\].\
The dagger $\dag$ satisfies $X^{\dag} = X$ on objects and $f^{\dag\dag} = f$ on morphisms. It comes with a number of definitions. A map $f$ in [$\mathbf{D}$]{} is called a dagger mono(morphism) if $f^{\dag} {\mathrel{\circ}}f = {\ensuremath{\mathrm{id}_{}}}$ and a dagger epi(morphism) if $f{\mathrel{\circ}}f^{\dag} = {\ensuremath{\mathrm{id}_{}}}$. Hence $f$ is a dagger mono if and only if $f^{\dag}$ is a dagger epi. A map $f$ is a dagger iso(morphism) when it is both dagger monic and dagger epic; in that case $f^{-1} =
f^{\dag}$ and $f$ is sometimes called unitary (in analogy with Hilbert spaces). An endomorphism $p\colon X\rightarrow X$ is called self-adjoint if $p^{\dag} = p$.
The zero object $0\in{\ensuremath{\mathbf{D}}}$ is by definition both initial and final. Actually, in the presence of $\dag$, initiality implies finality, and vice-versa. For an arbitrary object $X\in{\ensuremath{\mathbf{D}}}$, the unique map $X\rightarrow 0$ is then a dagger epi and the unique map $0\rightarrow X$ is a dagger mono. The “zero” map $0 = 0_{X,Y} =
(X\rightarrow 0 \rightarrow Y)$ satisfies $(0_{X,Y})^{\dag} =
0_{Y,X}$. Notice that $f {\mathrel{\circ}}0 = 0 = 0 {\mathrel{\circ}}g$. Usually there is no confusion between 0 as zero object and 0 as zero map. Two maps $f\colon X\rightarrow Z$ and $g\colon Y\rightarrow Z$ with common codomain are called orthogonal, written as $f{\mathrel{\bot}}g$, if $g^{\dag} {\mathrel{\circ}}f = 0$—or, equivalently, $f^{\dag} {\mathrel{\circ}}g = 0$.
Let us recall that a kernel of a morphism $f\colon X\rightarrow Y$ is a universal morphism $k\colon\ker(f)\rightarrow X$ with $f{\mathrel{\circ}}k =
0$. Universality means that for an arbitrary $g\colon Z\rightarrow X$ with $f{\mathrel{\circ}}g = 0$ there is a unique map $g'\colon Z\rightarrow
\ker(f)$ with $k {\mathrel{\circ}}g' = g$. Kernels are automatically (ordinary) monos. Just like we write $0$ both for a zero object and for a zero map, we often write $\ker(f)$ to denote either a kernel map, or the domain object of a kernel map.
Definition \[DagCatKerDef\] requires that kernels are dagger monos. This requirement involves a subtlety: kernels are closed under arbitrary isomorphisms but dagger monos are only closed under dagger isomorphisms. Hence we should be more careful in this requirement. What we really mean in Definition \[DagCatKerDef\] is that for each map $f$, among all its isomorphic kernel maps, there is at least one dagger mono. We typically choose this dagger mono as representant $\ker(f)$ of the equivalence class of kernel maps.
We shall write ${\ensuremath{\mathrm{KSub}}}(X)$ for the poset of (equivalence classes) of kernels with codomain $X$. The order $(M \rightarrowtail X) \leq
(N\rightarrowtail X)$ in ${\ensuremath{\mathrm{KSub}}}(X)$ is given by the presence of a (necessarily unique) map $M\rightarrow N$ making the obvious triangle commute. Intersections in posets like ${\ensuremath{\mathrm{KSub}}}(X)$, if they exist, are given by pullbacks, as in: $$\xymatrix{
\bullet\ar@{ >->}[r]\ar@{ >->}[d]\ar@{ >->}[dr]|{m{\mathrel{\wedge}}n} &
M\ar@{ >->}[d]^{m} \\
N\ar@{ >->}[r]_-{n} & X\rlap{.}
}$$
In presence of the dagger $\dag$, cokernels come for free: one can define a cokernel ${\ensuremath{\mathrm{coker}}}(f)$ as $\ker(f^{\dag})^{\dag}$. Notice that we now write $\ker(f)$ and ${\ensuremath{\mathrm{coker}}}(f)$ as morphisms. This cokernel ${\ensuremath{\mathrm{coker}}}(f)$ is a dagger epi. Finally, we define $m^{\perp} =
\ker(m^{\dag})$, which we often write as $m^\perp : M^\perp
\rightarrowtail X$ if $m\colon M \rightarrowtail X$. This notation is especially used when $m$ is a mono. In diagrams we typically write a kernel as $\smash{\[email protected]{\ar@{|>->}[r] & }}$ and a cokernel as $\[email protected]{\ar@{-|>}[r] & }$.
The following Lemma gives some basic observations.
\[KerLem\] In a dagger kernel category,
1. $\ker(\smash{X\stackrel{0}{\rightarrow} Y}) =
(\smash{X\stackrel{{\ensuremath{\mathrm{id}_{}}}}{\rightarrow}X})$ and $\ker(\smash{X\stackrel{{\ensuremath{\mathrm{id}_{}}}}{\rightarrow}X}) =
(\smash{0\stackrel{0}{\rightarrow}X})$; these yield greatest and least elements $1,0\in{\ensuremath{\mathrm{KSub}}}(X)$, respectively;
2. $\ker(\ker(f)) = 0$;
3. $\ker({\ensuremath{\mathrm{coker}}}(\ker(f))) = \ker(f)$, as subobjects;
4. $m^{\perp\perp} = m$ if $m$ is a kernel;
5. A map $f$ factors through $g^{\perp}$ iff $f{\mathrel{\bot}}g$ iff $g {\mathrel{\bot}}f$ iff $g$ factors through $f^{\perp}$; in particular $m \leq n^{\perp}$ iff $n\leq m^{\perp}$, for monos $m,n$; hence $(-)^{\perp} \colon {\ensuremath{\mathrm{KSub}}}(X)$ $\smash{\stackrel{\cong}{\longrightarrow}}$ ${\ensuremath{\mathrm{KSub}}}(X){\ensuremath{^{\mathrm{op}}}}$;
6. if $m\leq n$, for monos $m,n$, say via $m = n{\mathrel{\circ}}\varphi$, then:
1. if $m,n$ are dagger monic, then so is $\varphi$;
2. if $m$ is a kernel, then so is $\varphi$.
7. Booleanness amounts to $m{\mathrel{\wedge}}n = 0 \Leftrightarrow
m{\mathrel{\bot}}n$, *i.e.* disjointness is orthogonality, for kernels.
We skip the first two points because they are obvious and start with the third one. Consider the following diagram for an arbitrary $f\colon
X\rightarrow Y$: $$\xymatrix{
\ker(f)\ar@{ |>->}[rr]^-{k}\ar@{..>}[d]<-1ex>_-{k'} & &
X\ar[rr]^-{f}\ar@{-|>}[drr]_-{c} & & Y \\
\ker({\ensuremath{\mathrm{coker}}}(\ker(f)))\ar@{ |>->}[urr]_-{\ell}\ar@{..>}[u]<-1ex>_-{\ell'}
& & & & {\ensuremath{\mathrm{coker}}}(\ker(f)). \ar@{..>}[u]_{f'}
}$$
By construction $f{\mathrel{\circ}}k = 0$ and $c{\mathrel{\circ}}k=0$. Hence there are $f'$ and $k'$ as indicated. Since $f {\mathrel{\circ}}\ell = f' {\mathrel{\circ}}c {\mathrel{\circ}}\ell = f' {\mathrel{\circ}}0 = 0$ one gets $\ell'$. Hence the kernels $\ell$ and $k$ represent the same subobject.
For the fourth point, notice that if $m=\ker(f)$, then $$m^{\perp\perp}
=
\ker(\ker(m^{\dag})^{\dag})
=
\ker({\ensuremath{\mathrm{coker}}}(\ker(f)))
=
\ker(f)
=
m.$$
Next, $$\begin{array}{rcl}
\mbox{$f$ factors through $g^{\perp}$}
& \Longleftrightarrow &
g^{\dag} {\mathrel{\circ}}f = 0 \\
& \Longleftrightarrow &
f^{\dag} {\mathrel{\circ}}g = 0
\hspace*{\arraycolsep}\Longleftrightarrow\hspace*{\arraycolsep}
\mbox{$g$ factors through $f^{\perp}$.}
\end{array}$$
If, in the sixth point, $m = n {\mathrel{\circ}}\varphi$ and $m,n$ are dagger monos, then $\varphi^{\dag} {\mathrel{\circ}}\varphi = (n^{\dag} {\mathrel{\circ}}m)^{\dag} {\mathrel{\circ}}\varphi = m^{\dag} {\mathrel{\circ}}n {\mathrel{\circ}}\varphi = m^{\dag}
{\mathrel{\circ}}m = {\ensuremath{\mathrm{id}_{}}}$. And if $m = \ker(f)$, then $\varphi = \ker(f{\mathrel{\circ}}n)$, since: (1) $f {\mathrel{\circ}}n {\mathrel{\circ}}\varphi = f {\mathrel{\circ}}m = 0$, and (2) if $f{\mathrel{\circ}}n {\mathrel{\circ}}g = 0$, then there is a $\psi$ with $m {\mathrel{\circ}}\psi = n {\mathrel{\circ}}g$, and this gives a unique $\psi$ with $\varphi
{\mathrel{\circ}}\psi = g$, where uniqueness of this $\psi$ comes from $\varphi$ being monic.
Finally, Booleanness means that $m{\mathrel{\wedge}}n = 0$ implies $m^{\dag}
{\mathrel{\circ}}n = 0$, which is equivalent to $n^{\dag} {\mathrel{\circ}}m = 0$, which is $m{\mathrel{\bot}}n$ by definition. The reverse implication is easy, using that the meet ${\mathrel{\wedge}}$ of monos is given by pullback: if $m{\mathrel{\circ}}f = n {\mathrel{\circ}}g$, then $f = m^{\dag} {\mathrel{\circ}}m {\mathrel{\circ}}f =
m^{\dag} {\mathrel{\circ}}n {\mathrel{\circ}}g = 0 {\mathrel{\circ}}g = 0$. Similarly, $g=0$. Hence the zero object $0$ is the pullback $m{\mathrel{\wedge}}n$ of $m,n$.
Certain constructions from the theory of Abelian categories [@Freyd64] also work in the current setting. This applies to the pullback construction in the next result, but also, to a certain extent, to the factorisation of Section \[FactorisationSec\].
\[PullbackLem\] Pullbacks of kernels exist, and are kernels again. Explicitly, given a kernel $n$ and map $f$ one obtains a pullback: $$\begin{array}{rcrcl}
\raisebox{1.5em}{$\xymatrix{
M\ar[rr]^-{f'}\ar@{ |>->}[d]_{f^{-1}(n)}{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& & N\ar@{ |>->}[d]^{n} \\
X\ar[rr]_-{f} & & Y
}$}
& \quad\mbox{as}\quad &
f^{-1}(n)
& = &
\ker({\ensuremath{\mathrm{coker}}}(n) {\mathrel{\circ}}f).
\end{array}$$
If $f$ is a dagger epi, so is $f'$.
By duality there are of course similar results about pushouts of cokernels.
For convenience write $m$ for the dagger kernel $f^{-1}(n) =
\ker({\ensuremath{\mathrm{coker}}}(n) {\mathrel{\circ}}f)$. By construction, ${\ensuremath{\mathrm{coker}}}(n) {\mathrel{\circ}}f {\mathrel{\circ}}m = 0$, so that $f{\mathrel{\circ}}m$ factors through $\ker({\ensuremath{\mathrm{coker}}}(n)) = n$, say via $f'\colon M\rightarrow N$ with $n{\mathrel{\circ}}f' = f {\mathrel{\circ}}m$, as in the diagram. This yields a pullback: if $a\colon Z\rightarrow X$ and $b\colon Z\rightarrow N$ satisfy $f{\mathrel{\circ}}a = n{\mathrel{\circ}}b$, then ${\ensuremath{\mathrm{coker}}}(n) {\mathrel{\circ}}f {\mathrel{\circ}}a = {\ensuremath{\mathrm{coker}}}(n) {\mathrel{\circ}}n {\mathrel{\circ}}b = 0 {\mathrel{\circ}}b = 0$, so that there is a unique map $c\colon Z\rightarrow M$ with $m{\mathrel{\circ}}c = a$. Then $f'{\mathrel{\circ}}c = b$ because $n$ is monic.
In case $f$ is dagger epic, $f {\mathrel{\circ}}f^\dag {\mathrel{\circ}}n = n$. Hence there is a morphism $f''$ making following diagram commute, as the right square is a pullback: $$\xymatrix@R-3ex{
N \ar@{-->}_-{f''}[dr] \ar@(d,l)_-{f^\dag {\mathrel{\circ}}n}[dddr]
\ar@(r,ul)^-{{\ensuremath{\mathrm{id}_{}}}}[drrr] \\
& M {\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}\ar^-{f'}[rr] \ar@{ |>->}_-{m}[dd]
&& N \ar@{ |>->}^-{n}[dd] \\ \\
& X \ar_-{f}[rr] && Y.
}$$ Then $f'' = m^{\dag} {\mathrel{\circ}}m {\mathrel{\circ}}f'' = m^{\dag} {\mathrel{\circ}}f^{\dag} {\mathrel{\circ}}n = (f {\mathrel{\circ}}m)^{\dag} {\mathrel{\circ}}n = (n {\mathrel{\circ}}f')^{\dag} {\mathrel{\circ}}n =
f'^{\dag} {\mathrel{\circ}}n^{\dag} {\mathrel{\circ}}n = f'^{\dag}$. Hence $f'$ is dagger epic, too.
\[PullbackCor\] Given these pullbacks of kernels we observe the following.
1. The mapping $X\mapsto {\ensuremath{\mathrm{KSub}}}(X)$ yields an indexed category ${\ensuremath{\mathbf{D}}}{\ensuremath{^{\mathrm{op}}}}\rightarrow{\ensuremath{\mathbf{PoSets}}}$, using that each map $f\colon
X\rightarrow Y$ in [$\mathbf{D}$]{} yields a pullback (or substitution) functor $f^{-1}\colon {\ensuremath{\mathrm{KSub}}}(Y) \rightarrow {\ensuremath{\mathrm{KSub}}}(X)$. By the “pullback lemma”, see *e.g.* [@Awodey06 Lemma 5.10] or [@MacLane71 III, 4, Exc. 8], such functors $f^{-1}$ preserve the order on kernels, and also perserve all meets (given by pullbacks). This (posetal) indexed category ${\ensuremath{\mathrm{KSub}}}\colon
{\ensuremath{\mathbf{D}}}{\ensuremath{^{\mathrm{op}}}}\rightarrow{\ensuremath{\mathbf{PoSets}}}$ forms a setting in which one can develop categorical logic for dagger categories, see Subsection \[FibrationSubsec\].
2. The following diagram is a pullback, $$\[email protected]{
\ker(f)\ar[rr]\ar@{ |>->}[d]{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& & 0\ar@{ |>->}[d] \\
X\ar[rr]_-{f} & & Y
}$$
showing that, logically speaking, falsum—*i.e.* the bottom element $0\in{\ensuremath{\mathrm{KSub}}}(Y)$—is in general not preserved under substitution. Also, negation/orthocomplementation $(-)^{\perp}$ does not commute with substitution, because $1 = 0^{\perp}$ and $f^{-1}(1)
= 1$.
Being able to take pullbacks of kernels has some important consequences.
\[KernelCompLem\] Kernels are closed under composition—and hence cokernels are, too.
We shall prove the result for cokernels, because it uses pullback results as we have just seen. So assume we have (composable) cokernels $e, d$; we wish to show $e {\mathrel{\circ}}d = {\ensuremath{\mathrm{coker}}}(\ker(e{\mathrel{\circ}}d))$. We first notice, using Lemma \[PullbackLem\], $$\ker(e {\mathrel{\circ}}d)
=
\ker({\ensuremath{\mathrm{coker}}}(\ker(e)) {\mathrel{\circ}}d)
=
d^{-1}(\ker(e)),$$
yielding a pullback: $$\xymatrix{
& & A\ar[rr]^-{d'}\ar@{ |>->}[d]|{m = \ker(e{\mathrel{\circ}}d)}{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& &
B\ar@{ |>->}[d]^{\ker(e)} \\
K\ar@{ |>->}[rr]^-{\ker(d)}\ar@{-->}[urr]^{\varphi} & &
X\ar@{-|>}[rr]^-{d} & &
D\ar@{-|>}[rr]^-{e} & & E.
}$$
We intend to prove $e{\mathrel{\circ}}d = {\ensuremath{\mathrm{coker}}}(m)$. Clearly, $e
{\mathrel{\circ}}d {\mathrel{\circ}}m = e {\mathrel{\circ}}\ker(e) {\mathrel{\circ}}d' = 0 {\mathrel{\circ}}d' = 0$. And if $f\colon X\rightarrow Y$ satisfies $f{\mathrel{\circ}}m = 0$, then $f
{\mathrel{\circ}}\ker(d) = f {\mathrel{\circ}}m {\mathrel{\circ}}\varphi = 0$, so because $d =
{\ensuremath{\mathrm{coker}}}(\ker(d))$ there is $f'\colon D\rightarrow Y$ with $f' {\mathrel{\circ}}d
= f$. But then: $f' {\mathrel{\circ}}\ker(e) {\mathrel{\circ}}d' = f' {\mathrel{\circ}}d {\mathrel{\circ}}m =
f{\mathrel{\circ}}m = 0$. Then $f' {\mathrel{\circ}}\ker(e) = 0$, because $d'$ is dagger epi because $d$ is, see Lemma \[PullbackLem\]. This finally yields $f'' \colon E\rightarrow Y$ with $f'' {\mathrel{\circ}}e = f'$. Hence $f'' {\mathrel{\circ}}e {\mathrel{\circ}}d = f$.
As a result, the logic of kernels has intersections, preserved by substitution. More precisely, the indexed category ${\ensuremath{\mathrm{KSub}}}(-)$ from Corollary \[PullbackCor\] is actually a functor ${\ensuremath{\mathrm{KSub}}}\colon
{\ensuremath{\mathbf{D}}}{\ensuremath{^{\mathrm{op}}}}\rightarrow {\ensuremath{\mathbf{MSL}}}$ to the category [$\mathbf{MSL}$]{} of meet semi-lattices. Each poset ${\ensuremath{\mathrm{KSub}}}(X)$ also has disjunctions, by $m{\mathrel{\vee}}n = (m^{\perp} {\mathrel{\wedge}}n^{\perp})^{\perp}$, but they are not preserved under substitution/pullback $f^{-1}$. Nevertheless, $m
{\mathrel{\vee}}m^{\perp} = (m^{\perp} {\mathrel{\wedge}}m^{\perp\perp})^{\perp} =
(m^{\perp} {\mathrel{\wedge}}m)^{\perp} = 0^{\perp} = 1$.
The essence of the following result goes back to [@Crown75].
\[OrthmodularityProp\] Orthomodularity holds: for kernels $m\leq n$, say via $\varphi$ with $n{\mathrel{\circ}}\varphi = m$, one has pullbacks: $$\xymatrix{
M\ar@{ |>->}[rr]^-{\varphi}\ar@{=}[d]{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& &
N\ar@{ |>->}[d]^{n} & &
P\ar[d]\ar@{ |>->}[ll]_-{\varphi^{\perp}}{\save*!/ld-1.2pc/ld:(-1,1)@^{|-}\restore} \\
M\ar@{ |>->}[rr]_-{m} & & X & & M^{\perp}\ar@{ |>->}[ll]^-{m^{\perp}}
}$$
This means that $m {\mathrel{\vee}}(m^{\perp} {\mathrel{\wedge}}n) = n$.
The square on the left is obviously a pullback. For the one on the right we use a simple calculation, following Lemma \[PullbackLem\]: $$\begin{array}{rcll}
n^{-1}(m^{\perp})
& = &
\ker({\ensuremath{\mathrm{coker}}}(m^{\perp}) {\mathrel{\circ}}n) \\
& = &
\ker({\ensuremath{\mathrm{coker}}}(\ker(m^{\dag})) {\mathrel{\circ}}n) \\
& = &
\ker(m^{\dag} {\mathrel{\circ}}n)
& \mbox{since $m^{\dag}$ is a cokernel} \\
& \smash{\stackrel{(*)}{=}} &
\ker(\varphi^{\dag}) \\
& = &
\varphi^{\perp},
\end{array}$$
where the marked equation holds because $n{\mathrel{\circ}}\varphi = m$, so that $\varphi = n^{\dag} {\mathrel{\circ}}n {\mathrel{\circ}}\varphi = n^{\dag} {\mathrel{\circ}}m$ and thus $\varphi^{\dag} = m^{\dag} {\mathrel{\circ}}n$. Then: $$m {\mathrel{\vee}}(m^{\perp} {\mathrel{\wedge}}n)
\hspace*{\arraycolsep} = \hspace*{\arraycolsep}
(n {\mathrel{\circ}}\varphi) {\mathrel{\vee}}(n{\mathrel{\circ}}\varphi^{\perp}) \\
\hspace*{\arraycolsep}\smash{\stackrel{(*)}{=}}\hspace*{\arraycolsep}
n {\mathrel{\circ}}(\varphi {\mathrel{\vee}}\varphi^{\perp}) \\
\hspace*{\arraycolsep} = \hspace*{\arraycolsep}
n {\mathrel{\circ}}{\ensuremath{\mathrm{id}_{}}}\\
\hspace*{\arraycolsep} = \hspace*{\arraycolsep}
n.$$
The (newly) marked equation holds because $n{\mathrel{\circ}}(-)$ preserves joins, since it is a left adjoint: $n{\mathrel{\circ}}k \leq m$ iff $k
\leq n^{-1}(m)$, for kernels $k,m$.
The following notion does not seem to have an established terminology, and therefore we introduce our own.
\[ZeroMonoEpiDef\] In a category with a zero object, a map $m$ is called a zero-mono if $m{\mathrel{\circ}}f = 0$ implies $f=0$, for any map $f$. Dually, $e$ is zero-epi if $f {\mathrel{\circ}}e = 0$ implies $f=0$. In diagrams we write $\smash{\[email protected]{\ar@{ >->}|{\circ}[r] & }}$ for zero-monos and $\smash{\[email protected]{\ar@{->>}|{\circ}[r] & }}$ for zero-epis.
Clearly, a mono is zero-mono, since $m{\mathrel{\circ}}f = 0 = m{\mathrel{\circ}}0$ implies $f=0$ if $m$ is monic. The following points are worth making explicit.
\[ZeroMonoEpiLem\] In a dagger kernel category,
1. $m$ is a zero-mono iff $\ker(m) = 0$ and $e$ is a zero-epi iff ${\ensuremath{\mathrm{coker}}}(e) = 0$;
2. $\ker(m{\mathrel{\circ}}f) = \ker(f)$ if $m$ is a zero-mono, and similarly, ${\ensuremath{\mathrm{coker}}}(f {\mathrel{\circ}}e) = {\ensuremath{\mathrm{coker}}}(f)$ if $e$ is a zero-epi;
3. a kernel which is zero-epic is an isomorphism. [${\square}$]{}
We shall mostly be interested in zero-epis (instead of zero-monos), because they arise in the factorisation of Section \[FactorisationSec\]. In the presence of dagger equalisers, zero-epis are ordinary epis. This applies to [[$\mathbf{Hilb}$]{}]{}and [[$\mathbf{PInj}$]{}]{}. This fact is not really used, but is included because it gives a better understanding of the situation. A *dagger equaliser category* is a dagger category that has equalisers which are dagger monic.
\[EqualiserZeroEpiLem\] In a dagger equaliser category [$\mathbf{D}$]{} where every dagger mono is a kernel, zero-epis in [$\mathbf{D}$]{} are ordinary epis.
Assume a zero-epi $e\colon E\rightarrow X$ with two maps $f,g\colon
X\rightarrow Y$ satisfying $f{\mathrel{\circ}}e = g{\mathrel{\circ}}e$. We need to prove $f=g$. Let $m\colon M\rightarrowtail X$ be the equaliser of $f,g$, with $h = {\ensuremath{\mathrm{coker}}}(m)$, as in: $$\xymatrix{
E\ar@{->>}|{\circ}[rr]^-{e}\ar@{-->}[d]_{\varphi} & &
X \ar[rr]<.5ex>^-{f}\ar[rr]<-.5ex>_-{g} \ar@{-|>}[d]^(0.6){h={\ensuremath{\mathrm{coker}}}(m)} & & Y \\
M\ar@{ |>->}[urr]_-{m} & & Z
}$$
This $e$ factors through the equaliser $m$, as indicated, since $f{\mathrel{\circ}}e = g{\mathrel{\circ}}e$. Then $h {\mathrel{\circ}}e = h {\mathrel{\circ}}m {\mathrel{\circ}}\varphi = 0 {\mathrel{\circ}}\varphi = 0$. Hence $h = 0$ because $e$ is zero-epi. But $m$, being a dagger mono, is a dagger kernel. Hence $m =
\ker({\ensuremath{\mathrm{coker}}}(m)) = \ker(h) = \ker(0) = {\ensuremath{\mathrm{id}_{}}}$, so that $f=g$.
Indexed categories and fibrations {#FibrationSubsec}
---------------------------------
The kernel posets ${\ensuremath{\mathrm{KSub}}}(X)$ capture the predicates on an object $X$, considered as underlying type, in a dagger kernel category [$\mathbf{D}$]{}. Such posets are studied systematically in categorical logic, often in terms of indexed categories ${\ensuremath{\mathbf{D}}}{\ensuremath{^{\mathrm{op}}}}\rightarrow{\ensuremath{\mathbf{Posets}}}$ or even as a so-called fibration $\Big({\raisebox{.00in}
{\mbox{ ${{\raisebox{-.05in}{$\scriptstyle {\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})$}\atop
{\scriptscriptstyle\downarrow}}
\atop{\scriptstyle {\ensuremath{\mathbf{D}}}}}$}}}\Big)$, see [@Jacobs99a]. We shall occasionally borrow terminology from this setting, but will not make deep use of it. A construction that is definitely useful in the present setting is the “total” category ${\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})$. It has (equivalence classes of) kernels $M\rightarrowtail X$ as objects. Morphisms $\smash{(M\stackrel{m}{\rightarrowtail} X)}
\longrightarrow \smash{(N\stackrel{n}{\rightarrowtail} Y)}$ in ${\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})$ are maps $f\colon X \rightarrow Y$ in [$\mathbf{D}$]{} with $$\[email protected]{
M\ar@{ |>->}[d]_{m}\ar@{-->}[rr] & &
N\ar@{ |>->}[d]^{n\textstyle{\qquad\mbox{\textit{i.e.}~with}\qquad
m\leq f^{-1}(n).}} \\
X\ar[rr]_-{f} & & Y
}$$
We shall sometimes refer to this fibration as the “kernel fibration”. Every functor $F\colon {\ensuremath{\mathbf{D}}} \rightarrow {\ensuremath{\mathbf{E}}}$ in [[$\mathbf{DagKerCat}$]{}]{}induces a map of fibrations: $$\label{KerFibMapEqn}
\vcenter{\[email protected]{
{\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})\ar[rr]\ar[d] & & {\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{E}}})\ar[d] \\
{\ensuremath{\mathbf{D}}}\ar[rr]_-{F} & & {\ensuremath{\mathbf{E}}}
}}$$
because $F$ preserves kernels and pullbacks of kernels—the latter since pullbacks can be formulated in terms of constructions that are preserved by $F$, see Lemma \[PullbackLem\]. As we shall see, in some situations, diagram (\[KerFibMapEqn\]) is a pullback—also called a change-of-base situation in this context, see [@Jacobs99a]. This means that the map ${\ensuremath{\mathrm{KSub}}}(X) \rightarrow
{\ensuremath{\mathrm{KSub}}}(FX)$ is an isomorphism.
Let us mention one result about this category ${\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})$, which will be used later.
\[TotalInvolutionLem\] The category ${\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})$ for a dagger category [$\mathbf{D}$]{} carries an involution ${\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}}){\ensuremath{^{\mathrm{op}}}}\rightarrow
{\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})$ given by orthocomplementation: $$\begin{array}{rclcrcl}
\smash{(M\stackrel{m}{\rightarrowtail} X)}
& \longmapsto &
\smash{(M^{\perp} \stackrel{m\rlap{$^{\scriptscriptstyle\perp}$}}{\rightarrowtail} X)}
& \quad\mbox{and}\quad &
f
& \longmapsto &
f^{\dag}.
\end{array}$$
The involution is well-defined because a (necessarily unique) map $\varphi$ exists if and only if a (necessarily unique) map $\psi$ exists, in commuting squares: $$\label{InvolutionMapEqn}
\vcenter{\[email protected]{
M\ar@{ |>->}[dd]_{m}\ar@{-->}[r]^-{\varphi} & N\ar@{ |>->}[dd]^{n}
& &
\;\;N\rlap{$^{\perp}$}\;\;\ar@{ |>->}[dd]_{n^\perp}\ar@{-->}[r]^-{\psi}
& M\rlap{$^\perp$}\ar@{ |>->}[dd]^{m^\perp}
\\
& & \Longleftrightarrow \\
X\ar[r]_-{f} & Y & & Y\ar[r]_-{f^\dag} & X
}}$$
Given $\varphi$, we obtain $\psi$ because $f^{\dag}
{\mathrel{\circ}}n^{\perp}$ factors through $\ker(m^{\dag}) = m^{\perp}$ since $$m^{\dag} {\mathrel{\circ}}f^{\dag} {\mathrel{\circ}}n^{\perp}
=
\varphi^{\dag} {\mathrel{\circ}}n^{\dag} {\mathrel{\circ}}n^{\perp}
=
\varphi^{\dag} {\mathrel{\circ}}0
=
0.$$
The reverse direction follows immediately.
Main examples {#ExamplesSec}
=============
This section describes our four main examples, namely [[$\mathbf{Rel}$]{}]{}, [[$\mathbf{PInj}$]{}]{}, [[$\mathbf{Hilb}$]{}]{}and [[$\mathbf{PHilb}$]{}]{}, and additionally a general construction to turn a Boolean algebra into a dagger kernel category.
The category [[$\mathbf{Rel}$]{}]{}of sets and relations {#RelSubsec}
--------------------------------------------------------
Sets and binary relations $R\subseteq X\times Y$ between them can be organised in the familiar category [[$\mathbf{Rel}$]{}]{}, using relational composition. Alternatively, such a relation may be described as a Kleisli map $X\rightarrow \mathcal{P}(Y)$ for the powerset monad $\mathcal{P}$; in line with this representation we sometimes write $R(x) = \{y\in Y\,|\, R(x,y)\}$. A third way is to represent such a morphism in [[$\mathbf{Rel}$]{}]{}as (an equivalence class of) a pair of maps $(X
\stackrel{r_1}{\leftarrow} R \stackrel{r_2}{\rightarrow} Y)$ whose tuple $\langle r_{1}, r_{2}\rangle \colon R\rightarrow X\times Y$ of legs is injective.
There is a simple dagger on [[$\mathbf{Rel}$]{}]{}, given by reversal of relations: $R^{\dag}(y,x) = R(x,y)$. A map $R\colon X\rightarrow Y$ is a dagger mono in [[$\mathbf{Rel}$]{}]{}if $R^{\dag} {\mathrel{\circ}}R = {\ensuremath{\mathrm{id}_{}}}$, which amounts to the equivalence: $${\exists_{y\inY}.\,R(x,y) {\mathrel{\wedge}}R(x',y)}
\quad\Longleftrightarrow \quad
x=x'$$
for all $x,x'\in X$. It can be split into two statements: $${\forall_{x\inX}.\,{\exists_{y\inY}.\,R(x,y)}}
\quad\mbox{and}\quad
{\forall_{x,x'\inX}.\,{\forall_{y\inY}.\,R(x,y) {\mathrel{\wedge}}R(x',y) \Rightarrow x=x'}}.$$
Hence such a dagger mono $R$ is given by a span of the form $$\label{RelDagMonoEqn}
\left(\raisebox{1em}{$\[email protected]@R-1.5em{
& R\ar@{->>}[dl]_{r_{1}}\ar@{ >->}[dr]^{r_{2}} \\
X & & Y
}$}\right)$$
with an surjection as first leg and an injection as second leg. A dagger epi has the same shape, but with legs exchanged.
The empty set $0$ is a zero object in [[$\mathbf{Rel}$]{}]{}, and the resulting zero map $0\colon X\rightarrow Y$ is the empty relation $\emptyset
\subseteq X\times Y$.
The category [[$\mathbf{Rel}$]{}]{}also has kernels. For an arbitrary map $R\colon
X\rightarrow Y$ one takes $\ker(R) = {\{x\inX\;|\;\neg
{\exists_{y\inY}.\,R(x,y)}\}}$ with map $k\colon \ker(R) \rightarrow X$ in [[$\mathbf{Rel}$]{}]{}given by $k(x,x') \Leftrightarrow x=x'$. Clearly, $R {\mathrel{\circ}}k =
0$. And if $S\colon Z\rightarrow X$ satisfies $R{\mathrel{\circ}}S = 0$, then $\neg {\exists_{x\inX}.\,R(x,y) {\mathrel{\wedge}}S(z,x)}$, for all $z\in Z$ and $y\in
Y$. This means that $S(z,x)$ implies there is no $y$ with $R(x,y)$. Hence $S$ factors through the kernel $k$. Kernels are thus of the following form: $$\left(\raisebox{1em}{$\[email protected]@R-1.5em{
& K\ar@{=}[dl]\ar@{ >->}[dr] \\
K & & X
}$}\right) \quad\mbox{with}\quad
K = \{x\in X\,|\, R(x) = \emptyset\}.$$
[ **PROOF:** **ENDPROOF**]{}
So, kernels are essentially given by subsets: ${\ensuremath{\mathrm{KSub}}}(X) =
\mathcal{P}(X)$. Indeed, [[$\mathbf{Rel}$]{}]{}is Boolean, in the sense of Definition \[DagCatKerDef\]. A cokernel has the reversed shape.
Finally, a relation $R$ is zero-mono if its kernel is 0, see Lemma \[ZeroMonoEpiLem\]. This means that $R(x)\neq\emptyset$, for each $x\in X$, so that $R$’s left leg is a surjection.
\[RelMonosProp\] In [[$\mathbf{Rel}$]{}]{}there are proper inclusions: $$\mbox{kernel}
\;\subsetneq\;
\mbox{dagger mono}
\;\subsetneq\;
\mbox{mono}
\;\subsetneq\;
\mbox{zero-mono}.$$
Subsets of a set $X$ correspond to kernels in [[$\mathbf{Rel}$]{}]{}with codomain $X$.
There is of course a dual version of this result, for cokernels and epis.
We still need to produce (1) a zero-mono which is not a mono, and (2) a mono which is not a dagger mono. As to (1), consider $R\subseteq
\{0,1\} \times \{a,b\}$ given by $R = \{(0,a), (1,a)\}$. Its first leg is surjective, so $R$ is a zero-mono. But it is not a mono: there are two different relations $\{(*,0)\}, \{(*,1)\} \subseteq \{*\} \times
\{0,1\}$ with $R {\mathrel{\circ}}\{(*,0)\} = \{(*,a)\} = R{\mathrel{\circ}}\{(*,1)\}$.
As to (2), consider the relation $R\subseteq \{0,1\}\times\{a,b,c\}$ given by $R = \{(0,a), (0,b),$ $(1,b), (1,c)\}$. Clearly, the first leg of $R$ is a surjection, and the second one is neither an injection nor a surjection. We check that $R$ is monic. Suppose $S,T\colon X
\rightarrow \{0,1\}$ satisfy $R{\mathrel{\circ}}S = R{\mathrel{\circ}}T$. If $S(x,0)$, then $(R{\mathrel{\circ}}S)(x,a) = (R{\mathrel{\circ}}T)(x,a)$, so that $T(x,0)$. Similarly, $S(x,1) \Rightarrow T(x,1)$.
We add that the pullback $R^{-1}(n)$ of a kernel $n =
(N=N\rightarrowtail Y)$ along a relation $R\subseteq X\times Y$, as described in Lemma \[PullbackLem\], is the subset of $X$ given by the modal formula $\Box_{R}(n)(x) = R^{-1}(n)(x) \Leftrightarrow
({\forall_{y}.\,R(x,y) \Rightarrow N(y)})$. As is well-known in modal logic, $\Box_{R}$ preserves conjunctions, but not disjunctions. Interestingly, the familiar “graph” functor ${\cal
G}\colon {{\ensuremath{\mathbf{Sets}}}\xspace}\rightarrow {{\ensuremath{\mathbf{Rel}}}\xspace}$, mapping a set to itself and a function to its graph relation, yields a map of fibrations $$\label{SetsRelKerPbEqn}
\vcenter{\[email protected]{
{\ensuremath{\mathrm{Sub}}}({{\ensuremath{\mathbf{Sets}}}\xspace})\ar[d]\ar[rr] & & {\ensuremath{\mathrm{KSub}}}({{\ensuremath{\mathbf{Rel}}}\xspace})\ar[d] \\
{{\ensuremath{\mathbf{Sets}}}\xspace}\ar[rr]_-{\cal G} & & {{\ensuremath{\mathbf{Rel}}}\xspace}}}$$
[ **PROOF:** **ENDPROOF**]{}
which in fact forms a pullback (or a “change-of-base” situation, see [@Jacobs99a]). This means that the familiar logic of sets can be obtained from this kernel logic on relations. In this diagram we use that inverse image is preserved: for a function $f\colon X\rightarrow Y$ and predicate $N\subseteq Y$ one has: $$\begin{array}{rcl}
{\cal G}(f)^{-1}(N)
\hspace*{\arraycolsep} = \hspace*{\arraycolsep}
\Box_{{\cal G}(f)}(N)
& = &
{\{x\inX\;|\;{\forall_{y}.\,{\cal G}(f)(x,y) \Rightarrow N(y)}\}} \\
& = &
{\{x\inX\;|\;{\forall_{y}.\,f(x) = y \Rightarrow N(y)}\}} \\
& = &
{\{x\inX\;|\;N(f(x))\}} \\
& = &
f^{-1}(N).
\end{array}$$
The category [[$\mathbf{PInj}$]{}]{}of sets and partial injections {#PInjSubsec}
------------------------------------------------------------------
There is a subcategory [[$\mathbf{PInj}$]{}]{}of [[$\mathbf{Rel}$]{}]{}also with sets as objects but with “partial injections” as morphisms. These are special relations $F\subseteq X\times Y$ satisfying $F(x,y) {\mathrel{\wedge}}F(x,y')
\Rightarrow y=y'$ and $F(x,y){\mathrel{\wedge}}F(x',y)\Rightarrow x=x'$. We shall therefore often write morphisms $f\colon X\rightarrow Y$ in [[$\mathbf{PInj}$]{}]{} as spans with the notational convention $$\left(X \stackrel{f}{\longrightarrow} Y\right)
=
\left(\raisebox{1em}{$\[email protected]@R-1.5em{
& F\ar@{ >->}[dl]_{f_{1}}
\ar@{ >->}[dr]^{f_{2}} \\
X & & Y
}$}\right),$$
where spans $\smash{(X \stackrel{f_1}{\leftarrowtail} F
\stackrel{f_2}{\rightarrowtail} Y)}$ and $\smash{(X
\stackrel{g_1}{\leftarrowtail} G \stackrel{g_2}{\rightarrowtail}
Y)}$ are equivalent if there is an isomorphism $\varphi\colon
F\rightarrow G$ with $g_{i} {\mathrel{\circ}}\varphi = f_{i}$, for $i=1,2$—like for relations.
Composition of $\smash{X\stackrel{f}{\rightarrow} Y
\stackrel{g}{\rightarrow} Z}$ can be described as relational composition, but also via pullbacks of spans. The identity map $X\rightarrow X$ is given by the span of identities $X \leftarrowtail
X \rightarrowtail X$. The involution is inherited from [[$\mathbf{Rel}$]{}]{}and can be described as $\smash{\big(X\stackrel{f_1}{\leftarrowtail} F
\stackrel{f_2}{\rightarrowtail} Y\big)^{\dag}} =
\smash{\big(Y\stackrel{f_2}{\leftarrowtail} F
\stackrel{f_1}{\rightarrowtail} X\big)}$.
It is not hard to see that $f = \big(X\stackrel{f_1}{\leftarrowtail} F
\stackrel{f_2}{\rightarrowtail} Y\big)$ is a dagger mono—*i.e.* satisfies $f^{\dag} {\mathrel{\circ}}f =
{\ensuremath{\mathrm{id}_{}}}$—if and only if its first leg $f_{1}\colon F
\rightarrowtail X$ is an isomorphism. For convenience we therefore identify a mono/injection $m\colon M \rightarrowtail X$ in ${\ensuremath{\mathbf{Sets}}}$ with the corresponding dagger mono $\big(M\stackrel{{\ensuremath{\mathrm{id}_{}}}}{\leftarrowtail} M
\stackrel{m}{\rightarrowtail} X\big)$ in [[$\mathbf{PInj}$]{}]{}.
[ **PROOF:** **ENDPROOF**]{}
By duality: $f$ is dagger epi iff $f^{\dag}$ is dagger mono iff the second leg $f_{2}$ of $f$ is an isomorphism. Further, $f$ is a dagger iso iff $f$ is both dagger mono and dagger epi iff both legs $f_{1}$ and $f_{2}$ of $f$ are isomorphisms.
Like in [[$\mathbf{Rel}$]{}]{}, the empty set is a zero object, with corresponding zero map given by the empty relation, and $0^{\dag} = 0$.
For the description of the kernel of an arbitrary map $\smash{f =
\big(X\stackrel{f_1}{\leftarrowtail} F \stackrel{f_2}{\rightarrowtail}
Y\big)}$ in [[$\mathbf{PInj}$]{}]{}we shall use the *ad hoc* notation $\neg_{1}F \stackrel{\neg f_1}{\rightarrowtail} X$ for the negation of the first leg $f_{1}\colon F\rightarrowtail X$, as subobject/subset. It yields a map: $$\ker(f) = \left(\raisebox{1em}{$\[email protected]@R-1.5em{
& \neg_{1}F\ar@{=}[dl]\ar@{ >->}[dr]^{\neg f_{1}} \\
\neg_{1}F & & X
}$}\right)$$
It satisfies $f {\mathrel{\circ}}\ker(f) = 0$. It is a dagger mono by construction. Notice that kernels are the same as dagger monos, and are also the same as zero-monos. They all correspond to subsets, so that ${\ensuremath{\mathrm{KSub}}}(X) = \mathcal{P}(X)$ and [[$\mathbf{PInj}$]{}]{}is Boolean, like [[$\mathbf{Rel}$]{}]{}.
[ **PROOF:** **ENDPROOF**]{}
The next result summarises what we have seen so far and shows that [[$\mathbf{PInj}$]{}]{}is very different from [[$\mathbf{Rel}$]{}]{}(see Proposition \[RelMonosProp\]).
\[PInjMonosProp\] In [[$\mathbf{PInj}$]{}]{}there are proper identities: $$\mbox{kernel}
\;=\;
\mbox{dagger mono}
\;=\;
\mbox{mono}
\;=\;
\mbox{zero-mono}.$$
These all correspond to subsets.
The category [[$\mathbf{Hilb}$]{}]{}of Hilbert spaces {#HilbSubsec}
-----------------------------------------------------
Our third example is the category ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ of (complex) Hilbert spaces and continuous linear maps. Recall that a Hilbert space is a vector space $X$ equipped with an inner product, *i.e.* a function ${\ensuremath{\langle -\,|\,- \rangle}}\colon X \times X \to {\ensuremath{\mathbb{C}}}$ that is linear in the first and anti-linear in the second variable, satisfies ${\ensuremath{\langle x\,|\,x \rangle}}
\geq 0$ with equality if and only if $x=0$, and ${\ensuremath{\langle x\,|\,y \rangle}} =
\overline{{\ensuremath{\langle y\,|\,x \rangle}}}$. Moreover, a Hilbert space must be complete in the metric induced by the inner product by $d(x,y) =
\sqrt{{\ensuremath{\langle x-y\,|\,x-y \rangle}}}$.
The Riesz representation theorem provides this category with a dagger. Explicitly, for $f\colon X \to Y$ a given morphism, $f^\dag
\colon Y \to X$ is the unique morphism satisfying $${\ensuremath{\langle f(x)\,|\,y \rangle}}_Y = {\ensuremath{\langle x\,|\,f^\dag(y) \rangle}}_X$$ for all $x \in X$ and $y \in Y$. The zero object is inherited from the category of (complex) vector spaces: it is the zero-dimensional Hilbert space $\{0\}$, with unique inner product ${\ensuremath{\langle 0\,|\,0 \rangle}}=0$.
In the category ${{\ensuremath{\mathbf{Hilb}}}\xspace}$, dagger mono’s are usually called isometries, because they preserve the metric: $f^\dag {\mathrel{\circ}}f = {\ensuremath{\mathrm{id}_{}}}$ if and only if $$d(fx,fy)
= {\ensuremath{\langle f(x-y)\,|\,f(x-y) \rangle}}^{\frac{1}{2}}
= {\ensuremath{\langle x-y\,|\,(f^\dag {\mathrel{\circ}}f)(x-y) \rangle}}^{\frac{1}{2}}
= d(x,y).$$ Kernels are inherited from the category of vector spaces. For $f\colon X \to Y$, we can choose $\ker(f)$ to be (the inclusion of) ${\{x\inX\;|\;f(x)=0\}}$, as this is complete with respect to the restricted inner product of $X$. Hence kernels correspond to (inclusions of) closed subspaces. Being inclusions, kernels are obviously dagger monos. Hence ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ is indeed an example of a dagger kernel category. However, [[$\mathbf{Hilb}$]{}]{}is not Boolean. The following proposition shows that it is indeed different, categorically, from ${{\ensuremath{\mathbf{Rel}}}\xspace}$ and ${{\ensuremath{\mathbf{PInj}}}\xspace}$.
\[HilbMonosProp\] In [[$\mathbf{Hilb}$]{}]{}one has: $$\mbox{kernel}
\;=\;
\mbox{dagger mono}
\;\subsetneq\;
\mbox{mono}
\;=\;
\mbox{zero-mono}.$$
For the left equality, notice that both kernels and isometries correspond to closed subspaces. It is not hard to show that the monos in ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ are precisely the injective continuous linear functions, establishing the middle proper inclusion. Finally, ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ has equalisers by $\mathrm{eq}(f,g) = \ker(g-f)$, which takes care of the right equality.
As is well-known, the $\ell^2$ construction forms a functor $\ell^{2}
\colon {{\ensuremath{\mathbf{PInj}}}\xspace}\rightarrow {{\ensuremath{\mathbf{Hilb}}}\xspace}$ (but not a functor ${{\ensuremath{\mathbf{Sets}}}\xspace}\to {{\ensuremath{\mathbf{Hilb}}}\xspace}$), see *e.g.* [@Barr92; @HaghverdiS06]. Since it preserves daggers, zero object and kernels it is a map in the category [[$\mathbf{DagKerCat}$]{}]{}, and therefore yields a map of kernel fibrations like in (\[KerFibMapEqn\]). It does not form a pullback (change-of-base) between these fibrations, since the map ${\ensuremath{\mathrm{KSub}}}_{{{\ensuremath{\mathbf{PInj}}}\xspace}}(X) = \mathcal{P}(X) \rightarrow
{\ensuremath{\mathrm{KSub}}}_{{{\ensuremath{\mathbf{Hilb}}}\xspace}}(\ell^2(X))$ is not an isomorphism.
[ **PROOF:** **ENDPROOF**]{}
The category [[$\mathbf{PHilb}$]{}]{}: Hilbert spaces modulo phase {#PHilbSubsec}
------------------------------------------------------------------
The category ${\ensuremath{\mathbf{PHilb}}}$ of *projective Hilbert spaces* has the same objects as ${\ensuremath{\mathbf{Hilb}}}$, but its homsets are quotiented by the action of the circle group $U(1) =
{\{z\in{\ensuremath{\mathbb{C}}}\;|\;|z|=1\}}$. That is, continuous linear transformations $f,g\colon X \to Y$ are identified when $x=z\cdot y$ for some phase $z \in U(1)$.
Equivalently, we could write $PX = X_1 / U(1)$ for an object of ${\ensuremath{\mathbf{PHilb}}}$, where $X \in {\ensuremath{\mathbf{Hilb}}}$ and $X_1 =
{\{x\inX\;|\;\|x\|=1\}}$. Two vectors $x,y \in X_1$ are therefore identified when $x=z \cdot y$ for some $z \in U(1)$. Continuous linear transformations $f,g\colon X \to Y$ then descend to the same function $PX \to PY$ precisely when they are equivalent under the action of $U(1)$. This gives a full functor $P\colon {{\ensuremath{\mathbf{Hilb}}}\xspace}\to
{\ensuremath{\mathbf{PHilb}}}$.
The dagger of ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ descends to ${\ensuremath{\mathbf{PHilb}}}$, because if $f=z\cdot g$ for some $z \in U(1)$, then $${\ensuremath{\langle f(x)\,|\,y \rangle}}
= \bar{z} \cdot {\ensuremath{\langle g(x)\,|\,y \rangle}}
= \bar{z} \cdot {\ensuremath{\langle x\,|\,g^\dag(y) \rangle}}
= {\ensuremath{\langle x\,|\,\bar{z} \cdot g^\dag(y) \rangle}},$$ whence also $f^\dag = \bar{z} \cdot g^\dag$, making the dagger well-defined.
Also dagger kernels in ${\ensuremath{\mathbf{Hilb}}}$ descend to ${\ensuremath{\mathbf{PHilb}}}$. More precisely, the kernel $\ker(f) = {\{x\inX\;|\;f(x)=0\}}$ of a morphism $f\colon X \to Y$ is well-defined, for if $f=z \cdot f'$ for some $z
\in U(1)$, then $$\ker(f)
= {\{x\inX\;|\;z \cdot f'(x)=0\}}
= {\{x\inX\;|\;f'(x)=0\}}
= \ker(f').$$
\[PHilbMonosProp\] In [$\mathbf{PHilb}$]{} one has: $$\mbox{kernel}
\;=\;
\mbox{dagger mono}
\;\subsetneq\;
\mbox{mono}
\;=\;
\mbox{zero-mono}.$$
It remains to be shown that every zero-mono is a mono. So let $m\colon Y \to Z$ be a zero-mono, and $f,g\colon X \to Y$ arbitrary morphisms in ${\ensuremath{\mathbf{PHilb}}}$. More precisely, let $m,f$ and $g$ be morphisms in ${\ensuremath{\mathbf{Hilb}}}$ representing the equivalence classes $[m],
[f]$ and $[g]$ that are morphisms in ${\ensuremath{\mathbf{PHilb}}}$. Suppose that $[m
{\mathrel{\circ}}f] = [m {\mathrel{\circ}}g]$. Then $m {\mathrel{\circ}}f \sim m {\mathrel{\circ}}g$, say $m {\mathrel{\circ}}f = z \cdot (m {\mathrel{\circ}}g)$ for $z \in U(1)$. So $m {\mathrel{\circ}}(f-z \cdot g) = 0$, and $f-z \cdot g=0$ since $m$ is zero-mono. Then $f=z \cdot g$ and hence $f \sim g$, *i.e.* $[f]=[g]$. Thus $m$ is mono.
The full functor $P\colon{\ensuremath{\mathbf{Hilb}}} \to {\ensuremath{\mathbf{PHilb}}}$ preserves daggers, the zero object and kernels. Hence it is a map in the category [[$\mathbf{DagKerCat}$]{}]{}. In fact it yields a pullback (change-of-base) between the corresponding kernel fibrations. $$\label{HilbPHilbKerPbEqn}
\vcenter{\[email protected]{
{\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{Hilb}}}){\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}\ar[d]\ar[rr] & & {\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{PHilb}}})\ar[d] \\
{\ensuremath{\mathbf{Hilb}}}\ar[rr]_-{P} & & {\ensuremath{\mathbf{PHilb}}}
}}$$
From Boolean algebras to dagger kernel categories {#BAConstrSubsec}
-------------------------------------------------
The previous four examples were concrete categories, to which we add a generic construction turning an arbitrary Boolean algebra into a (Boolean) dagger kernel category.
To start, let $B$ with $(1,{\mathrel{\wedge}})$ be a meet semi-lattice. We can turn it into a category, for which we use the notation $\widehat{B}$. The objects of $\widehat{B}$ are elements $x\in B$, and its morphisms $x\rightarrow y$ are elements $f\in B$ with $f\leq x,y$, *i.e.* $f\leq x{\mathrel{\wedge}}y$. There is an identity $x\colon
x\rightarrow x$, and composition of $f\colon x\rightarrow y$ and $g\colon y\rightarrow z$ is simply $f{\mathrel{\wedge}}g\colon x\rightarrow z$. This $\widehat{B}$ is a dagger category with $f^{\dag} = f$. A map $f\colon x\rightarrow y$ is a dagger mono if $f^{\dag} {\mathrel{\circ}}f =
f{\mathrel{\wedge}}f = x$. Hence a dagger mono is of the form $x\colon
x\rightarrow y$ where $x\leq y$.
It is not hard to see that the construction $B\mapsto \widehat{B}$ is functorial: a morphism $h\colon B\rightarrow C$ of meet semi-lattices yields a functor $\widehat{h}\colon \widehat{B}\rightarrow \widehat{C}$ by $x\mapsto h(x)$. It clearly preserves $\dag$.
\[BAConstrProp\] If $B$ is a Boolean algebra, then $\widehat{B}$ is a Boolean dagger kernel category. This yields a functor ${\ensuremath{\mathbf{BA}}}\rightarrow
{{\ensuremath{\mathbf{DagKerCat}}}\xspace}$.
The bottom element $0\in B$ yields a zero object $0\in\widehat{B}$, and also a zero map $0\colon x\rightarrow y$. For an arbitrary map $f\colon x\rightarrow y$ there is a kernel $\ker(f) = \neg f {\mathrel{\wedge}}x$, which is a dagger mono $\ker(f) \colon \ker(f) \rightarrow x$ in $\widehat{B}$. Clearly, $f{\mathrel{\circ}}\ker(f) = f {\mathrel{\wedge}}\neg f {\mathrel{\wedge}}x
= 0 {\mathrel{\wedge}}x = 0$. If also $g\colon z\rightarrow x$ satisfies $f{\mathrel{\circ}}g = 0$, then $g\leq x,z$ and $f{\mathrel{\wedge}}g = 0$. The latter yields $g\leq \neg f$ and thus $g\leq \neg f {\mathrel{\wedge}}x = \ker(f)$. Hence $g$ forms the required mediating map $g\colon z\rightarrow
\ker(f)$ with $\ker(f) {\mathrel{\circ}}g = g$.
Notice that each dagger mono $m\colon m\rightarrow x$, where $m\leq
x$, is a kernel, namely of its cokernel $\neg m {\mathrel{\wedge}}x\colon
x\rightarrow (\neg m{\mathrel{\wedge}}x)$. For two kernels $m\colon
m\rightarrow x$ and $n\colon n\rightarrow x$, where $m,n\leq x$, one has $m\leq n$ as kernels iff $m\leq n$ in $B$. Thus ${\ensuremath{\mathrm{KSub}}}(x) =
{\mathop{\downarrow}}x$, which is again a Boolean algebra (with negation $\neg_{x}m = \neg m {\mathrel{\wedge}}x$). The intersection $m{\mathrel{\wedge}}n$ as subobjects is the meet $m{\mathrel{\wedge}}n$ in $B$. This allows us to show that $\widehat{B}$ is Boolean: if $m{\mathrel{\wedge}}n = 0$, them $m^{\dag}
{\mathrel{\circ}}n = m {\mathrel{\circ}}n = m{\mathrel{\wedge}}n = 0$.
[ **PROOF:** **ENDPROOF**]{}
The straightforward extension of the above construction to orthomodular lattices does not work: in order to get kernels one needs to use the and-then connective (${\mathrel{\&}}$, see Proposition \[SasakiProp\]) for composition; but ${\mathrel{\&}}$ is neither associative nor commutative, unless the lattice is Boolean [@Lehmann08]. However, at the end of [@Jacobs09a] a dagger kernel category is constructed out of an orthomodular lattice in a different manner, namely via the (dagger) Karoubi envelope of the associated Foulis semigroup. For more information about orthomodular lattices, see [@Kalmbach83], and for general constructions, see for instance [@Harding06].
Factorisation {#FactorisationSec}
=============
In this section we assume that [$\mathbf{D}$]{} is an arbitrary dagger kernel category. We will show that each map in [$\mathbf{D}$]{} can be factored as a zero-epi followed by a kernel, in an essentially unique way. This factorisation leads to existential quantifiers $\exists$, as is standard in categorical logic.
The image of a morphism $f\colon X\rightarrow Y$ is defined as $\ker({\ensuremath{\mathrm{coker}}}(f))$. Since it is defined as a kernel, an image is really an equivalence class of morphisms with codomain $X$, up to isomorphism of the domain. We denote a representing morphism by $i_f$, and its domain by ${\ensuremath{\mathrm{Im}}}(f)$. As with kernels, we can choose $i_f$ to be dagger mono. Both the morphism $i_f$ and the object ${\ensuremath{\mathrm{Im}}}(f)$ are referred to as the image of $f$. Explicitly, it can be obtained in the following steps. First take the kernel $k$ of $f^\dag$: $$\xymatrix{
\ker(f^{\dag})\ar@{ |>->}[r]^-{k} & Y\ar[r]^-{f^\dag} & X.
}$$ Then define $i_f$ as the kernel of $k^{\dag}$, as in the following diagram: $$\label{ImageEqn}
\vcenter{\xymatrix{
\llap{${\ensuremath{\mathrm{Im}}}(f) = \;$}
\ker(k^{\dag})\ar@{ |>->}[r]^-{i_f} & Y\ar@{-|>}[r]^-{k^{\dag}} & \ker(f^{\dag}). \\
X\ar[ur]_-{f}\ar@{-->}[u]^{e_f}
}}$$ The map $e_{f}\colon X\rightarrow {\ensuremath{\mathrm{Im}}}(f)$ is obtained from the universal property of kernels, since $k^{\dag} {\mathrel{\circ}}f =
(f^{\dag} {\mathrel{\circ}}k)^{\dag} = 0^{\dag} = 0$. Since $i_f$ was chosen to be dagger mono, this $e_f$ is determined as $e_{f} = {\ensuremath{\mathrm{id}_{}}}{\mathrel{\circ}}e_{f} = (i_{f})^{\dag} {\mathrel{\circ}}i_{f} {\mathrel{\circ}}e_{f} = (i_{f})^{\dag} {\mathrel{\circ}}f$.
So images are defined as dagger kernels. Conversely, every dagger kernel $m =
\ker(f)$ arises as an image, since $\ker({\ensuremath{\mathrm{coker}}}(m)) = m$ by Lemma \[KerLem\].
The maps that arise as $e_f$ in (\[ImageEqn\]) can be characterised.
\[FactorisationZeroEpiProp\] The maps in [$\mathbf{D}$]{} that arise of the form $e_f$, as in diagram (\[ImageEqn\]), are precisely the zero-epis.
We first show that $e_f$ is a zero-epi. So, assume a map $h\colon
\ker(k^{\dag}) \rightarrow Z$ satisfying $h {\mathrel{\circ}}e_{f} = 0$. Recall that $e_{f} = (i_{f})^{\dag} {\mathrel{\circ}}f$, so that: $$f^{\dag} {\mathrel{\circ}}(i_{f} {\mathrel{\circ}}h^{\dag})
=
(h {\mathrel{\circ}}(i_{f})^{\dag} {\mathrel{\circ}}f)^{\dag}
=
(h {\mathrel{\circ}}e_{f})^{\dag}
=
0^{\dag}
=
0.$$
This means that $i_{f} {\mathrel{\circ}}h^{\dag}$ factors through the kernel of $f^{\dag}$, say via $a\colon Z\rightarrow \ker(f^{\dag})$ with $k {\mathrel{\circ}}a = i_{f} {\mathrel{\circ}}h^{\dag}$. Since $k$ is a dagger mono we now get: $$a
=
k^{\dag} {\mathrel{\circ}}k {\mathrel{\circ}}a
=
k^{\dag} {\mathrel{\circ}}i_{f} {\mathrel{\circ}}h^{\dag}
=
0 {\mathrel{\circ}}h^{\dag}
=
0.$$
But then $i_{f} {\mathrel{\circ}}h^{\dag} = k {\mathrel{\circ}}a = k {\mathrel{\circ}}0 = 0
= i_{f} {\mathrel{\circ}}0$, so that $h^{\dag} = 0$, because $i_f$ is mono, and $h = 0$, as required.
Conversely, assume $g\colon X\rightarrow Y$ is a zero-epi, so that ${\ensuremath{\mathrm{coker}}}(g) = 0$ by Lemma \[ZeroMonoEpiLem\]. Trivially, $i_{g} =
\ker({\ensuremath{\mathrm{coker}}}(g)) = \ker(X\rightarrow 0) = {\ensuremath{\mathrm{id}_{X}}}$, so that $e_{g} =
g$.
The factorisation $f = i_{f} {\mathrel{\circ}}e_{f}$ from (\[ImageEqn\]) describes each map as a zero-epi followed by a kernel. In fact, these zero-epis and kernels also satisfy what is usually called the “diagonal fill-in” property.
\[DiagonalFillInLem\] In any commuting square of shape $$\xymatrix{
\cdot \ar@{->>}|{\circ}[r]\ar[d] &
\cdot\ar[d]^{\qquad\mbox{there is a (unique) diagonal}}
& \hspace*{10em} &
\cdot \ar@{->>}|{\circ}[r]\ar[d] &
\cdot\ar[d]\ar@{-->}[dl] \\
\cdot\ar@{ |>->}[r] & \cdot
& &
\cdot\ar@{ |>->}[r] & \cdot
}$$
making both triangles commute.
As a result, the factorisation (\[ImageEqn\]) is unique up to isomorphism. Indeed, kernels and zero-epis form a factorisation system (see [@BarrW85]).
Assume the zero-epi $e\colon E\rightarrow Y$ and kernel $m = \ker(h)
\colon M\rightarrowtail X$ satisfy $m{\mathrel{\circ}}f = g {\mathrel{\circ}}e$, as below, $$\[email protected]@R-.5pc{
E\ar@{->>}|{\circ}[r]^-{e}\ar[d]_{f} & Y\ar[d]^{g} \\
M\ar@{ |>->}[r]_-{m} & X\ar[r]_-{h} & Z
}$$
Then: $h{\mathrel{\circ}}g {\mathrel{\circ}}e = h {\mathrel{\circ}}m {\mathrel{\circ}}f = 0 {\mathrel{\circ}}f
= 0$ and $h{\mathrel{\circ}}g = 0$ because $e$ is zero-epi. This yields the required diagonal $d\colon Y\rightarrow M$ with $m {\mathrel{\circ}}d = g$ because $m$ is the kernel of $h$. Using that $m$ is monic we get $d
{\mathrel{\circ}}e = f$.
Factorisation standardly gives a left adjoint to inverse image (pullback), corresponding to existential quantification in logic. In this self-dual situation there are alternative descriptions.
Notice that this general prescription of quantifiers by categorical logic, when applied to our quantum setting, is of a different nature from earlier attempts at quantifiers for quantum logic [@Janowitz63; @Roman06], as it concerns multiple orthomodular lattices instead of a single one.
\[ExistentialProp\] For $f\colon X\rightarrow Y$, the pullback functor $f^{-1}\colon
{\ensuremath{\mathrm{KSub}}}(Y)\rightarrow {\ensuremath{\mathrm{KSub}}}(X)$ from Lemma \[PullbackLem\] has a left adjoint $\exists_f$ given as image: $$\xymatrix{
\Big(M\ar@{ |>->}[r]^-{m} & X\Big) \;\longmapsto \;
\Big({\ensuremath{\mathrm{Im}}}(f{\mathrel{\circ}}m) \ar@{ |>->}[rr]^-{\exists_{f}(m)=i_{f{\mathrel{\circ}}m}}
& & Y\Big)
}$$
Alternatively, $\exists_{f}(m) =
\Big((f^{\dag})^{-1}(m^{\perp})\Big)^{\perp}$.
The heart of the matter is that in the following diagram, the map $\varphi$ (uniquely) exists if and only if the map $\psi$ (uniquely) exists: $$\xymatrix@C+3ex{
M \ar@{ |>->}@(d,l)_-{m}[dr] \ar@{-->}^-{\varphi}[r]
\ar@{-->}@(ur,ul)^-{\psi}[rr]
& \cdot \ar@{ |>->}_-{f^{-1}(n)}[d] {\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}\ar[r]
& N \ar@{ |>->}^-{n}[d] \\
& X \ar_-{f}[r] & Y\rlap{.}
}$$ Thus one easily reads off: $$\begin{aligned}
m \leq f^{-1}(n)
& \Longleftrightarrow \text{ there is } \varphi \text{ such that }
m=f^{-1}(n) {\mathrel{\circ}}\varphi \\
& \Longleftrightarrow \text{ there is } \psi \text{ such that } f
{\mathrel{\circ}}m = n {\mathrel{\circ}}\psi \\
& \Longleftrightarrow \exists_f(m) \leq n.
\end{aligned}$$ For the alternative description: $$\begin{array}[b]{rcccl}
\Big((f^{\dag})^{-1}(m^{\perp})\Big)^{\perp} \leq n
& \Longleftrightarrow &
n^{\perp} \leq (f^{\dag})^{-1}(m^{\perp})
& \smash{\stackrel{(\ref{InvolutionMapEqn})}{\Longleftrightarrow}} &
m \leq f^{-1}(n).
\end{array}$$
This adjunction $\exists_{f} \dashv f^{-1}$ makes the kernel fibration $\Big({\raisebox{.00in}
{\mbox{ ${{\raisebox{-.05in}{$\scriptstyle {\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})$}\atop
{\scriptscriptstyle\downarrow}}
\atop{\scriptstyle {\ensuremath{\mathbf{D}}}}}$}}}\Big)$ an opfibration, and thus a bifibration, see [@Jacobs99a]. Recall the *Beck-Chevalley condition*: if the left square below is a pullback in ${\ensuremath{\mathbf{D}}}$, then the right one must commute. $$\label{eq:beckchevalley}\tag{BC}
\raise5ex\hbox{\xymatrix{
P {\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}\ar^-{q}[r] \ar_-{p}[d] & Y \ar^-{g}[d] \\
X \ar_-{f}[r] & Z
}} \qquad \Longrightarrow \qquad
\raise5ex\hbox{\xymatrix{
{\ensuremath{\mathrm{KSub}}}(P) \ar_-{\exists_p}[d] & {\ensuremath{\mathrm{KSub}}}(Y) \ar^-{\exists_g}[d]
\ar_-{q^{-1}}[l] \\
{\ensuremath{\mathrm{KSub}}}(X) & {\ensuremath{\mathrm{KSub}}}(Z) \ar^-{f^{-1}}[l]
}}$$ This condition ensures that $\exists$ commutes with substitution. If one restricts attention to the pullbacks of the form given in Lemma \[PullbackLem\], then Beck-Chevalley holds. In the notation of Lemma \[PullbackLem\], for kernels $k\colon K\rightarrowtail Y$ and $g\colon Y \rightarrowtail Z$: $$\begin{array}{rcll}
f^{-1}(\exists_g(k))
& = &
f^{-1}(g {\mathrel{\circ}}k)
& \mbox{because both $g,k$ are kernels} \\
& = &
p {\mathrel{\circ}}q^{-1}(k)
& \mbox{by composition of pullbacks} \\
& = &
\exists_p(q^{-1}(k)).
\end{array}$$
In [[$\mathbf{Hilb}$]{}]{}all pullbacks exist and Beck-Chevalley holds for all of them by [@Borceux94 II, Proposition 1.7.6] using [[$\mathbf{Hilb}$]{}]{}’s biproducts and equalisers.
The final result in this section brings more clarity; it underlies the relations between the various maps in the propositions in the previous section.
\[ZeroEpiEpiLem\] If zero-epis are (ordinary) epis, then dagger monos are kernels.
Recall that Lemma \[EqualiserZeroEpiLem\] tells that zero-epis are epis in the presence of equalisers.
Suppose $m\colon M\rightarrowtail X$ is a dagger mono, with factorisation $m = i {\mathrel{\circ}}e$ as in (\[ImageEqn\]), where $i$ is a kernel and a dagger mono, and $e$ is a zero-epi and hence an epi by assumption. We are done if we can show that $e$ is an isomorphism. Since $m = i {\mathrel{\circ}}e$ and $i$ is dagger monic we get $i^{\dag} {\mathrel{\circ}}m = i^{\dag} {\mathrel{\circ}}i {\mathrel{\circ}}e = e$. Hence $e^{\dag} {\mathrel{\circ}}e =
(i^{\dag} {\mathrel{\circ}}m)^{\dag} {\mathrel{\circ}}e = m^{\dag} {\mathrel{\circ}}i {\mathrel{\circ}}e =
m^{\dag} {\mathrel{\circ}}m = {\ensuremath{\mathrm{id}_{}}}$ because $m$ is dagger mono. But then also $e {\mathrel{\circ}}e^{\dag} = {\ensuremath{\mathrm{id}_{}}}$ because $e$ is epi and $e{\mathrel{\circ}}e^{\dag}
{\mathrel{\circ}}e = e$.
\[ImageDomainEx\] In the category [[$\mathbf{Rel}$]{}]{}the image of a morphism $(X
\stackrel{r_1}{\leftarrow} R \stackrel{r_2}{\rightarrow} Y)$ is the relation $\smash{i_{R} = (Y' \stackrel{=}{\leftarrow} Y'
\rightarrowtail Y)}$ where $Y' = {\{y\inY\;|\;{\exists_{x}.\,R(x,y)}\}}$ is the image of the second leg $r_2$ in [$\mathbf{Sets}$]{}. The associated zero-epi is $e_{R} =
(X \stackrel{r_1}{\leftarrow} R \stackrel{r_2}{\twoheadrightarrow}
Y')$. Existential quantification $\exists_{R}(M)$ from Proposition \[ExistentialProp\] corresponds to the modal diamond operator (for the reversed relation $R^{\dag}$): $$\exists_{R}(M)
=
{\{y\inY\;|\;{\exists_{x\inM}.\,R(x,y)}\}}
=
\diamondsuit_{R^{\dag}}(M)
=
\neg\Box_{R^{\dag}}(\neg M).$$
[ **PROOF:** **ENDPROOF**]{}
It is worth mentioning that the “graph” map of fibrations (\[SetsRelKerPbEqn\]) between sets and relations is also a map of opfibrations: for a function $f\colon X\rightarrow Y$ and a predicate $M\subseteq X$ one has: $$\begin{array}{rcl}
\exists_{{\cal G}(f)}(M)
& = &
{\{y\;|\;{\exists_{x}.\,{\cal G}(f)(x,y) {\mathrel{\wedge}}M(x)}\}} \\
& = &
{\{y\;|\;{\exists_{x}.\,f(x) = y {\mathrel{\wedge}}M(x)}\}} \\
& = &
{\{f(x)\;|\;M(x)\}} \\
& = &
\exists_{f}(M),
\end{array}$$
where $\exists_f$ in the last line is the left adjoint to pullback $f^{-1}$ in the category [[$\mathbf{Sets}$]{}]{}.
In [[$\mathbf{PInj}$]{}]{}the image of a map $f = (X \stackrel{f_1}{\leftarrowtail} F
\stackrel{f_2}{\rightarrowtail} Y)$ is given as $i_{f} = (F
\stackrel{{\ensuremath{\mathrm{id}_{}}}}{\leftarrowtail} F \stackrel{f_2}{\rightarrowtail}
Y)$. The associated map $e_f$ is $(X \stackrel{f_1}{\leftarrowtail} F
\stackrel{{\ensuremath{\mathrm{id}_{}}}}{\rightarrowtail} F)$, so that indeed $f = i_{f}
{\mathrel{\circ}}e_{f}$. Notice that this $e_f$ is a dagger epi in [[$\mathbf{PInj}$]{}]{}.
In [[$\mathbf{Hilb}$]{}]{}, the image of a map $f\colon X \to Y$ is (the inclusion of) the closure of the set-theoretic image ${\{y\inY\;|\;{\exists_{x\inX}.\,y=f(x)}\}}$. This descends to ${\ensuremath{\mathbf{PHilb}}}$: the image of a morphism is the equivalence class represented by the inclusion of the closure of the set-theoretic image of a representative.
The functor $\ell^2 \colon {{\ensuremath{\mathbf{PInj}}}\xspace}\to {{\ensuremath{\mathbf{Hilb}}}\xspace}$ is a map of opfibrations: for a partial injection $f = (X \stackrel{f_1}{\leftarrowtail} F
\stackrel{f_2}{\rightarrowtail} Y)$ and a kernel $m\colon M
\rightarrowtail X$ in ${{\ensuremath{\mathbf{PInj}}}\xspace}$ one has: $$\begin{aligned}
\exists_{\ell^2(f)}(\ell^2(m))
& = \mathrm{Im}_{{{\ensuremath{\mathbf{Hilb}}}\xspace}}(\ell^2(f {\mathrel{\circ}}m)) \\
& = \mathrm{Im}_{{{\ensuremath{\mathbf{Hilb}}}\xspace}}(\ell^2(M) \times Y \ni (\varphi,y) \mapsto \sum_{x
\in (f {\mathrel{\circ}}m)^{-1}(y)} \varphi(x))) \\
& \cong \overline{{\{\varphi\in\ell^2(X)\;|\;\mathrm{supp}(\varphi) \subseteq
F \cap M\}}} \\
& = {\{\varphi\in\ell^2(X)\;|\;\mathrm{supp}(\varphi) \subseteq
F \cap M\}} \\
& \cong \ell^2(f_2 {\mathrel{\circ}}f_1^{-1}(m)) \\
& = \ell^2(\exists_f(m)).\end{aligned}$$
Also the full functor $P\colon {{\ensuremath{\mathbf{Hilb}}}\xspace}\to {{\ensuremath{\mathbf{PHilb}}}\xspace}$ is a map of opfibrations: for $f\colon X \to Y$ and a kernel $m\colon M
\rightarrowtail X$ in ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ one has: $$\begin{aligned}
\exists_{Pf}(Pm)
& = \mathrm{Im}_{{{\ensuremath{\mathbf{PHilb}}}\xspace}}(P(f {\mathrel{\circ}}m)) \\
& = \overline{{\{f(x)\;|\;x \in M\}}} \\
& = P(\overline{{\{f(x)\;|\;x \in M\}}}) \\
& = P(\mathrm{Im}_{{{\ensuremath{\mathbf{Hilb}}}\xspace}}(f {\mathrel{\circ}}m)) \\
& = P(\exists_f(m)).\end{aligned}$$
In the category $\widehat{B}$ obtained from a Boolean algebra the factorisation of $f\colon x\rightarrow y$ is the composite $x
\smash{\stackrel{f}{\longrightarrow}} f
\smash{\stackrel{f}{\longrightarrow}} y$. In particular, for $m\leq x$, considered as kernel $m\colon m\rightarrow x$ one has $\exists_{f}(m) =
(m{\mathrel{\wedge}}f\colon (m{\mathrel{\wedge}}f)\rightarrow x)$.
\[ManesEx\] In [@Manes89] the domain ${\ensuremath{\mathrm{Dom}}}(f)$ of a map $f\colon X\rightarrow
Y$ is the complement of its kernel, so ${\ensuremath{\mathrm{Dom}}}(f) = \ker(f)^{\perp}$, and hence a kernel itself. It can be described as an image, namely of $f^{\dag}$, since: $${\ensuremath{\mathrm{Dom}}}(f)
=
\ker(f)^{\perp}
=
\ker(\ker(f)^{\dag})
=
\ker({\ensuremath{\mathrm{coker}}}(f^{\dag}))
=
i_{f^{\dag}}.$$
It is shown in [@Manes89] that the composition $f
{\mathrel{\circ}}{\ensuremath{\mathrm{Dom}}}(f)$ is zero-monic—or “total”, as it is called there. This also holds in the present setting, since: $$f {\mathrel{\circ}}{\ensuremath{\mathrm{Dom}}}(f)
=
f^{\dag\dag} {\mathrel{\circ}}i_{f^{\dag}}
=
(i_{f^{\dag}} {\mathrel{\circ}}e_{f^{\dag}})^{\dag} {\mathrel{\circ}}i_{f^{\dag}}
=
(e_{f^{\dag}})^{\dag} {\mathrel{\circ}}(i_{f^{\dag}})^{\dag} {\mathrel{\circ}}i_{f^{\dag}}
=
(e_{f^{\dag}})^{\dag}.$$
This $e_{f^{\dag}}$ is zero-epic, by Proposition \[FactorisationZeroEpiProp\], so that $(e_{f^{\dag}})^{\dag}$ is indeed zero-monic. In case $f\colon
X\rightarrow X$ is a self-adjoint map, meaning $f^{\dag} = f$, then the image of $f$ is the same as the domain, and thus as the complement of the kernel.
There is one further property that is worth making explicit, if only in examples. In the kernel fibration over [[$\mathbf{Rel}$]{}]{}one finds the following correspondences. $${\ensuremath{\mathrm{KSub}}}(X)
\cong
{\cal P}(X)
\cong
{{\ensuremath{\mathbf{Sets}}}\xspace}(X, 2)
\cong
{{\ensuremath{\mathbf{Sets}}}\xspace}(X, {\cal P}(1))
\cong
{{\ensuremath{\mathbf{Rel}}}\xspace}(X, 1).$$ This suggests that one has “kernel classifiers”, comparable to “subobject classifiers” in a topos—or more abstractly, “generic objects”, see [@Jacobs99a]. But the naturality that one has in toposes via pullback functors $f^{-1}$ exists here via their left adjoints $\exists_f$. That is, we really have found a natural correspondence ${\ensuremath{\mathrm{KSub}}}(X) \cong {{\ensuremath{\mathbf{Rel}}}\xspace}(1,X)$ instead of ${\ensuremath{\mathrm{KSub}}}(X) \cong {{\ensuremath{\mathbf{Rel}}}\xspace}(X,1)$. Indeed, there are natural “characteristic” isomorphisms: $$\xymatrix@R-2pc{
\llap{${\ensuremath{\mathrm{KSub}}}(X)=\;$}{\cal P}(X)\ar[rr]^-{{\ensuremath{\mathrm{char}}}}_-{\cong} & & {{\ensuremath{\mathbf{Rel}}}\xspace}(1,X) \\
(M\subseteq X)\ar@{|->}[rr] & & {\{(*,x)\;|\;x\in M\}}.
}$$
Then, for $S\colon X\rightarrow Y$ in [[$\mathbf{Rel}$]{}]{}, $$\begin{array}{rcl}
S {\mathrel{\circ}}{\ensuremath{\mathrm{char}}}(M)
& = &
{\{(*,y)\;|\;{\exists_{x}.\,{\ensuremath{\mathrm{char}}}(M)(*,x) {\mathrel{\wedge}}S(x,y)}\}} \\
& = &
{\{(*,y)\;|\;{\exists_{x}.\,M(x) {\mathrel{\wedge}}S(x,y)}\}} \\
& = &
{\{(*,y)\;|\;\exists_{S}(M)(y)\}} \\
& = &
{\ensuremath{\mathrm{char}}}(\exists_{S}(M)).
\end{array}$$ Hence one could say that [[$\mathbf{Rel}$]{}]{}has a kernel “opclassifier”. This naturality explains our choice of ${{\ensuremath{\mathbf{Rel}}}\xspace}(1,X)$ over ${{\ensuremath{\mathbf{Rel}}}\xspace}(X,1)$: the latter formulation more closely resembles the subobject classifiers of a topos, but using the former, naturality can be formulated without using the dagger. Hence in principle one could even consider “opclassifiers” in categories without a dagger.
The same thing happens in the dagger categories $\widehat{B}$ from Subsection \[BAConstrSubsec\]. There one has, for $x\in B$, $$\xymatrix@R-2pc{
\llap{${\ensuremath{\mathrm{KSub}}}(x)=\;$}{\mathop{\downarrow}}x\ar[rr]^-{{\ensuremath{\mathrm{char}}}}_-{\cong} & &
\widehat{B}(1,x) \\
(m\leq x)\ar@{|->}[rr] & & (m\colon 1\rightarrow x)
}$$
As before, $f {\mathrel{\circ}}{\ensuremath{\mathrm{char}}}(m) = f{\mathrel{\wedge}}m =
\exists_{f}(m) = {\ensuremath{\mathrm{char}}}(\exists_{f}(m))$.
The category [$\mathbf{OMLatGal}$]{} of orthomodular lattices and Galois connections between them from [@Jacobs09a] also has such an opclassifier. There is no obvious kernel opclassifier for the category ${\ensuremath{\mathbf{Hilb}}}$. The category ${\ensuremath{\mathbf{PInj}}}$ is easily seen not to have a kernel opclassifier.
Images and coimages {#ImageCoimageSec}
===================
We continue to work in an arbitrary dagger kernel category [$\mathbf{D}$]{}. In the previous section we have seen how each map $f\colon
X\rightarrow Y$ in [$\mathbf{D}$]{} can be factored as $f = i_{f} {\mathrel{\circ}}e_{f}$ where the image $i_{f} = \ker({\ensuremath{\mathrm{coker}}}(f)) \colon{\ensuremath{\mathrm{Im}}}(f)\rightarrowtail
Y$ is a kernel and $e_f$ is a zero-epi. We can apply this same factorisation to the dual $f^{\dag}$. The dual of its image, $(i_{f^\dag})^{\dag} = {\ensuremath{\mathrm{coker}}}(\ker(f)) \colon X \twoheadrightarrow
{\ensuremath{\mathrm{Im}}}(f^{\dag})$, is commonly called the coimage of $f$. It is a cokernel and dagger epi by construction. Thus we have: $$\xymatrix@R-3ex@C+1ex{
X\ar@{->}[rr]^-{f}\ar@{->>}|{\circ}[dr]_{e_{f}} & & Y
& &
Y\ar@{->}[rr]^-{f^\dag}\ar@{->>}|{\circ}[dr]_{e_{f^\dag}} & & X \\
& {\ensuremath{\mathrm{Im}}}(f)\ar@{ |>->}[ur]_{i_{f}} &
& &
& {\ensuremath{\mathrm{Im}}}(f^{\dag})\ar@{ |>->}[ur]_{i_{f^\dag}} &
}$$
By combining these factorisations we get two mediating maps $m$ by diagonal fill-in (see Lemma \[DiagonalFillInLem\]), as in: $$\xymatrix@R-1ex@C+2ex{
X\ar@{->}[rr]^-{f}\ar@{->>}|{\circ}[dr]^{e_{f}}
\ar@{-|>}[ddr]_{(i_{f^\dag})^{\dag}} & & Y
&
Y\ar@{->}[rr]^-{f^\dag}\ar@{->>}|{\circ}[dr]^{e_{f^\dag}}
\ar@{-|>}[ddr]_{(i_{f})^{\dag}} & & X \\
& {\ensuremath{\mathrm{Im}}}(f)\ar@{ |>->}[ur]^{i_{f}} &
&
& {\ensuremath{\mathrm{Im}}}(f^{\dag})\ar@{ |>->}[ur]^{i_{f^\dag}} & \\
& {\ensuremath{\mathrm{Im}}}(f^{\dag})\ar@{ >->}|{\circ}[uur]_{(e_{f^\dag})^{\dag}}\ar@{..>}[u]|{m_f} &
&
& {\ensuremath{\mathrm{Im}}}(f)\ar@{ >->}|{\circ}[uur]_{(e_{f})^{\dag}}\ar@{..>}[u]|{m_{f^\dag}} &
}$$
We claim that $(m_{f})^{\dag} = m_{f^{\dag}}$. This follows easily from the fact that $(i_{f^\dag})^{\dag}$ is epi: $$(m_{f^{\dag}})^{\dag} {\mathrel{\circ}}(i_{f^\dag})^{\dag}
=
(i_{f^\dag} {\mathrel{\circ}}m_{f^{\dag}})^{\dag}
=
(e_{f})^{\dag\dag}
=
e_{f}
=
m_{f} {\mathrel{\circ}}(i_{f^\dag})^{\dag}.$$
Moreover, $m_f$ is both a zero-epi and a zero-mono.
[ **PROOF:** **ENDPROOF**]{}
As a result we can factorise each map $f\colon X\rightarrow Y$ in [$\mathbf{D}$]{} as: $$\label{ImageCoimageEqn}
\vcenter{\xymatrix{
X\ar@{-|>}[rr]^-{(i_{f^\dag})^{\dag}}_-{\textrm{coimage}}
& & {\ensuremath{\mathrm{Im}}}(f^{\dag})\ar@{ >->>}[r]|{\circ}^-{m_f}_{\begin{array}{c}
\labelstyle\textrm{zero-epi} \\[-.7pc]
\labelstyle\textrm{zero-mono} \end{array}}
& {\ensuremath{\mathrm{Im}}}(f)\ar@{ |>->}[rr]^-{i_f}_-{\textrm{image}} & & Y.
}}$$
This coimage may also be reversed, so that a map in [$\mathbf{D}$]{} can also be understood as a pair of kernels with a zero-mono/epi between them, as in: $$\xymatrix{
X
& & {\ensuremath{\mathrm{Im}}}(f^{\dag})\ar@{ >->>}[r]|{\circ}\ar@{ |>->}[ll]_-{i_{f^\dag}}
& {\ensuremath{\mathrm{Im}}}(f)\ar@{ |>->}[rr]^-{i_f} & & Y
}$$
The two outer kernel maps perform some “bookkeeping” to adjust the types; the real action takes place in the middle, see the examples below. The category [[$\mathbf{PInj}$]{}]{}consists, in a sense, of only these bookkeeping maps, without any action. This will be described more systematically in Definition \[KcKDef\].
\[ImageCoimageEx\] We briefly describe the factorisation (\[ImageCoimageEqn\]) in [[$\mathbf{Rel}$]{}]{}, [[$\mathbf{PInj}$]{}]{}and [[$\mathbf{Hilb}$]{}]{}, using diagrammatic order for convenience (with notation $f;g = g {\mathrel{\circ}}f$).
For a map $(X \stackrel{r_1}{\leftarrow} R \stackrel{r_2}{\rightarrow}
Y)$ in [[$\mathbf{Rel}$]{}]{}we take the images $X' \rightarrowtail X$ of $r_1$ and $Y'\rightarrowtail Y$ of $r_2$ in: $$\left(\raisebox{1em}{$\xymatrix@[email protected]{
& R\ar[dl]_{r_1}\ar[dr]^{r_2} \\
X & & Y
}$}\right)
=
\left(\raisebox{1em}{$\xymatrix@[email protected]{
& X'\ar@{ >->}[dl]\ar@{=}[dr] \\
X & & X'
}$}\right)
;
\left(\raisebox{1em}{$\xymatrix@[email protected]{
& R\ar@{->>}[dl]_{r_1}\ar@{->>}[dr]^{r_2} \\
X' & & Y'
}$}\right)
;
\left(\raisebox{1em}{$\xymatrix@[email protected]{
& Y'\ar@{=}[dl]\ar@{ >->}[dr] \\
Y' & & Y
}$}\right)$$
In [[$\mathbf{PInj}$]{}]{}the situation is simpler, because the middle part $m$ in (\[ImageCoimageEqn\]) is the identity, in: $$\left(\raisebox{1em}{$\xymatrix@[email protected]{
& F\ar@{ >->}[dl]_{f_1}\ar@{ >->}[dr]^{f_2} \\
X & & Y
}$}\right)
=
\left(\raisebox{1em}{$\xymatrix@[email protected]{
& F\ar@{ >->}[dl]_{f_1}\ar@{=}[dr] \\
X & & F
}$}\right)
;
\left(\raisebox{1em}{$\xymatrix@[email protected]{
& F\ar@{=}[dl]\ar@{ >->}[dr]^{f_2} \\
F & & Y
}$}\right).$$
In [[$\mathbf{Hilb}$]{}]{}, a morphism $f\colon X \to Y$ factors as $f=i {\mathrel{\circ}}m {\mathrel{\circ}}e$. The third part $i\colon I \to Y$ is given by $i(y)=y$, where $I$ is the closure $\overline{\{f(x) \;\colon x \in X\}}$. The first part $e\colon X \to E$ is given by orthogonal projection on the closure $E
= \overline{\{f^\dag(y) \;\colon y \in Y\}}$; explicitly, $e(x)$ is the unique $x'$ such that $x=x'+x''$ with $x' \in E$ and ${\ensuremath{\langle x''\,|\,z \rangle}}=0$ for all $z \in E$. Using the fact that the adjoint $e^\dag \colon E \to X$ is given by $e^\dag(x)=x$, we deduce that the middle part $m\colon E \to I$ is determined by $m(x) = (i
{\mathrel{\circ}}m)(x) = (f {\mathrel{\circ}}e^\dag)(x) = f(x)$. Explicitly, $$\left(X\stackrel{f}{\longrightarrow} Y\right)
=
\left(X\stackrel{e}{\longrightarrow} E\right)
;
\left(E\stackrel{m}{\longrightarrow} I\right)
;
\left(I\stackrel{i}{\longrightarrow} Y\right).$$
Categorical logic {#LogicSec}
=================
This section further investigates the logic of dagger kernel categories. We shall first see how the so-called Sasaki hook [@Kalmbach83] arises naturally in this setting, and then investigate Booleanness.
For a kernel $m\colon M\rightarrowtail X$ we shall write ${\mathfrak{E}(m)} =
m {\mathrel{\circ}}m^{\dag} \colon X\rightarrow X$ for the “effect” of $m$, see [@DvurecenskijP00]. This ${\mathfrak{E}(m)}$ is easily seen to be a self-adjoint idempotent: one has ${\mathfrak{E}(m)}^{\dag} = {\mathfrak{E}(m)}$ and ${\mathfrak{E}(m)} {\mathrel{\circ}}{\mathfrak{E}(m)} = {\mathfrak{E}(m)}$. The endomap ${\mathfrak{E}(m)}\colon X\rightarrow X$ associated with a kernel/predicate $m$ on $X$ maps everything in $X$ that is in $m$ to itself, and what is perpendicular to $m$ to $0$, as expressed by the equations ${\mathfrak{E}(m)} {\mathrel{\circ}}m = m$ and ${\mathfrak{E}(m)} {\mathrel{\circ}}m^{\perp} = 0$. Of interest is the following result. It makes the dynamical aspects of quantum logic described in [@CoeckeS04] explicit.
\[SasakiProp\] For kernels $m\colon M\rightarrowtail X$, $n\colon N\rightarrowtail X$ the pullback ${\mathfrak{E}(m)}^{-1}(n)$ is the Sasaki hook, written here as ${\mathrel{\supset}}$: $$m {\mathrel{\supset}}n
\smash{\;\stackrel{\textrm{def}}{=}\;}
{\mathfrak{E}(m)}^{-1}(n)
\;=\;
m^{\perp} {\mathrel{\vee}}(m{\mathrel{\wedge}}n).$$
The associated left adjoint $\exists_{{\mathfrak{E}(m)}} \dashv
{\mathfrak{E}(m)}^{-1}$ yields the “and then” operator: $$k {\mathrel{\&}}m
\smash{\;\stackrel{\textrm{def}}{=}\;}
\exists_{{\mathfrak{E}(m)}}(k)
\;=\;
m {\mathrel{\wedge}}(m^{\perp} {\mathrel{\vee}}k),$$
so that the “Sasaki adjunction” (see [@Finch70]) holds by construction: $$\begin{array}{rcl}
k {\mathrel{\&}}m \leq n
& \Longleftrightarrow &
k \leq m{\mathrel{\supset}}n.
\end{array}$$
Quantum logic based on this “and-then” ${\mathrel{\&}}$ connective is developed in [@Lehmann08], see also [@RomanR91; @RomanZ96]. This ${\mathrel{\&}}$ connective is in general non-commutative and non-associative[^1]. Some basic properties are: $m{\mathrel{\&}}m = m$, $1{\mathrel{\&}}m = m{\mathrel{\&}}1 = m$, $0{\mathrel{\&}}m = m{\mathrel{\&}}0 = 0$, and both $k{\mathrel{\&}}m\leq n$, $k^{\perp} {\mathrel{\&}}m \leq n$ imply $m\leq
n$ (which easily follows from the Sasaki adjunction).
Consider the following pullbacks. $$\xymatrix{
P\ar[d]_{p}\ar[r]^-{q}{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& N\ar@{ |>->}[d]^{n}
& &
Q\ar[d]_{r}\ar[r]^-{s}{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}&
P\rlap{$^{\perp}$}\ar@{ |>->}[d]^{(m{\mathrel{\wedge}}n)^{\perp} =\,\ker(p^{\dag} {\mathrel{\circ}}m^{\dag})} \\
M\ar@{ |>->}[r]_-{m} & X
& &
M\ar@{ |>->}[r]_-{m} & X
}$$
Then: $$\begin{array}[b]{rcl}
m^{\perp} {\mathrel{\vee}}(m{\mathrel{\wedge}}n)
& = &
\big(m {\mathrel{\wedge}}(m{\mathrel{\wedge}}n)^{\perp}\big)^{\perp} \\
& = &
\ker\big((m {\mathrel{\wedge}}(m{\mathrel{\wedge}}n)^{\perp})^{\dag}\big) \\
& = &
\ker\big(r^{\dag} {\mathrel{\circ}}m^{\dag}\big) \\
& = &
\ker\big(\ker({\ensuremath{\mathrm{coker}}}((m{\mathrel{\wedge}}n)^{\perp}) {\mathrel{\circ}}m)^{\dag} {\mathrel{\circ}}m^{\dag}\big) \\
& & \qquad\mbox{by definition of $r$ as pullback,
see Lemma~\ref{PullbackLem}} \\
& = &
\ker\big(\ker({\ensuremath{\mathrm{coker}}}(\ker(p^{\dag} {\mathrel{\circ}}m^{\dag})) {\mathrel{\circ}}m)^{\dag} {\mathrel{\circ}}m^{\dag}\big) \\
& = &
\ker\big(\ker(p^{\dag} {\mathrel{\circ}}m^{\dag} {\mathrel{\circ}}m)^{\dag} {\mathrel{\circ}}m^{\dag}\big) \\
& & \qquad\mbox{because $p^{\dag} {\mathrel{\circ}}m^{\dag}$ is a cokernel,
see Lemma~\ref{KernelCompLem}} \\
& = &
\ker({\ensuremath{\mathrm{coker}}}(p) {\mathrel{\circ}}m^{\dag}) \\
& = &
\big(m^{\dag}\big)^{-1}(p) \\
& = &
\big(m^{\dag}\big)^{-1}(m^{-1}(n)) \\
& = &
{\mathfrak{E}(m)}^{-1}(n).
\end{array}$$
[ **PROOF:** **ENDPROOF**]{}
As we have seen, substitution functors $f^{-1}$ in dagger kernel categories have left adjoints $\exists_f$. It is natural to ask if they also have right adjoints $\forall_f$. The next result says that existence of such adjoints $\forall_f$ makes the logic Boolean.
\[ForallProp\] Suppose there are right adjoints $\forall_f$ to $f^{-1} \colon {\ensuremath{\mathrm{KSub}}}(Y) \to {\ensuremath{\mathrm{KSub}}}(X)$ for each $f \colon X \to
Y$ in a dagger kernel category. Then each ${\ensuremath{\mathrm{KSub}}}(X)$ is a Boolean algebra.
[@Johnstone02 Lemma A1.4.13] For $k,l \in {\ensuremath{\mathrm{KSub}}}(X)$, define implication $(k \Rightarrow l) = \forall_k(k^{-1}(l)) \in
{\ensuremath{\mathrm{KSub}}}(X)$. Then for any $m \in {\ensuremath{\mathrm{KSub}}}(X)$: $$\begin{array}{rcl}
m \;\leq\; \forall_k(k^{-1}(l)) = (k \Rightarrow l)
& \Longleftrightarrow &
k^{-1}(m) \;\leq\; k^{-1}(l) \\
& \Longleftrightarrow &
m {\mathrel{\wedge}}k = k {\mathrel{\circ}}k^{-1}(m) \;\leq\; l,
\end{array}$$
where the last equivalence holds because $k{\mathrel{\circ}}-$ is left adjoint to $k^{-1}$, since $k$ is a kernel. Hence ${\ensuremath{\mathrm{KSub}}}(X)$ is a Heyting algebra, and therefore distributive. By Proposition \[OrthmodularityProp\] we know that it is also orthomodular. Hence each ${\ensuremath{\mathrm{KSub}}}(X)$ is a Boolean algebra.
These universal quantifiers $\forall_f$ do not exist in general because not all kernel posets ${\ensuremath{\mathrm{KSub}}}(X)$ are Boolean algebras. For a concrete non-example, consider the lattice ${\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbb{C}}}^2)$ in the category ${{\ensuremath{\mathbf{Hilb}}}\xspace}$—where ${\ensuremath{\mathbb{C}}}$ denotes the complex numbers. Consider the kernel subobjects represented by $$\kappa_1 \colon {\ensuremath{\mathbb{C}}} \to {\ensuremath{\mathbb{C}}}^2,
\qquad
\kappa_2 = (\kappa_1)^\perp \colon {\ensuremath{\mathbb{C}}} \to {\ensuremath{\mathbb{C}}}^2,
\qquad
\Delta = {\ensuremath{\protect\langle {\ensuremath{\mathrm{id}_{}}}\,,\,{\ensuremath{\mathrm{id}_{}}}\protect\rangle}} \colon {\ensuremath{\mathbb{C}}} \to {\ensuremath{\mathbb{C}}}^2.$$
Since we can write each $(z,w)\in{\ensuremath{\mathbb{C}}}^{2}$ as $(z,w) =
\Delta(z,z) + \kappa_{2}(w-z)$ we get $\Delta {\mathrel{\vee}}\kappa_2 = 1$ in ${\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbb{C}}}^2)$. This yields a counterexample to distributivity: $$\kappa_1 {\mathrel{\wedge}}(\Delta {\mathrel{\vee}}\kappa_2)
=
\kappa_1 {\mathrel{\wedge}}1
=
\kappa_1
\neq
0
=
0 {\mathrel{\vee}}0
=
(\kappa_1 {\mathrel{\wedge}}\Delta) {\mathrel{\vee}}(\kappa_1 {\mathrel{\wedge}}\kappa_2).$$
We now turn to a more systematic study of Booleanness. As we have seen, the categories [[$\mathbf{Rel}$]{}]{}, [[$\mathbf{PInj}$]{}]{}and $\widehat{B}$ (for a Boolean algebra $B$) are Boolean, but [[$\mathbf{Hilb}$]{}]{}and [[$\mathbf{PHilb}$]{}]{}are not. The following justifies the name “Boolean”.
\[BooleanLem\] A dagger kernel category is Boolean if and only if each orthomodular lattice ${\ensuremath{\mathrm{KSub}}}(X)$ is a Boolean algebra.
We already know that each poset ${\ensuremath{\mathrm{KSub}}}(X)$ is an orthomodular lattice, with bottom $0$, top $1$, orthocomplement $(-)^{\perp}$ (by Lemma \[KerLem\]), intersections ${\mathrel{\wedge}}$ (by Lemma \[KernelCompLem\]), and joins $m{\mathrel{\vee}}n = (m^{\perp} {\mathrel{\wedge}}n^{\perp})^{\perp}$. What is missing is distributivity $m {\mathrel{\wedge}}(n
{\mathrel{\vee}}k) = (m {\mathrel{\vee}}n) {\mathrel{\wedge}}(m {\mathrel{\vee}}k)$. We show that the latter is equivalent to the Booleanness requirement $m {\mathrel{\wedge}}n = 0
\Rightarrow m {\mathrel{\bot}}n$. Recall: $m{\mathrel{\bot}}n$ iff $n^{\dag}
{\mathrel{\circ}}m = 0$ iff $m\leq n^{\perp} = \ker(n^{\dag})$.
First, assume Booleanness. In any lattice one has $m \wedge (n \vee k) \geq (m \wedge n) \vee (m \wedge k)$. For the other inequality, notice that $$(m {\mathrel{\wedge}}(m {\mathrel{\wedge}}n)^{\perp}) {\mathrel{\wedge}}n
=
(m {\mathrel{\wedge}}n) {\mathrel{\wedge}}(m {\mathrel{\wedge}}n)^{\perp}
=
0.$$
Hence $m {\mathrel{\wedge}}(m {\mathrel{\wedge}}n)^{\perp} \leq n^{\perp}$. Similarly, $m {\mathrel{\wedge}}(m {\mathrel{\wedge}}k)^{\perp} \leq k^{\perp}$. So $$m {\mathrel{\wedge}}(m {\mathrel{\wedge}}n)^{\perp} {\mathrel{\wedge}}(m {\mathrel{\wedge}}k)^{\perp}
\leq
n^{\perp} {\mathrel{\wedge}}k^{\perp}
=
(n {\mathrel{\vee}}k)^{\perp},$$
and therefore $$m {\mathrel{\wedge}}(m {\mathrel{\wedge}}n)^{\perp} {\mathrel{\wedge}}(m {\mathrel{\wedge}}k)^{\perp}
{\mathrel{\wedge}}(n{\mathrel{\vee}}k)
=
0.$$
But then we are done by using Booleanness again: $$m {\mathrel{\wedge}}(n{\mathrel{\vee}}k)
\leq
((m {\mathrel{\wedge}}n)^{\perp} {\mathrel{\wedge}}(m {\mathrel{\wedge}}k)^{\perp})^{\perp}
=
(m {\mathrel{\wedge}}n) {\mathrel{\vee}}(m {\mathrel{\wedge}}k).$$
The other direction is easier: if $m{\mathrel{\wedge}}n = 0$, then $$\begin{array}{rcl}
m
\hspace*{\arraycolsep}=\hspace*{\arraycolsep}
m {\mathrel{\wedge}}1
& = &
m {\mathrel{\wedge}}(n{\mathrel{\vee}}n^{\perp}) \\
& = &
(m {\mathrel{\wedge}}n) {\mathrel{\vee}}(m {\mathrel{\wedge}}n^{\perp})
\quad\mbox{by distributivity} \\
& = &
0 {\mathrel{\vee}}(m {\mathrel{\wedge}}n^{\perp})
\hspace*{\arraycolsep}=\hspace*{\arraycolsep}
m {\mathrel{\wedge}}n^{\perp},
\end{array}$$
whence $m \leq n^{\perp}$.
The Booleanness property can be strengthened in the following way.
\[BooleanStrengthenedProp\] The Booleanness requirement $m{\mathrel{\wedge}}n = 0 \Rightarrow m\leq
n^{\perp}$, for all kernels $m,n$, is equivalent to the following: for each pullback of kernels: $$\begin{array}{rcrcl}
\raisebox{1.5em}{$\xymatrix{
P\ar[r]^-{p}\ar[d]_{q}{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& N\ar@{ |>->}[d]^{n} \\
M\ar@{ |>->}[r]_-{m} & X
}$}
& \qquad\mbox{one has}\qquad &
n^{\dag} {\mathrel{\circ}}m
& = &
p {\mathrel{\circ}}q^{\dag}.
\end{array}$$
It is easy to see that the definition of Booleanness is the special case $P=0$. For the converse, we put another pullback on top of the one in the statement: $$\xymatrix{
0\ar[r]\ar[d]{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& P^{\perp}\ar@{ |>->}[d]^{p^{\perp}} \\
P\ar@{ |>->}[r]^-{p}\ar@{ |>->}[d]_{q}{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& N\ar@{ |>->}[d]^{n} \\
M\ar@{ |>->}[r]_-{m} & X
}$$
We use that $p,q$ are kernels by Lemma \[PullbackLem\]. We see $m {\mathrel{\wedge}}(n {\mathrel{\circ}}p^{\perp}) = 0$, so by Booleanness we obtain: $$\begin{array}{rcl}
m \leq (n {\mathrel{\circ}}p^{\perp})^{\perp}
& = &
\ker\Big((n {\mathrel{\circ}}\ker(p^{\dag}))^{\dag}\Big) \\
& = &
\ker({\ensuremath{\mathrm{coker}}}(p) {\mathrel{\circ}}n^{\dag}) \\
& = &
(n^{\dag})^{-1}(p),
\end{array}$$
where the pullback is as described in Lemma \[PullbackLem\]. Hence there is a map $\varphi\colon M
\rightarrow P$ with $p {\mathrel{\circ}}\varphi = n^{\dag} {\mathrel{\circ}}m$. This means that $\varphi = p^{\dag} {\mathrel{\circ}}p {\mathrel{\circ}}\varphi = p^{\dag} {\mathrel{\circ}}n^{\dag}
{\mathrel{\circ}}m = (n {\mathrel{\circ}}p)^{\dag} {\mathrel{\circ}}m = (m {\mathrel{\circ}}q)^{\dag} {\mathrel{\circ}}m
= q^{\dag} {\mathrel{\circ}}m^{\dag} {\mathrel{\circ}}m = q^{\dag}$. Hence we have obtained $p {\mathrel{\circ}}q^{\dag} = n^{\dag} {\mathrel{\circ}}m$, as required.
\[KcKDef\] Let [$\mathbf{D}$]{} be a Boolean dagger kernel category. We write ${\ensuremath{\mathbf{D}}}_{kck}$ for the category with the same objects as [$\mathbf{D}$]{}; morphisms $X\rightarrow Y$ in ${\ensuremath{\mathbf{D}}}_{kck}$ are cokernel-kernel pairs $(c, k)$ of the form ${\ensuremath{\smash{\xymatrix@1{X \ar@{-|>}^-{c}[r] & \,\bullet\,
\ar@{ |>->}^-{k}[r] & Y}}}^{\rule[8.5pt]{0pt}{0pt}}}$. The identity $X\rightarrow X$ is ${\ensuremath{\smash{\xymatrix@1{X \ar@{-|>}^-{{\ensuremath{\mathrm{id}_{}}}}[r] & X \ar@{ |>->}^-{{\ensuremath{\mathrm{id}_{}}}}[r] & X}}}^{\rule[8.5pt]{0pt}{0pt}}}$, and composition of ${\ensuremath{\smash{\xymatrix@1{X \ar@{-|>}^-{c}[r] & M \ar@{ |>->}^-{k}[r]
& Y}}}^{\rule[8.5pt]{0pt}{0pt}}}$ and ${\ensuremath{\smash{\xymatrix@1{Y \ar@{-|>}^-{d}[r] & N \ar@{ |>->}^-{l}[r] & Z}}}^{\rule[8.5pt]{0pt}{0pt}}}$ is the pair $(q^{\dag} {\mathrel{\circ}}c,
l {\mathrel{\circ}}p)$ obtained via the pullback: $$\label{eq:kckcomposition}
\vcenter{\xymatrix{
& P\ar@{ |>->}[r]^-{p}\ar@{ |>->}[d]_{q}{\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}&
N\ar@{ |>->}[d]^{d^{\dag}}\ar@{ |>->}[r]^-{l} & Z \\
X\ar@{-|>}[r]_-{c} & M\ar@{ |>->}[r]_-{k} & Y
}}$$
To be precise, we identity $(c,k)$ with $(\varphi{\mathrel{\circ}}c,
k{\mathrel{\circ}}\varphi^{-1})$, for isomorphisms $\varphi$.
The reader may have noticed that this construction generalises the definition of [[$\mathbf{PInj}$]{}]{}. Indeed, now we can say ${{\ensuremath{\mathbf{PInj}}}\xspace}= {{\ensuremath{\mathbf{Rel}}}\xspace}_{kck}$.
\[KcKThm\] The category ${\ensuremath{\mathbf{D}}}_{kck}$ as described in Definition \[KcKDef\] is again a Boolean dagger kernel category, with a functor $D\colon{\ensuremath{\mathbf{D}}}_{kck}\rightarrow{\ensuremath{\mathbf{D}}}$ that is a morphism of [[$\mathbf{DagKerCat}$]{}]{}, and a change-of-base situation (pullback): $$\[email protected]{
{\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}}_{kck})\ar[d]\ar[rr] & & {\ensuremath{\mathrm{KSub}}}({\ensuremath{\mathbf{D}}})\ar[d] \\
{\ensuremath{\mathbf{D}}}_{kck}\ar[rr]^-{D} & & {\ensuremath{\mathbf{D}}}
}$$
Moreover, in ${\ensuremath{\mathbf{D}}}_{kck}$ one has: $$\mbox{kernel}
\;=\;
\mbox{dagger mono}
\;=\;
\mbox{mono}
\;=\;
\mbox{zero-mono},$$
and ${\ensuremath{\mathbf{D}}}_{kck}$ is universal among such categories.
The obvious definition $(c,k)^{\dag} = (k^{\dag}, c^{\dag})$ yields an involution on ${\ensuremath{\mathbf{D}}}_{kck}$. The zero object $0\in{\ensuremath{\mathbf{D}}}$ is also a zero object $0\in{\ensuremath{\mathbf{D}}}_{kck}$ with zero map ${\ensuremath{\smash{\xymatrix@1@C-2ex{X
\ar@{-|>}[r] & \;0\; \ar@{ |>->}[r] & Y}}}^{\rule[8.5pt]{0pt}{0pt}}}$ consisting of a cokernel-kernel pair. A map $(c,k)$ is a dagger mono if and only if $(c,k)^{\dag} {\mathrel{\circ}}(c,k) = (k^{\dag},k)$ is the identity; this means that $k={\ensuremath{\mathrm{id}_{}}}$.
The kernel of a map $(d,l) = ({\ensuremath{\smash{\xymatrix@1@C-2ex{Y \ar@{-|>}^-{d}[r] &
\;N\; \ar@{ |>->}^-{l}[r] & Z}}}^{\rule[8.5pt]{0pt}{0pt}}})$ in ${\ensuremath{\mathbf{D}}}_{kck}$ is $\ker(d,l) =
({\ensuremath{\smash{\xymatrix@1@C-2ex{N^{\perp} \ar@{-|>}^-{{\ensuremath{\mathrm{id}_{}}}}[r] & \;N^{\perp}\; \ar@{
|>->}^-{(d^{\dag})^{\perp}}[r] & Y}}}^{\rule[8.5pt]{0pt}{0pt}}})$, so that $\ker(d,l)$ is a dagger mono and $(d,l) {\mathrel{\circ}}\ker(d,l) = 0$. If also $(d,l) {\mathrel{\circ}}(c,k) = 0$, then $k {\mathrel{\wedge}}d^{\dag} = 0$ so that by Booleanness, $k
\leq (d^{\dag})^{\perp}$, say via $\varphi \colon M\rightarrow
N^{\perp}$ with $(d^{\dag})^{\perp} {\mathrel{\circ}}\varphi = k$. Then we obtain a mediating map $(c,\varphi) = ({\ensuremath{\smash{\xymatrix@1{X \ar@{-|>}^-{c}[r] &
\;M\; \ar@{ |>->}^-{\varphi}[r] & N^{\perp}}}}^{\rule[8.5pt]{0pt}{0pt}}})$ which satisfies $\ker(d,l) {\mathrel{\circ}}(c,\varphi) = ({\ensuremath{\mathrm{id}_{}}}, (d^{\dag})^{\perp}) {\mathrel{\circ}}(c,\varphi) = (c, (d^{\dag})^{\perp} {\mathrel{\circ}}\varphi) = (c,k)$. It is not hard to see that maps of the form $({\ensuremath{\mathrm{id}_{}}}, m)$ in ${\ensuremath{\mathbf{D}}}_{kck}$ are kernels, namely of the cokernel $(m^{\perp},{\ensuremath{\mathrm{id}_{}}})$.
The intersection of two kernels $({\ensuremath{\mathrm{id}_{}}},m) = ({\ensuremath{\smash{\xymatrix@1@C-2ex{M
\ar@{=}[r] & \;M\; \ar@{ |>->}^-{m}[r] & X}}}^{\rule[8.5pt]{0pt}{0pt}}})$ and $({\ensuremath{\mathrm{id}_{}}},n) =
({\ensuremath{\smash{\xymatrix@1@C-2ex{N \ar@{=}[r] & \;N\; \ar@{ |>->}^-{n}[r] & X}}}^{\rule[8.5pt]{0pt}{0pt}}})$ in ${\ensuremath{\mathbf{D}}}_{kck}$ is the intersection $m{\mathrel{\wedge}}n \colon P
\rightarrowtail X$ in [$\mathbf{D}$]{}, with projections $({\ensuremath{\smash{\xymatrix@1@C-2ex{P
\ar@{=}[r] & \;P\; \ar@{ |>->}[r] & M}}}^{\rule[8.5pt]{0pt}{0pt}}})$ and $({\ensuremath{\smash{\xymatrix@1@C-2ex{P
\ar@{=}[r] & \;P\; \ar@{ |>->}[r] & N}}}^{\rule[8.5pt]{0pt}{0pt}}})$. Hence if the intersection of $({\ensuremath{\mathrm{id}_{}}},m)$ and $({\ensuremath{\mathrm{id}_{}}},n)$ in ${\ensuremath{\mathbf{D}}}_{kck}$ is 0, then so is the intersection of $m$ and $n$ in [$\mathbf{D}$]{}, which yields $n^{\dag}
{\mathrel{\circ}}m = 0$. But then in ${\ensuremath{\mathbf{D}}}_{kck}$, $({\ensuremath{\mathrm{id}_{}}},n)^{\dag} {\mathrel{\circ}}({\ensuremath{\mathrm{id}_{}}},m) = (n^{\dag},{\ensuremath{\mathrm{id}_{}}}) {\mathrel{\circ}}({\ensuremath{\mathrm{id}_{}}},m) = 0$. Hence ${\ensuremath{\mathbf{D}}}_{kck}$ is also Boolean.
Finally, there is a functor ${\ensuremath{\mathbf{D}}}_{kck} \rightarrow {\ensuremath{\mathbf{D}}}$ by $X\mapsto X$ and $(c,k) \mapsto k{\mathrel{\circ}}c$. Composition is preserved by Proposition \[BooleanStrengthenedProp\], since for maps as in Definition \[KcKDef\], $$\begin{array}{rcl}
(d,l) {\mathrel{\circ}}(c,k)
\hspace*{\arraycolsep}=\hspace*{\arraycolsep}
(q^{\dag} {\mathrel{\circ}}c, l {\mathrel{\circ}}p)
& \longmapsto &
l {\mathrel{\circ}}p {\mathrel{\circ}}q^{\dag} {\mathrel{\circ}}c
\hspace*{\arraycolsep}=\hspace*{\arraycolsep}
(l {\mathrel{\circ}}d) {\mathrel{\circ}}(k {\mathrel{\circ}}c).
\end{array}$$
We have already seen that ${\ensuremath{\mathrm{KSub}}}(X)$ in ${\ensuremath{\mathbf{D}}}_{kck}$ is isomorphic to ${\ensuremath{\mathrm{KSub}}}(X)$ in [$\mathbf{D}$]{}. This yields the change-of-base situation.
We have already seen that kernels and dagger monos coincide. We now show that they also coincide with zero-monos. So let $(d,l)\colon Y
\rightarrow Z$ be a zero-mono. This means that $(d,l) {\mathrel{\circ}}(c,k) = 0
\Rightarrow (c,k) = 0$, for each map $(c,k)$. Using diagram (\[eq:kckcomposition\]), this means: $d^{\dag} {\mathrel{\wedge}}k = 0
\Rightarrow k=0$. By Booleanness, the antecedent $d^{\dag} {\mathrel{\wedge}}k =
0$ is equivalent to $k \leq (d^{\dag})^{\perp} = \ker(d)$, which means $d{\mathrel{\circ}}k = 0$. Hence we see that $d$ is zero-monic in [$\mathbf{D}$]{}, and thus an isomorphism (because it is already a cokernel).
Finally, let [$\mathbf{E}$]{} be a Boolean dagger kernel category in which zero-monos are kernels, with a functor $F\colon {\ensuremath{\mathbf{E}}}
\rightarrow {\ensuremath{\mathbf{D}}}$ in [[$\mathbf{DagKerCat}$]{}]{}. Every morphism $f$ in [$\mathbf{E}$]{} factors as $f=i_f {\mathrel{\circ}}e_f$ for a kernel $i_f$ and a cokernel $e_f$. Hence $G\colon {\ensuremath{\mathbf{E}}} \to {\ensuremath{\mathbf{D}}}_{kck}$ defined by $G(X) = F(X)$ and $G(f) =
(e_f,i_f)$ is the unique functor satisfying $F = D {\mathrel{\circ}}G$.
[ **PROOF:** **ENDPROOF**]{}
Ordering homsets {#HomsetOrderSec}
================
This section shows that homsets in dagger kernel categories automatically carry a partial order. However, this does not make the categories order enriched, because the order is not preserved by all morphisms.
\[OrderDef\] Let $f,g\colon X \rightarrow Y$ be parallel morphisms in a dagger kernel category. After factorising them as $f=i_f {\mathrel{\circ}}m_f {\mathrel{\circ}}(i_{f^\dag})^\dag$ and $g=i_g {\mathrel{\circ}}m_g {\mathrel{\circ}}(i_{g^\dag})^\dag$ like in (\[ImageCoimageEqn\]) we can define $f \leq g$ if and only if there are (necessarily unique, dagger monic) $\varphi\colon {\ensuremath{\mathrm{Im}}}(f)
\to {\ensuremath{\mathrm{Im}}}(g)$ and $\psi \colon {\ensuremath{\mathrm{Im}}}(f^{\dag}) \to {\ensuremath{\mathrm{Im}}}(g^{\dag})$, so that in the diagram $$\label{eq:orderdiagram}
\vcenter{\xymatrix@C+2ex@R-3ex{
& {\ensuremath{\mathrm{Im}}}(f^{\dag}) \ar^-{m_f}[r]
& {\ensuremath{\mathrm{Im}}}(f)\ar@{ |>->}^-{i_f}[dr] \ar@{-->}_-{\varphi}[dd]
& \\
X \ar@{-|>}^-{(i_{f^\dag})^{\dag}}[ur]
\ar@{-|>}_-{(i_{g^\dag})^{\dag}}[dr]
&&& Y \\
& {\ensuremath{\mathrm{Im}}}(g^{\dag}) \ar_-{m_g}[r] \ar@{-->}_-{\psi^{\dag}}[uu]
& {\ensuremath{\mathrm{Im}}}(g) \ar@{ |>->}_-{i_g}[ur]
}}$$ one has $$\psi^{\dag} {\mathrel{\circ}}(i_{g^\dag})^{\dag} = (i_{f^\dag})^{\dag}
\quad
\varphi{\mathrel{\circ}}m_f = m_g {\mathrel{\circ}}\psi
\quad
\varphi^{\dag} {\mathrel{\circ}}m_g = m_f {\mathrel{\circ}}\psi^{\dag}
\quad
i_g {\mathrel{\circ}}\varphi = i_f.$$
The relation $\leq$ is a partial order on each homset of a dagger kernel category, with the zero morphism as least element.
Reflexivity is easily established by taking $\varphi={\ensuremath{\mathrm{id}_{}}}$ and $\psi={\ensuremath{\mathrm{id}_{}}}$ in (\[eq:orderdiagram\]). For transitivity, suppose that $f \leq g$ via $\varphi$ and $\psi$, and that $g \leq h$ via $\alpha$ and $\beta$. Then the four conditions in the previous definition are fulfilled by $\alpha {\mathrel{\circ}}\varphi$ and $\psi {\mathrel{\circ}}\beta$, so that $f \leq h$. Finally, for anti-symmetry, suppose that $f \leq g$ via $\varphi$ and $\psi$, and that $g \leq f$ via $\alpha$ and $\beta$. Then $i_{f} {\mathrel{\circ}}\alpha {\mathrel{\circ}}\varphi =
i_{g} {\mathrel{\circ}}\varphi = i_{f}$, so that $\alpha {\mathrel{\circ}}\varphi =
{\ensuremath{\mathrm{id}_{}}}$. Similarly, $\beta {\mathrel{\circ}}\psi = {\ensuremath{\mathrm{id}_{}}}$. By Lemma \[KerLem\], $\alpha$ is a dagger mono so that $\alpha^{\dag}
= \alpha^{\dag} {\mathrel{\circ}}\alpha {\mathrel{\circ}}\varphi = \varphi$. Similarly, $\beta^{\dag} = \psi$, and thus: $$\begin{array}{rcl}
f
\hspace*{\arraycolsep}=\hspace*{\arraycolsep}
i_{f} {\mathrel{\circ}}m_{f} {\mathrel{\circ}}(i_{f^\dag})^{\dag}
\hspace*{\arraycolsep}=\hspace*{\arraycolsep}
i_{f} {\mathrel{\circ}}\alpha {\mathrel{\circ}}\varphi {\mathrel{\circ}}m_{f} {\mathrel{\circ}}(i_{f^\dag})^{\dag}
& = &
i_{g} {\mathrel{\circ}}m_{g} {\mathrel{\circ}}\psi {\mathrel{\circ}}(i_{f^\dag})^{\dag} \\
& = &
i_{g} {\mathrel{\circ}}m_{g} {\mathrel{\circ}}\beta^{\dag} {\mathrel{\circ}}(i_{f^\dag})^{\dag} \\
& = &
i_{g} {\mathrel{\circ}}m_{g} {\mathrel{\circ}}(i_{g^\dag})^{\dag} \\
& = &
g.
\end{array}$$
Finally, for any $f$ we have $0 \leq f$ by taking $\varphi =
\psi = 0$ in (\[eq:orderdiagram\]).
\[OrderPreservationLem\] If $f\leq g$, then:
1. $(k{\mathrel{\circ}}f) \leq (k{\mathrel{\circ}}g)$ for a kernel $k$;
2. $(f{\mathrel{\circ}}c) \leq (g{\mathrel{\circ}}c)$ for a cokernel $c$;
3. $f^{\dag} \leq g^{\dag}$.
The first two points are obvious. The third one then follows because $(m_{f})^{\dag} = m_{f^{\dag}}$ as shown in Section \[ImageCoimageSec\].
\[OrderEx\] We describe the situation in [[$\mathbf{PInj}$]{}]{}, [[$\mathbf{Rel}$]{}]{}and [[$\mathbf{Hilb}$]{}]{}, using the factorisations from Example \[ImageCoimageEx\].
Two parallel maps $\smash{f = (X \stackrel{f_1}{\leftarrowtail} F
\stackrel{f_2}{\rightarrowtail} Y)}$ and $\smash{g = (X
\stackrel{g_1}{\leftarrowtail} G \stackrel{g_2}{\rightarrowtail}
Y)}$ in [[$\mathbf{PInj}$]{}]{}satisfy $f\leq g$ if and only if there are $\varphi,\psi\colon F\rightarrow G$ in: $$\xymatrix@C+2ex@R-4ex{
& F\ar@{=}[r] & F\ar@{ |>->}[dr]^-{f_2}\ar@{-->}[dd]_{\varphi} & \\
X\ar@{-|>}[ur]^-{(f_{1})^{\dag}}\ar@{-|>}[dr]_{(g_{1})^{\dag}} & & & Y \\
& G\ar@{=}[r]\ar@{-->}[uu]_{\psi^{\dag}} & G\ar@{ |>->}[ur]_{g_2}
}$$
This means $\varphi=\psi$ and $g_{i} {\mathrel{\circ}}\varphi =
f_{i}$, for $i=1,2$, so that we obtain the usual order (of one partial injection extending another).
Next, $R\leq S$ for $\smash{R = (X \stackrel{r_1}{\leftarrow} R
\stackrel{r_2}{\rightarrow} Y)}$ and $\smash{S = (X
\stackrel{s_1}{\leftarrow} S \stackrel{s_2}{\rightarrow}
Y)}$ in [[$\mathbf{Rel}$]{}]{}means: $$\xymatrix@C+2ex@R-4ex{
& {\ensuremath{\mathrm{Im}}}(r_{1})\ar[r]^-{R} & {\ensuremath{\mathrm{Im}}}(r_{2})\ar@{ |>->}[dr]\ar@{-->}[dd]_{\varphi} & \\
X\ar@{-|>}[ur]\ar@{-|>}[dr] & & & Y \\
& {\ensuremath{\mathrm{Im}}}(s_{1})\ar[r]_-{S}\ar@{-->}[uu]_{\psi^{\dag}} & {\ensuremath{\mathrm{Im}}}(s_{1})\ar@{ |>->}[ur]
}$$
Commutation of the triangles means ${\ensuremath{\mathrm{Im}}}(r_{1})\subseteq{\ensuremath{\mathrm{Im}}}(s_{1})$ and ${\ensuremath{\mathrm{Im}}}(r_{2})\subseteq{\ensuremath{\mathrm{Im}}}(s_{2})$. The equations for the square in the middle say that: $$R(x,y) \Leftrightarrow S(x,y)
\quad\mbox{for all}\quad
\left\{\begin{array}{l}
(x,y)\in{\ensuremath{\mathrm{Im}}}(r_{1})\times{\ensuremath{\mathrm{Im}}}(s_{2}) \\
(x,y)\in{\ensuremath{\mathrm{Im}}}(r_{2})\times{\ensuremath{\mathrm{Im}}}(s_{1}).
\end{array}\right.$$
This means $R\subseteq S$, as one would expect.
The order on the homsets of the category ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ can be characterized as follows [@Heunen10a Example 5.1.10]: $f \leq g$ for $f,g \colon X \to Y$ if and only if $g = f+f'$ for some $f' \colon X \to Y$ with ${\ensuremath{\mathrm{Im}}}(f)$ and ${\ensuremath{\mathrm{Im}}}(f^\dag)$ orthogonal to ${\ensuremath{\mathrm{Im}}}(f')$ and ${\ensuremath{\mathrm{Im}}}((f')^\dag)$, respectively. To see this, suppose that $g=f+f'$ as above. Then ${\ensuremath{\mathrm{Im}}}(g)$ is the direct sum of ${\ensuremath{\mathrm{Im}}}(f)$ and ${\ensuremath{\mathrm{Im}}}(f')$, and likewise ${\ensuremath{\mathrm{Im}}}(g^\dag) =
{\ensuremath{\mathrm{Im}}}(f^\dag) \oplus {\ensuremath{\mathrm{Im}}}((f')^\dag$. Moreover, $m_g$ is the direct sum of $m_f$ and $m_{f'}$. Therefore, taking $\psi =
\varphi = \kappa_1$ makes diagram commute, so that $f \leq g$. Conversely, suppose that $f \leq g$, so that diagram commutes. Then the cotuple $[\varphi,
\varphi^\perp] \colon {\ensuremath{\mathrm{Im}}}(f) \oplus {\ensuremath{\mathrm{Im}}}(f)^\perp \to {\ensuremath{\mathrm{Im}}}(g)$ is an isomorphism, and so is the cotuple $[\psi, \psi^\perp]$. Since $\varphi^\dag {\mathrel{\circ}}m_g = m_f {\mathrel{\circ}}\psi^\dag$, there is a morphism $n$ making the following diagram commute: $$\xymatrix@R-2ex{
{\ensuremath{\mathrm{Im}}}(f^\dag)^\perp \ar@{ |>->}_-{\ker(\psi^\dag)=\psi^\perp}[d]
\ar@{-->}^-{n}[rr]
&& {\ensuremath{\mathrm{Im}}}(f)^\perp \ar@{ |>->}^{\varphi^\perp = \ker(\varphi^\dag)}[d] \\
{\ensuremath{\mathrm{Im}}}(g^\dag) \ar^-{m_g}[rr] \ar@{-|>}_-{\psi^\dag}[d]
&& {\ensuremath{\mathrm{Im}}}(g) \ar@{-|>}^{\varphi^\dag}[d] \\
{\ensuremath{\mathrm{Im}}}(f^\dag) \ar_-{m_f}[rr]
&& {\ensuremath{\mathrm{Im}}}(f).
}$$ Now, taking $$\xymatrix{
f' = \Big( X \ar^-{(i_{g^\dag})^\dag}[r]
& {\ensuremath{\mathrm{Im}}}(g^\dag) \ar^-{(\psi^\perp)^\dag}[r]
& {\ensuremath{\mathrm{Im}}}(f^\dag)^\perp \ar^-{n}[r]
& {\ensuremath{\mathrm{Im}}}(f)^\perp \ar^-{\varphi^\perp}[r]
& {\ensuremath{\mathrm{Im}}}(g) \ar^-{i_g}[r]
& Y \Big)
}$$ fulfills $g=f+f'$, and ${\ensuremath{\mathrm{Im}}}(f)$ and ${\ensuremath{\mathrm{Im}}}(f^\dag)$ are orthogonal to ${\ensuremath{\mathrm{Im}}}(f')$ and ${\ensuremath{\mathrm{Im}}}((f')^\dag)$, respectively.
In Hilbert spaces there is a standard correspondence between self-adjoint idempotents and closed subsets. Recall that an endomap $p\colon X\rightarrow X$ is self-adjoint if $p^{\dag} = p$ and idempotent if $p {\mathrel{\circ}}p = p$. In the current, more general, setting this works as follows, using the order on homsets.
\[KernelEndomapProp\] The “effect”[^2] mapping $m\mapsto {\mathfrak{E}(m)}
\,\smash{\stackrel{\textrm{def}}{=}}\, m {\mathrel{\circ}}m^{\dag}$ from Section \[LogicSec\] yields an order isomorphism: $$\begin{array}{rcl}
{\ensuremath{\mathrm{KSub}}}(X)
& \cong &
{\{p\colon X\rightarrow X\;|\;p^{\dag} = p\leq {\ensuremath{\mathrm{id}_{}}}\}} \\
& \cong &
{\{p\colon X\rightarrow X\;|\;p^{\dag} = p{\mathrel{\circ}}p = p\leq {\ensuremath{\mathrm{id}_{}}}\}} \\
& \smash{\stackrel{(*)}{\cong}} &
{\{p\colon X\rightarrow X\;|\;p^{\dag} = p{\mathrel{\circ}}p = p\}},
\end{array}$$
where the marked isomorphism holds if zero-epis are epis (like in [[$\mathbf{Hilb}$]{}]{}).
Clearly, ${\mathfrak{E}(m)} = m {\mathrel{\circ}}m^{\dag}$ is a self-adjoint idempotent. It satisfies ${\mathfrak{E}(m)} \leq {\ensuremath{\mathrm{id}_{}}}$ via: $$\xymatrix@C+2ex@R-4ex{
& M\ar@{=}[r] & M\ar@{ |>->}[dr]^-{m}\ar@{-->}[dd]_{m} & \\
X\ar@{-|>}[ur]^-{m^{\dag}}\ar@{=}[dr] & & & X \\
& X\ar@{=}[r]\ar@{-->}[uu]_{m^{\dag}} & X\ar@{=}[ur]
}$$
where the kernel $m\colon M\rightarrowtail X$ is a dagger mono so that ${\ensuremath{\mathrm{Im}}}({\mathfrak{E}(m)}) = M$.
This mapping ${\mathfrak{E}(-)}\colon {\ensuremath{\mathrm{KSub}}}(X) \rightarrow {\{p\;|\;p^{\dag} =
p \leq {\ensuremath{\mathrm{id}_{}}}\}}$ is surjective: if $p\colon X\rightarrow X$ is a self-adjoint with $p\leq{\ensuremath{\mathrm{id}_{}}}$ then we first note that the factorisation from (\[ImageCoimageEqn\]) yields $p = i_{p} {\mathrel{\circ}}m_{p} {\mathrel{\circ}}(i_{p})^{\dag}$. By Definition \[OrderDef\] there are $\varphi,\psi\colon {\ensuremath{\mathrm{Im}}}(p)\rightarrow X$ with $\psi^{\dag} =
(i_{p})^{\dag}$, $\varphi {\mathrel{\circ}}m_{p} = \psi$, $\varphi^{\dag} =
m_{p} {\mathrel{\circ}}\psi^{\dag}$ and $\varphi = i_{p}$. This yields $\psi=i_{p}$ and $m_{p}={\ensuremath{\mathrm{id}_{}}}$. Hence $p = i_{p} {\mathrel{\circ}}(i_{p})^{\dag} = {\mathfrak{E}(i_{p})}$, so that $p$ is automatically idempotent. This establishes the second isomorphism.
The mapping ${\mathfrak{E}(-)}$ preserves and reflects the order. If $m\leq n$ in ${\ensuremath{\mathrm{KSub}}}(X)$, say via $\varphi\colon M\rightarrow N$ with $n{\mathrel{\circ}}\varphi
= m$, then ${\mathfrak{E}(m)} \leq {\mathfrak{E}(n)}$ via: $$\xymatrix@C+2ex@R-4ex{
& M\ar@{=}[r] & M\ar@{ |>->}[dr]^-{m}\ar@{-->}[dd]_{\varphi} & \\
X\ar@{-|>}[ur]^-{m^{\dag}}\ar@{-|>}[dr]_{n^{\dag}} & & & X \\
& N\ar@{=}[r]\ar@{-->}[uu]_{\varphi^{\dag}} & N\ar@{ |>->}[ur]_{n}
}$$
Conversely, if ${\mathfrak{E}(m)} \leq {\mathfrak{E}(n)}$, say via $\varphi\colon
M\rightarrow N$ and $\psi\colon M\rightarrow N$, then $n{\mathrel{\circ}}\varphi = m$ so that $m\leq n$ in ${\ensuremath{\mathrm{KSub}}}(X)$.
Finally, if zero-epis are epis, we write for a self-adjoint idempotent $p$, $$i_{p} {\mathrel{\circ}}e_{p}
=
p
=
p {\mathrel{\circ}}p
=
p^{\dag} {\mathrel{\circ}}p
=
(e_{p})^{\dag} {\mathrel{\circ}}(i_{p})^{\dag} {\mathrel{\circ}}i_{p} {\mathrel{\circ}}e_{p}
=
(e_{p})^{\dag} {\mathrel{\circ}}e_{p},$$
and obtain $i_{p} = (e_{p})^{\dag}$. Hence $p = {\mathfrak{E}(i_{p})}$ and thus $p\leq {\ensuremath{\mathrm{id}_{}}}$.
Completeness and atomicity of kernel posets {#ComplAtomSec}
===========================================
In traditional quantum logic, orthomodular lattices are usually considered with additional properties, such as completeness and atomicity [@Piron76]. This section considers how these requirements on the lattices ${\ensuremath{\mathrm{KSub}}}(X)$ translate to categorical properties. For convenience, let us recall the following standard order-theoretical definitions.completeness
For elements $x,y$ of a poset, we say that $y$ covers $x$ when $x<y$ and $x \leq z < y$ implies $z=x$ (where $z<y$ if and only if $z \leq y$ and $z \neq y$). An element $a$ of a poset with least element 0 is called an *atom* when it covers 0. Equivalently, an atom cannot be expressed as a join of strictly smaller elements. Consequently, $0$ is not an atom. A poset is called *atomic* if for any $x \neq 0$ in it there exists an atom $a$ with $a \leq x$. Finally, a lattice is *atomistic* when every element is a join of atoms [@DaveyP90].
\[prop:atoms\] For an arbitrary object $I$ in a dagger kernel category, the following are equivalent:
1. ${\ensuremath{\mathrm{id}_{I}}}=1$ is an atom in ${\ensuremath{\mathrm{KSub}}}(I)$;
2. ${\ensuremath{\mathrm{KSub}}}(I)=\{0,1\}$;
3. each nonzero kernel $x\colon I \rightarrowtail X$ is an atom in ${\ensuremath{\mathrm{KSub}}}(X)$.
For the implication $(1) \Rightarrow (2)$, let $m$ be a kernel into $I$. Because $m \leq {\ensuremath{\mathrm{id}_{I}}}$ and the latter is an atom, we have that $m=0$ or $m$ is isomorphism. Thus ${\ensuremath{\mathrm{KSub}}}(I)=\{0,1\}$.
To prove $(2) \Rightarrow (3)$, suppose that $m \leq x$ for kernels $m\colon M \rightarrowtail X$ and $x\colon I \rightarrowtail X$. Say $m = x {\mathrel{\circ}}\varphi$ for $\varphi\colon M \rightarrowtail I$. Then $\varphi$ is a kernel by Lemma \[KerLem\]. Since ${\ensuremath{\mathrm{KSub}}}(I)=\{0,1\}$, either $\varphi$ is zero or $\varphi$ is isomorphism. Hence either $m=0$ or $m=x$ as subobjects. So $x$ is an atom. Finally, $(3) \Rightarrow (1)$ is trivial.
\[def:KSubsimple\] If $I$ satisfies the conditions of the previous lemma, we call it a *${\ensuremath{\mathrm{KSub}}}$-simple* object. (Any simple object in the usual sense of category theory is ${\ensuremath{\mathrm{KSub}}}$-simple.)
Similarly, let us call $I$ a *${\ensuremath{\mathrm{KSub}}}$-generator* if $f=g\colon X \to Y$ whenever $f {\mathrel{\circ}}x = g {\mathrel{\circ}}x$ for all kernels $x\colon I \rightarrowtail X$. (Any ${\ensuremath{\mathrm{KSub}}}$-generator is a generator in the usual sense of category theory.)
\[GeneratorEx\] The objects $1 \in {{\ensuremath{\mathbf{PInj}}}\xspace}$, $1 \in {{\ensuremath{\mathbf{Rel}}}\xspace}$, ${\ensuremath{\mathbb{C}}} \in {{\ensuremath{\mathbf{Hilb}}}\xspace}$ and ${\ensuremath{\mathbb{C}}} \in {\ensuremath{\mathbf{PHilb}}}$ are ${\ensuremath{\mathrm{KSub}}}$-simple ${\ensuremath{\mathrm{KSub}}}$-generators.
The two-element orthomodular lattice $2$ is a generator in the category [$\mathbf{OMLatGal}$]{} from [@Jacobs09a], because maps $2\rightarrow X$ correspond to elements in $X$. But $2$ is not a ${\ensuremath{\mathrm{KSub}}}$-generator: these maps $2\rightarrow X$ are not kernels.
Because $1 \in {\ensuremath{\mathbf{Rel}}}$ is a ${\ensuremath{\mathrm{KSub}}}$-simple ${\ensuremath{\mathrm{KSub}}}$-generator, one might expect a connection between Definition \[def:KSubsimple\] and the “kernel opclassifiers” discussed at the end of Section \[FactorisationSec\]. There is, however, no apparent such connection. For example, the object $1$ in the category ${\ensuremath{\mathbf{PInj}}}$ is a ${\ensuremath{\mathrm{KSub}}}$-simple ${\ensuremath{\mathrm{KSub}}}$-generator, but not a “kernel opclassifier”.
\[lem:atomic\] Suppose that a dagger kernel category ${\ensuremath{\mathbf{D}}}$ has a ${\ensuremath{\mathrm{KSub}}}$-simple ${\ensuremath{\mathrm{KSub}}}$-generator $I$. Then beneath any nonzero element of ${\ensuremath{\mathrm{KSub}}}(X)$ lies a nonzero element of the form $x\colon I \rightarrowtail
X$. Hence ${\ensuremath{\mathrm{KSub}}}(X)$ is atomic, and its atoms are the nonzero kernels $x \colon I \rightarrowtail X$.
Suppose $m\colon M\rightarrowtail X$ is a nonzero kernel. Since $I$ is a ${\ensuremath{\mathrm{KSub}}}$-generator, there must be a kernel $x\colon I\rightarrowtail
M$ with $m{\mathrel{\circ}}x \neq 0$. By Proposition \[prop:atoms\] this $m{\mathrel{\circ}}x$ is an atom. It satisfies $m{\mathrel{\circ}}x \leq m$, so we are done.
\[lem:atomistic\] If a dagger kernel category has a ${\ensuremath{\mathrm{KSub}}}$-simple ${\ensuremath{\mathrm{KSub}}}$-generator $I$, then ${\ensuremath{\mathrm{KSub}}}(X)$ is atomistic for any object $X$.
Any atomic orthomodular lattice is atomistic [@Kalmbach83].
The categorical requirement of a simple generator is quite natural in this setting, as it is also used to prove that a certain class of dagger kernel categories embeds into ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ [@Heunen09b].
We now turn to completeness, by showing that the existence of directed colimits ensures that kernel subobject lattices are complete. This, too, is a natural categorical requirement in the context of infinite-dimensionality [@Heunen08a]. Recall that a *directed colimit* is a colimit of a directed poset, considered as a diagram. The following result can be obtained abstractly in two steps: directed colimits in [$\mathbf{D}$]{} yield direct colimits in slice categories ${\ensuremath{\mathbf{D}}}/X$, see [@Borceux94 Vol. 2, Prop. 2.16.3]. The reflection ${\ensuremath{\mathrm{KSub}}}(X) \hookrightarrow
{\ensuremath{\mathbf{D}}}/X$ induced by factorisation transfers these directed colimits to ${\ensuremath{\mathrm{KSub}}}(X)$. However, in the proof below we give a concrete construction.
If a dagger kernel category ${\ensuremath{\mathbf{D}}}$ has directed colimits, then ${\ensuremath{\mathrm{KSub}}}(X)$ is a complete lattice for every $X \in {\ensuremath{\mathbf{D}}}$.
A lattice is complete if it has directed joins (see [@Johnstone82 Lemma I.4.1], or [@Kalmbach86 Lemma 2.12]), so we shall prove that ${\ensuremath{\mathrm{KSub}}}(X)$ has such directed joins. Let $(m_{i}\colon M_{i}\rightarrowtail
X)_{i\in I}$ be a directed collection in ${\ensuremath{\mathrm{KSub}}}(X)$. For $i\leq j$ we have $m_{i} \leq m_{j}$ and thus $m_{j} {\mathrel{\circ}}m_{j}^{\dag} {\mathrel{\circ}}m_{i} = m_{i}$.
Let $M$ be the colimit in [$\mathbf{D}$]{} of the domains $M_i$, say with coprojections $c_{i}\colon M_{i}\rightarrow M$. The $(m_{i}\colon
M_{i}\rightarrowtail X)_{i\in I}$ form a cocone by assumption, so there is a unique map $m\colon M\rightarrow X$ with $m{\mathrel{\circ}}c_{i} =
m_{i}$. The kernel/zero-epi factorisation (\[ImageEqn\]) yields: $$\xymatrix{
m = \big(M\ar@{->>}[r]|-{\circ}^-{e} & N\ar@{ |>->}[r]^-{n} & X\Big)
}$$
We claim that $n$ is the join in ${\ensuremath{\mathrm{KSub}}}(X)$ of the $m_i$.
- $m_{i} \leq n$ via $e{\mathrel{\circ}}c_{i}\colon M_{i} \rightarrow N$ satisfying $n {\mathrel{\circ}}(e{\mathrel{\circ}}c_{i}) = m {\mathrel{\circ}}c_{i} = m_{i}$.
- If $m_{i} \leq k$, then $k{\mathrel{\circ}}k^{\dag} {\mathrel{\circ}}m_{i} = m_{i}$. Also, the maps $k_{i} = k^{\dag} {\mathrel{\circ}}m_{i} \colon M_{i}\rightarrow K$ form a cocone in [$\mathbf{D}$]{} because the $m_i$ are directed and $k$ is monic: if $i\leq j$, then, $$k{\mathrel{\circ}}k_{j} {\mathrel{\circ}}m_{j}^{\dag} {\mathrel{\circ}}m_{i}
=
k {\mathrel{\circ}}k^{\dag} {\mathrel{\circ}}m_{j} {\mathrel{\circ}}m_{j}^{\dag} {\mathrel{\circ}}m_{i}
=
k {\mathrel{\circ}}k^{\dag} {\mathrel{\circ}}m_{i}
=
k{\mathrel{\circ}}k_{i}.$$
As a result there is a unique $\ell\colon M\rightarrow K$ with $\ell {\mathrel{\circ}}c_{i} = k_{i}$. Then $k {\mathrel{\circ}}\ell = m$ by uniqueness since: $$k{\mathrel{\circ}}\ell {\mathrel{\circ}}c_{i}
=
k{\mathrel{\circ}}k_{i}
=
k {\mathrel{\circ}}k^{\dag} {\mathrel{\circ}}m_{i}
=
m_{i}
=
m {\mathrel{\circ}}c_{i}.$$
Hence we obtain $n\leq k$ by diagonal-fill-in from Lemma \[DiagonalFillInLem\] in: $$\vcenter{\xymatrix{
M\ar@{->>}[r]|-{o}^-{e}\ar[d]_{\ell} &
N\ar@{ |>->}[d]^{n}\ar@{..>}[dl] \\
K\ar@{ |>->}[r]_-{k} & X
}}$$
The categories ${{\ensuremath{\mathbf{PInj}}}\xspace}$, ${{\ensuremath{\mathbf{Rel}}}\xspace}$, ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ and ${\ensuremath{\mathbf{PHilb}}}$ have directed colimits, and therefore their kernel subobject lattices are complete orthomodular lattices. Since they also have appropriate generators, see Example \[GeneratorEx\], each ${\ensuremath{\mathrm{KSub}}}(X)$ in ${{\ensuremath{\mathbf{PInj}}}\xspace}$, ${{\ensuremath{\mathbf{Rel}}}\xspace}$, ${{\ensuremath{\mathbf{Hilb}}}\xspace}$ or ${\ensuremath{\mathbf{PHilb}}}$ is a complete atomic atomistic orthomodular lattice.
Any atom of a Boolean algebra $B$ is a ${\ensuremath{\mathrm{KSub}}}$-simple object in the dagger kernel category $\widehat{B}$ from Proposition \[BAConstrProp\]. But $\widehat{B}$ has a ${\ensuremath{\mathrm{KSub}}}$-generator only if $B$ is atomistic. In that case the greatest element 1 is a ${\ensuremath{\mathrm{KSub}}}$-generator. For if $f {\mathrel{\circ}}a = g {\mathrel{\circ}}a$ for all $a \leq 1 {\mathrel{\wedge}}x = x$ and $f,g \leq x {\mathrel{\wedge}}y$, then, writing ${\mathop{\downarrow}}_{A}x =
{\{a\in\mathrm{Atoms}(B)\;|\;a\leq x\}}$ we get: $$\begin{aligned}
\textstyle f
=
f {\mathrel{\wedge}}x
=
f {\mathrel{\wedge}}\big(\bigvee{\mathop{\downarrow}}_{A}x\big)
& =
\textstyle\bigvee{\{f{\mathrel{\wedge}}a\;|\;a\in\mathrm{Atoms}(B), a\leq x\}} \\
& =
\textstyle\bigvee{\{g{\mathrel{\wedge}}a\;|\;a\in\mathrm{Atoms}(B), a\leq x\}} \\
& =
\textstyle g {\mathrel{\wedge}}\big(\bigvee{\mathop{\downarrow}}_{A}x\big)
=
g {\mathrel{\wedge}}x
=
g.\end{aligned}$$
Conclusions and future work {#ConclSec}
===========================
The paper shows that a “dagger kernel category” forms a simple but powerful notion that not only captures many examples of interest in quantum logic but also provides basic structure for categorical logic. There are many avenues for extension and broadening of this work, by including more examples (*e.g.* effect algebras [@DvurecenskijP00]) or more structure (like tensors). Also, integrating probabilistic aspects of quantum logic is a challenge.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Thanks are due to an anonymous referee for pointing out the reference [@Crown75]. It already contains several early ideas that are rediscovered and elaborated in the current (categorical) setting. The category [$\mathbf{OMLatGal}$]{} that plays a central role in [@Jacobs09a] is also already in [@Crown75].
[^1]: The “and-then” connective ${\mathrel{\&}}$ should not be confused with the multiplication of a quantale [@Rosenthal90], since the latter is always associative.
[^2]: The name “effect” was chosen because of connections to effect algebras [@DvurecenskijP00]. For example, in the so-called standard effect algebra of a Hilbert space [@FoulisGB94], an effect corresponds a positive operator beneath the identity.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $A$ be any af[f]{}ine surject[i]{}ve endomorphism of a solenoid [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} over the circle $S^1$ which is not an inf[i]{}nite-order translat[i]{}on of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}. We prove the existence of a cylinder absolute winning ([CAW]{}) subset $F \subset {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$ with the property that for any $x \in F$, the orbit closure $\overline{\{ A^{\ell} x \mid \ell \in {\ensuremath{\mathbb{N}}}\}}$ does not contain any periodic orbits. The class of inf[i]{}nite solenoids considered in this paper provides, to our knowledge, some of the f[i]{}rst examples of non-Federer spaces where absolute games can be played and won. Dimension maximality and incompressibility of [CAW]{} sets is also discussed for a number of possibilit[i]{}es in addit[i]{}on to their winning nature for the games known from before.'
author:
- 'L. Singhal'
title: Cylinder absolute games on solenoids
---
[ oldtitletitle title[oldtitle]{}]{} [ @oldtitletitle title[@oldtitle]{}]{}
Introduction {#S:intro}
============
Let ¶ be an (f[i]{}nite or inf[i]{}nite) ordered set of prime numbers with $p_1 < p_2 < \cdots$ and $X_{{\ensuremath{\mathcal{P}}}, n}$ be the restricted product space $$\label{E:spaceX}
{\ensuremath{\mathbb{R}}}^n \times \prod{}^{\prime}_{p \in {\ensuremath{\mathcal{P}}}}\, {\ensuremath{\mathbb{Q}}}_{p}^n,$$ where $\prod'$ denotes that for each element $x \in X_{{\ensuremath{\mathcal{P}}}, n}$, the entries $x_p \in {\ensuremath{\mathbb{Z}}}^n_p$ for all but f[i]{}nitely many $p$’s. A *¶-solenoid* of topological dimension $n$ is the quo[t]{}[i]{}ent space ${\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}(n) := \nicefrac{X_{{\ensuremath{\mathcal{P}}}, n}}{\Delta ( R^n )}$, where $R$ is the ring ${\ensuremath{\mathbb{Z}}}[ \{ p^{-1} \mid p \in {\ensuremath{\mathcal{P}}}\} ]$ whose $n$-fold product is embedded diagonally in $X_{{\ensuremath{\mathcal{P}}}, n}$ as a uniform la[t]{}[t]{}[i]{}ce. We call the quo[t]{}[i]{}ent map $X_{{\ensuremath{\mathcal{P}}}, n} \rightarrow {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}(n)$ to be $\Pi$. Solenoids are compact, connected metrizable abelian groups. They have somet[i]{}mes been called “fractal versions of tori” [@Sem12c]. When ¶ is a f[i]{}nite set of cardinality $l - 1$, the Hausdorf[f]{} dimension of $X_{{\ensuremath{\mathcal{P}}}, n}$ (and therefore of ${\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}(n)$ too) under the natural metric given by is $nl$. This also implies that the dimension is inf[i]{}nite when ¶ is so, as the increasing sequence of f[i]{}nite products associated with the f[i]{}nite truncat[i]{}ons of ¶ are isometrically embedded inside $X_{{\ensuremath{\mathcal{P}}}, n}$.\
The set of endomorphisms of ${\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}:= {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}(1)$ is precisely the ring $R$ whose elements act mul[t]{}[i]{}plica[t]{}[i]{}vely componentwise. An *af[f]{}ine transformat[i]{}on* $A : {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}\rightarrow {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$ is meant to denote the map $$\label{E:affine}
\mathbf{x} \mapsto ( m / n ) \mathbf{x} + \mathbf{a}$$ where $m / n \in R \setminus \{ 0 \}$ and $\mathbf{a} \in {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$. It is well known that when $A = m /n$ is a surject[i]{}ve endomorphism of the solenoid, it acts ergodically on [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} if[f]{} $m/n \notin \{ 0, \pm 1 \}$ [@Wil76 Proposit[i]{}on 1.4]. We also learn from @Ber85 [@Ber85 Theorem 3.2] that for every compact group $G$, any semigroup of its af[f]{}ine transformat[i]{}ons lying above an ergodic semigroup of surject[i]{}ve endomorphisms is ergodic as well. This gives us a suf[f]{}icient condit[i]{}on for the transformat[i]{}on $A \mathbf{x} = ( m / n ){\ensuremath{\mathbf{x}}}+ \mathbf{a}$ to be ergodic, namely that $m/n \neq \pm 1$. Ergodicity of the act[i]{}on guarantees that almost all orbits of $A$ are dense in [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}. However, just like @Dan88 [@Dan88], this work is concerned with understanding the complementary set. Given an af[f]{}ine transformat[i]{}on $A$, we will like to know the set of points of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} whose $A$-orbits remain away from periodic $B$-orbits for all $B \in R \setminus \{ \pm 1 \}$.\
When ¶ is the set consist[i]{}ng of all the primes in [$\mathbb{N}$]{}, the space [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} is called the *full solenoid* over $S^1$ with the f[i]{}eld [$\mathbb{Q}$]{} being its ring of endomorphisms. Let $B$ be a non-zero rat[i]{}onal number. The growth of the number of $B$-periodic orbits as a funct[i]{}on of the period is determined by the entropy of the act[i]{}on on [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}. Lat[t]{}er has been computed by @Yuz [@Yuz] f[i]{}rst and recovered by @LW88 in [@LW88] who explained it to be the sum of the Euclidean and the $p$-adic contribut[i]{}ons. In fact, they achieve it for all automorphisms of solenoids over higher-dimensional tori as well. We remark that each such epimorphism li[f]{}[t]{}s uniquely to a homomorphism from ${\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}:= X_{{\ensuremath{\mathcal{P}}}, 1}$ to itself, which we shall con[t]{}[i]{}nue to denote by the same ra[t]{}[i]{}onal number. For an af[f]{}ine transformat[i]{}on, we however have a choice involved in terms of a representat[i]{}ve for the translat[i]{}on part $\mathbf{a}$.\
Let $y \in {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$ be arbitrary and $A$ be a (surject[i]{}ve) af[f]{}ine transformat[i]{}on of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} as in with either $m/n \neq 1$ or $\mathbf{a} = \mathbf{0}$. We intend to show that the set of points $\mathbf{x} \in {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ whose forward orbit under the map $\mathbf{x} \mapsto A\mathbf{x}$ maintains some posi[t]{}[i]{}ve distance from the $1$-uniformly discrete subset $\Pi^{-1} ( \{ y \} ) \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ is *cylinder absolute winning* ([CAW]{}) in a similar sense as @FSU15 [@FSU15]. Once we have this, we can take intersect[i]{}on of countably many of these sets to conclude about $A$-orbits which avoid neighbourhoods of all periodic orbits of surject[i]{}ve endomorphisms. This strategy is in same taste and builds upon the work of @Dan88 [@Dan88] on orbits of semisimple toral automorphisms.\
Our setup has two players in which one of them (Alice) will be blocking open cylinder subsets of [$X_{{\ensuremath{\mathcal{P}}}}$]{} at every stage of a two-player game. To elaborate, one such cylinder is given by $$\label{E:cyl}
C ( \mathbf{x}, {\ensuremath{\varepsilon}}, i ) := \begin{cases}
{\ensuremath{\mathbb{R}}}\times \prod_{j < i} {\ensuremath{\mathbb{Q}}}_{p_j} \times B ( x_i, p_i {\ensuremath{\varepsilon}}) \times \prod{}^{\prime}_{j > i} {\ensuremath{\mathbb{Q}}}_{p_{j}} &\textrm{ if } i > 0 \textrm{ and}\\
B ( x_0, {\ensuremath{\varepsilon}}) \times \prod^{\prime}_{j > 0} {\ensuremath{\mathbb{Q}}}_{p_j} &\textrm{ otherwise,}
\end{cases}$$ where $\mathbf{x} = ( x_0, \ldots, x_i, \ldots ) \in {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$. For us, $B ( x_i, r )$ will be always be the set of points in ${\ensuremath{\mathbb{Q}}}_{p_i}$ whose distance from $x_i$ is strictly less than $r$ while $\overline{B} ( x_i, r )$ will also include those whose distance from $x_i$ is exactly $r$. We explain this game in §\[S:games\] a[f]{}[t]{}er a brief tour of some of its older and related versions. Our aim is to prove the following statement in this paper:
\[Th:main\] Let [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} be a solenoid over the circle $S^1$ and $$\{ A_j : {\ensuremath{\mathbf{x}}}\mapsto ( m_j / n_j ) {\ensuremath{\mathbf{x}}}+ {\ensuremath{\mathbf{a}}}_j \mid j \in {\ensuremath{\mathbb{N}}}\}$$ be any subset of af[f]{}ine surject[i]{}ve endomorphisms of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} such that
1. none of the $A_j$’s is a non-trivial translat[i]{}on of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}, and
2. the collect[i]{}on of rat[i]{}onal numbers $\{ m_j / n_j \}_{j \in {\ensuremath{\mathbb{N}}}}$ lying below the family $\{ A_j \}$ belong to some f[i]{}nite ring extension of [$\mathbb{Z}$]{}.
Then, there exists a cylinder absolute winning subset $F \subset {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$ such that for any $x \in F$ and $j \in {\ensuremath{\mathbb{N}}}$, the orbit closure $\overline{ \{ A^{k}_{j} x \mid k \in {\ensuremath{\mathbb{N}}}\} }$ contains no periodic $B$-orbit for all $B \in R \setminus \{ \pm 1 \}$.
[This is done in §\[S:morph\]. In the last sect[i]{}on, we illustrate how this informat[i]{}on can be used to infer something about the $f$-dimensional Hausdorf[f]{} measure of $F$. We also discuss the strong game winning and incompressible nature of [CAW]{} subsets when ¶ is f[i]{}nite.]{}\
{#SS:Weil}
It is plausible that some of our results given here may also be obtained from the more general framework discussed by @Wei13 [@Wei13]. We take some t[i]{}me to expound the import of his main result.\
Let $( \overline{X}, d )$ be a proper metric space (i. e. all closed balls are compact) and $X$ a closed subset of $\overline{X}$. Consider a family of subsets $\{ R_{\lambda} \subset \overline{X} \mid \lambda \in \Lambda \}$ which are called *resonant sets* and a family of contract[i]{}ons $\{ \psi_{\lambda} \mid\ ] 0, 1 ] \rightarrow 2^{\overline{X}} \mid \lambda \in \Lambda \}$, indexed by some (same) countable set $\Lambda$. It is required that $R_{\lambda} \subset \psi_{\lambda} ( t_1 ) \subset \psi (t_2)$ for all $0 < t_1 < t_2$ and $\lambda$. This datae is writ[t]{}en in a concise form as $\mathcal{F} = ( \Lambda, R_{\lambda}, \psi_{\lambda} )$. The set of *badly approximable points in $S$* with respect to the family $\mathcal{F}$ is def[i]{}ned as $$\label{E:BAF}
{\ensuremath{\mathrm{BA}}}_X (\mathcal{F}) := \big\{ x \in X \mid \exists c = c(x) > 0 \textrm{ such that } x \notin \bigcup_{\lambda \in \Lambda} \psi_{\lambda} (c) \big\}.$$ Next, each $R_{\lambda}$ is assigned a *height* $h_{\lambda}$ with $\inf_{\lambda} h_{\lambda} > 0$. The standard contract[i]{}on $\psi_{\lambda}$ is then determined as $\psi_{\lambda} (c) := \mathcal{N}_{c / h_{\lambda}} (R_{\lambda})$, where $\mathcal{N}_{{\ensuremath{\varepsilon}}} (S)$ denotes the set of all points of $\overline{X}$ in the ${\ensuremath{\varepsilon}}$-vicinity of elements of $S$. We further assume that our resonant sets $R_{\lambda}$ are *nested* with respect to the height funct[i]{}on $h$, i. e., $R_{\lambda} \subseteq R_{\beta}$ for every $\lambda, \beta \in \Lambda$ such that $h_{\lambda} \leq h_{\beta}$ and that the values taken by $h$ form a discrete subset of $]\,0,\,\infty\,[$. For any collect[i]{}on $\mathcal{S}$ of subsets of $\overline{X}$, the set $X$ is said to be *$b_*$-di[f]{}fuse with respect to $\mathcal{S}$* for some $0 < b_* < 1$ if there exists some $r_0 > 0$ such that for all balls $\overline{B} ( x, r ),\ x \in X,\ 0 < r < r_0$ and $S \in \mathcal{S}$, there exists a sub-ball $$\label{E:diffuse}
\overline{B} ( y, b_* r ) \subset \overline{B} ( x, r ) \setminus \mathcal{N}_{b_* r} (S) \textrm{ with } y \in X.$$ The family $\mathcal{F}$ is *locally contained in $\mathcal{S}$* if for any $\overline{B} ( x, r ),\ x \in X$ with $r < r_0$ and $\lambda \in \Lambda$ with $h_{\lambda} \leq 1/r$, there is an $S \in \mathcal{S}$ such that $B \cap R_{\lambda} \subset S$. Concrete realizat[i]{}ons of this abstract formalism include the case when $\overline{X}$ is the Euclidean space ${\ensuremath{\mathbb{R}}}^n$ and $\mathcal{S}$ consists of all af[f]{}ine hyperplanes in $\overline{X}$. Some examples of *hyperplane dif[f]{}use* sets are supports of absolutely decaying measures such as the Cantor sets and the Sierpiński triangle.
Let $X \subset \overline{X}$ be closed and $b_*$-dif[f]{}use with respect to a collect[i]{}on $\mathcal{S}$ of subsets of $\overline{X}$. Also, $\mathcal{F}$ is a family with nested resonant sets $R_{\lambda}$ and discrete heights, locally contained in $\mathcal{S}$. The set ${\ensuremath{\mathrm{BA}}}_X (\mathcal{F})$ def[i]{}ned in is Schmidt winning subset of $X$.
Many of the terms used above will be explained in § \[S:games\]. Actually, it is shown in [@Wei13] that ${\ensuremath{\mathrm{BA}}}_X (\mathcal{F})$ is *absolute winning with respect to $\mathcal{S}$*. This covers many important examples like the set of $( s_1, \ldots, s_n )$-badly approximable vectors in ${\ensuremath{\mathbb{R}}}^n,\ {\ensuremath{\mathcal{C}}}^2$ and ${\ensuremath{\mathbb{Z}}}_p^2$, the set of sequences in the Bernoulli-shift which avoid all periodic sequences and the set of orbits of toral endomorphisms which stay away from periodic orbits. Curiously for us, we do not f[i]{}nd any discussion on solenoidal endomorphisms in his work.\
In order to be able to use his results, we would have to show that (a) the space [$X_{{\ensuremath{\mathcal{P}}}}$]{} is $b_*$-dif[f]{}use with respect to the collect[i]{}on $\mathcal{C}$ of open cylinders in [$X_{{\ensuremath{\mathcal{P}}}}$]{} for an appropriate value of $b_*$, and (b) the family of pre-images $A^{-j} B ( {\ensuremath{\mathbf{y}}}+ {\ensuremath{\mathbf{z}}}, t )$ is locally contained in $\mathcal{C}$. We have endeavoured to provide a more direct proof here. In this sense, our work may be considered as an addit[i]{}on to the list of examples given in [@Wei13]. We must also point out that the assumpt[i]{}on of having a Federer measure with $X$ as its support in [@Wei13 Proposit[i]{}on 2.1] need not be true when $X = {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}},\ {\ensuremath{\mathcal{P}}}$ is inf[i]{}nite and $\mu$ is the Haar measure on [$X_{{\ensuremath{\mathcal{P}}}}$]{} (cf. ).
Metric and measure structure on solenoids {#S:sol}
=========================================
Before we discuss the game, it is imperat[i]{}ve that we say a few words about how balls in our space $X_{{\ensuremath{\mathcal{P}}}, n}$ “look like.” A metric on $X_{{\ensuremath{\mathcal{P}}}, n}$ is given by $$\label{E:solmet}
d ( \mathbf{x}, \mathbf{z} ) := \max \left\{ {\ensuremath{\left\lvert\, x_0 - z_0 \,\right\rvert}},\ \sup_{p \in {\ensuremath{\mathcal{P}}}} \big\{ p^{-1} {\ensuremath{\left\lvert\, x_p - z_p \,\right\rvert}}_{p} \big\} \right\}$$ where [$\left\lvert\, \cdot \,\right\rvert$]{} is the usual Euclidean metric on ${\ensuremath{\mathbb{R}}}^n$ and ${\ensuremath{\left\lvert\, \cdot \,\right\rvert}}_{p}$ refers to the $p$-adic ultrametric on ${\ensuremath{\mathbb{Q}}}_{p}^n$ such that the diameter of $( p^{-1} {\ensuremath{\mathbb{Z}}}_{p} )^n$ is $p$. Clearly, distance between any two dis[t]{}[i]{}nct points of $\Delta (R^n)$ is at least $1$ or in other words, the injec[t]{}[i]{}vity radius for the quo[t]{}[i]{}ent map $\Pi : X_{{\ensuremath{\mathcal{P}}}, n} \rightarrow {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}(n)$ equals $1$. More generally, as per [@BFK11], a subset $Z$ of any metric space $X$ is said to be *$\delta$-uniformly discrete* if the distance between any two distinct points of $Z$ is at least $\delta$. In this terminology, the set $\Delta ( R^n )$ described above is $1$-uniformly discrete in $X_{{\ensuremath{\mathcal{P}}}, n}$.\
The definit[i]{}on of the metric in ensures that balls $B ( {\ensuremath{\mathbf{x}}}, r ) \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ are direct products of their coordinatewise project[i]{}ons when (i) ¶ is a f[i]{}nite set, or (ii) when $r < 1$. Moreover for $r < 1,\ 1 \leq p_i r < p_i$ for all $i$ large enough. As the coordinates $x_i$ of [$\mathbf{x}$]{} belong to ${\ensuremath{\mathbb{Z}}}_{p_i}$ for $i \gg 1$, the project[i]{}ons $B ( x_i, p_i r ) = {\ensuremath{\mathbb{Z}}}_{p_i}$ for all but f[i]{}nitely many $i$’s whereby we get that any such ‘open’ ball $B ( {\ensuremath{\mathbf{x}}}, r )$ is actually open in [$X_{{\ensuremath{\mathcal{P}}}}$]{} while the ‘closed’ ball $\overline{B} ( {\ensuremath{\mathbf{x}}}, r )$ is a compact neighbourhood of [$\mathbf{x}$]{}. It is easy to see that the diameter of $B ( {\ensuremath{\mathbf{x}}}, r )$ is $2r$ but in most (all but one) of the places $p \in {\ensuremath{\mathcal{P}}}$, the $p$-adic diameters of the project[i]{}ons will be strictly less than $pr$ as distances in the non-archimedean f[i]{}elds are a discrete set. For a given $r > 0$, there might not be any integer $j$ such that $p_i^j = p_i r$. This seemingly minor issue is very crucial in our next set of calculat[i]{}ons. If $j_i \in {\ensuremath{\mathbb{Z}}}$ is such that $p_i^{j_i} \leq r < p_i^{j_i + 1}$, then we call $\lfloor r \rfloor_i := p_i^{j_i}$. It should be noted that ${\ensuremath{\left\lfloor\, p_i^m r \,\right\rfloor}}_i = p_i^m {\ensuremath{\left\lfloor\, r \,\right\rfloor}}_i$ for all $m \in {\ensuremath{\mathbb{Z}}}$ and more generally, ${\ensuremath{\left\lfloor\, t r \,\right\rfloor}}_i \leq {\ensuremath{\left\lfloor\, p_i {\ensuremath{\left\lfloor\, t \,\right\rfloor}}_i r \,\right\rfloor}}_i = p_i {\ensuremath{\left\lfloor\, t \,\right\rfloor}}_i {\ensuremath{\left\lfloor\, r \,\right\rfloor}}_i\ \forall r, t > 0$. The lemma below may be of independent interest to the reader:
\[L:prod\] Let $x > 1$ and $P_{{\ensuremath{\mathcal{P}}}} (x)$ denote the product $$\prod_{i \in {\ensuremath{\mathbb{N}}}} {\ensuremath{\left\lfloor\, \frac{p_i}{x} \,\right\rfloor}}_{i} = \prod_{\substack{i \in {\ensuremath{\mathbb{N}}},\\ p_i < x}} {\ensuremath{\left\lfloor\, \frac{p_i}{x} \,\right\rfloor}}_{i}.$$ Then, $$\ln P_{{\ensuremath{\mathcal{P}}}} (x) \leq - {\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x\,\right)}} + O \big( ( \ln x )^2 \big), \textrm{ where } {\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x\,\right)}} := \sum_{\substack{p \leq x\\ p \in {\ensuremath{\mathcal{P}}}}} \ln p.$$
Let $p_i < x$ and $m_i \in {\ensuremath{\mathbb{N}}}$ be the unique integer for which $$\dfrac{1}{p_i^{m_i}} \leq \dfrac{p_i}{x} < \dfrac{1}{p_i^{m_i - 1}}\quad \Leftrightarrow\quad p_i^{m_i} < x \leq p_i^{m_i + 1}.$$ We know that $m_i = k$ if and only if $x^{1 / ( k + 1 )} \leq p_i < x^{1/k}$. This observat[i]{}on leads to decomposing $P_{{\ensuremath{\mathcal{P}}}} (x)$ as a double product $$P_{{\ensuremath{\mathcal{P}}}} (x) = \prod_{k = 1}^{\ell} \prod_{\substack{p \in {\ensuremath{\mathcal{P}}},\\ x^{1 / ( k + 1 )} \leq p < x^{1/k}}} p^{-k}$$ where $\ell \in {\ensuremath{\mathbb{N}}}$ is such that $x^{1 / ( \ell + 1 )} \leq 2 < x^{1 / \ell}$ (i. e., $\log x / \log 2 - 1 \leq \ell < \log x / \log 2$). Taking negat[i]{}ve logarithms on both sides, we have $$\begin{aligned}
- \ln P_{{\ensuremath{\mathcal{P}}}} (x) &= \sum_{k = 1}^{\ell} k \sum_{\substack{p \in {\ensuremath{\mathcal{P}}},\\ x^{1 / ( k + 1 )} \leq p < x^{1/k}}} \ln p\notag\\
&\geq \sum_{k = 1}^{\ell} k \left( {\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x^{1/k}\,\right)}} - {\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x^{1/ ( k + 1 )}\,\right)}} - \frac{1}{k} \ln x \right)\\
&= \sum_{k = 1}^{\ell} {\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x^{1/k}\,\right)}} - \ell \left( {\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x^{1/ ( \ell + 1 )}\,\right)}} + \ln x \right).\notag
\end{aligned}$$ As ${\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x^{1 / ( \ell + 1 )}\,\right)}} \leq {\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,2\,\right)}} \leq \ln 2$ and $\ell \ll \ln x$, we are done.
When ¶ is the full subset of primes, ${\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x\,\right)}} \sim x$ as $x \rightarrow \infty$. If ¶ consists of all primes in an arithmet[i]{}c progression with common dif[f]{}erence $k$ and $( a, k ) = 1$ where $a$ is one of the terms in the progression, then [$\theta_{{\ensuremath{\mathcal{P}}}} \left(\,x\,\right)$]{} goes roughly as $x / \varphi (k)$ as $x \rightarrow \infty$. Here, $\varphi$ stands for the Euler’s tot[i]{}ent funct[i]{}on. For more on this, the reader is redirected to @MV07’s book [@MV07]. We conclude that for all subsets ¶ of primes which come from arithmet[i]{}c progressions (and in part[i]{}cular the full subset), there exist some $0 < c_1 = c_1 (k) < 1$ such that $$\label{E:Pp1byr}
P_{{\ensuremath{\mathcal{P}}}} \left( \frac{1}{r} \right) \leq r^{c_1 \frac{1/r}{\ln (1/r)}} \textrm{ for all } 0 < r \ll 1.$$ This short exercise serves dual purpose. On one hand, it is a roundabout way of establishing the inf[i]{}niteness of the Hausdorf[f]{} dimension of [$X_{{\ensuremath{\mathcal{P}}}}$]{} when ¶ is as above using the *mass distribut[i]{}on principle*. A f[i]{}nite non-zero measure $\mu$ whose support is a bounded subset of a metric space $M$ is called a *mass distribut[i]{}on* on $M$. We require our *dimension funct[i]{}ons* (also known as *gauge funct[i]{}ons*) $f : {\ensuremath{\mathbb{R}}}_{\geq 0} \rightarrow {\ensuremath{\mathbb{R}}}_{\geq 0}$ to be increasing in some neighbourhood $[ 0, a_f )$, cont[i]{}nuous on $( 0, a_f )$, right cont[i]{}nuous at $0$ and $f (r) = 0$ if[f]{} $r$ equals zero [@Fal90 pg. 33].
\[P:massdis\] Let $\mu$ be a mass distribut[i]{}on on a second countable metric space $M$ such that for some dimension funct[i]{}on $f$ and $\delta_0 > 0$, $$\mu (U) \leq c_2 f ( {\ensuremath{\left\lvert\, U \,\right\rvert}} )$$ for some f[i]{}xed $c_2 > 0$ and all subsets $U$ with ${\ensuremath{\left\lvert\, U \,\right\rvert}} \leq \delta_0$. Then, $\mathcal{H}^f (M) \geq \mu (M) / c_2$. If $f$ is the power rule $r \mapsto r^s$, we can say that $\dim M \geq s$.
Now, let $\mu$ be the restrict[i]{}on to $[ 0, 1 ] \times \prod_{p \in {\ensuremath{\mathcal{P}}}} {\ensuremath{\mathbb{Z}}}_p$ of the Haar measure $\nu$ on [$X_{{\ensuremath{\mathcal{P}}}}$]{} which is the product of the Haar measures on [$\mathbb{R}$]{} and on each ${\ensuremath{\mathbb{Q}}}_p,\ p \in {\ensuremath{\mathcal{P}}}$. We normalize it so that $\mu$ is a probability measure on [$X_{{\ensuremath{\mathcal{P}}}}$]{}. Any ball $B ( {\ensuremath{\mathbf{x}}}, r )$ with radius $0 < r \ll 1$ will then have $$\label{E:nonFed}
\mu \big( B ( {\ensuremath{\mathbf{x}}}, r ) \big) \leq 2r \times \prod_{i \in {\ensuremath{\mathbb{N}}}} \lfloor p_i r \rfloor_i = 2r \cdot P_{{\ensuremath{\mathcal{P}}}} ( 1/r ) \leq 2 r^{c_1 \frac{1/r}{\ln ( 1/r )} + 1}$$ by when ¶ is the set of all primes in any inf[i]{}nite arithmet[i]{}c progression with $( a, k ) = 1$. Af[t]{}er some more work, the above proposit[i]{}on will give us that the Hausdorf[f]{} dimension of the space [$X_{{\ensuremath{\mathcal{P}}}}$]{} is inf[i]{}nite in such cases.\
On the other hand, Lemma \[L:prod\] will help us again in §\[S:hausdorff\] when we examine the dimension-theoret[i]{}c largeness of various winning subsets of cylinder absolute games. In our version, Alice shall be dealing with the family [$\mathcal{C}$]{} of closed subsets exactly one of whose co-ordinates $x_i,\ i \in \{ 0, \ldots, l - 1 \}$ is a fixed constant. The resulting ${\ensuremath{\varepsilon}}$-neighbourhood $C ( {\ensuremath{\mathbf{x}}}, {\ensuremath{\varepsilon}}, i )$ of such a set $P \in {\ensuremath{\mathcal{C}}}$ as also de[f]{}[i]{}ned in will be called an (open) *cylinder*. Given a cylinder $C = C ( {\ensuremath{\mathbf{x}}}, {\ensuremath{\varepsilon}}, i )$, we say that the *radius of $C$* is ${\ensuremath{\varepsilon}}$ if $i = 0$ and the minimum such ${\ensuremath{\varepsilon}}'$ for which $C ( {\ensuremath{\mathbf{x}}}, {\ensuremath{\varepsilon}}', i ) = C ( {\ensuremath{\mathbf{x}}}, {\ensuremath{\varepsilon}}, i )$ otherwise. The index $i$ is called the *constraining coordinate of $C$*. We emphasize that both Alice and Bob are fully aware of the radii of the balls chosen by the lat[t]{}er at any stage of our game by reading the real coordinate.
In[f]{}[i]{}nite games on complete metric spaces {#S:games}
================================================
Let $M$ be a complete metric space and $F$ be a [f]{}[i]{}xed subset of $M$. In the original game introduced by @Sch66 [@Sch66], Alice and Bob are two players who each take turns to pick closed balls in $M$ in the following manner: We have $\alpha, \beta \in ( 0, 1 )$ to be two real numbers such that $1 - 2 \alpha + \alpha \beta > 0$. The game begins with Bob choosing any closed ball $B_0 = \overline{B} ( b_0, r ) \subseteq M$ subsequent to which Alice has to make a choice of some $A_1 = \overline{B} ( a_1, \alpha r )$ such that $A_1 \subset B_0$. After this, Bob picks $B_1 = \overline{B} ( b_1, \beta \alpha r ) \subset A_1$ and the game goes on till in[f]{}[i]{}nity. We thus get a decreasing sequence of closed, non-empty subsets of a complete metric space $$\label{E:seq}
M \supset B_0 \supset A_1 \supset B_1 \supset A_2 \supset \ldots$$ Alice is declared the winner if $\cap_j B_j = \cap_j A_j = \{ a_{\infty} \} \subset F$. A set $F$ is called *$( \alpha, \beta )$-winning* if Alice has a strategy to win the above game regardless of Bob’s moves. Further, it is *$\alpha$-winning* if it is $( \alpha, \beta )$-winning for all $\beta \in ( 0, 1 )$ and *Schmidt winning* if it is $\alpha$-winning for some $\alpha \in ( 0, 1 )$.\
For various applica[t]{}[i]{}ons of pract[i]{}cal interest, one [f]{}[i]{}nds out that Alice need not bother herself too much about choosing the balls $A_j$’s as long as she is able to block out neighbourhoods of certain undesirable points. This is true for example when $M$ is the real line and $F$ is the set BA of badly approximable numbers as discussed in [@Sch66] where Alice needs to be far from ra[t]{}[i]{}onal numbers with small denominators. Moreover if she is careful enough about her strategy, she has to worry about very few of such ra[t]{}[i]{}onals – at t[i]{}mes just one of them and hence, she need only shift the game outside of a ball B centered at some $p/q \in {\ensuremath{\mathbb{Q}}}\cap B_j$ at the $j$-th stage. This was formalized by @McM10 [@McM10] who called the new variant to be *absolute winning games*. Let $\beta \in ( 0, 1/3 )$ and now Alice chooses open balls $A_j$’s with radius $(A_j) \leq \beta \cdot \mathrm{radius}\,(B_{j - 1})$. Then, Bob is supposed to pick some closed ball $B_j \subset B_{j - 1} \setminus A_j$ with $\mathrm{radius}\,( B_j ) \geq \beta \cdot \mathrm{radius}\,( B_{j - 1} )$. The sequence of sets we now have is $$\label{E:abs}
B_0 \supset B_0 \setminus A_1 \supset B_1 \supset B_1 \setminus A_2 \supset \ldots$$ Note that the countable intersec[t]{}[i]{}on $\cap_j B_j$ might be bigger than a singleton set now and we have to set the winning condi[t]{}[i]{}on to be $\cap_j B_j \cap F \neq \phi$. The set $F$ is said to be *$\beta$-absolute winning* if Alice has a strategy to win in this situa[t]{}[i]{}on and it is called *absolute winning* if it is $\beta$-absolute winning for all $\beta \in ( 0, 1/3 )$.\
In the same paper [@McM10], @McM10 also gave the concept of a *strong winning set.* We again have two parameters $\alpha, \beta \in\,]\,0,\,1\,[$ but now Alice is allowed more opt[i]{}ons in the form of balls $A_{i + 1} \subset B_i$ such that ${\ensuremath{\left\lvert\, A_{i + 1} \,\right\rvert}} \geq \alpha {\ensuremath{\left\lvert\, B_i \,\right\rvert}}$ for all $i \in {\ensuremath{\mathbb{N}}}$ while for Bob, ${\ensuremath{\left\lvert\, B_i \,\right\rvert}} \geq \beta {\ensuremath{\left\lvert\, A_i \,\right\rvert}}$ for $i > 1$. It cont[i]{}nues to be mandatory that $A_i \subset B_{i - 1}$ and $B_i \subset A_i$ for all $i$. A subset $F$ for which Alice has a winning strategy in this game is called an *$( \alpha, \beta )$-strong winning set.* The subset $F$ is said to be *$\alpha$-strong winning* if it is $( \alpha, \beta )$-strong winning for all $\beta \in\,]\,0,\,1\,[$ and *strong winning* if it is $\alpha$-strong winning for some $\alpha > 0$. For Euclidean spaces, a strong winning subset is Schmidt winning too and retains its strong winning property under quasisymmetric mappings [@McM10 Theorem 1.2].\
The absolute game has an obvious drawback that if $F$ is the set of badly approximable vectors in ${\ensuremath{\mathbb{R}}}^n$ for any $n > 1$, then Bob can force the game to be always centered on the hyperplane ${\ensuremath{\mathbb{R}}}^{n - 1} \times \{ 0 \}$ and Alice is not able to win trivially. Therefore, it was proposed in [@BFK+12] that she be allowed to block out a neighbourhood of some $k$-dimensional a[f]{}[f]{}[i]{}ne subspace of ${\ensuremath{\mathbb{R}}}^n$ at each stage of the game. Taking this into considera[t]{}[i]{}on, they gave a family of games played on the Euclidean space ${\ensuremath{\mathbb{R}}}^n$ called *$k$-dimensional $\beta$-absolute games* ($0 < \beta < 1/3,\ 0 \leq k < n$) where Bob having chosen $B_0 = B ( \mathbf{b}_0, r_0 ) \subset {\ensuremath{\mathbb{R}}}^n$, Alice picks some a[f]{}[f]{}[i]{}ne subspace $V_1$ of dimension $k$ and for some $0 < \varepsilon_1 \leq \beta r_1$ removes the $\varepsilon_1$-neighbourhood of $V_1$, namely $A_1 = V_1^{(\varepsilon_1)}$ from $B_0$. This is followed by Bob picking a closed ball $B_1 \subseteq B_0 \setminus A_1$ with $\mathrm{radius}\,(B_1) \geq \beta r_0$ and the game proceeds in a similar fashion. In general, the parameter $\varepsilon_j$ is allowed to depend on $j$ subject only to $0 < \varepsilon_j \leq \beta\cdot\mathrm{radius} (B_j)$. Alice wins if $\cap_j B_j \cap F \neq \phi$. As before, $F \subseteq {\ensuremath{\mathbb{R}}}^n$ is *$k$-dimensional $\beta$-absolute winning* if Alice can win the $k$-dimensional $\beta$-absolute game over $F$ irrespec[t]{}[i]{}ve of Bob’s strategy. It is called *$k$-dimensional absolute winning* if it is $k$-dimensional $\beta$-absolute winning for all $\beta \in ( 0, 1/3 )$. It is clear from the de[f]{}[i]{}ni[t]{}[i]{}ons that for $0 \leq k_1 < k_2 < n$, if a set $F \subseteq {\ensuremath{\mathbb{R}}}^n$ is $k_1$-dimensional $\beta$-absolute winning, then it is $k_2$-dimensional $\beta$-absolute winning too. Also, $0$-dimensional $\beta$-absolute winning is the same as $\beta$-absolute winning.\
All of this culminated in the axioma[t]{}[i]{}za[t]{}[i]{}on by @FSU15 [@FSU15] where $M$ is a complete metric space, is a non-empty collec[t]{}[i]{}on of closed subsets of $M$ and $F \subseteq M$ is [f]{}[i]{}xed before the start of play. For $0 < \beta < 1$, the set $F$ is called *$( {\ensuremath{\mathcal{H}}}, \beta )$-absolute winning* if Alice can ensure the intersec[t]{}[i]{}on $\cap_j B_j \cap F \neq \phi$ by removing neighbourhoods $A_j = H_j^{(\varepsilon_j)}$ for some $H_j \in {\ensuremath{\mathcal{H}}}$ and $0 < \varepsilon_j \leq \beta\cdot\mathrm{radius}\,(B_{j - 1})$ at every $j$-th stage of the game. We follow @KL15 [@KL15] to declare Bob the winner by default if at any (f[i]{}nite) stage of the game, he is le[f]{}[t]{} with no legal choice of the ball $B_j$ to make. In the course of the game, Alice will have to make sure that such an event does not ever occur. This is keeping in mind the example of a Schmidt game illustrated in [@KW10 Proposit[i]{}on 5.2] where Bob is not able to win because he has no opt[i]{}on of $B_j$ left. The reader is caut[i]{}oned at this point that in [@FSU15 Def[i]{}nit[i]{}on C.1], the authors resort to the opposite convent[i]{}on of Alice winning the game if it ends abruptly.\
Ever since [@Sch66] came out, Schmidt games have been played and won over subsets of various metric spaces. We were unable to f[i]{}nd any reasonable survey art[i]{}cle covering the developments in the area. It will also be impossible to give here a comprehensive account of all the progress that has been made by dif[f]{}erent people and groups. We will have to contend ourselves by pointing to only a few representat[i]{}ve works. @Dan86 [@Dan86] formulated and proved results about the winning nature of the set of points in a homogeneous space $G / \Gamma$ of a semisimple Lie group $G$ whose orbits under a one-parameter subgroup act[i]{}on are bounded. @Ara94 [@Ara94] showed that the set of points on any non-constant $C^1$ curve $\sigma$ on the unit tangent sphere $S_p$ of any point $p$ on a complete, non-compact Riemannian manifold $M$ with constant negat[i]{}ve curvature and f[i]{}nite Riemannian volume which lead to bounded geodesic orbits is Schmidt winning.\
When $\Gamma \subset G$ is an irreducible lat[t]{}ice of a connected, semisimple $G$ with no compact factors, @KM96 [@KM96] established that the subset of points in $G / \Gamma$ with bounded $H$-orbits is of full Hausdorff dimension whenever $H$ is a nonquasiunipotent one-parameter subgroup of $G$. @KW10 [@KW10] allowed for Alice’s and Bob’s choices of subsets to be more flexible than just metric balls and used this to set[t]{}le that the set of $\mathbf{s}$-badly approximable vectors in ${\ensuremath{\mathbb{R}}}^n$ is winning for any f[i]{}xed $\mathbf{s} \in {\ensuremath{\mathbb{R}}}^n_+$. This was part of an ef[f]{}ort to understand Schmidt’s conjecture on the intersect[i]{}on of the sets of weighted badly approximable vectors for dif[f]{}erent weights which was f[i]{}nally resolved by @BPV11 [@BPV11]. It was shown by @EGL [@EGL] that the set of points on any $C^1$ curve which are badly approximable by rat[i]{}onals coming from a number f[i]{}eld $\mathbb{K}$ is Schmidt winning. [More generally, it is possible to def[i]{}ne a hyperplane absolute game on any $C^1$ manifold. For example, it was recently proved in [@AGGL] that for any one parameter $\mathrm{Ad}$-semisimple subsemigroup $\{ g_t \}_{t \geq 0}$ of the product $G$ of f[i]{}nitely many copies of $\mathrm{SL}_2 ({\ensuremath{\mathbb{R}}})$’s, the set of points $x$ belonging to any lat[t]{}ice quot[i]{}ent $G / \Gamma$ of $G$ and with bounded $\{ g_t \}$-orbit in $G / \Gamma$ is hyperplane absolute winning.]{}\
In our se[t]{}[t]{}[i]{}ng, $M$ shall be [$X_{{\ensuremath{\mathcal{P}}}}$]{} (or $\Sigma_{{\ensuremath{\mathcal{P}}}}$ if you prefer), is the family [$\mathcal{C}$]{} of subsets described in §\[S:sol\] and an example of the target set $F$ is given below. A less contrived one will be available in the next sec[t]{}[i]{}on. A *cylinder $\beta$-absolute game* begins with Bob choosing a closed ball $B_0 = \overline{B} ( \mathbf{x}_0, r_0 )$. Subsequent to this, Alice blocks an open cylinder $C_1$ whose radius has to be less than or equal to $\beta r_0$. The cylinders seem to us to be the appropriate replacement for the hyperplane neighbourhoods of [@BFK+12] in metric spaces like solenoids. Recall that the exact value of the radius can be read of[f]{} from the real coordinate. Moreover, $\mathrm{radius}\,(B_j)$ is required to be at least $\beta\cdot\mathrm{radius}\,(B_{j - 1})$ for all $j \in {\ensuremath{\mathbb{N}}}$. The game of our interest goes as $$\label{E:caw}
B_0 \supset B_0 \setminus C_1 \supset B_1 \supset B_1 \setminus C_2 \supset \cdots$$ and $F$ is said to be *cylinder $\beta$-absolute winning* if Alice can devise a method to win this game, i. e., $\bigcap_j B_j \cap F \neq \phi$. It is [*cylinder absolute winning*]{} if there exists a $0 < {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}= {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}(F) \leq 1/3$ such that $F$ is cylinder $\beta$-absolute winning for all $\beta \in\ ]\,0,\,{\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}\,[$. The supremum of such ${\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}$’s is christened the *[CAW]{} dimension* of $F$.
\[P:ccwin\] A countable intersect[i]{}on of [CAW]{} subsets of [$X_{{\ensuremath{\mathcal{P}}}}$]{} with winning dimension $\geq \beta_0$ each is a [CAW]{} set with winning dimension $\geq \beta_0$.
The basic idea of the proof remains the same as in [@Sch66 Theorem 2] and is being skipped here. We next give a theorem largely inspired by one of @Dan86 [@Dan86].
\[Th:cgame\] Let $N$ be a countable indexing set and $\{ A_{( n, t )} \subseteq C_{( n, t )} \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\mid n \in N,\ t \in ( 0, 1 ) \}$ be a family of set pairs where $C_{( n, t )}$’s are restricted to be open cylinders in [$X_{{\ensuremath{\mathcal{P}}}}$]{} with the same f[i]{}xed constraining coordinate $i$. If for any compact $K \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ and $\mu \in ( 0, 1 )$, there exist $R \geq 1,\ {\ensuremath{\varepsilon}}\in ( 0, 1 )$ and a sequence $( R_n )$ of posit[i]{}ve reals with the following propert[i]{}es:
1. if $n \in N$ and $t \in ( 0, {\ensuremath{\varepsilon}})$ are such that $A_{( n, t )} \cap K \neq \phi$, then $R_n \leq R$ and the radius $r ( C_{( n, t )} )$ of the cylinder $C_{(n, t )}$ is at most $t R_n$,
2. if $n_1, n_2 \in N$ and $t \in ( 0, {\ensuremath{\varepsilon}})$ are such that both $A_{( n_i, t )}$ intersect $K$ non-trivially and the radius bounds of the associated cylinders are comparable, i. e., $\mu R_{n_1} \leq R_{n_2} \leq \mu^{-1} R_{n_1}$, then either $n_1 = n_2$ or $d\,( A_{( n_1, t )},\ A_{( n_2, t )} ) \geq {\ensuremath{\varepsilon}}( R_{n_1} + R_{n_2} )$.
Then, $F = \bigcup_{\delta > 0} \big( {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\setminus \cup_{n = 1}^{\infty} A_{( n, \delta )} \big)$ is a cylinder absolute winning set with winning dimension at least $\beta_0$ where $\beta_0 := 1 / p_{i}$ if $i > 0$ and $1/3$ otherwise.
Given any $0 < \beta < \beta_0$, let $B_0 = \overline{B}(\mathbf{x}, r_0 )$ be the init[i]{}al closed ball of radius $r_0$ chosen by Bob to kick start the cylinder $\beta$-absolute game. Without loss of generality, we may assume that $r_0 < 1/2$ as well as that the balls $B_i$ chosen by Bob have radii $r_i \rightarrow 0$ (Alice can force this by removing some largest possible cylinder which is legally allowed at each turn). We let $R,\ {\ensuremath{\varepsilon}}$ and $( R_n )$ take the values dictated by our hypothesis for $K = B_0$ and $\mu = \beta^2/2$. Then, let $k_0 \in {\ensuremath{\mathbb{N}}}$ be the smallest such that $\mu^{k_0} < \min \{ {\ensuremath{\varepsilon}}\mu r_0^{-1}, R^{-1} \}$ and $\delta := \mu^{k_0 + 1} r_0 < {\ensuremath{\varepsilon}}$. For $k \geq 1$, mark $h_k \rightarrow \infty$ to be any strictly increasing subsequence such that $$\label{E:h_k}
\beta \mu^k r_0 < \mathrm{radius}\,( B_{h_k} ) =: r_k \leq \mu^k r_0.$$ This is well-def[i]{}ned as $r_{k + 1} \geq \beta r_k$ for all $k \in {\ensuremath{\mathbb{N}}}$ and $\mu^{k + 1} < \beta \mu^k$. We claim that Alice is able to play in such a manner that the closed ball $B_{h_k}$ does not intersect any $A_{( n, \delta )}$ with $R_n \geq \mu^{k - k_0}$. The limit point $b_{\infty} = \cap_{k = 0}^{\infty} B_k = \cap_{k = 0}^{\infty} B_{h_k}$ shall then be in $F$ and the proof of the theorem will be done (as $\beta < \beta_0$ is arbitrary).\
Our claim is vacuously true for $k = 0$ as $R_n \leq R < \mu^{-k_0}$ for all $A_{( n, \delta )}$ intersect[i]{}ng $B_0$ non-trivially by our assumpt[i]{}on. Thereaf[t]{}er, supposing that the claim holds for $k$, we show it to be true for $k + 1$. Since the sets $A_{( n, \delta )}$ with the corresponding cylinder radii bounds $R_n \geq \mu^{k - k_0}$ have already been taken care of, we only need to show that Alice can now ensure $B_{h_{k + 1}}$ does not intersect $A_{( n, \delta )}$ for any $n \in N$ such that $\mu^{k + 1 - k_0} \leq R_n < \mu^{k - k_0}$. As hinted before, she has to worry about exactly one such subset. For, if both $A_{(n_1, \delta )} \cap B_{h_k},\ A_{( n_2, \delta )} \cap B_{h_k} \neq \phi$ and $R_{n_1}, R_{n_2} \in [\,\mu^{k + 1 - k_0},\,\mu^{k - k_0} )$, then the second condit[i]{}on of the theorem says that $d ( A_{( n_1, \delta )},\ A_{( n_2, \delta )} ) \geq {\ensuremath{\varepsilon}}( R_{n_1} + R_{n_2} ) \geq 2{\ensuremath{\varepsilon}}\mu^{k + 1 - k_0}$ while $| B_{h_k} | \leq 2 r_k \leq 2\mu^k r_0$ and we have a contradict[i]{}on.\
If $n \in N$ is the unique index for which $A_{( n, \delta )} \cap B_{h_k} \neq \phi$ and $R_n \in [\,\mu^{k + 1 - k_0}, \mu^{k - k_0} )$ where $B_{h_k} = \overline{B} ( {\ensuremath{\mathbf{x}}}_k, r_k )$, Alice chooses $C_{h_k + 1}$ to be the open cylinder $C_{( n, \delta )}$ and since $$\label{E:Aradius}
\mathrm{radius}\,( C_{h_k + 1} ) \leq \delta R_n < \mu^{k_0 + 1} r_0\cdot\mu^{k - k_0} < \beta\cdot r_k,$$ this const[i]{}tutes a legal move. It only remains to be argued that Bob has some choice of $B_{h_k + 1} \subset B_{h_k} \setminus C_{h_k + 1}$ lef[t]{} (in fact, plenty of them). If the constraining coordinate $i$ of $C_{h_k + 1}$ equals zero, we only need to find a point in the closed ball $\overline{B} \big( x_{k, 0}, ( 1 - \beta ) r_k \big) \subset {\ensuremath{\mathbb{R}}}$ which is at a Euclidean distance $\beta r_k$ from some open ball $B ( y, \beta r_k )$ containing the project[i]{}on $\pi_0 ( C_{h_k + 1} )$ of $C_{( n, \delta )}$ in the archimedean coordinate. This is clearly possible as long as $\beta < 1/3$.\
Else if $i > 0$, let $\overline{B} ( x_{k, i}, \lfloor p_i r_k \rfloor_i ),\ B ( y, p_i r'_{k + 1} ) \subset {\ensuremath{\mathbb{Q}}}_{p_i}$ be the respect[i]{}ve images of $B_{h_k}$ and $C_{h_k + 1}$ under the project[i]{}on $\pi_i$ onto the $i$-th coordinate. As $B_{h_k} \cap C_{h_k + 1} \neq \phi$ by our assumption, we get that $\overline{B} ( x_{k, i}, \lfloor p_i r_k \rfloor_i )\,\cap\,B ( y, r'_{k + 1} ) \neq \phi$ too. Being balls in an ultrametric space, one of them then has to be contained in the other and because $\beta < 1 / p_i$, we have $$\label{E:ultra}
r'_{k + 1} \leq \frac{1}{p_i} \lfloor r_k \rfloor_i$$ and thereby $B ( y, p_i r'_{k + 1} ) \subsetneq \overline{B} ( x_{k, i}, p_i \lfloor r_k \rfloor_i ) = \overline{B} ( y, p_i \lfloor r_k \rfloor_i )$. Bob picks a point $z_i \in \overline{B} ( y, p_i \lfloor r_k \rfloor_i )$ whose distance from $y$ is equal to $p_i \lfloor r_k \rfloor_i$ and if $$\label{E:cproj}
( x_{k, 0}, \ldots, x_{k, i - 1}, x_{k, i + 1}, \ldots ) = \pi^{\perp}_i ( \mathbf{x}_k ),$$ where $\pi^{\perp}_i : X \rightarrow {\ensuremath{\mathbb{R}}}\times \prod^{\prime}_{j \neq i} {\ensuremath{\mathbb{Q}}}_{p_j}$ is the complementary project[i]{}on of $\pi_i$, he def[i]{}nes $$\label{E:nextball}
B_{h_k + 1} = \overline{B} \big( ( x_{k, 0}, \ldots x_{k, i - 1}, z_i, x_{k, i + 1}, \ldots ), \beta r_k \big).$$ This has diameter $2 \beta r_k$, is contained in $B_{h_k}$ and avoids the cylinder $C_{h_k + 1}$.
Given any posit[i]{}ve lower bound on the [CAW]{} dimension, we can boost it to an absolute quant[i]{}ty (depending on ¶ alone) for f[i]{}nite solenoids.
\[L:windim\] Let ${\ensuremath{\mathcal{P}}}= \{ p_1 < \cdots < p_{l - 1} \}$ be f[i]{}nite. Any [CAW]{} subset $S$ of [$X_{{\ensuremath{\mathcal{P}}}}$]{} has winning dimension $\geq \min {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}:= \{ 1/3, 1 / p_{l - 1} \}$.
Assume $0 < \beta < {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}$. We are guaranteed that there exists some $0 < \beta' < \beta$ such that $S$ is cylinder $\beta'$-absolute winning. Changing the game parameter from $\beta'$ to $\beta$ only enlarges the set of choices available to Alice while Bob cont[i]{}nues to have some legal choice lef[t]{} as long as $\beta < 1/3$ and $\beta < 1 / p_{l - 1} \leq 1 / p_i$, if $i > 0$ is the constraining coordinate of the cylinder blocked by Alice in the previous move. Also, all of his valid moves in the cylinder $\beta$-absolute game remain so in the $\beta'$-game. Alice just needs to pretend that the game parameter is $\beta'$ and follow her winning strategy for the same.
Non-dense orbits of solenoidal maps {#S:morph}
===================================
As already men[t]{}[i]{}oned in §\[S:intro\], [an af[f]{}ine endomorphism $A: {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}\rightarrow {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$ is of the form ${\ensuremath{\mathbf{x}}}\mapsto (m/n){\ensuremath{\mathbf{x}}}+ {\ensuremath{\mathbf{a}}}$ for some $m/n \in R$ and ${\ensuremath{\mathbf{a}}}\in {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$.]{} Here, $R$ is the set of endomorphisms of the solenoid ${\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}= \nicefrac{{\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}}{\Delta(R)}$ given by the ring $R = {\ensuremath{\mathbb{Z}}}\big[ \{ 1/ p_i \mid p_i \in {\ensuremath{\mathcal{P}}}\} \big]$. [The af[f]{}ine transformat[i]{}on $A$ is invert[i]{}ble if[f]{} $n/m \in R$ too.]{}\
Next, the cylinder $\beta$-absolute game on [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} can be shi[f]{}[t]{}ed to a game played on [$X_{{\ensuremath{\mathcal{P}}}}$]{} once the radii of the balls $B_i \subset {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$ become small enough (say $< 1/2$). This can be forced on Bob in [f]{}[i]{}nitely many steps after the beginning of the game.\
Pick some $y \in {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$ and let $A$ be any [f]{}[i]{}xed af[f]{}ine transformat[i]{}on of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} with its linear part $m/n \in R \setminus \{ 0, \pm 1 \}$. We abuse nota[t]{}[i]{}on and call any of its li[f]{}[t]{}s from ${\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\rightarrow {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ to be $A$ too. Note that any such lif[t]{} is an invert[i]{}ble self map of [$X_{{\ensuremath{\mathcal{P}}}}$]{} as long as $A: {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}\rightarrow {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$ is surject[i]{}ve. Further, let $F_A (y)$ denote the set of points $\mathbf{x} \in {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ whose image $x := \Pi ( \mathbf{x} )$ has its $A$-orbit not entering some $\delta ( x, y, A )$-neighbourhood of $y$. The goal for Alice is to avoid ${\ensuremath{\varepsilon}}$-neighbourhoods of the grid points $\Pi^{-1} ( \{ y \} ) = \mathbf{y} + \Delta(R)$ (for some $\mathbf{y} \in \Pi^{-1} ( \{ y \} )$) which are all at least a unit distance away from each other. Otherwise said, the set $F_{A} (y)$ that Alice should aim for is $$\label{E:winset}
\bigcup_{t > 0} \left( {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\setminus \bigcup_{j = 0}^{\infty} A^{-j} \big( \Delta (R) + B ( \mathbf{y}, t ) \big) \right)$$ and $\nicefrac{F_{A} (y)}{\Delta(R)}$ shall be the image set for the game played on [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}. By our assump[t]{}[i]{}on about $A$, there exists $i \geq 0$ such that $| \nicefrac{m}{n} |_{p_i} =: \lambda_i > 1$. We let $\lambda_A$ to be $\sup_i \lambda_i$. This is f[i]{}nite, at[t]{}ained for some $i = i_0$ and strictly greater than $1$. In part[i]{}cular for any $\mathbf{x}_1, \mathbf{x}_2 \in {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$, we have that $$\label{E:dlb}
d (\, A^{-j} \mathbf{x}_1,\, A^{-j} \mathbf{x}_2 \,) \geq \lambda_A^{-j} d (\, \mathbf{x}_1,\, \mathbf{x}_2 \,) \textrm{ for all } j \geq 0$$ [as translat[i]{}on by any [$\mathbf{a}$]{} is an isometry of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} (and [$X_{{\ensuremath{\mathcal{P}}}}$]{}). Equivalently for any two subsets $F_1, F_2 \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$, $$d (\, A^{-j} F_1,\, A^{-j} F_2 \,) \geq \lambda_A^{-j} d (\, F_1,\, F_2 \,)\ \forall j \geq 0.$$]{} The constraining coordinate of all the cylinders removed by Alice will be some f[i]{}xed $i_0$ for which $\lambda_{i_0} = \lambda_A$. Let $0 < \mu < 1$ and $\ell \in {\ensuremath{\mathbb{N}}}$ be the smallest for which $\lambda_A^{-\ell} < \mu$. If $a = m^{-\ell}$, then $(\nicefrac{m}{n})^{-j}R \subseteq aR$ for all $j \in \{ 0, \ldots, \ell \}$. Not unlike $R$, the points of $aR$ too const[i]{}tute a $\delta$-uniformly discrete set for some $0 < \delta = \delta (a, {\ensuremath{\mathcal{P}}}) \leq 1$. We also choose $b \geq 1$ given by $$b = \max_{0 \leq j \leq \ell} \sup_{{\ensuremath{\mathbf{x}}}\in \overline{B ( \mathbf{0}, 1 )}} d \big( A^{-j} {\ensuremath{\mathbf{x}}}, \mathbf{0} \big)$$ and let $$\label{E:t0}
t_0 = \min \dfrac{1}{3b} \big\{ d (\,\mathbf{y} - A^{-j}\mathbf{y},\,a\mathbf{z}\,) > 0 \mid 0 \leq j \leq \ell,\ \mathbf{z} \in \Delta(R) \big\}.$$ This belongs to $]\,0,\,1/3\,]$ and thereby for any $\mathbf{z}_1, \mathbf{z}_2 \in \Delta(R)$ such that $\mathbf{y} + \mathbf{z}_1 \neq A^{-j} ( \mathbf{y} + \mathbf{z}_2 )$ for some $0 \leq j \leq \ell$, we have $$\begin{aligned}
\label{E:dcalc}
d \left( B( \mathbf{y} + \mathbf{z}_1, t_0 ), A^{-j} ( B( \mathbf{y} + \mathbf{z}_2, t_0 ) ) \right) &\geq d \left( B( \mathbf{y} + \mathbf{z}_1, bt_0 ), B( A^{-j} (\mathbf{y} + \mathbf{z}_2), bt_0 ) \right)\notag\\
&\geq 3bt_0 - 2bt_0 = bt_0 \geq t_0.
\end{aligned}$$ If $j_1 \leq j_2$ are any two exponents such that $\mu^{k + 1} < \lambda_A^{-j_2} \leq \lambda_A^{-j_1} \leq \mu^k$ for some $k \in {\ensuremath{\mathbb{N}}}$, then $j_2 - j_1 \leq \ell$ by the very def[i]{}nit[i]{}on of $\ell$. Hence, for $0 < t < \mu t_0 / 2$ and any $\mathbf{z}_1, \mathbf{z}_2 \in \Delta(R)$ for which $\mathbf{y} + \mathbf{z}_1 \neq A^{- ( j_2 - j_1 )} ( \mathbf{y} + \mathbf{z}_2 )$, we get that $$\label{E:farset}
d \left( A^{-j_1} B ( \mathbf{y} + \mathbf{z}_1, t ), A^{-j_2} ( B ( \mathbf{y} + \mathbf{z}_2, t ) ) \right) > \mu^{k + 1} t_0 = \frac{\mu t_0}{2} ( \mu^k + \mu^k )$$ while $A^{-j} B ( \mathbf{y} + \mathbf{z}, t ) \subset C \big( A^{-j} ( \mathbf{y} + \mathbf{z} ), \lambda_A^{-j}t, i_0 \big) \subseteq C \big( A^{-j} ( \mathbf{y} + \mathbf{z} ), \mu^k t, i_0 \big)$ for $\lambda_A^{-j} \leq \mu^k$. We let $$\label{E:index}
N = \{\,(\,j,\,\mathbf{y} + \mathbf{z}\,) \mid j \in {\ensuremath{\mathbb{N}}},\ \mathbf{z} \in aR\,\}$$ be the countable indexing set in our Theorem \[Th:cgame\]. The f[i]{}rst hypothesis therein is satisf[i]{}ed by taking $R = 1$ and for $n = ( j, \mathbf{y} + \mathbf{z} )$, let[t]{}ing $$A_{( n, t )} = A^{-j} B ( \mathbf{y} + \mathbf{z}, t )\ \subset\ C_{( n, t )} := C \big( A^{-j} ( \mathbf{y} + \mathbf{z} ), \lambda_A^{-j}t, i_0 \big)$$ which suggests that we should take $R_n = \lambda_A^{-j}$. Clearly, the second hypothesis has been shown to hold here in . Hence, we infer that $F_{A} (y) = \cup_{t > 0} \big( {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\setminus \bigcup_{j = 0}^{\infty} A^{-j} \big( R + B ( \mathbf{y}, t ) \big) \big)$ is a [CAW]{} subset of [$X_{{\ensuremath{\mathcal{P}}}}$]{} with winning dimension as in the statement of Theorem \[Th:cgame\] and so is its image in [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}. Because Proposit[i]{}on \[P:ccwin\], we can extend this result to the set of points whose $A$-orbits avoid some neighbourhoods of countably many points $\{ {\ensuremath{\mathbf{y}}}_k \}_{k \in {\ensuremath{\mathbb{N}}}} \subset {\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$.\
If $A ({\ensuremath{\mathbf{x}}}) = (m/n){\ensuremath{\mathbf{x}}}+ {\ensuremath{\mathbf{a}}}$ is such that $m/n = -1$, then $A^2$ is the ident[i]{}ty endomorphism. In this case, Alice only needs to move the game away from the countable set $\{ {\ensuremath{\mathbf{y}}}_k \} \cup \{ {\ensuremath{\mathbf{a}}}- {\ensuremath{\mathbf{y}}}_k \}$. This is trivial. The situat[i]{}on is even simpler when $A$ is just the ident[i]{}ty map. Now, let $Y$ be the set consist[i]{}ng of all those points of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} which have a periodic orbit for some $B \in R \setminus \{ \pm 1 \}$. This is countable and leads us to conclude:
\[Th:solmain\]
Let $\{ A_j : {\ensuremath{\mathbf{x}}}\mapsto ( m_j / n_j ) {\ensuremath{\mathbf{x}}}+ {\ensuremath{\mathbf{a}}}_j \}_{j \in {\ensuremath{\mathbb{N}}}}$ be any subset of af[f]{}ine surject[i]{}ve endomorphisms of the solenoid [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} such that
1. none of the $A_j$’s is a non-trivial translat[i]{}on of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}, and
2. the collect[i]{}on of rat[i]{}onal numbers $\{ m_j / n_j \}_{j \in {\ensuremath{\mathbb{N}}}}$ belong to some f[i]{}nite extension of [$\mathbb{Z}$]{}.
Then, the set of points whose orbit closure under the act[i]{}on of any of the $A_j$’s does not contain any periodic $B$-orbit for all $B \in R \setminus \{ \pm 1 \}$ is cylinder absolute winning.
If $\{ A_j \} \subset {\ensuremath{\mathbb{Z}}}\big[ \{ 1 / p_1, \ldots, 1 / p_n \mid p_i \in {\ensuremath{\mathcal{P}}}\} \big]$, then the winning dimension of each of the subsets $F_{A_j}$ is at least $\min \{ 1/3, \min \{ 1 / p_i \mid 1 \leq i \leq n \} \} > 0$. This is also a lower bound on the [CAW]{} dimension of the intersect[i]{}on $\cap_j F_{A_j}$ invoking Proposit[i]{}on \[P:ccwin\] once again.
Note that even though $R$ is a countable set, we cannot further this argument to take intersect[i]{}ons over any arbitrarily chosen sequences of af[f]{}ine surject[i]{}ve endomorphisms of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}. This is because the lower bound on the winning dimension of the [CAW]{} subsets of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} corresponding to each $A$ is dependent on $A$ itself in terms of $i_0$ for which $\lambda_{i_0} = \lambda_A$. However, for f[i]{}nite ¶, each such $\beta_0$ is at least $\min \{ 1/3, \min \{ 1 / p_i \mid p_i \in {\ensuremath{\mathcal{P}}}\} \}$. We can then remove the second condit[i]{}on in Theorem \[Th:solmain\] to get
\[Th:sol\] Let ¶ be a f[i]{}nite set of rat[i]{}onal primes [and $\{ A_j \}$ be any sequence of af[f]{}ine surject[i]{}ve endomorphisms of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{} such that none of the $A_j$’s is a translat[i]{}on.]{} The set of points whose orbit closure under the act[i]{}on of any $A_j$ does not contain any periodic $B$-orbit for all $B \in R \setminus \{ \pm 1 \}$ is [CAW]{} with winning dimension at least $\min \{ 1/3, 1 / p_{l - 1} \}$. Here, $p_{l - 1}$ is the largest prime in ¶.
[In part[i]{}cular, this is true of the collect[i]{}on of all surject[i]{}ve endomorphisms of [$\Sigma_{{\ensuremath{\mathcal{P}}}}$]{}.]{}
{#S:hausdorff}
Let ¶ be f[i]{}nite. We start by discussing the implicat[i]{}ons of [CAW]{} property of a subset $F$ for a strong game played on [$X_{{\ensuremath{\mathcal{P}}}}$]{} with $F$ as its target.
\[P:CAWstrong\] A [CAW]{} subset of [$X_{{\ensuremath{\mathcal{P}}}}$]{} is $\alpha$-strong winning for all $\alpha < {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}$.
Without loss of generality, we may take the [CAW]{} dimension of $F$ to be ${\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}$ due to Lemma \[L:windim\]. This means our target set $F$ is $\beta$-CAW for all $0 < \beta < {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}$. Now, suppose that $\alpha \in\ ]\,0,\,{\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}\,[$ and $\gamma \in\ ]\,0,\,1\,[$ are any f[i]{}xed (strong) game parameters for Alice and Bob, respect[i]{}vely.\
Given a ball $B_0 = \overline{B} ( \mathbf{x}, r )$ chosen by Bob at any stage of the strong game, Alice checks the cylinder $C$ with ${\ensuremath{\operatorname{radius} \left(\,C\,\right)}} \leq \alpha\gamma r$ to be removed by her in accordance with her winning strategy for $F$ when playing the cylinder $( \alpha\gamma )$-absolute game. If $B \cap C = \phi$, she chooses any $A \subset B$ allowed by the rules of the strong game. Assume this to not be the case for the rest of this proof.\
If the constraining coordinate $i$ of $C$ is archimedean, Alice has no problem in choosing a Euclidean ball $A \subset \pi_0 (B) \setminus \pi_0 (C)$ with ${\ensuremath{\operatorname{radius} \left(\,A\,\right)}} \geq \alpha r$ as $\alpha \leq {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}\leq 1/3$. Else again when $i > 0$, we have $\pi_i ( C ) \subsetneq \pi_i (B)$ because $${\ensuremath{\operatorname{radius} \left(\,\pi_i (C)\,\right)}} \leq p_i {\ensuremath{\left\lfloor\, {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}r \,\right\rfloor}}_i \leq {\ensuremath{\left\lfloor\, r \,\right\rfloor}}_i \text{ while } {\ensuremath{\operatorname{radius} \left(\,\pi_i (B)\,\right)}} \geq p_i {\ensuremath{\left\lfloor\, r \,\right\rfloor}}_i.$$ We can moreover take the center $x_i$ of $\pi_i (B)$ to be the same as that of $\pi_i (C)$. Let $z_i \in \pi_i (B) \setminus \pi_i (C)$. Then, ${\ensuremath{\left\lvert\, z_i - x_i \,\right\rvert}}_{p_i} = p_i {\ensuremath{\left\lfloor\, r \,\right\rfloor}}_i$ and the ultrametric also gives us that $\overline{B}_{{\ensuremath{\mathbb{Q}}}_{p_i}} ( z_i, {\ensuremath{\left\lfloor\, r \,\right\rfloor}}_i ) \subset \pi_i (B) \setminus \pi_i (C)$. In either case, the pre-images $\pi_0^{-1} (A)$ or $\pi_i^{-1} \big( \overline{B}_{{\ensuremath{\mathbb{Q}}}_{p_i}} ( z_i, {\ensuremath{\left\lfloor\, r \,\right\rfloor}}_i ) \big)$ contain a ball of [$X_{{\ensuremath{\mathcal{P}}}}$]{} of radius at least ${\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}r \geq \alpha r$ which lies inside $B \setminus C$. Alice chooses one such $A_1$ to be her next move. Bob’s choice of any $B_1 \subset A_1$ with ${\ensuremath{\operatorname{radius} \left(\,B_1\,\right)}} \geq \gamma\cdot{\ensuremath{\operatorname{radius} \left(\,A_1\,\right)}} \geq \alpha\gamma r$ immediately af[t]{}er is also a valid move in cylinder $(\alpha\gamma)$-absolute game.
It should be ment[i]{}oned here that the relat[i]{}onship between winning sets for strong games and quasisymmetric homeomorphisms of [$X_{{\ensuremath{\mathcal{P}}}}$]{} is not clear to us. Nor do we have the analogous statement of Proposit[i]{}on \[P:CAWstrong\] for the full solenoid.
Incompressibility {#SS:incom}
-----------------
The next result is about the incompressible behaviour of cylinder absolute winning subsets of [$X_{{\ensuremath{\mathcal{P}}}}$]{}. A set $S \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ is *strongly af[f]{}inely incompressible* if for any non-empty open subset $U$ and any sequence of invert[i]{}ble af[f]{}ine homomoprhisms $( \Psi_i )_{i \in {\ensuremath{\mathbb{N}}}}$, the set $\cap_{i \in {\ensuremath{\mathbb{N}}}} \Psi_i^{-1} S \cap U$ has the same Hausdorf[f]{} dimension as $U$ [@BFK+12; @Dan89]. It is our claim that [CAW]{} subsets of [$X_{{\ensuremath{\mathcal{P}}}}$]{} are strongly af[f]{}inely incompressible for f[i]{}nite ¶. We show this by proving a lower bound on the [CAW]{} dimension of $\Psi^{-1} S \cap U$ in terms of $\operatorname{windim} S$ for any af[f]{}ine map $\Psi$ of [$X_{{\ensuremath{\mathcal{P}}}}$]{}. Together with Proposit[i]{}on \[P:ccwin\], this will give us that the intersect[i]{}on of any countably many pre-images of a [CAW]{} subset under invert[i]{}ble af[f]{}ine homomorphisms is [CAW]{} too.
Let ¶ be f[i]{}nite, $U \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ open and $S$ be any [CAW]{} subset. Also, let $\Psi : {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\rightarrow {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ be an invert[i]{}ble af[f]{}ine homomorphism. Then, the set $\Psi^{-1} S \cup ( {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\setminus U )$ is also [CAW]{} with winning dimension at least ${\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}$.
As in Proposit[i]{}on \[P:CAWstrong\], we may take the [CAW]{} dimension of $S$ to be ${\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}$ without loss of generality. Let us f[i]{}rst make some reduct[i]{}ons to simpler situat[i]{}ons. If the diameters ${\ensuremath{\left\lvert\, B_k \,\right\rvert}}$ of balls chosen by Bob don’t go to zero as $k \rightarrow \infty$, then $\cap_k B_k$ contains an open ball inside it. As $S$ is a winning subset, it has to be dense and in turn its pre-image $\Psi^{-1} S$ is also dense in [$X_{{\ensuremath{\mathcal{P}}}}$]{}. Second, if $B_k \cap ( {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\setminus U ) \neq \phi$ for inf[i]{}nitely many $k$, then they form a decreasing sequence of closed subsets of the compact ball $B_1$. Their intersect[i]{}on cannot be empty and hence $\cap_k B_k$ contains a point of ${\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\setminus U$ result[i]{}ng in Alice’s victory. It is safe to exclude both of these events from the rest of the proof. We can moreover take that $B_0 \subset U$.\
Let $0 < \beta < {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}$ and $\Psi ({\ensuremath{\mathbf{x}}}) = D{\ensuremath{\mathbf{x}}}+ {\ensuremath{\mathbf{a}}}$ as explained before. Following [@BFK+12], Alice will run a ‘hypothet[i]{}cal’ Game 2 (in her mind) where the target set is $S$ and a dif[f]{}erent game parameter $\beta'$ which is some posit[i]{}ve power of $\beta$. She carefully decides and projects some of Bob’s moves in the $\big( \Psi^{-1} (S) \cup ( {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\setminus U ), \beta \big)$-game to construct choices made by a hypothetical Bob II in Game 2. Since there is a winning strategy for the lat[t]{}er by hypothesis, she channels the winning moves in this second game via the inverse map $\Psi^{-1}$ to win over $\Psi^{-1} (S)$. We take $$\lambda_{\Psi} := \max \{\,\max_{p \in {\ensuremath{\mathcal{P}}}} {\ensuremath{\left\lvert\, D \,\right\rvert}}_p,\,{\ensuremath{\left\lvert\, D \,\right\rvert}}\,\}$$ which makes sense as $D$ is a rat[i]{}onal number. As $D \neq 0$, we have $1 \leq \lambda_{\Psi} < \infty$ for any $\Psi$ and any ¶. It is clear that $$d \big( \Psi ({\ensuremath{\mathbf{x}}}), \Psi ({\ensuremath{\mathbf{y}}}) \big) \leq \lambda_{\Psi} d ( {\ensuremath{\mathbf{x}}}, {\ensuremath{\mathbf{y}}})$$ as $d$ is a translat[i]{}on-invariant metric on [$X_{{\ensuremath{\mathcal{P}}}}$]{}. We re-label the choices made by Bob such that ${\ensuremath{\operatorname{radius} \left(\,B_0\,\right)}} < 1 / \lambda_{\Psi}$. Let $n \in {\ensuremath{\mathbb{N}}}$ be the smallest posit[i]{}ve natural number for which $$\lambda_{\Psi}\lambda_{\Psi^{-1}} ( \beta + 1 ) \beta^{n - 2} < 1,\quad \beta' = \beta^n\quad \textrm{and}\quad \eta := ( \beta + 1 ) \beta^{n - 1}.$$ Alice waits for the stages $0 = j_1 < j_2 < \cdots$ in the original $\big( \Psi^{-1} (S) \cup ( {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\setminus U ), \beta \big)$-cylinder absolute game when $$\beta^{n} \leq {\ensuremath{\operatorname{radius} \left(\,B_{j_k}\,\right)}} / {\ensuremath{\operatorname{radius} \left(\,B_{j_{k - 1}}\,\right)}} < \beta^{n - 1}$$ for the f[i]{}rst t[i]{}me. Not[i]{}ce that this is well-def[i]{}ned and exists because we assumed ${\ensuremath{\left\lvert\, B_k \,\right\rvert}} \rightarrow 0$ and the radius of Bob’s choice at $( k + 1 )$-th step cannot shrink by a factor of more than $\beta$ compared to that of his choice at the $k$-th step for all $k$. Imitat[i]{}ng [@BFK+12], denote $$B_{j_k} = \overline{B} ( {\ensuremath{\mathbf{x}}}_k, r_k ) \textrm{ and } B'_k = \overline{B} ( \Psi ({\ensuremath{\mathbf{x}}}_k), r'_k ) \textrm{ where } r'_k = \lambda_{\Psi} r_k$$ so that $\Psi ( B_{j_k} ) \subset B'_k$ for all $k$ by our def[i]{}nit[i]{}on of $\lambda_{\Psi}$. Then, $\Psi ( \cap_k B_k ) = \Psi ( \cap_k B_{j_k} ) \subset \cap_k B'_k$ while the intersect[i]{}ons $\cap_k B_k$ and $\cap_k B'_k$ are both singleton sets (we are in the case when ${\ensuremath{\left\lvert\, B_k \,\right\rvert}} \rightarrow 0 \Rightarrow {\ensuremath{\left\lvert\, B'_k \,\right\rvert}} \rightarrow 0$ as $k \rightarrow \infty$). Thus, we see that the non-empt[i]{}ness of $\cap_k B'_k \cap S$ will imply that of $\cap_k B_k \cap \Psi^{-1} (S)$.\
If $C'_{k + 1} = C ( {\ensuremath{\mathbf{y}}}'_{k + 1}, {\ensuremath{\varepsilon}}'_{k + 1}, i_{k + 1} )$ is the cylinder to be removed by Alice in Game 2 where ${\ensuremath{\varepsilon}}'_{k + 1} \leq \beta' r'_{k}$, she chooses $C_{j_k + 1}$ as $$C \left( \Psi^{-1} ( {\ensuremath{\mathbf{y}}}'_{k + 1} ), \lambda_{\Psi}\lambda_{\Psi^{-1}}\eta r_{k}, i_{k + 1} \right) \supset \Psi^{-1} \big( C ( {\ensuremath{\mathbf{y}}}'_{k + 1}, {\ensuremath{\varepsilon}}'_{k + 1}, i_{k + 1} ) \big)$$ to be blocked next in the $\beta$-cylinder absolute game with target set $\Psi^{-1} (S)$. By design, $\lambda_{\Psi}\lambda_{\Psi^{-1}}\eta < \beta$ and it only remains to show that Bob has some choice of $B_{j_k + 1} \subset B_{j_k} \setminus C_{j_k + 1}$ lef[t]{} with ${\ensuremath{\operatorname{radius} \left(\,B_{j_k + 1}\,\right)}} \geq \beta\cdot{\ensuremath{\operatorname{radius} \left(\,B_{j_k}\,\right)}}$. As we have seen in the proof of Theorem \[Th:cgame\], this is clearly not a problem as $\beta < {\ensuremath{\beta_{{\ensuremath{\mathcal{P}}}}}}\leq \min \{ 1/3, 1 / p_{i_{k + 1}} \}$ when $i_{k + 1} > 0$ or otherwise. All of Bob’s subsequent choices in Game 1, including $B_{j_{k + 1}} = \overline{B} ( {\ensuremath{\mathbf{x}}}_{k + 1}, r_{k + 1} )$, obey $${\ensuremath{\left\lvert\, x_{k, 0} - x_{k + 1, 0} \,\right\rvert}} \leq r_k - r_{k + 1}$$ in the archimedean coordinate and $${\ensuremath{\left\lvert\, x_{k, i} - x_{k + 1, i} \,\right\rvert}}_{p_i} \leq p_i r_k\ \forall i > 0.$$ Then, $$\begin{aligned}
{\ensuremath{\left\lvert\, \Psi ( {\ensuremath{\mathbf{x}}}_{k + 1} )_0 - \Psi ( {\ensuremath{\mathbf{x}}}_{k} )_0 \,\right\rvert}} &= {\ensuremath{\left\lvert\, Dx_{k + 1, 0} + a_0 - (Dx_{k, 0} + a_0) \,\right\rvert}}\\
&\leq \lambda_{\Psi} {\ensuremath{\left\lvert\, x_{k + 1, 0} - x_{k, 0} \,\right\rvert}} \leq \lambda_{\Psi} ( r_k - r_{k + 1} ) = r'_k - r'_{k + 1},\notag
\end{aligned}$$ and for all $i > 0,\ {\ensuremath{\left\lvert\, \Psi ( {\ensuremath{\mathbf{x}}}_{k + 1} )_i - \Psi ( {\ensuremath{\mathbf{x}}}_{k} )_i \,\right\rvert}}_{p_i} \leq \lambda_{\Psi} p_i r_k = p_i r'_k$ similarly. The conclusion cannot be escaped that the corresponding ball $B'_{k + 1} = \overline{B} \big( \Psi({\ensuremath{\mathbf{x}}}_{k + 1}), r'_{k + 1} \big) \subset B'_k$ in Game 2. It is also outside of $C'_{k + 1}$ when $i_{k + 1} > 0$ as $${\ensuremath{\left\lvert\, x_{k + 1, i_{k + 1}} - \Psi^{-1} ( {\ensuremath{\mathbf{y}}}'_{k + 1} )_{i_{k + 1}} \,\right\rvert}}_{i_{k + 1}} \geq p_{i_{k + 1}} \lambda_{\Psi}\lambda_{\Psi^{-1}}\eta r_k$$ which gives that $$\begin{aligned}
{\ensuremath{\left\lvert\, \Psi ( {\ensuremath{\mathbf{x}}}_{k + 1} )_{i_{k + 1}} - y'_{k + 1, i_{k + 1}} \,\right\rvert}}_{i_{k + 1}} &\geq p_{i_{k + 1}} \lambda_{\Psi}\eta r_k = p_{i_{k + 1}} \eta r'_k\notag\\
&= p_{i_{k + 1}} ( \beta' + \beta^{n - 1} ) r'_k\\
&\geq p_{i_{k + 1}} ( {\ensuremath{\varepsilon}}'_{k + 1} + r'_{k + 1} )\notag
\end{aligned}$$ by Alice’s choice of marking for $B_{j_{k + 1}}$. The computat[i]{}ons are not very dif[f]{}erent when the constraining coordinate of $C'_{k + 1}$ is archimedean.
For general ¶, we are only able to show the largeness of countable intersect[i]{}ons of pre-images under translations of [$X_{{\ensuremath{\mathcal{P}}}}$]{}.
\[P:translation\] Let $( {\ensuremath{\mathbf{a}}}_k )_{k \in {\ensuremath{\mathbb{N}}}} \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ be any arbitrary sequence and $S$ be a [CAW]{} subset with winning dimension $\beta_0$. Then, so is $S \cap \bigcap_{k \in {\ensuremath{\mathbb{N}}}} ( S + {\ensuremath{\mathbf{a}}}_k )$.
Each of the translat[i]{}ons $\Psi_k ({\ensuremath{\mathbf{x}}}) := {\ensuremath{\mathbf{x}}}- {\ensuremath{\mathbf{a}}}_k$ is an isometry and in part[i]{}cular, does not change the shape of the balls in [$X_{{\ensuremath{\mathcal{P}}}}$]{}. Given any such single $\Psi$, we argue that $\Psi^{-1} (S)$ has the same [CAW]{} dimension as $S$. Alice simply translates back her choices for the Game 2 described above by $-{\ensuremath{\mathbf{a}}}$ when $\Psi ({\ensuremath{\mathbf{x}}}) = {\ensuremath{\mathbf{x}}}+ {\ensuremath{\mathbf{a}}}$ and projects Bob’s succeeding choice forward by $\Psi$. Note that as $\lambda_{\Psi_k} = \lambda_{\Psi_k^{-1}} = 1$ for all $k$, she should take $\beta' = \beta$ for any $0 < \beta < \beta_0$. One should also replace $\eta = 1$ in the previous calculat[i]{}ons. The countable intersect[i]{}on property then follows by Proposit[i]{}on \[P:ccwin\].
{#SS:Hau}
Lastly, we will try to understand the sizes of [CAW]{} sets in terms of Hausdorf[f]{} dimensions and measures. Towards this goal, we will require an est[i]{}mate on the number of legal choices that Bob has at any stage of the game.
\[L:nodisjointballs\] Let $0 < \beta \ll 1$. Then, the maximum number of pairwise disjoint balls of radius $\beta r$ contained in any closed ball $\overline{B} ( \mathbf{x}, r ) \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}= {\ensuremath{\mathbb{R}}}\times \prod^{\prime}_{j > 0} {\ensuremath{\mathbb{Q}}}_{p_j}$ which do not intersect an open cylinder $C ( \mathbf{y}, \beta r, i )$ and also maintain a distance at least $\beta r$ from each other is given by $$N_{{\ensuremath{\mathcal{C}}}} ( \beta ) \gg \beta^{- \frac{{\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,1 / \beta\,\right)}}}{\ln ( 1 / \beta )}}$$ where ${\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,t\,\right)}} = \sum_{p \in {\ensuremath{\mathcal{P}}},\ p \leq t} \ln p$.
Because , every closed ball is the Cartesian product of its coordinatewise projections. In any non-archimedean coordinate $j$ such that $p_j \leq 1 / \beta$, there are at least $( p_j {\ensuremath{\left\lfloor\, \beta \,\right\rfloor}}_j )^{-1}$ pairwise disjoint balls contained in the project[i]{}on $\pi_j \big( \overline{B} ( \mathbf{x}, r ) \big) = \overline{B} ( x_j, \lfloor p_j r \rfloor_j ) $ whose radius equals $\lfloor p_j \beta r \rfloor_j$. Each of them also maintain a distance of at least $p_j^2 {\ensuremath{\left\lfloor\, \beta r \,\right\rfloor}}_j$ from each other which means that the pre-images of any two such sub-balls in [$X_{{\ensuremath{\mathcal{P}}}}$]{} are $\geq p_j {\ensuremath{\left\lfloor\, \beta r \,\right\rfloor}}_j > \beta r$ away. The lower bound $( p_j {\ensuremath{\left\lfloor\, \beta \,\right\rfloor}}_j )^{-1}$ equals ${\ensuremath{\left\lfloor\, 1 / \beta \,\right\rfloor}}_j$ unless $\beta$ is an integral power of $p_j$ in which case it is $p_j^{-1} {\ensuremath{\left\lfloor\, 1 / \beta \,\right\rfloor}}_j$. Note that this can happen for at most one prime for any given $\beta$ and we have already assumed $\beta < 1 / p_j$. In the real coordinate, this number is $\gg \beta^{-1}$ even when we ask that the balls are at least $\beta r$ apart. The pre-images under the project[i]{}on map $\pi_{\beta} : {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}\rightarrow {\ensuremath{\mathbb{R}}}\times \prod_{p_j \leq 1 / \beta} {\ensuremath{\mathbb{Q}}}_{p_j}$ of any product of these sub-balls of $\pi_j \big( \overline{B} ( \mathbf{x}, r ) \big)$ for $j = 0$ or $p_j \leq 1/\beta$ are pairwise disjoint, each contain at least one sub-ball of $\overline{B} ( \mathbf{x}, r )$ of radius $\beta r$ and the minimum distance between any two of those pre-images is $\geq \beta r$.\
When asking for only those sub-balls that do not intersect the open cylinder $C ( \mathbf{y}, \beta r, i )$, it is necessary and suf[f]{}icient that we restrict ourselves to only those from the above chosen collect[i]{}on whose images in $\pi_i \big( QB^{-} ( \mathbf{x}, r ) \big)$ do not intersect $\pi_i \big( C ( \mathbf{y}, \beta r, i ) \big)$. Otherwise said, all coordinates but $i$ are not af[f]{}ected. If $i = 0$, the number of such balls in $\pi_0 \big( QB^- ( \mathbf{x}, r ) \big) = \overline{B} ( x_0, r )$ that do not intersect $\pi_0 \big( C ( \mathbf{y}, \beta r, 0 ) \big) = B ( y_0, \beta r )$ is still $\gg \beta^{-1}$ albeit with a smaller constant. Else, it is at least $( p_i \beta )^{-1} - 1 > \beta^{-1} / 2p_i$. The Cartesian product of these disjoint balls in ${\ensuremath{\mathbb{Q}}}_{p_j}$’s and [$\mathbb{R}$]{} then gives us that $$\label{E:NCbeta}
N_{{\ensuremath{\mathcal{C}}}} ( \beta ) \gg \prod_{\substack{p_j \in {\ensuremath{\mathcal{P}}},\\ p_j < 1 / \beta}} {\ensuremath{\left\lfloor\, \beta^{-1} \,\right\rfloor}}_j \geq \beta^{- \frac{{\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,1 / \beta\,\right)}}}{\ln ( 1 / \beta )}}\ \forall 0 < \beta \ll 1$$ by a calculat[i]{}on similar to the one in Lemma \[L:prod\].
As discussed in §\[S:sol\], ${\ensuremath{\theta_{{\ensuremath{\mathcal{P}}}} \left(\,t\,\right)}} \sim t$ when $t \rightarrow \infty$ and ¶ is the full set of primes. More generally, it shows an asymptot[i]{}c linear growth with $t$ when ¶ is any (inf[i]{}nite) set of all primes in an arithmet[i]{}c progression. We record that the constant implied by the Vinogradov notat[i]{}on in is independent of the constraining coordinate of the cylinder $C$. For f[i]{}nite ¶, we use a slightly dif[f]{}erent lower bound.
\[L:ndbfin\] Let ${\ensuremath{\left\lvert\, {\ensuremath{\mathcal{P}}}\,\right\rvert}} = l - 1$ and $p_{l - 1}$ be the largest prime in ¶. Then, $$N_{{\ensuremath{\mathcal{C}}}} (\beta) \gg \beta^{-l}\ \forall 0 < \beta \ll 1 / p_{l - 1},$$ where the implied constant may depend on the primes $p_1, \ldots, p_{l - 1}$ and $l$.
We only need to replace the lower bound for the number of disjoint sub-balls in each non-archimedean coordinate by $\beta^{-1} / p_j$ and the rest of the argument remains the same.
Suppose $F$ is a cylinder absolute winning subset of $X$ and that Alice always plays according to a winning strategy if it is available. We shall now construct a subset $F^* \subseteq F$ which corresponds to the points obtained when Bob is only to allowed to choose one of the $N_{{\ensuremath{\mathcal{C}}}} (\beta)$-many sub-balls described in Lemmata \[L:nodisjointballs\] or \[L:ndbfin\] at each stage of the game. This resembles closely a device from @Kri06 [@Kri06] (see also [@Sch66 Theorem 6]).
\[P:hdim\] Let $f$ be any dimension funct[i]{}on such that $$\limsup_{\delta \rightarrow 0} \frac{\log f (\delta)}{\log \delta} < \liminf_{\beta \rightarrow 0} \frac{\log N_{{\ensuremath{\mathcal{C}}}} (\beta)}{{\ensuremath{\left\lvert\, \log \beta \,\right\rvert}}}.$$ Then, the $f$-dimensional Hausdorf[f]{} measure of any [CAW]{} subset of [$X_{{\ensuremath{\mathcal{P}}}}$]{} is greater than zero.
Let $\beta_0, \delta_0 > 0$ be small enough so that $\beta_0$ is less than the winning dimension of our [CAW]{} set $F$, $\delta_0 < 1$ and $$\sup_{\delta < \delta_0} \big( \log f (\delta) / \log \delta \big) < \inf_{\beta \leq \beta_0} \big( \log N_{{\ensuremath{\mathcal{C}}}} (\beta) / {\ensuremath{\left\lvert\, \log \beta \,\right\rvert}} \big).$$ They exist by virtue of our hypothesis about $f$. Now, let $\Lambda := \{ 0, 1, \ldots, N_{{\ensuremath{\mathcal{C}}}} ( \beta_0 ) - 1 \}^{{\ensuremath{\mathbb{N}}}}$, the sequence space each of whose element $\lambda = ( \lambda_k )_{k \in {\ensuremath{\mathbb{N}}}}$ corresponds to a sequence of choices made by Bob when he is only allowed to choose from one of the $N_{{\ensuremath{\mathcal{C}}}} ( \beta_0 )$-many disjoint sub-balls inside $B_{k - 1}$. The choices made by him at the $k$-th stage are labelled $B ( \lambda_1, \lambda_2, \ldots, \lambda_k )$. If $\lambda \neq \lambda'$, they di[f]{}[f]{}er in some entry $k_0$ and the corresponding balls $B ( \lambda_1, \ldots, \lambda_{k_0} )$ and $B ( \lambda'_1, \ldots, \lambda'_{k_0} )$ are disjoint. This implies that the points obtained at inf[i]{}nity, $\mathbf{a}_{\infty} ( \lambda ) = \cap_{k \rightarrow \infty} B ( \lambda_1, \ldots, \lambda_k ) \neq \mathbf{a}_{\infty} ( \lambda' ) = \cap_{k \rightarrow \infty} B ( \lambda'_1, \ldots, \lambda'_k ) \in F$ under the belief that Alice is following a winning strategy for the target set $F$ with game parameter $\beta_0$. Let $$\label{E:Fstar}
F^* := \{ \mathbf{a}_{\infty} ( \lambda ) \mid \lambda \in \Lambda \} \subseteq F.$$ [We show that ${\ensuremath{\mathcal{H}}}^f (F^*) > 0$]{} and this shall in turn give us our desired statement. For this, the space $F^*$ is mapped in a cont[i]{}nuous fashion (via the biject[i]{}on with $\Lambda$) onto $[ 0, 1 ]$ using the $N_{{\ensuremath{\mathcal{C}}}} ( \beta_0 )$-adic expansion of real numbers, namely $\mathbf{a}_{\infty} (\lambda) \mapsto 0.\lambda_1 \lambda_2 \cdots$. Call this map $\psi$ and let $( U_n )_{n \in {\ensuremath{\mathbb{N}}}}$ be any $\delta$-cover of $F^*$ for some $\delta < \delta_0$. Without loss of generality, let $U_n \subset F^*$ for all $n$. Plainly, $\big( \psi ( U_n ) \big)_{n \in {\ensuremath{\mathbb{N}}}}$ is a cover for $[ 0, 1 ]$ and since diameter is an outer measure on [$\mathbb{R}$]{}, we get that $$1 \leq \sum_{n \in {\ensuremath{\mathbb{N}}}} {\ensuremath{\left\lvert\, \psi ( U_n ) \,\right\rvert}}.$$ Def[i]{}ne $$j_n = \left\lfloor \dfrac{\log ( 2 {\ensuremath{\left\lvert\, U_n \,\right\rvert}} )}{\log \beta_0} \right\rfloor$$ so that $j_n > 0$ for all ${\ensuremath{\left\lvert\, U_n \,\right\rvert}}$ small enough and furthermore, ${\ensuremath{\left\lvert\, U_n \,\right\rvert}} < \beta_0^{j_n}$. Thus, $U_n$ intersects non-trivially with at most one of the balls $B ( \lambda_1, \ldots, \lambda_{j_n} )$ as any two such are at least $\beta_0^{j_n}$ apart. Further, being a subset of $F^*$ it is completely contained inside some $B ( \lambda_1, \ldots, \lambda_{j_n} )$. The lat[t]{}er itself is mapped by $\psi$ into the interval of length $N_{{\ensuremath{\mathcal{C}}}} (\beta_0)^{-j_n}$ of $I$ consist[i]{}ng of numbers whose $N_{{\ensuremath{\mathcal{C}}}} (\beta_0)$-adic expansion begins with $0.\lambda_1 \cdots \lambda_{j_n}$. We conclude that ${\ensuremath{\left\lvert\, \psi ( U_n ) \,\right\rvert}} \leq N_{{\ensuremath{\mathcal{C}}}} (\beta_0)^{-j_n}$ and thereby, $$\begin{aligned}
\label{E:hdim}
1 &\leq \sum_{n \in {\ensuremath{\mathbb{N}}}} {\ensuremath{\left\lvert\, \psi ( U_n ) \,\right\rvert}} \leq \sum_{n \in {\ensuremath{\mathbb{N}}}} N_{{\ensuremath{\mathcal{C}}}} ( \beta_0 )^{-j_n}\\
&= \sum_{n \in {\ensuremath{\mathbb{N}}}} N_{{\ensuremath{\mathcal{C}}}} ( \beta_0 )^{-\big\lfloor \frac{\log ( 2 {\ensuremath{\left\lvert\, U_n \,\right\rvert}} )}{\log \beta_0} \big\rfloor} \leq N_{{\ensuremath{\mathcal{C}}}} ( \beta_0)\cdot 2^{ \frac{\log N_{{\ensuremath{\mathcal{C}}}} (\beta_0)}{{\ensuremath{\left\lvert\, \log \beta_0 \,\right\rvert}}}} \sum_{n \in {\ensuremath{\mathbb{N}}}} {\ensuremath{\left\lvert\, U_n \,\right\rvert}}^{ \frac{\log N_{{\ensuremath{\mathcal{C}}}} (\beta_0)}{{\ensuremath{\left\lvert\, \log \beta_0 \,\right\rvert}}}}.\notag
\end{aligned}$$ As ${\ensuremath{\left\lvert\, U_n \,\right\rvert}} \leq \delta < \delta_0 < 1$ for all $n \in {\ensuremath{\mathbb{N}}}$ and $$\frac{\log N_{{\ensuremath{\mathcal{C}}}} (\beta_0)}{{\ensuremath{\left\lvert\, \log \beta_0 \,\right\rvert}}} > \sup_{\delta < \delta_0} \frac{\log f (\delta)}{\log \delta} \geq \frac{\log f ({\ensuremath{\left\lvert\, U_n \,\right\rvert}})}{\log {\ensuremath{\left\lvert\, U_n \,\right\rvert}}}$$ by our assumpt[i]{}on, we have that $$\label{E:poslb}
\sum_{n \in {\ensuremath{\mathbb{N}}}} f ({\ensuremath{\left\lvert\, U_n \,\right\rvert}} ) \geq \big( N_{{\ensuremath{\mathcal{C}}}} (\beta_0) \big)^{-1} 2^{\log N_{{\ensuremath{\mathcal{C}}}} (\beta_0) / \log \beta_0}$$ for any arbitrary $\delta$-cover $( U_n )$ of $F^*$. Thus, the inf[i]{}mum of the sums on the lef[t]{} side of taken over all $\delta$-coverings of $F^*$ is a posit[i]{}ve number independent of $\delta$. Let[t]{}ing $\delta \rightarrow 0$ from the right, we conclude that the $f$-dimensional Hausdorf[f]{} measure of $F^*$ is strictly posit[i]{}ve. In part[i]{}cular, this proves our claim.
Let $\Sigma$ be the full solenoid over $S^1$. Then, the Hausdorf[f]{} dimension of any [CAW]{} subset of $\Sigma$ is inf[i]{}nite.
We know $N_{{\ensuremath{\mathcal{C}}}} (\beta)$ rises faster than $\beta^{c / ( \beta \log \beta )}$ as $\beta \rightarrow 0$ for some absolute constant $c > 0$, when ${\ensuremath{\mathcal{P}}}$ is the set of all rat[i]{}onal primes. Take $f$ to be the power funct[i]{}on $r \mapsto r^n$ for some $n \in {\ensuremath{\mathbb{N}}}$. The condit[i]{}on in Proposit[i]{}on \[P:hdim\] is sat[i]{}sf[i]{}ed then and we get that the $n$-dimensional Hausdorf[f]{} measure of any [CAW]{} subset of $\Sigma$ is posit[i]{}ve. F[i]{}nally, we let $n \rightarrow \infty$.
The same is true for [CAW]{} subsets of ${\ensuremath{\Sigma_{{\ensuremath{\mathcal{P}}}}}}$, when ¶ is an inf[i]{}nite set consist[i]{}ng of all primes in some arithmet[i]{}c progression. It will also be interest[i]{}ng to study the class of exact dimension funct[i]{}ons for the spaces [$X_{{\ensuremath{\mathcal{P}}}}$]{} and their [CAW]{} subsets [@Fal90]. For f[i]{}nite ¶, we are sat[i]{}sf[i]{}ed with a statement about maximality of Hausdorf[f]{} dimension.
\[P:findim\] Let ${\ensuremath{\left\lvert\, {\ensuremath{\mathcal{P}}}\,\right\rvert}} < \infty$ and $F \subset {\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ be a $\beta$-CAW set. Then, $$\dim F \geq \frac{\log N_{{\ensuremath{\mathcal{C}}}} (\beta)}{{\ensuremath{\left\lvert\, \log \beta \,\right\rvert}}}.$$
The proof is the same as that for Proposit[i]{}on \[P:hdim\] till with $\beta$ replacing $\beta_0$ everywhere.
Once again if we let $\beta \rightarrow 0$ for a [CAW]{} set, we get
Any [CAW]{} subset of ${\ensuremath{X_{{\ensuremath{\mathcal{P}}}}}}$ with ${\ensuremath{\left\lvert\, {\ensuremath{\mathcal{P}}}\,\right\rvert}} = l - 1$ has Hausdorf[f]{} dimension equal to $l$. In part[i]{}cular, the collect[i]{}on of points described in Cororllary \[Th:sol\] has full dimension.
The proofs of Proposit[i]{}ons \[P:hdim\] and \[P:findim\] are suggest[i]{}ve that while there can be Schmidt winning subsets in the example metric space of @KW10 [@KW10] ment[i]{}oned in §\[S:games\] which are of Hausdorf[f]{} dimension zero (in fact, countable), any absolute winning subset thereof shall have to be of full dimension.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks Anish Ghosh for sugges[t]{}[i]{}ng the problem, lots of advice, encouragement and reading various preliminary versions of the manuscript. I am grateful to Jinpeng An, S. G. Dani and Sanju Velani for being a pat[i]{}ent audience and asking many interest[i]{}ng quest[i]{}ons. The calculat[i]{}ons in Lemma \[L:prod\] together with the reference [@MV07] were explained to me by Divyum Sharma.\
Parts of this work f[i]{}rst appeared in a slightly dif[f]{}erent avatar in the author’s PhD thesis submit[t]{}ed to the Tata Inst[i]{}tute of Fundamental Research, Mumbai in 2017. For this durat[i]{}on of our invest[i]{}gat[i]{}ons, f[i]{}nancial support from CSIR, Govt. of India under SPM-07/858(0199)/2014-EMR-I is duly acknowledged.
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abstract: 'Photometric properties of stellar populations that evolve according to chemical evolution models developed by Carigi, Coln, & Peimbert (1999) and Carigi & Peimbert (2001) are explored. Models explain the oxygen abundance and gas mass fraction of irregular galaxies NGC 1560, I Zw 18, NGC 2366, and a typical irregular galaxy. The photometric predictions help to narrow down the range of possible chemical evolution models. Observed colors of I Zw 18 imply an age of $10^8 - 10^9$ yr for its dominant population, while $10^9 - 10^{10}$ yr is an older age preferred by the rest of the galaxy sample. Observed colors are in general, within errors, close to predicted. There is a tendency for the observed $(B-V)$ color to imply higher metallicities at a fixed age than $(U-B)$ (equivalently, $(U-B)$ implies younger ages than $(B-V)$).'
author:
- Leticia Carigi
- Gustavo Bruzual
title: Photometric Constraints on Chemical Evolution Models of Irregular Galaxies
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
Irregular galaxies are metal poor systems whose oxygen deficiency can be explained by an IMF with a larger fraction of low-mass stars relative to the solar vicinity IMF, but similar to that found in globular clusters (Carigi, Coln, & Peimbert 1999, CCP; Carigi & Peimbert 2001, CP). In both papers, the chemical evolution of three irregular galaxies and an average galaxy called “typical irregular galaxy" are discussed by models for three ages: $10^8$, $10^9$, and $10^{10}$ yr and different factors $r$ that indicate the relative excess of sub-stellar objects ($m<0.1$ ). The lack of abundance determinations for other elements besides O, or an estimation of gas mass fraction, prevented the authors to discriminate which of these models ($r$, age) is most appropriate.
This work explores whether existing photometric data for these galaxies and predictions from population synthesis models (Carigi & Bruzual 2000) can indicate the chemical history of these irregular galaxies, under the CCP and CP chemical evolution models.
Results and Discussion
======================
Carigi & Bruzual (2000) adopt IMF (or factor $r$) and $Z(t)$ from closed-box models with continuous or bursting star formation rates.
($U-B$) and ($B-V$) colors evolved according to different spectral evolution models and color data suggest that both, [**NGC 1560**]{} and [**NGC 2366**]{} have been forming stars for the last 10 Gyr. The case [**I Zw 18**]{} is different, its observed colors are consistent with an age between $10^8$ and $10^9$ yr, certainly lower than for NGC 1560 and NGC 2366.
For the [**Typical Irregular Galaxy**]{} the $(U-B)$ color is bluer for a given $(B-V)$ than expected from continuous star formation models. This may also indicate that star formation in these systems occurs in bursts. After each burst of star formation, a galaxy becomes bluer (especially in $(U-B)$) due to the increase in the number of massive main sequence. CCP models with bursting SFR can explain the bluest galaxies in the sample, as well as the O abundance of this system.
We compares observed colors of several [**Local Group dwarf irregular galaxies**]{}, with models for NGC 1560, the best known galaxy in our sample. Again, observed colors are consistent with the predictions from chemical evolution models $10^9$ and $10^{10}$ yr old, although some galaxies show $(B-V)$ colors redder than expected.
Conclusions
===========
In this work we have explored the color behavior of stellar populations that evolve according to chemical evolution models. Our main conclusions are:
$\diamond$ At a given age, the color predicted for a typical irregular galaxy is independent of the fraction of sub-stellar objects. Therefore, color data helps to discriminate model age but not $r$.
$\diamond$ Based on available photometric data, models with fast chemical enrichment (age less than $10^9$ yr) are discarded for most irregular galaxies. These populations are too young to reach the observed colors. The exception is I Zw 18, whose favored age is between $10^8$ and $10^9$ yr.
$\diamond$ Although the age of the dominant stellar population in these galaxies is unknown, photometric models can be used to arrange these systems on an age sequence based on observed colors. We have found a tendency of unknown origin for the $(B-V)$ color to be matched by stellar populations of higher $Z$ than the $(U-B)$ color, at constant age. Equivalently, $(U-B)$ implies younger ages, at a given metallicity, than $(B-V)$.
$\diamond$ At a given age, color and abundance ratios are almost independent of the fraction of sub-stellar objects.
Carigi, L. & Bruzual, G. 2000, in preparation Carigi, L., Coln, P., & Peimbert, M. 1999, , 514, 787 (CCP) Carigi, L. & Peimbert, M. 2001, submitted Rev. Mex. Astron. Astrofis. (CP)
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---
author:
- 'V. D’Elia$^{1,2}$, J. P. U. Fynbo$^{3}$, S. Covino$^{4}$, P. Goldoni$^{5,6}$, P. Jakobsson$^{7}$, F. Matteucci$^{8}$, S. Piranomonte$^{1}$, J. Sollerman$^{3,9}$, C.C. Thöne$^{4}$, S.D. Vergani$^{10,11}$, P.M. Vreeswijk$^{3}$, D.J. Watson$^{3}$, K. Wiersema$^{12}$, T. Zafar$^{3}$, A. de Ugarte Postigo$^{4}$, H. Flores$^{11}$, J. Hjorth$^{3}$, L. Kaper$^{13}$, A.J. Levan$^{14}$, D. Malesani$^{3}$, B. Milvang-Jensen$^{3}$, E. Pian$^{8,15}$, G. Tagliaferri$^{4}$, N.R. Tanvir$^{12}$'
title: 'VLT/X-shooter spectroscopy of the GRB090926A afterglow [^1]'
---
[The aim of this paper is to study the environment and intervening absorbers of the gamma-ray burst GRB 090926A through analysis of optical spectra of its afterglow.]{} [We analyze medium resolution spectroscopic observations ($R=10 000$, corresponding to 30 km s$^{-1}$, S/N$=15 - 30$ and wavelength range $3 000-25 000$) of the optical afterglow of GRB090926A, taken with X-shooter at the VLT $\sim 22$ hr after the GRB trigger.]{} [The spectrum shows that the ISM in the GRB host galaxy at $z =
2.1071$ is rich in absorption features, with two components contributing to the line profiles. In addition to the ground state lines, we detect , , , and excited absorption features. No host galaxy emission lines, molecular absorption features nor diffuse interstellar bands are detected in the spectrum. The line of sight of GRB090926A presents four weak intervening absorption systems in the redshift range $ 1.24 < z < 1.95$.]{}
The Hydrogen column density associated to GRB090926A is $\log
N_{\rm H}/{\rm cm}^{-2} = 21.60 \pm 0.07$, and the metallicity of the host galaxy is in the range \[X/H\] $=
3.2\times10^{-3}-1.2\times10^{-2}$ with respect to the solar values, i.e., among the lowest values ever observed for a GRB host galaxy. A comparison with galactic chemical evolution models has suggested that the host of GRB090926A is likely to be a dwarf irregular galaxy. No emission lines were detected, but we note that a H$\alpha$ flux in emission of $9\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ (i.e., a star formation rate of $2~M_\odot$yr$^{-1}$), which is typical of many GRB hosts, would have been detected in our spectra, and thus emission lines are well within the reach of X-shooter. We put an upper limit to the molecular fraction of the host galaxy ISM, which is $f < 7\times10^{-7}$. The continuum has been fitted assuming a power-law spectrum, with a spectral index of $\beta = 0.89^{+0.02}_{-0.02}$. The best fit does not essentially require any intrinsic extinction since $E_{B-V} < 0.01$ mag adopting a SMC extinction curve.
We derive information on the distance between the host absorbing gas and the site of the GRB explosion. The distance of component I is found to be $2.40 \pm 0.15$ kpc, while component II is located far away from the GRB, possibly at $\sim 5$ kpc. These values are compatible with that found for other GRBs.
Introduction
============
The study of the inter stellar medium (ISM) of $z\gs1$ galaxies has traditionally relied upon Lyman-break galaxies (LBGs) at $z=3-4$ (see e.g. Steidel et al. 1999), K-band selected galaxies (Savaglio et al. 2004) and galaxies which happen to be along the lines of sight to bright background quasars (or QSOs). However, LBGs are characterized by pronounced star-formation and their inferred chemical abundances may relate to these regions rather than being representative of typical high-redshift galaxies. Weak metal line systems along the line of sight to quasars probe mainly galaxy haloes, rather than their bulges or discs (Fynbo et al. 2008). Taking advantage of ultra-deep Gemini multi-object spectrograph observations, Savaglio et al. (2004, 2005) studied the ISM of a sample of faint $K$-band selected galaxies at $1.4 < z < 2.0$, finding MgII and FeII abundances much higher than in QSO systems but similar to those in GRB host galaxies. Such studies can hardly be extended to higher redshift with the present generation of 8m class telescopes, because of the faintness of high-redshift galaxies. Since the discovery that gamma-ray bursts (GRBs) are extragalactic, we now can avail of an independent tool to study the ISM of high-redshift galaxies.
Metallicities measured in GRB host galaxies vary from less than $10^{-2}$ to nearly solar values and are on average larger than those found along QSO sightlines (see e.g., Fynbo et al. 2006, Prochaska et al. 2007). This result supports the notion that GRBs originate in dwarf and/or low-mass galaxies. Since star formation occurs in molecular clouds, the latter are expected to be the GRB birthplaces. In this scenario, absorption from ground-state and vibrationally excited levels of the H$_2$ molecules are expected, but these are typically not observed in GRB afterglow spectra (Vreeswijk et al. 2004, Tumlinson et al. 2007). The non-detection of H$_2$ molecules (with the exception of GRB080607, see Prochaska et al. 2009, Sheffer et al. 2009) can possibly be explained by photo-dissociation by the intense UV flux from the GRB afterglow.
Indeed, the main difference between QSO and GRB absorption spectroscopy is that QSOs are nearly stationary in their emission, while GRBs are the most variable and violent phenomena in the Universe. Thus, while QSOs have the time to ionize the ISM along their lines of sight, the physical, dynamical and chemical status of the circumburst medium in the star-forming region hosting GRB progenitors can be modified by the explosive event, through shock waves and ionizing photons. The transient nature of GRBs is manifested through the detection of fine structure and other excited levels of the atom and the ions , , and . These features are routinely identified in GRB spectra, and are most probably excited by the intense UV flux from the afterglow, since strong variation is observed when multi-epoch spectroscopy is available. This variation is not consistent with a pure infrared excitation or collisional processes (Prochaska, Chen & Bloom 2006, Vreeswijk et al. 2007, D’Elia et al. 2009a). Thus, assuming UV pumping as the responsible mechanism for the production of these lines, the distance of the gas to the GRB can be computed. This distance comes out to be of the order of a few hundred pc (see D’Elia et al. 2009b for GRB080330 and Ledoux et al. 2009 for GRB050730) or even in the kpc scale (see Vreeswijk et al. 2007 for GRB060418 and D’Elia et al. 2009a for the naked-eye GRB080319B).
GRB spectroscopy is also suitable for studying systems lying along the line of sight to GRBs. Surprisingly, the number density of strong intervening absorbers in GRB spectra is more than twice larger than along QSO sightlines (Prochter et al. 2006, Vergani et al. 2009), while absorbers do not show any statistical difference (Sudilovsky et al. 2007, Tejos et al. 2007). The reason for the excess in GRB spectra is still unclear, and a larger sample is needed to properly address this issue (Porciani et al. 2007, Cucchiara et al. 2009).
All these issues can now be systematically addressed using the X-shooter spectrograph. This is the first second-generation instrument at the ESO’s Very Large Telescope (VLT) at Paranal Observatory (Chile). It is a single target spectrograph capable of obtaining a medium resolution spectrum ($R=\lambda/\Delta\lambda=4000
- 14,000$) covering the spectral range 3000 - 24,800 Å in a single exposure thanks to the splitting of the light into three arms: ultraviolet/blue (UVB), visual (VIS) and near-infrared (NIR). The high spectrograph efficiency allowed us to obtain good quality observations of the GRB090313 afterglow (de Ugarte Postigo et al. 2010), although it being observed two days after the burst and under unfavourable conditions. We refer the reader to D’Odorico et al. (2006) for a more complete description of the X-shooter specifications and capabilities.
We discuss here the case of GRB090926A, observed by X-shooter the 27th of September 2009. We investigate both the local medium surrounding the GRB, and the intervening systems. We derive the metallicity by comparing the column densities of Hydrogen and metals, we search for molecular absorption and galactic emission lines, and we constrain the distance between the GRB and the absorber by comparing the ratios between the ground and excited level column densities with photoexcitation codes, under the assumption of indirect UV pumping production of the excited levels. An analysis of the intervening absorbers lying along the line of sight to the GRB afterglow and a search for diffuse interstellar bands (DIBs) is also presented.
The paper is organized as follows. Section $2$ makes a short summary of the GRB090926A detection and observations from the literature; Section $3$ presents the X-shooter observations and data reduction; Sections $4$ and $5$ are devoted to the study of the features from the host galaxy, Sect. $6$ derives the extinction curve shape for this GRB; Section $7$ presents the analysis of the other absorbing systems identified in the GRB090926A line of sight; finally in Sect. $8$ the results are discussed and conclusions are drawn. We assume a cosmology with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm m} = 0.3$, $\Omega_\Lambda = 0.7$. Hereafter, the \[\] refers to element abundances relative to Solar values and X refers to any chemical element.
GRB090926A
==========
GRB090926A was discovered by [*Fermi*]{} on September 26, 2009, at 04:20:26 UT, and was detected by both the GBM (Bissaldi 2009) and the LAT (Uehara et al. 2009) instruments. [*Swift*]{} repointed to the target 13 hr later and found the X-ray counterpart (Vetere et al. 2009). The afterglow was later reported to be detected at optical wavelengths with the Skynet/PROMPT telescopes (Haislip et al. 2009) and with UVOT (Gronwall & Vetere 2009); the reported PROMPT magnitude was $R\sim18$, although the observations took place almost 20 hr post burst. The redshift was secured by X-shooter, which observed the afterglow 2 hr after these optical detections. The preliminary reported value was $z=2.1062$ (Malesani et al. 2009). FORS2 spectroscopic observations and GROND photometric follow-up of the GRB090926A afterglow have been presented in Rau et al. (2010, hereafter R10): we will compare our results with those of R10 in the following.
Observations and data reduction
===============================
In the framework of the Science Verification phase program, we observed the afterglow of GRB090926A with X-shooter (D’Odorico et al. 2006), mounted at the VLT-UT2 telescope. The observations consist of 4 different exposures of 600 s each (see Table 1). The exposures were taken using the nodding along the slit technique with an offset of 5 arcsec between exposures in a standard ABBA sequence. The sequence began on September 27th, 2007 at 02:23:14 UT, $\sim 22$ hr after the GRB trigger. The magnitude of the afterglow at the time of the observation was reported by R10 to be $R = 18.7$, which translates into a $6600$Å flux of $0.8 \times 10^{-16}$ erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$. The total net exposure time of our observations is 40 min. The slit width was set to 0.9 in the VIS and NIR arms and 1.0 in the UVB arm. The UVB and VIS CCD detectors were rebinned to $1 \times 2$ pixels (binned in the spectral direction but not in the spatial one) to reduce the readout noise.
We processed the spectra using a preliminary version of the X-shooter data reduction pipeline (Goldoni et al. 2006). The pipeline performed the following actions: The raw frames were first bias subtracted and cosmic ray hits were detected and removed using the method developed by van Dokkum (2001). The orders were extracted and rectified in wavelength space using a wavelength solution previously obtained from calibration frames. The resulting rectified orders were shifted and coadded to obtain the final 2D spectrum. In the overlapping regions, orders were merged by weighting them according to the errors propagated during the entire reduction process. From the resulting 2D merged spectrum, a one dimensional spectrum was extracted at the source position. The one dimensional spectrum with the corresponding error file and bad pixel map is the final product of the reduction.
The resolution is $R \sim 10,000$ and the achieved spectral range is $\sim 3000$ to $\sim 24,800$ Å. The data below $\sim 3020$ and above $\sim 24,000$ Å are, however, totally dominated by noise. To perform flux calibration we extracted a spectrum from a staring observation of the flux standard BD +17${\deg}$4708 (Bohlin & Gilliland 2004). In this case we subtracted the sky emission lines using the Kelson (2003) method. This spectrum was divided by the flux table of the same star from the CALSPEC HST database (Bohlin 2007, http://www.stsci.edu/hst/observatory/cdbs/calspec.html) to produce the response function. The response was then interpolated where needed in the atmospheric absorption bands in VIS and NIR and applied to the spectrum of the source. No telluric correction was applied, so that prominent atmospheric bands, especially in the NIR arm, can still be seen (Fig. 1). We searched for variability in the absorption features, but we found none (the equivalent widths in the four spectra are constant at the $1.5 \sigma$ level), so we summed the four, flux-calibrated observations (see Fig. 1). This lack of variability is not surprising, since the spectra have been acquired nearly 1 day after the burst (see Sect. 4.3 for details). The signal-to noise ratio (per pixel) ranges from $\sim 10$ to $\sim 30$ in the co-added spectrum. In order to perform line fitting (see section 4) the spectrum was normalized to the continuum, which was evaluated by fitting the data with cubic splines, after the removal of the absorption features. Finally, the noise spectrum, used to determine the errors on the best fit line parameters, was calculated from the real, background-subtracted spectrum using line-free regions. This takes into account both statistical and systematic errors in the pipeline processing and background subtraction.
The host galaxy atomic absorption features
==========================================
The gas residing in the GRB host galaxy is responsible for many of the features observed in the GRB090926A afterglow spectrum. Metallic features are apparent, from neutral (, , ), low ionization (, , , , , , , , ) and high ionization (, , , , ) species. In addition, strong absorption from the fine structure levels of , , , and from the metastable level of is identified, suggesting that the intense radiation field from the GRB excites such features. Table 2 gives a summary of all the absorption lines due to the host galaxy gas. The analysis of the spectral features has been performed with FITLYMAN (Fontana & Ballester 1995). This program is able to simultaneously fit several absorption lines, linking the redshifts, column densities and Doppler parameters if required. FITLYMAN takes into account the atomic masses when a thermal model of the Doppler broadening is adopted (as we did), enabling to link the Doppler parameters of different species. The probed ISM of the host galaxy is resolved into two components separated by $48$ km s$^{-1}$ which contribute to the absorption system. The wealth of metal-line transitions allows us to precisely determine the redshift of the GRB host galaxy. This yields a vacuum-heliocentric value of $z=2.1071 \pm 0.0001$, setting the reference point to the red component of the main system; this component is the only one for which there is significant absorption from all the excited levels of the intervening gas (see Sect. 4.1 and 4.3). This redshift value supersedes that reported by Malesani et al. (2009), which was based on archival calibration data and using an older X-shooter pipeline version. The host galaxy environment will be described in the next sub-sections together with a study of the excited lines aimed at estimating the distance from the GRB of the circumburst gas.
Line fitting procedure
----------------------
In general, the analysis of the GRB environment is not straightforward due to the complexity of the absorption lines profile, which in several cases cannot be fitted with a single line profile. This means that many components contribute to the gas in the GRB environment. In other words, several layers of gas which may be close to or far from each other, appear mixed together in the spectrum (in velocity space). The presence of several components is thus indicative of clumpy gas in the GRB environment, composed of different absorbing regions each with different physical properties. This behaviour is particularly evident for GRB afterglows observed at high resolution (see e.g. Fiore et al. 2005, Thöne et al. 2008).
For what concerns GRB090926A, the and lines have the wider velocity range. This behaviour is common to many GRBs (see e.g. D’Elia et al. 2007, Piranomonte et al. 2008), where these lines are the ones that are most clearly detected and used to guide the identification of the different components constituting the circumburst matter. A two-component model provides a good fit for the and lines (Fig. 2). Thus, the redshifts and the Doppler parameters of these components were fixed in order to fit the other species present in the spectrum with the same model. This modeling adequately fits all other absorption lines at the GRB redshift. All species feature absorption in the red component, and most of them have absorption also in the blue component. Figures 3 and 4 show the two-component model fit to all the absorption lines at the GRB redshift. Asterisks mark the fine structure levels. The lines [ $\lambda\lambda$2166, 2217 and 2316]{} (Fig. 4, left) are produced by the second excited level ($^4F_{9/2}$). The blue component could not be derived because of blending with Ly$\alpha$ forest features (Fig. 4, right).
The column densities for all the elements and ions of the host galaxy absorbing gas, estimated using this two-component model, are reported in Table 2. The upper limits reported are at the 90% confidence level. The red and blue components are marked as I and II, respectively. The reference zero point of the velocity shifts has been placed at $z = 2.1071$, coincident with the redshift of the red component. Component I of needs a slight wavelength red-shift in order to be adequately fitted (Fig. 3, bottom left). Anyway, the resulting column density of this component is consistent with the value obtained linking the and central wavelengths. The features of some species, such as , , and , could be saturated at least in component I. In this case, the column densities in Table 2 should be regarded as lower limits to the real values. The column density of has been calculated using the $\lambda$1304 transition only (see Sect. 4.3). It is interesting to note that among the excited absorption features in the spectrum, produced by fine structure and/or metastable levels, , and show significant absorption in both components I and II (the absorption in component II from is marginal), while and only show this in the red, reference component I. We fixed the zero point to the redshift of component I because the fine structure levels require a high UV flux in order to remain populated (see Sect. 4.3), implying that component I is possibly closer to the GRB explosion site than II. As described in the introduction, the observation of excited states in GRB absorption spectra is a quite common feature. Sub-section 4.3 is thus devoted to these features and to the information that can be extracted from their analysis.
Species Observed transitions I (0 km s$^{-1}$) II ($-48$ km s$^{-1}$)
----------------- ------------------------------ ------------------- ------------------------
$^2S_{1/2}$ Ly$\alpha$, Ly$\beta$ $21.60 \pm 0.07$ -
$^2P^{0}_{1/2}$ $\lambda$1036, $\lambda$1334 $14.48 \pm 0.04$ $14.04 \pm 0.04$
$^2P^{0}_{3/2}$ $\lambda$1037, $\lambda$1335 $>14.45 \pm 0.04$ $14.01 \pm 0.05$
$^2S_{1/2}$ $\lambda$1548, $\lambda$1550 $14.43 \pm 0.02$ $13.89 \pm 0.03$
$^2S_{1/2}$ $\lambda$1238, $\lambda$1242 $14.08 \pm 0.03$ $13.66 \pm 0.07$
$^3P_{2}$ $\lambda$1039, $\lambda$1302 $>14.75 \pm 0.03$ $ < 13.8 $
$^3P_{1}$ $\lambda$1304 $14.49 \pm 0.03$ $ < 13.7 $
$^3P_{0}$ $\lambda$1306 $14.37 \pm 0.04$ $ < 13.7 $
$^2S_{1/2}$ $\lambda$1031, $\lambda$1037 $>14.60 \pm 0.16$ Blend
$^1S_0$ $\lambda$2852 $12.74 \pm 0.01$ $11.98 \pm 0.05$
$^2S_{1/2}$ $\lambda$1239, $\lambda$1240 $>14.05 \pm 0.01$ $13.05 \pm 0.02$
$\lambda$2796, $\lambda$2803
$^1S_0$ $\lambda$1670 $13.20 \pm 0.03$ $12.66 \pm 0.04$
$^2S_{1/2}$ $\lambda$1854, $\lambda$1862 $13.24 \pm 0.03$ $ < 12.3 $
$^2P^{0}_{1/2}$ $\lambda$1020, $\lambda$1190 $14.41 \pm 0.03$ $13.98 \pm 0.07$
$\lambda$1193, $\lambda$1260
$\lambda$1304, $\lambda$1526
$\lambda$1808
$^2P^{0}_{3/2}$ $\lambda$1194, $\lambda$1197 $13.96 \pm 0.03$ $13.42 \pm 0.09$
$\lambda$1264, $\lambda$1309
$\lambda$1533, $\lambda$1309
$^1S_{0}$ $\lambda$1206 $13.50 \pm 0.11$ Blend
$^2S_{1/2}$ $\lambda$1393, $\lambda$1402 $13.97 \pm 0.03$ $13.61 \pm 0.04$
$^4S^{0}_{3/2}$ $\lambda$1250, $\lambda$1253 $14.89 \pm 0.06$ $ < 14.3 $
$\lambda$1259
$^4S^{0}_{3/2}$ $\lambda$1062, $\lambda$1253 $14.45 \pm 0.19$ $ < 14.2 $
$^1S_{0}$ $\lambda$4227 $12.15 \pm 0.07$ $ < 11.9 $
$^2S_{1/2}$ $\lambda$3969 $13.21 \pm 0.03$ $ < 12.8 $
$a^7S_{3}$ $\lambda$2576 $ < 12.3 $ $ < 12.3 $
$a^6D_{9/2}$ $\lambda$1081, $\lambda$1096 $14.03 \pm 0.03$ $13.46 \pm 0.04$
$\lambda$1144, $\lambda$2344
$\lambda$2348, $\lambda$2374
$\lambda$2382, $\lambda$2586
$\lambda$2600
$a^6D_{7/2}$ $\lambda$2333, $\lambda$2396 $12.52 \pm 0.08$ $ < 12.2 $
$\lambda$2612, $\lambda$2626
$a^4F_{9/2}$ $\lambda$1559, $\lambda$2332 $ < 13.6 $ $ < 13.6 $
$^5D_{4}$ $\lambda$1122 $14.54 \pm 0.07$ $14.38 \pm 0.10$
$^2D_{5/2}$ $\lambda$1317, $\lambda$1370 $13.51 \pm 0.06$ $13.09 \pm 0.30$
$\lambda$1454, $\lambda$1703
$\lambda$1709, $\lambda$1741
$\lambda$1751
$^4F_{9/2}$ $\lambda$2166, $\lambda$2217 $13.54 \pm 0.02$ $12.47 \pm 0.36$
$\lambda$2316
: **Absorption line column densities for the two components of the main system.**
All values of the column densities are logarithmic (in cm$^{-2}$). Reported lower limits are due to possible line saturation.
Metallicities
-------------
The GRB090926A redshift was high enough to allow the Hydrogen Ly$\alpha$ and Ly$\beta$ lines to enter the X-shooter spectral window. Ly$\alpha$ can be clearly seen in Fig. 1 (upper panel) at $\sim 4000$ Å. We used the two Lyman features in order to constrain the Hydrogen column density. Figure 5 shows the Lyman-$\alpha$ and Lyman-$\beta$ profiles. Both features are damped and we can not separate the two components identified in the metal-line fits. The Hydrogen column density computed has $\log (N_{\rm H}/{\rm cm}^{-2}) =
21.60 \pm 0.07$ cm$^{-2}$, for a reduced $\chi^2$ value of $1.33$, and is virtually insensitive to the adopted b parameter. From this $N_{\rm
H}$ value, we can compute the metallicity for the host of GRB090926A, using the metallic column densities reported in Table 2. Due to the wide spectral coverage of X-shooter, this can be done for a large number of elements. We proceeded as follows: first of all, since the Lyman features cannot be resolved into components, we summed up line by line the values in table 2 (e.g., the two components) to obtain the total column density for each species; second, we summed the total column densities of the transitions belonging to the same atom (different ionization and excitation states), in order to evaluate the atomic column densities. These values have been divided by $N_{\rm H}$, and compared to the corresponding solar values given in Asplund et al. (2009). The results are listed in Table 3. Column 2 reports the total abundance of each atom, while columns 3 and 4 report the absolute and solar-scaled $N_X$/$N_{\rm H}$ ratios, respectively, with $X$ being the corresponding element in column 1. Lower limits are reported whenever saturation does not allow us to securely fit the metallic column densities. We derive very low metallicity values with respect to the solar ones, between $4.2\times10^{-3}$ and $1.4\times10^{-2}$. The very low value derived for is due to the fact that we could not fit low ionization lines, but the species only, meaning that this value is not truly representative for all the N ionization states.
Element $X$ $\log N_X /{\rm cm}^{-2}$ $\log N_X$/$N_{\rm H}$ $[X/{\rm H}]$
------------- --------------------------- ------------------------ -----------------
C $>15.05\pm0.10$ $>-6.55\pm0.10$ $>-2.94\pm0.10$
N $ 14.22\pm0.08$ $ -7.38\pm0.08$ $ -3.16\pm0.08$
O $>15.31\pm0.10$ $>-6.29\pm0.10$ $>-2.95\pm0.10$
Mg $>14.11\pm0.08$ $>-7.49\pm0.08$ $>-3.02\pm0.08$
Al $ 13.59\pm0.10$ $ -8.01\pm0.10$ $ -2.38\pm0.10$
Si $ 14.80\pm0.08$ $ -6.80\pm0.08$ $ -2.31\pm0.08$
S $ 14.89\pm0.10$ $ -6.71\pm0.10$ $ -1.85\pm0.10$
Ca $ 13.25\pm0.08$ $ -8.35\pm0.08$ $ -2.66\pm0.08$
Fe $ 14.86\pm0.09$ $ -6.74\pm0.09$ $ -2.19\pm0.09$
Ni $ 13.92\pm0.13$ $ -7.68\pm0.13$ $ -1.91\pm0.13$
: **Metallicities**
Excited levels
--------------
The level structure of an atom or ion is characterized by a principal quantum number $n$, which defines the atomic level, and by the spin-orbit coupling (described by the quantum number $j$), which splits these levels into fine structure sub-levels. In GRB absorption spectra, several excited features are detected at the GRB redshift, due to the population of both $n>1$ and/or $n=1$ fine structure levels. As mentioned before, component I of the main system in the spectrum of GRB090926A shows several absorptions from excited states. In particular, the first fine structure level of the ground state ($a^6D$), the $^2P^{0}_{3/2}$, $^2P^{0}_{3/2}$, $^3P_{1}$ and $^3P_{0}$ fine structure levels and the $^4F_{9/2}$ metastable level are present. Moreover, excited states of , and (marginally) are detected also in component II (see Table 2 for details).
There are basically two mechanisms to excite the gas of the GRB host galaxy to such states. The first is through collisional effects (if the electron density is sufficiently high, i.e., $\ge 10^5$ cm$^{-3}$); the second is through the absorption of electromagnetic radiation. In the latter case, the absorbed photons can be infrared, through direct population of the fine-structure levels of the ground state and other excited levels with a low value of $n$, or UV, through the population of higher levels followed by the de-population into the states responsible for the absorption features. Multi-epoch spectroscopy together with proper modeling of the atomic level population has proven to be a powerful tool to discriminate between these two processes. The strong variability in the column density of the and excited levels observed in GRB060418 (Vreeswijk et al. 2007) and GRB080319B (D’Elia et al. 2009a) ruled out collisional processes and direct infrared pumping as being responsible for the excitation. A collisional origin of the excitation can be ruled out even if multi-epoch spectroscopic data are missing. In fact, collisions populate higher energy levels less than lower energy ones. For instance, GRB080330 (D’Elia et al. 2009b) and GRB050730 (Ledoux et al. 2009) exhibit a $a^4F_{9/2}$ excited state column density larger than that of several fine structure levels of the $a^6D$ ground state. This means that radiative processes are at work also for these GRBs. For GRB090926A we do have multi-epoch spectroscopy, but the time lag between different observations (a few minutes, see Table 1) is too short when compared to the time delay between the GRB explosion and the epoch at which our data were taken. Supported by these results on atomic excitation mechanisms, we assume that indirect UV pumping by the fading afterglow is at work also for GRB090926A.
As a first step, we assume a steady state flux coming from the GRB, which in general is not a good approximation. Nevertheless, the GRB090926A afterglow was observed by X-shooter almost one day after the GRB explosion, and its flux level was decaying slowly at this stage. Under this assumption, we compute the ratio between the first excited level and the ground state for , and . We then use these values as input for the plot in Fig. 7 of Prochaska, Chen & Bloom (2006) in order to estimate the flux experienced by the absorbing gas. This plot has been produced using the model from Silva & Viegas (2002), which further assumes optically thin clouds. We first note that for component I the and ratios are not compatible, i.e., the flux levels inferred from these ions are not consistent. This is possily due to a slight saturation of the $\lambda$1526 line (see Fig. 3, top left), which was formerly used together with the $\lambda$1304 transition to evaluate the column density. Therefore, we recomputed the column density using just the $\lambda$1304 line, as described in Sect. 4.1. The flux levels with the new $N_{\ion{Si}{II}}$ for component I are now fully compatible. We further note that the ratio is far from being compatible with that of and . This confirms the saturation of the $\lambda$1302 transition (Fig. 3, middle), which can just be used to set a lower limit to the ground state column density. The ratio for component II is close to that for I. Thus, we derive a distance of the gas responsible for the absorption of the two components I and II of $d = 1.8 \pm 0.2$ kpc.
To check the results from the steady state approximation, we must compare our observed column densities to those predicted by a photo-excitation code for the time when the spectroscopic observations have been acquired. The photo-excitation code is that used by Vreeswijk et al. (2007) and D’Elia et al. (2009), to which we refer the reader for further details. Basically, it solves the detailed balance equation in a time-dependent way for a set of transitions involving the levels of a given species (e.g. and ). The equation depends on the flux level experienced by the absorbing gas. This flux is of course a function of the distance $d$ of the gas from the GRB explosion site, which is a free parameter of the computation and the quantity we want to calculate. The other free parameters are the initial column densities of the levels involved, which are assumed to be in the ground state before the GRB flux reaches the gas, and the Doppler parameter of the gas itself. The afterglow spectral index has been calculated directly from the merged, flux-calibrated X-shooter spectrum, and is $\beta \sim 0.9$ (see Sect. 6). The flux behaviour before the X-shooter observation has been estimated using the data in R10. In detail, if the flux in the $R$ band is $F_R=F_R(t_*) \times
(t/t_*)^{-\alpha}$, we have $F_R(t_*)=1.1\times 10^{-27}$ erg cm$^{-2}$ s$^{-1}$ Hz$^{-1}$ at $t_*=72.9$ ks, with a decay index $\alpha$ of 1.6 (-2.7) before (after) $t_*$. No spectral variation is assumed during the time interval between the burst and our observation (this assumption is strongly supported by the R10 data). The initial column densities of the ground states have been computed from the observed column densities of all the levels of each ion. The exact values are: $\log (N_\ion{Si}{II}/{\rm cm}^{-2}) = 14.44 \pm 0.04$ and $\log (N_\ion{Fe}{II}/{\rm cm}^{-2}) = 14.04 \pm 0.03$. Finally, the Doppler parameter values used as input of this model are $30$ and $90$ km s$^{-1}$, i.e. the values that best fit the and absorptions corresponding to components I and II, respectively.
We first modeled the photo-excitation. Fig. 6 (top) shows the model which best fits the data and the two theoretical curves compatible within the error bars for the first fine structure level column density. The distance of component I from the GRB explosion site results $d=2.6\pm0.3$ kpc. The same calculation has been performed using the atomic data. The results are displayed in Fig. 6 (bottom), and the estimated distance is $d=2.25\pm0.15$ kpc, which is consistent with that estimated using the data. In addition, our distances are less than 30% away from that estimated using the code by Silva & Viegas (2002), and compatible with it at the $2\sigma$ level, confirming that the steady state and optically thin approximations are appropriate for component I of GRB090926A. Regarding component II, we do not have excited features, so we can just set a lower limit to the GRB/absorber distance, which is $2.1$ ($2.2$) kpc adopting a Doppler parameter of $30$ ($90$) km s$^{-1}$. The fine structure features are instead present in component II. The lack of excited levels in components featuring fine structure ones is not atypical, see e.g. component III of GRB050730 identified in D’Elia et al. (2007). Nevertheless, the large errors for this feature and the strong dependence of the model from the Doppler parameter make the distance estimation for this component quite uncertain. We run our code using the Doppler parameters that best fit Components I ($30$ km s$^{-1}$) and II ($90$ km s$^{-1}$). The distance of component II from the GRB is found to be $4.4\pm0.6$ kpc for $b=30$ km s$^{-1}$, and $5.8\pm0.8$ kpc for $b=90$ km s$^{-1}$. Despite these values are consistent with the lower limits estimated through data, we caution that a direct comparison to check the distance is missing.
Other features at the host redshift
===================================
In this section we turn our attention to the search for galactic emission lines (Sect. 5.1) and absorption features not due to matter in the atomic gas phase. The latter can be produced by molecules (Sect. 5.2) or DIBs (Sect. 5.3).
Emission lines from the host galaxy
-----------------------------------
We searched for most of the stronger emission lines like \[\] $\lambda$ 3727, \[\] $\lambda$ 5007 and the Balmer lines from the host galaxy in the X-shooter spectrum, but found none. We did this by subtracting the spectral PSF. To derive upper limits for e.g., H$\alpha$, we then added artificial emission lines of increasing strength to the data, until the line was easily detectable. For H$\alpha$, which in this case is observed at $2~\mu$m, we find that an emission line of $9\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ with an intrinsic velocity width of 100 km s$^{-1}$ would have been detected (barring slit losses). This means that we would have detected emission lines from a host galaxy with a star formation rate $\gtrsim 2 M_\odot~\mathrm{yr}^{-1}$ (Kennicutt 1998). We also note that there is no Ly$\alpha$ emission in the trough of the DLA. The 5$\sigma$ detection limit in a 6Å wide extraction window is $3.0 \times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ corresponding to $\mathrm{SFR} < 1 M_\odot$ yr$^{-1}$. However, this limit is much more senstitive to uncertainties in radiative transfer of Ly$\alpha$ photons and to host galaxy dust extinction (see, e.g., Hayes et al. 2010). Although there is no evidence for dust extinction from the SED of the afterglow radiative transfer can still completely remove the Ly$\alpha$ emission from the line-of-sight (e.g., Atek et al. 2008). We note that many GRB host galaxies do have higher SFR than this limit (e.g., Christensen et al. 2004), and we therefore expect to be able to detect emission lines from many high-z GRB hosts using X-shooter in the future.
Molecular absorption features
-----------------------------
Molecular gas is expected to be found in star-forming environments, but the search for its absorption features has often given negative results (see e.g. Tumlinson et al. 2007). Fynbo et al. (2006) interpreted an absorption feature of the GRB060206 afterglow spectrum as a possible H$_2$ detection. Prochaska et al. (2009) and Sheffer et al. (2009) reported the presence of strong H$_2$ and CO absorption features in the spectrum of the GRB080607 afterglow. These are the first (and by now unique) positive detections of molecules in GRB host galaxies. For GRB090926A, we searched for the strongest features of the CO molecule, e.g., $\lambda\lambda$1510, 1478 and 1447, but we found no evidence of absorption at these wavelengths. Quantitatively, the upper limit for the column density depends on the adopted Doppler value $b$. Assuming it is in the range $30 - 90$ km s$^{-1}$ (i.e., the values that best fits components I and II, respectively, see Sect. 4.1), the resulting upper limit for the column density lies in the interval $\log (N_{\rm CO}/{\rm cm}^{-2}) < 14 - 14.3$. We also searched for H$_2$ absorption. The strongest H$_2$ transitions of the Lyman-Werner bands have a wavelength $<3000$ Å at the redshift of GRB090926A. Nevertheless, using the H$_2$ L2R0 transition, which falls around $3350$ Å, we can set a $90\%$ upper limit of $\log (N_{{\rm H}_2}/{\rm cm}^{-2}) < 14.9$ for a $b$ parameter fixed at $30$ km s$^{-1}$ and $\log (N_{{\rm H}_2}/{\rm
cm}^{-2}) < 15.3$ for $b$ fixed at $90$ km s$^{-1}$. Finally, we searched for other absorptions, due to the , and molecules, but also in these cases we found none.
Diffuse interstellar bands
--------------------------
DIBs (see Jenniskens & Desert 1994 and references therein) are broad absorption features in the near-infrared to UV wavelength range. They were observed for the first time more than 70 years ago in the Milky Way (Merril 1934), and more recently in near galaxies (see e.g., Cox & Cordiner 2008, Cordiner et al. 2008). Although they have been known for a long time and hundreds of features have already been identified, the nature of the carrier of such transitions is still uncertain. The identification is difficult since most of them are not correlated with each other. The most promising candidates are large molecules, which are thought to be polycyclic aromatic hydrocarbons. We searched for the ten strongest and most common DIBs in the GRB090926A afterglow spectrum, in the rest-frame wavelength range $\lambda = 4000 -
7000$ Å, but we found no positive detection. A conservative $2\sigma$ upper limit to the rest frame equivalent width of these systems (taking into account that they have very different FWHM) is ${\rm EW} < 0.3$ Å. This limit is set by the signal-to-noise ratio of the infrared arm of our X-shooter spectrum, i.e., is the “noise EW”.
The extinction curve shape
==========================
The shape of extinction curves at high redshift is a powerful tool to derive information about dust formation and possibly about the various processes affecting dust absorption and destruction close to GRB sites. In most cases, inferences about dust extinction curves are obtained by photometric observations of GRB afterglows which unfortunately is biased by a strong degeneracy between afterglow spectral slope and extinction. Moreover, in particular for very high redshift events, uncertainties in photometry of single bands which can also present specific calibration problems may affect the whole analysis (see for instance discussion in Stratta et al. 2007 and Zafar et al. 2010 about GRB050904). For early time afterglows even the possible non-absolute simultaneity of the available photometric information has to be properly considered to avoid spurious results.
For most GRB afterglows, when accurate multi-band photometry is available (e.g., Covino et al. 2008, Schady et al. 2010, Kann et al. 2010), the derived extinction curve is in fair agreement with what observed locally in the Small Magellanic Cloud (SMC, Pei 1992) although often, due to the limited wavelength resolution, this simply means that the observed extinction curve has to be chromatic (i.e. wavelength dependent) and featureless. A few remarkable exceptions have been recorded in an intervening system along the line of sight of GRB060418 (Vreeswijk et al. 2007), for GRB070802 (Krühler et al. 2008; Elíasdóttir et al. 2009) and GRB080607 (Prochaska et al. 2009), where the characteristic absorption feature at about 2175Å, prominent in the Milky Way extinction curve, has also been detected.
Clearly, when spectroscopic information is available (e.g., Liang & Li 2010), better constrained results can be derived, disentangling the extinction curve and spectral slope effects. The case of GRB090926A is a good example of the X-shooter capabilities in this context. The flux calibrated spectrum (Fig.1) has been analyzed after removing wavelength intervals affected by telluric lines of strong absorptions. The Ly$\alpha$ range has been included in the analysis due to the importance of its wavelength range for extinction determination using the Hydrogen column density reported in Table2. We then rebin the spectrum in bins of approximately 50Å$\;$ by a sigma-clipping algorithm to avoid the effect of residuals absorption systems. The flux-calibration of X-shooter spectra ranging from the UV to the $K$ band is not an easy task and the reduction pipeline (Goldoni et al. 2006) is still under active development. In order to obtain an acceptable fit we had to introduce additional systematic uncertainties at about $\sim 2.5$% level added in quadrature, which possibly reflects normalization biases of the three arms and/or still not fully modeled slit losses along the whole wavelength range. Assuming a power-law spectrum, we obtained $\beta = 0.89 \pm 0.02$ (errors at 1$\sigma$) where the afterglow spectrum is modeled, as customary in GRB literature, as $F_\nu \propto \nu^{-\beta}$. We try to fit the data using different extinction curves, namely, SMC, LMC, Milky Way and Starburst. The best fit is obtained assuming a SMC extinction curve with $E_{B-V} < 0.01$ mag at $3\sigma$ ($\chi^2/{\rm dof} = 1.14$ for 484 dof). Other extinction curve recipes were not required by the data.
The GRB090926A line of sight
============================
The absorption lines due to the gas belonging to the GRB host galaxy are the dominant features of our spectrum, but they are not the only ones. A detailed analysis of the data reveals that at least four other absorbers are present along the line of sight to GRB090926A. Three of these systems, those with the highest redshifts ($z=1.75 - 1.95$), show absorption from the [ $\lambda\lambda$1548, 1550]{} doublet, to which corresponds a well defined [ $\lambda$1215]{} line inside the Ly$\alpha$ forest. All these systems exhibit a very simple line profile, that can be fitted by a single Voigt function (Fig. 7). The Ly$\alpha$ of the system at $z=1.7986$ requires a Doppler parameter that is about three times that for the doublet, in order to be adequately fit. Anyway, if we assume a double component model for this Lyman-$\alpha$ feature, and fix the $b$ parameter of one component to that of the doublet, the estimated $N_{\rm H}$ column density is not significantly different from that computed using the single component model. The last of the GRB090926A intervening systems has a lower redshift ($z=1.2456$) and features absorption from the [ $\lambda \lambda$2796, 2803]{} doublet and marginally ($2.7 \sigma$) from the [ $\lambda$2852]{} line. This is a rather weak system, and the rest frame equivalent width for the [ $\lambda$2796]{} line is just EW$_{\rm r}=0.19\pm0.06~\AA$ ($2\sigma$ confidence). Again, a single component Voigt profile describes well the absorption features (Fig. 7).
Table 4 summarizes the rest frame equivalent widths, column densities and redshifts calculated for these intervening systems.
Species Transition Redshift $\log$ (N/cm$^{-2}$) EW$_{\rm r}$ (Å)$^a$
--- --------- ------------ ---------- ---------------------- ----------------------
1 CIV 1548, 1550 1.9466 $ 13.70 \pm 0.03 $ $ 0.15\pm 0.04 $
1 HI Ly$\alpha$ 1.9466 $ 14.64 \pm 0.04$ $ $
2 CIV 1548, 1550 1.7986 $ 13.63 \pm 0.03 $ $ 0.11\pm 0.03$
2 HI Ly$\alpha$ 1.7986 $ 14.56 \pm 0.07 $ $ $
3 CIV 1548, 1550 1.7483 $ 13.90 \pm 0.02 $ $ 0.21\pm 0.03$
3 HI Ly$\alpha$ 1.7483 $ 14.98 \pm 0.41$ $ $
4 MgII 2796, 2803 1.2456 $ 12.39 \pm 0.05 $ $ 0.19\pm0.06$
4 MgI 2852 1.2456 $ 11.47 \pm 0.13 $ $ $
: **Redshifts, absorption line column densities and equivalent widths for the intervening systems.**
$^a$ Rest frame equivalent widths for [ $\lambda$2796]{} and [ $\lambda$1548]{}; EW errors are given at the $2\sigma$ confidence level.
Conclusions and discussion
==========================
In this paper we present intermediate resolution ($R=10,000$) spectroscopy of the optical afterglow of GRB090926A, observed using the X-shooter spectrograph at the VLT $\sim 22$ hr after the trigger.
From the detection of Hydrogen and metal absorption features, we find that the heliocentric redshift of the host galaxy is $z=2.1071$. The spectrum shows that the ISM of the GRB host galaxy has at least two components contributing to this main absorption system at $z =
2.1071$. Such components, whose line centres are separated by $\sim
48$ km s$^{-1}$, are identified in this paper as I and II, according to their decreasing velocity values. The total width of the two components is $\sim 250$ km s$^{-1}$. We stress that the identification of just two components may be due to the X-shooter resolution. In fact, GRB afterglows observed with higher spectral resolution feature more complex environments either if the host galaxy absorber has a similar width (e.g., 7 components for GRB050922C, see Piranomonte et al. 2008) or even in case of smaller widths (GRB080319B has six components for a total width of $\sim 120$ km s$^{-1}$, see D’Elia et al. 2009a).
The absorption lines appear both as neutral metal-absorption, low ionization and high ionization species. In addition, strong absorption from the fine structure and metastable levels of several species are detected. The distances between the GRB and the two absorbers have been estimated using the code by Silva & Viegas (2002), in a steady state and optically thin approximation, using the ratios between the ground state and the first excited levels of different species to infer the flux level experienced by the absorbing gas. For both components we find that the absorber is located at a distance of $d=1.8\pm0.2$ kpc.
The value of this distance can be refined by comparing the column densities of the ground and excited levels to those predicted by a time dependent photo-excitation code. Using and we find that the absorbing gas of component I is located at $d = 2.40 \pm 0.15$ kpc from the GRB, a value which is in not far from that estimated assuming a steady state approximation. For component II, this distance is larger, $\sim 5$ kpc, but this value has been obtained using only, because we just have a lower limit for the fine structure column density. Despite this lower limit gives a distance consistent to the result, a safe cross check can not be performed.
The GRB090926A/absorber distance is compatible with that found for the 4 other GRBs for which a similar analysis has been performed, i.e., $1.7$ kpc for GRB060418 (Vreeswijk et al. 2007), $2-6$ kpc for GRB080319B (D’Elia et al. 2009a), $280$ pc for GRB080330 (D’Elia et al. 2009b) and $440$ pc for GRB050730 (Ledoux et al. 2009). This is a further confirmation that the power of a GRB affects a region of gas which is at least a few hundreds pc in size.
Several high ionization lines are detected in the GRB090926A spectrum (see Table 2). Our column density is within the range of the GRB sample studied by Prochaska et al. (2008), who claim that this ion can be located very close to the GRB explosion site. Fox et al. (2008) estimated a lower limit to the distance from the absorber using the non detection of the fine structure feature and a photoexcitation code similar to that used in this paper. They conclude that for GRB050730 this ion is located at distances greater than $400$ pc. An upper limit of the same order of magnitude can be roughly estimated also for GRB090926A, since we compute a tighter limit for the column density of $\log
(N_\ion{S}{IV}/{\rm cm}^{-2}) < 13.3$, but the flux of our GRB is about three times smaller than that of GRB050730 (compared at the time of the acquisition of the spectra). In addition, Fox et al. (2008) report the detection in six of the seven GRB spectra analyzed in their work of CIV high velocity components at 500-5000km/s. These gas can belong to foreground clouds but also be associated to the Wolf-Rayet wind of the GRB progenitors. We do not find any of such absorptions in the spectra of GRB090926A.
The redshift of GRB090926A allows us to determine the Hydrogen column density, which has $\log (N_{\rm H}/{\rm cm}^{-2}) = 21.60 \pm
0.07$. This value is lower than that found by R10 ($\log (N_{\rm
H}/{\rm cm}^{-2}) = 21.79 \pm 0.07$), but the $2\sigma$ regions overlap, and each value is consistent with the other at the $2.7\sigma$ level. Using $N_{\rm H}$ we evaluate the GRB090926A host galaxies metallicity. The values we find are in the range $10^{-3}-10^{-2}$ with respect to the solar abundances (see Table 3). R10 report a metallicity calculated using , , , and . The first three values are in perfect agreement with our estimates for S, Si and Al. Their value is consistent with our Oxygen lower limit (set mainly by the saturated features). However, the authors note that their extremely low value ($\sim -3.08$) is possibly due to saturation and it could consequently lead to an underestimate of the column density, and thus this should be regarded as a lower limit as well. Another way to derive a metallicity using Oxygen, is to evaluate the ground state column density through the fine structure one, in the hypothesis of a steady state UV pumping as the responsible for this excitation. This has proven to be a good approximation for component I, the only one featuring . Under this assumption, we obtain $\log (N_{\ion{O}{I}}/{\rm cm}^{-2}) \sim 16.05$ and a metallicity of \[O/H\] $= -2.18\pm0.16$, which is consistent with the values obtained using other elements. Finally, the R10 metallicity obtained using the features (\[Fe/H\] $\sim -2.93$) is not consistent with our Iron value of \[Fe/H\] $= -2.19\pm0.07$. This is because we considered the [$\lambda$1122]{} line which brings a relevant contribution to the total Iron column density for GRB090926A. Due to this contribution, we do not detect an overabundance of Silicon with respect to Iron, contrary to what reported by R10.
For what concerns our results, the extremely low value derived for (\[N/H\] $\sim -3.31$) is most certainly due to the fact that we could not fit low ionization lines, but the species only. For (\[Ca/H\] $\sim -2.71$) we can not exclude a possible contamination by sky lines. Despite this, we still have a metallicity in the range $4.2\times10^{-3}$ - $1.4\times10^{-2}$ with respect to solar. The average, logarithmic metallicity is \[X/H\] $= -2.14 \pm
0.09$ and is consistent with that computed by the R10 data excluding Oxygen and Iron. This value lies at the lower end of the GRB distribution (Savaglio 2006, Prochaska et al. 2007, Savaglio, Glazebrook & Le Borgne, 2009), in fact, only GRB050730 and GRB050922C have metallicity values below $10^{-2}$.
A powerful way to infer the nature and the age of objects whose morphology is unknown is to use abundances and abundance ratios (see Matteucci 2001). This method is based on the fact that galaxies of different morphological type are characterized by different star formation histories and these strongly influence the \[$X$/Fe\] versus \[Fe/H\] behaviour. For very large star formation rates, as expected in spheroids (bulges and ellipticals) in their evolution phases, the \[$\alpha$/Fe\] ratios are overabundant relative to the Sun in a large range of \[Fe/H\] values, and this is because in a regime of strong star formation the large number of Type II SNe acting in the early phases of galaxy evolution increases the \[Fe/H\] in the gas up to large values (almost solar and solar) on timescales too short for the Type Ia SNe to substantially enrich the gas in Fe. Therefore, the \[$\alpha$/Fe\] ratios will reflect the production ratio of SNe II which is larger than in the Solar birthplace. In fact, the bulk of Fe is supposed to have been produced by Type Ia SNe. On the other hand, objects evolving slowly with a low star formation rate, with respect to normal ellipticals, such as irregular galaxies, would show low \[$\alpha$/Fe\] ratios even at low \[Fe/H\] values. This is because in this case SNe Ia have time to pollute the gas much before it reaches the Fe solar value. Therefore, if we compare the measured abundance ratios in the host of GRB090926A with predictions from detailed chemical evolution models we can in principle understand the nature of the host. This is done in Fig. 8 where models for ellipticals and for irregular galaxies are compared with the measured abundances. These models, which are discussed in Fan et al. (2010, in preparation), show that the host of GRB090926A is probably an irregular galaxy with baryonic mass $10^{8}~M_{\odot}$ and evolving with a star formation efficiency (the inverse of the timescale of star formation) of $0.05
{\rm Gyr}^{-1}$. In fact the measured \[$\alpha$/Fe\] ratios agree better with the prediction for the irregular and are instead too low for an elliptical galaxy of baryonic mass $10^{10}~M_{\odot}$, shown for comparison. The model for the irregular galaxy takes into account metal-enhanced galactic winds, induced by SN feedback, which produce the loops in the predicted abundance ratios.
This kind of host galaxy, associated with the low metallicity of the GRB090926A ISM, agrees with the tentative evidence for a trend of declining ISM metallicity with decreasing galaxy luminosity in the star-forming galaxy population at $z = 2-4$ (Chen et al. 2009).
We also searched for other features at the host galaxy redshift. No emission lines were detected for the GRB090926A host, but our lower limits ensure that we will be able to detect emission lines from many high-redshift GRB hosts using X-shooter in the future. Similarly, we find no evidence for molecular absorption at the GRB redshift. The lower limit on H$_2$ translates into an upper limit for the Hydrogen molecular fraction of $f= 2N_{{\rm H}_2}/(2N_{{\rm H}_2}+N_{\rm H}) <
(3.0-7.4)\times10^{-7}$. The absence of molecules is not surprising, since Hydrogen molecular fractions $\log f > -4.5$ are detected in just $10\%$ of the QSO-DLA population. In addition, GRB environments with metallicities below $0.1 Z_\odot$ (such as that of GRB090926A) and low dust content can explain the lack of H$_2$ (see Ledoux et al. 2009). As a confirmation of this scenario, the only positive detection of molecules in GRB host galaxies has been reported in the GRB080607 afterglow, for which a solar or even super-solar metallicity has been inferred (Prochaska et al. 2009). Finally, DIBs have not been detected either.
The GRB090926A continuum has been fitted assuming a power-law spectrum. The best fit spectral index is $\beta =
0.89^{+0.02}_{-0.02}$ ($1\sigma$). This value is consistent with that obtained in R10 by fitting the GROND data ($0.98^{+0.06}_{-0.07}$), and is close to that obtained by the same authors fitting the optical/IR and the X-ray data together ($\sim 1.03$). Our best fit does not essentially allow for any intrinsic extinction since $E_{B-V}
< 0.01$ mag at $3\sigma$ adopting an SMC extinction curve. Other extinction curve recipes are not required by the data. This intrinsic extinction limit is consistent with that reported by R10.
The GRB090926A sightline has also been analyzed. The signal-to-noise level of our spectrum allowed us to be sensitive to lines with equivalent width (observed frame) as weak as $0.06-0.15$ Å (depending on the spectral region), at the $2\sigma$ confidence level. The redshift path analyzed for the search of () systems ranges from $z=2.107$ to $z=0.35$ ($z=1.44$). Four intervening systems between z = 1.95 and z = 1.24 have been identified. All systems have rest frame equivalent widths very small, below $0.3$ Å. Two of these systems, those marked as 1 and 3 in Table 4, have also been reported by R10, and their equivalent widths are consistent with ours. The line of sight toward GRB090926A is thus very clean, if compared to that of other GRBs. In fact, Prochter et al. (2006) and Vergani et al. (2009) claimed an excess of strong (EW $>1$ Å) absorbers along GRB sight lines with respect to QSO’s. On the other hand, the number of and weak systems is consistent along the line of sight of the two classes of objects (Tejos et al. 2007, 2009). The reason for the strong discrepancy is still uncertain, despite several possibilities have been already ruled out (e.g Porciani et al. 2007, D’Elia et al. 2010). The only proposed explanation which is still a possible candidate for this discrepancy is a multiband magnification bias in GRB sightlines (Vergani et al. 2009). Anyway, more observations and analysis are needed in order to solve this issue.
We thank an anonymous referee for several helpful comments in improving the quality and clarity of the paper. The Dark Cosmology Centre is funded by the Danish National Research Foundation.
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[^1]: Based on observations collected at the European Southern Observatory, ESO, the VLT/Kueyen telescope, Paranal, Chile, during the science verification phase, proposal code: 060-9427(A).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The study of $CP$ violation in the $B$ system allows us to perform quantitative tests of the $CP$ symmetry in the Standard Model. Many precise measurements of the sides and angles of the Unitarity Triangle used to test the theory are made possible by the abundant experimental data accumulated at the [$B$ factories]{} and the Tevatron. I review the Standard Model description of $CP$ violation and the key measurements which allow us to use $CP$ violation studies as a probe for New Physics.'
author:
- 'Gabriella Sciolla\'
bibliography:
- 'sample.bib'
title: Beauty in the Standard Model and Beyond
---
[ address=[Massachusetts Institute of Technology, 26-443\
77 Massachusetts Avenue\
Cambridge, MA 02139]{} ]{}
Introduction
============
$CP$ violation plays a fundamental role in the explanation of the matter-dominated universe [@sakarov]. In the Standard Model, $CP$ violation occurs in weak interactions due to the complex phase in the quark mixing matrix, the Cabibbo-Kobayashi-Maskawa (CKM) matrix [@KM]. This description of $CP$ violation, known as the CKM mechanism, provides an elegant and simple explanation of this phenomenon, and is in agreement with the experimental measurements in the kaon and $B$ sectors. However, the CKM mechanism fails to account for the observed baryon-to-photon density ratio in the Universe. This suggests that other sources of $CP$ violation must exist besides the CKM mechanism, and that $CP$ violation studies may be used as probes for New Physics. The key for these studies is to measure $CP$ violation in channels that are theoretically very well understood in the Standard Model, and look for deviations from the expectation.
A convenient tool for these studies is given by the Unitarity Triangle (UT), illustrated in Fig. \[UT\].
![The Unitarity Triangle.[]{data-label="UT"}](ut.eps){height="0.16\textheight"}
All sides and angles of the UT can be measured in the study of $B$ decays: the time-dependent $CP$ asymmetries measure the angles, while the sides can be determined by the measurements of the semileptonic $B$ decays and the $B$ mixing. Since one of the sides of the UT is normalized to a known quantity, only two measurements are necessary to define the triangle (e.g. the two sides). Any additional measurement (e.g. an angle) can therefore be used to test the CKM mechanism: any inconsistency can be interpreted as a sign of New Physics. Alternatively, we can look for New Physics by measuring the same quantity (an angle or a side) through channels that have different sensitivity to New Physics. It is important to note that precision and redundancy are essential for testing the theoretical predictions.
The asymmetric [$B$ factories]{} at SLAC and KEK were specifically designed for such measurements. In these machines, electrons and positrons collide at $\sqrt{s}\approx$ 10 GeV and produce an $\Upsilon(4S)$ resonance which decays into a $B\overline{B}$ pair. The clean environment, typical of $e^+e^-$ colliders, allows the two experiments, and Belle, to reconstruct $B$ decays with very high purity and reconstruction efficiency.
The two Tevatron experiments, CDF and [DØ]{}, study $B$ hadrons produced in $p\overline{p}$ collisions at $\sqrt{s}\approx$ 2 TeV. The harsher experimental environment is compensated by the large boost of the $B$ mesons in the laboratory frame and the fact that all $B$ hadrons can be produced. The Tevatron $B$ physics program is therefore complementary to the programs at the [$B$ factories]{}.
MEASUREMENT OF THE ANGLES
===========================
At the [$B$ factories]{}, the angles of the Unitarity Triangle can be precisely determined through the measurement of the time dependent $CP$ asymmetry, $A_{CP}(t)$: $$A_{CP}(t) \equiv \frac{N(\Bzb(t)\to f_{CP}) - N(\Bz(t)\to f_{CP})} {N(\Bzb(t)\to f_{CP}) + N(\Bz(t)\to f_{CP})},
\label{acpt}$$ where $N(\Bzb(t)\to f_{CP})$ is the number of that decay into the $CP$-eigenstate $f_{CP}$ after a time $t$. If only one amplitude contributes to the decay, $ A_{CP}(t) $ can be written as $$A_{CP}(t) = - \eta_f \Im(\lambda)\sin(\Dm t) ,
\label{acpt3}$$ where $\Delta m$ is the difference in mass between $B$ mass eigenstates and $\eta_f$ is the $CP$ eigenvalue of the final state. For some decays, $\Im(\lambda)$ is directly and simply related to an angle of the UT. For example, in the decay $B\to\jpsi K^0$, $\Im \lambda = \sin2\beta$.
The measurement of $A_{CP}(t)$ utilizes decays of the $\Upsilon (4S)$ into two neutral $B$ mesons, of which one is completely reconstructed into a $CP$ eigenstate, while the decay products of the other identify its flavor at decay time. The time $t$ between the two $B$ decays is determined by reconstructing the two $B$ decay vertices. The $CP$ asymmetry amplitudes are determined from an unbinned maximum likelihood fit to the decay time distributions separately for events tagged as and .
![Feynman diagrams that mediate the decays used to measure the angle $\beta$: a) $\Bz\to\mbox{charmonium}+K^0$; b) penguin dominated $B$ decays.[]{data-label="fey"}](diagram){height="0.2\textheight"}
Measurement of the angle $\beta$
--------------------------------
The decays $\Bz\to\mbox{charmonium}+K^0$, known as “golden modes” for the measurement of the angle $\beta$, are dominated by a tree level diagram $b\to c\overline{c}s$ with internal $W$ boson emission (Fig. \[fey\]a). Besides the theoretical simplicity, these modes are advantageous because of their relatively large branching fractions ($\sim 10^{-4}$) and the presence of the narrow $\jpsi$ resonance in the final state, which provides a powerful rejection of combinatorial background. The $CP$ eigenstates considered for this analysis are $\jpsi K_S$, $\psi$(2S)$K_S$, $\chi_{c1}K_S$, $\eta_cK_S$ and $\jpsi K_L$.
The results for the measurements of $CP$ violation in $\Bz\to\mbox{charmonium}+K^0$ are illustrated in Fig. \[sin2b\] (left). The asymmetry between the distributions of events tagged as $B^0$ and events tagged as $\Bzb$, clearly visible in a) and c), is a striking manifestation of $CP$ violation in the $B$ system. The corresponding time dependent $CP$ asymmetry is shown in b) and d). measures $\sin 2\beta=0.722\pm 0.040\pm 0.023$ [@sin2bbabar]. When combining this result with the corresponding measurement from the Belle experiment $\sin 2\beta=0.652\pm 0.039\pm 0.020$ [@sin2bbelle], we obtain $\sin 2\beta=0.685\pm 0.032$ [@hfag]. This implies that the angle $\beta$ is known to a precision of 1 degree.
![Left: measurement of $\sin 2\beta$ in the “golden modes”. Plot a) shows the time distributions for events tagged as (full dots) or (open squares) in $CP$ odd (charmonium $K_S$) final states. Plot b) shows the corresponding raw $CP$ asymmetry with the projection of the unbinned maximum likelihood fit superimposed. Plots c) and d) show the corresponding distributions for $CP$ even ($\jpsi K_L$) final states. Right: measurements of $\sin2\beta$ in penguin dominated modes. []{data-label="sin2b"}](babar-sin2b-2.eps "fig:"){height="0.35\textheight"} ![Left: measurement of $\sin 2\beta$ in the “golden modes”. Plot a) shows the time distributions for events tagged as (full dots) or (open squares) in $CP$ odd (charmonium $K_S$) final states. Plot b) shows the corresponding raw $CP$ asymmetry with the projection of the unbinned maximum likelihood fit superimposed. Plots c) and d) show the corresponding distributions for $CP$ even ($\jpsi K_L$) final states. Right: measurements of $\sin2\beta$ in penguin dominated modes. []{data-label="sin2b"}](sPengS_CP "fig:"){height="0.4\textheight"}
In the Standard Model, final states dominated by $b\to s \overline{s} s $ or $b\to s \overline{d} d $ decays offer a clean and independent way of measuring $\sin2\beta$ [@sPenguin]. Examples of these final states are $\phi K^0$, $\eta 'K^0$, $f_0K^0$, $\pi^0 K^0$, $\omega K^0$, $K^+K^-K_S$ and $K_S K_S K_S$. These decays are mediated by the gluonic penguin diagram illustrated in Fig. \[fey\]b. With contributions from physics beyond the Standard Model, new particles such as squarks and gluinos could participate in the loop and affect the time dependent asymmetries [@phases].
The decay $\Bz\to \phi K_S$ is ideal for these studies. In the Standard Model, this decay is an almost pure $b\to s \overline{s} s $ penguin decay, and its $CP$ asymmetry is expected to coincide with the one measured in charmonium + $K^0$ decays within a few percent [@phases]. Experimentally, this channel is also very clean, thanks to the powerful background suppression due to the narrow $\phi$ resonance. Unfortunately, the branching fraction for this mode is quite small ($\approx 8\times 10^{-6}$), therefore the measurement is limited by a large statistical error.
The decay $\Bz\to \eta^{\prime}K_S$ is favored by a larger branching fraction ($\approx 6\times 10^{-5}$). In the Standard Model, this decay is also dominated by penguin diagrams; other contributions are expected to be small [@etaprimetheory].
A summary of the measurements of $A_{CP}(t)$ in penguin modes [@phiks; @etaprimeKs; @kspi0; @f0k0; @omegaKs; @sin2bbelle] by the and Belle experiments is reported in Fig. \[sin2b\] (right). The naive averaging of all the penguin modes [@hfag] results in a 2.5$\sigma$ deviation from the value of $\sin 2\beta$ measured in the golden mode. However, this discrepancy has to be interpreted with caution since each mode can be affected by new physics in different ways.
Measurement of the angles $\alpha$ and $\gamma$
------------------------------------------------
The most accurate determination of the angle $\alpha$ comes from the measurement of the time-dependent $CP$ asymmetry in $\Bz\to\rho^+\rho^-$ decays. In the SM, these decays are dominated by a $b\to u \overline{u} d$ tree diagram. In the assumption that no other diagram contributes to the final state, $\Im\lambda = \sin2\alpha$. Penguin diagrams can contribute to this final state, but their contribution is thought to be small because of the small branching fraction measured for the $\Bz\to\rho^0\rho^0$ decay [@rho0rho0]. Since $\rho$ is a vector meson, the $\rho^+\rho^-$ final state is characterized by three possible angular momentum states, and therefore it is expected to be an admixture of $CP=+1$ and $CP=-1$ states. However, polarization studies [@rhorho_babar; @rhorho_belle] indicate that this final state is almost completely longitudinally polarized, and therefore almost a pure $CP=+1$ eigenstate. The parameter $\sin2\alpha$ is therefore measured from the amplitude of the time dependent $CP$ asymmetry, using the same technique described for the measurement of the angle $\beta$.
Other final states, such as $\Bz\to\pi^+\pi^-$ and $B\to\rho\pi$ [@alphababar; @alphabelle], provide additional constraints on the angle $\alpha$. Combining all and Belle results, we measure $\alpha =(105^{+15}_{-9})^{\circ}$ [@CKMFitter].
The angle $\gamma$ is measured exploiting the interference between the decays $B^+\to D^0K^+$ and $B^+\to\overline{D}{}^0K^+$, where both $D^0$ and $\overline{D}{}^0$ decay to the same final state. This measurement can be performed in three different ways: utilizing decays of $D$ mesons to $CP$ eigenstates [@GWL]; utilizing doubly Cabibbo-suppressed decays of the $\overline{D}$ meson [@ADS]; exploiting the interference pattern in the Dalitz plot of $D\to K_S\pi^+\pi^-$ decays [@GGSZ]. Currently, the last analysis provides the best measurement of the angle $\gamma$. Combining all results from and Belle, we measure $\gamma=(65\pm 20)^{\circ}$ [@utfit].
MEASUREMENT OF THE SIDES
==========================
The left side of the Unitarity Triangle is determined by the ratio of the CKM elements $|V_{ub}|$ and $|V_{cb}|$. Both elements are measured in the study of semi-leptonic $B$ decays. The measurement of $|V_{cb}|$ is already very precise, with errors of the order of 2% [@hfag]. The determination of $|V_{ub}|$ is more challenging, mainly due to the large background coming from $b\to c\ell\nu$ decays, about 50 times more likely to occur than $b\to u\ell\nu$ transitions.
Two approaches, inclusive and exclusive, can be used to determine $|V_{ub}|$. In inclusive analyses of $B\to X_u\ell\nu$, the $b\to c\ell\nu$ background is suppressed by cutting on a number of kinematical variables. This implies that only partial rates can be directly measured, and theoretical assumptions are used to infer the total rate and extract $|V_{ub}|$. The theoretical error associated with these measurements is $\approx 4\%$. Averaging all inclusive measurements from the , Belle, and CLEO experiments we determine $|V_{ub}|=(4.45\pm 0.33 ) \times 10^{-3}$ [@hfag], where the error includes statistical, systematic and theoretical errors.
In exclusive analyses, $|V_{ub}| $ is extracted from the measurement of the branching fraction $B\to \pi\ell\nu$. These analyses are usually characterized by a good signal/background ratio, but lead to measurements with large statistical errors due to the the small branching fractions of the mode studied. In addition, the theoretical errors, dominated by the uncertainties in the form factor calculation, are $\approx 12\%$. Both experimental and theoretical errors are expected to decrease in the future, making this approach competitive with the inclusive method.
The right side of the Unitarity Triangle is determined by the ratio of the CKM elements $|V_{td}|$ and $|V_{ts}|$. This ratio can be determined with small theoretical uncertainly from the measurement of ratio of the $B^0$ and $B_s$ mixing frequencies. While the $B^0$ mixing parameter $\Delta m_d$ has been measured very precisely by many experiments [@hfag], the $B_s$ mixing parameter $\Delta m_s$ had escaped detection until recently, due to the difficulty in detecting its very fast oscillations. This spring, the Tevatron experiments succeeded in this endeavor, and published evidence for $B_s$ oscillations [@D0mixing; @CDFmixing], as described in detail in [@cdftalk]. The value of $\Delta m_s $ measured by CDF is $17.33^{+0.42}_{-0.21}\pm 0.07 \mathrm{ps}^{-1}$. Combining this measurement with the world average for $\Delta m_d$, one can extract $|V_{td}/V_{ts}|=0.208^{+0.008}_{-0.007}$.
CONCLUSION
============
Precise and redundant measurements of the sides and angles of the Unitarity Triangles have provided a crucial test of $CP$ violation in the Standard Model. The constraints on the ($\rho ,\eta$) plane due to the measurements described in this article are illustrated in Fig. \[rhoeta\]. The comparison shows excellent agreement between all measurements, as predicted by the CKM mechanism.
Measurements of time-dependent $CP$ violation asymmetries in penguin-dominated modes are sensitive to contributions from physics beyond the Standard Model. These measurements are still heavily dominated by statistical errors and will benefit greatly from a factor two increase in statistics that both and Belle are planning to achieve by 2008.
![ Constraints on the apex of the Unitarity Triangle resulting from the various measurements of its sides and angles. []{data-label="rhoeta"}](rhoeta){height="0.5\textheight"}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We calculate the bound state properties of $J/\psi$ in a hot and dense QCD plasma using phenomenological potentials augmented by inputs from perturbative QCD. The temperature and density region of study will be relevant in future heavy ion collision experiments at FAIR. We find that the effect of baryon density on the dissociation of $J/\psi$ is small in this regime. However we indicate that if there is a critical end point in the QCD phase diagram then strong density fluctuation will dissociate charmonia near hadronization. The measurement of $J/\psi$ suppression can therefore signify the existence of the critical point unambiguously.'
author:
- Purnendu
- Subhashis
bibliography:
- 'spectral\_refs.bib'
title: '$J/\psi$ in a hot baryonic plasma'
---
The aim of the ongoing relativistic heavy ion collision experiments is to create a deconfined phase of strongly interacting matter dubbed as quark gluon plasma (QGP). Almost 30 years ago, Matsui and Satz argued that the screening of the confining potential at high temperature will lead to the dissolution of heavy quark bound states in the plasma and their depleted production may be used for a forensic study of the hot and dense medium created in such collisions [@Matsui:1986dk]. One of the significant results of SPS heavy ion program was the observation of anomalous $J/\psi$ suppression. For $\sqrt{s_{NN}} = 17.3$ GeV Pb+Pb and In+In collisions, the relative yield of $J/\psi$ was found to be suppressed compared to estimates based on cold nuclear matter (CNM) effects alone beyond a centrality threshold [@Alessandro:2004ap; @*Arnaldi:2007zz]. High statistics data for quarkonia suppression in Au+Au collisions at $\sqrt{s_{NN}} = 200$ GeV at RHIC [@Adare:2006ns] and in Pb+Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV at LHC [@Pillot:2011zg; @*Chatrchyan:2012np; @*Chatrchyan:2012lxa] have been checked against screening based models and it lends strong support for the creation of a deconfined partonic phase [@Grandchamp:2002wp; @*Zhao:2011cv; @*Gunji:2007uy; @*Strickland:2011mw].
The bulk matter created at RHIC and LHC have low baryon densities, $\mu_B
\simeq 0$. Upcoming Facility for Antiproton and Ion Research (FAIR) at GSI will collide heavy ions in the energy range $\sqrt{s_{NN}} \sim 6-10$ GeV. Numerical simulations employing different dynamical models show that a medium with high baryon density and relatively low temperature is likely to be created at such collision energies [@Arsene:2006vf]. CBM collaboration at FAIR, in particular, has a dedicated heavy flavor program and is expected to throw light on the possible in-medium modifications of open and hidden charm spectra. At zero baryon density, the excited states of charmonium family - $\chi_c$ and $\psi^\prime$ melt just above $T_{pc}^0$ where $T_{pc}^0 \simeq 170$ MeV is the *pseudo-critical* temperature of QCD in a baryon symmetric medium. The ground state may survive upto $\sim 1.5\, T_{pc}^0$ and the maximum temperature achievable at FAIR will be below it whereas the chemical potential $\mu_q$ could be as high as $\sim 2\,T_{pc}^0$. Should we expect to see a density driven melting of $J/\psi$ at low energy collisions then? The purport of the present paper is to find an answer.
Heavy quark spectroscopy is well described by non-relativistic potential models [@Lucha:1991vn] at zero temperature. In statistical QCD, the choice of the potential is debated and it is unclear whether free energy, internal energy or a linear combination thereof is the right candidate for it. Nevertheless, potential models have been extensively used at finite temperature to calculate various in-medium properties of quarkonia, see [@Mocsy:2013syh; @Rapp:2009my] for recent reviews. Apart from simplicity, the advantage of the potential model is that a slew of information can be extracted from it at no cost. Such studies complements first principle calculation from lattice gauge theory and seems indispensable now for baryon rich phase of QCD where progress in lattice computation is hindered by hitherto unsolved sign problem.
As alluded earlier, our aim here is to understand the in-medium modification of charmonia in a hot baryonic plasma which might be produced at low energy heavy ion collisions. We scan the region of phase diagram where $T/T_{pc}^0 \sim (1 - 1.5)$, and $\mu_q/T_{pc}^0
\sim \left(1 - 2\right)$ where $T$ and $\mu_q$ are equilibrium temperature and chemical potential of the system respectively. It is assumed that $\mu_u = \mu_d = \mu_q$. How does the finite baryon chemical potential influence the charmonium dissociation? A large chemical potential implies increased screening or weak binding of charmonia in the medium. A substantial background temperature is responsible for, apart from decrease in binding, rupture of resonances through partonic breakup processes. In the extreme case of cold and dense quark matter $\mu_q/T \to \infty$, partonic dissociation shuts off and sharp nature of Fermi surfaces may lead to nontrivial modification of heavy quark bound states [@Kapusta:1988fi]. We relegate this issue for discussion elsewhere.
First quantitative assessment of quarkonia dissociation within potential model in a hot QCD plasma was made by Karsch, Mehr and Satz [@Karsch:1987pv]. For the $Q\bar{Q}$ free energy, following choice was adopted, $$\label{pot1}
\mathcal{F} = \frac{\sigma}{m_D} \left(1 - e^{-m_D r}\right) -
\alpha \frac{ e^{-m_D r}}{r}\,.$$ Here $\sigma$ is the string tension and $\alpha = C_F \alpha_s$. $C_F = (N_c^2 - 1)/(2 N_c)$ and $\alpha_s$ is the coupling constant of QCD. $N_c = 3$ is the number of color. $m_D$ is the electric screening mass. The long range part of the free energy can be realized in Gribov-Zwanziger-Sringl scenario of confinement [@Gribov:1977wm; @*Zwanziger:1989mf; @*Stingl:1985hx] involving a $D = 2$ gluon condensate [@Megias:2007pq; @*Riek:2010fk].
Since the free energy contains an entropy contribution at finite temperature, $\mathcal{F} = \mathcal{U} - T S$, it is not the potential *per se*. So it was suggested to use the internal energy instead [@Shuryak:2004tx]. Lattice based internal energy were employed in several [@Shuryak:2004tx; @Alberico:2005xw; @*Cabrera:2006wh] investigations. Soon it was realized that since the entropy changes rapidly across the transition temperature, internal energy computed on lattice provides more binding than the vacuum potential. In [@Mocsy:2007jz; @*Mocsy:2007yj] the authors constructed a model for “maximally binding” potential by fitting the lattice data for free energy at short and long distances. Since a first principle derivation is not possible, we follow here a simpler approach suggested in [@Dumitru:2009ni]. The internal energy is obtained here by subtracting entropy (and number density) contribution at all distances from the free energy in Eq. , $$\begin{aligned}
\label{pot2}
\mathcal{U} &=& \mathcal{F} - T \frac{\partial \mathcal{F}}{\partial T}
- \mu_q \frac{\partial \mathcal{F}}{\partial \mu_q} \nonumber \\
&=&\frac{2 \sigma}{m_D} \left(1 - e^{-m_D r}\right) -
e^{-m_D r} \left(\sigma r + m_D + \frac{\alpha}{r}\right). \end{aligned}$$ Running of $\alpha_s$ is neglected in arriving at . The problem of overshooting the vacuum potential is not eliminated in but it is minimal near $T_{pc}^0$ where most of the bound states are supposed to melt [@Dumitru:2009ni].
Recently, Laine and collaborators [@Laine:2006ns; @*Laine:2007gj; @*Burnier:2007qm] have shown that the real time static $\bar{Q}Q$ potential has an imaginary part and describes dissociation of quarkonium through scattering via exchange of a spacelike gluon (see also [@Beraudo:2007ky]). We equate the real part of potential to the internal energy in Eq. and augment it by a spin independent relativistic correction, the later is needed for a accurate description of charmonium spectrum [@Mocsy:2007jz; @Bali:2000gf]. To wit, real part of the potential reads $\Re \left\{V\right\} = \mathcal{U} - 0.8\sigma/(m_Q^2 r)$. The imaginary part is calculated in hard loop approximation [@Laine:2006ns; @*Laine:2007gj; @*Burnier:2007qm], $$\label{pot3}
\Im\left[V\right] = - 2 i \alpha T \int_0^\infty\, \frac{ds
s}{\left(s^2 + 1\right)^2}
\left(1 - \frac{\sin{m_D r s}}{{m_D r s}}\right)\,.$$ For the parameters in the potential, we take $\sigma = 0.223$ GeV^2^ and $\alpha = 0.385$. The electric screening mass is written as, $
m_D^2 = 4 \pi \alpha_s \kappa_1^2 \left(1 + N_f/6\right) T_s^2
$ where, $$\label{eqforts}
T_s^2 = T^2 \left(1 + \kappa_2 \frac{3N_f}{\pi^2 \left(6 + N_f\right)}
\frac{\mu_q^2}{T^2} \right)\,.$$ We call $T_s$ an effective screening temperature. It can be thought of as the equilibrium temperature of a plasma without a net baryon excess that produces the same amount of electric screening as the plasma with temperature $T$ and chemical potential $\mu_q$. The encapsulation of the combined effect of temperature and density in $T_s$ makes comparison with corresponding result at zero density easier. $\kappa_1$ and $\kappa_2$ are parameters to take care of nonpertubative effects in the transition region. We take $\kappa_1 = 1.4$ as follows from comparing leading order result of screening mass with that from a fit to long distance part of lattice $Q\bar{Q}$ free energy [@Kaczmarek:2005ui]. Determination of $\kappa_2$ is little subtle. On general ground, it is expected that $\kappa_2 \simeq 1$ [@Simonov:2007xc]. This is also consistent with lattice result in [@Doring:2005ih] for $T \geq 1.5 T_{pc}$. Curiously enough the lattice simulations seem to suggest a divergent behavior of $\kappa_2$ and hence a diverging screening mass close to $T_{pc}$. Later we shall argue that this divergence in the screening mass is a reflection of the proximity to a critical end point and discuss the correlated consequences. For the moment being, however, we neglect this divergence and set $\kappa_2 = 1$ in what follows.
 Evolution of temperature and chemical potential in the central hotspot for most central $(b = 0)$ [Au + Au]{} collision at $\sqrt{s_{NN}} = 7.62$ GeV.](T_mu_Ts_urqmd){width=".7\columnwidth"}
Let us now proceed to evaluate bound state properties. For definiteness, we focus on the central hotspot and consider most central $(b = 0)$ collisions. We take the evolution of central baryon density and energy density from URQMD simulation at $\sqrt{s_{NN}} = 7.62$ GeV [@Arsene:2006vf]. The local temperature and chemical potential are then extracted using a “*fuzzy bag*” equation of state [@Pisarski:2006yk] as parameterized in [@Kapusta:2010ke]. The fuzziness here merely represent the $T^2$ correction in pressure and could be understood in terms of $D = 2 $ gluon condensate akin to the potential in Eq. [@Megias:2009mp]. We assume that the system equilibrates at time $\tau_0 = 3.5$ fm/c which corresponds to little more than the passing time of two nuclei $\tau = 2 R_A/\sqrt{\gamma^2 - 1}$. The evolution is followed until $\tau_f = 6.5$ fm/c when the temperature falls below hadronization temperature $T_{pc}^0 = 170$ MeV. For brevity, we take here the same hadronization temperature as in zero baryon density case. The resulting evolution of temperature and quark chemical potential is shown in Fig. \[fig1\]. With $N_f = 2$, $T_s \simeq
T\sqrt{1 + 0.076 \, \mu_q^2/T^2}$ from and it obviates from and that the medium effects on the quarkonia spectroscopy are thus essentially determined by the background temperature and the density has little role to play except in extreme conditions. This is transpired in Fig. \[fig1\] wherein it is shown that $T_s$ remains close to $T$ for the entire evolution of the system.
The potential embodied in and is now fed into Schrödinger equation and complex energy eigenvalue $E = M
-i\frac{\Gamma}{2}$ is solved for. The binding energy of the resonance is obtained as $\epsilon =
2 m_c + \Re\left\{V\left(r \to \infty\right)\right\} - M$, where $m_c = 1.3$ GeV is the charm quark mass. A resonance is effectively dissociated when binding energy and decay width come at par $\epsilon = \Gamma$.
The evolution of binding energy and decay width of $J/\psi$ in the central hotspot are shown in Fig. \[fig2\]. The decay width remains lower than the binding energy even at earliest time of evolution when most extreme condition of temperature and density are met. As the system cools and dilutes, the screening and decay width become weaker and correlation between $Q\bar{Q}$ pair grows until the system hadronizes.
We delineate in Fig. \[fig3\] the evolution of $J/\psi$ spectral function in the central hotspot. Close to threshold, the spectral function $\rho_v \left(\omega\right)$ is related to the forward correlator, $$\rho_v \left(\omega\right) = \lim_{\vec{r},\vec{r}^\prime \to 0}
C_v^{>}\left(\omega, \vec{r}, \vec{r}^\prime\right)\, + \mathcal{O}\left(
e^{-\frac{2m_c}{T}}\right).$$ A nice algorithm has been presented in [@Burnier:2007qm] for the numerical evaluation of the spectral function which we followed here.
The dissolution of a resonance is signaled by the disappearance of the corresponding peak from the spectral function. As seen from the figure, the $J/\psi$ peak is not smeared out even at the initial time. The ground state remains strongly correlated throughout the evolution.
 evolution of $J/\psi$ spectral density](spectral_fig){width=".7\columnwidth"}
The strong correlation between quark-antiquark pair does not imply the survival of the resonance in the medium. Scattering with the particles in the heatbath will destroy this correlation and put the quark and antiquark in separate trajectories. The pertinent observable here is the survival probability, $S = \exp\left(- \int_{\tau_0}^{\tau_f} d\tau\,
\Gamma \right)$ which measures the fraction of charmonium surviving the trek in the medium. Since the density effect on the screening properties of the medium is rather small the survival probability is essentially determined by the time of exposure of $J/\psi$ to the medium and the background temperature. For the energy range covered by FAIR, the changes in the local temperature and plasma life time with respect to collision energy do not change much so we do not expect appreciable change in the observed suppression when collision energy is varied. At higher collision energies, competing effect from the regeneration of charmonium in the medium becomes important [@Zhou:2013aea]. In fact, the regeneration of charmonium is arguably the reason for similar $J/\psi$ suppression at RHIC and SPS. Combined together, the direct dissociation of primordial charmonium and the regeneration in the medium is expected to result in a rather flat suppression pattern of $J/\psi$ from low to moderate collision energy where baryon density could have had any effect. If an appreciable deviation of in-medium charmonium suppression from the baseline measurement at SPS is observed here then it is possibly a hint for a new physics.
What this new physics could be? Theoretically it has been argued that there is a critical end point (CEP) in the QCD phase diagram where the line of first order phase transition terminates at a second order point [@Stephanov:1998dy]. The conjectured critical point belongs to the universality class of 3D Ising model. The exact coordinate of the CEP on the phase diagram is currently unknown but lattice calculations have provided some hazy clue about its location [@Fodor:2004nz; @*Datta:2012pj]. If such a critical point exists it will lead to enhanced susceptibilities which can be measured through event-by-event analysis of fluctuation of conserved charges. Since the fermionic contribution to the electric screening mass is proportional to the quark number susceptibility $m_{D,q}^2 \propto \chi_q$ [@McLerran:1987pz; @*Chakraborty:2001kx], an enhancement in susceptibility is also expected to lead to an increased screening mass. Precisely this behavior is observed in lattice simulation at finite chemical potential near $T_{pc}^0$ [@Doring:2005ih; @Ejiri:2009hq]. We assume that the critical behavior of succeptibility is also shared by the screening mass and write $(m_D^i)^2 = 2 \pi \alpha_s \chi_q^i/3$, where $m_D^i$ and $\chi_q^i$ are irregular part of the electric screening masses and quark number susceptibility respectively. The divergent part of the quark number number susceptibility is gleaned from [@Kapusta:2012zb], $$\label{chi_crit}
\chi_q^i = \frac{9 n_c^2}{\left(\delta + 1\right) P_c}
\left[\frac{1}{3} \frac{\left(\delta - 1\right)}{\left(2 - \gamma\right)}
t^\gamma + 5 \delta \left|\eta\right|^{\delta - 1}\right]^{-1}\,.$$ Here, $t = (T - T_c)/T_c$ and $\eta = (n - n_c)/n_c$, $n$ is the quark number density and $P$ is the pressure. $T_c$, $P_c$ and $n_c$ are the critical values of the respective variables. The critical exponents are $\gamma = 1.24$ and $\delta = 4.815$. The regular part of the screening mass has been mentioned in . So we can now see how the presence of a critical point inflict upon the survival of $J/\psi$ in the deconfined medium.
 evolution of $J/\psi$ spectral function in presence of a critical point. We chose $(T_c, \mu_{c,q)} = (159.8, 138.57)$ MeV. ](spectral_fig_crit){width=".70\columnwidth"}
In Fig. \[fig4\] we display the spectral function of $J/\psi$ near the critical point. As the critical point is approached, the spectral strength of the $J/\psi$ is significantly reduced. The loss of quark-antiquark correlation in this case is brought about by the increase of screening mass near the critical point. This should be contrasted with high $T$ behavior in Fig. \[fig3\] where disappearance of the spectral peak is caused both by increased screening (reduction in strength) as well as increase in decay width (smearing of peak). This leads to an interesting picture of charmonium survival in a baryonic plasma depending on whether or not the critical point is hit or missed during the course of evolution. If the evolution of the system cross the phase boundary away from the critical point then the singlet quark-antiquark correlation goes on increasing till hadronization. On the other hand, the critical point will be hit if the evolution of the system proceeds in proximity since CEP acts as attractor of hydrodynamical trajectories [@Nonaka:2004pg; @*Asakawa:2008ti]. As the critical point is approached, the $Q\bar{Q}$ correlation goes on dwindling due to increased screening and thermal excitation can easily break it off. The divergence of the susceptibilities imply that the trajectories in the $(T,\mu_q)$ plane linger near the CEP. More the bound state stays close to the CEP more likely it is to be broken by scattering in the background medium. It should be emphasized that the critical point presents a difficult condition for the hidden charm states to be realized close to hadronization. By the time the hadronization is complete, the signature of the CEP is imprinted in the near absence of charmonia in the medium and it is unlikely that subsequent hadronic evolution will mask it. The sudden drop of $J/\psi$ yield at the critical point will therefore provide a clean and robust signal for its existence.
Summarizing, we have discussed in detail the bound state properties of $J/\psi$ in a hot baryonic medium. This provides the requisite input for an all embracing investigation of charmonium production at low collision energy. Work along this direction is under progress and will be reported elsewhere. Furthermore, we have argued that if the evolution of the medium proceeds through a critical point then strong density fluctuation will remove the charmonium states from the spectrum before hadronization. This opens up an interesting possibility to locate the critical end point through the measurement of charmonium suppression.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'Marcasite class of compounds provide facile platform to explore novel phenomena of fundamental and technological importance, such as unconventional superconductivity or high performance electrocatalyst. We report the synthesis and experimental investigation of a new marcasite CoSeAs in this letter. Experimental investigation of the new material using neutron scattering measurements reveal weak magnetic correlation of cobalt ions below $T$ = 36.2 K. The modest isotropic exchange interaction between cobalt moments, inferred from random phase approximation analysis, hints of magnetically unstable environment. It is a desirable characteristic to induce unconventional superconductivity via chemical pressure application.'
author:
- 'Y. Chen$^{1}$'
- 'G. Yumnam$^{1}$'
- 'A. Dahal$^{1}$'
- 'J. Rodriguez$^{2,3}$'
- 'G. Xu$^{2}$'
- 'T. Heitmann$^{4}$'
- 'D. K. Singh$^{1,*}$'
title: Magnetic order and instability in newly synthesized CoSeAs marcasite
---
Transition metals are at the core of almost every novel phenomena in magnetism.[@Zaanen; @Carlin] Among the many chemical groups of transition metal compounds, marcasite phase is one of the most intriguing lattice groups with FeS$_{2}$-type crystal structure.[@marcasite; @marcasite2] These materials are of strong technological importance, especially for application in photovoltaics and in the design of efficient electrocatalyst. Some of the notable examples include the demonstration of high performance electrocatalyst in FeX$_{2}$ ($X$ = S, Se) and high absorption coefficient photovoltaic property in CoSe$_{2}$.[@Chen; @Zhang; @Li] Marcasites are also known to manifest superconducting phenomena of both conventional and unconventional origins.[@Hull; @Amsler] The proposed observation of unconventional superconductivity in marcasite phase FeBi$_{2}$ is attributed to the competing instability of underlying magnetism in the system.[@Amsler] The chemical structure of a marcasite is described by the distorted octahedrons of six anions (ligands) enveloping the cation of 3d or 4d transition metal ion.[@Endo] Despite the presence of transition metal as the key constituting element in the stoichiometric composition, most of them are either diamagnetic or paramagnetic with semiconducting electrical characteristic.[@Hulliger] The Jahn-Teller distortion in transition metal ion coordination octahedron is arguably responsible for the non-magnetic ground state in majority of the transition metal marcasites.[@Hull] We have synthesized a new compound CoSeAs in this series. Detailed experimental investigations of CoSeAs using elastic and inelastic neutron scattering measurements suggest the development of long range magnetic order below $T_c$ = 36.2 K. It sets a new precedent in 1:1:1 stoichiometric composition of the corresponding lattice group. Furthermore, we find that Co ions are correlated with the weak nearest neighbor exchange interaction, $J$ = 0.25 (4) meV, which makes the system susceptible to a transition to non-magnetic or different phase of matter, such as superconductivity, under modest external effect.
CoSeAs stands at the cross-roads of CoSe and CoAs compounds that crystallize in MnP-type tetragonal structure. While CoSe is argued to manifest a combination of ferromagnetic and spin glass properties,[@Efrain; @Paglione] CoAs is considered non-magnetic.[@Selte] However, both materials exhibit the metallic characteristic. Sharing magnetic traits of both compounds, CoSeAs is on the verge of magnetic instability. CoSeAs crystallizes in the FeS$_{2}$-type marcasite structure with weak metallic characteristic, bordering to the semiconducting phase at low temperature. We have synthesized the polycrystalline samples of CoSeAs using repetitive solid-state reaction method in evacuated quartz tubes. The starting materials were 99.998% Co, 99.999% Se (Alfa Aesar) and 99.997% As (Sigma-Aldrich).[@NIST] CoSe was synthesized from the stoichiometric composition of Co and Se. The mixture was grinded, pelletized and loaded into a quartz tube, subsequently evacuated and sealed, then annealed at 900$^{o}$ C for two days. After confirming the pure structure of CoSe (with no oxidation) using X-ray diffraction (XRD) measurement, stoichiometric amount of As was added to CoSe powder. The mixture was grinded, pelletized and sintered in evacuated quartz tube at 900$^{o}$ C for another two days. As shown in Fig. 1, the XRD pattern clearly manifests the high quality of as synthesized polycrystalline CoSeAs. X-ray peaks are well indexed by the $Pnnm$ space group body centered orthorhombic FeS$_{2}$-type marcasite structure, as shown in the inset in Fig. 1, with lattice parameters of $\textit{a}$ = 4.756 $\AA$, $\textit{b}$ =5.756 $\AA$ and $\textit{c}$ =3.570 $\AA$.
{width="8.8"}
There is not much known about CoSeAs. The knowledge of electrical conducting properties and a theoretical understanding of the density of states at the Fermi surface are necessary to characterize the new material. We have performed first principles electronic structure calculations for CoSeAs based on the density functional theory by employing the plane-wave basis set, as implemented in the Quantum-ESPRESSO.[@Baroni] The projector augmented wave method was used with Troullier-Martins norm-conserving pseudopotential with nonlinear core correction. The calculations were performed with (without) spin-orbit interaction by using fully (scalar) relativistic pseudopotentials. The exchange correlation functional was treated within the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE-GGA).[@Perdew] The correlation effects of Co 3$d$ electrons were included via GGA+U method within the simplified rotational invariant scheme of Cococcioni et al.[@Coco] The value of on-site Coulombic interaction term (U) was set to a well-tested value of $U$ = 4.5 eV. A well converged kinetic-energy cutoff of 80 Ry was used with a Monkhorst-Pack sampling of 16$\times$16$\times$16. The experimental lattice parameter obtained from X-ray diffraction measurements were used as the initial configuration of the atoms for the DFT calculations. The magnetic configuration of the Co atoms was initialized to be ferromagnetic. Note that the final magnetic configuration of Co atoms is independent of the initial magnetization direction. A strict self-consistent energy convergence criterion of 10$^{-8}$ Ry was imposed. As shown in Fig. 2c, the calculated spin-resolved density of states (DOS) show that the characteristic Co-$d$ states are embedded well below the fermi energy for both majority (Fig. 2a) and minority (Fig. 2b) spin carriers, indicating weak-metallic or semiconducting characteristic due to the embedded Co d-orbitals. We note that the minority-spin carriers exhibit a much larger DOS at the Fermi level than the majority-spin carriers, which is a typical characteristic of ferromagnetic material. This is further substantiated by the spin-resolved hole Fermi surface, as shown in Fig. 2b, where the minority-carrier hole Fermi surface is more populated even though it has a much smaller volume compared to that of the majority-carriers (Fig. 2a). It gives rise to non-degenerate energies of electrons with opposite spins.
The high quality polycrystalline sample of CoSeAs was electrically characterized using a closed cycle refrigerator based cryostat with a base temperature of $T \simeq$ 5 K. As shown in Fig. 2d, the system exhibits very weak metallic-to-semiconducting behavior as temperature is reduced, also consistent with the DFT calculations. Information about the underlying magnetic properties is obtained from detailed elastic and inelastic neutron scattering measurements. Neutron measurements were performed on a $~$4.4g polycrystalline sample of CoSeAs at the multi-axis crystal spectrometer (MACS) with fixed final neutron energy of $E_f$ = 3.7 meV at the NIST Center for Neutron Research. Additional measurements were performed on the spin-polarized triple-axis spectrometer (SPINS) at NCNR and a position sensitive detector (PSD) powder diffractometer at the University of Missouri Research Reactor with fixed final neutron energy of 37 meV using graphite monochromator. Elastic measurements at SPINS employed a flat pyrolytic graphite (PG) analyzer with cold BeO filter in front, while the measurements on the powder diffractometer was performed using the tighter collimations before the monochromator and a PG filter. Inelastic measurements on MACS were performed in the focussed analyzer configuration with fixed $E_F$ = 3.7 meV and the energy resolution of $\simeq$ 0.25 meV. The sample was loaded in liquid $^{4}$He cooled cryostat with the lowest accessible temperature of $T$ = 1.7 K.
{width="8.7"}
{width="18"}
Elastic scattering measurements on CoSeAs powder are used to infer the underlying static magnetic correlation between Co ions. We show the representative scans at two temperatures in Fig. 3a. Additional Bragg peaks arise as the sample is cooled to low temperature, indicating the development of magnetic order in the system. Compared to the nuclear peak intensities, shown in Fig. 3b, magnetic peaks are significantly weaker. Furthermore, elastic measurements required long counting time to obtain the statistically significant magnetic peak intensities. Together, they hint of small ordered moment of Co ions in the system.[@Harriger] Experimental data is well described by the resolution limited Gaussian lineshape. The estimated full width half maximum (FWHM) of magnetic peak is comparable to the instrument resolution of MACS spectrometer, suggesting the existence of long range magnetic order in CoSeAs. The magnetic order is found to persist to reasonably high temperature. In Fig. 3d, we show the plot of order parameter as a function of temperature at the magnetic wave vector q = 0.595 $\AA$$^{-1}$. Fitting of experimental data using a power law, given by $I$ $\propto$ (1-T/T$_{c}$)$^{-\beta}$,[@Singh] yields a transition temperature of $T_c$ = 36.2 K to magnetic ordered state. The estimated value of power law exponent is $\beta$ = 0.357(4).
The magnetic wave vectors are identified to be both integer and rational fractions of reciprocal lattice units e.g. (100) and (1/4 1/4 1/4). The nature of magnetic correlation is deduced by performing detailed numerical modeling of experimental data. The experimentally observed structure factor, estimated from the Gaussian fit of the elastic data, are compared with the numerically calculated structure factor for model spin configurations. Structure factor is calculated using, $ F_{M} =\sum_{j} S_{\perp j} p_{j} e^{ iQr_{j} }e^{-W_{j}} $,[@Shirane] where $ S_{\perp} =\hat{Q}\times(S\times\hat{Q})$ is the spin component perpendicular to the Q, $ p = (\frac{\gamma r_{0}}{2})gf(Q) $, $ (\frac{\gamma r_{0}}{2}) $= 0.2695 $ \times 10^{-12} cm$, g is the Lande splitting factor and was taken to be g = 2, $ f(Q) $ is the magnetic form factor, and $e^{-W_{j}} $ is the Debye-Waller factor and was taken to be 1.[@Shirane; @Dahal2] Simulated intensities are powder averaged by multiplying with an appropriate factor of (1/sin($\theta$).cos(2$\theta$)).[@Shirane] Best fit to experimental data is obtained for magnetic moments arrangement comprised of two magnetic sublattices: (a) Co ions occupying the vertices of the orthorhombic lattice are ferromagnetically aligned along the $b$-axis and arranged in a density wave configuration with quadrupled magnetic unit cell, and (b) Co moment at the body-centered position are arranged in density wave configuration with quadrupled magnetic unit cell (see details in Supplementary Materials). As shown in Fig. 3c, the numerically simulated powder profile for the aforementioned spin structure, shown in Fig. 3e, well describes the experimentally observed diffraction pattern. The proposed spin configuration also gives rise to peak intensity at (001) rlu, which was not resolved in the experimental data. The estimated ordered moment is 0.26(5)$\mu_B$. Such a small value of the ordered moment reflects the nearly compensated spin polarities in individual anion octahedron. This is consistent with the general observation in marcasite phase magnetic material of weak or no magnetic order due to the strong screening of magnetic moment by conduction electrons. Experimental findings hint of the weakly correlated Co ions in CoSeAs.
To gain insight about the strength of exchange interaction between Co ions in CoSeAs, we have performed detailed inelastic measurements. In Fig. 4a, we show the color map of inelastic spectrum, obtained on MACS spectrometer, at $T$ = 1.7 K. A q-independent band of inelastic excitation tends to develop below E $\simeq$ 3 meV at low temperature. The excitation at higher q follows the Co form factor, thus gradually weakens. Inelastic data is background subtracted and thermally balanced by multiplying the Intensity by a factor of $\pi$(1-exp(-$E$/$k_B$$T$)). The absence of any dispersion in experimental data suggests an isotropic nearest neighbor interaction in the system. Further quantitative information is obtained by analyzing the dynamic susceptibility $\chi$$^{''}$(Q, E), given by
$$\begin{aligned}
{S(Q, \omega)}&=&{\gamma_0}^{2}(\frac{k_i}{k_f}){f(Q)}^{2}\frac{1}{1- {e}^{-h\omega/{k_B}T}}(\frac{{\chi}^{^{\prime\prime}}(Q, \omega)}{\pi})\end{aligned}$$
where $\gamma_0$$^{2}$ = 0.073$/$$\mu_B$$^{2}$, $k$$_{i}$ and $k$$_{f}$ represent initial and final neutron wave vectors and $f(Q)$ is the form factor of magnetic ion (in this case Co ion). In Fig. 4c, we plot $\chi$$^{''}$(Q, E) as a function of energy at $T$ = 1.7 K at a representative Q = 1.4 $\AA$$^{-1}$. Clearly, a broad peak in $\chi$$^{''}$(Q, E), centered around $E$ $\simeq$ 0.75 meV, is observed. At higher temperature, above $T_c$, the excitation becomes indistinguishable from the background, indicating magnetic nature of inelastic peak. A dispersive excitation is detected at higher temperature, which can be associated to the phonon excitation in the system. However, the excitation does not prevail to high q values, typical of phonon excitation. One explanation to such discrepancy may be due to the coupling between phonon excitation and the dynamic magnetic interaction, which follows the magnetic form factor. Hence, the dispersive excitation disappears at higher q value.
We have fitted the data using random phase approximation (RPA) model.[@Broholm] Previously, the RPA model has been successful in describing inelastic phenomena in transition metal ion correlated systems.[@Birgeneau; @Dahal] Fitting using RPA model is based on the assumptions that the only appreciable interaction is nearest neighbor interaction between Co-ions, $J$, and the interaction is isotropic in nature. The small ordered moment, despite the long counting time, and a nearly Q-independent excitation in MACS measurement conform to the applicability of RPA model to estimate the exchange interaction between magnetic ions. Under RPA model, $\chi$$^{''}$(Q,E) is described by,
$$\begin{aligned}
{\chi}^{^{\prime\prime}}(Q, \omega)&=&\sum_{\pm}\frac{\omega \chi_0 \Gamma_{Q^\pm}}{{\Gamma_{Q^\pm}}^{2}+\omega^2}\end{aligned}$$
where $\Gamma$$_{Q^\pm}$ = $\Gamma$\[1$\mp$$\chi_0$ $J$\] and $\chi_0$ is static susceptibility. Clearly, the RPA model describes the dynamic properties of CoSeAs very well. Fitted value of $J$ = 0.25(4) meV at $T$ = 1.7 K indicates weak exchange interaction between Co-ions. Obtained values of $\Gamma$ is plotted as a function of wave vector Q in Fig. 4d. $\Gamma$, representing the full width at half maximum of dynamic correlation or the inverse of relaxation time $\tau$, seems to be independent of the wave vector Q. The dynamic susceptibility at $T$ = 135 K is barely distinguishable from the background. The small deviation from the background at low energy is most likely arising due to the paramagnetic fluctuation of Co ions at higher temperature.
{width="9.2cm"}
The analysis of inelastic data reveals the large value of $\Gamma$, $\simeq$ 0.8 meV, for such a modest exchange coupled magnet. It suggests that Co ions are fluctuating with short relaxation time. Such behavior are usually observed in magnetically unstable systems.[@Ueland] Perhaps, the stoichiometric composition of CoSeAs is on the verge of a transition to another phase of material.
In summary, we have synthesized a new marcasite phase material CoSeAs. Neutron scattering investigation of polycrystalline CoSeAs reveals the development of long range magnetic order below $T_c$ = 36.2 K. Given the fact that only a very few transition metal marcasites are known to manifest magnetic order, this is an important finding. Moreover, the weak nearest neighbor exchange coupling between Co ions makes it an interesting candidate material for the exploration of unconventional superconductivity using chemical doping method. Chemical pressure can further distort the anion octahedron, encompassing the transition metal ion, to induce a transition to new phase of material. Further research works are highly desirable in this pursuit. Besides the exploration of a possible superconducting state in chemically doped CoSeAs, we envision two possible applications of the new compound as photovoltaic absorber and in the design of robust electrocatalyst. There is an increasing trend in the use of marcasites and pyrites for photovoltaic application in recent years.[@Zhang; @Wu] Future researches on the study of optical properties of CoSeAs thin film can elucidate its possible application in photovoltaics. More recently, an analogous marcasite CoSe$_{2}$ was demonstrated to preserve electrocatalytic integrity after long hours of usage in the acidic media.[@Zhang2] Similar studies on the crystalline specimen of CoSeAs can reveal new electrocatalytic properties in this compound.
DKS thankfully acknowledges the support by the Department of Energy, Office of Science, Office of Basic Energy Sciences under the grant no. DE-SC0014461. This work utilized facilities supported by the Department of Commerce.
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---
author:
- 'Van-Nham Phan'
- Holger Fehske
- 'Klaus W. Becker'
date: 'Received: date / Revised version: date'
subtitle: 'Many-body corrections beyond the random phase approximation'
title: 'Linear response within the projection-based renormalization method'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#I}
============
The most popular approach to evaluate linear response coefficients for many-body systems is probably the standard random phase approximation (RPA), which is used in quite different fields in physics. Besides its physical merits including the fulfillment of conservation laws the popularity of the RPA results from is conceptual simplicity in its derivation as well as from its numerical practicability. Attempts to go beyond the RPA have turned out to be extremely demanding. One of the options to improve the RPA are methods based on conserving approximations (see, e.g., Ref. [@BSW89]), thereby following an approach which was introduced by Baym and Kadanoff [@BK61; @KB62]. Conserving approximations are consistent with microscopic conservation laws for particle number, energy or momentum. Other work is based on the time-dependent Hartree-Fock approximation which uses a frequency-dependent local field factor in a modified RPA expression [@GV08]. However, this method turns out to be rather complex and physically unsatisfactory. Attempts to find a numerical solution of the basic integral equation could not be reached without further approximations (see discussion in Ref. [@GV08] and references therein). Recently, Vilk and Tremblay extended the RPA by including vertex corrections, taken into account correlation and exchange effects [@VT97]. Comparing their [*ansatz*]{} for double occupancies in the Hubbard model, the authors found good quantitative agreement with results from Monte Carlo simulations for single-particle and two-particle properties [@BSW94; @VCT94]. Another way to improve the RPA is the so-called self-consistent RPA [@JSDB05], which is based on a non-perturbative variational scheme. This approach has been adopted to the investigation of various nontrivial models but is however limited to small systems [@JSDB05; @KS94; @HMDS02].
For these reasons it is important to develop new many-particle techniques having the ability to include correlation effects. One approach that overcomes some of the shortcomings of the RPA is the projection-based renormalization method (PRM) [@BHS02; @SHB09b]. In the recent past the PRM has been successfully applied to several physical problems such as superconductivity [@HB03], quantum phase transitions in coupled electron-phonon systems [@SHBWF05; @SHB06; @PFB11; @PBF13], exciton and plasmaron formation [@PFB11; @PF12], BCS-BEC transition [@PBF10], electronic phase separation [@ESBF12], valence transitions [@PMB10], or the Kondo lattice problem [@SB13]. In the present work, adding time- and wave-vector-dependent external fields, we demonstrate how the PRM can be combined with linear response theory in order to calculate response functions for generic correlation models. In particular we derive an explicit analytical expression for the dynamical spin susceptibility of the Hubbard model. For the two-dimensional (2D) case the PRM phase boundaries between the paramagnetic and antiferromagnetic respectively ferromagnetic phases are determined for weak-to-intermediate Hubbard interactions. An elaborate weak-coupling approach is of particular importance in low spatial dimensions since in 1D and 2D also weakly interacting systems tend to be strongly correlated.
The paper is organized as follows. In the Sect. \[II\], we recapitulate the RPA to the Hubbard model. Section \[III\] introduces the PRM approach, which is applied to the Hubbard model in Sect. \[IV\], focusing on the response to an external magnetic field. Thereby the renormalization equations for the model parameters, the transformations of the operators and various expectations values are derived. Details can be found in Appendices A and B. Section \[IV.4\] provides our main analytical result: the explicit expression for the dynamical spin susceptibility. Selected numerical results for the 2D Hubbard model can be found in Sec. \[V\], in particular the ground-state phase diagram in the $U$-$n$ plane and the wave-vector- and frequency-dependence of the magnetic susceptibility. We conclude in Sect. \[VI\].
Standard RPA approach to the Hubbard model {#II}
==========================================
The Hubbard model is a paradigmatic model for the study of correlation effects in itinerant electron systems. Independently proposed by Gutzwiller [@Gu63], Hubbard [@Hu63], and Kanamori [@Ka63] in 1963, it was originally designed to describe the ferromagnetism of transition metals. Successively, the model has been studied in the context of antiferromagnetism, metal-insulator transition, and high temperature superconductivity. The Hubbard Hamiltonian is given by $$\begin{aligned}
\label{1}
\mathcal H &=& \bar{t} \sum_{\langle i, j \rangle\sigma} c_{i\sigma}^\dag c^{}_{j\sigma}
+ \frac{U}{2} \sum_{i \sigma} n^{}_{i\sigma} n^{}_{i, -\sigma}\,.
$$ Here $c_{i\sigma}^\dag$ $(c_{i\sigma}^{})$ is a fermionic creation (annihilation) operator of a spin $\sigma\;(=\uparrow,\downarrow)$ electron, and $n^{}_{i\sigma} = c_{i\sigma}^\dag c_{i\sigma}^{}$. $U$ denotes the on-site Coulomb interaction and $\bar{t}$ are the electron transfer matrix elements between nearest-neighbor Wannier sites $i$ and $j$. The physics of the model is governed by the competition between itinerancy ($\bar t$; delocalization, kinetic energy) and short-range Coulomb repulsion ($U$; localization, magnetic order), where the fermionic nature of the charge carriers is of great importance (Pauli exclusion principle). Besides the parameter ratio $U/\bar{t}$, the particle density $n$, the temperature $T$, and the spatial dimension $D$ (geometry of the lattice) are crucial.
Although a tremendous amount of work has been devoted to the Hubbard model, in order to determine its ground-state, spectral and thermodynamic properties, exact results are rare and only a few special cases and limits are ultimately understood. In 1D, the algebraic and thermodynamic Bethe [*ansatz*]{} enables an exact treatment of the model [@Tak99; @EFGKK05]. However the Bethe [*ansatz*]{} technique does not provide a complete framework since it generally does not allow the evaluation of the response functions. For $D>1$ approximations are unavoidable anyway. There usually the weak- $(U/W \ll 1)$ and strong-coupling $(U/W \gg 1$) limits of the model were studied, with uncertain extrapolations to the region $U/W \sim 1$. Here $W$ is the bare electronic bandwidth. For a $D$-dimensional hypercubic lattice we have $W=4D\bar{t}$.
In consideration of the magnetic behavior of a Hubbard model system the response to an applied external field is of particular importance. Adding a small magnetic field that periodically oscillates in space and time the Hamiltonian takes the form $$\begin{aligned}
\label{3}
\mathcal H(t) = \mathcal H_{kin} + \mathcal H_U + {\mathcal H}_h (t)\,,\end{aligned}$$ where $$\begin{aligned}
\label{4}
\mathcal H_{kin} &=& \sum_{\mathbf k \sigma} {\varepsilon}_{{\mathbf{k}}} \, c_{{\mathbf{k}}\sigma}^\dag c^{}_{{\mathbf{k}}\sigma}, \end{aligned}$$ is the kinetic energy of electrons in momentum space. In the numerical evaluation below the Hubbard model is considered on a square, where the dispersion is given by $$\label{4a}
{\varepsilon}_{{\mathbf{k}}}=2\bar{t}(\cos k_x+\cos k_y) -\mu$$ with chemical potential $\mu$. The last two terms in Eq. (\[3\]) respectively read $$\begin{aligned}
\mathcal H_U& = &\frac{U}{2N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma} c_{{\mathbf{k}} \sigma}^\dag\, c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} \,
c_{{\mathbf{k}}', -\sigma}^\dag \, c^{}_{{\mathbf{k}}' -{\mathbf{p}}, -\sigma} \, , \end{aligned}$$ and $$\begin{aligned}
\label{5}
{\mathcal H}_h(t) &=&
- \sum_i h(t) \cos{({\mathbf{q}} \cdot {\mathbf{R}}_i)} \, s^z_i = - \frac{ h(t)}{2}\big(s^z_{{\mathbf{q}}} + s^z_{-{\mathbf{q}}} \big)
\nonumber \, . \\
&&
\end{aligned}$$ Note that the wave vector ${\mathbf{q}}$ is imposed by the external field. $s_{{\mathbf{q}}}^z$ is the component of the spin operator in field direction: $$s^z_{{\mathbf{q}}} = \sum_i e^{i {\mathbf{q}} \cdot {\mathbf{R}}_i} s_i^z= \sum_{{\mathbf{k}} \sigma} \frac{\sigma}{2} c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} - {\mathbf{q}}, \sigma} \, .
\label{6}$$ Then the linear response of the spin expectation value $\langle s^z_{-{\mathbf{q}}} \rangle(t)$ with respect to $h(t) \sim \textrm{Re} \;e^{-i\omega t}$ is given by $$\begin{aligned}
\label{7}
\langle s^z_{-q} \rangle(t) &=& -{i} \int_0^\infty dt' \langle [s_{-{\mathbf{q}}}^z(t') , {\mathcal H}_h(t- t')] \rangle
\nonumber \\
&=& N \chi({\mathbf{q}}, \omega) \frac{h(t)}{2} \, , \end{aligned}$$ where $$\begin{aligned}
\label{8}
\chi({\mathbf{q}}, \omega) &=& \frac{i}{N}\int_0^\infty dt'\langle [s^z_{-{\mathbf{q}}}(t') , s^z_{{\mathbf{q}}} ] \rangle \,
e^{i (\omega + i \eta) t'} \end{aligned}$$ ($\eta = 0^+$) is the formal expression for the dynamical magnetic susceptibility. Here, the expectation value is formed with Hamiltonian $\mathcal H$ \[Eq. (\[1\])\] in the absence of the external perturbation. Due to the Coulomb part $\mathcal H_U$ in $\mathcal H$ a straightforward evaluation of $\chi({\mathbf{q}}, \omega)$ turns out to be difficult. To proceed, in a first step, let us introduce fluctuation operators $$\begin{aligned}
\label{9}
: c_{{\mathbf{k}} \sigma}^\dag\, c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma}: &=&
c_{{\mathbf{k}} \sigma}^\dag\, c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} -
\langle c_{{\mathbf{k}} \sigma}^\dag\, c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} \rangle(t) \, , \\
&& \nonumber
\end{aligned}$$ where the expectation value $\langle \cdots \rangle(t)$ on the right-hand side is formed with $\mathcal H(t)$ and therefore becomes time dependent. It can be simplified by help of the operator identity $$\begin{aligned}
c_{{\mathbf{k}} \sigma}^\dag\, c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} &=&
\frac{1}{2} \sum_{\tilde{\sigma}} c^\dag_{{\mathbf{k}} \tilde \sigma} c^{}_{{\mathbf{k}} + {\mathbf{p}}, \tilde{\sigma}}
+ \sigma \sum_{\tilde \sigma }\frac{\tilde \sigma}{2} \,
c_{{\mathbf{k}} \tilde{\sigma} }^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}}, \tilde \sigma}
\nonumber \\
&&
\end{aligned}$$ to give $$\begin{aligned}
\label{10}
\langle c_{{\mathbf{k}} \sigma}^\dag\, c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma}\rangle(t) &=&
\frac{\delta_{{\mathbf{p}},0}}{2} \sum_{\tilde{\sigma}}
\langle c^\dag_{{\mathbf{k}} \tilde \sigma} c^{}_{{\mathbf{k}} , \tilde{\sigma}} \rangle
+\sigma \delta_{{\mathbf{p}}, - {\mathbf{q}}} \langle s^z_{{\mathbf{k}}, {\mathbf{q}}}\rangle(t) \nonumber \\
&+& \sigma \delta_{{\mathbf{p}}, {\mathbf{q}}} \langle s^z_{{\mathbf{k}}, -{\mathbf{q}}}\rangle(t)
\, .
\end{aligned}$$ Here ${\mathbf{p}}$ is either ${\mathbf{p}} =\pm {\mathbf{q}}$ or ${\mathbf{p}} = 0$. We have also introduced the ${\mathbf{k}}$-resolved spin operator $$\label{11}
s^z_{{\mathbf{k}},{\mathbf{q}}} = \sum_{\tilde \sigma} \frac{\tilde \sigma}{2} c^\dag_{{\mathbf{k}} \tilde \sigma}c^{}_{{\mathbf{k}} - {\mathbf{q}}, {\tilde \sigma}} \, .$$ The expectation value $\langle c^\dag_{{\mathbf{k}}\sigma} c^{}_{{\mathbf{k}} \sigma} \rangle$ has no linear contribution in $ h(t)$ and can be considered as time independent.
Using Eqs. (\[9\]) and (\[10\]), the Hamiltonian $\mathcal H(t)$ can be rewritten as $$\begin{aligned}
\label{12}
\mathcal H(t) = {\mathcal H}_{0} + \hat {\mathcal H}_h(t) + \mathcal H_{f}(t)\,,\end{aligned}$$ where $$\begin{aligned}
\label{13}
&& {\mathcal H}_{0} = \sum_{{\mathbf{k}} \sigma} \Big( \varepsilon^{}_{{\mathbf{k}}} + \frac{U}{2} \langle n \rangle
\Big) \, c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} \sigma}\,, \\
& & \hat {\mathcal H}_h(t)
= - \frac{\hat h_{{\mathbf{q}}}(t)}{2} \, \Big( s^z_{{\mathbf{q}}} + s^z_{-{\mathbf{q}}}\Big) \nonumber \,,\end{aligned}$$ and $$\begin{aligned}
\label{14}
\frac{{\hat h}_{{\mathbf{q}}}(t)}{2} = \frac{ h(t)}{2} + \frac{U}{N} \langle s^z_{ -{\mathbf{q}}} \rangle(t) \, .\end{aligned}$$ In Eq. (\[14\]) we have introduced an effective field ${\hat h}_{{\mathbf{q}}}(t)$ that contains an internal field proportional to $U$. Also the kinetic energy ${\mathcal H}_{0}$ has acquired a Hartree shift proportional to $U$. Finally, the part $\mathcal H_f(t)$ reads $$\begin{aligned}
\label{15}
\mathcal H_{f}(t) = \frac{U}{2N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma}
: c_{{\mathbf{k}} \sigma}^\dag\, c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma}: \,
:c_{{\mathbf{k}}', -\sigma}^\dag \, c^{}_{{\mathbf{k}}' -{\mathbf{p}}, -\sigma} : \, ,\quad \end{aligned}$$ where the $t$-dependence enters via the fluctuation operators.
The standard RPA expression for $\chi({\mathbf{q}}, \omega)$ is obtained by neglecting the fluctuating part $\mathcal H_{f}(t)$ completely, i.e., $\mathcal H(t)$ reduces to $$\begin{aligned}
\label{16}
\mathcal H_{RPA}(t) &=& {\mathcal H}_{0} + \hat {\mathcal H}_h(t) \, .
\end{aligned}$$ The linear response of $ \langle s^z_{-q} \rangle(t)$ to the effective field $\hat h_{{\mathbf{q}}}(t)$ becomes $$\begin{aligned}
\label{17}
\langle s^z_{-q} \rangle(t) &=& -{i} \int_0^\infty dt' \langle [s_{-{\mathbf{q}}}^z(t') , \hat{\mathcal H}_h(t- t')] \rangle_{0}
\nonumber \\
&=& N \chi^{}_{0} ({\mathbf{q}}, \omega) \, \frac{\hat h_{{\mathbf{q}}}(t)}{2} \, ,
\end{aligned}$$ where the expectation value is now formed with the unperturbed Hamiltonian $\mathcal H_{0}$. By help of relation (\[14\]) one arrives at $$\begin{aligned}
\label{18}
\langle {s}^z_{-{\mathbf{q}}}\rangle (t) &=& \frac{N \chi^{}_{0} ({\mathbf{q}}, \omega)}{1-
\displaystyle U \chi^{}_{0} ({\mathbf{q}}, \omega) } \, \frac{ h(t)}{2}
\, .
\end{aligned}$$ Here $\chi_{0} ({\mathbf{q}}, \omega)$ is the dynamical susceptibility of the unperturbed system ${\mathcal H}_{0}$ $$\begin{aligned}
\label{19}
\chi^{}_{0} ({\mathbf{q}}, \omega) &=& \frac{i}{N} \int_0^\infty dt'
\langle [s^z_{-{\mathbf{q}}}(t'), s^z_{{\mathbf{q}}} ] \rangle_{0} \, e^{i (\omega + i\eta)t'} \nonumber \\
&=& \frac{1}{2N} \sum_{{\mathbf{k}}} \frac{f(\varepsilon_{{\mathbf{k}} + {\mathbf{q}}}) -f(\varepsilon_{{\mathbf{k}}})}{\varepsilon_{{\mathbf{k}}}- \varepsilon_{{\mathbf{k}} + {\mathbf{q}}} + \omega + i \eta}
\end{aligned}$$ with $\eta =0^+$. Note that Eqs. (\[17\])-(\[19\]) are the usual RPA equations.Thus, the dynamical RPA susceptibility is defined by the prefactor in Eq. (\[18\]): $$\label{20}
\chi^{}_{RPA} ({\mathbf{q}}, \omega) = \frac{ \chi^{}_{0} ({\mathbf{q}}, \omega)}{1-
\displaystyle {U} \chi^{}_{0} ({\mathbf{q}}, \omega) } \, .$$
PRM formalism {#III}
=============
Our aim is to evaluate the dynamical susceptibility $\chi({\mathbf{q}}, \omega)$ beyond the standard RPA by including fluctuation processes, which are induced by the fluctuation part $\mathcal H_f(t)$ of the Coulomb interaction. To this end, we combine linear response theory with the PRM. The PRM in the original version without time-dependent external field starts by separating a given many-particle Hamiltonian into an unperturbed part $\mathcal H_0$ and a time-independent perturbation $\mathcal H_f$. Since $\mathcal H_f$ and $\mathcal H_0$ do not commute, the perturbation induces transitions between the eigenstates of $\mathcal H_0$. The basic idea of the PRM is to eliminate successively all transitions due to $\mathcal H_f$ so that finally only the unperturbed, yet renormalized Hamiltonian (now called $\tilde{\mathcal H}_0$) remains.
In the present case the many-particle Hamiltonian (\[12\]) is time dependent, $ {\mathcal H}(t) = \mathcal H_{0} + \hat{\mathcal H}_{h}(t) + \mathcal H_f(t)$, since $\hat{\mathcal H}_{h}(t)$ is time-dependent due to the external field. As before, our aim is to evaluate the response of the expectation value $\langle s_{-\mathbf q}\rangle(t)$ up to linear order in the external field. However, the fluctuation term $\mathcal H_f(t)$ of the Coulomb interaction should now be taken into account. Since both $ \mathcal H_{0}$ and $\hat{\mathcal H}_{h}(t)$ do not commute with $\mathcal H_f(t)$, the latter Hamiltonian will henceforth be considered as perturbation. In particular, $\mathcal H_f(t)$ again induces transitions between the eigenstates of $\mathcal H_{0}$. In the PRM these transitions will be eliminated by a sequence of unitary transformations, which are performed in small steps $\Delta \lambda$ by proceeding from large to small transition energies. Let $\mathcal H_\lambda$ be the Hamiltonian after all transitions with energies larger than some cutoff $\lambda$ have already been integrated out. The transformation from cutoff $\lambda$ to a somewhat reduced cutoff $\lambda -\Delta \lambda$ formally reads $$\label{21}
{\mathcal H}_{\lambda - \Delta \lambda}(t) = e^{X_{\lambda, \Delta \lambda}} {\mathcal H}_\lambda(t) \,
e^{-X_{\lambda, \Delta \lambda}} \, .$$ Here $X_{\lambda, \Delta \lambda}= - X^\dag_{\lambda, \Delta \lambda}$ is the generator of the unitary transformation from $\lambda$ to $\lambda - \Delta \lambda$, whereas $\mathcal H_\lambda(t)$, $$\label{22}
{\mathcal H}_{\lambda}(t) = \mathcal H_{0,\lambda} + \hat{\mathcal H}_{h,\lambda}(t) +
\mathcal H_{f,\lambda}(t) \, ,$$ represents the renormalized Hamiltonian after all transitions (in the eigenbasis of ${\mathcal H}_{0,\lambda}$) with energies larger than $\lambda$ have been eliminated from ${\mathcal H}_f(t)$. Similarly, ${\mathcal H}_{\lambda - \Delta \lambda}(t)$ denotes the Hamiltonian with the somewhat reduced cutoff $\lambda - \Delta \lambda$. Due to transformation (\[21\]) the parameters of $\mathcal H_\lambda$ become renormalized, and also new terms can in principle be generated. For the generator $X_{\lambda, \Delta \lambda}$ we chose $$\begin{aligned}
\label{23}
X_{\lambda, \Delta \lambda}(t) &=& \frac{1}{\mathbf L_{0,\lambda}} \mathbf Q_{\lambda -\Delta \lambda}
\, \mathcal H_{f,\lambda}(t) \, ,
\end{aligned}$$ which agrees with the lowest order result for $X_{\lambda, \Delta \lambda}$ in the absence of an external field [@BHS02]. Here $\mathbf L_{0,\lambda}$ is the Liouville operator of the ’unperturbed’ Hamiltonian $\mathcal H_{0,\lambda}$. It is defined by the commutator $\mathbf L_{0,\lambda}\, {\mathcal A} =
[\mathcal H_{0,\lambda}, {\mathcal A}]$, applied to any operator variable ${\mathcal A}$. Moreover, $\mathbf Q_{\lambda -\Delta \lambda}$ is a generalized projection operator. It projects on all transition operators in $\mathcal H_{f,\lambda}$ (in the basis of $\mathcal H_{0,\lambda}$) with transition energies larger than $\lambda -\Delta \lambda$. Note that $X_{\lambda, \Delta \lambda}$ also depends on time $t$ since $\mathcal H_{f,\lambda}(t)$ depends on $t$ via the external field.
The elimination procedure starts from the original Hamiltonain $\mathcal H(t)$ (where the largest cutoff energy is called $\lambda= \Lambda$) and proceeds in steps $\Delta \lambda$ until $\lambda=0$ is reached. This limit provides the desired effective Hamiltonian\
$\tilde{\mathcal H}(t):=\mathcal H_{\lambda \rightarrow 0}(t)$ with ${\mathcal H}_{\lambda \rightarrow 0}(t) = \mathcal H_{0, \lambda\rightarrow 0}+
\mathcal H_{h,\lambda \rightarrow 0}(t)$. Note that the elimination of the transitions leads to renormalized parameters in $\tilde{\mathcal H}(t)$. Thus, after all transitions from ${\mathcal H}_f(t)$ have been used up, the final Hamiltonian $\tilde{\mathcal H}(t)$ is diagonal or at least quasi-diagonal and allows to evaluate expectation values. As a matter of course the parameters in $\tilde{\mathcal H}(t)$ depend on their values in the original model $\mathcal H(t)$.
Having in mind small renormalization steps $\Delta \lambda$, the transformation (\[21\]) from $\lambda$ to $\lambda - \Delta \lambda$ can be restricted to an expansion up to second order in $U$ (and linear order in $h(t)$). Then $\mathcal H_{\lambda-\Delta \lambda}(t)$ reads $$\begin{aligned}
\label{24}
&& {\mathcal H}_{\lambda - \Delta \lambda} = e^{X_{\lambda, \Delta \lambda}} \mathcal H_\lambda
e^{-X_{\lambda, \Delta \lambda}} \nonumber \\
&& \quad = \mathcal H_{0,\lambda} +
\hat{\mathcal H}_{h,\lambda} +
\mathbf P_{\lambda -\Delta \lambda} \mathcal H_{f,\lambda}
+ [X_{\lambda, \Delta \lambda}, \hat {\mathcal H}_{h, \lambda} ] \nonumber \\
&& \quad +
[X_{\lambda, \Delta \lambda}, {\mathcal H}_{f,\lambda}] -
\frac{1}{2} [X_{\lambda, \Delta \lambda},
\mathbf Q_{\lambda -\Delta \lambda} {\mathcal H}_{f,\lambda}]
\nonumber \\
&& \quad + \frac{1}{2} [ X_{\lambda, \Delta \lambda},
[ X_{\lambda, \Delta \lambda}, \hat {\mathcal H}_{h,\lambda}] ]
+ \cdots \, ,
\end{aligned}$$ where relation (\[23\]) was used and the explicit $t$-dependence is suppressed. $\mathbf P_{\lambda -\Delta \lambda}=
1- \mathbf Q_{\lambda -\Delta \lambda}$ is the projector on all low-energy transitions with energies smaller than $\lambda- \Delta \lambda$. The commutators in Eq. (\[24\]) give rise to renormalization contributions to $\mathcal H_{\lambda- \Delta \lambda}(t)$. Having in mind an application of linear response theory, Eq. has to be evaluated up to linear order in the external field. Finally, comparing the result of the evaluated right-hand side with the generic form of $\mathcal H_\lambda$ one is led to renormalization equations, which relate the parameters of the Hamiltonian at cutoff $\lambda$ with those at cutoff $\lambda - \Delta \lambda$.
As is discussed below one also has to evaluate expectation values, which are formed with Hamiltonian $\mathcal H$. Exploiting the unitary invariance of operators below a trace we can write $$\begin{aligned}
\label{25}
\langle {\mathcal A} \rangle =
\frac{{\mbox{Tr}} {\mathcal A} e^{-\beta {\mathcal H}}}{ {\mbox{Tr}} e^{-\beta {\mathcal H}}} =
\langle {\mathcal A}(\lambda) \rangle_{{\mathcal H}_\lambda}=
\langle \tilde{{\mathcal A}}\rangle_{\tilde{{\mathcal H}}} \, , \end{aligned}$$ where the same unitary transformation as before is applied to operator $\mathcal A$, i.e. ${\mathcal A}(\lambda) = e^{{X}_\lambda}{\mathcal A}e^{-{X}_\lambda}$. Here ${X}_\lambda$ is generator of the unitary transformation between cutoff $\Lambda$ and $\lambda$, and $\tilde{{\mathcal A}}=
{\mathcal A}(\lambda \rightarrow 0)$. Thus, additional renormalization equations for ${\mathcal A}(\lambda)$ are required.\
Let us mention that Wegner and coworkers [@We94; @Keh06] have introduced a theoretical approach related to the PRM. This approach is based on the application of continuous unitary transformations instead of discrete ones as in the present case. To our knowledge it was not applied up to now to the investigation of many-body corrections beyond the random phase approximation discussed in the present work. However, correlation and fluctuation processes can be discussed in this method as well, compare for instance references [@ZaDo11; @FrKe10; @KDU12], or [@VMM13]. The relationship between the continuous method and the PRM for the case without time-dependent field is studied in references [@PBF10] and [@HSB08]. There it was shown that the continuous method can be derived within the PRM framework in the limit of small $\Delta \lambda \rightarrow 0$ using a particular choice for the complement part ${\mathbf P}_{\lambda- \Delta \lambda}X_{\lambda, \Delta \lambda}$ of generator (\[23\]).
PRM for the Hubbard model {#IV}
=========================
[***Ansatz***]{} for Hamiltonian $H_\lambda(t)$ {#IV.1}
-----------------------------------------------
We are now in the position to apply the general formalism of Section \[III\] to the Hubbard model. Thereby, the influence of the fluctuation term $\mathcal H_f (t)$ will be investigated. Following the ideas of the PRM, we have to start from an [*ansatz*]{} for the renormalized Hamiltonian $\mathcal H_\lambda(t)$. A perturbative evaluation of transformation (\[21\]) suggests the use of the following expression for ${\mathcal H}_{\lambda}(t)$ (see Appendix A), where $$\begin{aligned}
\label{26}
{\mathcal H}_{0,\lambda} &=& \sum_{{\mathbf{k}} \sigma} {\varepsilon}^{}_{{\mathbf{k}},\lambda} \,
c_{{\mathbf{k}}\sigma}^\dag c^{}_{{\mathbf{k}} \sigma} \,,
\label{27}\\
\hat {\mathcal H}_{h, \lambda}(t) &=&
-\sum_{{\mathbf{k}} \sigma} \Big[ \Big( \frac{\hat h_{{\mathbf{q}}}(t)}{2} + u_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}},\lambda}(t) \Big)\,
\frac{\sigma}{2} c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} - {\mathbf{q}}, \sigma}
\label{27a}\\
&&\qquad\qquad+ \textrm{H.c.}\Big] \nonumber
\\
&& -
\frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma} \Big[
v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}} , \lambda}(t) \frac{\sigma}{2}
\nonumber
\\
&& \times
:c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma}: \,
:c_{{\mathbf{k}}', -\sigma}^\dag c^{}_{{\mathbf{k}}'- {\mathbf{p}}, -\sigma} : + \textrm{H.c.} \Big] \,,\nonumber \\
\label{28}
{\mathcal H}_{f, \lambda}(t) &=&
\frac{U}{2N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma}
\mathbf P_{\lambda} \big( : c^\dag_{{\mathbf{k}} \sigma} \,c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma}: \, :c^\dag_{{\mathbf{k}}', -\sigma} \,
c^{}_{{\mathbf{k}}' - {\mathbf{p}}, -\sigma} : \big) \, .
\nonumber\\&&
\end{aligned}$$ Due to the renormalization all coefficients in $ {\mathcal H}_{0,\lambda}$ and $\hat {\mathcal H}_{h, \lambda}(t)$ depend on $\lambda$. Moreover, in $\hat {\mathcal H}_{h,\lambda}(t)$ new operator contributions are generated.
The coefficients $u_{{\mathbf{k}},{\mathbf{k}} - {\mathbf{q}},\lambda}(t)$ and $ v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}} , \lambda}(t)$ are expected to depend linearly on the external field and are therefore explicitly time-dependent. From hermiticity follows that they obey the relations: $$\begin{aligned}
u_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}},\lambda}&=& u^{*}_{{\mathbf{k}} +{\mathbf{q}}, {\mathbf{k}},\lambda},\\
v_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}&=&
- v^*_{{\mathbf{k}}'- {\mathbf{p}}, {\mathbf{k}}'; {\mathbf{k}} +{\mathbf{p}} +{\mathbf{q}},{\mathbf{k}},\lambda}\nonumber\\ &=&
- v_{{\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}; {\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, \lambda}\,.\end{aligned}$$
Finally $\mathbf P_{\lambda}$ in $\mathcal H_{f,\lambda}(t)$ projects on the low-energy excitations smaller than $\lambda$. One finds $$\begin{aligned}
\label{29}
&& {\mathcal H}_{f, \lambda}(t) = \frac{U}{2N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma} \Big( \Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda } \,\\
&& \quad \times
c^\dag_{{\mathbf{k}} \sigma} \,c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} \, c^\dag_{{\mathbf{k}}', -\sigma} \,
c^{}_{{\mathbf{k}}' - {\mathbf{p}}, -\sigma} \nonumber \\
&& \quad - \Theta_{{\mathbf{k}}, {\mathbf{k}} + {\mathbf{p}},\lambda} \,
c^\dag_{{\mathbf{k}} \sigma} \,c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} \, \langle c^\dag_{{\mathbf{k}}', -\sigma} \,
c^{}_{{\mathbf{k}}' - {\mathbf{p}}, -\sigma}\rangle \nonumber \\
&&\quad - \Theta_{{\mathbf{k}}', {\mathbf{k}}' - {\mathbf{p}},\lambda} \,
\langle c^\dag_{{\mathbf{k}} \sigma} \,c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} \rangle \,
c^\dag_{{\mathbf{k}}', -\sigma}
c^{}_{{\mathbf{k}}' - {\mathbf{p}}, -\sigma}
+ \textrm{const.} \Big) \nonumber \, ,
\end{aligned}$$ which shows that the operators on the right-hand side have different transition energies. Here, we have defined two $\Theta$-functions: $$\begin{aligned}
\label{30}
&& \Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda } = \\
&& \quad = \Theta\big( \lambda -|\varepsilon_{{\mathbf{k}}, \lambda} - \varepsilon_{{\mathbf{k}} + {\mathbf{p}}, \lambda}
+ \varepsilon_{{\mathbf{k}}', \lambda} - \varepsilon_{{\mathbf{k}}' -{\mathbf{p}}, \lambda} |
\big)\,, \nonumber \\
&& \Theta_{{\mathbf{k}}, {\mathbf{k}} + {\mathbf{p}},\lambda} =
\Theta\big( \lambda -|\varepsilon_{{\mathbf{k}}, \lambda} - \varepsilon_{{\mathbf{k}} + {\mathbf{p}}, \lambda}|\big) \, .
\end{aligned}$$ They guarantee that only transitions with excitation energies smaller than $\lambda$ are kept in $\mathcal H_{f,\lambda}(t)$. Finally we use relation (\[9\]) to regroup $\mathcal H_{f,\lambda}(t)$: $${\mathcal H}_{f, \lambda}(t) = {\mathcal H}^\alpha_{f, \lambda}(t) + {\mathcal H}^\beta_{f, \lambda}(t)$$ with $$\begin{aligned}
\label{31}
{\mathcal H}^\alpha_{f, \lambda}(t) &=& \frac{U}{2N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma}
\Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda } \\
&&\times :c^\dag_{{\mathbf{k}} \sigma} \,c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma}: \, :c^\dag_{{\mathbf{k}}', -\sigma} \,
c^{}_{{\mathbf{k}}' - {\mathbf{p}}, -\sigma}: \,, \nonumber\\[0.3cm]
{\mathcal H}^\beta_{f, \lambda}(t) &=& - {2U} \sum_{{\mathbf{k}} \sigma} \Big[
\varphi_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{q}}, \lambda}
\frac{\sigma}{2} \, c^\dag_{{\mathbf{k}} \sigma}
c_{{\mathbf{k}} + {\mathbf{q}}, \sigma} \\
&&+ \varphi_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda} \frac{\sigma}{2} \, c^\dag_{{\mathbf{k}} \sigma}
c_{{\mathbf{k}} - {\mathbf{q}}, \sigma}
\Big] \nonumber \, \nonumber .
\end{aligned}$$ $ {\mathcal H}^\alpha_{f, \lambda}(t)$ only depends on fluctuation operators, whereas $ {\mathcal H}^\beta_{f, \lambda}(t)$ is linear to the external field via the expectation value $ \langle s^z_{{\mathbf{k}}', -{\mathbf{q}}} \rangle$. Here we have defined $$\begin{aligned}
\label{32}
\varphi_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda} &=& \frac{1}{N} \sum_{{\mathbf{k}}'}
\Big(\Theta_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' + {\mathbf{q}},\lambda} - \Theta_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}},\lambda} \Big)
\langle s^z_{{\mathbf{k}}', -{\mathbf{q}}} \rangle \nonumber \\
&=&\varphi_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}(t) \,.\end{aligned}$$ Note that also higher-order fluctuation contributions both to $\mathcal H_{f,\lambda}(t)$ and $X_{\lambda, \Delta \lambda}(t)$ could be considered. Their inclusion would extend the range of validity of the present approach, which is restricted to the range from small to intermediate coupling $U/W \lesssim 1$, to larger values. However, they would further complicate the evaluation of transformation (\[24\]) and will be neglected.
Similarly, from Eq. (\[23\]) one finds the following expression for the generator $X_{\lambda, \Delta \lambda}$: $$\label{33}
X_{\lambda, \Delta \lambda}(t) = X^\alpha_{\lambda, \Delta \lambda}(t)
+ X^\beta_{\lambda, \Delta \lambda}(t)$$ with $$\begin{aligned}
\label{34}
X^\alpha_{\lambda, \Delta \lambda}(t) &=& \frac{1}{2N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma}
A_{{\mathbf{k}}, {\mathbf{k}} + {\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) \\
&&\times
: c^\dag_{{\mathbf{k}} \sigma} \,c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} \, c^\dag_{{\mathbf{k}}', -\sigma} \,c^{}_{{\mathbf{k}}' - {\mathbf{p}}, -\sigma}: \,,
\nonumber \\
X^\beta_{\lambda, \Delta \lambda}(t)
&=&
- 2\sum_{{\mathbf{k}} \sigma} \Big(
{B}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{q}}}(\lambda, \Delta \lambda) \frac{ \sigma}{2} c^\dag_{{\mathbf{k}} \sigma}
c_{{\mathbf{k}} + {\mathbf{q}}, \sigma}
\label{34a}
\\
&& +
{B}_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}}(\lambda, \Delta \lambda) \frac{ \sigma}{2} c^\dag_{{\mathbf{k}} \sigma}
c_{{\mathbf{k}} - {\mathbf{q}}, \sigma}
\Big) \nonumber \, .
\end{aligned}$$ The coefficients in Eqs. (\[34\]), (\[34a\]) are defined by $$\begin{aligned}
\label{35}
&&A_{{\mathbf{k}}, {\mathbf{k}} + {\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) =\\
&&\qquad\qquad =\frac{\Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}} (\lambda, \Delta \lambda) }
{\varepsilon_{{\mathbf{k}},\lambda} - \varepsilon_{{\mathbf{k}} + {\mathbf{p}},\lambda} + \varepsilon_{{\mathbf{k}}',\lambda}
- \varepsilon_{{\mathbf{k}}' -{\mathbf{p}}, \lambda}
} \, U \nonumber \, . \nonumber
\end{aligned}$$ and $$\begin{aligned}
\label{36}
&& {B}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{q}}}(\lambda, \Delta \lambda)(t)=\frac{1}{N} \sum_{{\mathbf{k}}'} \Big(A_{{\mathbf{k}}, {\mathbf{k}} + {\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{q}}}(\lambda, \Delta \lambda)
\nonumber \\
&& \hspace*{3cm}- \hat A_{{\mathbf{k}}, {\mathbf{k}}+ {\mathbf{q}}}(\lambda, \Delta \lambda) \Big) \langle s^z_{{\mathbf{k}}', {\mathbf{q}}}\rangle \, ,
\end{aligned}$$ where $$\begin{aligned}
\label{37}
&&\hat A_{{\mathbf{k}}, {\mathbf{k}} + {\mathbf{q}}}(\lambda, \Delta \lambda) = \frac{\Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{q}}} (\lambda, \Delta \lambda) }{\varepsilon_{{\mathbf{k}},\lambda} - \varepsilon_{{\mathbf{k}} + {\mathbf{q}},\lambda} }
\, U \, .
\end{aligned}$$ In $$\begin{aligned}
\label{38}
\Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}} (\lambda, \Delta \lambda) &=&
\Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}\\
&& \times
\big( 1 - \Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda - \Delta \lambda}
\big) \nonumber
\end{aligned}$$ and $$\begin{aligned}
&& \Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{q}}} (\lambda, \Delta \lambda) = \Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{q}}, \lambda}
\big( 1- \Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{q}}, \lambda- \Delta \lambda}
\big) \end{aligned}$$ the products of $\Theta$-functions assure that only excitations between $\lambda$ and $\lambda- \Delta \lambda$ are eliminated by the unitary transformation (\[21\]).
Renormalization equations {#IV.2}
-------------------------
Integrating out all transitions induced by $\mathcal H_{f,\lambda}(t)$, the parameters of $\mathcal H_\lambda(t)$ will be renormalized. Only the Coulomb coupling $U$ remains $\lambda$-independent (apart from the $\Theta$-functions in Eq. (\[30\])). The $\lambda$-dependence of the parameters will be derived with transformation (\[24\]) for an additional step from $\lambda$ to $\lambda - \Delta \lambda$. The result of the explicit evaluation has to be compared with the generic expression for $ {\mathcal H}_{\lambda - \Delta \lambda}(t)$, which is obtained by replacing $\lambda$ in $\mathcal H_{\lambda}$ \[Eqs. (\[26\])-(\[30\])\]. In this way one obtains the desired renormalization equations, which connect the $\lambda$-dependent parameters of $\mathcal H_\lambda$ with those at cutoff $\lambda - \Delta \lambda$. According to Appendix B we find $$\begin{aligned}
\label{39}
&&\varepsilon_{{\mathbf{k}}, \lambda -\Delta \lambda} - \varepsilon_{{\mathbf{k}}, \lambda} =
\delta\varepsilon^{(1)}_{{\mathbf{k}}, \lambda} - \frac{1}{2} \delta \varepsilon^{(2)}_{{\mathbf{k}}, \lambda} \,,\\
\label{40}
&&u_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}},\lambda -\Delta \lambda}(t) -
u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\lambda}(t)
=
\sum_{n=1}^4\delta u^{(n)}_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda}(t) \,, \\
\label{41}
&& v_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda -\Delta \lambda}(t) -
v_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}(t) \nonumber \\
&& \hspace*{2.5cm} =
\sum_{n=1}^3
\delta v^{(n)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}(t)
\, .
\end{aligned}$$ The renormalization contributions on the right-hand sides of these equations are of order $U$ and $U^2$. They are given in Appendix B: $ \delta\varepsilon^{(1)}_{{\mathbf{k}}, \lambda}$, $\delta \varepsilon^{(2)}_{{\mathbf{k}}, \lambda}$ by Eqs. (\[B10\]) and (\[B24\]); $\delta u^{(n)}_{{\mathbf{k}}, {\mathbf{q}}, \lambda}(t)$, $(n=1 \ldots 4)$ by Eqs. (\[B2\]), (\[B12\]), (\[B17\]), and (\[B20\]); and finally $ \delta v^{(n)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}(t)$, $(n=1,2,3)$, by Eqs. (\[B3\]), (\[xyz1\]), and (\[xyz2\]). In order to reduce the operator structure of $\mathcal H_{\lambda- \Delta \lambda}(t)$ to operators which appear in $\mathcal H_\lambda(t)$ an additional factorization of higher operator terms has been performed. Therefore, the expectation values $\langle c_{{\mathbf{k}} \sigma}^\dag c_{{\mathbf{k}} \sigma }\rangle$ and $\langle c_{{\mathbf{k}} \sigma}^\dag c_{{\mathbf{k}} \pm {\mathbf{q}}, \sigma }\rangle$ enter the renormalization contributions.
The renormalization equations have to be solved numerically, starting from the initial parameters of the original model ${\mathcal H}(t)$, i.e., $$\begin{aligned}
\label{42}
&& {\varepsilon}_{{\mathbf{k}},\Lambda} = {\varepsilon}_{{\mathbf{k}}} + U \langle n \rangle/2 \, ,\\
&&u_{{\mathbf{k}},{\mathbf{k}}-{\mathbf{q}}, \Lambda}(t) = 0\,,\\
&&
v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}} , \Lambda}(t)=0 \, .\label{42a}
\end{aligned}$$ Suppose, the expectation values on the right-hand side of Eqs. (\[39\])-(\[41\]) are known, the renormalization procedure from $\Lambda$ to $\lambda=0$ leads to the fully renormalized Hamiltonian $$\tilde{\mathcal H}(t)=
\mathcal H_{0,\lambda =0} + \hat{\mathcal H}_{h,\lambda=0}(t)$$ with $$\begin{aligned}
\label{43}
&&{\mathcal H}_{0,\lambda=0} = \sum_{{\mathbf{k}} \sigma} \tilde{\varepsilon}_{{\mathbf{k}}} \,
c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} \sigma} \,, \\
\label{44}
&& \hat {\mathcal H}_{h,\lambda=0}(t) = -
\sum_{{\mathbf{k}} \sigma}\Big[\Big(\frac{\hat h_{{\mathbf{q}}}(t)}{2} +\tilde u_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}}(t)\Big) \,
\frac{\sigma}{2} c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} - {\mathbf{q}}, \sigma} \nonumber \\
&& \phantom{ \hat {\mathcal H}_{h,\lambda=0}(t)} \qquad \qquad \qquad+ \textrm{H.c.}\Big]
\\
&& \phantom{ \hat {\mathcal H}_{h,\lambda=0}(t)}- \frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma} \big[
\tilde v_{{\mathbf{k}}, {\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(t) \nonumber \\
&& \phantom{ \hat {\mathcal H}_{h,\lambda=0}(t)}\times \frac{\sigma}{2}
: c^\dag_{{\mathbf{k}}\sigma} c_{{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, \sigma} c^\dag_{{\mathbf{k}}' , -\sigma}
c_{{\mathbf{k}}' -{\mathbf{p}}, -\sigma}: + \textrm{H.c.} \big]
\, . \nonumber
\end{aligned}$$ The tilde symbols denote the fully renormalized quantities at $\lambda=0$. All excitations from $\mathcal H_{f,\lambda}(t)$ have been eliminated, leading to the renormalization of $\mathcal H_{0,\lambda}$ and $\hat{\mathcal H}_{h,\lambda}(t)$. The final Hamiltonian $\tilde{\mathcal H}(t)$ describes a system of free renormalized conduction electrons in a renormalized effective field. Thereby, the quantities $\tilde u_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}}(t)$ and $\tilde v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}}(t)$ depend linearly on the external field and are time-dependent. This follows from the renormalization equations (\[39\])-(\[41\]), using the expressions for the renormalization contributions from Appendix A, together with the initial conditions (\[42\]). Therefore, relying on linear response theory with respect to the effective field, any expectation value can be evaluated.
Expectation values {#IV.3}
------------------
### Occupation numbers $\langle n_{{\mathbf{k}}\sigma} \rangle$ {#IV.3.1}
The yet unknown expectation values on the right-hand side of the renormalization equations (\[39\])- (\[41\]) can be evaluated self-consistently as follows. Let us first consider the averaged occupation number $\langle n_{{\mathbf{k}}\sigma}\rangle$ for fixed spin $\sigma$. Using Eq. (\[25\]), $\langle n_{{\mathbf{k}}\sigma}\rangle$ can be rewritten as $$\label{45}
\langle n_{{\mathbf{k}}\sigma} \rangle =
\langle c^\dag_{{\mathbf{k}}\sigma}c^{}_{{\mathbf{k}}\sigma}\rangle=\langle \tilde{c}^\dag_{{\mathbf{k}} \sigma}\tilde{c}^{}_{{\mathbf{k}} \sigma}\rangle_{\tilde{\mathcal{H}}}\, .$$ In principle, the last expectation value has to be formed with the time-dependent Hamiltonian $\tilde{\mathcal{H}}(t)$. However, restricting ourselves to first order in $h(t)$, $\tilde{\mathcal{H}}(t)$ can be replaced by the time-independent Hamiltonian $\tilde{\mathcal H}_{0}$. $\tilde{c}^{\dag}_{{\mathbf{k}}\sigma}$ is the fully renormalized creation operator $\tilde{c}^{\dag}_{{\mathbf{k}}\sigma}=c^{\dag}_{{\mathbf{k}}\sigma}(\lambda\rightarrow 0)$, where $c^{\dag}_{{\mathbf{k}}\sigma}(\lambda)$ is defined by $c^{\dag}_{{\mathbf{k}}\sigma}(\lambda)
= e^{X_\lambda} c^\dag_{{\mathbf{k}} \sigma} e^{-X_\lambda}$.
For $c^{\dag}_{{\mathbf{k}}\sigma}(\lambda)$ an appropriate [*ansatz*]{} is necessary. We choose $$\begin{aligned}
\label{46}
c^\dag_{{\mathbf{k}} \sigma}(\lambda) &=& x^{}_{\mathbf{k},\lambda}c^\dag_{{\mathbf{k}} \sigma} \\
&&+ \!\frac{1}{2N}\sum_{{\mathbf{p}}{\mathbf{k}}'}
y^{}_{{\mathbf{k}}{\mathbf{p}}{\mathbf{k}}',\lambda}c^\dag_{\mathbf{k-p},\sigma}\!:\!c^\dag_{{\mathbf{k}}',-\sigma}c^{}_{{\mathbf{k}}'-{\mathbf{p}},-\sigma}\!:
\, , \nonumber\end{aligned}$$ where the operator structure of (\[46\]) is again taken over from the lowest order expansion in $X_\lambda$ of the unitary transformation.
In analogy to Eq. (\[21\]) renormalization equations for the coefficients $x^{}_{\mathbf{k},\lambda}$ and $y^{}_{{\mathbf{k}}{\mathbf{p}}{\mathbf{k}}',\lambda}$ can be derived by evaluating the renormalization step from $\lambda$ to $\lambda- \Delta \lambda$. One finds $$\label{47}
y^{}_{\mathbf{kp}{\mathbf{k}}',\lambda-\Delta\lambda}-y^{}_{\mathbf{kp}{\mathbf{k}}',\lambda}=x^{}_{\mathbf{k},\lambda}
A_{\mathbf{k-p},{\mathbf{k}};{\mathbf{k}}',{\mathbf{k}}'-{\mathbf{p}}}(\lambda,\Delta\lambda) \, .$$ This equation connects $y^{}_{\mathbf{kp}{\mathbf{k}}',\lambda-\Delta\lambda}$ at cutoff $\lambda -\Delta \lambda$ with the coefficients $x^{}_{\mathbf{k},\lambda}$, $y^{}_{\mathbf{kp}{\mathbf{k}}',\lambda}$ at cutoff $\lambda$. Similarly, also an renormalization equation for $x_{{\mathbf{k}}, \lambda}$ can be found. Alternatively one may start from the anti-commutator relation of $c^\dag_{{\mathbf{k}}\sigma}(\lambda)$ and $c^{}_{{\mathbf{k}}\sigma}(\lambda)$. Taking the expectation value with $\mathcal H_\lambda$, we get $$\label{48}
\langle [c^\dag_{{\mathbf{k}}\sigma}(\lambda),c^{}_{{\mathbf{k}}\sigma}(\lambda)]_+\rangle_{\mathcal{H}_\lambda}=1 \, ,$$ or $$\begin{aligned}
\label{49}
&& |x^{}_{\mathbf{k},\lambda}|^2+\frac{1}{4N^2}\sum_{\mathbf{p}{\mathbf{k}}'}|y^{}_{{\mathbf{k}}{\mathbf{p}}{\mathbf{k}}',\lambda}|^2S^{c}_{\mathbf{k}\mathbf{p}{\mathbf{k}}'}=1 \, .\end{aligned}$$ Here, a factorization approximation for $$S^{c}_{\mathbf{kp}{\mathbf{k}}'}=\langle n_{\mathbf{k}-{{\mathbf{p}}}}\rangle(\langle n_{\mathbf{k}'}\rangle-\langle n_{\mathbf{k}'-{\mathbf{p}}}\rangle
+\langle n_{\mathbf{k}'-{\mathbf{p}}}\rangle(2-\langle n_{\mathbf{k}'}\rangle)$$ was used. Eq. (\[49\]) connects the coefficients $x^{}_{\mathbf{k},\lambda}$ and $y^{}_{\mathbf{kp}{\mathbf{k}}',\lambda}$ for any value of $\lambda$. The equation for $x_{{\mathbf{k}}, \lambda- \Delta \lambda}$ is found from the sum rule (\[49\]), when $\lambda$ is replaced by $\lambda-\Delta\lambda$, i.e., $$\begin{aligned}
\label{50}
&& |x^{}_{\mathbf{k},\lambda-\Delta\lambda}|^2=1-\frac{1}{4N^2}\sum_{\mathbf{p}{\mathbf{k}}'}|y^{}_{\mathbf{kp}{\mathbf{k}}',\lambda-\Delta\lambda}|^2S^{c}_{\mathbf{kp}{\mathbf{k}}'}\,.\end{aligned}$$ Together with Eq. (\[47\]) this equation relates $x^{}_{\mathbf{k},\lambda-\Delta\lambda}$ with $x^{}_{\mathbf{k},\lambda}$ and $y^{}_{{\mathbf{k}}{\mathbf{p}}{\mathbf{k}}',\lambda}$. Thus Eqs. (\[50\]), (\[47\]) connect the parameter values at $\lambda$ with those at $\lambda -\Delta \lambda$. Integrating Eqs. (\[47\]), (\[50\]) between $\Lambda$ and $\lambda =0$ (thereby using $x^{}_{\mathbf{k},\Lambda}=1$ and $y^{}_{\mathbf{kp}{\mathbf{k}}',\Lambda}=0$), we obtain $$\begin{aligned}
\label{51}
&& \tilde{c}^\dagger_{\mathbf{k}\sigma}=\tilde{x}^{}_{\mathbf{k}}c^\dagger_{\mathbf{k}\sigma}\\
&& \qquad +\frac{1}{2N}\sum_{\mathbf{p}{\mathbf{k}}'}
\tilde{y}^{}_{\mathbf{kp}{\mathbf{k}}'}c^\dagger_{\mathbf{k-p},\sigma}
:c^\dagger_{\mathbf{k}',-\sigma}c^{}_{\mathbf{k}'-{\mathbf{p}},-\sigma}: \, , \nonumber \end{aligned}$$ where the tildes again denote the fully renormalized quantities. Thus, for $\langle n_{\mathbf k \sigma} \rangle$ the final result is $$\begin{aligned}
\label{52}
&& \langle n_{\mathbf{k} \sigma} \rangle=
|\tilde{x}^{}_{\mathbf{k}}|^2f(\tilde{\varepsilon}_{{\mathbf{k}}}) \\
&& \quad +\frac{1}{N^2}\sum_{\mathbf{p}{\mathbf{k}}'}\left|
\tilde{y}^{}_{\mathbf{kp}{\mathbf{k}}'}\right|^2
f(\tilde{\varepsilon}_{{\mathbf{k}}-{\mathbf{p}}})f(\tilde{\varepsilon}_{{\mathbf{k}}'})[1-f(\tilde{\varepsilon}_{{\mathbf{k}}'-{\mathbf{p}}})]
\nonumber \, ,\end{aligned}$$ which is independent of $\sigma$ in the paramagnetic state. $f(\tilde{\varepsilon}_{{\mathbf{k}}})$ is the Fermi function.
### Transformation of spin operators {#IV.3.2}
To evaluate the dynamical spin susceptibility we need the transformed spin operator, $\tilde s^z_{{\mathbf{q}}} = s^z_{{\mathbf{q}}, \lambda \rightarrow 0}$, where $s_{{\mathbf{q}},\lambda}^z= e^{X_\lambda} s_{{\mathbf{q}}}^z e^{-X_\lambda}$. For the $\lambda$-dependence we use the following [*ansatz*]{}, which corresponds to [*ansatz*]{} (\[28\]) for $\hat{\mathcal H}_{h,\lambda}$. According to Appendix A we write $$\begin{aligned}
\label{53}
s^z_{{\mathbf{q}},\lambda} &=&
\sum_{{\mathbf{k}} \sigma} \alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}},\lambda} \,
\frac{\sigma}{2} c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} - {\mathbf{q}}, \sigma}
\\
&&+
\frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma}
\beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}
\nonumber \\
&&\times
\frac{\sigma}{2}
:c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma }
c_{{\mathbf{k}}', -\sigma}^\dag c^{}_{{\mathbf{k}}'- {\mathbf{p}}, -\sigma} :
\nonumber
\end{aligned}$$ with $\lambda$-dependent coefficients $\alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}},\lambda}$, $\beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}$. Their initial values at $\lambda =\Lambda$ are $$\begin{aligned}
\label{54}
\alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}},\Lambda} = 1 \, , \quad
\beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \Lambda} =0 \, .
\end{aligned}$$ The renormalization equations for the coefficients are found from the transformation step $s^z_{{\mathbf{q}}, \lambda-\Delta \lambda}= e^{X_{\lambda, \Delta \lambda}}
s^z_{{\mathbf{q}}, \lambda}e^{-X_{\lambda, \Delta \lambda}}$, where in linear response theory $X_{\lambda, \Delta \lambda}$ can be replaced by its part $X^{(\alpha)}_{\lambda, \Delta \lambda}$. A closer inspection shows that the renormalization equations can be taken over from the equations for $u_{{\mathbf{k}},{\mathbf{k}}-{\mathbf{q}},\lambda}$ and $ v_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}$. We get $$\begin{aligned}
\label{55}
&& \alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}},\lambda-\Delta \lambda} -\alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}},\lambda} =
\delta \alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}},\lambda}^{(1)} \nonumber \\
&& \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda- \Delta \lambda} -
\beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}\ \nonumber \\
&& \qquad \qquad=\delta \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}^{(1)}\,,
\end{aligned}$$ where $$\begin{aligned}
\label{56}
&& \delta \alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}, \lambda}^{(1)} =
\frac{1}{N} \sum_{{\mathbf{k}}'} \big(\alpha_{{\mathbf{k}}' + {\mathbf{q}},{\mathbf{k}}', \lambda}
A_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' +{\mathbf{q}}}(\lambda, \Delta \lambda) \nonumber \\
&& \qquad - \alpha_{{\mathbf{k}}', {\mathbf{k}}' -{\mathbf{q}}, \lambda}
A_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}; {\mathbf{k}}' -{\mathbf{q}}, {\mathbf{k}}'}(\lambda, \Delta \lambda)
\big) \langle c_{{\mathbf{k}}',-\sigma}^\dag c_{{\mathbf{k}}', -\sigma}\rangle \nonumber \\
&& \qquad \qquad = -\frac{1}{N} \sum_{{\mathbf{k}}'} \alpha_{{\mathbf{k}}', {\mathbf{k}}' -{\mathbf{q}}, \lambda}
A_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}; {\mathbf{k}}' -{\mathbf{q}}, {\mathbf{k}}'}(\lambda, \Delta \lambda) \nonumber \\
&& \qquad \qquad\times \big(
\langle c_{{\mathbf{k}}',-\sigma}^\dag c_{{\mathbf{k}}', -\sigma}\rangle
- \langle c_{{\mathbf{k}}' -{\mathbf{q}},-\sigma}^\dag c_{{\mathbf{k}}' -{\mathbf{q}}, -\sigma}\rangle
\big)\,,\end{aligned}$$ and $$\begin{aligned}
\label{57}
&& \delta \beta^{(1)}_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}= \\
&& \qquad
\alpha_{{\mathbf{k}} + {\mathbf{p}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}},\lambda}
A_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) \nonumber \\
&& \qquad - \alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}, \lambda}
A_{{\mathbf{k}} -{\mathbf{q}}, {\mathbf{k}}+{\mathbf{p}}-{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) \nonumber
\, .\end{aligned}$$ Note that in Eq. (\[55\]) no equivalent to $\delta u^{(2)}_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}}$ enters, since the latter contribution was caused by the commutator $[X_{\lambda,\Delta \lambda}, \mathcal H_{f,\lambda}]$ (compare Appendix A). Furthermore, renormalizations from the second part in [*ansatz*]{} (\[53\]), being proportional to $\sim \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}$, have been neglected.
Let us finally stress that the renormalization contributions to $\alpha_{{\mathbf{k}},{\mathbf{k}}-{\mathbf{q}},\lambda}$ and $ \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}',{\mathbf{k}}' -{\mathbf{p}} \lambda} $ vanish for wave vector ${\mathbf{q}}=0$. This is in accord with the rotational invariance of the system in spin space.
Dynamical magnetic susceptibility {#IV.4}
---------------------------------
The dynamical ${\mathbf{q}}$- and $\omega$-dependent magnetic susceptibility $\chi({\mathbf{q}}, \omega)$ is defined by the linear response of the averaged spin $\langle s^z_{-q} \rangle(t)$ to the small external field $h(t)$. Since $\hat{\mathcal H}_{h,\lambda=0}$ itself is proportional to $h(t)$: $$\label{58}
\langle s^z_{-q} \rangle (t)
= - i \int_0^\infty dt'
\langle [\tilde s_{-{\mathbf{q}}}^z(t') , \hat{ \mathcal H}_{h,\lambda=0}(t- t')] \rangle_{\tilde{\mathcal H}_0} \,.$$ Here we have again used the unitary invariance (\[25\]) of operator expressions under a trace. The expectation value in Eq. (\[58\]) is formed with the renormalized one-particle Hamiltonian $\tilde{\mathcal H}_0 = \mathcal H_{0,\lambda =0} = \sum_{{\mathbf{k}} \sigma}
\tilde{\varepsilon}_{{\mathbf{k}}} c_{{\mathbf{k}} \sigma}^\dag c_{{\mathbf{k}} \sigma}$. Both $\tilde s_{-{\mathbf{q}}}^z$ and $\hat{ \mathcal H}_{h,\lambda=0}(t)$ are the fully renormalized quantities. They are given by $$\begin{aligned}
\label{59}
\tilde s^z_{{\mathbf{q}}} &=& \nonumber
\sum_{{\mathbf{k}} \sigma} \tilde \alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}} \,
\frac{\sigma}{2} c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} - {\mathbf{q}}, \sigma}
\\
&&+
\frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma}
\tilde \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}
\nonumber \\
&&\times
\frac{\sigma}{2}
:c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma }
c_{{\mathbf{k}}', -\sigma}^\dag c^{}_{{\mathbf{k}}'- {\mathbf{p}}, -\sigma} : \, ,
\end{aligned}$$ where $ \tilde \alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}}$ and $ \tilde \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}$ are independent of $h(t)$, and $$\begin{aligned}
\label{60}
\hat {\mathcal H}_{h,\lambda=0}(t) &=& -
\sum_{{\mathbf{k}} \sigma} \Big(\frac{\hat h_{{\mathbf{q}}}(t)}{2}+ \tilde u_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}}(t)\Big) \,
\frac{\sigma}{2} c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} - {\mathbf{q}}, \sigma}
\nonumber\\
&&-
\frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma}
\tilde v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}}(t)
\nonumber \\
&&\times
\frac{\sigma}{2}
:c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma }
c_{{\mathbf{k}}', -\sigma}^\dag c^{}_{{\mathbf{k}}'- {\mathbf{p}}, -\sigma} :
\nonumber \\
&&+ \textrm{H.c.} \, . \
\end{aligned}$$ Thus, Eq. (\[58\]) reduces to $$\begin{aligned}
\label{61}
&& \langle s^z_{-q} \rangle (t) = \sum_{{\mathbf{k}} }
\tilde{\alpha}^*_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}} \, \chi_{{\mathbf{k}}}^0({\mathbf{q}}, \omega) \,
\Big( \frac{\hat h_{{\mathbf{q}}}(t)}{2}+ \tilde u_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}}(t) \Big)
\nonumber \\
&& \qquad\qquad+ \frac{1}{ N^2} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} }
\tilde \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}^* \,
\chi^0_{{\mathbf{k}}, {\mathbf{k}}', {\mathbf{p}}}({\mathbf{q}}, \omega) \nonumber \\
&& \hspace*{3cm} \times \tilde v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}}(t) \, ,\end{aligned}$$ where we have introduced ${\mathbf{k}}$-resolved susceptibilities $$\begin{aligned}
\label{62}
&& {\chi}^0_{{\mathbf{k}}}({\mathbf{q}},\omega) = \nonumber\\
&& \frac{i}{2} \int_0^\infty
\langle [ (c_{{\mathbf{k}}\sigma}^\dag c^{}_{{\mathbf{k}} -{\mathbf{q}},\sigma})^\dag (t'), c_{{\mathbf{k}}\sigma}^\dag c^{}_{{\mathbf{k}} -{\mathbf{q}},\sigma} ]
\rangle_{\tilde{\mathcal H}_0}
\, e^{i ( \omega + i\eta)t'} \, dt' \nonumber \\
&& \nonumber \\
&& = \frac{1}{2}\frac{f(\tilde{\varepsilon}_{{\mathbf{k}} -{\mathbf{q}}}) -f(\tilde{\varepsilon}_{{\mathbf{k}}})}{(\tilde{\varepsilon}_{{\mathbf{k}}} - \tilde{\varepsilon}_{{\mathbf{k}}- {\mathbf{q}}}) -( \omega + i\eta)} \, ,\end{aligned}$$ and $$\begin{aligned}
\label{63}
&& {\chi}^0_{{\mathbf{k}},{\mathbf{k}}', {\mathbf{p}}}({\mathbf{q}},\omega) = \nonumber\\
&& \quad \frac{i}{2} \int_0^\infty \langle [
\big( :c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma }
c_{{\mathbf{k}}', -\sigma}^\dag c^{}_{{\mathbf{k}}'- {\mathbf{p}}, -\sigma} :)^\dag (t') \nonumber \\
&& \qquad
:c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma }
c_{{\mathbf{k}}', -\sigma}^\dag c^{}_{{\mathbf{k}}'- {\mathbf{p}}, -\sigma} :
] \rangle_{\tilde{\mathcal H}_0} \,
e^{i( \omega + i\eta)t'} dt' \nonumber \\
&& \nonumber \\
&& \quad =\frac{1}{2}\frac{N_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}}}}{{\omega}_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}}} -(\omega + i\eta)}
\end{aligned}$$ with $$\begin{aligned}
\omega_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}}} &=&
\tilde{\varepsilon}_{{\mathbf{k}}} - \tilde{\varepsilon}_{{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}} + \tilde{\varepsilon}_{{\mathbf{k}}'}
- \tilde{\varepsilon}_{{\mathbf{k}}' -{\mathbf{p}}} \,, \\
N_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}}} &=&
f(\tilde{\varepsilon}_{{\mathbf{k}}'- {\mathbf{p}}}) \big[ 1- f(\tilde{\varepsilon}_{{\mathbf{k}}'})\big]
\big[ f(\tilde{\varepsilon}_{{\mathbf{k}}+ {\mathbf{p}} -{\mathbf{q}}}) - f(\tilde{\varepsilon}_{{\mathbf{k}}}) \big] \nonumber
\\
&\!\!\!\!\!\!\!\!+&\!\!\!\!\!\!\!\!f(\tilde{\varepsilon}_{{\mathbf{k}}})\big[1- f(\tilde{\varepsilon}_{{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}})\big]
\big[ f(\tilde{\varepsilon}_{{\mathbf{k}}'- {\mathbf{p}}}) - f(\tilde{\varepsilon}_{{\mathbf{k}}'}) \big]
\, .
\end{aligned}$$ Note that in both susceptibilities (\[62\]) and (\[63\]) renormalized energies $\tilde\varepsilon_{{\mathbf{k}}}$ enter and not the unrenormalized energies $\varepsilon_{{\mathbf{k}}}$ as in Eq. (\[19\]).
According to Sect. \[IV.2\] and Appendix B the quantities $\tilde u_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}}(t)$ and $\tilde v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}}(t)$ depend linearly on the external field $h(t)$ via the ${\mathbf{k}}$-resolved spin operator expectation values $\langle s^z_{{\mathbf{k}}, -{\mathbf{q}}}\rangle(t)$. In order to simplify the further calculation we shall trace back these quantities to the expectation value $\langle s^z_{-{\mathbf{q}}}\rangle(t)$ of the full spin operator. This is done by assuming that the coefficients $ B^{(n)}_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}(\lambda,\Delta \lambda)$, $(n=2,3,4)$ and $D^{(n)}_{\bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} {\mathbf{q}}}(\lambda, \Delta \lambda)$, $(n=2,3)$ defined in Appendix B are almost independent of the wave vector $\bar {{\mathbf{p}}}$. For example, we use for the renormalization contribution $ \delta u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\lambda}^{(2)}(t)$ \[Eq. (\[B12\])\]: $$\begin{aligned}
\label{64}
&& \delta u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\lambda}^{(2)}(t) \approx
- \frac{U}{2N^2} \sum_{{\mathbf{p}}, \bar{{\mathbf{p}}}}
B_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}^{(2)}(\lambda, \Delta \lambda) \frac{\langle s^z_{-{\mathbf{q}}}\rangle(t)}{N} \;\,\end{aligned}$$ with $\langle s^z_{-{\mathbf{q}}}\rangle= \sum_{{\mathbf{k}}'} \langle s^z_{{\mathbf{k}}', -{\mathbf{q}}}\rangle $. Thus, from $u_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda}(t)$ as well as from $v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}, \lambda}(t)$ a common factor $\langle s^z_{-{\mathbf{q}}}\rangle(t)/N$ can be extracted: $$\begin{aligned}
\label{65}
&& u_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda}(t) = u^0_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda} \,
\frac{\langle s^z_{-{\mathbf{q}}}\rangle(t)}{N}\,,
\\
&& v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}, \lambda}(t) =
v^0_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}, \lambda}
\frac{\langle s^z_{-{\mathbf{q}}}\rangle(t)}{N} \nonumber\, ,\\
\end{aligned}$$ where we introduced time-independent quantities $ u^0_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda}$ and $ v^0_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}, \lambda}$. They obey the following renormalization equations: $$\begin{aligned}
\label{66}
&& u^0_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}},\lambda -\Delta \lambda} -
u^0_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\lambda} = \delta u^{0(1)}_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda}
+ \sum_{n=2}^4 \delta u^{0(n)}_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda} \,,\nonumber\\ \\
\label{67}
&& v^0_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda -\Delta \lambda} -
v^0_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda} = \nonumber \\
&& \quad\delta v^{0(1)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}
+ \sum_{n=2}^3
\delta v^{0(n)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}
\, ,
\end{aligned}$$ with the time-independent renormalization contributions given in Appendix C. Due to Eq. (\[42\]) the initial conditions for $ u^0_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda}$ and $ v^0_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}, \lambda}$ at cutoff $\Lambda$ are $$\begin{aligned}
\label{68}
&& u^0_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\Lambda} = v^0_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \Lambda} =0 \, .
\end{aligned}$$
Having Eq. (\[66\]), (\[67\]) we are in a position to rewrite relation (\[61\]) in a time-independent form: $$\begin{aligned}
\label{69}
&& 1 = \frac{1}{N} \sum_{{\mathbf{k}} }
\tilde{\alpha}^*_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}} \, \chi_{{\mathbf{k}}}^0({\mathbf{q}}, \omega) \,
\big( U + \frac{1}{\chi({\mathbf{q}}, \omega)}+ \tilde u^0_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}} \big)
\nonumber \\
&& \qquad\qquad+ \frac{1}{ N^3} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} }
\tilde \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}^* \,
\chi^0_{{\mathbf{k}}, {\mathbf{k}}', {\mathbf{p}}}({\mathbf{q}}, \omega) \nonumber \\
&& \hspace*{3cm} \times \tilde v^0_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}} \,,\end{aligned}$$ where, on the right-hand side, we have used the relation $$\label{70}
\frac{\hat h_{{\mathbf{q}}}(t)}{2} = \Big(U + \frac{1}{\chi({\mathbf{q}}, \omega)}\Big) \frac{\langle s^z_{-{\mathbf{q}}}\rangle(t)}{N} \, .$$ The coefficients $\tilde u^0_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}} $ and $ \tilde v^0_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}} $ are again fully renormalized. They are found by solving the renormalization equations (\[66\]), (\[67\]) due to the initial conditions (\[68\]). Eq. (\[69\]) is an implicit equation for the dynamical susceptibility $\chi({\mathbf{q}}, \omega)$ which also enters the quantities $\delta u_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}, \lambda}^{0(1)}$ and $ \delta v^{0(1)}_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}$ (see Appendix C).
Solving for $\chi({\mathbf{q}}, \omega)$ we obtain our final analytical result: $$\begin{aligned}
\label{71}
\chi({\mathbf{q}}, \omega) &=& \displaystyle \frac{\displaystyle \frac{1}{N}\sum_{{\mathbf{k}} }
\tilde{\alpha}^*_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}} \, \chi_{{\mathbf{k}}}^0({\mathbf{q}}, \omega) }{1- \displaystyle \frac{U}{N}\sum_{{\mathbf{k}} }
\tilde{\alpha}^*_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}} \, \chi_{{\mathbf{k}}}^0({\mathbf{q}}, \omega)
-\Delta({\mathbf{q}}, \omega)} \, ,
$$ with $$\begin{aligned}
\label{72}
&&\Delta({\mathbf{q}}, \omega)= \frac{1}{N} \sum_{{\mathbf{k}} }
\tilde{\alpha}^*_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}} \, \chi_{{\mathbf{k}}}^0({\mathbf{q}}, \omega)\tilde u^0_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}}\\
&&+ \frac{1}{N^3} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} }
\tilde \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}^* \,
\chi^0_{{\mathbf{k}}, {\mathbf{k}}', {\mathbf{p}}}({\mathbf{q}}, \omega) \tilde v^0_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}}\nonumber\, .\end{aligned}$$ The PRM result (\[71\]) for the dynamical magnetic susceptibility of the Hubbard model represents an extension of the standard RPA expression. Besides the new coefficients $\tilde{\alpha}_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}}$ in the numerator and denominator an extra term $\Delta({\mathbf{q}}, \omega)$ occurs in the denominator, which generalizes the overall shape of an RPA expression. Noteworthy the quantities $\tilde u^0_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}}$ and $\tilde v^0_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}}$ in $\Delta({\mathbf{q}}, \omega)$ themselves depend on $\chi({\mathbf{q}}, \omega)$ (Appendix \[appC\]). Thus, $\chi({\mathbf{q}}, \omega)$ has to be solved self-consistently from Eqs. (\[71\]), (\[72\]). Expression (\[71\]) reduces to the standard RPA results when all renormalization effects are disregarded. Then, $\tilde {\alpha}_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}}$ and $ \tilde \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}$ keep their original values $1$ and $0$, whereas $\tilde u^0_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}} $ and $ \tilde v^0_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}}}$ remain zero and also $\Delta({\mathbf{q}}, \omega)$ vanishes. Then $\chi({\mathbf{q}}, \omega)$ becomes $\chi_{RPA}({\mathbf{q}}, \omega)$ given by Eq. (\[20\]) with $ \chi_0({\mathbf{q}}, \omega)$ being the unrenormalized dynamical magnetic susceptibility of free electrons \[Eq. (\[19\])\].\
In the next section $\chi({\mathbf{q}}, \omega)$ will be evaluated numerically. Before, let us study the special case ${\mathbf{q}}=0$. Since the total spin $s^z_{{\mathbf{q}}=0}$ commutes with the total Hamiltonian, i.e. $[{\mathcal H}, s^z_{{\mathbf{q}}}]=0$, the dynamical susceptibility vanishes, which can immediately be seen from the general expression (\[8\]) for $\chi({\mathbf{q}}=0, \omega)$ ($\omega$ finite). The same conclusion can also be drawn from the PRM formalism. Since the total spin $s^z_{{\mathbf{q}}=0}$ commutes with the Hamiltonian it also commutes with the generator $X_{\lambda, \Delta \lambda}$. Therefore, the coefficients in representation (\[59\]) for $\tilde s^z_{{\mathbf{q}}, \lambda}$ will not be renormalized for ${\mathbf{q}}=0$. Thus, we have $\tilde{\alpha}_{{\mathbf{k}}, {\mathbf{k}}}=1$ and $\tilde{\beta}_{{\mathbf{k}}, {\mathbf{k}}+ {\mathbf{p}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}=0$. Similarly, the coefficients $\tilde u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}}^{0}$ and $\tilde v^0_{{\mathbf{k}}, {\mathbf{k}}+ {\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}$ in expression (\[60\]) for $\hat{\mathcal H}_{h,\lambda =0}(t)$ also vanish for ${\mathbf{q}}=0$. Thus, the quantity $\Delta ({\mathbf{q}}, \omega)$ in the denominator of Eq. (\[71\]) vanishes and the susceptibility $\chi({\mathbf{q}}=0, \omega)$ takes the standard RPA form $$\begin{aligned}
\label{73}
&& \chi({\mathbf{q}}=0, \omega) = \displaystyle \frac{\displaystyle \chi^0(0, \omega) }
{1-U\chi^0(0, \omega)}\, .\end{aligned}$$ Here, according to Eq. (\[62\]), the susceptibility $$\label{73a}
\chi^0({\mathbf{q}}, \omega)= \frac{1}{N}\sum_{{\mathbf{k}}} \chi_{{\mathbf{k}}}^0({\mathbf{q}}, \omega)\,,$$ which contains the renormalized energies $\tilde{\varepsilon}_{{\mathbf{k}}}$ and not the unrenormalized energies $\varepsilon_{{\mathbf{k}}}$. From Eqs. (\[73\]) and (\[73a\]) one immediately concludes that for ${{\mathbf{q}}}=0$ the real and imaginary parts of $ \chi({\mathbf{q}}, \omega)$ vanish (for any finite $\omega$). Compare also Section \[V.3\] below. On the other hand, when the limit $\omega\to 0$ is taken first, the imaginary part of $\chi({{\mathbf{q}}}, \omega)$ vanishes for any ${\mathbf{q}}$. This follows from the analytical properties of ${\rm Im} \chi({\mathbf{q}}, \omega)$ or from Eqs. (\[71\]) and (\[73a\]). In contrast, the real part ${\rm Re}\chi({\mathbf{q}}, \omega=0)$ stays finite and reduces to the static ${\mathbf{q}}$-dependent susceptibility $$\begin{aligned}
\label{73b}
\chi({\mathbf{q}}) = {\rm Re}\chi({\mathbf{q}}, \omega=0) \, .
\end{aligned}$$ In the limit ${\mathbf{q}} \rightarrow 0$, $\chi({\mathbf{q}})$ gives the uniform static susceptibility. Last but not least, keep in mind that the renormalization equations derived so far, exclusively apply to the paramagnetic phase.
Numerical Results and Discussion {#V}
================================
The set of self-consistency Eqs. (\[39\])-(\[42a\]), (\[71\]), and (\[72\]) has to be solved numerically in momentum space. Due to additional internal ${\mathbf{k}}$-sums in the renormalization contributions, we restrict ourselves to a square lattice with $N=24\times 24$ sites using periodic boundary conditions. In contrast, for the non-interacting susceptibility $\chi_{{\mathbf{k}}}^0({\mathbf{q}}, \omega)$ a larger mesh in momentum space of $2000 \times 2000$ points is used close to the Fermi surface due to the large variations in this quantity. Choosing reasonable initial values for the various expectation values, the renormalization starts from the cutoff $\Lambda$ of the original model and proceeds in energy steps $\Delta \lambda=0.5 \bar{t}$ until $\lambda =0$ is reached. Then the expectation values are recalculated. Convergence is assumed to be achieved if all quantities are determined with a relative error less than 10$^{-5}$. We have convinced ourselves that a larger lattice size as well as a smaller value of $\Delta \lambda$ will not modify the presented results. In what follows we measure all energies in units of $\bar{t}$.
Band renormalization {#V.1}
--------------------
Correlation effects, which are included in the PRM scheme, lead to a momentum-dependent renormalization $\varepsilon_{{\mathbf{k}}}\to \tilde{\varepsilon}_{{\mathbf{k}}}$ of the band structure in the paramagnetic phase. This differs from standard Hartree-Fock [@Pe66; @Hi85a], Gutzwiller [@Gu63; @BR70], or slave-boson treatments [@KR86; @DFB92], where band renormalization either not at all or in terms of a momentum independent band narrowing takes place. To illustrate the PRM renormalization of the quasiparticle band, in Fig. \[fig:9\] the difference $\delta\varepsilon_{{\mathbf{k}}}=\varepsilon_{{\mathbf{k}}}-\tilde{\varepsilon}_{{\mathbf{k}}}$ is shown for a square lattice Brillouin zone for the particle density $n=0.9$ and three different $U$ values. The overall bandwidth is reduced by a factor of 0.22, 0.34 and 0.45 for $U=2$, 3 and 4, respectively. Beyond that, we find that the momentum dependence of $\delta\varepsilon_{{\mathbf{k}}}$ increases with $U$ and is largest near the center ${{\mathbf{k}}}=(0,0)$ and at the corners ${{\mathbf{k}}}=(\pi,\pi)$ of the Brillouin zone. This should have strong impact on the uniform and staggered (static) spin susceptibilities which are studied in the next subsection.
Static spin susceptibility {#V.2}
--------------------------
### Temperature dependence {#V.2.1}
Let us first consider the half-filled band case $n=1$ and track the behavior of the static susceptibility $\chi({\mathbf{q}};T)$ as the temperature $T$ is lowered. The PRM results for $\chi({\mathbf{q}};T)$ are shown in Fig. \[fig:1\] for two fixed wave vectors ${\mathbf{q}}=0$ (uniform susceptibility, panel (a)) and ${\mathbf{q}}={\mathbf{Q}}=(\pi,\pi)$ (staggered susceptibility, panel (b)). As one can see, the logarithmic singularity in the density of states at the band center, $\rho(E) \propto \ln (\varepsilon/4\bar{t})$ for $\varepsilon\to 0$, leads to a divergence of the noninteracting susceptibilities $\chi^0(0;T)\propto -\ln (T/\bar{t})$ and $\chi^0({\mathbf{Q}};T)\propto -[\ln (T/\bar{t})]^2$. This indicates a magnetic instability of the corresponding PRM susceptibilities at some finite $T$ for any $U$. Therefore, since the divergence of $\chi^0({\mathbf{Q}}; T)$ is stronger than that of $\chi^0(0;T)$, the PRM predicts a transition to a magnetic phase with strong antiferromagnetic fluctuations, which sets in at a higher temperature. The larger the $U$-values, the higher are the transition temperatures. For very small $U$ the RPA and PRM results are nearly identical. As follows from the preceding section, the PRM renormalization of the uniform susceptibility $\chi(0;T)$ is solely caused by the one-particle energies $\tilde{\varepsilon}_{{\mathbf{k}}}$ (compare Eq. (\[73\])), which are barely changed in this limit; cf. Fig. \[fig:9\]. On the other hand, the renormalization of $\chi({\mathbf{Q}};T)$ is affected by the coefficients $\tilde{\alpha}_{\bf k, \bf k- {{\mathbf{q}}}} $ as well as by the second contribution $\Delta({\mathbf{Q}})$ in the denominator. This term shifts the zero of the denominator in Eq. (\[71\]), with the result that antiferromagnetic transition temperature is reduced, as it should be if the correlations/fluctuations of the Hubbard system are treated better. Acceptably the antiferromagnetic critical temperature stays larger than the ferromagnetic one.
Of course, in 2D the occurrence of a finite transition temperature is an artefact of the approximations in the PRM, which is also known from the standard RPA. According to the Mermin-Wagner theorem, for a 2D model with continuous symmetry long-range order can only occur for $T=0$ [@MW66; @Hi85a]. Indeed, from unbiased numerical approaches [@Hi85a] it was shown that long-range (antiferromagnetic) order is expected at $T=0$ only for half-filling. Clearly, the PRM in the present version does not overcome this shortcoming. However, as seen from Fig. \[fig:1\], it gives the right tendency. One may expect that higher order fluctuation terms, not included at present in [*ansatz*]{} (\[27\])-(\[28\]) for $\mathcal H_{\lambda}(t)$ and (\[33\])-(\[34a\]) for $X_{\lambda, \Delta \lambda}(t)$ improve the situation further.
### $U$-dependence {#V.2.2}
Figure \[fig:2\] shows how (a) the uniform and (b) the staggered spin susceptibilities vary in the paramagnetic phase as the Hubbard interaction $U$ is enhanced at zero temperature. Now we are off half-filling, $n=0.9$. We again find divergencies in both susceptibilities, signaling a tremendous increase of ferromagnetic and antiferromagnetic correlations. If compared to the RPA results (dashed lines), the PRM (solid lines) shifts the critical value of $U$ to lower (higher) values in the former (latter) case. This is easily understood since the renormalization of $\chi({\mathbf{q}})$ at ${\mathbf{q}}=0$ comes (solely) from the PRM band narrowing yielding a higher density of states ($\chi^0(0)$) and consequently a lower $U_c$ than within an RPA treatment. On the other hand, for $\chi({\mathbf{Q}})$ the term $\Delta({\mathbf{Q}})>0$ is more important, which enhances $U_c$, i.e., suppresses the range where the state with long-range antiferromagnetic correlations pops up.
### Magnetic phase diagram {#V.2.3}
Tracing the divergencies of the uniform and staggered susceptibilities in the $n$-$U$ model-parameter plane at $T=0$, a ground-state phase diagram of the 2D Hubbard model can be derived. Thereby, only paramagnetic states with increasingly strong ferromagnetic and antiferromagnetic correlations can be detected. The such kind determined phase diagrams agree with those obtained from the corresponding order parameter self-consistency equations [@Mah00; @DFKTI94; @ZIBF11]. Of course, the paramagnetic-ferromagnetic and paramagnetic-antiferromagnetic phase boundaries have to be determined separately. This has been undertaken in Fig. \[fig:4\]. Since our PRM treatment of the Hubbard model is a weak-to-intermediate coupling approach, the calculations were restricted to densities not too far away from half filling (the instabilities appear at large values of $U$ otherwise). In the whole density range studied ($0.7\leq n\leq 1$), the antiferromagnetic instability sets in first, i.e. an antiferromagnetic state is established before ferromagnetic order can be established. This corroborates previous Hartree-Fock, RPA, and slave-boson results [@Pe66; @KR86; @DFB92]. Quantitative deviations from the RPA phase boundaries exist however (cf., in Fig. \[fig:4\], the corresponding transition lines). A tricritical point, where the ferromagnetic and antiferromagnetic instabilities intersect, is expected to appear at a density slightly smaller than $n=0.7$. It is not our aim, however, to map out the phase diagram in more detail; simply because it is now commonly accepted—owing to numerical studies but not rigorously proven—that long-range ordered phases will not be stable in the 2D Hubbard model away from half filling. Thus, as noticed long-time ago, it appears that approximative solutions to the simple 2D Hubbard model might do better in describing the magnetic features of real quasi-2D materials than the (still not available) exact solution [@Hi85a]. In this respect primarily meaningful will be the improved treatment of correlations by the PRM in the paramagnetic phase.
Dynamic spin susceptibility {#V.3}
---------------------------
The phase diagram Fig. \[fig:4\] from Sec. \[V.2\] shows that transitions from the PM to the AFM and from the PM to the FM states approach each other, when the density $n$ gets close to half-filling ($n=1$). Thereby the respective critical $U$ values go to zero. In Fig. \[fig:5\] the PRM results for the imaginary and real parts of $\chi({{\mathbf{q}}}, \omega)$ are displayed for the density $n=0.985$ very close to 1 and a small coupling $U=2$. Curves are shown for different ${{\mathbf{q}}}$ values along the diagonal direction ${{\mathbf{q}}}= (q_x, q_x)$ in the Brillouin zone. The steps between subsequent $q_x$ curves are chosen as $\pi/12$, where the lowest $q_x$ value is $\pi/12$. Note that ${\rm Im}\chi({{\mathbf{q}}},\omega)$ for $q_x =0$ (${{\mathbf{q}}}=0$) vanishes due to rotational symmetry of the total spin density $s^z_{{\mathbf{q}}=0}$. A strong paramagnon peak structure is found in ${\rm Im}\chi({{\mathbf{q}}}, \omega)$ at $\omega=0$ around the antiferromagnetic wave vector ${{\mathbf{Q}}}=(\pi, \pi)$. Also in the real part ${\rm Re}\chi({{\mathbf{q}}}, \omega)$ a strong peak structure appears for the same ${{\mathbf{q}}}$ and $\omega$ values. However, no such strong structure is found at small ${\mathbf{q}}$. Since ${\rm Re}\chi({{\mathbf{q}}}, \omega)$ agrees for $\omega=0$ with the static susceptibility $\chi({{\mathbf{q}}}) = \chi({{\mathbf{q}}}, \omega=0)$ one concludes that antiferromagnetic fluctuations at ${{\mathbf{Q}}}$ dominate ferromagnetic fluctuations (with ${{\mathbf{q}}} \ll 1)$ already for small deviations from half-filling.\
Next, let us discuss the circumstances at the density $n=0.7$, which is slighter above the density where the ferromagnetic and antiferromagnetic instabilities are expected to intersect (compare Fig. \[fig:4\]). Fig. \[fig:6\] shows again the real and imaginary part of $\chi({{\mathbf{q}}}, \omega)$, now for $U=11.5$, which is approximately the critical $U$ value for the PM-AFM transition. Indeed, for the antiferromagnetic wave vector ${{\mathbf{Q}}}$ there is again a paramagnon structure at $\omega \approx 0$ in $\chi({{\mathbf{q}}}, \omega)$. However, as shown in the real part of $\chi({{\mathbf{q}}},\omega )$, also ferromagnetic fluctuations (for ${{\mathbf{q}}} \ll 1$ and $\omega=0$) are considerably enhanced compared to the case of Fig. \[fig:5\]. From the $U$ dependence of the uniform static susceptibility $\chi({{\mathbf{q}}}=0)$ (not shown), one finds that it tremendously increases for slightly increasing $U$ and diverges at the critical value $U_{crit}^{PM-FM} \approx 12.5$. This divergence would correspond to a transition to a ferromagnetic phase, if it had not been before the transition to the antiferromagnetic phase. The red dashed curves in both panels of Fig. \[fig:6\] result from a standard RPA calculation for an extremely small ${\mathbf{q}}$ value, ${\mathbf{q}}= (0.01, 0.01)$, which is expected to almost agree with PRM results. Due to the finite lattice of $24 \times 24$ sites used in the PRM calculation the PRM curves in the figures are restricted to not too small ${\mathbf{q}}$ values.
Finally, let us compare the PRM with the standard RPA. As already discussed in section \[II\] the RPA arises when all renormalization effects are neglected. Panels (a)-(c) of Fig. \[fig:7\] show the imaginary part ${\rm Im}\chi({{\mathbf{q}}},\omega)$ as a function of $\omega$ for an intermediate density $n=0.8$ and three different $U$ values, (a) $U=0$, (b) $U=2$, and (c) $U=6$. When ${{\mathbf{q}}}\to {{\mathbf{Q}}}$ the RPA curves (red dashed curves) for $U=6$ exhibit a relatively narrow peak at low frequencies, which is again interpreted as paramagnon peak. In contrast, the PRM curves at the same ${\mathbf{q}}$ and $U$ are much less pronounced. The different behavior is easily understood from the phase diagram in Fig. \[fig:4\],
since $U=6$ is much closer to the critical RPA value $U_{crit}^{RPA}\simeq 6.75$ than to the critical value $U^{PRM}_{crit}\simeq 8.8$ from the PRM approach. The corresponding PRM result for $U=8.8$ is shown in panel (d) which clearly shows a pronounced paramagnon peak as expected.
In all panels (a) to (d) the curves at ${{\mathbf{Q}}}= (\pi, \pi)$ are more pronounced around $\omega=0$ than those for ${\mathbf{q}}$ values close to the center of the Brillouin zone. From this feature one should not draw the conclusion that antiferromagnetic fluctuations are always more important than ferromagnetic ones. As was already mentioned, for $\omega$ finite ${\rm Im}\chi({{\mathbf{q}}}, \omega)$ always vanishes at ${\mathbf{q}}=0$, i.e. at the ferromagnetic wave vector. Therefore, as was done in section \[IV\], a comparison of antiferromagnetic and ferromagnetic fluctuations can only be drawn from the values of the real part $ {\rm Re}\chi({{\mathbf{q}}}, \omega)$ of the dynamical susceptibility at $\omega=0$, which is equivalent to the static ${\mathbf{q}}$ dependent susceptibility $\chi({\mathbf{q}})= {\rm Re}\chi({{\mathbf{q}}}, \omega=0)$.
Summary {#VI}
=======
Combining linear response theory with the projector-based renormalization method, we presented a theoretical approach for the evaluations of the susceptibilities that generalizes the standard RPA scheme. In this way important many-body correlations beyond the RPA level were included. To exemplify the advancement the theory was applied to the two-dimensional paradigmatic Hubbard model, for which an analytical expression for the dynamical spin susceptibility $\chi({{\mathbf{q}}},\omega)$ was derived that improves the RPA result. While the uniform spin susceptibility, where ${{\mathbf{q}}}=0$, still exhibits the standard RPA form, renormalized quasiparticle energies enter the band susceptibility contributions $\chi^0({{\mathbf{q}}},\omega)$. At any finite wave-vector, however, the shape of $\chi({{\mathbf{q}}},\omega)$ changes. Besides momentum-dependent prefactors in $\chi^0({{\mathbf{q}}},\omega)$, an extra term occurs in the denominator of $\chi({{\mathbf{q}}},\omega)$, which has to be determined self-consistently. This term particularly changes the pole structure of $\chi({{\mathbf{q}}},\omega)$ in the limit $\omega\to 0$. As a result, the magnetic phase diagram, which can be derived from the instabilities of the static spin susceptibilities at wave-vectors ${{\mathbf{q}}}=0$ or ${{\mathbf{q}}}={{\mathbf{Q}}}$, is modified quantitatively. The same holds for the paramagnon spectrum obtained from the dynamical response. Most notably, the better (PRM) treatment of the Coulomb interaction effects severely reduce the exaggeration of the paramagnons in comparison to the RPA, i.e., the tendency of the Hubbard system to develop long-range ferromagnetic or antiferromagnetic order at certain band fillings is weakened.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank S. Sykora for valuable discussions. This research was funded by Vietnam National Foundation for Science and Technology Development under Grant No. 103.02-2012.52 and by Deutsche Forschungsgemeinschaft through SFB 652, B5.
$\lambda$-dependence of operator quantities {#appA}
===========================================
As an example let us justify the [*ansatz*]{} (\[27\]) for $\hat {\mathcal H}_{h,\lambda}(t)$. We consider the transformation (\[21\]) for $\hat{\mathcal H}_{h}(t)$ for a small step from the original cutoff $\Lambda$ to the somewhat reduced cutoff $\Lambda - \Delta \lambda$, $$\begin{aligned}
\label{A1}
\hat{\mathcal H}_{h,\Lambda - \Delta \lambda}&=& e^{X_{\Lambda, \Delta \lambda}}\,
\hat{\mathcal H}_{h}\, e^{-X_{\Lambda, \Delta \lambda}} \\
&=& \hat{\mathcal H}_{h} + [X_{\Lambda, \Delta \lambda}, \hat{\mathcal H}_{h}]
+ \cdots \nonumber\end{aligned}$$ Evaluating the commutator in Eq. (\[A1\]) and extracting the one-particle and two-particle contributions one is immediately led to the following structure $$\begin{aligned}
\label{A2}
&& \hat{\mathcal H}_{h,\lambda}(t) = \\
&& \quad -
\sum_{{\mathbf{k}} \sigma} \Big[\Big( \frac{\hat h_{{\mathbf{q}}}(t)}{2} +
u_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}},\lambda}(t)\Big) \,
\frac{\sigma}{2} c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} - {\mathbf{q}}, \sigma} + \textrm{H.c.} \Big] \nonumber
\\
&& \quad -
\frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma} \big[
v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}} , \lambda}(t)
\nonumber \\
&& \quad \times
\frac{\sigma}{2}
:c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma }
c_{{\mathbf{k}}', -\sigma}^\dag c^{}_{{\mathbf{k}}'- {\mathbf{p}}, -\sigma} : + \textrm{H.c.} \big]
\nonumber \, ,
\end{aligned}$$ where the prefactors $u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\lambda}(t)$ and $ v_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}} , \lambda}(t)$ depend on $\lambda$ and also on the external field $h(t)$.
Similarly, for an [*ansatz*]{} of the transformed spin $s^z_{{\mathbf{q}}, \lambda}$ one starts from $$\begin{aligned}
\label{A3}
s^z_{{\mathbf{q}},\Lambda- \Delta \lambda}&=& e^{X_{\Lambda, \Delta \lambda}} s^z_{{\mathbf{q}},\Lambda}
e^{- X_{\Lambda, \Delta \lambda}} \\
&=& s^z_{{\mathbf{q}},\Lambda} + [X_{\Lambda, \Delta \lambda}, s^z_{{\mathbf{q}},\Lambda}]
+ \cdots\,, \nonumber
\end{aligned}$$ where $s^z_{{\mathbf{q}},\Lambda}= s^z_{{\mathbf{q}}}$. In lowest order perturbation theory one obtains $$\begin{aligned}
\label{A4}
s^z_{{\mathbf{q}},\Lambda- \Delta \lambda} &=&
\sum_{{\mathbf{k}} \sigma} \frac{\sigma}{2} \,
c^\dag_{{\mathbf{k}} \sigma} \, c^{}_{{\mathbf{k}} -{\mathbf{q}}, \sigma} \\
&& +
\frac{1}{2N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} }
A^{}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}} (\Lambda, \Delta \lambda)
\nonumber \\
&& \times \sum_{\sigma} \frac{\sigma}{2} \Big(
\, c^\dag_{{\mathbf{k}}, \sigma}\, c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma} \,
:c^\dag_{{\mathbf{k}}',- \sigma}\, c^{}_{{\mathbf{k}}' -{\mathbf{p}},-\sigma}: \nonumber \\
&& -
c^\dag_{{\mathbf{k}}+ {\mathbf{q}}, \sigma}\, c^{}_{{\mathbf{k}} +{\mathbf{p}},\sigma} \,
:c^\dag_{{\mathbf{k}}', -\sigma}\, c^{}_{{\mathbf{k}}' - {\mathbf{p}}, -\sigma}: \nonumber \\
&& +
:c^\dag_{{\mathbf{k}}, -\sigma}\, c^{}_{{\mathbf{k}} +{\mathbf{p}}, -\sigma}: \,
c^\dag_{{\mathbf{k}}', \sigma}\, c^{}_{{\mathbf{k}}' -{\mathbf{p}} -{\mathbf{q}},\sigma} \nonumber \\
&& -
:c^\dag_{{\mathbf{k}}, -\sigma}\, c^{}_{{\mathbf{k}} + {\mathbf{p}},-\sigma}: \,
c^\dag_{{\mathbf{k}}' +{\mathbf{p}}, \sigma}\, c^{}_{{\mathbf{k}}' - {\mathbf{p}}, \sigma}
\Big) \nonumber \end{aligned}$$ which fulfills $s_{{\mathbf{q}},\lambda}^z= (s_{-{\mathbf{q}}, \lambda}^z)^\dag$. Thus, one arrives at $$\begin{aligned}
\label{A5}
s^{z}_{{\mathbf{q}},\lambda} &=&
\sum_{{\mathbf{k}} \sigma} \alpha_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}, \lambda} \, \frac{\sigma}{2}
c^\dag_{{\mathbf{k}} \sigma} \, c^{}_{{\mathbf{k}} -{\mathbf{q}}, \sigma} \\
&&+
\frac{1}{N} \sum_{{\mathbf{k}} \sigma} \beta_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}',{\mathbf{k}}' -{\mathbf{p}} \lambda}
\nonumber \\
&&\times
\frac{\sigma}{2}
:c^\dag_{{\mathbf{k}}, \sigma}\, c^{}_{{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, \sigma} \,
c^\dag_{{\mathbf{k}}', -\sigma}\, c^{}_{{\mathbf{k}}' -{\mathbf{p}},-\sigma} :\,, \nonumber
\end{aligned}$$ where again the one-particle and two-particle contributions were extracted. This result agrees with Eq. (\[53\]).
Evaluation of commutators {#appB}
=========================
In this appendix we evaluate the commutators from transformation (\[24\]), which are responsible for the renormalization of $\mathcal H_\lambda(t)$.
Commutator $ [X_{\lambda, \Delta \lambda}(t), {\mathcal H}_{h, \lambda}(t) ]$ {#appB.1)}
-----------------------------------------------------------------------------
Since $\mathcal H_{h,\lambda}(t)$ is linear in the external field, the generator $X_{\lambda, \Delta \lambda}(t)$ can be limited to the part $X^\alpha_{\lambda, \Delta \lambda}(t)$ in Eq. (\[33\]). For the part of $\hat{\mathcal H}_{h, \lambda}(t)$ proportional to $u$ , we find $$\begin{aligned}
\label{B1}
&& [X^\alpha_{\lambda, \Delta \lambda}(t), \hat{\mathcal H}_{h, \lambda}(t)]\big|_u = \\
&& \quad - \sum_{{\mathbf{k}}\sigma} \Big\{
\Big(\frac{\hat h_{{\mathbf{q}}}(t)}{2}+u_{{\mathbf{k}},{\mathbf{k}}-{\mathbf{q}}, \lambda}\Big) \nonumber \\
&& \quad \times
[X^\alpha_{\lambda, \Delta \lambda}(t), \frac{\sigma}{2} c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} -{\mathbf{q}},\sigma} ] + \textrm{H.c.} \Big\}
\nonumber \\
&& = -\sum_{{\mathbf{k}} \sigma} \Big\{
\Big(\frac{\hat h_{{\mathbf{q}}}(t)}{2}+ \delta u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}^{(1)}\Big) \frac{\sigma}{2}
c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} -{\mathbf{q}}, \sigma} + \textrm{H.c.} \Big\}
\nonumber \\
&& \quad - \frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma} \Big\{
\delta v^{(1)}_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}
\nonumber \\
&& \quad \times
\frac{\sigma}{2}
:c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma }
c_{{\mathbf{k}}', -\sigma}^\dag c^{}_{{\mathbf{k}}'- {\mathbf{p}}, -\sigma} : + \textrm{H.c.} \Big\}
\nonumber \, . \\
&& \nonumber\end{aligned}$$ Here the factors $\delta u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}^{(1)}$, and $\delta v^{(1)}_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}$ are given by $$\begin{aligned}
\label{B2}
&& \delta u_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}, \lambda}^{(1)} = \\
&& - \frac{1}{N} \sum_{{\mathbf{k}}'} \Big(\frac{\hat h_{{\mathbf{q}}}(t)}{2}+ u_{{\mathbf{k}}', {\mathbf{k}}' -{\mathbf{q}}, \lambda} \Big)
A_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}; {\mathbf{k}}' -{\mathbf{q}}, {\mathbf{k}}'}(\lambda, \Delta \lambda) \nonumber \\
&& \times \big(
\langle c_{{\mathbf{k}}',-\sigma}^\dag c^{}_{{\mathbf{k}}', -\sigma}\rangle
- \langle c_{{\mathbf{k}}' -{\mathbf{q}},-\sigma}^\dag c^{}_{{\mathbf{k}}' -{\mathbf{q}}, -\sigma}\rangle
\big) \nonumber\end{aligned}$$ and $$\begin{aligned}
\label{B3}
&& \delta v^{(1)}_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}= \\
&& \quad \Big( \frac{\hat h_{{\mathbf{q}}}(t)}{2}+u_{{\mathbf{k}} + {\mathbf{p}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}},\lambda}\Big)
A_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) \nonumber \\
&& \quad- \Big( \frac{\hat h_{{\mathbf{q}}}(t)}{2}+u_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}, \lambda} \Big)
A_{{\mathbf{k}} -{\mathbf{q}}, {\mathbf{k}}+{\mathbf{p}}-{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) \,.\nonumber \end{aligned}$$ The second contribution to the above commutator arises from the term in $\mathcal H_{h,\lambda}(t)$ linear in $v$ $$\begin{aligned}
\label{B4}
&& [X^\alpha_{\lambda, \Delta \lambda}(t), \hat{\mathcal H}_{h,\lambda}(t)]\big|_v
=- \frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma} \Big\{
v_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'- {\mathbf{p}},\lambda} \nonumber
\\
&& \quad \times
[X^\alpha_{\lambda, \Delta \lambda}(t), \frac{\sigma}{2}
:c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}},\sigma} c^\dag_{{\mathbf{k}}', -\sigma}
c^{}_{{\mathbf{k}}' -{\mathbf{p}},-\sigma}: ]
\nonumber \\
&& \quad + \textrm{H.c.} \Big\} \, .\end{aligned}$$ Here, the evaluation has to be followed up by a decomposition into one-particle and two-particle contributions. This leads to renormalization contributions to $u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}$ and $v_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}$. However, they turn out to be small and can be neglected. They are at least by a factor of order $O(U/ \Delta \varepsilon)$ smaller than the contributions (\[B2\]), (\[B3\]), where $\Delta \varepsilon$ denotes an energy difference of the order of the conduction electron band width.
Commutator $[X_{\lambda, \Delta \lambda}(t), {\mathcal H}_{f,\lambda}]$ {#appB.2}
------------------------------------------------------------------------
Due to decompositions (\[33\]) and (\[31\]) of $X_{\lambda, \Delta \lambda}$ and $\mathcal H_{f,\lambda}$ one has to evaluate three different contributions to order $h(t)$.
### Commutator $ [X^\alpha_{\lambda, \Delta \lambda}(t),
{\mathcal H}^\alpha_{f,\lambda}]$ {#appB.2.1}
The first commutator leads to renormalization contributions of order $U^2$: $$\begin{aligned}
\label{B5}
&&[X^\alpha_{\lambda, \Delta \lambda}, {\mathcal H}^\alpha_{f,\lambda}] =
\frac{U}{2N^2} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{k}}'', {\mathbf{p}} {\mathbf{p}}' \sigma}
\Big\{
F_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{k}}'', {\mathbf{p}} {\mathbf{p}}'}(\lambda, \Delta \lambda) \nonumber \\
&& \times
c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} + {\mathbf{p}} +{\mathbf{p}}', \sigma} \, :c^\dag_{{\mathbf{k}}' , -\sigma} c^{}_{{\mathbf{k}}' -{\mathbf{p}}',
-\sigma}: \,
: c^\dag_{{\mathbf{k}}'',-\sigma} c^{}_{{\mathbf{k}}'' -{\mathbf{p}}, -\sigma}: \nonumber \\
&& + \textrm{H.c.} \Big\} \, ,
$$ where $$\begin{aligned}
\label{B6}
&& F_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{k}}'' {\mathbf{p}} {\mathbf{p}}'}(\lambda, \Delta \lambda) = \\
&& \quad
A_{{\mathbf{k}}, {\mathbf{k}}+ {\mathbf{p}}', {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}'}(\lambda, \Delta \lambda) \,
\Theta_{{\mathbf{k}}+ {\mathbf{p}}' , {\mathbf{k}} + {\mathbf{p}} + {\mathbf{p}}', {\mathbf{k}}'', {\mathbf{k}}'' -{\mathbf{p}}, \lambda} \nonumber \\
&& \quad -
A_{{\mathbf{k}} +{\mathbf{p}}, {\mathbf{k}}+ {\mathbf{p}} +{\mathbf{p}}', {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}'}(\lambda, \Delta \lambda) \,
\Theta_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} , {\mathbf{k}}'', {\mathbf{k}}'' -{\mathbf{p}}, \lambda} \, .
\nonumber\end{aligned}$$ As before, the operator structure in (\[B5\]) has to be reduced to operators which appear in ${\mathcal H}_\lambda(t)$. Besides a term proportional to $c^\dag c$, also contributions with four creation and annihilation operators can be extracted from Eq. (\[B5\]). The first contribution reads $$\begin{aligned}
\label{B7}
&&
[X^\alpha_{\lambda, \Delta \lambda}, {\mathcal H}^\alpha_{f,\lambda}] =
\frac{U}{2N^2} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{k}}'' {\mathbf{p}} {\mathbf{p}}' \sigma}
F_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{k}}''; {\mathbf{p}} {\mathbf{p}}'}(\lambda, \Delta \lambda) \nonumber \\
&&
\times\Big\{ \langle c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} + {\mathbf{p}} +{\mathbf{p}}', \sigma}\rangle \,
\langle c_{{\mathbf{k}}' -{\mathbf{p}}', -\sigma} c^{\dag}_{{\mathbf{k}}'', -\sigma} \rangle \,
:c^{\dag}_{{\mathbf{k}}' ,-\sigma} c^{}_{{\mathbf{k}}'' -{\mathbf{p}},-\sigma}: \nonumber \\
&& + \langle c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} + {\mathbf{p}} +{\mathbf{p}}', \sigma}\rangle \,
\langle c^{\dag}_{{\mathbf{k}}',-\sigma} c^{}_{{\mathbf{k}}''- {\mathbf{p}},-\sigma} \rangle
:c^{}_{{\mathbf{k}}' -{\mathbf{p}}', -\sigma} c^{\dag}_{{\mathbf{k}}'' , -\sigma}: \nonumber \\
&&+ \langle c^\dag_{{\mathbf{k}}', -\sigma} c^{}_{{\mathbf{k}}'' -{\mathbf{p}}, -\sigma} \rangle \,
\langle c^{}_{{\mathbf{k}}'- {\mathbf{p}}',-\sigma} c^\dag_{{\mathbf{k}}'',-\sigma} \rangle \,
:c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} + {\mathbf{p}} +{\mathbf{p}}', \sigma}:
\nonumber \\
&& + \textrm{H.c.} \Big\} \, ,\end{aligned}$$ where there are two options for the expectation values according to relation (\[10\]): $$\label{B8}
\langle c_{{\mathbf{k}} \sigma}^\dag\, c^{}_{{\mathbf{k}} - {{\mathbf{p}}}, \sigma} \rangle =
\frac{\delta_{{{\mathbf{p}}},0}}{2} \langle n_{{\mathbf{k}}} \rangle
+ \sigma \delta_{{{\mathbf{p}}}, \pm {\mathbf{q}}} \langle s^z_{{\mathbf{k}}, \pm {\mathbf{q}}}\rangle
\, ,$$ with either ${{\mathbf{p}}} =0$ or ${{\mathbf{p}}}= \pm {\mathbf{q}}$. For the choice ${\mathbf{p}}=0 $ the commutator (\[B7\]) leads to a renormalization of the electronic one-particle energy $\varepsilon_{{\mathbf{k}}, \lambda}$, $$\begin{aligned}
\label{B9}
&&
[X^\alpha_{\lambda, \Delta \lambda}, {\mathcal H}^\alpha_{f,\lambda}]\big|_{(p=0)} =
\sum_{{\mathbf{k}},\sigma} \delta \varepsilon^{(1)}_{{\mathbf{k}},\lambda } \,
c_{{\mathbf{k}} \sigma}^\dag c^{}_{{\mathbf{k}} \sigma} \, ,
$$ where $\delta \varepsilon^{(1)}_{{\mathbf{k}},\lambda}$ is given by $$\begin{aligned}
\label{B10}
&& \delta \varepsilon^{(1)}_{{\mathbf{k}},\lambda} = \\
&& \quad \frac{U}{4N^2} \sum_{{\mathbf{k}}' {\mathbf{p}} } \Big[
F_{{\mathbf{k}}', {\mathbf{k}}, {\mathbf{k}} + {\mathbf{p}}; {\mathbf{p}}, -{\mathbf{p}}}(\lambda, \Delta \lambda) \,
\langle n_{{\mathbf{k}}'}\rangle \, (2- \langle n_{{\mathbf{k}} +{\mathbf{p}}} \rangle) \nonumber \\
&& \quad +
F_{{\mathbf{k}}, {\mathbf{k}}', {\mathbf{k}}' +{\mathbf{p}}; {\mathbf{p}}, -{\mathbf{p}}}(\lambda, \Delta \lambda) \,
\langle n_{{\mathbf{k}}'} \rangle \,
(2- \langle n_{{\mathbf{k}}' + {\mathbf{p}}} \rangle) \nonumber \\
&& \quad -
F_{{\mathbf{k}}', {\mathbf{k}} -{\mathbf{p}}, {\mathbf{k}}; {\mathbf{p}}, - {\mathbf{p}}}(\lambda, \Delta \lambda) \,
\langle n_{{\mathbf{k}}'}\rangle \langle n_{{\mathbf{k}} -{\mathbf{p}}} \rangle
\Big] \nonumber\end{aligned}$$ with $\langle c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} \sigma}\rangle =\langle n_{{\mathbf{k}}} \rangle/2$. The second choice ${\mathbf{p}} = \pm {\mathbf{q}}$ leads to a renormalization of the effective field. The contribution to $u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}}(t)$ is $$\begin{aligned}
\label{B11}
[X^\alpha_{\lambda, \Delta \lambda}, {\mathcal H}^\alpha_{f,\lambda}]_{(p=\pm q)} &=& - \sum_{{\mathbf{k}} \sigma}
\Big\{ \delta u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\lambda}^{(2)}(t)
\frac{\sigma}{2} \, c^\dag_{{\mathbf{k}} \sigma} c^{}_{{\mathbf{k}} -{\mathbf{q}}, \sigma} \nonumber \\
&&\qquad\quad+ \textrm{H.c.} \Big\} \end{aligned}$$ with $$\begin{aligned}
\label{B12}
&& \delta u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\lambda}^{(2)}(t) = - \frac{U}{2N^2} \sum_{{\mathbf{k}}' {\mathbf{p}}}
B^{(2)}_{{\mathbf{p}} {\mathbf{k}}'; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda) \langle s^z_{{\mathbf{k}}', -{\mathbf{q}}}\rangle(t) \, ,
\nonumber \\
&& \end{aligned}$$ where the pre-factor $ B^{(2)}_{{\mathbf{p}} {\mathbf{k}}'; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda)$ is of order $U/\Delta \varepsilon$: $$\begin{aligned}
\label{B13}
&&B^{(2)}_{{\mathbf{p}} {\mathbf{k}}' ; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda)= \\
&& - F_{{\mathbf{p}}, {\mathbf{k}}, {\mathbf{k}}'; {\mathbf{k}}' -{\mathbf{k}} + {\mathbf{q}}, -({\mathbf{k}}' -{\mathbf{k}} + {\mathbf{q}})}(\lambda, \Delta \lambda)
\langle n_{{\mathbf{p}}} \rangle \nonumber \\
&& - F_{{\mathbf{k}}', {\mathbf{k}}, {\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}; {\mathbf{p}}, -{\mathbf{p}} +{\mathbf{q}}}(\lambda, \Delta \lambda)
\big( 2- \langle n_{{\mathbf{k}}+{\mathbf{p}} -{\mathbf{q}}} \rangle \big) \nonumber \\
&& - F_{{\mathbf{p}}, {\mathbf{k}}', {\mathbf{k}}; {\mathbf{k}} -{\mathbf{k}}' - {\mathbf{q}}, -({\mathbf{k}} -{\mathbf{k}}' - {\mathbf{q}})}(\lambda, \Delta \lambda)
\langle n_{{\mathbf{p}}} \rangle \nonumber \\
&&
+F_{{\mathbf{k}}', {\mathbf{k}} -{\mathbf{p}}, {\mathbf{k}}; {\mathbf{p}}, -{\mathbf{p}} +{\mathbf{q}}}(\lambda, \Delta \lambda)
\langle n_{{\mathbf{k}}-{\mathbf{p}}} \rangle \nonumber \\
&&
+ F_{{\mathbf{k}}, {\mathbf{k}}' -{\mathbf{p}}, {\mathbf{k}}'; {\mathbf{p}}, -{\mathbf{p}} - {\mathbf{q}}}(\lambda, \Delta \lambda)
\langle n_{{\mathbf{k}}' -{\mathbf{p}}} \rangle \nonumber \\
&&
- F_{{\mathbf{k}}, {\mathbf{k}}', {\mathbf{k}} + {\mathbf{p}} +{\mathbf{q}}; {\mathbf{p}}, -{\mathbf{p}} -{\mathbf{q}}}(\lambda, \Delta \lambda)
\big(2- \langle n_{{\mathbf{k}}' + {\mathbf{p}} + {\mathbf{q}}} \rangle \big)\,. \nonumber
$$ Note that a common factor $\langle s^z_{{\mathbf{k}}', -{\mathbf{q}}}\rangle(t)$ was already extracted in Eq. (\[B12\]).\
The remaining contribution to commutator (\[B5\]) with four creation and annihilation operators leads to a renormalization of $\mathcal H_{f,\lambda}$ of order $U^2/ \Delta\varepsilon$. Such contributions have been left out from the very beginning.
### Commutator $[X^\beta_{\lambda, \Delta \lambda}(t),
{\mathcal H}^\alpha_{f,\lambda}]$ {#appB.2.2}
The evaluation of this commutator leads to renormalizations of $u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}$ and $v_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}},\lambda}$: $$\begin{aligned}
\label{B14}
&&[X^\beta_{\lambda, \Delta \lambda}(t),
{\mathcal H}^\alpha_{f,\lambda}] = - \sum_{{\mathbf{k}} \sigma} \Big(\delta u^{(3)}_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}
\frac{\sigma}{2}\,
c^\dag_{{\mathbf{k}},\sigma} c^{}_{{\mathbf{k}} -{\mathbf{q}}, \sigma} + \mathrm{H.c.} \Big) \nonumber \\
&& \quad \qquad -\frac{1}{N} \sum_{{\mathbf{k}} {\mathbf{k}}'{\mathbf{p}} \sigma} \Big[
\delta v^{(2)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}},\lambda} \nonumber\\
&& \quad \qquad\times \frac{\sigma}{2}
: c^\dag_{{\mathbf{k}}\sigma} c^{}_{{\mathbf{k}} + {\mathbf{p}} -{\mathbf{q}}, \sigma}\, c^\dag_{{\mathbf{k}}',-\sigma} c^{}_{{\mathbf{k}}' -{\mathbf{p}},-\sigma}:
+ \mathrm{H.c.}
\Big]\end{aligned}$$ with $$\begin{aligned}
\label{B15}
&& \delta u^{(3)}_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda} = - \frac{2U}{N}
\sum_{{\mathbf{k}}'} \Theta_{{\mathbf{k}}', {\mathbf{k}}' + {\mathbf{q}}; {\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}, \lambda}
B_{{\mathbf{k}}' + {\mathbf{q}}, {\mathbf{k}}'}(\lambda, \Delta \lambda) \nonumber \\
&&\qquad \times \big(
\langle n_{{\mathbf{k}}' + {\mathbf{q}}, -\sigma}\rangle - \langle n_{{\mathbf{k}}', -\sigma}\rangle\big) \nonumber \\
\label{B16}
&& \delta v^{(2)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}},\lambda} = \nonumber \\
&& \qquad2U \Big( \Theta_{{\mathbf{k}} -{\mathbf{q}}, {\mathbf{k}}+ {\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}},\lambda}
B_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}}(\lambda, \Delta \lambda) \nonumber \\
&& \qquad-
\Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} ; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}},\lambda}
B_{{\mathbf{k}} +{\mathbf{p}}, {\mathbf{k}}+ {\mathbf{p}} - {\mathbf{q}}}(\lambda, \Delta \lambda)
\Big)
\, , \end{aligned}$$ where $n_{{\mathbf{k}} \sigma} = \langle c^\dag_{{\mathbf{k}}, \sigma} c^{}_{{\mathbf{k}} \sigma} \rangle $. Inserting (\[36\]) for $B_{{\mathbf{k}}, {\mathbf{k}} + {\mathbf{q}}}(\lambda, \Delta \lambda)$, we can again extract a common factor $\langle s^z_{\bar{{\mathbf{p}}}, - {\mathbf{q}}}\rangle $ from equations (\[B15\]), (\[B16\]). We find $$\begin{aligned}
\label{B17}
&& \delta u^{(3)}_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda} = - \frac{2U}{N^2}
\sum_{{\mathbf{p}}, \bar{{\mathbf{p}}}} B^{(3)}_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}
(\lambda, \Delta \lambda) \langle s^z_{\bar{{\mathbf{p}}}, -{\mathbf{q}}}\rangle \,, \\\label{xyz1}
&& \delta v^{(2)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}}, \lambda} =
\frac{2U}{N} \sum_{\bar{{\mathbf{p}}}} D^{(2)}_{\bar{{\mathbf{p}}} ; {\mathbf{k}} {\mathbf{k}}' {\mathbf{p}}{\mathbf{q}}}(\lambda, \Delta \lambda)
\langle s^z_{\bar{{\mathbf{p}}}, -{\mathbf{q}}}\rangle\nonumber\\&&
\end{aligned}$$ with $$\begin{aligned}
&& B^{(3)}_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda) =
\Theta_{{\mathbf{p}}, {\mathbf{p}} + {\mathbf{q}}; {\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}}
\big(\langle n_{{\mathbf{p}} + {\mathbf{q}}, -\sigma}\rangle -\langle n_{{\mathbf{p}}, -\sigma} \rangle\big) \nonumber \\
&& \qquad\times
\big( A_{{\mathbf{p}} + {\mathbf{q}}; {\mathbf{p}}; \bar{{\mathbf{p}}}, \bar{{\mathbf{p}}} +{\mathbf{q}}}(\lambda,\Delta \lambda) -
A_{{\mathbf{p}} + {\mathbf{q}}, {\mathbf{p}}}(\lambda, \Delta \lambda)\big)
\end{aligned}$$ and $$\begin{aligned}
&& D^{(2)}_{\bar{{\mathbf{p}}} ; {\mathbf{k}} {\mathbf{k}}' {\mathbf{p}}{\mathbf{q}}}(\lambda, \Delta \lambda) =
\Theta_{{\mathbf{k}}- {\mathbf{q}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}},\lambda} \\
&& \qquad \times
\Big( A_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}; \bar{{\mathbf{p}}}, \bar{{\mathbf{p}}} + {\mathbf{q}}}(\lambda, \Delta \lambda)
- A_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}}(\lambda, \Delta \lambda) \Big) \nonumber \\
&& \qquad -
\Theta_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}}, {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}},\lambda} \nonumber\\
&& \qquad \times
\Big( A_{{\mathbf{k}} +{\mathbf{p}}, {\mathbf{k}} +{\mathbf{p}} - {\mathbf{q}}; \bar{{\mathbf{p}}}, \bar{{\mathbf{p}}} + {\mathbf{q}}}(\lambda, \Delta \lambda)
- A_{{\mathbf{k}}+ {\mathbf{p}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}}(\lambda, \Delta \lambda) \Big)\,.\nonumber
\end{aligned}$$
### Commutator $[X^\alpha_{\lambda, \Delta \lambda}(t),
{\mathcal H}^\beta_{f,\lambda}] $ {#appB.2.3}
In analogy to the last commutator we obtain the renormalization contributions $$\begin{aligned}
\label{B18}
&& \delta u^{(4)}_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda} = \frac{2U}{N}
\sum_{{\mathbf{k}}'} A_{{\mathbf{k}}' -{\mathbf{q}}, {\mathbf{k}}'; {\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}} (\lambda, \Delta \lambda) \,
\varphi_{{\mathbf{k}}', {\mathbf{k}}' -{\mathbf{q}}, \lambda} \nonumber \\
&&\qquad \times \big(
\langle n_{{\mathbf{k}}', -\sigma} \rangle - \langle n_{{\mathbf{k}}' -{\mathbf{q}}, -\sigma}\rangle\big) \end{aligned}$$ and $$\begin{aligned}
\label{B19}
&& \delta v^{(3)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}},\lambda} = \\
&& \quad-2U \Big( A_{{\mathbf{k}} -{\mathbf{q}}, {\mathbf{k}}+ {\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) \,
\varphi_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \lambda} \nonumber \\
&& \quad-
A_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}} ; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda,\Delta \lambda) \,
\varphi_{{\mathbf{k}} +{\mathbf{p}}, {\mathbf{k}}+ {\mathbf{p}} - {\mathbf{q}},\lambda}
\Big)
\, , \nonumber\end{aligned}$$ where the quantity $\varphi_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}=
\varphi_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}(t)$ was already defined in Eq. (\[32\]). Due to this relation we can extract a common factor $\langle s^z_{\bar{{\mathbf{p}}}, -{\mathbf{q}}}\rangle $ in Eqs. (\[B18\]) and (\[B19\]), and obtain: $$\begin{aligned}
\label{B20}
&& \delta u^{(4)}_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda} = \frac{2U}{N^2}
\sum_{{\mathbf{p}}, \bar{{\mathbf{p}}}} B^{(4)}_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda)
\langle s^z_{\bar{{\mathbf{p}}}, -{\mathbf{q}}}\rangle\,, \\\label{xyz2}
&& \delta v^{(3)}_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}},\lambda} =
-\frac{2U}{N} \sum_{\bar{{\mathbf{p}}}} D^{(3)}_{\bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} {\mathbf{q}}}
\langle s^z_{\bar{{\mathbf{p}}}, -{\mathbf{q}}}\rangle \,,\qquad \end{aligned}$$ where $$\begin{aligned}
&& B^{(4)}_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda) =
A_{{\mathbf{p}} -{\mathbf{q}}, {\mathbf{p}}; {\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}} (\lambda, \Delta \lambda) \\
&& \times
\big( \Theta_{{\mathbf{p}}, {\mathbf{p}}- {\mathbf{q}}; \bar{{\mathbf{p}}}, \bar{{\mathbf{p}}}+ {\mathbf{q}}, \lambda} -
\Theta_{{\mathbf{p}}, {\mathbf{p}}- {\mathbf{q}}, \lambda}
\big)\big(\langle n_{{\mathbf{p}}, -\sigma}\rangle -\langle n_{{\mathbf{p}} -{\mathbf{q}}, -\sigma}\rangle\big)\,, \nonumber \\
&& D^{(3)}_{\bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} {\mathbf{q}}} =
A_{{\mathbf{k}} -{\mathbf{q}}, {\mathbf{k}}+ {\mathbf{p}}-{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}}}(\lambda, \Delta \lambda) \nonumber \\
&& \quad \times
\big( \Theta_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}}, \bar{{\mathbf{p}}}, \bar{{\mathbf{p}}}+ {\mathbf{q}},\lambda}
- \Theta_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{q}},\lambda}
\big) \nonumber \\
&& \quad -
A_{{\mathbf{k}}, {\mathbf{k}}+ {\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}}}(\lambda, \Delta \lambda) \nonumber \\
&& \quad \times
\big( \Theta_{{\mathbf{k}} + {\mathbf{p}}, {\mathbf{k}}+ {\mathbf{p}} - {\mathbf{q}}, \bar{{\mathbf{p}}}, \bar{{\mathbf{p}}}+ {\mathbf{q}},\lambda}
- \Theta_{{\mathbf{k}} + {\mathbf{p}}, {\mathbf{k}}+ {\mathbf{p}} - {\mathbf{q}},\lambda}
\big) \nonumber \, .\nonumber\end{aligned}$$
Commutator $ [X_{\lambda, \Delta \lambda}, [ X_{\lambda, \Delta \lambda},
{\mathcal H}_{0,\lambda}] ]$ {#B.3}
---------------------------------------------------------------------------
First, by use of Eq. (\[23\]) this commutator can be transformed to $- [X_{\lambda, \Delta \lambda},
\mathbf Q_{\lambda -\Delta \lambda} {\mathcal H}_{f,\lambda} ]$, where $X_{\lambda, \Delta \lambda}$ and ${\mathcal H}_{f,\lambda}$ consist of two parts. For the commutator part with $X^\alpha_{\lambda, \Delta \lambda}$ and $\mathcal H^\alpha_{f,\lambda}$ one starts from expression (\[34\]) for $X^\alpha_{\lambda, \Delta \lambda}$ and $$\begin{aligned}
\label{B21}
&& \mathbf Q_{\lambda -\Delta \lambda}{\mathcal H}^\alpha_{f, \lambda} =
\frac{U}{2N} \sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} \sigma} \Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}
\nonumber \qquad\\
&& \qquad \times
\big( 1 - \Theta_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda - \Delta \lambda}
\big)
\nonumber \\
&& \qquad \times :c^\dag_{{\mathbf{k}} \sigma} \,c^{}_{{\mathbf{k}} + {\mathbf{p}}, \sigma} \, c^\dag_{{\mathbf{k}}', -\sigma} \,c^{}_{{\mathbf{k}}' - {\mathbf{p}}, -\sigma}: \, . \end{aligned}$$ Note that both $X^\alpha_{\lambda, \Delta \lambda}$ and $ \mathbf Q_{\lambda -\Delta \lambda}{\mathcal H}^\alpha_{f, \lambda}$ contain a product of two $\Theta$ functions which restrict the allowed excitations to the small energy shell between $\lambda$ and $\lambda -\Delta \lambda$. Therefore only those contributions to the commutator are important, for which identical excitation energies enter the two $\Theta$ function products. We get $$\begin{aligned}
\label{B22}
&& -[X^\alpha_{\lambda, \Delta \lambda}, \mathbf Q_{\lambda- \Delta \lambda}\mathcal H^\alpha_{f,\lambda}]] = \\
&& \qquad -
\frac{U}{2N^2}
\sum_{{\mathbf{k}} {\mathbf{k}}' {\mathbf{p}}, \sigma}
\Big\{
G_{{\mathbf{k}}, {\mathbf{k}}', {\mathbf{k}}'+ {\mathbf{p}}, {\mathbf{p}}, -{\mathbf{p}}}(\lambda, \Delta \lambda) \nonumber \\
&&\qquad \times
c^\dag_{{\mathbf{k}}\sigma} c^{}_{{\mathbf{k}}\sigma}\, :c^\dag_{{\mathbf{k}}',-\sigma} c^{}_{{\mathbf{k}}'+{\mathbf{p}}, -\sigma}:
\, :c^\dag_{{\mathbf{k}}' + {\mathbf{p}}, -\sigma} c^{}_{{\mathbf{k}}', -\sigma}: \nonumber \\
&& \qquad+ \textrm{H.c.} \Big\} \, , \nonumber\end{aligned}$$ where we have introduced $$\begin{aligned}
\label{B23}
&& G_{{\mathbf{k}}, {\mathbf{k}}', {\mathbf{k}}' + {\mathbf{p}}, {\mathbf{p}}, -{\mathbf{p}}}(\lambda, \Delta \lambda) = \\
&& \qquad
A_{{\mathbf{k}}, {\mathbf{k}}- {\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}'+ {\mathbf{p}}}(\lambda, \Delta \lambda) -
A_{{\mathbf{k}}+ {\mathbf{p}}, {\mathbf{k}}; {\mathbf{k}}', {\mathbf{k}}'+ {\mathbf{p}}}(\lambda, \Delta \lambda) \, .
\nonumber
\nonumber \\
&& \nonumber\end{aligned}$$ After reducing (\[B22\]) to the one-particle part, we are led to the following renormalization of $ \varepsilon_{{\mathbf{k}},\lambda }$: $$\begin{aligned}
\label{B24}
&& \delta \varepsilon^{(2)}_{{\mathbf{k}},\lambda } = \\
&& \quad
\frac{U}{4N^2} \sum_{{\mathbf{k}}' {\mathbf{p}}' } \Big[
G_{{\mathbf{k}}', {\mathbf{k}}, {\mathbf{k}}+ {\mathbf{p}}; {\mathbf{p}} , -{\mathbf{p}}}(\lambda, \Delta \lambda) \langle n_{{\mathbf{k}}'}\rangle
\big(2- \langle n_{{\mathbf{k}} +{\mathbf{p}}}\rangle\big)
\nonumber \\
&&\quad + G_{{\mathbf{k}}, {\mathbf{k}}' , {\mathbf{k}}' + {\mathbf{p}}; {\mathbf{p}} , -{\mathbf{p}}}(\lambda, \Delta \lambda)
\langle n_{{\mathbf{k}}'}\rangle \big (2- \langle n_{{\mathbf{k}}' +{\mathbf{p}}}\rangle \big)
\nonumber \\
&& \quad- G_{{\mathbf{k}}', {\mathbf{k}} -{\mathbf{p}}, {\mathbf{k}}; {\mathbf{p}} , -{\mathbf{p}}}(\lambda, \Delta \lambda) \langle n_{{\mathbf{k}}'}\rangle
\langle n_{{\mathbf{k}} -{\mathbf{p}}}\rangle \nonumber
\Big]\,. \end{aligned}$$ Possible contributions to $u_{{\mathbf{k}}, {\mathbf{k}} - {\mathbf{q}}, \lambda}$ and $v_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}},\lambda}$ cancel. The commutator part of (\[B22\]) with four fermion operators leads to higher order contributions to $\mathcal H_{f,\lambda}$, which have been neglected. The remaining ’mixed’ contributions $(\alpha \beta)$, $(\beta \alpha)$ to the commutator $[X_{\lambda, \Delta \lambda}, \mathbf Q_{\lambda- \Delta \lambda}\mathcal H_{f,\lambda}]]$, which are of order $h(t)$, are negligible as well, since the energies of the two $\Theta$-function products do not coincide.
Finally we consider the last commutator in transformation (\[24\]), $[ X_{\lambda, \Delta \lambda},
[ X_{\lambda, \Delta \lambda}, \hat {\mathcal H}_{h,\lambda}] ] $. Here only the four fermion part $X^\alpha_{\lambda, \Delta \lambda}$ contributes in first order in $h(t)$. Renormalization contributions to $u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}}, \lambda}$ and $v_{{\mathbf{k}}, {\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}'-{\mathbf{p}}, \lambda}$ are again expected to be small due to the appearance of the two $\Theta$-function products from the two generators $X^\alpha_{\lambda, \Delta \lambda}$.
Renormalization contributions of Eqs. (\[66\]) and (\[67\]) {#appC}
============================================================
In Sect. V.D the general result (\[61\]) for $\langle s^z_{-{\mathbf{q}}}\rangle(t)$ was simplified by tracing back the ${\mathbf{k}}$-resolved expectation values $\langle s^z_{{\mathbf{k}}, -{\mathbf{q}}}\rangle(t)$ to the compact variable $\langle s^z_{-{\mathbf{q}}}\rangle(t)$. The resulting time-independent renormalization equations are given in Eqs. (\[66\]), (\[67\]). The renormalization contributions on the right-hand sides read $$\begin{aligned}
\label{C1}
&& \delta u_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}, \lambda}^{0(1)}= \\
&& \qquad - \frac{1}{N} \sum_{{\mathbf{k}}'} \Big( U + \frac{1}{\chi({\mathbf{q}}, \omega)}
+ u^0_{{\mathbf{k}}', {\mathbf{k}}' -{\mathbf{q}}, \lambda} \Big) \nonumber \\
&& \qquad \times A_{{\mathbf{k}}, {\mathbf{k}}-{\mathbf{q}}; {\mathbf{k}}' -{\mathbf{q}}, {\mathbf{k}}'}(\lambda, \Delta \lambda) \nonumber \\
&& \qquad \times \big(
\langle c_{{\mathbf{k}}',-\sigma}^\dag c_{{\mathbf{k}}', -\sigma}\rangle
- \langle c_{{\mathbf{k}}' -{\mathbf{q}},-\sigma}^\dag c_{{\mathbf{k}}' -{\mathbf{q}}, -\sigma}\rangle
\big) \,,\nonumber \\
\label{C2}
&& \sum_{n=2}^4 \delta u_{{\mathbf{k}}, {\mathbf{k}} -{\mathbf{q}},\lambda}^{0(n)} = \\
&& \qquad - \frac{U}{2N^2} \sum_{{\mathbf{p}} \bar{{\mathbf{p}}}}
\big( B^{(2)}_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda)
+ 4 B^{(3)}_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda) \nonumber \\
&& \qquad -4 B^{(4)}_{{\mathbf{p}} \bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{q}}}(\lambda, \Delta \lambda)
\big) \nonumber
\end{aligned}$$ and $$\begin{aligned}
\label{C3}
&& \delta v^{0(1)}_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda}= \\
&&
\Big( U+ \frac{1}{\chi({\mathbf{q}}, \omega)} + u^0_{{\mathbf{k}} + {\mathbf{p}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}},\lambda} \Big)
A_{{\mathbf{k}}, {\mathbf{k}}+{\mathbf{p}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) \nonumber \\
&& - \Big( U+ \frac{1}{\chi({\mathbf{q}}, \omega)} + u^0_{{\mathbf{k}},{\mathbf{k}} -{\mathbf{q}}, \lambda} \Big)
A_{{\mathbf{k}} -{\mathbf{q}}, {\mathbf{k}}+{\mathbf{p}}-{\mathbf{q}}; {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}}(\lambda, \Delta \lambda) \, ,
\nonumber\end{aligned}$$ $$\begin{aligned}
\label{C4}
&& \sum_{n=2,3} \delta v^{0(n)}_{{\mathbf{k}},{\mathbf{k}} +{\mathbf{p}} -{\mathbf{q}}, {\mathbf{k}}', {\mathbf{k}}' -{\mathbf{p}}, \lambda} = \\
&& \quad\frac{2U}{N} \sum_{\bar{{\mathbf{p}}}}\Big( D^{(2)}_{\bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} {\mathbf{q}}}(\lambda, \Delta \lambda)
- D^{(3)}_{\bar{{\mathbf{p}}}; {\mathbf{k}} {\mathbf{k}}' {\mathbf{p}} {\mathbf{q}}}(\lambda, \Delta \lambda)
\Big) \nonumber\,,\end{aligned}$$ where $$\label{C5}
\frac{\hat h_{{\mathbf{q}}}(t)}{2} = \Big(U + \frac{1}{\chi({\mathbf{q}}, \omega)}\Big) \frac{\langle s^z_{-{\mathbf{q}}}\rangle(t)}{N} \,.$$
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---
abstract: 'Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over an uncountable algebraically closed field of characteristic zero can be obtained by solving a so-called tropicalized differential system. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions. We show that the Fundamental Theorem can be extended to the case of systems of partial differential equations by introducing vertex sets of Newton polygons.'
author:
- Sebastian Falkensteiner
- 'Cristhian Garay-López'
- Mercedes Haiech
- Marc Paul Noordman
- Zeinab Toghani
- 'Fran[ç]{}ois Boulier'
bibliography:
- 'tpdag.bib'
title: The Fundamental Theorem of Tropical Partial Differential Algebraic Geometry
---
Introduction
============
Given an algebraically closed field of characteristic zero $K$, we consider the partial differential ring $(R_{m,n},D)$, where $$R_{m,n}=K[[t_1,\ldots,t_m]]\{x_1,\ldots,x_n\}$$ and $D=(\tfrac{\partial}{\partial t_k}\::\:k=1,\ldots,m)$ for $n,m\geq1$ (see Section \[S\_PDAG\] for definitions). Up to now, tropical differential algebra has been limited to the study of the relation between the set of solutions ${\operatorname{Sol}}(G)\subseteq K[[t]]^n$ of differential ideals $G$ in $R_{1,n}$ and their corresponding [*tropicalizations*]{}, which are certain polynomials $p$ with coefficients in a tropical semiring $\mathbb{T}[t]:=(\mathbb{Z}_{\geq0}\cup\{\infty\},+,\min{})$ and with a set of solutions ${\operatorname{Sol}}(p) \subseteq \mathcal{P}(\mathbb{Z}_{\geq0})^n$, see [@grigoriev2017tropical] and [@AGT16]. These elements $S\in {\operatorname{Sol}}(p)$ can be found by looking at [*evaluations*]{} $p(S)\in \mathbb{T}[t]$ where the usual tropical vanishing condition holds.
In this paper, we consider the case $m>1$. On this account, we work with elements in $\mathbb{Z}_{\ge 0}^m$, which requires new techniques. We show that considering the Newton polygons and their vertex sets is the appropriate method for formulating and proving our generalization of the Fundamental Theorem of Tropical Differential Algebraic Geometry. We remark that in the case of $m=1$ the definitions and properties presented here coincide with the corresponding ones in [@AGT16] and therefore, this work can indeed be seen as a generalization.
The problem of finding power series solutions of systems of partial differential equations has been extensively studied in the literature, but is very limited in the general case. In fact, we know from [@DenefLipshitz Theorem 4.11] that there is already no algorithm for deciding whether a given linear partial differential equation with polynomial coefficients has a solution or not. The Fundamental Theorem, as it is stated in here, helps to find necessary conditions for the support of possible solutions.
The structure of the paper is as follows. In Section \[S\_PDAG\] we cover the necessary material from partial differential algebra. In Section \[sec:the tropical semi-ring\] we introduce the semiring of supports $\mathcal{P}(\mathbb{Z}_{\geq0}^m)$, the semiring of vertex sets $\mathbb{T}[[t_1,\ldots,t_m]]$ and the vertex homomorphism ${\operatorname{Vert}}\colon\mathcal{P}(\mathbb{Z}_{\geq0}^m)\longrightarrow\mathbb{T}[[t_1,\ldots,t_m]]$. In Section \[S\_DRPS\] we introduce the support and the tropicalization maps. In Section \[S\_TDP\] we define the set of tropical differential polynomials $\mathbb{T}_{m,n}$, the notion of tropical solutions for them, and the tropicalization morphism ${\operatorname{trop}}\colon R_{m,n}\to\mathbb{T}_{m,n}$. The main result is Theorem \[fundamental theorem\], which is proven in Section \[S\_TFT\]. The proof we give here differs essentially from that one in [@AGT16] for the case of $m=1$. In Section \[Section\_ExLim\] we give some examples to illustrate our results.
In the following we will use the conventions that for a set $S$ we denote by $\mathcal{P}(S)$ its power set, and by $K$ we denote an algebraically closed field of characteristic zero.
Partial differential algebraic geometry {#S_PDAG}
=======================================
Here we recall the preliminaries for partial differential algebraic geometry. The reference book for differential algebra is [@Kolchin73].
A **partial differential ring** is a pair $(R,D)$ consisting of a commutative ring with unit $R$ and a set $D=\{\delta_1,\ldots,\delta_m\}$ of $m > 1$ **derivations** which act on $R$ and are pairwise commutative. We denote by $\Theta$ the free commutative monoid generated by $D$. If $J=(j_1,\ldots,j_m)$ is an element of the monoid $\mathbb{Z}^m_{\geq0}=(\mathbb{Z}^m_{\geq0},+,0)$, we denote $\Theta(J) = \delta_1^{j_1} \cdots \delta_m^{j_m}$ the **derivative operator** defined by $J$. If $\varphi$ is any element of $R$, then $\Theta(J)\varphi$ is the element of $R$ obtained by application of the derivative operator $\Theta(J)$ on $\varphi$. Let $(R,D)$ be a partial differential ring and $x_1,\ldots,x_n$ be $n$ **differential indeterminates**. The monoid $\Theta$ acts on the differential indeterminates, giving the infinite set of the **derivatives** which are denoted by $x_{i,J}$ with $1 \leq i \leq n$ and $J \in \mathbb{Z}^m_{\geq0}$. Given any $1 \leq k \leq m$ and any derivative $x_{i,J}$, the action of $\delta_k$ on $x_{i,J}$ is defined by $\delta_k(x_{i,J})=x_{i,J+e_k}$ where $e_k$ is the $m$-dimensional vector whose $k$-th coordinate is $1$ and all other coordinates are zero. One denotes $R\{x_1,\ldots,x_n\}$ the ring of the polynomials, with coefficients in $R$, the indeterminates of which are the derivatives. More formally, $R\{x_1,\ldots, x_n\}$ consists of all $R$-linear combinations of differential monomials, where a differential monomial in $n$ independent variables of order less than or equal to $r$ is an expression of the form
$$\label{differential_monomial}
E_M:=\prod_{\substack{1\leq i\leq n\\ ||J||_\infty\leq r}}x_{i,J}^{M_{i,J}} $$
where $J=(j_1,\ldots,j_m)\in \mathbb{Z}_{\geq0}^m$, $||J||_\infty:=\text{max}_i\{j_i\}=\text{max}(J)$ and $M=(M_{i,J}) \in (\mathbb{Z}_{\ge0})^{n\times(r+1)^m}$.
The pair $(R\{x_1,\ldots,x_n\},D)$ then constitutes a **differential polynomial ring**. A differential polynomial $P\in R\{x_1,\ldots,x_n\}$ induces an evaluation map from $R^n$ to $R$ given by $$P\colon R^n \to R,\quad (\varphi_1,\ldots,\varphi_n)\mapsto P|_{x_{i,J}=\Theta(J)\varphi_i},$$ where $P|_{x_{i,J}=\Theta(J)\varphi_i}$ is the element of $R$ obtained by substituting $\Theta(J)\varphi_i$ for $x_{i,J}$ .
A **zero** or **solution** of $P\in R\{x_1,\ldots,x_n\}$ is an $n$-tuple $\varphi=(\varphi_1,\ldots,\varphi_n)\in R^n$ such that $P(\varphi)=0$. An $n$-tuple $\varphi \in R^n$ is a solution of a system of differential polynomials $\Sigma \subseteq R\{x_1,\ldots,x_n\}$ if it is a solution of every element of $\Sigma$. We denote by ${\operatorname{Sol}}(\Sigma)$ the solution set of the system $\Sigma$.
A **differential ideal** of $R\{x_1,\ldots,x_n\}$ is an ideal of that ring which is stable under the action of $\Theta$. A differential ideal is said to be **perfect** if it is equal to its radical. If $\Sigma \subseteq R\{x_1,\ldots,x_n\}$, one denotes by $[\Sigma]$ the **differential ideal generated by $\Sigma$** and by $\{ \Sigma \}$ the **perfect differential ideal generated by $\Sigma$**, which is defined as the intersection of all perfect differential ideals containing $\Sigma$.
For $m,n\geq1$, we will denote by $R_m$ the partial differential ring $$(K[[t_1,\ldots,t_m]],D)$$ where $D=\{\tfrac{\partial}{\partial t_1},\ldots,\tfrac{\partial}{\partial t_m}\}$, and the partial differential ring $(R_m\{x_1,\ldots,x_n\},D)$ will be denoted by $R_{m,n}$ . The proof of the following proposition can be found in [@BH19].
\[prop\_Ritt\_Raudenbush\] For any $\Sigma \subseteq R_{m,n}$, there exists a finite subset $\Phi$ of $\Sigma$ such that ${\operatorname{Sol}}(\Sigma)={\operatorname{Sol}}(\Phi)$.
The semirings of supports and vertex sets {#sec:the tropical semi-ring}
=========================================
In this part we introduce and give some properties on our main idempotent semirings, namely the semiring of supports $\mathcal{P}(\mathbb{Z}_{\geq0}^m)$, the semiring of vertex sets $\mathbb{T}[[t_1,\ldots,t_m]]$ and the map ${\operatorname{Vert}}\colon\mathcal{P}(\mathbb{Z}_{\geq0}^m)\rightarrow\mathbb{T}[[t_1,\ldots,t_m]]$ which is a homomorphism of semirings.
First, we recall some preliminaries on partially ordered sets (or posets). Let $(P,\leq)$ be a poset and consider $S\subseteq P$. We say that $S$ is an **antichain** if any two distinct elements in the subset are incomparable. We denote by $\min(S)$ the set of minimal elements of $S$, i.e. $\min(S)=\{a\in S\::\:b\leq a,\: b\in S\Rightarrow b=a\}$. Observe that $\min(S)$ is always an antichain, and that $S$ is an antichain if and only if $\min(S) = S$. If we have multiple preorders $\leq_1,\ldots,\leq_m$ defined on $P$, then we denote by $\min_{\leq_i}(S)$ the antichain of minimal elements of $S$ with respect to $\leq_i$. Recall that a commutative semiring $S$ is a tuple $(S,+,\times,0,1)$ such that $(S,+,0)$ and $(S,\times,1)$ are commutative monoids and additionally, for all $a,b,c \in S$ it holds that
1. $a\times (b+c)=a\times b+a\times c$;
2. $0\times a=0$.
A semiring is called **idempotent** if $a + a = a$ for all $a \in S$. A map $f\colon S_1\longrightarrow S_2$ between semirings is a morphism if it induces morphisms at the level of monoids.
Let $M=(M,*,e)$ be a commutative monoid. $M$ admits a natural preorder, called the monoid algebraic preorder: for every $x,y \in M$, we have $x\leq_* y $ if and only if there exists $z \in M$, such that $x*z=y$. This preorder is algebraic in the sense that for all $z \in M$ and $x\leq_* y$ it follows that $x*z\leq_* y*z$. For $m\geq1$, we denote by $\mathcal{P}(\mathbb{Z}^m_{\geq0})$ the idempotent semiring whose elements are the subsets of $\mathbb{Z}^m_{\geq0}$ equipped with the union $X\cup Y$ as sum and the Minkowski sum $X+Y=\{x+y\::\:x\in X, y\in Y\}$ as product. We call it the **semiring of supports**. For $n\in\mathbb{Z}_{\geq1}$ and $X \in \mathcal{P}(\mathbb{Z}_{\geq 0}^m)$, the notation $nX$ will indicate $\underbrace{X+ \cdots + X}_{n \ \text{times}}$. By convention we set $0X = \{(0,\ldots,0)\}$.
The corresponding preorders $\leq_\cup$ and $\leq_+$ defined on $\mathcal{P}(\mathbb{Z}^m_{\geq0})$ are in fact partial orders; for instance, we have that $X\leq_\cup Y$ if and only if $X\subseteq Y$. For $S\subseteq \mathcal{P}(\mathbb{Z}^m_{\geq0})$ we denote by $\min_\cup(S)$ or $\min_+(S)$ the corresponding antichain of minimal elements of $S$. We define the **Newton polygon** $\mathcal{N}(X)\subseteq\mathbb{R}^m_{\geq0}$ of $X\in\mathcal{P}(\mathbb{Z}^m_{\geq0})$ as the convex hull of $X+\mathbb{Z}^m_{\geq0}$. We call $x \in X$ a **vertex** if $x \notin \mathcal{N}(X \setminus \{x\})$, and we denote by ${\operatorname{Vert}}X$ the set of vertices of $X$.
Note that ${\operatorname{Vert}}X$ is a subset of the minimal basis of the monomial ideal generated by $X$ (see e.g. [@CLS92]), but in general they are not equal. Moreover, there exist $X$ and $Y$ generating different monomial ideals such that ${\operatorname{Vert}}X = {\operatorname{Vert}}Y$ as Example \[Example\_VertSet\] shows.
\[N\_equal\_impies\_Vert\_equal\] Let $S,T \in \mathcal{P}(\mathbb{Z}_{\geq0}^m)$ such that $\mathcal{N}(S)= \mathcal{N}(T)$. Then ${\operatorname{Vert}}S={\operatorname{Vert}}T$.
Let $s \in {\operatorname{Vert}}S$ and we assume that $s \in \mathcal{N}(T \setminus\{s\})$. Then there are $t_i \in T \setminus \{s\}$, $w_i \in \mathbb{Z}_{\geq 0}^m$ and positive $\lambda_i \in \mathbb{R}$ adding up to 1 such that $$s = \sum_{i} \lambda_i (t_i + w_i).$$ Since $t_i \in \mathcal{N}(S)$, we can write the $t_i$ as $$t_i = \sum_{j}\mu_{i,j} (s_{i,j} + z_{i,j}),$$ where $s_{i,j} \in S$, $z_{i,j} \in \mathbb{Z}_{\ge 0}^m$ and $\mu_{i,j} \in \mathbb{R}$ are positive and adding up to 1. Thus, $$s = \sum_{i,j} \lambda_{i}\mu_{i,j} (s_{i,j} + z_{i,j} + w_i) = \sum_{i,j}\lambda_i \mu_{i,j} s_{i,j} + v,$$ where $v$ is a vector with non-negative coefficients. By excluding in the sum those summands $s_{i,j}$ which are equal to $s$, we obtain $$s = cs + \sum_{\substack{i,j\\s_{i,j} \neq s}} \lambda_i\mu_{i,j}s_{i,j} + v$$ where $c = \sum_{i,j: s_{i,j = s}} \lambda_i \mu_{i,j} \in [0,1]$. If $c < 1$ we can solve the equation above for $s$ to get $$s = \sum_{\substack{i,j\\s_{i,j} \neq s}} \frac{\lambda_i\mu_{i,j}}{1 - c} s_{i,j} + \frac{v}{1-c}.$$ The coefficients for the $s_{i,j}$ are positive and sum to 1, so the summation in the right hand side gives an element of $\mathcal{N}(S \setminus\{s\})$. Since $\mathcal{N}(S\setminus\{s\})$ is closed under adding elements of $\mathbb{R}_{\geq 0}^m$, and the coefficients of $v/(1-c)$ are non-negative, we then find that $s \in \mathcal{N}(S \setminus\{s\})$ in contradicting to the assumption that $s$ is a vertex of $S$. If $c = 1$, then all $s_{i,j}$ are equal to $s$ and we get $s = s + v$. Therefore, $v = 0$ and $t_i = s$ for each $i$, and in particular $s \in T \setminus \{s\}$, which is a contradiction. So we conclude that $s \notin \mathcal{N}(T \setminus \{s\})$ and $s$ is a vertex of $T$.
Let $X \in \mathcal{P}(\mathbb{Z}_{\geq0}^m)$. Then $\mathcal{N}({\operatorname{Vert}}X) = \mathcal{N}(X)$.
By Dickson’s lemma [@CLS92 chap. 2, Thm 5], there is a finite subset $S \subseteq X$ with $X \subseteq S + \mathbb{Z}_{\geq 0}^m$. For such $S$, it holds that $\mathcal{N}(X) = \mathcal{N}(S)$ and by \[N\_equal\_impies\_Vert\_equal\], we get ${\operatorname{Vert}}X = {\operatorname{Vert}}S$. Therefore, replacing $X$ by $S$, we may assume that $X$ is finite.
We proceed by induction on $\#X$. Indeed, if $X = \emptyset$, the statement is obvious. Let $X$ be an arbitrary finite set. If every element of $X$ is a vertex of $X$, then $\mathcal{N}(X) = \mathcal{N}({\operatorname{Vert}}X)$ is trivially true. Else, take $x \in X \setminus {\operatorname{Vert}}X$ and let $Y = X \setminus\{x\}$. Then $\mathcal{N}(X) = \mathcal{N}(Y)$ by definition, so applying \[N\_equal\_impies\_Vert\_equal\] again we obtain ${\operatorname{Vert}}X = {\operatorname{Vert}}Y$. Since $\# Y < \#X$, we may apply the induction hypothesis to $Y$, and get that $\mathcal{N}(X) = \mathcal{N}(Y) = \mathcal{N}({\operatorname{Vert}}Y) = \mathcal{N}({\operatorname{Vert}}X)$.
\[equal\_Vert\_is\_equal\_N\] For $X, Y \in \mathcal{P}(\mathbb{Z}_{\geq0}^m)$ we have ${\operatorname{Vert}}X = {\operatorname{Vert}}Y$ if and only if $\mathcal{N}(X) = \mathcal{N}(Y)$.
\[lemma\_Properties\_vert\] For $X,Y\in\mathcal{P}(\mathbb{Z}_{\geq0}^m)$, we have $$\begin{array}{ccl}
{\operatorname{Vert}}({\operatorname{Vert}}(X)\cup{\operatorname{Vert}}(Y)) & = & {\operatorname{Vert}}({\operatorname{Vert}}(X)\cup Y) \\
& =& {\operatorname{Vert}}(X\cup{\operatorname{Vert}}(Y)) \\
&=& {\operatorname{Vert}}(X\cup Y)
\end{array}$$ and $$\begin{array}{ccl}
{\operatorname{Vert}}({\operatorname{Vert}}(X)+{\operatorname{Vert}}(Y)) & = & {\operatorname{Vert}}({\operatorname{Vert}}(X)+ Y) \\
& =& {\operatorname{Vert}}(X+{\operatorname{Vert}}(Y)) \\
&=& {\operatorname{Vert}}(X+ Y).
\end{array}$$
Let $*$ be either $\cup$ or $+$. We have the following diagram of inclusions $$\xymatrix{
&{\operatorname{Vert}}(X)* Y\ar[dr]&\\
{\operatorname{Vert}}(X)*{\operatorname{Vert}}(Y)\ar[dr]\ar[ur]\ar[rr]&&X*Y\\
&X* {\operatorname{Vert}}(Y)\ar[ur]&\\
}$$ We show that these four sets generate the same Newton polygon. For this, it is enough to show that $X*Y\subseteq \mathcal{N}({\operatorname{Vert}}(X)*{\operatorname{Vert}}(Y))$.
For $* = \cup$, we have $X \subseteq \mathcal{N}({\operatorname{Vert}}X) \subseteq \mathcal{N}({\operatorname{Vert}}(X) \cup {\operatorname{Vert}}(Y))$ and similarly $Y \subseteq \mathcal{N}({\operatorname{Vert}}(X) \cup {\operatorname{Vert}}(Y))$. Hence, $X \cup Y \subseteq \mathcal{N}({\operatorname{Vert}}(X) \cup {\operatorname{Vert}}(Y))$.
Now suppose that $*=+$. Let $t\in X+Y$, and write $t = x + y$ with $x \in X$ and $y \in Y$. Using the inclusions $X \subseteq \mathcal{N}({\operatorname{Vert}}X)$ and $Y \subseteq \mathcal{N}({\operatorname{Vert}}Y)$, there are $x_{i}\in{\operatorname{Vert}}(X)$, $y_{j}\in {\operatorname{Vert}}(Y)$, $u_{i}, v_{j}\in \mathbb{Z}_{\geq0}^m$ and $\alpha_{i},\beta_{j} \in \mathbb{R}_{\ge 0}$ satisfying $\sum_{i}\alpha_{i} = 1$ and $\sum_{j}\beta_{j} = 1$ such that $$t=\sum_i \alpha_{i}(x_{i} + u_{i})+\sum_j \beta_{j}(y_{j} + v_{j}).$$ Rewriting this gives $$t = \sum_{i,j} \alpha_{i}\beta_{j}(x_{i} + y_{j} + u_{i} + v_{j}).$$ For each pair $i,j$, the expression between parentheses is an element of ${\operatorname{Vert}}(X) + {\operatorname{Vert}}(Y) + \mathbb{Z}_{\geq 0}$ and the coefficients are non-negative and sum up to 1. This shows that $t \in \mathcal{N}({\operatorname{Vert}}(X) + {\operatorname{Vert}}(Y))$, which ends the proof of the inclusions.
\[Example\_VertSet\] An element $X\in \mathcal{P}(\mathbb{Z}_{\geq0}^m)$ generates a monomial ideal which contains a unique minimal basis $B(X)$ (see e.g. [@CLS92]). In general, ${\operatorname{Vert}}(X)\subset B(X)$ and this inclusion may be strict. Consider the set $X=\{A_1=(1,4),A_2=(2,3),A_3=(3,3),A_4=(4,1)\} \subseteq \mathbb{Z}_{\geq 0}^2$. The Newton polygon $\mathcal{N}(X)$ can be visualized as in Figure \[Figure\_NP\] and $\mathrm{Vert}(X) = \{A_1, A_4\}$ which is a strict subset of $B(X)=\{A_1,A_2,A_4\}.$
![The Newton polygon of $X$. The vertex set of $X$ is $\{A_1, A_4\}$.[]{data-label="Figure_NP"}](VS.pdf)
We deduce from \[equal\_Vert\_is\_equal\_N\] that the map $\mathrm{Vert}\colon\mathcal{P}(\mathbb{Z}^m_{\geq0})\longrightarrow \mathcal{P}(\mathbb{Z}^m_{\geq0})$ is a projection operator in the sense that $\mathrm{Vert}^2=\mathrm{Vert}$.
We denote by $\mathbb{T}[[t_1,\ldots,t_m]]$ the image of the operator $\mathrm{Vert}$, and call its elements either **vertex sets** or **tropical formal power series**. For $S,T \in \mathbb{T}[[t_1,\ldots,t_m]]$, we define $$S\oplus T=\mathrm{Vert}(S\cup T) \quad \textrm{ and }\quad \ S\odot T=\mathrm{Vert}(S+T).$$
\[lemma:commutative semiring\] The set $(\mathbb{T}[[t_1,\ldots,t_m]],\oplus,\odot)$ is a commutative idempotent semiring, with the zero element $\emptyset$ and the unit element $\{(0,\ldots,0)\}$.
The only things to check are associativity of $\oplus$, associativity of $\odot$ and the distributive property. The associativity of $\oplus$ and $\odot$ follows from the equalities $$S\oplus(T\oplus U)=\mathrm{Vert}(S\cup T\cup U)= (S\oplus T)\oplus U$$ and $$S\odot(T\odot U)=\mathrm{Vert}(S+ T+ U)= (S\odot T)\odot U$$ which are consequences of \[lemma\_Properties\_vert\]. The distributivity follows from $$S\odot(T\oplus U)={\operatorname{Vert}}((S+T)\cup U)={\operatorname{Vert}}((S+T)\cup (S+U))=(S\odot T)\oplus (S\odot U).\qedhere$$
\[Lemma\_linearity\] The map ${\operatorname{Vert}}$ is a homomorphism of semirings. In particular, for any finite family $\{X_i\}_i$ of elements $X_i\in\mathcal{P}(\mathbb{Z}_{\geq0}^m)$, we have ${\operatorname{Vert}}(\sum_iX_i) = \bigodot_{i\in I}{\operatorname{Vert}}(X_i)$, ${\operatorname{Vert}}(\bigcup_iX_i) = \bigoplus_{i\in I}{\operatorname{Vert}}(X_i)$ and ${\operatorname{Vert}}(nT)={\operatorname{Vert}}(T)^{\odot n}$.
Follows directly from \[lemma\_Properties\_vert\] and \[lemma:commutative semiring\].
The differential ring of power series and the support map {#S_DRPS}
=========================================================
We consider the differential ring $R_m$ from Section \[S\_PDAG\], and the semirings $\mathcal{P}(\mathbb{Z}^m_{\geq0})$, $\mathbb{T}[[t_1,\ldots,t_m]]$ from Section \[sec:the tropical semi-ring\]. In this part we introduce the support and the tropicalization maps, which are related by the following commutative diagram $$\xymatrix{R_m\ar[r]^{\text{Supp}}\ar[dr]_{\text{trop}}&\mathcal{P}(\mathbb{Z}^m_{\geq0})\ar[d]^{{\operatorname{Vert}}}\\
&\mathbb{T}[[t_1,\ldots,t_m]]\\}$$
If $J=(j_1, \ldots, j_m)$ is an element of $\mathbb{Z}^m_{\geq0}$, we will denote by $t^J$ the monomial $t_1^{j_1} \cdots t_m^{j_m}$. An element of $R_m$ is of the form $\varphi = \sum_{J \in \mathbb{Z}^m_{\geq0}} a_J t^J$ with $a_J \in K.$
1. The **support** of $\varphi = \sum a_J t^J \in R_m$ is defined as $$\mathrm{Supp}(\varphi) = \{J \in \mathbb{Z}^m_{\geq0} \ |\ a_J \neq 0\}.$$
2. For a fixed integer $n$, the mapping from $R_m^n$ to the $n$-fold product $\mathcal{P}(\mathbb{Z}^m_{\geq0})^n$ will also be denoted by $\mathrm{Supp}$: $$\begin{array}{cccc}
R_m^n & \to & \mathcal{P}(\mathbb{Z}^m_{\geq0}) ^n \\
\varphi=(\varphi_1, \ldots, \varphi_n) & \mapsto & \mathrm{Supp}(\varphi)= (\mathrm{Supp}(\varphi_1), \ldots, \mathrm{Supp}(\varphi_n)).
\end{array}$$
3. The **set of supports** of $T\subseteq R_m^n$ is its image under the map Supp: $$\mathrm{Supp}(T) = \{\mathrm{Supp}(\varphi) \ | \ \varphi \in T\} \subseteq \mathcal{P}(\mathbb{Z}^m_{\geq0}) ^n$$
4. The mapping that sends each series in $R_m$ to the vertex set of its support is called the **tropicalization** map $$\begin{array}{cccc}
\mathrm{trop} \colon &R_m & \to &\mathbb{T}[[t_1,\ldots,t_m]] \\
& \varphi & \mapsto & {\operatorname{Vert}}(\mathrm{Supp}(\varphi))
\end{array}$$
The **support** of $\varphi = \sum a_J t^J \in R_m$ is defined as $$\mathrm{Supp}(\varphi) = \{J \in \mathbb{Z}^m_{\geq0} \ |\ a_J \neq 0\}.$$ For a fixed integer $n$, the map which sends $\varphi=(\varphi_1, \ldots, \varphi_n) \in R_m^n$ to $\mathrm{Supp}(\varphi)= (\mathrm{Supp}(\varphi_1), \ldots, \mathrm{Supp}(\varphi_n)) \in \mathcal{P}(\mathbb{Z}^m_{\geq0})^n$ will also be denoted by $\mathrm{Supp}$. The **set of supports** of a subset $T\subseteq R_m^n$ is its image under the map Supp: $$\mathrm{Supp}(T) = \{\mathrm{Supp}(\varphi) \ | \ \varphi \in T\} \subseteq \mathcal{P}(\mathbb{Z}^m_{\geq0}) ^n$$
The mapping that sends each series in $R_m$ to the vertex set of its support is called the **tropicalization** map $$\begin{array}{cccc}
\mathrm{trop} \colon &R_m & \to &\mathbb{T}[[t_1,\ldots,t_m]] \\
& \varphi & \mapsto & {\operatorname{Vert}}(\mathrm{Supp}(\varphi))
\end{array}$$
\[lem:trop\_morphism\_of\_semi\_ring\] The tropicalization map is a non-degenerate valuation in the sense of [@Giansiracusa-Giansiracusa Definition 2.5.1]. This is, it satisfies
1. ${\operatorname{trop}}(0)=\emptyset$, ${\operatorname{trop}}(\pm1)=\{(0,\ldots,0)\}$,
2. ${\operatorname{trop}}(\varphi\cdot\psi)={\operatorname{trop}}(\varphi)\odot{\operatorname{trop}}(\psi)$,
3. ${\operatorname{trop}}(\varphi + \psi) \oplus {\operatorname{trop}}(\varphi) \oplus {\operatorname{trop}}(\psi) = {\operatorname{trop}}(\varphi) \oplus {\operatorname{trop}}(\psi)$,
4. ${\operatorname{trop}}(\varphi)=\emptyset$ implies that $\varphi=0$.
The first point is clear. For the second point, note that the Newton polygon has the well-known homomorphism-type property $$\mathcal{N}({\operatorname{Supp}}(\varphi \cdot \psi))= \mathcal{N}({\operatorname{Supp}}(\varphi))+\mathcal{N}({\operatorname{Supp}}(\psi))=\mathcal{N}({\operatorname{Supp}}(\varphi)+{\operatorname{Supp}}(\psi)).$$ Hence, the vertices of the left hand side coincide with the vertices of the right hand side. This gives ${\operatorname{trop}}(\varphi\cdot\psi) = {\operatorname{Vert}}(\mathcal{N}({\operatorname{Supp}}(\varphi) + {\operatorname{Supp}}(\psi)))$. That this is equal to ${\operatorname{trop}}(\varphi)\odot{\operatorname{trop}}(\psi)$ follows from \[lemma\_Properties\_vert\]. The third point follows from the observation that ${\operatorname{Supp}}(\varphi + \psi) \subseteq {\operatorname{Supp}}(\varphi) \cup {\operatorname{Supp}}(\psi)$ and \[Lemma\_linearity\]. The last point follows from the fact that the empty set is the only set with empty Newton polygon.
For $J = (j_1, \ldots, j_m) \in \mathbb{Z}^m_{\geq 0}$, we define the **tropical derivative operator** $\Theta_{{\operatorname{trop}}}(J)\colon\mathcal{P}(\mathbb{Z}^m_{\geq0})\to\mathcal{P}(\mathbb{Z}^m_{\geq0})$ as $$\Theta_{{\operatorname{trop}}}(J)T:=\left\{ (t_1-j_1, \ldots, t_m -j_m) \ \left| \
\begin{array}{cccc} (t_1, \ldots, t_m) \in T, \\
t_i-j_i \ge 0 \text{ for all }i
\end{array}
\right. \right\}$$
For example, if $T$ is the grey part in Figure \[F\_DS2\] left and $J=(1,2)$, then informally $\Theta_{{\operatorname{trop}}}(J)T$ is a translation of $T$ by the vector $-J$ and then keeping only the non-negative part. It is represented by the grey part in Figure \[F\_DS2\] right.
![The operator $\Theta_{{\operatorname{trop}}}(J)$ for $J=(1,2)$ applied to $T$.[]{data-label="F_DS2"}](DS.pdf)
Since $K$ is of characteristic zero, for all $\varphi \in R_m$ and $J \in \mathbb{Z}^m_{\geq0}$, we have $$\label{dif_comm}
\mathrm{Supp}(\Theta(J)\varphi) = \Theta_{{\operatorname{trop}}}(J)\mathrm{Supp}(\varphi)$$
Consider a differential monomial $E_M$ as in and $S=(S_1,\ldots,S_n)\in\mathcal{P}(\mathbb{Z}_{\geq0}^m)^n$. We can now define the evaluation of $E_M$ at $S$ as $$\label{expression_evaluation}
E_M(S) = \sum_{\substack{1\leq i\leq n\\ ||J||_\infty\leq r}} M_{i,J}\Theta_{{\operatorname{trop}}}(J)S_i\in\mathcal{P}(\mathbb{Z}_{\geq0}^m).$$
\[lem:monomial equalitymod\] Given $\varphi=(\varphi_1,\ldots,\varphi_n) \in R_m^n$ and a differential monomial $E_M$, we have ${\operatorname{trop}}(E_M(\varphi))={\operatorname{Vert}}(E_M({\operatorname{Supp}}(\varphi)))$
By applying ${\operatorname{Vert}}$ to equation , we have $$\label{eq-help1}
{\operatorname{trop}}(\Theta(J)\varphi_i) = {\operatorname{Vert}}(\Theta_{{\operatorname{trop}}}(J)\mathrm{Supp}(\varphi_i)).$$ Using the multiplicativity of trop, equation and \[Lemma\_linearity\], we obtain $$\begin{aligned}
{\operatorname{trop}}(E_M(\varphi))&=\bigodot_{i,J}{\operatorname{trop}}(\Theta(J)\varphi_i)^{\odot M_{i,J}}\\ &=\bigodot_{i,J}{\operatorname{Vert}}(\Theta_{{\operatorname{trop}}}(J)\mathrm{Supp}(\varphi_i))^{\odot M_{i,J}}\\ &={\operatorname{Vert}}(E_M(\text{Supp}(\varphi))).\qedhere\end{aligned}$$
If $P=\sum_M\alpha_ME_M \in R_{m,n}$ and $\varphi=(\varphi_1,\ldots,\varphi_n)\in R_m^n$, then we can consider the upper support $US(P,\varphi)$ of $P$ at $\varphi$ : $$US(P,\varphi)=\bigcup_M\left({\operatorname{Supp}}(\alpha_M)+{\operatorname{Supp}}(E_M(\varphi)) \right)\in\mathcal{P}(\mathbb{Z}_{\geq0}^m).$$
We now compute the vertex set of $US(P,\varphi)$ by applying the operation ${\operatorname{Vert}}$ and \[Lemma\_linearity\] to the above expression to find $$\begin{aligned}
{\operatorname{Vert}}\Bigl(&\bigcup_M(\text{Supp}(\alpha_M)+\text{Supp}(E_M(\varphi)) \Bigr) =\bigoplus_M{\operatorname{Vert}}(\text{Supp}(\alpha_M)+\text{Supp}(E_M(\varphi)) \\
&=\bigoplus_M{\operatorname{trop}}(\alpha_M)\odot{\operatorname{trop}}(E_M(\varphi))=\bigoplus_M{\operatorname{trop}}(\alpha_M)\odot{\operatorname{Vert}}(E_M(\text{Supp}(\varphi))),
\end{aligned}$$ since ${\operatorname{trop}}(E_M(\varphi))={\operatorname{Vert}}(E_M({\operatorname{Supp}}(\varphi)))$ by Lemma \[lem:monomial equalitymod\]. This motivates the definition of tropical differential polynomials in the next section.
If $P=\sum_M\alpha_ME_M \in R_{m,n}$ and $\varphi=(\varphi_1,\ldots,\varphi_n)\in R_m^n$, then we can consider the upper support $US(P,\varphi)$ of $P$ at $\varphi$ as $$US(P,\varphi)=\bigcup_M\left({\operatorname{Supp}}(\alpha_M)+{\operatorname{Supp}}(E_M(\varphi)) \right)\in\mathcal{P}(\mathbb{Z}_{\geq0}^m).$$
We now compute the vertex set of $US(P,\varphi)$ by applying the operation ${\operatorname{Vert}}$ and \[Lemma\_linearity\] to the above expression to find $$\begin{aligned}
{\operatorname{Vert}}\Bigl(&US(P,\varphi) \Bigr)=\bigoplus_M{\operatorname{trop}}(\alpha_M)\odot{\operatorname{trop}}(E_M(\varphi))\\ &=\bigoplus_M{\operatorname{trop}}(\alpha_M)\odot{\operatorname{Vert}}(E_M(\text{Supp}(\varphi))),
\end{aligned}$$ since ${\operatorname{trop}}(E_M(\varphi))={\operatorname{Vert}}(E_M({\operatorname{Supp}}(\varphi)))$ by Lemma \[lem:monomial equalitymod\]. This motivates the definition of tropical differential polynomials in the next section.
Tropical differential polynomials {#S_TDP}
=================================
In this section we define the set of tropical differential polynomials $\mathbb{T}_{m,n}$ and the corresponding tropicalization morphism ${\operatorname{trop}}\colon R_{m,n}\to\mathbb{T}_{m,n}$. Let us remark that in the case of $m=1$ the definitions and properties presented here coincide with the corresponding ones in [@AGT16]. Moreover, later in Section \[Section\_ExLim\] we illustrate in \[example:NewtonPolygon\] the reason for the particular definitions given here.
\[def:definition ValS\] For a set $S \in \mathcal{P}(\mathbb{Z}^m_{\geq0})$ and a multi-index $J \in \mathbb{Z}^m_{\geq0}$ we define $$\mathrm{Val}_J(S) = {\operatorname{Vert}}(\Theta_{{\operatorname{trop}}}(J)S).$$
Note that for $\varphi \in R_m$ and any multi-index $J$ this means that $${\operatorname{Val}}_J({\operatorname{Supp}}(\varphi)) = {\operatorname{trop}}(\Theta(J)\varphi).$$ In particular, $\mathrm{Val}_J(S) = \emptyset$ if and only if $\Theta(J)\varphi = 0$. It follows from \[Lemma\_linearity\] that $${\operatorname{Vert}}(E_M(S))=\bigodot_{\substack{1\leq i\leq n\\ ||J||_\infty\leq r}} \mathrm{Val}_J(S_i)^{\odot M_{i,J}}.$$
A **tropical differential monomial** in the variables $x_1, \ldots, x_n$ of order less or equal to $r$ is an expression of the form $$\epsilon_M=\bigodot_{\substack{1\leq i\leq n\\ ||J||_\infty\leq r}}x_{i,J}^{\odot M_{i,J}}$$ where $M=(M_{i,J}) \in (\mathbb{Z}_{\ge0})^{n\times(r+1)^m}$.
A tropical differential monomial $\epsilon_M$ induces an evaluation map from $\mathcal{P}(\mathbb{Z}^m_{\geq0})^n$ to $\mathbb{T}[[t_1,\ldots,t_m]]$ by $$\epsilon_M(S_1,\ldots,S_n) ={\operatorname{Vert}}(E_M(S ))= \bigodot_{i,J} \mathrm{Val}_J(S_i)^{\odot M_{i,J}}$$ where $\mathrm{Val}_J(S_i)$ is given in \[def:definition ValS\] and $E_M(S)$ as in . Let us recall that, by \[lemma:commutative semiring\], we can also write $$\epsilon_M(S_1, \ldots, S_n)= \mathrm{Vert}\biggl(\sum_{i,J}\mathrm{Val}_J(S_i)^{\odot M_{i,J}}
\biggr).$$
\[def:tropical diff poly\] A **tropical differential polynomial** in the variables $x_1, \ldots, x_n$ of order less or equal to $r$ is an expression of the form $$p = p(x_1, \ldots, x_n) = \bigoplus_{M \in \Delta } a_M \odot \epsilon_M$$ where $a_M \in \mathbb{T}[[t_1,\ldots,t_m]], a_M \neq \emptyset$ and $\Delta$ is a finite subset of $(\mathbb{Z}_{\ge0})^{n\times(r+1)^m}$. We denote by $\mathbb{T}_{m,n}=\mathbb{T}[[t_1,\ldots,t_m]]\{x_1,\ldots,x_n\}$ the set of tropical differential polynomials.
A tropical differential polynomial $p$ as in Definition \[def:tropical diff poly\] induces a mapping from $\mathcal{P}(\mathbb{Z}^m_{\geq0})^n$ to $\mathbb{T}[[t_1,\ldots,t_m]]$ by $$p(S) = \bigoplus_{M \in \Delta } a_M \odot \epsilon_M(S) = \mathrm{Vert}\Bigl( \bigcup_{M \in \Delta} (a_M + \epsilon_M(S))\Bigr)$$ The second equality follows again from \[lemma:commutative semiring\]. A differential polynomial $P \in R_{m,n}$ of order at most $r$ is of the form $$P= \sum_{M\in \Delta} \alpha_M E_M$$ where $\Delta$ is a finite subset of $(\mathbb{Z}_{\ge0})^{n\times(r+1)^m}$, $\alpha_M \in K[[t_1, \ldots, t_m]]$ and $E_M$ is a differential monomial as in . Then the **tropicalization** of $P$ is defined as $$\mathrm{trop}(P) = \bigoplus_{M \in \Delta }{\operatorname{trop}}(\alpha_M) \odot \epsilon_M\in \mathbb{T}_{m,n}$$ where $\epsilon_M$ is the tropical differential monomial corresponding to $E_M$.
Let $G \subseteq R_{m,n}$ be a differential ideal. Its **tropicalization** $\mathrm{trop}(G)$ is the set of tropical differential polynomials $\{\mathrm{trop}(P) \ | \ P \in G\} \subseteq \mathbb{T}_{m,n}$.
\[lem:monomial equality\] Given a differential monomial $E_M$ and $\varphi= (\varphi_1, \ldots, \varphi_n) \in K[[t_1, \ldots, t_m]]^n$, we have that $${\operatorname{trop}}(E_M(\varphi)) = \epsilon_M({\operatorname{Supp}}(\varphi)).$$
Follows from notations and \[lem:monomial equalitymod\].
Consider $P=\sum_M\alpha_ME_M \in R_{m,n}$ and $\varphi=(\varphi_1,\ldots,\varphi_n)\in R_m^n$. Set $p={\operatorname{trop}}(P)$ and $S={\operatorname{Supp}}(\varphi)$, then $p(S)={\operatorname{Vert}}(US(P,\varphi))$.
The following tropical vanishing condition is a natural generalization of the case $m=1$, but now the evaluation $p(S)$ consists of a vertex set instead of a single minimum.
Let $p = \bigoplus_{M \in \Delta } a_M \odot \epsilon_M$ be a tropical differential polynomial. An $n$-tuple $S \in \mathcal{P}(\mathbb{Z}^m_{\geq0})^n$ is said to be a **solution** of $p$ if for every $J \in p(S)$ there exists $M_1, M_2 \in \Delta$ with $M_1 \neq M_2$ such that $J \in a_{M_1} \odot \epsilon_{M_1}(S)$ and $J \in a_{M_2} \odot \epsilon_{M_2}(S)$. Note that in the particular case of $p(S) = \emptyset$, $S$ is a solution of $p$.
For a family of differential polynomials $H \subseteq \mathbb{T}_{m,n}$, $S$ is called a **solution** of $H$ if and only if $S$ is a solution of every tropical polynomial in $H$. The set of solutions of $H$ will be denoted by $\mathrm{Sol}(H)$.
\[easy-direction\] Let $G$ be a differential ideal in the ring of differential polynomials $R_{m,n}$. If $\varphi \in {\operatorname{Sol}}(G)$, then ${\operatorname{Supp}}(\varphi) \in {\operatorname{Sol}}({\operatorname{trop}}(G))$.
Let $\varphi$ be a solution of $G$ and $S = {\operatorname{Supp}}(\varphi)$. Let $P= \sum_{M \in \Delta} \alpha_M E_M \in G$ and $p = {\operatorname{trop}}(P) = \bigoplus_{M \in \Delta}a_M \odot \epsilon_M,$ where $a_M = {\operatorname{trop}}(\alpha_M)$. We need to show that $S$ is a solution of $p$. Let $J \in p(S)$ be arbitrary. By the definition of $\oplus$, there is an index $M_1$ such that $$J \in a_{M_1} \odot \epsilon_{M_1}(S).$$ Hence, by \[lem:monomial equality\], and multiplicative property of trop \[lem:trop\_morphism\_of\_semi\_ring\] $$J \in {\operatorname{Vert}}({\operatorname{Supp}}(\alpha_{M_1} E_{M_1}(\varphi))).$$ Since $P(\varphi) = 0$, there is another index $M_2 \neq M_1$ such that $$J \in {\operatorname{Supp}}(\alpha_{M_2} E_{M_2}(\varphi)),$$ because otherwise there would not be cancellation. Since $J$ is a vertex of $p(S)$, it follows that $J$ is a vertex of every subset of $\mathcal{N}(p(S))$ containing $J$ and in particular of $\mathcal{N}(\mathrm{Supp}(\alpha_{M_2} E_{M_2}(\varphi)))$. Therefore, $$J \in a_{M_2} \odot \epsilon_{M_2}(S)$$ and because $J$ and $P$ were chosen arbitrary, $S$ is a solution of $G$.
The Fundamental Theorem {#S_TFT}
=======================
Let $G \subset R_{m,n}$ be a differential ideal. Then \[easy-direction\] implies that ${\operatorname{Supp}}({\operatorname{Sol}}(G)) \subseteq {\operatorname{Sol}}({\operatorname{trop}}(G))$. The main result of this paper is to show that the reverse inclusion holds as well if the base field $K$ is uncountable.
\[fundamental theorem\] Let $K$ be an uncountable, algebraically closed field of characteristic zero. Let $G$ be a differential ideal in the ring $R_{m,n}$. Then $${\operatorname{Supp}}({\operatorname{Sol}}(G)) = {\operatorname{Sol}}({\operatorname{trop}}(G)).$$
The proof of the Fundamental Theorem will take the rest of the section and is split into several parts. First let us introduce some notations. If $J=(j_1, \ldots, j_m)$ is an element of $\mathbb{Z}^m_{\geq0}$, we define by $J!$ the component-wise product $j_1! \cdots j_m !$. The bijection between $K^{\mathbb{Z}^m_{\geq0}}$ and $R_m$ given by $$\begin{array}{cccc}
\psi \colon & K^{\mathbb{Z}^m_{\geq0}} & \to & R_m \\
& \underline{a}=(a_J)_{J \in \mathbb{Z}^m_{\geq0}} & \mapsto & \displaystyle \sum_{J \in \mathbb{Z}^m_{\geq0}} \frac{1}{J!}a_J t^J
\end{array}$$ allows us to identify points of $R_m$ with points of $K^{\mathbb{Z}^m_{\geq0}}$. Moreover, if $I \in \mathbb{Z}^m_{\geq0}$, the mapping $\psi$ has the following property: $$\Theta(I)\psi(\underline{a}) = \sum_{J \in \mathbb{Z}^m_{\geq0}} \frac{1}{J!}a_{I+J} t^J$$ which implies $$\underline{a} = (\Theta(I)\psi(\underline{a})|_{t=0})_{I \in \mathbb{Z}^m_{\geq0}}.$$
Fix for the rest of the section a finite set of differential polynomials $\Sigma=\{P_1, \ldots, P_s\} \subseteq G$ such that $\Sigma$ has the same solution set as $G$ (this is possible by \[prop\_Ritt\_Raudenbush\]). For all $\ell \in \{1, \ldots, s\}$ and $I \in \mathbb{Z}_{\geq 0}^m$ we define $$F_{\ell, I} = ( \Theta(I)P_\ell)|_{t_1 = \cdots = t_m = 0} \, \in K\big[x_{i, J} : 1 \leq i \leq n, J \in \mathbb{Z}_{\geq 0}^m\big]$$ and $$A_{\infty} = \{(a_{i,J}) \in K^{n \times (\mathbb{Z}_{\geq 0}^m)} : F_{\ell, I}(a_{i,J}) = 0 \textrm{ for all } 1 \leq \ell\leq s, I \in \mathbb{Z}_{\geq 0}^m \}.$$ The set $A_{\infty}$ corresponds to the formal power series solutions of the differential system $\Sigma = 0$ as the following lemma shows.
Let $\varphi \in K[[t_1, \ldots, t_m]]^n$ with $\varphi = (\varphi_1, \ldots, \varphi_n)$, where $$\varphi_i = \sum_{J \in \mathbb{Z}_{\geq 0}^m} \frac{a_{i,J}}{J!} t^J.$$ Then $\varphi$ is a solution of $\Sigma = 0$ if and only if $(a_{i,J}) \in A_\infty$.
This statement follows from formula $$P_\ell(\varphi_1,\ldots, \varphi_n) = \sum_{I \in \mathbb{Z}_{\geq 0}^m} \frac{F_{\ell,I}((a_{i,J})_{i,J})}{I!}t^I,$$ which is often called Taylor formula. See [@Seidenberg58] for more details.
For any $S = (S_1, \ldots, S_n) \in \mathcal{P}(\mathbb{Z}_{\geq 0}^m)^n$ we define $$A_{\infty, S} = \{ (a_{i,J}) \in A_\infty : a_{i,J} = 0 \textrm{ if and only if } J \notin S_i\}.$$ This set corresponds to power series solutions of the system $\Sigma = 0$ which have support exactly $S$. In particular, $S \in {\operatorname{Supp}}({\operatorname{Sol}}(G))$ if and only if $A_{\infty, S} \neq \emptyset$.
The sets $A_{\infty}$ and $A_{\infty, S}$ refer to infinitely many coefficients. We want to work with a finite approximation of these sets. For this purpose, we make the following definitions. For each integer $k \geq 0$, choose $N_k \geq 0$ minimal such that for every $\ell \in \{1, \ldots, s\}$ and $||I||_\infty \leq k$ it holds that $$F_{\ell, I} \in K[x_{i, J} : 1 \leq i \leq n, ||J||_\infty \leq N_k].$$ Note that for $k_1 \leq k_2$ it follows that $N_{k_1} \leq N_{k_2}.$ Then we define $$\begin{aligned}
A_k = \{(a_{i,J}) \in &K^{n \times \{1, \ldots, N_k\}^m} : F_{\ell, I}(a_{i,J}) = 0 \\
&\quad\quad\textrm{ for all } 1 \leq \ell\leq s, ||I||_\infty \leq k \}\end{aligned}$$
and $$A_{k,S} = \{ (a_{i,J}) \in A_k : a_{i,J} = 0 \textrm{ if and only if } J \notin S_i\}.$$
\[finite-approximation\] Let $S \in \mathcal{P}(\mathbb{Z}_{\geq 0}^m)^n$. If $A_{\infty, S} = \emptyset$, then there exists $k \geq 0$ such that $A_{k, S} = \emptyset$.
Assume that $A_{k,S} \neq \emptyset$ for every $k \geq 0$; we show that this implies $A_{\infty, S} \neq \emptyset$. We follow the strategy of the proof of [@DenefLipshitz Theorem 2.10]: first we use the ultrapower construction to construct a larger field $\mathbb{K}$ over which a power series solution with support $S$ exists, and then we show that this implies the existence of a solution with the same support and with coefficients in $K$. For more information on ultrafilters and ultraproducts, the reader may consult [@BDLD79].
For each integer $k \geq 0$, choose an element $(a^{(k)}_{i,J})_{1 \leq i \leq n, ||J||_{\infty} \leq N_k} \in A_{k,S}$. Fix a non-principal ultrafilter $\mathcal{U}$ on the natural numbers $\mathbb{N}$ and consider the ultrapower $\mathbb{K}$ of $K$ along $\mathcal{U}$. In other words, $\mathbb{K} = (\prod_{r \in \mathbb{N}} K) / \sim$ where $x \sim y$ for $x = (x_r)_{r \in \mathbb{N}}$ and $y = (y_r)_{r \in \mathbb{N}}$ if and only if the set $\{r \in \mathbb{N} : x_r = y_r\}$ is in $\mathcal{U}$. We will denote the equivalence class of a sequence $(x_r)$ by $[(x_r)]$. We consider $\mathbb{K}$ as a $K$-algebra via the diagonal map $K \to \mathbb{K}$. Now for each $i$ and $J$, we may define $a_{i,J} \in \mathbb{K}$ as $$a_{i,J} = [ (a_{i,J}^{(k)} : k \in \mathbb{N}) ]$$ where we set $a_{i,J}^{(k)} = 0$ for the finitely many values of $k$ with $||J||_{\infty} > N_k$. For all $\ell$ and $I$, we have that $F_{\ell, I}((a^{(k)}_{i,J})_{i,J}) = 0$ for $k$ large enough, and so $F_{\ell, I}((a_{i,J})_{i,J}) = 0$ in $\mathbb{K}$, because the set of $k$ such that $F_{\ell, I}((a^{(k)}_{i,J})_{i,J}) \neq 0$ is finite. Moreover, for $J \in S_i$ we have, by hypothesis, $a_{i,J}^{(k)} \neq 0$ for all sufficiently large $k$, so $a_{i,J} \neq 0$ in $\mathbb{K}$. On the other hand, for $J \notin S_i$ we have $a_{i,J}^{(k)} = 0$ for all $k$, so also $a_{i,J} = 0$. Now consider the ring $$R = K\left[\begin{array}{l}
x_{i,J}: 1 \leq i \leq n, J \in \mathbb{Z}^m_{\geq 0}\\
x_{i,J}^{-1} : 1 \leq i \leq n, J \in S_i
\end{array}\right] / \left(\begin{array}{l}
F_{\ell, I} : 1 \leq \ell \leq s, I \in \mathbb{Z}_{\geq 0}^m\\
x_{i,J}: 1 \leq i \leq n, J \notin S_i
\end{array}\right)$$ The paragraph above shows that the map $R \to \mathbb{K}$ defined by sending $x_{i,J}$ to $a_{i,J}$ is a well-defined ring map. In particular, $R$ is not the zero ring. Let $\mathfrak{m}$ be a maximal ideal of $R$. We claim that the composition $K \to R \to R/\mathfrak{m}$ is an isomorphism. Indeed, $R/\mathfrak{m}$ is a field, and as a $K$-algebra it is countably generated, since $R$ is. Therefore, it is of countable dimension as $K$-vector space (it is generated as $K$-vector space by the products of some set of generators as a $K$-algebra). If $t \in R/\mathfrak{m}$ were transcendental over $K$, then by the theory of partial fraction decomposition, the elements $1/(t - \alpha)$ for $\alpha \in K$ would form an uncountable, $K$-linearly independent subset of $R/\mathfrak{m}$. This is not possible, so $R/\mathfrak{m}$ is algebraic over $K$. Since $K$ is algebraically closed, we conclude that $K = R/\mathfrak{m}$. Now let $b_{i,J} \in K$ be the image of $x_{i,J}$ in $R/\mathfrak{m}=K$. Then by construction, the set $(b_{i,J})$ satisfies the conditions $F_{\ell,I}((b_{i,J})) = 0$ for all $\ell$ and $I$, and $b_{i, J} = 0$ if and only if $J \notin S_i$. So $(b_{i,J})$ is an element of $A_{\infty, S}$, and in particular $A_{\infty, S} \neq \emptyset$.
We now prove the remaining direction of the Fundamental Theorem by contraposition. Let $S = (S_1, \ldots, S_n)$ in $\mathcal{P}(\mathbb{Z}_{\geq 0}^m)^n$ be such that $A_{\infty, S} = \emptyset$, i.e. there is no power series solution of $\Sigma = 0$ in $K[[t_1, \ldots, t_m]]^n$ with $S$ as the support. Then by \[finite-approximation\] there exists $k \geq 0$ such that $A_{k,S} = \emptyset$. Equivalently, $$V\left(\begin{array}{l}
F_{\ell, I} : 1 \leq \ell \leq s, ||I||_{\infty} \leq N_k \\
x_{i,J} : 1 \leq i \leq n, J \notin S_i, ||J||_{\infty} \leq k
\end{array}\right) \subseteq V\Biggl(\prod_{\substack{1 \leq i \leq n\\ J \in S_i\\ ||J||_{\infty} \leq N_k}} x_{i,J} \Biggr).$$ By Hilbert’s Nullstellensatz, there is an integer $M \geq 1$ such that $$E := \bigg(\prod_{\substack{1 \leq i \leq n\\ J \in S_i\\||J||_{\infty} \leq N_k}} x_{i,J}\bigg)^M \in \left\langle\begin{array}{l}
F_{\ell, I} : 1 \leq \ell \leq s, ||I||_{\infty} \leq N_k \\
x_{i,J} : 1 \leq i \leq n, J \notin S_i, ||J||_{\infty} \leq k
\end{array}\right\rangle.$$ Therefore, there exist $G_{\ell, I}$ and $H_{i,J}$ in $K[x_{i,J} : 1 \leq i \leq n, ||J||_{\infty} \leq N_k]$ such that $$E = \sum_{\substack{1 \leq \ell \leq s\\ ||I||_{\infty} \leq k}} G_{\ell, I}F_{\ell, I} + \sum_{\substack{1 \leq i \leq n\\ J \notin S_i \\ ||J||_{\infty} \leq N_k}} H_{i,J} x_{i,J}.$$ Define the differential polynomial $P$ by $$P = \sum_{\substack{1 \leq \ell \leq s\\ ||I||_{\infty} \leq k}} G_{\ell, I}\Theta(I)(P_\ell).$$ Then $P$ is an element of the differential ideal generated by $P_1, \ldots, P_s$, so in particular $P \in G$. Since $F_{\ell, I} = \Theta(I)(P_\ell) |_{t = 0}$, there exist $h_i \in R_{m,n}$ such that $$P = E - \sum_{\substack{1 \leq i \leq n\\ J \notin S_i \\ ||J||_{\infty} \leq N_k}} H_{i,J} x_{i,J} \,+\, t_1 h_1 + \ldots + t_m h_m.$$ Notice that $E$ occurs as a monomial in $P$, since it cannot cancel with other terms in the sum above. By construction we have ${\operatorname{trop}}(E)(S) = \{(0,\ldots, 0)\}$. However, we have $(0, \ldots, 0) \notin {\operatorname{trop}}(H_{i,J} x_{i,J})(S)$ because $J \notin S_i$, and we have $(0, \ldots, 0) \notin {\operatorname{trop}}(t_i h_i)(S)$ because the factor $t_i$ forces the $i$th coefficient of each element of ${\operatorname{trop}}(t_i h_i)(S)$ to be at least $1$. Hence, the vertex $\{(0,\ldots, 0)\}$ in ${\operatorname{trop}}(P)(S)$ is attained exactly once, in the monomial $E$, and therefore, $S$ is not a solution of ${\operatorname{trop}}(P)$. Since $P \in G$, it follows that $S \notin {\operatorname{Sol}}({\operatorname{trop}}(G))$, which proves the statement.
Examples and remarks on the Fundamental Theorem {#Section_ExLim}
===============================================
In this section we give some examples to illustrate the results obtained in the previous sections. Moreover, we show that some straight-forward generalizations of the Fundamental Theorem from [@AGT16] and our version, Theorem \[fundamental theorem\], do not hold. Also we give more directions for further developments.
Let us consider the system of differential polynomials $$\begin{aligned}
\Sigma=\{&P_1=x_{1,(1,0)}^2 - 4\,x_{1,(0,0)}\,,\
P_2=x_{1,(1,1)}\,x_{2,(0,1)} - x_{1,(0,0)} + 1\,,\\
&P_3=x_{2,(2,0)} - x_{1,(1,0)}\}\,\end{aligned}$$ in $R_{2,2}$. By means of elimination methods in differential algebra such as the ones implemented in the MAPLE `DifferentialAlgebra` package, it can be proven that $$\begin{aligned}
{\operatorname{Sol}}(\Sigma)=\{
&\varphi_1(t_1,t_2)=2\,c_0\,t_1+c_0^2+\sqrt{2}\,c_0\,t_2+t_1^2+\sqrt{2}\,t_1\,t_2+\frac{1}{2}\,t_2^2,\\
&\varphi_2(t_1,t_2)=c_2\,t_1+c_1+\frac{1}{2}\,\sqrt{2}\,(c_0^2-1)\,t_2+c_0\,t_1^2 \\
& \quad +\sqrt{2}\,c_0\,t_1\,t_2 +\frac{1}{2}\,c_0\,t_2^2 \\
& \quad +\frac{1}{3}\,t_1^3+\frac{1}{2}\,\sqrt{2}\,t_1^2\,t_2+\frac{1}{2}\,t_1\,t_2^2+\frac{1}{12}\,\sqrt{2}\,t_2^3 \},\end{aligned}$$ where $c_0,c_1,c_2 \in K$ are arbitrary constants. By setting $c_0=c_2=0,c_1 \neq 0$, we obtain for example that $$(\{(2,0),(1,1),(0,2)\},\{(0,0),(0,1),(3,0),(2,1),(1,1),(0,3)\})$$ is in ${\operatorname{Supp}}({\operatorname{Sol}}(\Sigma))$.
Now we illustrate that by our results necessary conditions and relations on the support can be found. Let $(S_1,S_2) \in \mathcal{P}(\mathbb{Z}_{\ge 0}^2)^2$ be a solution of ${\operatorname{trop}}([\Sigma])$. Let us first consider $${\operatorname{trop}}(P_1)(S_1,S_2)={\operatorname{Vert}}(2 \cdot \Theta_{{\operatorname{trop}}}(1,0)S_1 \, \cup \, S_1).$$ If we assume that $(0,0) \in S_1$, then $(0,0)$ is a vertex of $S_1$. By the definition of a solution of a tropical differential polynomial, $(0,0)$ must be a vertex of the term $2\cdot \Theta_{{\operatorname{trop}}}(1,0)S_1$ as well, so we then know that $(1,0) \in S_1$. Conversely, if $(1,0) \in S_1$, then $(0,0) \in S_1$ follows. This is what we expect since the corresponding monomials in $\varphi_1$ vanish if and only if $c_0=0$.\
Now consider $$\begin{aligned}
&{\operatorname{trop}}(\Theta(1,0)P_1)(S_1,S_2)=\\
&{\operatorname{Vert}}(\Theta_{{\operatorname{trop}}}(1,0)S_1+\Theta_{{\operatorname{trop}}}(2,0)S_1\,\cup\,\Theta_{{\operatorname{trop}}}(1,0)S_1).
\end{aligned}$$ If we assume that $(0,0)$ is not a vertex of this expression, which implies that $(1,0) \notin S_1$, and $(k,0)$ is a vertex for some $k \ge 1$, then we obtain from the two tropical differential monomials that necessarily $(k,0)=(2k-1,0)$. This is fulfilled only for $k=1$ and hence, $(2,0) \in S_1$.
A natural way for defining $\odot$ and $\oplus$ in Section \[sec:the tropical semi-ring\] would be to simply take the Newton polygon and not take its vertex set, as we do. If we do this, then some intermediate results and in particular Proposition \[easy-direction\], do not hold anymore as the following example shows.
\[example:NewtonPolygon\] Let $\{e_1,\ldots,e_4\}$ be the standard basis for $\mathbb{Z}^4_{\geq0}$. We consider the differential ideal in $R_{4,1}=K[[t_1,\ldots,t_4]]\{x\}$ generated by $$P = x_{e_3}x_{e_4} + (-t_1^2 + t_2^2) x_{e_1+e_3} = \frac{\partial x}{\partial t_3}\cdot \frac{\partial x}{\partial t_4} + (-t_1^2 + t_2^2) \frac{\partial^2 x}{\partial t_1 \partial t_3}$$ and the solution $\varphi = (t_1 + t_2)t_3 + (t_1 - t_2)t_4$. Then $${\operatorname{Supp}}(\varphi)=\{e_1+e_3, e_2+e_3, e_1+e_4, e_2+e_4\}.$$ On the other hand, for $S \in \mathcal{P}(\mathbb{Z}^4_{\geq0})$ we obtain $$\begin{aligned}
{\operatorname{trop}}(P)(S)={\operatorname{Vert}}(&{\operatorname{Vert}}(\Theta_{{\operatorname{trop}}}(e_3)S+\Theta_{{\operatorname{trop}}}(e_4)S) \\ &\cup \mathrm{Vert}(2e_1+\Theta_{{\operatorname{trop}}}(e_1+e_3)S) \\ &\cup \mathrm{Vert}(2e_2+\Theta_{{\operatorname{trop}}}(e_1+e_3)S).\end{aligned}$$
If we set $S={\operatorname{Supp}}(\varphi)$, we obtain $$\begin{aligned}
{\operatorname{trop}}(P)(S)={\operatorname{Vert}}(&{\operatorname{Vert}}(\{2e_1,e_1+e_2,2e_2\})\cup \{2e_1\} \, \cup \{2e_2\}).\end{aligned}$$ Since $${\operatorname{Vert}}(\{2e_1,e_1+e_2,2e_2\})=\{2e_1,2e_2\},$$ every $J \in {\operatorname{trop}}(P)(S)$, namely $2e_1$ and $2e_2$, occurs in three monomials in ${\operatorname{trop}}(P)(S)$ and $S$ is indeed in ${\operatorname{Sol}}({\operatorname{trop}}(P))$. Note that in the Newton polygon the point $e_1+e_2$, which is not a vertex, comes from only one monomial in ${\operatorname{trop}}(P)(S)$. Therefore, it is necessary to consider the vertices instead of the whole Newton polygon such that for instance Proposition \[easy-direction\] holds.
\[example:CountableField\] The Fundamental Theorem for systems of partial differential equations over a countable field such as $\overline{\mathbb{Q}}$ does in general not hold anymore by the following reasoning. According to [@DenefLipshitz Corollary 4.7], there is a system of partial differential equations $G$ over $\mathbb{Q}$ having a solution in $\mathbb{C}[[t_1, \ldots, t_m]]$ but no solution in $\overline{\mathbb{Q}}[[t_1, \ldots, t_m]]$. Taking $K = \overline{\mathbb{Q}}$ as base field, we have ${\operatorname{Sol}}({\operatorname{trop}}(G)) \neq \emptyset$ because ${\operatorname{Sol}}({\operatorname{trop}}(G))={\operatorname{Supp}}({\operatorname{Sol}}(G))$ is non-empty in $\mathbb{C}$, but ${\operatorname{Supp}}({\operatorname{Sol}}(G)) = \emptyset$.
In this paper we focus on formal power series solutions. A natural extension would be to consider formal Puiseux series instead. The following example shows that with the natural extension of our definitions to Puiseux series, the fundamental theorem does not hold, even for $m = n = 1$.
\[example:Puiseux\] Let us consider $R_{1,1}=K[t]\{x\}$ and the differential ideal generated by the differential polynomial $$P=2tx_{(1)}-x_{(0)}=2t \cdot \frac{\partial x}{\partial t}-x.$$ There is no non-zero formal power series solution $\varphi$ of $P=0$, but $\varphi=ct^{1/2}$ is for any $c \in K$ a solution. In fact, $\{\varphi\}$ is the set of all formal Puiseux series solutions.
On the other hand, let $S \in \mathcal{P}(\mathbb{Z}_{\ge0})$. Then every point $J$ in $$\begin{aligned}
{\operatorname{trop}}(P)(S)&={\operatorname{Vert}}({\operatorname{Vert}}(\{1\}+(\Theta_{{\operatorname{trop}}}(1)S) \cup {\operatorname{Vert}}(S))\end{aligned}$$ occurs in both monomials except if $0 \in J$. Hence, for every $S \in {\operatorname{Sol}}({\operatorname{trop}}(P))$ we know that $0 \notin S$. For every $I \ge 0$ we have that $$\Theta(I)P=2tx_{(I+1)}+(2I-1)x_{(I)} \in G$$ and $$\begin{aligned}
{\operatorname{trop}}(\Theta(I)P)(S)&={\operatorname{Vert}}({\operatorname{Vert}}(\{(1)\}+(\Theta_{{\operatorname{trop}}}((I+1)S) \cup {\operatorname{Vert}}(\Theta_{{\operatorname{trop}}}(I)S)).\end{aligned}$$ Similarly to above, every $J \in {\operatorname{trop}}(\Theta(I)P)(S)$ occurs in both monomials except if $I \in J$. Therefore, $I \notin S$ and so the only $S \in \mathcal{P}(\mathbb{Z}_{\ge0})$ with $S \in {\operatorname{Sol}}({\operatorname{trop}}([P]))$ is $S = \emptyset$. Hence, ${\operatorname{Sol}}({\operatorname{trop}}([P]))=\{\emptyset\}={\operatorname{Supp}}({\operatorname{Sol}}([P])).$
Now we want to consider formal Puiseux series solutions instead of formal power series solutions. First, $\{{\operatorname{Supp}}(\varphi)\}=\{\emptyset,\{1/2\}\}$. Now let us set for $S \in \mathbb{Q}^m$ and $J=(j_1,\ldots,j_m) \in \mathbb{Z}_{\ge 0}^m$, the set $\Theta_{{\operatorname{trop}}}(J)S$ defined as $$\left\{ (s_1-j_1, \ldots, s_m -j_m) \ \left| \
\begin{array}{cccc} (s_1, \ldots, s_m) \in S, \\
\forall 1 \le i \le m, \ s_i<0 \text{ or } s_i-j_i \notin \mathbb{Z}_{<0}
\end{array}
\right. \right\}$$ This is the natural definition, since only in the case when the exponent of a monomial is a non-negative integer, the derivative can be equal to zero. We have that $\Theta_{{\operatorname{trop}}}(J)({\operatorname{Supp}}(\psi)) = {\operatorname{Supp}}(\Theta(J)\psi)$ for all Puiseux series $\psi$. For ${\operatorname{Val}}_J$ and the operations $\odot$ and $\oplus$ the definitions remain unchanged.
Let $Q \in [P]$. Then $$Q=\sum_{k \in \mathcal{I}}Q_k \cdot \Theta(I_k)P$$ for some index-set $\mathcal{I}$ and $Q_k \in R_{m,n}$. For every $I_k$ we know that ${\operatorname{Supp}}(\varphi)=\{(1/2)\} \in {\operatorname{Sol}}({\operatorname{trop}}(\Theta(I_k)P))$. Let $\alpha \in \mathbb{Q} \cap (0,1)$. Then for every $J \in {\operatorname{trop}}(\Theta(I_k)P) \in \mathbb{Z}_{\ge 0}$ we have that $\Theta_{{\operatorname{trop}}}(J)\{(1/2)\}=\Theta_{{\operatorname{trop}}}(J)\{(\alpha)\}+\{(1/2-\alpha)\}$. Thus, $\{\alpha\} \in {\operatorname{Sol}}({\operatorname{trop}}(\Theta(I_k)P))$. Since $${\operatorname{trop}}(G_k \cdot \Theta(I_k)P)={\operatorname{trop}}(G_k)\odot {\operatorname{trop}}(\Theta(I_k)P),$$ the solvability remains by multiplication with $G_k$. Therefore, $\{\alpha\} \in {\operatorname{Sol}}({\operatorname{trop}}(G_k \cdot \Theta(I_k)P))$ and consequently, $\{\alpha\} \in {\operatorname{Sol}}({\operatorname{trop}}([P]))$. However, $\{\alpha\} \notin {\operatorname{Supp}}({\operatorname{Sol}}([P]))$ for $\alpha \neq 1/2$.
We remark that $P$ is an ordinary differential polynomial and by similar computations as here, the straight-forward generalization from formal power series to formal Puiseux series fails for the Fundamental Theorem in [@AGT16] as well.
We conclude this section by emphasizing that the Fundamental Theorem may help to find necessary conditions on the support of solutions of systems of partial differential equations, but in general it cannot be completely algorithmic. In fact, according to [@DenefLipshitz Theorem 4.11], already determining the existence of a formal power series solution of a linear system with formal power series coefficients is in general undecidable.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work was started during the Tropical Differential Algebra workshop, which took place on December 2019 at Queen Mary University of London. We thank the organizers and participants for valuable discussions and initiating this collaboration. In particular, we want to thank Fuensanta Aroca, Alex Fink, Jeffrey Giansiracusa and Dima Grigoriev for their helpful comments during this week.
<span style="font-variant:small-caps;">Research Institute for Symbolic Computation (RISC)\
Johannes Kepler University Linz, Austria.</span>
Web: [risc.jku.at/m/sebastian-falkensteiner](risc.jku.at/m/sebastian-falkensteiner)\
<span style="font-variant:small-caps;">Centro de Investigación en Matemáticas, A.C. (CIMAT)\
Jalisco S/N, Col. Valenciana CP. 36023 Guanajuato, Gto, México.</span>\
e-mail: `[email protected]`
<span style="font-variant:small-caps;">Institut de recherche mathématique de Rennes, UMR 6625 du CNRS\
Université de Rennes 1, Campus de Beaulieu\
35042 Rennes cedex (France)</span>\
e-mail: `[email protected]`
<span style="font-variant:small-caps;">Bernoulli Institute, University of Groningen, The Netherlands</span>\
Web: <https://www.rug.nl/staff/m.p.noordman/>
<span style="font-variant:small-caps;">School of Mathematical Sciences, Queen Mary University of London</span>\
e-mail: `[email protected]`
<span style="font-variant:small-caps;">Univ. Lille, CNRS, Centrale Lille, Inria, UMR 9189 - CRIStAL - Centre de Recherche en Informatique Signalet Automatique de Lille, F-59000 Lille, France</span>\
Web: [pro.univ-lille.fr/francois-boulier](pro.univ-lille.fr/francois-boulier)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We explore the topological properties of double-Weyl semimetals with cold atoms in optical lattices. We first propose to realize a tight-binding model of simulating the double-Weyl semimetal with a pair of double-Weyl points by engineering the atomic hopping in a three-dimensional optical lattice. We show that the double-Weyl points with topological charges of $\pm2$ behave as sink and source of Berry flux in momentum space connecting by two Fermi arcs and they are stabilized by the $C_{4h}$ point-group symmetry. By applying a realizable $C_4$ breaking term, we find that each double-Weyl point splits into two single-Weyl points and obtain rich phase diagrams in the parameter space spanned by the strengths of an effective Zeeman term and the $C_4$ breaking term, which contains a topological and a normal insulating phases and two topological Weyl semimetal phases with eight and four single-Weyl points, apart from the double-Weyl semimetal phase. Furthermore, we demonstrate with numerical simulations that (i) the mimicked double- and single-Weyl points can be detected by measuring the atomic transfer fractions after a Bloch oscillation; (ii) the Chern number of different quantum phases in the phase diagram can be extracted from the center shift of the hybrid Wannier functions, which can be directly measured with the time-of-flight imaging; (iii) the band topology of the $C_4$-symmetric Bloch Hamiltonian can be detected simply from measuring the spin polarization at the high symmetry momentum points with a condensate in the optical lattice. The proposed system would provide a promising platform for elaborating the intrinsic exotic physics of double-Weyl semimetals and the related topological phase transitions.'
author:
- 'Xue-Ying Mai'
- 'Dan-Wei Zhang'
- Zhi Li
- 'Shi-Liang Zhu'
title: 'Exploring topological double-Weyl semimetals with cold atoms in optical lattices'
---
introduction
============
Recently, topological Weyl semimetals have attracted a broad interest due to their wide range of exotic properties that are distinct from those of topological insulators [@Kane; @Qi; @Balents; @Burkov; @Delplace; @Zhao; @Taas; @WeylTheo1; @SMExp1; @SMExp2; @Zyuzin; @Parameswaran]. Most importantly, the long-sought Weyl fermions, which are massless spin-1/2 particles in quantum field theory but have never been observed as fundamental particles in nature, can emerge as gapless quasiparticle excitations near band touching points dubbed as Weyl points in Weyl semimetals. The topological nature of the Weyl points in three-dimensional momentum space supports the existence of nontrivial Fermi arc surface states. The Weyl fermions in the bulk and the Fermi arc states in the surfaces are expected to give rise to exotic phenomena in Weyl semimetals, such as anomalous electromagnetic responses [@Zhao; @Zyuzin; @Parameswaran]. A significant advance has been theoretically and experimentally made for exploring Weyl physics not only in real materials [@WeylTheo1; @SMExp1; @SMExp2], but also in some artificial systems, such as photonic and acoustic crystals [@Lu2013; @Lu2015; @Xiao2015; @Chen2016]. Interestingly, a new three-dimensional topological semimetal state, dubbed double-Weyl semimetal, has been theoretically proposed in solids that possess certain point-group symmetries [@Multi; @Srsi; @Xu; @Shivamoggi; @Jian; @Detect; @Thermo; @Lepori]. The standard Weyl semimetal has linear dispersion in all the three momentum directions near the single-Weyl points with topological charge of $\pm1$. In contract, the dispersion of a double-Weyl semimetal is quadratic in two dimensions and linear in the third dimension near the gapless points with topological charge of $\pm2$, which are thus named as double-Weyl points. Several materials have been proposed to be potential candidates of double-Weyl semimetals [@Multi; @Srsi; @Xu; @Shivamoggi; @Jian; @Detect; @Thermo; @Lepori], and the double-Weyl points have recently been observed in photonic crystals [@Chen2016]. Some important properties of this newly predicted topological state are rarely explored, such as the symmetry-breaking effects and the topological phase transition. Thus, other experimentally tunable systems for exploring the exotic topological properties double-Weyl semimetals are highly desirable.
On the other hand, ultracold atoms in optical lattices play an important role in advancing our understanding of condensed matter physics [@ColdAtom1]. Remarkably, as recent experimental advances in realizing spin-orbit coupling and artificial gauge field for neutral atoms [@GaugeRMP; @GaugeRPP; @SOC-Review], these systems provide a powerful platform with unparalleled controllability towards studying topological states of matter. For instance, the celebrated Harper-Hofstadter [@HHModel] model and Haldane model [@HaldaneModel] have been realized [@Miyake; @Bloch2013a; @Jotzu; @Bloch2015; @Shao] experimentally in optical lattices, where the Chern number has also been successfully probed. The chiral edge states have been experimentally observed in one-dimensional optical lattices subjected to a synthetic magnetic field and an artificial dimension [@Cold-Edge1; @Cold-Edge2]. The topological (geometric) pumping has been demonstrated with cold atoms in optical superlattices [@Pumping1; @Pumping2; @Pumping3]. The two-dimensional spin-orbit coupling for Bose-Einstein condensates has been realized in optical lattices and the band topology has also been measured [@2DSOC]. Then an important question is raised: can we realize other predicted topological phases that are rare in solid-state materials in the cold atom systems? Several proposals have been suggested to realize exotic topological insulting states [@Liu2013; @Liu2014; @Duan2014; @Osterloh; @Mazza; @Goldman] and topological nodal semimetals with single-Weyl points or nodal loops [@Jiang; @Dubcek; @ZDW2015; @He; @Xu1; @ZDW2016; @Xu2; @Shastri] in optical lattices. Notably, it was proposed to simulate the double-Weyl semimetals with ultracold atoms in optical lattices in the presence of synthetic non-Abelian SU(2) gauge potentials [@Lepori]. Other feasible schemes for mimicking tunable double-Weyl semimetal states and detecting their intrinsic topological properties in cold atomic systems are still badly awaited.
In this paper, we explore the topological double-Weyl semimetals with cold atoms in optical lattices. We first propose to realize a tight-binding model of simulating the double-Weyl semimetal with tunable double-Weyl points by engineering the atomic hopping in a three-dimensional cubic optical lattice. We show that a pair of double-Weyl points with nontrivial monopole charges behave as sink and source of Berry fluxes in momentum space and they are stabilized by the $C_{4h}$ point-group symmetry. We further investigate the topological properties of the double-Weyl semimetal by calculating $k_z$-dependent Chern number and the gapless edge states. By applying a realizable $C_4$ breaking term, we find that each double-Weyl point splits into two single-Weyl points and obtain a rich phase diagram in the parameter space spanned by the strengths of an effective Zeeman potential and the $C_4$ breaking term, which contains a topological insulator phase, a normal band insulator phase, and two topological Weyl semimetal phases with eight and four single-Weyl points apart from the double-Weyl semimetal phase. Finally, we demonstrate with numerical simulations that (i) the analogous double- and single-Weyl points can be detected by measuring the atomic transfer fractions after a Bloch oscillation; (ii) the $k_z$-dependent Chern number of different quantum phases in the phase diagram can be extracted from the center shift of the hybrid Wannier functions, which are based on time-of-flight imaging; (iii) the band topology of the $C_4$-symmetric Bloch Hamiltonian can be detected simply from measuring the spin polarization at the high symmetry momentum points with a condensate in the optical lattice. The proposed cold-atom system provides a promising platform for elaborating the intrinsic exotic physics of double-Weyl semimetals and the related topological phase transitions.
The paper is organized as follows. Section II introduces the tight-binding model and optical-lattice system for realizing double-Weyl semimetals with double-Weyl points. In Section III, with the numerical calculation of the Chern number and the chiral edge states, we elaborate the topological properties of the simulated double-Weyl semimetals and obtain a rich phase diagram containing other topological quantum phases by applying a symmetry breaking term. In Section IV, we propose realistic schemes to detect the simulated Weyl points and the characteristic topological invariant with cold atoms in the optical lattice. Finally, a short conclusion is given in Sec.V.
model and system
================
We consider a non-interacting (pseudo)spin-1/2 degenerate fermionic gas in a three-dimensional cubic optical lattice, where the spins are encoded by two atomic internal states labeled as $|\uparrow\rangle$ and $|\downarrow\rangle$. The tight-binding Hamiltonian of the cold atom system is considered to be $$\begin{aligned}
\hat{H}&=&\frac{t}{2}\sum_{\boldsymbol{r}}\left(\hat{a}_{\boldsymbol{r}+\boldsymbol{x},\uparrow}^{\dag}\hat{a}_{\boldsymbol{r},\downarrow}
-\hat{a}_{\boldsymbol{r}+\boldsymbol{y},\uparrow}^{\dag}\hat{a}_{\boldsymbol{r},\downarrow}+\text{H.c.}\right) \nonumber \\
&&-\frac{it}{4}\sum_{\boldsymbol{r}}\left[\hat{a}_{\boldsymbol{r}+(\boldsymbol{x}+\boldsymbol{y}),\uparrow}^{\dag}\hat{a}_{\boldsymbol{r},\downarrow}
-\hat{a}_{\boldsymbol{r}+(\boldsymbol{x}-\boldsymbol{y}),\uparrow}^{\dag}\hat{a}_{\boldsymbol{r},\downarrow}+\text{H.c.} \right]\nonumber \\
&&-\frac{t}{2}\sum_{\boldsymbol{r},\boldsymbol{\eta}}\left(\hat{a}_{\boldsymbol{r}+\boldsymbol{\eta},\uparrow}^{\dag}\hat{a}_{\boldsymbol{r},\uparrow}-\hat{a}_{\boldsymbol{r}-\boldsymbol{\eta},\downarrow}^{\dag}\hat{a}_{\boldsymbol{r},\downarrow}+\text{H.c.}\right)\\ &&+m_z\sum_{\boldsymbol{r}}\left(\hat{a}_{\boldsymbol{r},\uparrow}^{\dag}\hat{a}_{\boldsymbol{r},\uparrow}-\hat{a}_{\boldsymbol{r},\downarrow}^{\dag}\hat{a}_{\boldsymbol{r},\downarrow}\right), \nonumber\end{aligned}$$ where $\hat{a}_{\boldsymbol{r},\sigma}$ ($\hat{a}_{\boldsymbol{r},\sigma}^{\dag}$) is the annihilation (creation) operator on site $\boldsymbol{r}$ for the fermion with spin $\sigma=\{\uparrow,\downarrow\}$, $\boldsymbol{\eta} =
\boldsymbol{x,y,z}$ denote the hopping directions, $m_z$ is the strength of an effective Zeeman potential, and the hopping strength is set $t=1$ as the energy unit hereafter. By defining the two-component annihilation operator at site $\boldsymbol{r}$ as $\hat{a}_{\boldsymbol{r}}=(\hat{a}_{\boldsymbol{r},\uparrow},\hat{a}_{\boldsymbol{r},\downarrow})^{T}$, Hamiltonian (1) can be rewritten as $$\begin{aligned}
\hat{H}&=&\sum_{\boldsymbol{r},\boldsymbol{\eta}}\left(\hat{a}_{\boldsymbol{r}+\boldsymbol{\eta}}^{\dag}U_{\eta}\hat{a}_{\boldsymbol{r}}+\text{H.c.}\right)+m_z\sum_{\boldsymbol{r}}\hat{a}_{\boldsymbol{r}}^{\dag}\sigma_z\hat{a}_{\boldsymbol{r}}\\
&&+\sum_{\boldsymbol{r}}\left[\hat{a}_{\boldsymbol{r}+(\boldsymbol{x}+\boldsymbol{y})}^{\dag}U_{xy}\hat{a}_{\boldsymbol{r}}-\hat{a}_{\boldsymbol{r}+(\boldsymbol{x}-\boldsymbol{y})}^{\dag}U_{xy}\hat{a}_{\boldsymbol{r}}+\text{H.c.}\right]\nonumber,\end{aligned}$$ where the hopping matrices along the three axis are $U_x=\frac{1}{2}(\sigma_x-\sigma_z)$, $U_y=-\frac{1}{2}(\sigma_x+\sigma_z)$ and $U_z=-\frac{1}{2}\sigma_z$, and along the $xy$ direction is $U_{xy}=-\frac{1}{4}\sigma_y$, with $\sigma_{x,y,z}$ being the Pauli matrices acting on the spin states.
Here the atomic hoppings $U_{\eta}$ and $U_{xy}$ between two lattice sites along the corresponding direction can be spin-conserved hopping (the $\sigma_z$ term) or spin-flip hopping (the $\sigma_x$ and $\sigma_y$ terms), which can be achieved by the laser-assisted tunnelling technique with well-designed Raman coupling between the two spin states [@GaugeRMP; @GaugeRPP; @SOC-Review]. First, one can use a moderate magnetic field to distinguish the spin states with the Zeeman splitting, which allows one to correlate tunnelling in a spatial direction with rotations in internal spin states and state-dependent tunnelling phases. Second, the natural hopping along each direction is suppressed by titling the cubic optical lattice with a homogeneous energy gradient along the $x,y,z$-directions, with the large tilt potential $\Delta
_{\eta}\gg t_{N}$ (such that the hopping probability $\left(t_{N}/\Delta _{\eta}\right) ^{2}$ induced by the natural tunneling is negligible) and $t_N$ denoting the natural tunneling rate. The tilt potential can be created through the natural gravitational field or the gradient of a dc- or ac-Stark shift, and here we require different linear energy shifts per site $\Delta_{x}\neq\Delta_{y}\neq\Delta_z\neq\Delta_{x}\pm\Delta_{y}$ in order to distinguish between the tunnellings directed along different directions. Finally, the hopping terms can be restored and engineered by application of two-photon Raman coupling with the laser beams of proper configurations through the laser-frequency and polarization selections [@GaugeRMP; @GaugeRPP; @SOC-Review]. In principle, arbitrary $2\times2$ hopping matrices including the required $U_{\eta,xy}$ can be generated in this way with well-designed laser configurations [@Osterloh; @Mazza; @Goldman]. Since several protocols for implementing similar atomic hopping matrices and the tunable Zeeman potential have been theoretically proposed or experimentally realized [@2DSOC; @Liu2013; @Liu2014; @Duan2014; @Osterloh; @Mazza; @Goldman] and a different model of double-Weyl semimetals in optical lattices has been presented, here we leave some details of realization out and focus on exploration and detection of their novel topological properties in the following.
For this lattice system under the periodic boundary condition, the tight-binding Hamiltonian can be rewritten as $\hat{H}=\sum_{\boldsymbol{k},\sigma\sigma'}\hat{a}^{\dag}_{\boldsymbol{k}\sigma}[\mathcal{H}(\boldsymbol{k})]_{\sigma\sigma'}\hat{a}_{\boldsymbol{k}\sigma}$, where $\hat{a}_{\boldsymbol{k}\sigma}=1/\sqrt{V}\sum_{\boldsymbol{r}}e^{-i\boldsymbol{k\cdot
r}}\hat{a}_{\boldsymbol{r}\sigma}$ is the annihilation operator in momentum space $\boldsymbol{k}=(k_x,k_y,k_z)$, and $\mathcal{H}(\boldsymbol{k})=\boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\hat{\sigma}}$ is Bloch Hamiltonian. Here $\boldsymbol{d}(\boldsymbol{k})=(d_x,d_y,d_z)$ denotes the Bloch vectors: $d_x = \cos k_x-\cos k_y$, $d_y = \sin k_x\sin k_y$, and $d_z = m_z-\cos k_x-\cos k_y-\cos k_z$, with the lattice spacing $a\equiv1$ and $\hbar\equiv1$ hereafter. The Bloch Hamiltonian is thus given by $$\begin{aligned}
\label{HK}
\nonumber \mathcal{H}(\boldsymbol{k})&=&(\cos k_x-\cos k_y)\sigma_x+\sin k_x\sin k_y\sigma_y\\
&&+(m_z-\cos k_x-\cos k_y-\cos k_z)\sigma_z.\end{aligned}$$ The energy spectrum of the system is given by $E_{\pm}(\boldsymbol{k})=\pm|\boldsymbol{d}(\boldsymbol{k})|$. The bulk gap closes when $d_{x}(\boldsymbol{k})=d_{y}(\boldsymbol{k})=d_{z}(\boldsymbol{k})=0$. By solving the equations, we can obtain a pair of twofold degenerate points that are double-Weyl points $\boldsymbol{W}_{\pm}=(0,0,\pm\arccos(m_{z}-2))$ for $1<m_z<3$ and another pair of double-Weyl points $\boldsymbol{W}_{\pm}=(\pi,\pi,\pm\arccos(m_{z}+2))$ for $-3<m_z<-1$. For instance, the energy spectrum as a function of $k_y$ and $k_z$ with fixed $k_x=0$ for $m_z=2$ is shown in Fig. \[DoubleWeyl\](a), where two double-Weyl points locate at $(0,0,\pm\pi/2)$.
![(Color online) (a) The band dispersion of the double-Weyl semimetal in the $k_y$-$k_z$ plane with $k_x=0$ and $m_z=2$. (b) The vector distribution of the Berry curvature $\boldsymbol{F}(\boldsymbol{k})$ in the $k_y$-$k_z$ plane. The double-Weyl point $\boldsymbol{W}_+=(0,0,+\frac{\pi}{2})$ denoted by yellow dot is a sink in momentum space and the other point $\boldsymbol{W}_-=(0,0,-\frac{\pi}{2})$ denoted by red dot is a source in the momentum space.[]{data-label="DoubleWeyl"}](weyl-points.pdf){width="8.5cm"}
We consider the nodes $\boldsymbol{W}_{\pm}=(0,0,\pm\arccos(m_{z}-2))$ to further show that they are double-Weyl points. Expanding the Bloch Hamiltonian near the two nodes with $\boldsymbol{q}=(q_x,q_y,q_z)\equiv\boldsymbol{k}-\boldsymbol{W}_{\pm}$ yields the low-energy effective Hamiltonian $$\mathcal{H}_{\pm}\approx\frac{1}{2}(q_y^2-q_x^2)\sigma_x+q_xq_y\sigma_y+\chi v_zq_z\sigma_z,$$ where $\chi=\pm1$ respectively for the two nodes $\boldsymbol{W}_{\pm}$ and $v_z=\sqrt{1-(m_z-2)^{2}}$. The effective Hamiltonian shows that the dispersion near the nodes is quadratic in $k_x$ and $k_y$ and linear in $k_z$. One can rewrite the low-energy effective Hamiltonian as $$\mathcal{H}_{\pm}=\epsilon\left(
\begin{array}{cc}
\chi\cos\theta & -\sin\theta e^{i2\varphi}\\
-\sin\theta e^{-i2\varphi} & -\chi\cos\theta\\
\end{array},
\right)$$ where $v_{\parallel}=\frac{1}{2}$, $q_{\parallel}^{2}=q_x^{2}+q_y^{2}$, $\epsilon=\sqrt{(v_zq_z)^{2}+v_{\parallel}^{2}(q_x^{2}+q_y^{2})^{2}}$, $\cos\theta=v_zq_z/\epsilon$, $\sin\theta=v_{\parallel}q_{\parallel}^{2}/\epsilon$, $\sin\varphi=q_x/q_{\parallel}$, and $\cos\varphi=q_y/q_{\parallel}$. The eigenstates of the lowest band near the two nodes with index $\chi=\pm1$ are respectively given by $|u_{0}\rangle=(\sin\frac{\theta}{2}e^{i2\varphi},\cos\frac{\theta}{2})^T$ and $|u_{0}\rangle=(\cos\frac{\theta}{2}e^{i2\varphi},-\sin\frac{\theta}{2})^T$. The Chern number (topological charge) $C_{\chi}$ of the nodes can thus be computed by integrating the Berry curvature over an arbitrary Fermi sphere $S$ that encloses each node [@Detect]: $$\begin{aligned}
C_{\chi}=\frac{1}{2\pi}\oint_{\boldsymbol{S}}d\boldsymbol{S}\cdot\boldsymbol{F}=-2\chi,\end{aligned}$$ where the Berry curvature $\boldsymbol{F}=\nabla\times\boldsymbol{A}$ and $\boldsymbol{A}=(A_{\theta},A_{\phi})$ is the Berry connection given by $A_{\theta}=i\langle u_0|\partial_\theta|u_0\rangle=0$ and $A_\varphi=i\langle
u_0|\partial_\varphi|u_0\rangle=-2\chi\sin^{2}\frac{\theta}{2}$. The above results reveal that the two nodes have opposite topological charges of $\pm2$ and quadratic dispersion, in contract to the standard Weyl points in Weyl semimetals that have topological charges of $\pm1$ and linear dispersion, so they are named double-Weyl points. Thus the system is in the double-Weyl semimetal phase when the Fermi level lies in the vicinity of the double-Weyl points.
In momentum space, the gauge field associated with the Berry curvature near the neighborhood of Weyl node behaves like a magnetic field originating from a magnetic monopole. Here the opposite chirality of the paired double-Weyl points can also be viewed as a monopole-antimonopole pair in the momentum space. To show this point, we calculate the Berry curvature as a function of the momentum $\boldsymbol{k}$: $\boldsymbol{F}(\boldsymbol{k})=\nabla\times\boldsymbol{A}(\boldsymbol{k})$ with the Berry connection $\boldsymbol{A}(\boldsymbol{k})=i\langle
u_0(\boldsymbol{k})|\nabla_{\boldsymbol{k}}|u_0(\boldsymbol{k})\rangle$ defined by the wave function $|u_0(\boldsymbol{k})\rangle$ in the lowest band. For the two bands system, the lowest-band Berry curvature in the momentum space is given by [@anomalous] $$\begin{aligned}
F^{a}=\epsilon_{abc}F_{bc}=\epsilon_{abc}\left[\frac{1}{2d^{3}}\boldsymbol{d}\cdot\left(\frac{\partial\boldsymbol{d}}{\partial k_b}\times\frac{\partial\boldsymbol{d}}{\partial k_c}\right)\right],\end{aligned}$$ where the three components are obtained as $F^{x}=(\sin k_x\cos
k_x\cos k_y\sin k_z-\sin k_x\sin k_z)/N(\boldsymbol{k})$, $F^{y}=(\cos k_x\cos k_y\sin k_y\sin k_z-\sin k_y\sin
k_z)/N(\boldsymbol{k})$, and $F^{z}=(2\sin^{2}k_y+2\sin^{2}k_x\cos^{2}k_y + (\cos
k_z-m_z)(\sin^{2}k_x\cos k_y+\cos
k_x\sin^{2}k_y))/N(\boldsymbol{k})$, with $N(\boldsymbol{k})=2[(\cos k_x-\cos
k_y)^{2}+\sin^{2}k_x\sin^{2}k_y+(m_z-\cos k_x-\cos k_y-\cos
k_z)^{2}]^{3/2}$. The vector distribution of the Berry curvature $\boldsymbol{F}(\boldsymbol{k})$ in the $k_x=0$ plane are plotted in Fig. \[DoubleWeyl\](b), which clearly shows that the double-Weyl points located at $\boldsymbol{W}_{\pm}=(0,0,\pm\frac{\pi}{2})$ behave as sink and source of the Berry flux.
Finally in this section, we note that the double-Weyl points are stabilized by the $C_{4h}$ symmetry of the system, which consists of $C_4$ point-group symmetry and mirror symmetry $P$ with the Bloch Hamiltonian obeying [@Multi; @Shivamoggi] $$\begin{aligned}
C_4\mathcal{H}(\boldsymbol{k})C_{4}^{-1}=\mathcal{H}(P\boldsymbol{k})\end{aligned}$$ where $C_4=e^{-i\frac{\pi}{2}\sigma_z}$ is a point-group operator for the fourfold rotation about the $z$ axis and $P$ is a matrix transfering $(k_x,k_y,k_z)$ to $(k_y,-k_x,-k_z)$. Here we can define $f(\boldsymbol{k})=d_x(\boldsymbol{k})-id_y(\boldsymbol{k})$ and $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$, such that the Bloch Hamiltonian of the two-band model can be rewritten as $\mathcal{H}(\boldsymbol{k})=f(\boldsymbol{k})\sigma_++f^{*}(\boldsymbol{k})\sigma_-+d_z(\boldsymbol{k})\sigma_z$. Thus the transform of $\mathcal{H}(\boldsymbol{k})$ under $C_4$ leads to $C_4H(\boldsymbol{k})C_4^{-1}=f(\boldsymbol{k})e^{-i\frac{\pi}{2}\sigma_z}\sigma_+e^{i\frac{\pi}{2}\sigma_z}+f^{*}
(\boldsymbol{k})e^{-i\frac{\pi}{2}\sigma_z}\sigma_-e^{i\frac{\pi}{2}\sigma_z}+d_z(\boldsymbol{k})\sigma_z$. Since the Bloch vectors of the double-Weyl semimetal satisfy $f(P\boldsymbol{k})=-f(\boldsymbol{k})$, $f^{*}(P\boldsymbol{k})=-f^{*}(\boldsymbol{k})$ and $d_z(P\boldsymbol{k})=d_z(\boldsymbol{k})$, the system preserves $C_{4h}$ symmetry. When the $C_{4h}$ symmetry are broken, the double-Weyl points will be destroyed and the system is no longer in the double-Weyl semimetal phase. Thus it would be valuable to study the symmetry-breaking effects and the related phase transition in this tunable system.
![(Color online) (a) The energy spectrum and edge states of the reduced chain with lattice sites $L_y=60$ under open boundaries for $k_z=0$. There are two chiral states per surface and the inset shows the density distributions of four typical edge modes. (b) The energy spectrum for $k_z=0.5\pi$. (c) The energy spectrum for $k_z=0.6\pi$. Other parameter $m_z=2$.[]{data-label="ChainEnergy"}](energy.pdf){width="8.5cm"}
topological properties of the simulated double-Weyl semimetals
==============================================================
To further study the topological properties of this system, we consider the Bloch Hamiltonian with dimension reduction method: considering $k_{z}$ as a good quantum number and then reduce the three-dimensional system to a set of two-dimensional subsystems with $k_{z}$ as a parameter. For a fixed $k_{z}$, the reduced Bloch Hamiltonian $\mathcal{H}_{k_{z}}(k_{x},k_{y})$ is given by $$\begin{aligned}
\nonumber \mathcal{H}_{k_{z}}(k_{x},k_{y})&=&(\cos k_{x}-\cos k_{y})\sigma_{x}+\sin k_{x}\sin k_{y}\sigma_{y}\\
&&+(M_{z}-\cos k_{x}-\cos k_{y})\sigma_{z},\end{aligned}$$ where $M_{z}=m_{z}-\cos k_{z}$. The bands of these subsystems with fixed $k_{z}$ are all gapped when $k_{z}\neq \pm k_{z}^{c}$ with $k_{z}^{c}=\arccos(m_{z}-2)$. Under this condition, $\mathcal{H}_{k_{z}}(k_{x},k_{y})$ describes effective two-dimensional Chern insulators since the $k_z$-dependent Chern number is given by $$\label{Ckz}
C_{k_z}=\frac{1}{4\pi} \int_{-\pi}^{\pi}dk_x\int_{-\pi}^{\pi}dk_y~\boldsymbol{\hat{d}}\cdot\left(\partial_{k_x}\boldsymbol{\hat{d}}\times\partial_{k_y}\boldsymbol{\hat{d}}\right),$$ where $\boldsymbol{\hat{d}}\equiv
\boldsymbol{d}/|\boldsymbol{d}|$. In the parameter regime $1<|m_z|<3$, we obtain that $C_{k_z}=2$ for the planes with $-k_{z}^{c}<k_z<k_{z}^{c}$ and $C_{k_z}=0$ for other cases. With similar dimension reduction method, such a two-dimensional Chern insulator can be regarded as a fictitious one-dimensional chain subjected to an external parameter $k_x$ since we can consider $k_x$ as a good quantum number. The tight-binding Hamiltonian of such a one-dimensional chain along the $y$ axis can be written as
$$\begin{aligned}
\hat{H}_{y}(k_x,k_z)&=& \nonumber-\frac{1}{2}\sum_{i_y}\left(\hat{a}^{\dag}_{i_y,\uparrow}\hat{a}_{i_y+1,\uparrow}-\hat{a}^{\dag}_{i_y,\downarrow}\hat{a}_{i_y+1,\downarrow}+\text{H.c.}\right)\\ \nonumber&&-\frac{1}{2}\sum_{i_y}[(1+\sin k_x)\hat{a}^{\dag}_{i_y,\uparrow}\hat{a}_{i_y+1,\downarrow}+\\ \nonumber&&~~~~~~~~(1-\sin k_x)\hat{a}^{\dag}_{i_y,\uparrow}\hat{a}_{i_y-1,\downarrow}+\text{H.c.}]\\
\nonumber&&+\sum_{i_y}M_{zx}(\hat{a}^{\dag}_{i_y,\uparrow}\hat{a}_{i_y,\uparrow}-\hat{a}^{\dag}_{i_y,\downarrow}\hat{a}_{i_y,\downarrow})\\
&&+\sum_{i_y}\cos
k_x(\hat{a}^{\dag}_{i_y,\uparrow}\hat{a}_{i_y,\downarrow}+\hat{a}^{\dag}_{i_y,\downarrow}\hat{a}_{i_y,\uparrow}),\end{aligned}$$
where $M_{zx}=M_z-\cos k_x=m_z-\cos k_x-\cos k_x$. The tight-binding Hamiltonian can be used to study the nontrivial edges states in the system.
We numerically calculate the energy spectrum $E(k_x)$ of the reduced chain with length $L_y=60$ under open boundary conditions in different $k_z$ planes for fixed $m_z=2$, which corresponds to $k_{z}^{c}=0.5\pi$. As shown in Fig. \[ChainEnergy\](a) for $k_z=0$, two symmetric bulk bands with an energy gap accompanies two chiral in-gap states per surface. The two chiral states have degeneracies and connect the separated bands, which is consistent with bulk-edge correspondence in this case with the bulk Chern number $C_{k_z}=2$. The two chiral states gradually spread into the bulk when their energies are closer to the bulk bands. The density distributions of some edge modes are shown in the inset in Fig. \[ChainEnergy\](a) for typical $k_x$. Increasing $|k_z|$ to the critical points $k_{z}^{c}$, the two degeneracies of surface states move to the center and then merge at the double-Weyl points at $k_z=\pm k_{z}^{c}$, with the energy spectrum of $k_z=k_{z}^{c}=0.5\pi$ being shown in Fig. \[ChainEnergy\](b). When $k_z$ inters the region $|k_z|>k_{z}^{c}$, the energy spectrum is again gapped but there is no chiral edge state since $C_{k_z}=0$ in this region, with the case of $k_z=0.6\pi$ shown in Fig. \[ChainEnergy\](c). The change of topological invariant $C_{k_z}$ from 2 to 0 indicates a double-Weyl point of monopole charge 2.
![(Color online) (a) Fermi arcs connecting the mimicked double-Weyl points $W_{\pm}$ with monopole charges $\mp2$ denoted by white dots. The black line denotes Fermi arcs formed by gapless zero-energy edge modes. (b) Fermi arcs when $\delta=1$. The two double-Weyl points split into four single-Weyl points when $\delta\sigma_x$ term is added, and the Fermi arcs (black lines) terminate at four points denoted by white dots. $W_{1,+}$ and $W_{2,+}$ have monopole charge $-1$, $W_{1,-}$ and $W_{2,-}$ have monopole charge $+1$. (c) and (d) The band dispersions $E(k_x,k_z)$ and $E(k_x,k_y)$ for $\delta=1$ with fixed $k_y=0$ and $k_z=\frac{\pi}{3}$, respectively. The other parameter is $m_z=1.5$.[]{data-label="FermiArc"}](fermi-arc.pdf){width="8.5cm"}
We further study the Fermi-arc zero modes (with energy $E=0$) by numerically calculating the energy spectrum $E(k_x,k_z)$ of the surface states, with the results being plotted in Fig. \[FermiArc\](a) for typical parameter $m_z=1.5$. In this case, the two double-Weyl points locate at $\boldsymbol{W}_{\pm}=(0,0,\pm\frac{2\pi}{3})$, and they are connecting by two Fermi arcs with $E=0$ plotted with the black lines. We also find that if $m_z$ are approaching to the critical values $\pm 1$ or $\pm 3$, the Fermi arcs shrink since the two double-Weyl nodes move to each other, and they vanish entirely at the phase boundaries when the two Weyl points merge.
We proceed to study the effects of symmetry breaking in the double-Weyl semimetals. To this end, we can add a term $\mathcal{H}_P=\delta \sigma_x$ to the Bloch Hamiltonian in Eq. (\[HK\]) to break its $C_4$ symmetry, the resultant Hamiltonian $\tilde{\mathcal{H}}=\mathcal{H}+\mathcal{H}_P$ becomes $$\begin{aligned}
\label{HKP}
\nonumber \tilde{\mathcal{H}}&=&(\cos k_x-\cos k_y +\delta)\sigma_x+\sin k_x\sin k_y\sigma_y\\
&&+(m_z-\cos k_x-\cos k_y-\cos k_z)\sigma_z.\end{aligned}$$ In the optical lattice, the $\mathcal{H}_P$ term corresponds to the tunable in-site coupling between the two spin states described by $H_P=\delta\sum_{\boldsymbol{r}}\hat{a}^{\dag}_{\boldsymbol{r},\uparrow}\hat{a}_{\boldsymbol{r},\downarrow}+\text{H.c.}$, which can be realized by additional Raman coupling. In this case, we find that the bulk gap can be closed if $|\delta|\leqslant2$. In particular, for $1-m_z\leqslant\delta\leqslant3-m_z$, we can find four single Weyl points located at$\boldsymbol{W}_{1,\pm}=(-\arccos(1-\delta),0,\pm\arccos(m_z-2+\delta))$ and $\boldsymbol{W}_{2,\pm}=(\arccos(1-\delta),0,\pm\arccos(m_z-2+\delta))$. For $m_z+1\leqslant\delta\leqslant m_z+3$, there are also four single Weyl points at $(k_x,k_y,k_z)=(\pi,\pm\arccos(1-\delta),\pm\arccos(m_z-2+\delta))$. Similarly, when $-2\leqslant\delta\leqslant0$, the single Weyl points are at $(k_x,k_y,k_z)=(0,\pm\arccos(1+\delta),\pm\arccos(m_z-2+\delta))$ for $m_z-3\leqslant\delta\leqslant m_z-1$ and at $(k_x,k_y,k_z)=(\pm\arccos(-1-\delta),\pi,\pm\arccos(m_z+2+\delta))$ for $-3-m_z\leqslant\delta\leqslant -1-m_z$.
To reveal the topological nature of these gapless points more clearly, we first expand the Hamiltonian near the four points $\boldsymbol{W}_{\lambda,\mu}$ with $\lambda=1,2$ and $\mu=+,-$. We obtain the corresponding low-energy effective Hamiltonian $\mathcal{H}_{\lambda,\mu}$ (the other three cases proceed similarly): $$\mathcal{H}_{\lambda,\mu}\approx(-1)^{\lambda}\alpha[q_y\sigma_y -(q_x-q_z)\sigma_x]+\mu \alpha_zq_z\sigma_z,$$ where $\alpha=\sqrt{1-(1-\delta)^{2}}$ and $\alpha_z=\sqrt{1-(m_z-2+\delta)^{2}}$ are the effective Fermi velocities, and $\boldsymbol{q}=(q_x,q_y,q_z)=\boldsymbol{k}-\boldsymbol{W}_{1,\pm}$ or $\boldsymbol{q}=\boldsymbol{k}-\boldsymbol{W}_{2,\pm}$ for the four points. Thus, the dispersion near the gapless points is linear along the three momentum directions, indicating that these nodes are single-Weyl points. We then consider the evolution of the Fermi arcs as increasing $\delta$ from 0 to 2, and find that each double-Weyl point of monopole charge $+2(-2)$ in the double-Weyl semimetal ($1<m_z<3$ and $\delta=0$) splits into two pairs of single-Weyl points with monopole charge $+1(-1)$ connected by two disconnected Fermi arcs. When $\delta=2$ the two pairs of single-Weyl points merge and the Fermi arcs disappear. For example, there are two Fermi arcs which terminate at the four single-Weyl points for $\delta=1$ and $m_z=1.5$, as shown in Fig. \[FermiArc\](b). The corresponding energy spectra on the $k_x-k_z$ ($k_y=0$) and $k_x-k_y$ ($k_z=0$) planes are respectively shown in Fig.3 (c) and (d), which indicate the linear dispersion of the four single-Weyl points along each momentum direction. Therefore in this parameter region ($1<m_z<3$ and $0<\delta<2$), the system is in the single-Weyl semimetal phase with four single-Weyl points, which can be obtained for the other three parameter regions.
![(Color online) The phase diagram of the Hamiltonian in Eq. (\[HKP\]). TI denotes the topological insulating phase (dark red), NI denotes the normal band insulating phase (green), WSM$_8$ the Weyl semimetal phase with eight single-Weyl points (dark blue), WSM$_4$ denotes the Weyl semimetal phase with four single-Weyl points (blue), and DWSM is the double-Weyl semimetal phase (yellow lines).[]{data-label="phase-diagram"}](phase-diagram.pdf){width="7cm"}
By similar analysis of the gapless points and the topological properties, we obtain the phase diagram for the Hamiltonian in Eq. (\[HKP\]) in the parameter space spanned by $m_z$ and $\delta$, as shown in Fig. \[phase-diagram\]. In the phase diagram, apart from the double-Weyl semimetal phase (denoted by DWSM) and the single-Weyl semimetal phase with four Weyl points (denoted by WSM$_4$), there are other three different phases: a normal band insulating phase (denoted by NI) with $C_{k_z}=0$ when $m_z$ or $\delta$ is large enough to open a trivial energy gap, a weak topological insulating phase (denoted by TI) with $C_{k_z}=2$ and chiral edge states for all the range of $k_z$ when $-m_z-1<\delta<1-m_z$ and $m_z-1<\delta<m_z+1$, and a single-Weyl semimetal phase with eight Weyl points (denoted by WSM$_8$). For the WSM$_8$ phase with $1<\delta<2$, the eight (four pairs) single-Weyl points locate at $\boldsymbol{W}_{1,\pm}=(-\arccos(1-\delta),0,\pm\arccos(m_z-2+\delta))$, $\boldsymbol{W}_{2,\pm}=(\arccos(1-\delta),0,\pm\arccos(m_z-2+\delta))$, $\boldsymbol{W'}_{1,\pm}=(\pi,-\arccos(\delta-1),\pm\arccos(m_z+2-\delta))$, and $\boldsymbol{W'}_{2,\pm}=(\pi,\arccos(\delta-1),\pm\arccos(m_z+2-\delta))$. Figure \[WSM8\](a) depicts the position and monopole charge of these eight single-Weyl points in momentum space. The corresponding $k_z$-dependent Chern number $C_{k_z}$ as a function of $k_z$ is plotted in Fig. \[WSM8\](b). We find that $C_{k_z}=2$ when $k_z\epsilon(-k^{w},-k^{w'})$ with $k^{w}=\arccos(m_z-2+\delta)$ and $k^{w'}=\arccos(m_z+2-\delta)$. When $k_z$ sweeps through two single-Weyl points with the total monopole charge being $+2(-2)$, the Chern number will increase (decrease) 2, and thus $C_{k_z}=0$ within the region $(-k^{w'},k^{w'})$. The Chern number increases from 0 to 2 when $k_z$ sweeps through two points $\boldsymbol{W'}_{1,+}$ and $\boldsymbol{W'}_{2,+}$, so $C_{k_z}=2$ for $k_z\epsilon(k^{w'},k^{w})$. Finally $C_{k_z}=0$ when $k_z>k^{w}$ or $k_z<-k^{w'}$.
![(Color online)(a) Illustration of the eight single-Weyl points. The monopole charges of $\boldsymbol{W'}_{1,-}$, $\boldsymbol{W'}_{2,-}$, $\boldsymbol{W}_{1,+}$, $\boldsymbol{W}_{2,+}$ are -1, and the monopole charges of $\boldsymbol{W'}_{1,+}$, $\boldsymbol{W'}_{2,+}$, $\boldsymbol{W}_{1,-}$, $\boldsymbol{W}_{2,-}$ are +1. (b) The Chern number as function of $k_z$. Here $k^{w}=\arccos(m_z-2+\delta)$ and $k^{w'}=\arccos(m_z+2-\delta)$.[]{data-label="WSM8"}](single-weyl.pdf){width="8cm"}
experimental detection schemes
==============================
At this stage, we have introduced the optical lattice system for simulation of the double-Weyl semimetal states and explored the relevant topological properties and the phase diagram. In this section, we propose practical methods for their experimental detection. We first show that the simulated Weyl points can be probed by measuring the atomic Zener tunneling to the excited band after a Bloch oscillation, and then propose two feasible schemes to obtain the $k_z$-dependent Chern number from the shift of hybrid Wannier center and from the the spin polarization in momentum space, respectively.
![(Color online) (a) The distribution $\xi_x(k_y,k_z)$. Two maximum transfer positions in $k_y$-$k_z$ plane correspond to the positions of the double-Weyl points. (b) The distribution $\xi_z(k_x,k_y)$. The maximum dip inside the ring profile indicates $k_x=k_y=0$ for the points. In (a) and (b), $\delta=0$ and $m_z=2$. (c) $\xi_x(k_z)$ for different parameter $m_z$ with $\delta=0$ and fixed $k_y=0$. The maximum transfer positions of $\xi_x(k_z)$ correspond to the expected $k_z$ positions of the paired double-Weyl points, which are denoted by the white dashed line. (d) The distribution $\xi_z(k_x,k_y)$ with two rings. (e) The distribution $\xi_y(k_x,k_z)$ with four maximum peaks. (f) The distribution $\xi'_x(k_y,k_z)$ with two rings. The patterns in (d-f) with $\delta=0.5$ and $m_z=2$ reveal the positions of four single-Weyl points $(\pm\pi/3,0,\pm\pi/3)$. The other parameter is $F=0.2$ in (a-f).[]{data-label="LZ"}](LZ.pdf){width="8.5cm"}
Detection of the Weyl points
----------------------------
Here we propose to use the atomic Bloch-Zener oscillation in the optical lattice to detect the double- and single-Weyl points in this system. One can prepare noninteracting fermionic atoms in the lower band initially and apply a constant force $F$ along $\eta$ axis, which push the atoms moving along $k_{\eta}$ direction in momentum space and gives rise to Bloch oscillations. Then one can obtain the momentum distribution of the transfer fraction in the upper band from time-of-flight measurement after performing a Bloch oscillation. For the system with the double-Weyl points $\boldsymbol{W}_{\pm}=(0,0,\pm k_z^c)$, the transfer fractions $\xi_{\eta}$ along different directions are given by [@Tarruell; @Lim] $$\begin{aligned}
\label{PLZ}\nonumber
&&\xi_{x}(k_y,k_z)=P_{LZ}^{x}(k_y,k_z),\\
&&\xi_{y}(k_x,k_z)=P_{LZ}^{y}(k_x,k_z),\\\nonumber
&&\xi_{z}(k_x,k_y)=2P_{LZ}^{z}(k_x,k_y)[1-P_{LZ}^{z}(k_x,k_y)],\end{aligned}$$ where the Landau-Zener transition probabilities are $P_{LZ}^{x}=e^{-\pi\Delta_{x}^{2}(k_y,k_z)/4v_{x}F}$, $P_{LZ}^{y}=e^{-\pi\Delta_{y}^{2}(k_x,k_z)/4v_{y}F}$ and $P_{LZ}^{z}=e^{-\pi\Delta_{z}^{2}(k_x,k_y)/4v_{z}F}$, with $v_{x}=v_{y}=v_{\parallel}$ and the energy gaps $\Delta_x=2E_+(k_x=0,k_y,k_z)$, $\Delta_y=2E_+(k_x,k_y=0,k_z)$ and $\Delta_z=2E_+(k_x,k_y,k_z=k_{z}^{c})$ for the Landau-Zener events along the $k_{\eta}$ directions.
We numerically calculate the transfer fractions $\xi_{\eta}$, with the results for typical parameter shown in Fig. \[LZ\]. For the case of $\delta=0$ and $m_z=2$ in Fig. \[LZ\](a), there are two maximum transfer positions of quasimomentum distribution $\xi_x(k_y,k_z)$ in the $k_y$-$k_z$ plane. The positions correspond to the expected $k_y=0$ and $k_z=\pm k_z^c$ for the paired double-Weyl points. As the energy gap decreases, the transition probability in a Landau-Zener tunnelling increases exponentially. When the band gap closes, the transfer fraction $\xi_x(k_y,k_z)$ will increase dramatically and thus the band crossing points can be clearly identified in experiments. Due to the $C_4$ symmetry of the Bloch Hamiltonian when $\delta=0$, one can find that $\Delta_y(k_x,k_z)=\Delta_x(k_y,k_z)$ and the distribution of the transfer fraction $\xi_y(k_x,k_z)$ is the same as the one of $\xi_x(k_y,k_z)$ shown in Fig. \[LZ\](a) by replacing $k_y$ with $k_x$. When the atoms move along $k_z$ direction, there are two subsequent Landau-Zener tunnelling. In Fig. \[LZ\](b), the distribution of the transfer fraction $\xi_z(k_x,k_y)$ shows the ring-type profile. The position with the value $\xi_z=0$ indicates $k_x=k_y=0$ for the double-Weyl points. In Fig. \[LZ\](c), for $\delta=0$ and fixed $k_y=0$, the maximum transfer positions of quasimomentum distributions $\xi_x(k_z)$ correspond well to the expected $k_z$ positions of the mimicked double-Weyl points as plotted by the dashed line. Therefore, by probing the momentum distribution $\xi_x(k_z,k_y)$ and $\xi_z(k_x,k_y)$ from the standard time-of-flight measurement after a Bloch oscillation, the positions of the double-Weyl points in momentum space can be well revealed.
The method is applicable for detecting the single-Weyl points created by the symmetry breaking when $\delta\neq0$ in this system. We consider the typical case of $\delta=0.5$ and $m_z=2$, with four single-Weyl points in $k_x$-$k_z$ plane. In this case, there are two subsequent Landau-Zener transitions along $k_x$ direction and thus the transfer fraction $\xi_{x}$ in Eq. (\[PLZ\]) becomes $$\begin{aligned}
\xi^{'}_{x}(k_y,k_z)=2P^{x}_{LZ}(k_y,k_z)[1-P^{x}_{LZ}(k_y,k_z)],\end{aligned}$$ while $\xi_{y}$ and $\xi_{z}$ remain the same expressions. The numerical results of $\xi_{z}$, $\xi_{y}$ and $\xi'_{x}$ are respectively shown in Figs. \[LZ\](d,e,f). One can find that both the distributions $\xi_z(k_x,k_y)$ and $\xi'_x(k_y,k_z)$ have two rings and the positions inside each ring with $\xi_z=\xi'_x=0$ indicate four band crossing points located at $(\pm\frac{\pi}{3},0,\pm\frac{\pi}{3})$ as expected for the single-Weyl points in this case. The four peaks of transfer fraction $\xi_y(k_x,k_z)$ shown in Fig. \[LZ\](e) also reveal the exact positions of the four gapless points. We note that the Bloch-Zener method can not tell the trivial (accidental) gapless points and the non-trivial Weyl points in the bands. However, the double- and single-Weyl points in our model system can be distinguished from the different patterns of $\xi_{\eta}$, as shown in Fig. \[LZ\]. To detect the topology of the gapless points, one may further perform the interference between two atomic gases traveling across the points in momentum space revealed by the Bloch-Zener method to extract the Berry phases and thus the Chern numbers [@Duca]. Below we present two different approaches to measure the band topology in our model system.
{width="14cm"}
Detection of the Chern number from the shift of hybrid Wannier center
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We now proceed to propose a realistic scheme to directly measure the Chern number of the double-Weyl semimetals and other topological states in optical lattices, based on the particle pumping approach and hybrid Wannier functions in the band theory [@Pumping1; @Pumping2; @Pumping3; @Wanglei; @Marzari; @Vanderbilt]. With the dimension reduction method, the three-dimensional system can be treated as a collection of $k_z$-modified two-dimensional Chern insulators with the $k_z$-dependent Chern number defined in $k_x$-$k_y$ plane as different slices of out-of-plane quasimomentum $k_z$. Such a two-dimensional insulating subsystem can be further viewed as a fictitious one-dimensional insulator subjected to an external parameter $k_x$. Thus, its Chern number can be defined by the polarization $P(k_x,k_z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}
\boldsymbol{A}(\boldsymbol{k})dk_y$ for the geometry of the underlying band structure. According to the modern theory of polarization [@Marzari; @Vanderbilt], the Chern number defined in $k_x$-$k_y$ space can be obtained from the change in polarization induced by adiabatically changing the parameter $k_x$ by $2\pi$: $C_{k_z}=\int_{-\pi}^{\pi}\frac{\partial
P(k_x,k_z)}{\partial k_x}dk_x$. For measuring $P(k_x,k_z)$, one can use another fact that the polarization can alternatively written as the center of mass of the Wannier function constructed for the single occupied band.
In this system, the polarization $P(k_x,k_z)$ can be expressed by means of the centers of the hybrid Wannier functions, which are localized in the $y$ axis retaining Bloch character in the $k_x$ and $k_z$ dimensions. The variation of the polarization and thus the Chern number are directly related to the shift of the hybrid Wannier center along the $y$ axis in the lattice. The shift of hybrid Wannier center by adiabatically changing $k_x$ is proportional to the Chern number, which is a manifestation of topological pumping with $k_x$ being the adiabatic pumping parameter. In this system, the hybrid Wannier center of a one-dimensional insulating chain along $y$ axis described by the Hamiltonian $\tilde{H}=H+H_P$ can be written as $$\begin{aligned}
\langle n_y(k_x,k_z)\rangle=\frac{\sum_{i_y}i_y\rho_{i_y}(k_x,k_z)}{\sum_{i_y}\rho_{i_y}(k_x,k_z)},\end{aligned}$$ where $\rho_{i_y}(k_{x},k_{z})$ denotes the density distribution of the hybrid Wannier function as a function of the parameter $k_{x}$ and $k_{z}$ with $i_{y}$ being the lattice-site index in the one-dimensional chain, and takes the following form $$\begin{aligned}
\rho_{i_y}(k_x,k_z)=\sum_{\text{occ}}|k_x,k_z\rangle_{i_y}{}_{i_y}\langle k_x,k_z|,\end{aligned}$$ where $|k_{x},k_{z}\rangle_{i_y}$ denotes the hybrid wave function of the system at site $i_y$ and the notation occ denotes the occupied states. In cold atom experiments, the atomic density $\rho_{i_y}(k_{x},k_{z})$ can be directly measured by the hybrid time-of-flight images, which is referring to an *in situ* measurement of the density distribution of the atomic cloud in the $y$ direction during free expansion along the $x$ and $z$ directions. In the measurement, the optical lattice is switched off along the $x$ and $z$ directions while keeping the system unchanged in the $y$ direction. One can map out the crystal momentum distribution along $k_x$ and $k_z$ in the time-of-flight images and a real space density resolution in the $y$ direction can be done at the same time. Thus one can directly extract the Chern number from this hybrid time-of-fight images in the cold atom system.
To demonstrate the feasibility of the proposed method, we numerically calculate $\langle n_y(k_x,k_z)\rangle$ in a tight-binding chain of length $L_y=60$ under the open boundary condition at half filling for some typical parameters, with the results shown in Fig. \[HWF\]. For $\delta=0$ and $m_z=2$ in Figs. \[HWF\](a) and \[HWF\](b), the system is in the double-Weyl semimetal phase with $k_{z}^{c}=\pm\pi/2$, and we find that as $k_x$ changing from $-\pi$ to $\pi$, $\langle n_y(k_x)\rangle$ exhibits two discontinuous jumps of one unit cell within the region $k_z\epsilon(-0.5\pi,0.5\pi)$ and the jumps disappear outside this region. To be more clearly, we also plot $\langle n_y(k_x)\rangle$ for $k_z=0$ and $k_z=0.6\pi$ as two examples in Fig. \[HWF\](b). The double one-unit-cell jumps driven by $k_x$ indicates that two particles is pumped across the system, as expected for $\mathcal{C}_{k_z}=2$. For $\delta=0$ and $m_z=0.5$ in Figs. \[HWF\](d) and \[HWF\](e), we find that two discontinuous jumps of $\langle n_y(k_x)\rangle$ for all $k_z$ as $k_x$ changing from $-\pi$ to $\pi$, indicating that the system is in the topological insulating phase with $C_{k_z}=2$. For $m_z=0.5$ and $\delta=1.7$ in Figs. \[HWF\](g) and \[HWF\](h), the system is in the Weyl semimetal phase with eight single-Weyl points and we find that when $k_z\epsilon(-0.46\pi,-0.20\pi)$ and $k_z\epsilon(0.20\pi,0.46\pi)$, $\langle n_y(k_x)\rangle$ shows two discontinuous jumps of one-unit-cell by varying $k_x$ from $-\pi$ to $\pi$, consistent with $C_{k_z}=2$ in these $k_z$ regions as shown in Fig. \[WSM8\]. When the system is in the normal band insulating phase for $m_z=3.2$ and $\delta=0$, as expected, there is no jump of the hybrid Wannier center $\langle n_y(k_x)\rangle$ for all $k_z$ by changing adiabatic pumping parameter $k_x$ from $-\pi$ to $\pi$, as shown in Figs. \[HWF\](j) and \[HWF\](k). We also obtain the results of $\langle n_y(k_x)\rangle$ when the system is in the Weyl semimetal phase with four single-Weyl points, similar with those in Fig. \[HWF\](a). This establishes a direct and clear connection between the shift of the hybrid Wannier center and the topological invariant of the system in different phases.
In order to simulate the realistic experiment, we add a weak harmonic trap to this finite-site lattice with the open boundary. The trapping potential in the chain can be effectively described as $H_t=V_t\sum_{i_y}(i_y-\frac{L_y}{2})^{2}\hat{a}^{\dag}_{i_y}\hat{a}_{i_y}$, where $V_t$ is the trap strength. Within a local density approximation, as long as the lower band is still filled at the center of the trap, the shifts of the hybrid Wannier center can be expected to be nearly the same as those without the trap potential. If the band gap $E_g<V_t(i_y-\frac{L_y}{2})^{2}$, the lower band is only partially filled near the two edges and then this pumping argument is no longer well applicable. In practical experiments, one can turn the trap strength to $V_t\sim4E_g/L_y^{2}$ or emphasize the shift of hybrid Wannier center in the central region. With numerical simulations, we demonstrate that the results of $\langle n_y(k_x,k_z)\rangle$ preserve with a deviation less than $2\%$ except the regions near the band crossing points for $V_t=3\times10^{-4}$ in Fig. \[HWF\](c) and \[HWF\](l), $V_t=6\times10^{-4}$ in Fig. \[HWF\](f), and $V_t=2\times10^{-4}$ in Fig. \[HWF\](i). They are consistent with the estimates in the local-density analysis.
Detection of the band topology from the spin polarization in momentum space
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Below we propose an alternative method to probe the band topology of the $C_4$-symmetric Bloch Hamiltonian with $\delta=0$ from the spin polarization in momentum space, which can be implemented with bosonic atoms in the optical lattice. When the system has $C_4$ symmetry in the $xy$ plane, we can treat $k_z$ as an effective parameter and reduce it to a collection of effective two-dimensional subsystems, whose Chern number $C_{k_z}$ for a fixed $k_z$ can be determined by the following equation [@Multi; @Shivamoggi]
$$\begin{aligned}
\label{symmetry}
e^{i\frac{\pi}{2}C_{k_z}}=\prod_{n\epsilon \text{occ}}\gamma_n(0,0,k_z)\gamma_n(\pi,\pi,k_z)\chi_n(0,\pi,k_z).\end{aligned}$$
Here $\gamma_n$ and $\chi_n$ are the $C_4$ and $C_2$ eigenvalues on the $n$-th Bloch band at high-symmetry momentum points in $k_x$-$k_y$ plane, respectively.
For our two bands system, $C_4=e^{-i\frac{\pi}{2}\sigma_z}=-i\sigma_z$ and $C_2=C_{4}^{2}=-1$, such that the term $\chi_n(0,\pi,k_z)=-1$ in Eq. (\[symmetry\]) can be dropped from the expression. The Chern number $C_{k_z}$ of the lower band for different $k_z$ can thus be determined by the simple relation $$\begin{aligned}
\label{symmetry-Ckz}
e^{i\frac{\pi}{2}C_{k_z}}=S_-(0,0,k_z)S_-(\pi,\pi,k_z)\end{aligned}$$ where $S_-(0,0,k_z)$ and $S_-(\pi,\pi,k_z)$ are the eigenvalues of the $\sigma_z$ operator on the lower band. Thus to obtain $C_{k_z}$ for a given $k_z$, one only needs to measure the eigenvalues of $\sigma_z$ in the two high symmetry points in $k_x$-$k_y$ plane ${\boldsymbol{\Lambda_i}}=\{\boldsymbol{\Gamma}=(0,0,k_z),\boldsymbol{M}=(\pi,\pi,k_z)\}$. This can simplify the experimental detection of the topological invariant of the Bloch bands. The high symmetry points $\boldsymbol{\Lambda_i}$ satisfy that $P\boldsymbol{\Lambda_i}=\boldsymbol{\Lambda_i}$. Thus, the constraints at these points give $f(\boldsymbol{\Lambda_i})=-f(\boldsymbol{\Lambda_i})$ and $f^{*}(\boldsymbol{\Lambda_i})=-f^{*}(\boldsymbol{\Lambda_i})$, which imply that $f(\boldsymbol{k})$ and $f^{*}(\boldsymbol{k})$ vanish. So at the high symmetry points $\boldsymbol{\Lambda_i}$, the Bloch Hamiltonian can be written as $$\begin{aligned}
\mathcal{H}(\boldsymbol{\Lambda_i})=d_z(\boldsymbol{\Lambda_i})\sigma_z,\end{aligned}$$ where the energy of the two bands $E_{\pm}(\boldsymbol{\Lambda_i})=\pm|d_z(\boldsymbol{\Lambda_i})|$. Since the Bloch Hamiltonian commutes with the symmetry operator, i.e., $[C_4,\mathcal{H}(\boldsymbol{\Lambda_i})]=0$, the Bloch states of the two bands $|u_{\pm}(\boldsymbol{\Lambda_i})\rangle$ are also the eigenstates of $C_4$. Therefore, one can obtain the Chern number of the lower band for different $k_z$ by measuring the spin polarization $\langle\sigma_z\rangle$ near the high symmetry points in momentum space, which can be written as $$\label{polarization}
\langle\sigma_z(\boldsymbol{\Lambda_i})\rangle=\frac{n_\uparrow(\boldsymbol{\Lambda_i})-n_\downarrow(\boldsymbol{\Lambda_i})}{n_\uparrow(\boldsymbol{\Lambda_i})+n_\downarrow(\boldsymbol{\Lambda_i})}.$$ Here $n_{\uparrow,\downarrow}(\boldsymbol{\Lambda_i})$ denotes the atomic density of spin states $|\uparrow,\downarrow\rangle$ at the high symmetry points in $k_x$-$k_y$ plane for a fixed $k_z$. Since this detection protocol only requires measurement of the atomic density distribution in momentum space, it can be applied to bosonic atoms, typically Bose-Einstein condesates, in the optical lattice system with the topological bands.
In the experiment with a condensate in the optical lattice, the spin polarization at the two high symmetry momenta can be written as $$\langle\sigma_z(\boldsymbol{\Lambda_i})\rangle\approx S_-(\boldsymbol{\Lambda_i})f(E_-,T)
+S_+(\boldsymbol{\Lambda_i})f(E_+,T),$$ where $f(E_{\pm},T)=1/[e^{(E_{\pm}(\boldsymbol{\Lambda_i})-\mu)/k_BT}-1]$ is the Bose-Einstein statistics with $\mu$ and $T$ respectively being the chemical potential and temperature, and $S_{\pm}(\boldsymbol{\Lambda_i})$ are the eigenvalues of $\sigma_z$ on the lower and upper bands at $\boldsymbol{\Lambda_i}$. Since $S_+(\boldsymbol{\Lambda_i})=-S_-(\boldsymbol{\Lambda_i})$, one has $\langle\sigma_z(\boldsymbol{\Lambda_i})\rangle\approx S_-(\boldsymbol{\Lambda_i})[f(E_-,T)-f(E_+,T)]$. Thus by preparing a cloud of bosonic atoms with the temperature satisfying $f(E_-(\boldsymbol{\Lambda_i}),T)>f(E_+(\boldsymbol{\Lambda_i}),T)$, one can obtain $$\begin{aligned}
\text{sgn}[\langle\sigma_z(\boldsymbol{\Lambda_i})\rangle]=\text{sgn}[S_-(\boldsymbol{\Lambda_i})].\end{aligned}$$ Therefore, the spin polarization $\langle\sigma_z(\boldsymbol{\Lambda_i})\rangle$ can be precisely measured with a condensate at low temperature.
In practical experiments, one can prepare the atoms in the spin-up state and adiabatically load the condensate into the $\boldsymbol{\Lambda_i}$ points. Then one can perform the spin-resolved time-of-flight expansion, which projects the Bloch states onto free momentum states according to the plane-wave expansion with a complete basis of plane waves $\{\psi_{m,n}^{\uparrow}(\boldsymbol{\Lambda_i}),\psi_{p,l}^{\downarrow}(\boldsymbol{\Lambda_i})\}$. The Bloch state of lowest band can be expressed as $|u_-(\boldsymbol{\Lambda_i})\rangle=\sum_{m,n}a_{m,n}\psi_{m,n}^{\uparrow}(\boldsymbol{\Lambda_i})|\uparrow\rangle+\sum_{p,l}b_{p,l}\psi_{p,l}^{\downarrow}(\boldsymbol{\Lambda_i})|\downarrow\rangle$, where $a_{m,n}$ and $b_{p,l}$ are coefficients. The spin polarization for the Bloch eigenstates of the lower band at high symmetry points is given by $\langle\sigma_z(\boldsymbol{\Lambda_i})\rangle=\langle u_-(\boldsymbol{\Lambda_i})|\sigma_z|u_-(\boldsymbol{\Lambda_i})\rangle
=\sum_{m,n}|a_{m,n}\psi_{m,n}^{\uparrow}(\boldsymbol{\Lambda_i})|^{2}-\sum_{p,l}|b_{p,l}\psi_{p,l}^{\downarrow}(\boldsymbol{\Lambda_i})|^{2}$, which gives rise to the expression in Eq. (\[polarization\]). Finally one can obtain $n_{\uparrow,\downarrow}(\boldsymbol{\Lambda_i})$ by the time-of-flight measurement, and thus obtain the $k_z$-dependent Chern number of the Bloch bands from Eq. (\[symmetry-Ckz\]) with $S_-(\boldsymbol{\Gamma})S_-(\boldsymbol{M})=\text{sgn}[\langle\sigma_z(\boldsymbol{\Gamma})\rangle]\text{sgn}[\langle\sigma_z(\boldsymbol{M})\rangle]$ in this case.
Finally, the fact that the topology of the $C_4$-symmetric bands can be determined by only the Bloch states at the symmetric momenta can greatly simplify the experimental detection of the topological bands. Similar protocol has been implemented to detect the topology of the inversion-symmetric bands with Bose-Einstein condensates in two-dimensional optical lattices [@2DSOC]. In our three-dimensional system, one can extract $C_{k_z}$ from the proposed measurements for various $k_z$ and $m_z$, corresponding to the line of $\delta=0$ in the phase diagram. If all two-dimensional slices have $C_{k_z}=0$, the system is in the trivial insulator phase, while it is in the topological insulator phase if $C_{k_z}=2$ for all $k_z$. The change of $C_{k_z}$ by two along $k_z$ axis indicates the presence of double-Weyl points and the system is in the double-Weyl semimetal phase.
Conclusions
===========
In summary, we have proposed an optical lattice system for simulation and exploration of double-Weyl semimetals. We have investigated the topological properties of the double-Weyl semimetal phase and obtained a rich phase diagram with several other quantum phases, which include a topological insulator phase and two single-Weyl semimetal phases. Furthermore, with numerical simulations, we have proposed practical methods for the experimental detection of the mimicked Weyl points and the characteristic topological invariants with cold atoms in the lattice system. The proposed system would provide a promising platform for elaborating the intrinsic exotic physics of double-Weyl semimetals and the related topological phase transitions that are elusive in nature.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In this paper we analyze the diffuse X-ray coronae surrounding the elliptical galaxy NGC5846, combining measurements from two observatories, ROSAT and ASCA. We map the gas temperature distribution and find a central cool region within an approximately isothermal gas halo extending to a radius of about 50 kpc, and evidence for a temperature decrease at larger radii. With a radially falling temperature profile, the total mass converges to $9.6\pm1.0 \times
10^{12}$ at $\sim$230 kpc radius. This corresponds to a total mass to blue light ratio of $53\pm5$ /. As in other early type galaxies, the gas mass is only a few percent of the total mass.
Using the spectroscopic measurements, we also derive radial distributions for the heavy elements silicon and iron and find that the abundances of both decrease with galaxy radius. The mass ratio of Si to Fe lies between the theoretical predictions for element production in SN Ia and SN II, suggesting an important role for SN Ia, as well as SN II, for gas enrichment in ellipticals. Using the SN Ia yield of Si, we set an upper limit of $0.012h_{50}^2$ SNU for the SN Ia rate at radii $>50$ kpc, which is independent of possible uncertainties in the iron L-shell modeling. We compare our observations with the theoretical predictions for the chemical evolution of ellipticals, taken from Matteucci & Gibson (1995). We conclude that the metal content in stars, if explained by the star formation duration, requires a significant decline in the duration of star formation with galaxy radius, ranging from $\sim1$ Gyr at the center to $\sim0.01$ Gyr at 100 kpc radius. Alternatively, the decline in metallicity with galaxy radius may be caused by a similar drop with radius in the efficiency of star formation. Based on the Si and Fe measurements presented in this paper, we conclude that the latter scenario is preferred, unless a dependence of the SN Ia rate on stellar metallicity is invoked.
author:
- 'A. Finoguenov$^{1,2}$, C. Jones$^1$, W. Forman$^1$ and L. David$^1$'
title: |
[Stellar Metallicities and SN Ia Rates in the Early-type Galaxy NGC5846]{}\
[from ROSAT and ASCA Observations]{}
---
Introduction
============
X-ray observations have shown that early type galaxies are gas rich systems with gas masses up to several 10$^{10}$ $M_{\odot}$ and temperatures around $10^{7}$ K (e.g. Forman 1979; Forman, Jones & Tucker 1985). The heavy element abundances in these hot coronae and their radial dependencies should reflect input to the corona from evolving stars and supernovae. Since the flow time of the gas in the corona is long (billions of years), spatially resolved X-ray spectroscopy of the gas can provide constraints on abundances in the stellar component and the rate of supernovae (Loewenstein and Mathews 1991, hereafter LM). Measuring the elemental abundances in the extended coronae of galaxies is complimentary to measuring abundances from the integrated optical light of stars, which is characteristic of the central region in a galaxy.
With the spectral resolution of the ROSAT PSPC, only the iron abundance could be reliably measured in the X-ray coronae of early-type galaxies (Forman 1993, Davis and White 1996). With the higher spectral resolution of ASCA, we can now separate the spectral features of several heavy elements, which allows us to distinguish between the impact of stellar mass loss and SN Ia input.
To explore the abundances in the hot gas around early-type galaxies, we analyzed both ASCA and ROSAT observations of NGC5846, the brightest galaxy ($L_B=1.82\times10^{11}$) in the LGG393 group (Garcia 1993 and references therein). The NGC5846 group is also known as cV50 and CfA 150. NGC5846 is classified as an S0$_1$(0) galaxy by Sandage and Tammann (1987), although it is more likely an elliptical of E0 type, as classified in RC2 ( Wrobel & Heeschen 1991). The galaxy’s velocity relative to the Local Group is 1674 km s$^{-1}$, corresponding to a distance of 33.5 Mpc (0) which implies a scale of $\sim10$ (9.74) kpc per arcminute. An optical study of the NGC5846 group yielded a velocity dispersion of $\sigma=381$ km s$^{-1}$ and M/L=163 in solar units (Haynes & Giovanelli 1991). For NGC5846, the internal velocity dispersion is $\sigma=230$ km s$^{-1}$ ( Fisher, Illingworth & Franx 1995). As one of only a few nearby, bright elliptical galaxies, NGC5846 is also bright in X-rays, so that the X-ray properties of the corona and the underlying mass distribution can be studied with the ROSAT and ASCA observatories. The total X-ray luminosity of NGC5846 inside a 10 radius is $7\times10^{41}$ , in the 0.2–2.0 keV band. NGC5846 was previously studied in X-rays by Biermann (1989) using observations, who assuming an isothermal coronae, derived a gravitating mass inside a 120 kpc radius of $7\pm2\times10^{12}$ and a total gas mass of $8.5\times10^{10}$.
In the sections which follow, we describe the ROSAT and ASCA observations of NGC5846. Section 2 discusses the analysis of both the imaging and spectroscopic data. Section 3 briefly describes our results for the spiral galaxy NGC5850, which is detected in both the ROSAT and ASCA images. In section 4 we discuss the mass determination around NGC5846 and our heavy element abundance measurements. Finally, we described in the Appendix our approach to analyzing extended sources when observed with the broad, energy dependent PSF of the ASCA X-ray telescopes.
ROSAT and ASCA Observations and Analysis
=========================================
For imaging analysis and to map the temperature distribution of the emission around NGC5846, we used ROSAT PSPC observations carried out during 25 July – 8 August 1992 and a second observation performed during 18 January 1993. We used the software described in Snowden (1994) and references therein to determine the “good time” intervals (GTI’s), to estimate the non-X-ray background, and to construct exposure maps for flat fielding.
We observed NGC5846 with ASCA during 7–8 February 1994 for a total of 30 ksec. In our analysis, we used data from the SIS0 and SIS1 detectors. The four-CCD SIS mode, which we used, provides a FOV of 20$\times$20 with an energy resolution of $\sim$75 eV at 1.5 keV (preflight value). A detailed description of the ASCA observatory, as well as the SIS detectors, can be found in Tanaka, Inoue & Holt (1994) and Burke (1991). Standard data screening was carried out with FTOOLS version 3.6. In our spectral analysis, we account for the effects of the broad ASCA PSF, as described in [*Appendix*]{} of this paper, and include the geometrical projection effect of the three-dimensional distribution of emitting gas. GIS data were not used in our analysis, due to their lower energy resolution.
Finally, we compare the results of a three-dimensional approach to modeling the ROSAT and ASCA observations. In such an approach, where all the derived spectra are fitted simultaneously and where correlations between different regions are high, a simple minimization of the $\chi^2$ term is shown to be an ill-conditioned task (Press 1992, p.780). Among the regularization algorithms developed to solve this problem (Press 1992, p.801), we chose the assumption that a linear function is a good representation of the radial distribution of the gas temperature and abundance. The impact of the regularization is limited to changing the best-fit values only within their 68% confidence level intervals. Thus, the procedure results in a linear solution for the parameters of interest in the regions where they could not be resolved either statistically or spatially. Further details of this minimization procedure for ROSAT and ASCA observations are described in Finoguenov and Ponman (1998). We adapted the XSPEC analysis package to perform the actual fitting.
Imaging Analysis
-----------------
### ROSAT PSPC
Since ROSAT has superior spatial resolution compared to ASCA, we used the ROSAT PSPC images to determine both the X-ray surface brightness and the gas temperature distribution for NGC5846. For surface brightness analysis, we used the broad energy band (Snowden bands R47). We first generated an auxiliary map, in which we identified point-sources and sharp emission features. We used this map to determine the smoothing kernel for R47 and also for the gas temperature analysis, so that the sharp features would not be strongly smoothed. In addition to smoothing the background subtracted image, an identical smoothing was applied to the exposure map. With these two images, we generated a vignetting corrected, background subtracted image (counts sec$^{-1}$ cm$^{-2}$ arcmin$^{-2}$) for all regions where the exposure map exceeded 1% of the map’s maximum. For NGC5846 this procedure results in smoothing the emission of the core and all point sources with a gaussian of $\sigma$=30, the inner corona with $\sigma$=1 and the outskirts ($>5$) of the galaxy, with $\sigma$=2. We use this image to determine the center of the X-ray emission.
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In Figure \[opt\_imh\] we present an optical image of the NGC5846 field with the ROSAT broadband (R47) contours superposed. X-ray contours extend to a radius of 200 kpc, far beyond the detected optical light of NGC5846. While there are numerous serendipitous sources in the image, in addition to NGC5846, the only galaxy detected is NGC5850 (type SBb, Sandage & Tammann 1988), which lies beyond the N5846 group.
Using the ROSAT PSPC images, we extracted a radial surface brightness profile around NGC5846, corrected it by the corresponding exposure map profiles, and added a 4% systematic error. We use a $\beta$-model of the form $$S(r)=S(0) \; (1+({r / r_a})^2)^{-3\beta+{1\over 2}}$$ to characterize the surface brightness profile. We use the region 20–40 for background estimation. Our results, listed in Table 1, provide a better determination of $r_a$ and $\beta$, but are consistent with the analysis (Biermann 1989). The best fit and the surface brightness profile are presented in Figure \[rad\_pro\]. Since the ROSAT PSPC countrate is affected by variations in the gas temperature and abundance, we take into account our determinations of the projected temperature and abundance in our analysis of the surface brightness profile, which we also use for gas mass calculations. As shown in Table 1, changes occur primarily in the derived values of the core radius ($\pm10$%), with only $\pm2$% changes in the $\beta$ index.
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To analyze the gas temperature distribution, we generated a hardness ratio map. We choose two semi-independent energy bands, (R45) and (R67) (see Finoguenov 1998 for a discussion of the particular choice of these bands). We used the point source map and minimal smoothing scale $\sigma=1$ to avoid small-scale artifacts, which would otherwise arise from the energy dependence of the ROSAT PSPC PSF (Hasinger 1994). Diffuse emission from NGC5846 within a radius of 5 was smoothed with $\sigma=2$ and the rest with $\sigma=10$.
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In Figure \[te\_map\] we present a hardness ratio image defined as (R67–R45)/(R67+R45) along with the broad-band X-ray surface brightness contours. This hardness ratio provides a robust estimator of the temperature in the range 0.5–1.5 keV, even if and abundances vary (Finoguenov 1998). As Figure \[te\_map\] shows, the X-ray emission appears cool at the galaxy center, with the temperature increasing smoothly with radius to $\sim1$ keV at $\sim$3 from the center. At radii beyond 5 the emission has a cooler temperature ($\sim0.8$ keV).
### ASCA SIS
The ASCA SIS imaging capability is limited compared to the ROSAT PSPC, due to the broader ASCA PSF. Nevertheless, analysis of the ASCA images is important for several respects, in particular to measure the distribution of harder ($>2$ keV) emission. To analyze the ASCA observations, we first determine the alignment between the ROSAT and ASCA images. We use the results of the ROSAT profile fitting to simulate the ASCA image in the range and then find the relative alignment with the true ASCA image. Our determined ASCA position for the NGC5846 center is 15$^h$ 06$^m$ 29.9$^s$; +01 36 21 (RA; Dec., Equinox J2000) compared with the ROSAT position 15$^h$ 06$^m$ 29.2$^s$; +01 36 17, corrected for the ROSAT XRT/detector boresight offset (Briel 1993). The relative misalignment between ASCA and ROSAT is within the expected uncertainties in both ASCA (Gotthelf 1996) and ROSAT (Briel 1993) aspect determination. For both ASCA and ROSAT, the X-ray center is consistent with the optical center of the galaxy, 15$^h$ 06$^m$ 29.26$^s$; +01 36 20.7.
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The ASCA image also detects an X-ray source at the position of NGC5850. We also examined the ASCA image in the harder band, and found only a peak at the galaxy center, with no detected surrounding diffuse emission (see source profile in Fig.\[hard\_pro\]). For our analysis, we used the 3–6 keV image to determine the normalization for the SIS blank sky background. Finally, we note that in analyzing the 0.4–0.6 keV ASCA image, in the outer regions of the SIS chips, we found a possible additional background component with a very soft spectrum.
Spectral Analysis of NGC5846 {#sec:spe}
-----------------------------
In the spectral modeling of NGC5846 from both the ROSAT and ASCA observations, we used a single temperature MEKAL model (Mewe 1985, Liedahl 1995), modified by absorption, with the column density fixed to the galactic value of $4\times10^{20}$ cm$^{-2}$ (Stark 1992) and the source redshift of 0.0058. The solar values for Si and Fe are defined as 3.55e-5 and 4.68e-5 by number, compared to hydrogen (Anders & Grevesse 1989). Fitting was done over the energy ranges of 0.2–2.0 keV for ROSAT and 0.7–2.2 keV for ASCA. For PSPC spectral analysis elements other then Fe are fixed to the ASCA values.
Our choice of the MEKAL plasma code leads to systematically lower temperatures (at the 20% level), compared to the Raymond-Smith code (Raymond 1977). Also, as was shown in a study by Matsushita (1998), if Fe abundance is decoupled from the other heavy elements, differences in the derived Fe abundances are only 20–30%. If there is excess absorption at the galaxy center, only the innermost ROSAT abundance measurements would be affected by fitting the spectra with fixed galactic absorption. Omitting the low-energy channels in the analysis of ASCA spectra results in the absence of sensitivity to both the O abundance level and small variations in galactic column. Due to the complications of cooling, as well as absorption, we do not present the abundance results for the central region.
To perform a spectral analysis for both ASCA and ROSAT, an ARF matrix was calculated, using a 3-dimensional model of the source, based on the surface brightness profile, derived from ROSAT data and taking the PSF into account (see Appendix). Although the count rate in the 0.5–2.0 keV ROSAT PSPC band is strongly affected by variations in temperature and abundance for values typical for NGC5846, a problem would arise only if these changes are within one data bin, chosen for further analysis. Otherwise changes only affect the normalization, which we remove in our calculation of the scattering matrix, by normalizing the redistribution of counts on the input image.
To exclude serendipitous sources from the ROSAT spectra of NGC5846, we omitted a circular region around each detected source. The radius of the circle was chosen to match the radial dependence of the PSF, with 1 the smallest radii for on-axis sources. To generate the ASCA spectra for NGC5846, we subtract the contribution from background sources by normalizing the “blank sky” reference spectrum to our ASCA image in the 3–6 keV band.
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Figure \[te\_pro\] shows good agreement between the radial temperatures derived from ASCA and those from ROSAT, particularly outside the cooling region. Cooling is important in the central 20 kpc. Outside this region, the coronae is approximately isothermal within 50 kpc. Outside 50 kpc, the ASCA data suggest a decrease in temperature, which also is seen in the ROSAT map of the projected temperature (Fig.\[te\_map\]). In Fig.\[te\_pro\] we show a fit to our radial temperature measurements, which we use in Section 4.1 to derive the total mass profile.
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With the energy resolution of ASCA and ROSAT, we can derive the distribution of iron and silicon, outside the central cooling region. In Figure \[ab\_pro\] we present the radial profiles for the abundances of Si and Fe, derived from ASCA data (also given in Table 2), and the Fe profile, derived from ROSAT data. Abundance units are as stated in Section \[sec:spe\]. Both Si and Fe profiles decrease with radius. For comparison, in this figure we also present the stellar Fe abundance, calculated from the Fisher, Franx & Illingworth (1995, hereafter FFI) data on Fe$_{4668}$ using the modeling of line strength indices by Worthey (1994) for a galaxy age of 12 Gyr, chosen using the values H$_{\beta}=1.44$ and Fe$_{4668}=6.40$ from the FFI data on the galaxy nucleus. The “age-metallicity degeneracy”, as summarized by Kodama & Arimoto (1997), has little effect on results for giant ellipticals. Also, there is good agreement in the simulations by different researchers.
Although overall abundances of other elements, like Ne and Mg, could be constrained from our ASCA analysis of NGC5846, the emission from these lines is too weak to derive a radial distribution. Even placing respective upper limits is not straightforward, due to the overlap of emission from these elements with the poorly known Fe L-shell complex (e.g. Mushotzky 1996).
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While the Fe abundance level provides constraints on the duration of star formation and the SN Ia rates, comparison of Si and Fe abundances determines the role of different types of SNe in the chemical enrichment. Such an analysis for groups and clusters of galaxies was performed by Mushotzky (1996), Fukazawa (1998) and Finoguenov & Ponman (1998). Clusters reveal Si/Fe ratios favoring the prevalence of SNe II in the overall enrichment with Si/Fe increasing with cluster temperature (Fukazawa 1998). On the other hand, groups and the centers of clusters are characterized by a high input from SNe Ia (Fukazawa 1998, Finoguenov & Ponman 1998) with a significant dispersion in the Si/Fe ratio around the solar value (Finoguenov & Ponman 1998). The observed Si to Fe mass ratio for NGC5846 is similar to that found for groups, although the total iron mass converges to an IMLR (Iron Mass to Light Ratio, calculated as $M_{Fe}$/$L_B$ in /) of $0.9\pm0.2\times 10^{-4}$ (with $1\sigma$ uncertainty), which is much lower than in groups of galaxies (Finoguenov & Ponman 1998). This evidence for substantial enrichment of an elliptical galaxy by SNe Ia is an important indicator of post-formation SN Ia activity, since prevalence of SN II in stellar chemical enrichment is expected for systems with a short duration of star formation.
Analysis of the X-ray emission, in the 3–6 keV energy band, shows no extended “hard” emission around NGC5846, in excess of that associated with the 1 keV corona. As shown in Figure \[hard\_pro\], the hard emission, in excess of the best fit MEKAL model, is consistent with arising from an unresolved source located at the galaxy center. Nevertheless, following a suggestion from the referee, we determined the limits on a hard component, assuming a diffuse origin. For that, we have chosen one large region on the detector, calculated and subtracted the contributions from all the “soft” components we have identified. The residual spectrum was fit in the 3–6 keV energy band, assuming a bremsstrahlung spectrum of 5 keV temperature. The corresponding luminosity in the 0.5–4.5 keV band is $3.6\pm0.7\times10^{40}$, although vignetting effects were not taken into account, and could lead to an amplification of the measured flux by a factor of 1.1 for a 1 source size up to a factor of 1.4 for a 10 size. This value is comparable to the “expected” value of $6.\times10^{40}$, taken from the work of Matsumoto (1997, Ma97). However the flux associated with this component can be explained by the variations of the background, since the flux is only $\sim20$% above the calculated background level, compared to the expected 10% variations between the actual and predicted background fluxes. If a galactic population contributes hard X-ray emission, that emission should be concentrated within the confines of the optical galaxy. For NGC5846, the observed hard emission in the galaxy core corresponds to a luminosity of $6.3\pm2.0\times10^{39}$. The flux also could arise from a low-luminosity AGN.
NGC5850
========
As noted in section [*2.1.1*]{}, NGC5850 (SBb galaxy, Sandage & Tammann 1988, z=0.0084) is detected in both ROSAT and ASCA images. Although the extended nature of the emission is evident in the ROSAT image, the significance of the detection is insufficient to allow a separate spectral analysis of the bulge and arm components. With ROSAT, using the MEKAL model, we find $kT_e=0.32^{+0.41}_{-0.15}$ keV (the abundance is uncertain). The spectral fit yields $\chi_r^2=0.80$ for 24 degrees of freedom. We measure a flux for NGC5850 from 0.5 to 2.0 keV of $5.06\times10^{-14}$(which corresponds to a luminosity of $1.6\times10^{40}$, assuming a distance to NGC5850 of 51 Mpc). For the ASCA spectral analysis, we again use the MEKAL model and also fit the contribution from NGC5846 to the region around NGC5850. We find a gas temperature of $0.47^{+0.16}_{-0.11}$ keV, with an upper limit on the heavy element abundance of Z/Z$_\odot<0.04$. However, the quality of the fit is poor ($\chi_r^2=1.46$ for 47 d.o.f). An equally good fit is obtained for a power law spectrum with absorption. The constraints on the spectral parameters are =1.2$^{+8}_{-1}\times
10^{21}$ cm$^{-2}$ with $\alpha>2.3$ ($\chi_r^2=0.88$ with 24 degrees of freedom) and $<4\times 10^{21}$ cm$^{-2}$ with $\alpha=4.1^{+2.6}_{-0.9}$ ($\chi_r^2=1.45$ for 47 d.o.f) for ROSAT and ASCA data respectively. We note that the source spectrum is softer than for low-luminosity AGN found in the centers of some spirals (Iyomoto 1997), but comparable to the soft spectrum observed in the extended emission for spirals (Marston 1995; Read, Ponman and Strickland 1997).
Discussion
===========
Gravitating Mass Determination
-------------------------------
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In Figure \[mass\_pro\] we show the gravitating mass profiles derived using the equation of hydrostatic equilibrium and the data on temperature and density distributions. We choose the parameters for the density distribution, given in Table 1, corresponding to the analysis of the surface brightness profile, which take into account the effects of temperature and abundance changes with radius. The temperature distribution was modeled in two ways. First we used an isothermal gas with a mean temperature 0.85 keV (grey line). Second, we used the temperature variation from an analytic fit to the ASCA and ROSAT data (black line) in the form $ A_3 e^{A_1 log(r)^2 +
A_2 log(r)} $, chosen to provide an easy way to calculate the $ d log(T) / d
log(r) $ term in the mass determination. The maximum deviation in the mass between these two approaches is less then $\pm30$% at any radius. Nevertheless, with the decrease in gas temperature beyond 5 (50 kpc) suggested from the ROSAT and ASCA data, the mass profile converges to a M/$L_B$ value of $53\pm5$ / at a radius of 230 kpc (note that the temperature measurements extend only to 150 kpc). The total mass determined from data (Biermann 1991) is shown with a triangle and is in good agreement with our determinations. In Figure \[mass\_pro\] we also show the gas mass derived from spectral analysis of the ROSAT (grey lines) and ASCA (solid lines) data. The gas mass is typically only a few percent of the total mass, similar to that found in other elliptical galaxies (Forman, Jones & Tucker 1985).
Stellar metallicities, SN Ia rates and questions of star formation.
--------------------------------------------------------------------
It is generally agreed that early type galaxies are currently passively evolving stellar systems, where significant star formation was cut off at early epochs by a galactic wind ( Ciotti 1991, David 1991). The large amount of hot gas presently found in E’s is attributed to stellar mass loss, and as such, should be characterized by stellar metallicity and stellar velocity dispersion, with supernovae supplying additional elements and energy into the interstellar medium. Previous work, where this scheme was adopted, met a mismatch between stellar abundances, derived via modeling of optical measurements, and the tremendously low metallicities found in the X-ray gas, which imply both a low metal content in stars and a SN Ia rate which is lower then measured in optical searches ( Turatto 1994).
Perhaps the first step towards resolving this apparent discrepancy was the discovery of strong abundance gradients in the stellar content of ellipticals ( Carollo, Danziger and Buson 1993, hereafter CDB; FFI). Combining these results with the presence of gas inflow, LM were able to explain the low X-ray abundances, although the problem of low SN Ia rate persisted.
Triggered by numerous exciting achievements on the observational side, a number of theoretical models were developed to explain different aspects of the chemical evolution in ellipticals ( David 1991). However, detailed modeling of the radial behavior of the chemical content of the X-ray gas in elliptical galaxies has yet to be done. Nevertheless, there are two cornerstones that define this modeling. First is a set of models of elliptical galaxies with different mass, calculated by Matteucci (1994) and second is the suggestion by CDB that the optical abundance gradients result from the duration of star formation, which is shorter at larger radii, once a standard dissipative collapse model (Carlberg 1984) is adopted for the formation of ellipticals.
Assuming that NGC5846 is typical, we can use our measured profiles of silicon (an $\alpha$-element) and iron to constrain the enrichment models for gas-rich ellipticals. We build our model using the results of a classical wind model, presented in Table 1 of Matteucci & Gibson (1995, hereinafter MG) for three choices of the IMF, namely Salpeter (S), Arimoto & Yoshii (AY) (with A=0.02) and Kroupa (K). The models are constrained to reproduce the optical stellar abundance measurements as well as the X-ray determined abundance gradients at radii exceeding 2, where effects of a central cooling flow are negligible. We adopt the time of the galactic wind onset as a measure of the duration of star formation.
In detail, the modeling consists of adjusting the onset of the galactic wind to fit the derived stellar Fe and X-ray Si and Fe abundance measurements at all radii, using the predictions from one-zone models of MG. Enrichment from SN Ia is not considered, due to its present uncertainty, and X-ray abundances are assumed to be produced by stars, although we will later return to the predictions for SN Ia rate implied by the model. While a complete modeling including the gas mass exchanges, such as the one presented in Martinelli (1997), is beyond the scope of this paper, we will be able to draw a number of important conclusions from our simple model.
We parameterize the onset of the galactic wind as $$t_{gw} = t_{gw_o} \; exp({-({r / r_e})^{-\gamma}})$$
where $r_e$ is fixed to 6.3 (1/10 of the effective radius for NGC5846), and $t_{gw_o}$ and $\gamma$ are allowed to vary to match the actual abundance steepening.
The results of fitting the measured abundance profiles for NGC5846 with the three models including different IMFs are presented in Table 3. The best-fit curves for the predicted stellar abundances of Si and Fe are plotted in Fig.\[ab\_pro\]. From the slopes considered for the Initial Mass Function, the Kroupa IMF provides the best description of the Si data, while all the models provide only an approximate fit to the X-ray data on Fe. Yet, given the complexity with the modeling of SN Ia rate discussed below, we consider the results of this fitting as indicative, rather then an evidence for any particular choice of IMF slope.
Figure \[mod\_tgw\] suggests that a strongly decreasing duration of star formation with distance from the center of the galaxy causes the decline in stellar metallicity and SN Ia rate with radius. These results suggest that the duration of star formation in the outskirts of NGC5846 is extremely low, compared to the central value. At a radius of 100 kpc, the star formation duration would be only about $10^7$ yr. However, with such a short period of star formation, one should observe a SN II like element ratio, which is contrary to our measurements for Si and Fe, although a drastic difference between models and X-ray data in Fig.\[ab\_pro\] is seen mostly for X-ray data inside 60 kpc. While below, we consider a dependence of SN Ia rate on the metallicity of the progenitor star, an alternative way to explain both the low Si/Fe ratio and low Fe abundance is to introduce a prolonged, but inefficient star formation. We propose a scenario in which the declining metallicity results from a change in the efficiency of star formation with galaxy radius, while the duration of star formation does not vary significantly with radius. The constancy in the duration of star formation can be explained by the short time required for a galactic wind to terminate star formation everywhere in the galaxy. A small radial dependence in the time for star formation triggering is negligible compared to the typical duration of star formation $\sim3$ Gyrs, required for the SN Ia to play a significant role in the enrichment (Yoshii 1996).
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A drop in the efficiency of star formation with radius could result from a corresponding decrease in the density of gas clouds. This scenario would result in lower metallicities at larger radii, but similar abundance ratios throughout the galaxy, as we find for NCG5846. The observation that the abundance ratio of Mg to Fe does not change within an individual galaxy, but changes quite drastically from galaxy to galaxy (Worthey 1992) also supports this scenario. In addition, since this scenario just modifies the standard modeling by including an additional parameter, the efficiency of star formation as a function of radius, it should also meet other observational restrictions discussed in Matteucci (1994).
Considering the predictions for present-day SN Ia rates, illustrated in Fig. \[mod\_snr\], we note that for all the models, the resulting SN Ia rate in SNU is zero, except in the central region ($r<20$ kpc) of NGC5846. This result is quite different from that found, when a constant star formation rate with radius is assumed (as in LM). The behavior of the SN Ia rates in MG probably results from the choice of the evolutionary tracks from Van den Bergh & Bell (1985). As was shown by Bazan & Mathews (1990), these tracks correspond to shorter main sequence lifetimes for low metallicity stars due to their higher luminosities. In calculating the theoretical SN Ia rates, normalized to the blue luminosity, a low metal content in stars implies both [*faster*]{} evolution (and as a consequence a stronger decrease with time) of the SN Ia rate and enhanced stellar luminosities. We have verified this suggestion, by a detailed modeling of the SN Ia rate, as described in David (1991 and references therein), and found that the influence of the parameters described above results in suppressing the present SN Ia rate for low Z stars by $\sim 20$ times.
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Nevertheless, since the evolution of the stellar mass loss is based on the lifetime of stars on the main sequence, mass loss for the low metallicity stars at outskirts of the galaxy is also lower, and the ratio of SN Ia over stellar mass loss does not change significantly in our modeling, compared to estimates from LM. Therefore, the strict X-ray limits, usually calculated for the solar metal composition, are still a problem for the rate of SN Ia events found in optical searches (for a recent review of this issue, see Arimoto 1997). However, the X-ray limits on the SN Ia rate correspond to a different companion mass for the SN Ia progenitor. Thus one can, in principle, by changing the distribution of the probability of an SN Ia event as a function of mass ratio in a binary, satisfy both the X-ray and optical requirements. Another possibility, discussed in Timmes, Woosley & Weaver (1995) and Kobayashi (1998) is a dependence of the SN Ia rate on the metallicity of the progenitor star. In this respect, we would like to point out, that if a dependence of the SN Ia rate on stellar metallicity is assumed, the stellar Si/Fe ratio could be larger than that observed for the hot gas, altering our conclusions on the efficiency of star formation.
To illustrate the problem with SN Ia rates, we calculate the upper limits on the SN Ia rate, assuming that all the metals in the outskirts of NGC5846, found in our X-ray measurements, are due to SN Ia explosions. We then use all the specific values, calculated for the solar metal composition, so these limits can be directly compared to the SN Ia rate predictions for the galaxy center.
Following LM, the X-ray derived abundances can be expressed as
$$\label{fe_eq}
[Z/H]=[Z/H]_* + {f_{\rm SN} \; {\rm SNU} \; M_{{\rm
SN},_{Z}} \over {M_* / L_B} \; \; {\dot{M}_* / M_*}}\;,$$
where $f_{\rm SN}$ is the SN Ia rate in units of an SNU (1 SNU = one event per 100 yrs ${ L_B}$/10$^{10}$), equation has $M_{{\rm SN},_{Z}}$, but here is uses $M_{{\rm SN},_{Fe}}$ – the mass of iron released in each SN Ia event, and ${\dot{M}_* / M_*}$ is the stellar mass loss, which we adopt as 3 (for the Salpeter and Kroupa IMFs) or 5 (for the Arimoto & Yoshii IMF) $\times10^{-20}$ sec$^{-1}$ (following calculations by Mathews 1989). In such an approximation, we implicitly assume that the SN Ia rate changes with time, similar to the stellar mass loss. We adopt a central $M_*/L_B=11.3$/ by scaling the value of 19 from Lauer (1985), obtained for $H_o=84$km sec$^{-1}$ Mpc$^{-1}$. Z denotes the metal considered. In placing upper limits on the SN Ia rates, we neglect the stellar (first) term on the right side of the equation.
The upper limit on the SN Ia rate using our measurements for silicon in the 5–15 annulus is $0.012$ SNU ( 1 SNU = 1 event (100 yr)$^{-1}$ ${L_B}$/10$^{10}$), for a Salpeter IMF (1.35), and 0.021 SNU, for an IMF slope of 0.7. This calculation is independent of the modeling for the L-shell emission for iron and the effects of a central cooling flow or incomplete thermalization attributed to the central regions. Corresponding limits, using Fe, are 0.006 and 0.010 SNU for IMF slopes of 1.35 and 0.7, respectively.
The possibility of significant amounts of additional iron (or silicon) in the form of dust is low, since the dust mass inferred from recent observations is $\sim 7\times 10^3$ (Goudfrooij & Trinchieri 1998). While dust is not likely to play an important role in the abundance, it could be a major factor for the energy balance of the hot gas in the center of NGC5846 (Goudfrooij & Trinchieri 1998).
Recently ( Matsushita 1998), the X-ray emission associated with some ellipticals was suggested to have a group component, based on a “beta-model” analysis of the surface brightness profile. While our chemical enrichment modeling is insensitive to the presence of the surrounding group potential (LM), it is sensitive to ascribing the detected metals and gas to being a product of stars at the corresponding radius. Alternatively, detected elements could be treated as having been expelled from the galaxy center and held in the group potential. The observed X-ray gas would consist of the galaxy products: SN Ia output and stellar mass loss [*recently mixed*]{} with the intragroup gas, which should have low heavy element abundances, due only to the contribution of other galaxies in the group. The abundance gradient indicates the degree of mixing, while the constancy of the abundance ratio with radius, measured in X-rays, results because the SN Ia rate evolves like the stellar mass loss (LM).
However, this scenario contradicts the observed picture for NGC5846. The amount of iron should indicate the time of group formation. For the SN Ia rate at the galaxy center of 0.16 SNU, implied by optical observations (Turatto 1994), the observed iron mass in NGC5846 would be reproduced in less then 1 Gyr. Thus the group formation must be recent, which is surprising, considering the hierarchical clustering hypothesis. A possible solution would be if the group formed long ago, but the energy released by SN Ia continued to drive a galactic wind until the recent epoch. The calculation shows that the energy associated with the release of iron via SN Ia explosions is less then 10% of the thermal energy of the gas at any radius in NGC5846. It is dubious that such a small contribution recently played a dramatic role.
Finally, we comment on the energy balance in NGC5846. The kinetic energy of the stars is only 1/3 the observed thermal energy of the X-ray gas. The rest of the energy may have come from the gravitational infall of the gas. We can use our measured mass profile to verify this possibility. We find that sufficient energy can be produced if the gas falls in adiabatically from only 1/3 of the initial radius. An alternative explanation for the energy budget of the X-ray emitting ellipticals, was proposed by Mathews & Brighenti (1997), in which the temperature of the gas is due to the inflow of hot circumgalactic gas. We note, that a decline of the temperature with outer radius in NGC5846 contradicts the basic assumptions of this scenario, since it implies that the circumgalactic gas, if it exists in a considerable amount, is too “cold” to reproduce the measured temperature profile by simple gas mixing with “cold” gas of the galaxy, assumed in the model. Note, however, that in NGC5044 and NGC4472, such a drastic temperature drop is not observed (David 1994; Finoguenov & Ponman 1998; Forman 1993).
Conclusions
============
Applying methods of spatially resolved spectroscopy, we studied X-ray emission from hot gas in the elliptical galaxy NGC5846. We detect cooler gas both toward the galaxy center and in the outer parts of the galaxy. Such a decreasing temperature profile, if continued to larger radii, leads to a convergence of the total mass in NGC5846 to $9.6\pm1.0 \times
10^{12}$ at a radius of $\sim$230 kpc.
Our study of the heavy element distribution reveals radial abundance gradients in both Fe and Si, but a constant Si/Fe ratio at the level favoring a greater retention of SN Ia products, compared to SN II. Comparison of optical and X-ray data shows remarkable agreement in the radial distribution, with Fe abundance falling from a value of 1.3 times solar at the center to a value of 0.1 solar at 100 kpc. We apply the theoretical models of Matteucci & Gibson (1995) to quantify our data. These models are characterized by a decrease in the duration of star formation from a few Gyr in the central region of the galaxy to a duration of only $\sim0.01$ Gyr at 100 kpc. Alternatively, the decline in metallicity with galaxy radius may be caused by a variable efficiency of star formation as a function of the galactocentric distance.
The model prediction for the present-day SN Ia rate is non-zero only in the galaxy center and is caused by faster evolution of the low-metallicity stars found at the galaxy outskirts. Nevertheless, our calculation shows that tight X-ray limits, usually calculated for the solar element composition, remain a problem, since the SN Ia rate evolves like the stellar mass loss. To provide an acceptable explanation of X-ray and optical measurements, we suggest an introduction of a dependence of SN Ia rate on the progenitor star metallicity, as is suggested by Kobayashi (1998).
We also investigated alternative mechanisms for the low iron abundance in NGC5846. One alternative explanation for the low metallicity is that the primary emission arises from a group and NGC5846 only pollutes the surrounding intragroup media. However, under the assumption of standard SN Ia rates, the low iron mass detected leads to a recent time for group formation ($<1$ Gyr), which contradicts this scenario. Expelling the iron from the gravitational potential of NGC5846 via recent galactic winds, driven by SN Ia’s, also is unlikely, since the energy release by SN Ia’s is small, less then 10% of the thermal energy of the gas.
This research has made use of data obtained through the HEASARC Online Service, provided by the NASA/GSFC. C.J. and W.F. acknowledge support for this research from the Smithsonian Institution and the AXAF Science Center NASA Contract 16797840. A.F. was supported by the Smithsonian predoctoral fellowship program. Authors wish to thank Maxim Markevitch for useful discussions and the referee for helpful comments.
Anders E. and Grevesse N. 1989, [*Geochimica et Cosmochimica Acta*]{}, 53, 197 Arimoto N., Matsushita K., Ishimaru Y., Ohashi T., Renzini A., 1997, ApJ, 477, 128 å[208]{}[22]{}[89]{}[Biermann P.L., Kronberg P.P. and Schmutzler T.]{} Briel U.G., Hasinger G., Voges W. , 1993, The ROSAT Users’ Handbook Burke B.E., Mountain R.W., Harrison D.C., Bautz M.W., Doty J.P., Ricker G.R. and Daniels P.J., 1991, IEEE Trans., EED-38, 1069 Fukazawa Y., Makishima K., Tamura T., Ezawa H., Xu H. , 1998, PASJ, 50, 187 Gotthelf E., 1996 in Proceedings of III-d ASCA Symposium å[330]{}[123]{}[98]{}[Goudfrooij P. and Trinchieri G.]{} Iyomoto N., Makishima K., Fukazawa Y., Tashiro M. and Ishisaki Y. 1997. [*Frontiers Science Series*]{} No. 19, ed. F. Makino and K. Mitsuda, Tokyo. Kobayashi C., Tsujimoto T., Nomoto K., Hachisu I., Kato M., 1998, ApJ [*Letters*]{} (accepted); also preprint astro-ph/9806335 å[320]{}[41]{}[97]{}[Kodama T. and Arimoto N.]{} Matsushita K., Makishima K., Ikebe Ya., Rokutanda E., Yamasaki N., Ohashi T., 1998, ApJ [*Letters*]{}, 499, 13 å[288]{}[57]{}[94]{}[Matteucci F.]{} å[304]{}[11]{}[95]{}[Matteucci F. and Gibson B.K.]{} reference Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P. 1992, Numerical recipes in FORTRAN Sandage A. and Tammann G.A. 1987, Revised Shapley-Ames Catalog of Radial Velocities of Galaxies Timmes F. X., Woosley S. E. and Weaver T. A., 1995, ApJS, 98, 617
Incorporation of the ASCA Point Spread Function into Spectral Analyses {#incorporation-of-the-asca-point-spread-function-into-spectral-analyses .unnumbered}
======================================================================
In this chapter we will explore the effect produced by the broad energy dependent ASCA PSF on study of diffuse objects.
A general expression for the observed spectrum reads $$\eta(y,e^\prime) = \int_{E} r(e^\prime \mid e)
\beta(y) \int_{X} p\tild(y \mid x, e) \nu\tild(x,e) dx de$$
with the following definitions:
$\nu\tild(x,e)$ – emission from the source, where x stands for spatial coordinates and $e$ – energy
$p(y \mid x, e)$ – the Point Spread Function including telescope effective area
$y$ – refers to projected coordinates on the detector plane,
$r(e^\prime \mid e)$ – is the detector response matrix yielding the probability distribution of detecting a photon of energy $e$
$\beta(y)$ = 0 for “not valid” detector pixels like gaps or chips not used in the observation (different CCD clock mode), 1 - otherwise.
$\eta(y,e^\prime)$ – is the observed photon spectrum in units of ph sec$^{-1}$ cm$^{-2}$ keV$^{-1}$ arcmin$^{-2}$
where accumulated spectra are: $$\eta_j(e^\prime) = \int_{Y_j} \eta(y,e^\prime) dy$$ and $Y_j$ is a set of spatial regions in the image data, such that $Y_{j_n}\bigcap Y_{j_m} =\emptyset $ for any $j_n \neq j_m$. This is required for further use of standard likelihood estimators.
A [*key assumption*]{} is that spatial and spectral properties can be separated
$$\nu\tild(x,e) = \sum_i { \lambda_i(x)\nu_i(e)} \theta(x,X_i)\ \
,\ \ \ \ \ \int_{X_i} \lambda_i(x) dx = 1$$
where $\theta(x,X_i)=1$ if $x\in X_i$ and 0 otherwise, and $X_i$ is a set of spatial regions in the object image with $\bigcup_i X_i = X $ by definition. $\lambda_i(x)$ is the spatial distribution which we take from the surface brightness measurements of the ROSAT PSPC.
Now, we can separate the spectral and imaging parts:
$$\eta_j(e^\prime) = \int_{E} r(e^\prime \mid e) \sum_i \nu_i(e)
\int_{Y_j} \beta(y)\int_{X_i} p\tild(y \mid x, e) \lambda_i(x) dx dy de$$
The imaging part can be rewritten in a matrix form which further accounts for different telescope pointings:\
$$a_{ij}\tild(e) = \int_{Y_j} \beta(y)\int_{X_i} p\tild(y \mid x, e)
\lambda_i(x) dx dy \ \ ,\ \ \ \ \ a_{ij}\tild(e)\equiv a_{ij}\tild(e,k)$$
where $a_{ij}\tild(e,k)$ is a spatial correlation matrix, $k$ represents different telescope pointings.
We define $$A_{ij}(e) = \sum_k t_k a_{ij}\tild(e,k)$$ where $t_k$ is the time spent at each pointing. With this definition equation 5) becomes
$$\eta_j(e^\prime) = \int_{E} r(e^\prime \mid e) \sum_i \nu_i(e) A_{ij}(e)
\ de$$
The procedure of fitting the data with spectral programs like XSPEC is as follows:
1\) produce a set of models $ \nu_i(e)$\
2) mix them using calculated $A_{ij}(e)$\
3) convolve them with response matrix $r(e^\prime \mid e)$.
In the analysis presented in this paper, we used ASCA XRT ray-tracing azimuthly averaged PSF for the calculation of the scattering from the central part of the source emission (inside 5 radius) and the PSF data from ASCA GIS measurements (Takahashi , 1995) for the rest.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We provide lower bounds on the gonality of a graph in terms of its spectral and edge expansion. As a consequence, we see that the gonality of a random 3-regular graph is asymptotically almost surely greater than one seventh its genus.'
author:
- Neelav Dutta
- David Jensen
bibliography:
- 'Paperbib.bib'
title: Gonality of Expander Graphs
---
Introduction
============
In this paper, we study the relationship between the gonality of a graph $G$ and its expansion properties. In particular, we provide lower bounds on the gonality ${\operatorname{gon}}(G)$ in terms of the Cheeger constant $h(G)$. We refer the reader to Section \[Sec:Prelim\] for definitions of these graph invariants. Our main result is the following.
\[Thm:LowerBound\] For any $u \in (0, \frac{1}{2}]$, let $B_{u}(G)$ be the smallest degree of an effective divisor $D$ such that every connected component of $V(G) \smallsetminus {\operatorname{supp}}(D)$ has size at most $u \vert V(G) \vert $. Then $$gon(G)\geq \min \left\{ B_{u}(G), h(G)u \vert V(G) \vert \right\}.$$
Note that $B_u (G)$ is decreasing in $u$, while $h(G)u \vert V(G) \vert$ is increasing in $u$. As a consequence of Theorem \[Thm:LowerBound\], we obtain the following lower bound on the gonality of a random 3-regular graph.
\[Thm:RandomThreeRegular\] Let $G$ be a random 3-regular graph on $n$ vertices. Then $${\operatorname{gon}}(G) \geq 0.072n$$ asymptotically almost surely.
Note that the genus of a 3-regular graph on $n$ vertices is $\frac{1}{2}n+1$. Theorem \[Thm:RandomThreeRegular\] therefore implies that the expected gonality of a random 3-regular graph is at least one seventh its genus. This result is obtained by bounding the invariants $h(G)$ and $B_u (G)$ that appear in Theorem \[Thm:LowerBound\] for random 3-regular graphs. This yields a lower bound on the gonality of such graphs that depends on $u$, which is optimized for some $u$ between $0.35$ and $0.40$.
Our primary motivation for studying the gonality of graphs comes from the theory of algebraic curves. In [@Baker08], Baker develops a theory of specialization of divisors from curves to graphs. This theory allows one to establish results in algebraic geometry using combinatorial techniques, and conversely, to discover new combinatorics using algebraic geometry. Because the top-dimensional strata of $M_g^{{\operatorname{trop}}}$ correspond to 3-regular graphs, such graphs are considered to be the closest analogues of algebraic curves. It is therefore natural to compare invariants of random 3-regular graphs to the analogous invariants of general curves.
By [@Kempf71; @KleimanLaksov72], the gonality of any curve of genus $g$ is at most $\lfloor \frac{g+3}{2} \rfloor$. Moreover, the Brill-Noether theorem implies that equality holds for the general curve [@GriffithsHarris80]. In other words, the set of genus $g$ curves of gonality exactly $\lfloor \frac{g+3}{2} \rfloor$ is a dense open subset of the moduli of space of curves $M_g$. One could hope to address the analogous questions for 3-regular graphs. First, what is the maximum gonality of a 3-regular graph? Second, what is the expected gonality of a random 3-regular graph? Theorem \[Thm:RandomThreeRegular\] represents progress toward the second of these questions.
Theorem \[Thm:RandomThreeRegular\] is an improvement on several earlier results. In [@treewidth], de Bruyn and Gijswijt show that the gonality of a graph is bounded below by its treewidth. In [@KM93], the authors show that a random 3-regular graph on $n$ vertices almost surely has bisection width at least $\frac{10}{99}n$. As noted in [@NM08], this implies that such a graph almost surely has treewidth at least $\frac{5}{99}n$. In [@AminiKool], Amini and Kool use the results of [@treewidth] to produce a lower bound on graph gonality in terms of the algebraic connectivity. They then use this to show that the gonality of a random $k$-regular graph is bounded above and below by linear functions in $n$. While [@AminiKool] do not attempt to bound the implied constants, their approach cannot yield a bound higher than $\frac{5}{99}n$, due to their reliance on [@treewidth]. In a related direction, [@AminiKool] shows that the expected gonality of an Erdos-Renyi random graph on $n$ vertices is also bounded above and below by functions that are linear in $n$. An improvement of this result appears in [@randomgonality], where it is shown that the expected gonality of such graphs is in fact asymptotic to $n$.
The argument of the present paper is in some ways similar to the proof in [@treewidth] that gonality is bounded below by treewidth. The treewidth of a graph can be defined in terms of certain collections of connected subsets of the vertices, known as brambles. One example of a bramble is the set of all connected subsets of $V(G)$ of size at least $\frac{1}{2} \vert V(G) \vert$, which implies that the treewidth of a graph $G$, and hence its gonality, is at least $B_{\frac{1}{2}} (G)$. To improve on this, we note that [@treewidth] in fact uses the treewidth in two separate ways. First, the treewidth is a lower bound on the degree of an effective divisor that intersects every element of a bramble. Second, it provides a lower bound on the size of a cut that separates two elements of this bramble. If we consider instead the set of all connected subsets of size at least $u \vert V(G) \vert$ for some $u \leq \frac{1}{2}$ (which is not a bramble), then the minimum size of a cut that separates these subsets decreases, while the minimum size of an effective divisor that intersects every subset increases. By varying $u$, it is possible to obtain a higher bound.
The invariant $B_u (G)$ is somewhat mysterious, and much of the paper is devoted to finding bounds for it in terms of more familiar graph invariants, including the $u$-Cheeger constant $h_u (G)$ and the algebraic connectivity $\lambda_2$. We again refer the reader to Secion \[Sec:Prelim\] for definitions of these graph invariants. Specifically, we establish the following results.
\[Thm:CheegerBound\] Let $G$ be a $k$-regular graph. Then, for any $u$, we have $${\operatorname{gon}}(G) \geq \min \left\{ \frac{h_{u}(G)}{k+h_{u}(G)} \vert V(G) \vert , h(G)u \vert V(G) \vert \right\} .$$
\[Thm:SpectralBound\] Let $G$ be a graph, and let $d$ be the maximum valence of a vertex in $G$. Then $${\operatorname{gon}}(G) \geq \frac{\vert V(G) \vert}{2\lambda_2} \left[ -(7\lambda_2 + 9d) + 3\sqrt{9\lambda_2^2 + 14d\lambda_2 + 9d^2} \right] .$$
In the case of primary interest to us, namely that of random 3-regular graphs, the bound provided by Theorem \[Thm:CheegerBound\] is stronger than that of Theorem \[Thm:SpectralBound\]. Nevertheless, Theorem \[Thm:SpectralBound\] has at least two advantages. First, while Theorem \[Thm:CheegerBound\] applies only to regular graphs, Theorem \[Thm:SpectralBound\] applies to arbitrary graphs. Second, while computation of the $u$-Cheeger constant is NP-hard, the algebraic connectivity can be computed, to any degree of accuracy, in polynomial time. For this reason we expect that Theorem \[Thm:SpectralBound\] may be more useful for applications, as it is more efficient.
While this paper is primarily concerned with lower bounds on graph gonality, we make a few brief remarks on upper bounds. It follows from Baker’s specialization lemma that, if $G$ is a graph of genus $g$, then there exists a positive integer $e$ such that the refinement obtained by subdividing each edge of $G$ into $e$ edges has gonality at most $\lfloor \frac{g+3}{2} \rfloor$. We note that this argument uses the Kleiman-Laksov result on algebraic curves and to date there is no known purely combinatorial proof. The question of whether this statement holds without refinement – that is, whether the integer $e$ can always be taken to be 1 – remains open. Without passing to refinements, much less is known. As far as we are aware, the best known upper bound for the gonality of a graph is its genus. Specifically, if $G$ is a graph of genus $g$ and $E$ is any effective divisor of degree $g-2$, then by Riemann-Roch the divisor $K_G-D$ has rank at least 1. It follows that ${\operatorname{gon}}(G) \leq g$. To obtain another upper bound, note that the complement of an independent set has rank at least 1, and thus ${\operatorname{gon}}(G) \leq \vert V(G) \vert - \alpha (G)$, where $\alpha (G)$ is the size of the largest independent set in $G$. For a 3-regular graph $G$, however, this bound is smaller than $g$ if and only if $G$ is bipartite. In this case $\vert V(G) \vert - \alpha (G) = g-1$, so for bipartite 3-regular graphs this upper bound differs by only 1 from the bound obtained by Riemann-Roch.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The first author was supported by a gift from the Eaves family for summer undergraduate research. We thank the Eaves for their generosity. The second author was supported in part by NSF DMS-1601896.
Preliminaries {#Sec:Prelim}
=============
Divisor Theory of Graphs
------------------------
In this section, we outline the basic definitions and properties of divisors on graphs. For more details, we refer the reader to [@Divisors; @Baker08].
A *divisor* $D$ on a graph $G$ is a formal ${\mathbb{Z}}$-linear combination of vertices of $G$.
Divisors are alternatively known as “chip-configurations”, because we may view a divisor as consisting of stacks of chips on the vertices of $G$. We write $D(v)$ for the coefficient of the vertex $v$ in the divisor $D$, or alternatively the number of chips of $D$ at $v$. We define an equivalence relation on the set of divisors on $G$ using chip-firing moves, which we define here.
Let $D$ be a divisor on a graph $G$ and fix $v\in V(G)$. The divisor $D'$ obtained from $D$ by *firing* the vertex $v$ is defined by $$D'(w)=\left.
\begin{cases}
D(v) - {\operatorname{val}}(v) & \text{if } w = v \\
D(w) + \varepsilon(w,v) & \text{if } w \neq v
\end{cases}
\right.$$ where $\varepsilon(w,v)$ is the number of edges between $w$ and $v$.
If the vertices of a graph $G$ are labeled $v_1 , \ldots , v_n$, we define the *Laplacian* of $G$, denoted $L(G)$, to be the $n \times n$ matrix with entries $$L(G)_{ij} = \left.
\begin{cases}
-{\operatorname{val}}(v_{i}) & \text{if } i = j \\
\varepsilon(v_{i},v_{j}) & \text{if } i\neq j.
\end{cases}
\right.$$
Two divisors $D$ and $D'$ are said to be *linearly equivalent* if $D'$ can be obtained from $D$ by a sequence of chip-firing moves. Equivalently, $D$ and $D'$ are equivalent if their difference is contained in the image of the graph Laplacian $L(G){\mathbb{Z}}^{V(G)}$. This defines an equivalence relation on the set of divisors on $G$.
We now recall some definitions related to positivity of divisors on graphs.
A divisor $D$ is called *effective* if $D(v) \geq 0$ for all $v\in V(G)$.
The *complete linear series* of a divisor $D$ is $$\vert D \vert = \{D'\sim D \vert D' \text{ is effective}\}.$$ The *support* of an effective divisor $D$ is $${\operatorname{supp}}(D) = \{v\in V(G) \vert D(v) > 0\}.$$
Perhaps the most important property of a divisor on a graph is its Baker-Norine rank.
A divisor $D$ is said to have *rank at least* $r$ if, for every effective divisor $E$ of degree $r$, $D-E$ is equivalent to an effective divisor. The *rank* of $D$ is the largest integer $r$ such that $D$ has rank at least $r$.
Our main interest in this paper is a graph invariant known as the gonality.
The *gonality* of a graph $G$ is the smallest degree of a divisor on $G$ with positive rank.
While it is NP-hard to compute the gonality of a graph [@NPhard], the complexity arises from the sheer number of divisors that one would be required to check. If we are given a divisor, there is a relatively simple algorithm that determines whether it has positive rank. To follow the algorithm, one first needs the notion of a reduced divisor.
Let $v\in V(G)$. A divisor $D$ is said to be $v$-*reduced* if:
1. the divisor $D$ is “effective away from $v$” – that is, for all $w\in V(G) \smallsetminus \{ v \}$, $D(w)\geq 0$, and
2. for any $U\subset V(G)\smallsetminus \{ v \}$, the divisor obtained by firing all vertices in $U$ is not effective away from $v$.
Each divisor has a unique $v$-reduced representative. Moreover, if a divisor $D$ is equivalent to an effective divisor, then its $v$-reduced representative is effective. With these facts in hand, it becomes relatively easy to check that a divisor $D$ has positive rank. The divisor $D$ has positive rank if and only if the $v$-reduced divisor equivalent to $D-v$ is effective for all vertices $v$. Of course, it is crucial in this process that we be able to compute $v$-reduced divisors. To find $v$-reduced divisors quickly, we have the following algorithm, known as *Dhar’s Burning Algorithm*.
1. Start a fire at the vertex $v$, placing it in the set of burnt vertices.
2. For each burnt vertex, place each edge adjacent to it in the set of burnt edges.
3. If the number of burnt edges adjacent to a vertex $w$ exceeds the number of chips on $w$, place $w$ in the set of burnt vertices. Then repeat step 2. Otherwise, if no such vertex exists, proceed to step 4.
4. If the entire set of vertices burn, then $D$ is $v$-reduced. Otherwise, the divisor obtained by firing all unburnt vertices is effective away from $v$. Replace $D$ with this equivalent divisor and repeat the process.
Expander Graphs
---------------
In this paper we relate the gonality to the expansion properties of a graph. One of the most well-known measures of graph expansion is known variously as the Cheeger constant, isoperimetric number, or edge expansion constant. Throughout, if $G$ is a graph and $U\subseteq V(G)$, we define $\partial U$ to be the set of edges with exactly one endpoint in $U$.
[@Cheeger] The *Cheeger constant* of a graph $G$ is defined to be $$h(G):= \min_{0< \vert U \vert \leq \frac{1}{2} \vert V(G) \vert}\frac{\vert \partial U \vert}{\vert U \vert } .$$ More generally, for any $u \in (0,\frac{1}{2}]$, The $u$-*Cheeger constant* of $G$ is defined to be $$h_u(G) := \min_{0< \vert U \vert \leq u \vert V(G) \vert }\frac{ \vert \partial U \vert}{\vert U \vert} .$$ For values of $u$ greater than $\frac{1}{2}$, we vacuously define $h_u (G)$ to be infinity.
A second measure of graph expansion is the algebraic connectivity, which we define here.
The *algebraic connectivity* $\lambda_{2}$ of a graph $G$ is defined to be the second smallest[^1] eigenvalue of the graph Laplacian $L(G)$.
For $k$-regular graphs, the algebraic connectivity is related to the Cheeger constant by the well-known Cheeger inequalities: $$\frac{1}{2} \lambda_2 \leq h(G) \leq \sqrt{2k\lambda_2} .$$ One advantage of the algebraic connectivity is that the Cheeger constant is NP-hard to compute, whereas the algebraic connectivity of a graph with $n$ vertices can be computed to any degree of accuracy in $O(n^3)$ time.
The algebraic connectivity can be used to bound the size of vertex separators, which we define below.
[@Eig] Let $G$ be a graph. A set $C\subset V(G)$ is said to $\emph{separate}$ vertex sets $A,B \subset V(G)$ if
1. $A$, $B$ and $C$ partition $V(G)$ and
2. no vertex of $A$ is adjacent to a vertex of $B$.
[@Eig Theorem 2.8] \[Lemma:Eig\] Let $G$ be a graph and denote by $d$ the maximal valence of a vertex of $G$. If $C$ separates vertex sets $A$ and $B$, then $$\vert C \vert \geq \frac{4\lambda_2 \vert A \vert \vert B \vert}{d \vert V(G) \vert - \lambda_2 \vert A \cup B \vert}.$$
An important class of expander graphs is that of random regular graphs. We describe a model for such graphs, which can be found in [@Bollobas]. We fix $n$ sets $W_j$, each consisting of $k$ elements, with $kn$ even. A *configuration* is a partition of $W = \cup_{j=1}^n W_j$ into 2-element sets. We let $\Phi_{k,n}$ be the set of all configurations, and turn $\Phi_{k,n}$ into a probability space by taking every configuration to have equal probability of being selected. Each configuration in $\Phi_{k,n}$ has an associated $k$-regular, $n$-vertex graph, with an edge between two vertices $v_i$ and $v_j$ if and only if there is a set in the partition consisting of an element of $W_i$ and an element of $W_j$.
We say that a property of a random $k$-regular graph holds *asymptotically almost surely* if the probability that the property holds approaches one as $n$ approaches infinity.
Gonality and Edge Expansion
===========================
Our main observation is the following. If, for every representative of a divisor $D$ on a graph $G$, there exists a “large” connected component of $V(G) \smallsetminus {\operatorname{supp}}(D)$, then $D$ cannot have positive rank. This is made precise by Theorem \[Thm:LowerBound\].
Let $D$ be a divisor of positive rank and assume that $\deg(D) < B_{u}(G)$. For any representative $D' \in \vert D \vert$, we let $U_{D'}$ denote the union of all connected components of $V(G) \smallsetminus {\operatorname{supp}}(D')$ of size greater than $u \vert V(G) \vert$. Note that, by the definition of $B_u (G)$, for any representative $D'$, the set $U_{D'}$ is nonempty.
Choose $D_0 \in \vert D \vert$ such that $U_{D_0}$ is minimal. That is, for any $D' \in \vert D \vert$, $U_{D'}$ is not strictly contained in $U_{D_0}$. Now, pick a vertex $v \in U_{D_0}$. By [@treewidth Lemma 1.3], there exist firing sets $A_1 \subseteq A_2 \subseteq \cdots \subseteq A_n \subseteq V(G) \{ v \}$ such that the divisor obtained from $D_{i-1}$ by firing $A_i$ is an effective divisor $D_i$, and $D_n$ is $v$-reduced. Since by assumption $D$ has positive rank, $D_n(v) > 0$.
Since firing these sets $A_i$ eventually puts at least one chip in $U_{D_0}$, there must exist an index $i$ such that $U_{D_i} \neq U_{D_0}$. Let $i$ be the smallest such index. Since $U_{D_0}$ was chosen to be minimal, $U_{D_i} \not\subset U_{D_0} = U_{D_{i-1}}$. It follows that there exists some connected component $U \subset U_{D_i}$ that intersects the support of $D_{i-1}$. Since $U$ does not intersect the support of $D_i$, we conclude that $U \subset A_i$. It follows that $\vert A_i \vert \geq \vert U \vert \geq u \vert V(G) \vert$.
Let $U' \subset U_{D_0}$ be the connected component that contains $v$, and note that by assumption we have $U' \subset V(G) \smallsetminus A_i$. Since $\vert A_i \vert \leq \vert V(G) \smallsetminus U' \vert$, we have $\vert V(G) \smallsetminus A_i \vert \geq \vert U' \vert \geq u \vert V(G) \vert$. Taking the smaller of $A_i$ and $V(G) \smallsetminus A_i$, we therefore see that $\vert \partial A_i \vert \geq h(G) u \vert V(G) \vert$. Finally, since $D_i$ is effective, we must have $D_{i-1} (w) \geq {\mathrm{outdeg}}_w A_i$ for every vertex $w \in A_i$. Thus, $\deg(D) \geq \vert \partial A_i \vert$, and so $\deg (D) \geq h(G) u \vert V(G) \vert$.
The following lemma provides a bound on the invariant $B_u (G)$ introduced in Theorem \[Thm:CheegerBound\] for regular graphs.
\[Lem:CheegerRegular\] Let $G$ be a $k$-regular graph, and let $D$ be an effective divisor such that $$\deg(D) < \frac{h_{u}(G)}{k+h_{u}(G)}\vert V(G) \vert .$$ Then there exists a connected component of $V(G)\smallsetminus {\operatorname{supp}}(D)$ with size greater than $u \vert V(G) \vert$.
Assume that every connected component of $V(G) \smallsetminus {\operatorname{supp}}(D)$ has size less than or equal to $u \vert V(G) \vert$. Let $U$ be a connected component of $V(G) \smallsetminus {\operatorname{supp}}(D)$. Note that the number of edges with at least one end in $U$, here denoted $e(U)$, is $$e(U) = \frac{1}{2} (k \vert U \vert + \vert \partial U \vert ) \geq \frac{k+h_{u}(G)}{2} \vert U \vert.$$ To see this, note that the number of half-edges with one end in $U$ is exactly $k \vert U \vert$. Since the internal edges of $U$ contribute twice to this sum and the edges that leave $U$ contribute only once, we arrive at the equality above. The inequality is given by the definition of the $u$-Cheeger constant.
Let $e(V(G)\smallsetminus {\operatorname{supp}}(D))$ denote the number of edges in the complement of the support of $D$. We then have $$e(V(G)\smallsetminus {\operatorname{supp}}(D)) = \sum_U e(U),$$ where the sum is over connected components of $V(G) \smallsetminus {\operatorname{supp}}(D)$. By the above, therefore, we have $$e(V(G)\smallsetminus {\operatorname{supp}}(D)) \geq \sum_U \frac{k+h_{u}(G)}{2} \vert U \vert$$ $$= \frac{k+h_{u}(G)}{2} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert$$ $$> \frac{k+h_{u}(G)}{2}\left(1 - \frac{h_{u}(G)}{k+h_{u}(G)}\right)\vert V(G) \vert = \frac{k}{2} \vert V(G) \vert.$$
Since the total number of edges in $G$ is $\frac{k}{2} \vert V(G) \vert$, this is impossible. Therefore, there exists a connected component of the complement of ${\operatorname{supp}}(D)$ of size greater than $u \vert V(G) \vert$.
Together, Theorem \[Thm:LowerBound\] and Lemma \[Lem:CheegerRegular\] yield a lower bound on the gonality of regular graphs in terms of the Cheeger constant.
By Lemma \[Lem:CheegerRegular\], for any effective divisor $D$ with $\deg(D) < \frac{h_{u}(G)}{k+h_{u}(G)} \vert V(G) \vert$, there exists a connected component of $V(G) \smallsetminus {\operatorname{supp}}(D)$ of size greater than $u \vert V(G) \vert$. Since all effective divisors of this degree have this property, we see that $$\frac{h_{u}(G)}{k+h_{u}(G)} \vert V(G) \vert \leq B_u (G) .$$ The result then follows from Theorem \[Thm:LowerBound\].
By [@KM93], the Cheeger constant of a random 3-regular graph is asymptotically almost surely bounded below by $\frac{1}{4.95}$. Similarly, [@Kolesmald Theorem 3] provides lower bounds above which the $u$-Cheeger constant of a random regular graph is bounded asymptotically almost surely. For 3-regular graphs, this bound is optimized for $u \approx 0.36$, at which point it is $0.24$. By Theorem \[Thm:CheegerBound\], therefore, the gonality of a random 3-regular graph is asymptotically almost surely greater than or equal to $$\min \left\{ \frac{0.24}{3.24} \vert V(G) \vert, \frac{0.36}{4.95} \vert V(G) \vert \right\} = 0.072 \vert V(G) \vert .$$
While our main interest is in 3-regular graphs, we note the following amusing fact.
Let ${\operatorname{gon}}(k)$ be the supremum over all $\ell$ such that the gonality of a random $k$-regular graph on $n$ vertices is asymptotically almost surely at least $\ell n$. Then $$\lim_{k \to \infty} {\operatorname{gon}}(k) \geq \frac{1}{3} .$$
Let $i(k)$ be the supremum over all $\ell$ such that the Cheeger constant of a random $k$-regular graph is asymptotically almost surely at least $\ell k$. By [@Bollobas], $\lim_{k \to \infty} i(k) \geq \frac{1}{2}$. It follows that $$\lim_{k \to \infty} {\operatorname{gon}}(k) \geq \lim_{k \to \infty} \frac{i(k)k}{k+i(k)k} = \frac{\frac{1}{2}}{1+\frac{1}{2}} = \frac{1}{3} .$$
Gonality and Spectral Expansion
===============================
Determining the Cheeger constant of a graph is an NP-complete problem [@Kaibel04]. Using the algebraic connectivity, however, we can provide a lower bound on graph gonality that can be computed in polynomial time. This bound also does not require the graph to be regular, and can therefore be applied in a more general setting than Theorem \[Thm:CheegerBound\]. To obtain this bound, we first prove a proposition about divisors that act as separators.
\[Prop:Separator\] Let $D$ be a divisor on a graph $G$ such that all connected components of $V(G) \smallsetminus {\operatorname{supp}}(D)$ have size less than $\frac{1}{2}\vert V(G) \vert$. Then there exist sets $A, B \subset V(G)$ such that ${\operatorname{supp}}(D)$ separates $A$ and $B$ and $$\frac{1}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert \leq \vert A \vert \leq \frac{2}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert .$$
Let $A$ be the maximal union of connected components of $V(G) \smallsetminus {\operatorname{supp}}(D)$ such that $\vert A \vert \leq \frac{2}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert$, and let $B$ be the union of the remaining connected components. If $\frac{1}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert \leq \vert A \vert $, then we are done. We may therefore suppose that $\vert A \vert < \frac{1}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert$. In this case, we move a component $U$ of $B$ into $A$.
Note that, by our assumption that $A$ is maximal, we must have $\vert A \cup U \vert > \frac{2}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert$. Then, since each connected component of $V(G) \smallsetminus {\operatorname{supp}}(D)$ has size less than half the vertices of $G$, we see that $\frac{1}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert \leq \vert U \vert \leq \frac{1}{2} \vert V(G) \vert$. In that case, we may let $A$ = $U$, and let $B$ be the union of the remaining connected components.
We now use Proposition \[Prop:Separator\] in conjunction with Lemma \[Lemma:Eig\] to find a lower bound for the gonality entirely in terms of the algebraic connectivity, the maximum valence of a vertex in $G$, and $\vert V(G) \vert$.
Let $D$ be a divisor on $G$ of positive rank. By Theorem \[Thm:CheegerBound\], there exists a representative for $D$ such that every connected component of $V(G) \smallsetminus {\operatorname{supp}}(D)$ has size less than $\frac{1}{2} \vert V(G) \vert$. Therefore, by Proposition \[Prop:Separator\] we have that ${\operatorname{supp}}(D)$ separates vertex sets $A$ and $B$ such that $$\frac{1}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert \leq \vert A \vert \leq \frac{2}{3} \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert .$$ Let $x = \frac{\vert V(G) \vert}{\vert A \vert}$. By assumption, $\frac{1}{3} \leq x \leq \frac{2}{3}$, and $$\vert A \vert \vert B \vert = x(1-x) \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert ^2 .$$ On the interval $[\frac{1}{3}, \frac{2}{3}]$, the function $x(1-x)$ has a single critical point at $x = \frac{1}{2}$, but since this is a downward opening parabola we know this critical point is a maximum. Therefore, $x(1-x)$ attains its minimum value at the endpoints $x = \frac{1}{3}, \frac{2}{3}$. Thus we have that $$x(1-x) \vert V(G) \smallsetminus {\operatorname{supp}}(D) \vert^2 \geq \frac{2}{9}(\vert V(G) \vert - \vert {\operatorname{supp}}(D) \vert )^2 .$$ Employing Lemma \[Lemma:Eig\] we have $$|{\operatorname{supp}}(D)| \geq \frac{4\lambda_2 \vert A \vert \vert B \vert}{d \vert V(G) \vert -\lambda_2 \vert A\cup B \vert} \geq \frac{\frac{8 \lambda_2}{9}( \vert V(G) \vert - \vert {\operatorname{supp}}(D) \vert )^2}{d \vert V(G) \vert -\lambda_2( \vert V(G) \vert - \vert {\operatorname{supp}}(D) \vert )}.$$ Solving for $\vert {\operatorname{supp}}(D) \vert$, we obtain: $$\lambda_2 \vert {\operatorname{supp}}(D) \vert^2 + (7\lambda_2 + 9d)\vert V(G) \vert \vert {\operatorname{supp}}(D) \vert - 8\lambda_2 \vert V(G) \vert^2 \geq 0 .$$ We may now apply the quadratic formula. $$\vert {\operatorname{supp}}(D) \vert \geq \frac{1}{2\lambda_2} \left[ -(7\lambda_2 + 9d)\vert V(G) \vert + \sqrt{(7\lambda_2 + 9d)^2 \vert V(G) \vert^2 + 32\lambda_2^2 \vert V(G) \vert^2} \right] .$$ $$= \frac{\vert V(G) \vert}{2\lambda_2} \left[ -(7\lambda_2 + 9d) + 3\sqrt{9\lambda_2^2 + 14d\lambda_2 + 9d^2} \right] .$$
Note that the Laplacian of a graph can be constructed easily given a set of vertices and edge relations, and using the QR algorithm, eigenvalues of the Laplacian can be approximated to any degree of accuracy in $O(n^{3})$ time. This means that given an arbitrary graph, we can use Theorem \[Thm:SpectralBound\] to obtain a lower bound on the gonality in a relatively small amount of time.
Although it is not as good as our previous bound, we record here what Theorem \[Thm:SpectralBound\] can tell us about random 3-regular graphs.
Let $G$ be a random 3-regular graph. Then, for any $\epsilon > 0$, we have $${\operatorname{gon}}(G) \geq (0.0486 - \epsilon ) \vert V(G) \vert$$ asymptotically almost surely.
By [@Friedman08], for any $\epsilon > 0$, the algebraic connectivity of a random $k$-regular graph is asymptotically almost surely bounded below by $k - 2\sqrt{k-1} - \epsilon$. Taking $k = 3$, we evaluate the right hand side of the inequality in Theorem \[Thm:SpectralBound\] to obtain: $${\operatorname{gon}}(G) \geq \frac{\vert V(G) \vert}{2(3-2\sqrt{2})} \left[-(7(3-2\sqrt{2}) + 27) + 3\sqrt{9(3 - 2\sqrt{2})^{2} + 42(3-2\sqrt{2}) + 81} \right]$$ $$= 0.0486 \vert V(G) \vert .$$
An Example
==========
As an example, we compute these bounds for the Pappus graph pictured in Figure \[Fig:Pappus\]. The Pappus graph is notable for having the largest Cheeger constant of any 3-regular graph with at most 20 vertices. As such, it is a good candidate for testing our results.
(-5.05,-5.3) rectangle (5.1,5.25);
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/in [7/8,8/9,9/10,10/11,11/12,12/7]{}[ () – (); ]{}
As a first estimate of the gonality of the Pappus graph, we apply Theorem \[Thm:LowerBound\]. The $u$-Cheeger constants for the Pappus graph are provided in Table \[Table\]. Evaluating the expression $$\min \left\{ \frac{h_{u}(G)}{3+h_{u}(G)} \vert V(G) \vert ,h(G)u \vert V(G) \vert \right\}$$ for all $u$ in Table \[Table\], we find that the bound given by Theorem \[Thm:LowerBound\] is optimized for $u = \frac{1}{3}$, giving a bound of $\frac{1}{4}\vert V(G) \vert$. Thus ${\operatorname{gon}}(G) \geq 4.5$.
\[Table\]
$u$ $h_{u}(G)$
-------- ------------
$1/18$ $3$
$2/18$ $2$
$3/18$ $5/3$
$4/18$ $3/2$
$5/18$ $7/5$
$6/18$ $1$
$7/18$ $1$
$8/18$ $1$
$9/18$ $7/9$
: u-Cheeger constants of the Pappus graph[]{data-label="table"}
A better estimate is provided by Theorem \[Thm:SpectralBound\]. The second smallest eigenvalue of the Pappus graph is $\lambda_2 = 3-\sqrt{3}$. We may now evaluate the bound given by Theorem \[Thm:SpectralBound\]. $${\operatorname{gon}}(G) \geq \frac{18}{2(3-\sqrt{3})} \left[ -7(3-\sqrt{3} + 27)+3\sqrt{9(3-\sqrt{3})^{2} + 42(3-\sqrt{3}) + 81} \right] = 5.04.$$ Since the gonality is an integer, we now know that the gonality of the Pappus graph is at least 6. It is expected that this is the highest possible gonality of a graph of genus 10. We now show that the gonality of the Pappus graph is exactly 6.
Let $G$ be the Pappus graph. Then ${\operatorname{gon}}(G) = 6$.
Let $D$ be the divisor on the Pappus graph illustrated in Figure \[Fig:Divisor\]. Note that if we fire every vertex in the complement of the outer ring once, these chips will migrate to the outer ring. Therefore it suffices to show that we can get at least one chip to any vertex in the innermost ring. By symmetry it suffices to check that we can get a chip to a single vertex in the innermost ring. To do this, we will employ Dhar’s burning algorithm on $D$, with $v$ being a vertex in the innermost ring.
Now, the fire burns the edge connecting $v$ to an adjacent vertex in the inner ring. Each of these vertices is connected to two distinct vertices in the middle ring by one edge each. Since the support of our divisor is the middle ring, the fire must stop having only burned $v$ and the adjacent vertex, so we fire all other vertices once. This chip firing move puts a chip on $v$, which shows that $D$ has rank at least 1.
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[^1]: We note with mild exasperation that there is no apparent consistency in the combinatorics literature concerning the indices of the eigenvalues of the graph Laplacian. They are equally likely to be ordered smallest to largest as largest to smallest, or indexed starting with zero rather than one. Such issues nevertheless pale in comparison to the wealth of literature that refers simply to the “second eigenvalue” without indicating whether this means second smallest or second largest.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Kleene algebra (KA) is the algebra of regular events. Familiar examples of Kleene algebras include regular sets, relational algebras, and trace algebras. A Kleene algebra with tests (KAT) is a Kleene algebra with an embedded Boolean subalgebra. The addition of tests allows one to encode [while]{} programs as KAT terms, thus the equational theory of KAT can express (propositional) program equivalence. More complicated statements about programs can be expressed in the Hoare theory of KAT, which suffices to encode Propositional Hoare Logic.
That the equational theory of KAT reduces to the equational theory of KA has been shown by Cohen et al. Unfortunately, their reduction involves an exponential blowup in the size of the terms involved. Here we give an alternate feasible reduction.
author:
- |
James Worthington\
Mathematics Department, Cornell University\
Ithaca, NY 14853-4201 USA\
[[email protected]]{}
title: Feasibly Reducing KAT Equations to KA Equations
---
Introduction
============
The class of Kleene algebras is defined by equations and equational implications over the signature $\{0, 1, +, \cdot, ^*\}$. Some well-known examples of Kleene algebras include relational algebras, trace algebras, and sets of regular languages (see [@bib:be03] for more examples and applications). In fact, the set of regular languages over an alphabet $\Sigma$ is the free Kleene algebra on $\Sigma$. That is, given two KA terms $\alpha$ and $\beta$, $\alpha = \beta$ modulo the axioms of Kleene algebra if and only if $\alpha$ and $\beta$ denote the same regular set [@bib:ko94]. A Kleene algebra with tests is a Kleene algebra with an embedded Boolean subalgebra (the complementation function is only defined on Boolean terms).
Adding tests allows the encoding of [while]{} programs as KAT terms. As a result, the equational theory of KAT suffices to express (propositional) equivalence of [while]{} programs. Moreover, Propositional Hoare Logic can be encoded in the Hoare theory of KAT (equational implications of the form $r = 0 \rightarrow p = q$), and furthermore the Hoare theory of KAT reduces efficiently to the equational theory of KAT. Combining all of these reductions shows that the equational theory of KA can be used to express interesting properties of programs succinctly. See [@bib:ko97], [@bib:ko00], and [@bib:ko01] for details.
In [@bib:ko96], it is shown that the equational theory of KAT reduces to the equational theory of KA. Unfortunately, the reduction used can increase the size of the terms involved exponentially. We given alternate reduction, which increases the size of the terms by only a polynomial amount. This paper is organized as follows. In section 2, we provide the relevant definitions and recall the encoding of finite automata as Kleene algebra terms. In section 3, we prove some useful theorems of Kleene algebra used for reasoning about automata and give an overview of [*guarded string algebras*]{}. In section 4, we give a feasible reduction from a KAT term to an automaton encoded as a KA term. In section 5, we remark that the Hoare theory of KA(T) can be efficiently reduced to the equational theory of KA(T), and in section 6 we make an observation concerning automata constructed from KAT terms representing deterministic [while]{} programs.
Background
==========
In this section, we describe our proof system and recall some useful facts about KA(T). The axiomatization of Kleene algebra, results about matrices, and the encoding of automata as KA terms are from [@bib:ko94]. The definition of KAT is from [@bib:ko97].
Equational Logic
----------------
By “proof”, we mean a sequent in the equational implication calculus. Let $\alpha, \beta, \gamma, \delta$ be terms in the language of Kleene algebra. The equational axioms are: $$\begin{array}{l}
\alpha = \alpha\\
\alpha = \beta \rightarrow \beta = \alpha\\
\alpha = \beta \rightarrow \beta = \gamma \rightarrow \alpha = \gamma\\
\alpha = \beta \rightarrow \gamma = \delta \rightarrow \alpha + \gamma = \beta + \delta\\
\alpha = \beta \rightarrow \gamma = \delta \rightarrow \alpha \cdot \gamma = \beta \cdot \delta\\
\alpha = \beta \rightarrow \alpha^* = \beta^*.\\
\end{array}$$
We consider these Horn formulas to be implicitly universally quantified.\
Let $\Phi$ be a sequence of equations or equational implications, $e$ an equation, $\phi$ a Horn formula, and $\psi$ an equational axiom or an axiom of KA (given below). Let $\sigma$ be a substitution of terms for variables. The rules of inference are:
$$\vdash \sigma(\psi)~~~~~~~
e \vdash e ~~~~~~~~
\begin{array}{c}
\Phi \vdash \phi \\ \hline
\Phi,e \vdash \phi \\ \end{array}~~~~~~~~
\begin{array}{c}
\Phi,e \vdash \phi \\ \hline
\Phi \vdash e \rightarrow \phi \\ \end{array}~~~~~~~~
\begin{array}{c}
\Phi \vdash e ~~~~~~~\Phi \vdash e \rightarrow \phi \\ \hline
\Phi \vdash \phi \\ \end{array},$$
and the structural rules which allow us to treat a sequence of formulas as a set of formulas. For a proof that this is a complete deductive system, see [@bib:se72]. We also allow “substitution of equals for equals”. For example, from $a=b$, conclude $c(a+1)=c(b+1)$ in one step.
Kleene Algebra
--------------
We now state the axioms of Kleene algebra. The first are the idempotent semiring axioms. Note that we abbreviate $\alpha \cdot \beta$ as $\alpha \beta$.
1. $(a + b) + c = a + (b + c)$
2. $a + b = b + a$
3. $a + 0 = a$
4. $a + a = a$
5. $(ab)c = a(bc)$
6. $1a = a1 = a$
7. $a(b + c) = ab + ac$
8. $(a + b)c = ac + bc$
9. $0a = 0a = 0$
In any idempotent semiring, addition can be used to define a partial order: $$x \leq y \Leftrightarrow x + y = y.$$
For brevity, we add the symbol $\leq$ to the language.\
There are four axioms involving $^*$. The equational axioms are:\
10. $1 + xx^* = x^*$
11\. $1 + x^*x = x^*$\
There are also two equational implications:\
12. $b + ax \leq x \rightarrow a^* b \leq x$
13\. $b + xa \leq x \rightarrow ba^* \leq x$\
The equational implications guarantee unique least solutions to the linear inequalities $$b + aX \leq X$$ $$b + Xa \leq X$$
in the presence of the other axioms.
Kleene Algebra with Tests
-------------------------
A Kleene algebra with tests is a Kleene algebra with an embedded Boolean subalgebra; Boolean terms are called tests. Formally, a Kleene algebra with tests is a two-sorted structure $(K,B,+,\cdot,^*,^-,0,1)$ such that $(K,+,\cdot,^*,0,1)$ is a Kleene algebra and $(B,+,\cdot,^-,0,1)$ is a Boolean algebra. Note that complementation is only defined on tests.
We use the following axiomatization of Boolean algebra. Let $b,c,d$ be Boolean terms.
1. KA axioms 1 - 9
2. $\overline{0} = 1;~ \overline{1} = 0$
3. $b + 1 = 1$
4. $b\overline{b} = \overline{b}b= 0$
5. $\overline{\overline{b}} = b$
6. $bb=b$
7. $\overline{b + c} = \overline{b}\overline{c};~ \overline{bc} = \overline{b} + \overline{c}$
8. $bc= cb$
9. $b + cd = (b+c)(b+d)$
Any Boolean term $b$ satisfies $b \leq 1$. Since $1^* = 1$ and $^*$ is monotonic, the KA axioms imply $b^* = 1$. Note that any Kleene algebra can be viewed as a KAT with $\{0,1\}$ as the two-element Boolean subalgebra.
Matrices and Automata
---------------------
The Kleene algebra axioms imply that the set of $n \times n$ matrices over a KA also forms a KA. Addition and multiplication of matrices are defined in the usual way, 0 is interpreted as the $n \times n$ zero matrix, and 1 as $I_n$. Equality and the partial order $\leq$ are defined componentwise. To define the star of an $n \times n$ matrix, we first define the star of a $2 \times 2$ matrix:
$$\left[
\begin{array}{lr}
a & b \\
c & d \\ \end{array}
\right]^* =
\left[
\begin{array}{lr}
(a + bd^*c)^* & (a + bd^*c)bd^*\\
(d + ca^*b)ca^* & (d + ca^*b)^*\\ \end{array}
\right].$$
We then extend this definition to arbitrary square matrices inductively. Given a square matrix $E$, partition $E$ into four submatrices $$E = \left[
\begin{array}{l|r}
A & B \\ \hline
C & D \\ \end{array}
\right]$$
such that $A$ and $D$ are square. By induction, $A^*$ and $D^*$ exist. Let $F = A + BD^*C$. Then $$E^* = \left[
\begin{array}{c|c}
F^* & F^*BD^* \\ \hline
D^*CF^* & D^* + D^*CF^*BD^* \\ \end{array}
\right].$$ It is a consequence of the KA axioms that any partition may be chosen to compute $E^*$.\
In [@bib:co96], it is shown that the set of $n \times n$ matrices over a Kleene algebra with tests is a Kleene algebra with tests. The Boolean subalgebra is the set of matrices with Boolean terms on the diagonal and all other entries equal to 0.
At several points in the proof below, we will have to reason about non-square matrices. We would like to know whether the theorems of Kleene algebra hold when the primitive letters are interpreted as matrices of arbitrary dimension and the function symbols are treated polymorphically. In general, the answer is no. However, there is a large class of theorems for which this does hold, and they suffice for our purposes. See [@bib:ko98] for a thorough treatment of this issue.
We now recall how to use matrices over a KA to encode finite automata.
[*An*]{} automaton [*over a Kleene algebra K is a triple $(u,A,v)$ where $u$ and $v$ are $n$-dimensional vectors with entries from $\{0,1\}$ and $A$ is an $n \times n$ matrix over $K$. The vector $u$ encodes the start states of the automaton and is called the*]{} start vector. [*The vector $v$ encodes the accept states of the automaton and is called the*]{} accept vector. [*The matrix $A$ is called the*]{} transition matrix. [*The language accepted by $(u,A,v)$ is $u^{\rm{T}}A^*v$. The*]{} size [*of $(u,A,v)$ is the number of states, i.e., if $A$ is an $n \times n$ matrix, then the size of $(u,A,v)$ is $n$*]{}.
This definition is a bit general for the purposes at hand. Given an alphabet $\Sigma$, let $\mathcal{F}_{\Sigma}$ be the free Kleene algebra on generators $\Sigma$. Over $\mathcal{F}_{\Sigma}$, the definition of an automaton given above is essentially the same as the classical definition of a finite automaton. In the sequel, all automata are over some $\mathcal{F}_{\Sigma}$. Furthermore, most of the automata we consider have uncomplicated transition matrices.
[*Let $(u,A,v)$ be an automaton over $\mathcal{F}_{\Sigma}$. The automaton $(u,A,v)$ is*]{} simple
*if $A$ can be expressed as a sum*
$$A = J + \sum_{a \in \Sigma} a \cdot A_a$$
where $J$ and each $A_a$ is a 0-1 matrix.
The automaton $(u,A,v)$ is
$\epsilon$-free
*if $J$ is the zero matrix.*
The automaton $(u,A,v)$ is
deterministic [*if it is simple, $\epsilon$-free, and $u$ and all rows of each $A_a$ have exactly one 1.*]{}
Given an automaton $(u,A,v)$, we denote the transition relation encoded by $A$ as $\delta_A$, and the extended transition relation defined on (states,words) as $\hat{\delta}_A$. Given an $a \in \Sigma$, we denote the restriction of $\delta_A$ to only $a$-transitions by $\delta_A^a$. For transition matrices $A,B,C$, we denote the underlying state sets of the automata by $\mathcal{A,B,C}$. We now state the theorems of KA which we will use to reason about automata.
Useful Theorems of KA
=====================
The completeness result of [@bib:ko94] uses the fact that automata can be encoded as KA terms. To simplify proofs, we add several theorems of Kleene algebra involving automata to our list of allowable rules of inference. For each theorem we add, it will be clear that the hypotheses of the theorem are easy to check, so proofs constructed using these new rules of inference are verifiable in polynomial time. Several of the theorems about automata are based on the following theorems of Kleene algebra:
$$(x + y)^* = x^*(yx^*)^*$$ $$ay = yb \rightarrow a^*y = yb^*$$ $$x(yx)^* = (xy)^*x.$$
These are known as the [*denesting, bisimulation*]{}, and [*sliding*]{} rules, respectively. See [@bib:ko94] for a proof that these rules are consequences of the KA axioms.
We now provide an overview of [*guarded string algebras*]{}, which are models of the KAT axioms. For a more detailed introduction, see [@bib:ko96]. Guarded string algebras play the same role for KAT that regular languages do for KA; two KAT terms $t_1$ and $t_2$ are equivalent modulo the axioms of Kleene algebra with tests if and only if they denote the same set of guarded strings.
Let $P$ and $B$ be finite alphabets. Elements of $P$ are called atomic programs, and elements of $B$ are called primitive tests (to distinguish them from atomic elements of the Boolean algebra generated by $B$). Guarded strings are obtained from each word $w \in P^*$ by interspersing atoms of the free Boolean algebra on $B$ among the letters of $w$ (we require that a guarded string both begins and ends with an atom). Let $b_1,b_2,...,b_n$ be the elements of $B$. Recall that an atom $\alpha$ of the free Boolean algebra on $B$ is a product of the form
$$\alpha = c_1 c_2 \cdots c_n$$
where $c_i \in \{b_i, \overline{b_i}\}$ for each $i$. We require an ordering on the literals appearing in an atom so that there is a unique string denoting each atom. Let $A_B$ denote the set of atoms.
Given a guarded string $x$, let first($x$) be the leftmost atom of $x$, and last($x$) be the rightmost atom of $x$. We define a partial concatenation operation on guarded strings, denoted $\diamond$, as follows. Given two guarded strings, $x$ and $y$, let $x = x'\alpha$ and $y = \beta y'$, where $\alpha = $last($x$) and $\beta = $ first($y$).
Define $$x \diamond y = x'\alpha y', \text{ if } \alpha = \beta, \text{ undefined otherwise}.$$
We now give interpretations of the KAT operations on sets of guarded strings. Let $C$ and $D$ be sets of guarded strings. Define $$\begin{array}{l}
C + D = C \cup D\\
C \cdot D = \{x \diamond y~|~x \in C,~y \in D\}\\
C^0 = A_B\\
C^* = \bigcup_{n \geq 0}~C^n.\\
\end{array}$$
We must also interpret the complementation function. Let $C$ be a set of guarded strings such that $C \subseteq A_B$. Define $$\overline{C} = A_B - C.$$
Using these operations, we can define a function $G$ from KAT terms to sets of guarded strings inductively. The base cases are: $$\begin{array}{l}
G(0) = \emptyset\\
G(1) = \{\alpha~|~ \alpha \in A_B\}\\
G(b) = \{\alpha~|~ \alpha \rightarrow b \mbox{ is a propositional tautology}\}\\
G(p) = \{\alpha p \beta~|~ \alpha, \beta \in A_B\}.\\
\end{array}$$
In [@bib:ko96], the completeness of the guarded string model for the equational theory of KAT is shown by a reduction from the equational theory of KAT to the equational theory of KA. This is achieved by transforming a KAT term $t$ into a KAT-equivalent term $t'$ such that $R(t')=G(t)$. Unfortunately, the term $t'$ may be exponentially longer than $t$. We give an alternate construction. Given a term $t$, we construct an automaton $(u,A,v)$ such that $t = u^{\text{T}}A^*v$ modulo the axioms of KAT, and $(u,A,v)$ accepts precisely the set of guarded strings denoted by $t$. The automaton $(u,A,v)$ will be polynomial in the size of $t$.
We need a few additional theorems of Kleene algebra in our construction. The extra axioms satisfied by Boolean terms, particularly multiplicative idempotence and star-triviality, complicate the construction of the automaton. We overcome these difficulties by selectively applying the Boolean axioms to Boolean terms. That is, we first treat Boolean terms simply as words over an alphabet, and apply the lemmas below. However, these lemmas produce automata which are not simple. In the inductive construction in section 4.3 we then use the Boolean axioms to simplify the transition matrices. Note, however, that the two lemmas below are theorems of Kleene algebra, and do not require the Boolean axioms.
The KAT Concatenation Lemma
---------------------------
The [*KAT concatenation*]{} lemma is based on the following alternate way of constructing an automaton accepting the concatenation of two languages. The standard construction of such an automaton is to connect the accept states of the first automaton to the start states of the second with $\epsilon$-transitions. However, we could also do the following: for each state $i$ of $(u,A,v)$ with an outgoing $x$ transition to an accept state, and each state $j$ of $(s,B,t)$ with an incoming $y$ transition from a start state, add an $xy$ transition from $i$ to $j$. Note that we allow $x$ and $y$ to be arbitrary elements of a Kleene algebra, not just letters in $\Sigma$. This construction yields an automaton accepting $u^{\rm{T}}A^*v s^{\rm{T}}B^*t$, provided neither $(u,A,v)$ nor $(s,B,t)$ has a state which is both a start state and an accept state, which we can represent algebraically as $u^{\rm{T}}v = 0, ~s^{\rm{T}}t = 0$. This idea is the crux of the KAT concatenation lemma. The lemma itself looks rather complicated, so we explain how it will be used. In the construction in 5.2, we will have two $\epsilon$-free automata, $(u_1,A_1,v_1)$ and $(u_2,A_2,v_2)$. Each of these automata will be the disjoint union of two automata:
$$(u_i,A_i,v_i) =
\left(
\left[
\begin{array}{l}
o_i \\ \hline
s_i \end{array}
\right],
\left[
\begin{array}{l|r}
C_i & 0\\ \hline
0 & B_i \\ \end{array}
\right],
\left[
\begin{array}{l}
r_i \\ \hline
t_i \\ \end{array}
\right]
\right).$$
It will be the case that neither of them accept the empty word, i.e., $$o_i^{\text{T}}r_i = 0$$ $$s_i^{\text{T}}t_i = 0$$
for $i = 1,2$. The construction will require an automaton accepting
$$L =(o_1^{\text{T}}C_1^*r_1s_2^{\text{T}}B_2^*t_2) +
(s_1^{\text{T}}B_1^*t_1o_2^{\text{T}}C_2^*r_2) + (s_1^{\text{T}}B_1^*t_1s_2^{\text{T}}B_2^*t_2).$$
Let $\Phi$ be a sequence of equations or equational implications. The KAT concatenation lemma,
$$\begin{array}{c}
\Phi \vdash o_1^{\text{T}}r_1 =0 ~~~~~~\Phi \vdash o_2^{\text{T}}r_2= 0~~~~~~\Phi \vdash s_1^{\text{T}}t_1 = 0~~~~~~~
\Phi \vdash s_2^{\text{T}}t_2 = 0 \\ \hline
\Phi \vdash
\left[
\begin{array}{l}
o_1 \\
s_1 \\
0 \\
0 \\
\end{array}
\right]^{\text{T}}
\left[
\begin{array}{l|r}
\begin{array}{lr}
C_1 & 0 \\
0 & B_1 \\ \end{array}
&
\begin{array}{lr}
0 & C_1r_1 s_2^{\text{T}}B_2 \\
B_1t_1o_2^{\text{T}}C_2 & B_1t_1s_2^{\text{T}}B_2 \\ \end{array}
\\ \hline
\begin{array}{lr}
0 ~~& ~~0 \\
0 ~~&~~ 0 \\ \end{array}
&
\begin{array}{lr}
C_2~~~~~~~~ &~~~~~~ 0 \\
0 ~~~~~~~~& ~~~~~~B_2 \\ \end{array}
\end{array}
\right]^*
\left[
\begin{array}{l}
0 \\
0 \\
r_2 \\
t_2 \\ \end{array}
\right] = L
\end{array}$$
allows us to do this.
The proof is a straightforward calculation: $$\left[
\begin{array}{l}
o_1 \\
s_1 \\
0 \\
0 \\
\end{array}
\right]^{\text{T}}
\left[
\begin{array}{l|r}
\begin{array}{lr}
C_1 & 0 \\
0 & B_1 \\ \end{array}
&
\begin{array}{lr}
0 & C_1r_1 s_2^{\text{T}}B_2 \\
B_1t_1o_2^{\text{T}}C_2 & B_1t_1s_2^{\text{T}}B_2 \\ \end{array}
\\ \hline
\begin{array}{lr}
0 ~~& ~~0 \\
0 ~~&~~ 0 \\ \end{array}
&
\begin{array}{lr}
C_2~~~~~~~~ &~~~~~~ 0 \\
0 ~~~~~~~~& ~~~~~~B_2 \\ \end{array}
\end{array}
\right]^*
\left[
\begin{array}{l}
0 \\
0 \\
r_2 \\
t_2 \\ \end{array}
\right] =$$ $$o_1^{\text{T}}C_1^*C_1r_1s_2^{\text{T}}B_2B_2^*t_2 + s_1^{\text{T}}B_1^*B_1t_1o_2^{\text{T}}C_2C_2^*r_2 +
s_1^{\text{T}}B_1^*B_1t_1s_2^{\text{T}}B_2B_2^*t_2.$$
Using the hypotheses, it is easy to show that this sum is equal to $L$. The proofs involved are of the following form:
$$o_1^{\text{T}}C_1^*r_1 = o_1^{\text{T}}(1 + C_1^*C_1)r_1$$ $$= o_1^{\text{T}}r_1 + o_1^{\text{T}}C_1^*C_1r_1$$ $$= o_1^{\text{T}}C_1^*C_1r_1.$$
The KAT Asterate Lemma
----------------------
Let $(u,A,v)$ be a simple, $\epsilon$-free automaton and $\gamma$ be a regular expression. Suppose $u^{\rm{T}}A^*v = \gamma$. The standard construction of an automaton accepting $\gamma\gamma^*$ proceeds by adding $\epsilon$-transitions from the accept states of $(u,A,v)$ back to its start states. Suppose $(u,A,v)$ has no paths of length 0 or 1 from a start state to an accept state, which we can model algebraically as $u^{\rm{T}}v = 0,
u^{\rm{T}}Av = 0$. In this case, we can construct an automaton accepting $\gamma\gamma^*$ from $(u,A,v)$ with the following procedure: for each state $i$ with an outgoing $x$ transition to an accept state, and each state $j$ with an incoming $y$ transition from a start state, add an $xy$ transition from $i$ to $j$. This automaton, although not simple, accepts $\gamma\gamma^*$. This idea is the basis of the [*KAT asterate*]{} lemma.
Suppose $(u,A,v)$ is the disjoint union of two automata, $(o,C,r)$ and $(s,B,t)$. Also suppose that $o^{\text{T}}C^*r \leq 1$, and $s^{\text{T}}t + s^{\text{T}}Bt = 0$, which implies $s^{\text{T}}B^*t = s^{\text{T}}B^*BBt$. Under these conditions, we can apply the KAT asterate lemma:
$$\begin{array}{c}
\Phi \vdash o^{\text{T}}C^*r \leq 1~~~~~~\Phi \vdash s^{\text{T}}B^*t = s^{\text{T}}B^*BBt\\ \hline
\Phi \vdash
\left(
\left[
\begin{array}{l}
o \\ \hline
s \end{array}
\right]^{\text{T}}
\left[
\begin{array}{l|r}
C & 0\\ \hline
0 & B\\ \end{array}
\right]^*
\left[
\begin{array}{l}
r \\ \hline
t \\ \end{array}
\right]
\right)^*
=
\left[
\begin{array}{l}
1 \\ \hline
s \end{array}
\right]^{\text{T}}
\left[
\begin{array}{l|r}
1 & 0\\ \hline
0 & B+Bts^{\text{T}}B \\ \end{array}
\right]^*
\left[
\begin{array}{l}
1 \\ \hline
t \\ \end{array}
\right]
\end{array}.$$
Note that $B + Bts^{\text{T}}B$ algebraically encodes the alternate asterate construction.
Since $(u,A,v)$ is the disjoint union of $(o,C,r)$ and $(s,B,t)$, it is easy to show that $$u^{\text{T}}A^*v = o^{\text{T}}C^*r + s^{\text{T}}B^*t.$$
By KA axiom 10,
$$(u^{\text{T}}A^*v)^* = 1 + u^{\text{T}}A^*v(u^{\text{T}}A^*v)^*.$$
We can now substitute:
$$1 + u^{\text{T}}A^*v(u^{\text{T}}A^*v)^* = 1 + (o^{\text{T}}C^*r + s^{\text{T}}B^*t)(o^{\text{T}}C^*r + s^{\text{T}}B^*t)^*.$$
By the denesting rule of Kleene algebra, $$1 + (o^{\text{T}}C^*r + s^{\text{T}}B^*t)(o^{\text{T}}C^*r + s^{\text{T}}B^*t)^* =
1 + (o^{\text{T}}C^*r + s^{\text{T}}B^*t)(o^{\text{T}}C^*r)^*(s^{\text{T}}B^*t(o^{\text{T}}C^*r)^*)^*.$$
Since $o^{\text{T}}C^*r \leq 1, (o^{\text{T}}C^*r)^* = 1$. We can simplify:
$$1 + (o^{\text{T}}C^*r + s^{\text{T}}B^*t)(o^{\text{T}}C^*r)^*(s^{\text{T}}B^*t(o^{\text{T}}C^*r)^*)^* =
1 + (o^{\text{T}}C^*r + s^{\text{T}}B^*t)(s^{\text{T}}B^*t)^*.$$
By distributivity and axiom 10 again,
$$1 + (o^{\text{T}}C^*r + s^{\text{T}}B^*t)(s^{\text{T}}B^*t)^* = 1 + s^{\text{T}}B^*t(s^{\text{T}}B^*t)^*.$$
At this point, we have shown that $u^{\text{T}}A^*v = 1 + s^{\text{T}}B^*t(s^{\text{T}}B^*t)^*$. It remains to be shown that under the assumption $s^{\text{T}}B^*t = s^{\text{T}}B^*BBt$, $$~~~~~~~~~~s^{\text{T}}B^*t(s^{\text{T}}B^*t)^* = s^{\text{T}}(B + Bts^{\text{T}}B)^*t.~~~~~~~(1)$$
Reasoning algebraically,
$$s^{\text{T}}B^*t(s^{\text{T}}B^*t)^* = s^{\text{T}}B^*BBt(s^{\text{T}}B^*BBt)^*$$ $$= s^{\text{T}}B^*B(Bts^{\text{T}}B^*B)^*Bt$$ $$= s^{\text{T}}BB^*(Bts^{\text{T}}BB^*)^*Bt$$ $$= s^{\text{T}}B(B + Bts^{\text{T}}B)^*Bt.$$
The following equation is an easy consequence of the axioms of Kleene algebra: $$(B + Bts^{\text{T}}B)^* = 1 + Bts^{\text{T}}B(B + Bts^{\text{T}}B)^* + (B + Bts^{\text{T}}B)^*Bts^{\text{T}}B +
B(B + Bts^{\text{T}}B)^*B.$$
Multiplying the equation on the left by $s^{\text{T}}$, on the right by $t$, and simplifying using $s^{\text{T}}t=0$ and $s^{\text{T}}Bt = 0$ yields
$$s^{\text{T}}(B + Bts^{\text{T}}B)^*t = s^{\text{T}}B(B + Bts^{\text{T}}B)^*Bt.$$
This proves (1). We now add the trivial one-state automaton to the automaton $(s,B + Bts^{\text{T}}B,t)$, completing the proof of the KAT asterate lemma.
KAT Term to Automaton
=====================
In this section, we give the transducer which takes as input a KAT term $t$ and outputs an automaton accepting $G(t)$. Before constructing the automaton, it must convert $t$ into a well-behaved form.
Only Complement Primitive Tests
-------------------------------
The machine first uses the De Morgan laws and the Boolean axiom $\overline{\overline{b}} = b$ to transform a term $t$ into an equivalent term $t'$ in which the complementation symbol is only applied to atomic tests. If we interpret $t'$ as a regular expression, then $R(t') \subseteq (P \cup B \cup \overline{B})^*$, where $\overline{B} = \{\overline{b}~|~ b \in B\}$. The transducer works as follows. On input $t$, it copies $t$ onto its worktape and onto the output tape. Then, starting at the root of the syntax tree of $t$, it works it way down the tree until it finds a subtree containing only Boolean terms such that either some term is complemented twice, or a conjunction or disjunction appears under the complement symbol. It then applies the appropriate axiom to this subtree, overwrites its worktape contents, and then outputs the updated term. The machine then begins searching again at the root of the tree. When it scans the whole tree and does not have to apply any axioms, it stops. The transducer requires only polynomially many worktape cells. Furthermore, the increase in the size of the term is negligible. At the end of this stage, it has $t'$ written on its worktape.
New variables for atoms
-----------------------
For the remainder of the construction, it is advantageous to treat each atom as a single letter. Let $z = 2^{|B|}$. The machine generates $z$ many new variables, $x_1,x_2,...,x_z$. For each $i$, it outputs the equation $$x_i = \alpha_i$$ where $\alpha_i$ is the $i^{\text{th}}$ atom. The automaton constructed below uses the alphabet $P \cup \{x_1,x_2,...,x_z\}$. It is a routine matter to verify that two KAT terms denote same set of guarded strings if and only if they denote the same set of words after performing this substitution. For the rest of the construction, we use the terms “guarded strings” and “guarded strings after this substitution” interchangeably.
Constructing the Automaton
--------------------------
Now that the preprocessing of the term is complete, the machine constructs the automaton. The construction is inductive and resembles the construction the proof of Kleene’s theorem. However, the machine will maintain several invariants throughout the construction which were not necessary in the pure Kleene algebra case. At a given substage, let $(u,A,v)$ be the final automaton constructed. The automaton $(u,A,v)$ will satisfy:
- $(u,A,v)$ is simple and $\epsilon$-free.
- $(u,A,v)$ is the “disjoint union” of two automata, $(o,C,r)$ and $(s,B,t)$, or just $(o,C,r)$, or just $(s,B,t)$.
- $(s,B,t)$ accepts only words of length two or more, so., $s^{\text{T}}B^*t = s^{\text{T}}B^*BBt$.
- $(o,C,r)$ is a two state automaton accepting only one-letter words from the alphabet $\{x_1,x_2,...,x_z\}$.
- The first two states of $(u,A,v)$ are the states of $(o,C,r)$ (if $(o,C,r)$ is nonempty).
The base case of the induction is as follows. For an atomic term $a$, $\hat{a}$ denotes the automaton constructed. For an atom $x_i$ and a primitive test $b$, $x_i \leq b$ means that $x_i \rightarrow b$ is a propositional tautology. $$\begin{array}{l}
\hat{0} = (0,0,0)\\
\\
\hat{1} =
\left(
\left[
\begin{array}{c}
1 \\
0 \\ \end{array}
\right],
\left[
\begin{array}{lr}
0 & \sum_i x_i \\
0 & 0 \\ \end{array}
\right],
\left[
\begin{array}{c}
0 \\
1 \\ \end{array}
\right]
\right)\\
\\
\hat{b} =
\left(
\left[
\begin{array}{c}
1 \\
0 \\ \end{array}
\right],
\left[
\begin{array}{lr}
0 & \sum_{x_i \leq b} x_i\\
0 & 0 \\ \end{array}
\right],
\left[
\begin{array}{c}
0 \\
1 \\ \end{array}
\right]
\right)\\
\\
\hat{p} =
\left(
\left[
\begin{array}{c}
1 \\
0 \\
0 \\
0 \\ \end{array}
\right],
\left[
\begin{array}{cccc}
0 & \sum_i x_i & 0 & 0 \\
0 & 0 & p & 0\\
0 & 0 & 0 & \sum_i x_i\\
0 & 0 & 0 & 0\end{array}
\right],
\left[
\begin{array}{c}
0 \\
0 \\
0 \\
1 \\ \end{array}
\right]
\right)\\
\end{array}$$
For each automaton, the machine must prove that the language it accepts is KAT-equivalent to the appropriate atomic term. There are finitely many atomic terms, so the machine can store all of the necessary proofs in its finite control. Note that this expansion increases the size of a term by only a constant amount, although the constant is exponential in $|B|$. Cf. the proof that the Boolean algebra axioms entail all propositional tautologies.
We now treat the inductive step of the construction. The easiest automaton to construct is that for addition. Suppose we have two automata $(u_1,A_1,v_1)$ and $(u_2,A_2,v_2)$, such that $u_1^{\text{T}}A_1^*v_1 = \gamma$ and $u_2^{\text{T}}A_2^*v_2 = \delta$. By induction, $(u_1,A_1,v_1)$ is the disjoint union of $(o_1,C_1,r_1)$ and $(s_1,B_1,t_1)$, and $(u_2,A_2,v_2)$ is the disjoint union of $(o_2,C_2,r_2)$ and $(s_2,B_2,t_2)$. The machine first proves the equations
$$u_1^{\text{T}}A_1^*v_1 = o_1^{\text{T}}C_1^*r_1 + s_1^{\text{T}}B_1^*t_1$$ $$u_2^{\text{T}}A_2^*v_2 = o_2^{\text{T}}C_2^*r_2 + s_2^{\text{T}}B_2^*t_2.$$
It then outputs a proof that $$\gamma + \delta = (o_1^{\text{T}}C_1^*r_1 + o_2^{\text{T}}C_2^*r_2) + s_1^{\text{T}}B_1^*t_1 + s_2^{\text{T}}B_2^*t_2.$$
The machine can now construct a two-state automaton $(o,C,r)$ which accepts $(o_1^{\text{T}}C_1^*r_1 + o_2^{\text{T}}C_2^*r_2)$, then apply the addition construction from 4.1 to $(o,C,r),(s_1,B_1,t_1)$, and $(s_2,B_2,t_2)$. This yields an automaton $(u,A,v)$ which satisfies the invariants and accepts $\gamma + \delta$. Note that there are only finitely many possibilities for $(o_1,C_1,r_1)$ and $(o_2,C_2,r_2)$, so the machine can prove
$$o^{\text{T}}C^*r = o_1^{\text{T}}C_1^*r_1 + o_2^{\text{T}}C_2^*r_2$$
using data from its finite control.
The automaton for the product of two terms is more complicated. Again, let $(u_1,A_1,v_1)$ and $(u_2,A_2,v_2)$ be two automata such that $u_1^{\text{T}}A_1^*v_1 = \gamma$ and $u_2^{\text{T}}A_2^*v_2 = \delta$. As in the case for addition, we use the fact that each of these automata is the disjoint union of two automata:
$$u_1^{\text{T}}A_1^*v_1 = o_1^{\text{T}}C_1^*r_1 + s_1^{\text{T}}B_1^*t_1$$ $$u_2^{\text{T}}A_2^*v_2 = o_2^{\text{T}}C_2^*r_2 + s_2^{\text{T}}B_2^*t_2.$$
The machine can output a proof of the equations
$$\gamma \delta = (o_1^{\text{T}}C_1^*r_1 + s_1^{\text{T}}B_1^*t_1)(o_2^{\text{T}}C_2^*r_2 + s_2^{\text{T}}B_2^*t_2)$$ $$= (o_1^{\text{T}}C_1^*r_1o_2^{\text{T}}C_2^*r_2) +(o_1^{\text{T}}C_1^*r_1s_2^{\text{T}}B_2^*t_2) +
(s_1^{\text{T}}B_1^*t_1o_2^{\text{T}}C_2^*r_2) + (s_1^{\text{T}}B_1^*t_1s_2^{\text{T}}B_2^*t_2).$$
The term $(o_1^{\text{T}}C_1^*r_1o_2^{\text{T}}C_2^*r_2)$ is a sum of atoms after simplifying using the Boolean axioms. The machine can construct a two-state automaton $(o,C,r)$ accepting this sum. Since there are only finitely many choices for $o_1^{\text{T}}C_1^*r_1$ and $o_2^{\text{T}}C_2^*r_2$, all of the necessary proofs can be stored in the finite control of the machine.
Let $(s,B,t)$ be the automaton
$$\left(
\left[
\begin{array}{l}
o_1 \\
s_1 \\
0 \\
0 \\
\end{array}
\right],
\left[
\begin{array}{l|r}
\begin{array}{lr}
C_1 & 0 \\
0 & B_1 \\ \end{array}
&
\begin{array}{lr}
0 & C_1r_1 s_2^{\text{T}}B_2 \\
B_1t_1o_2^{\text{T}}C_2 & B_1t_1s_2^{\text{T}}B_2 \\ \end{array}
\\ \hline
\begin{array}{lr}
0 ~~& ~~0 \\
0 ~~&~~ 0 \\ \end{array}
&
\begin{array}{lr}
C_2~~~~~~~~ &~~~~~~ 0 \\
0 ~~~~~~~~& ~~~~~~B_2 \\ \end{array}
\end{array}
\right],
\left[
\begin{array}{l}
0 \\
0 \\
r_2 \\
t_2 \\ \end{array}
\right]
\right).$$
The machine first outputs proofs of the hypotheses of the KAT concatenation lemma. It can then output
$$s^{\text{T}}B^*t = (o_1^{\text{T}}C_1^*r_1s_2^{\text{T}}B_2^*t_2) +
(s_1^{\text{T}}B_1^*t_1o_2^{\text{T}}C_2^*r_2) + (s_1^{\text{T}}B_1^*t_1s_2^{\text{T}}B_2^*t_2),$$
which follows from the KAT concatenation lemma.
The machine now constructs a simple automaton $(s,B',t)$ by simplifying the transition matrix for $(s,B,t)$ using the Boolean axioms and outputs a proof of the equivalence of $(s,B,t)$ and $(s,B',t)$. It then adds the automata $(o,C,r)$ and $(s,B',t)$ together to get $(u,A,v)$, and outputs a proof of the equation $$u^{\text{T}}A^*v = \gamma\delta.$$
Finally, we come to the construction for $^*$. Let $(u,A,v)$ be an automaton such that $u^{\text{T}}A^*v = \gamma$. This automaton is the disjoint union of two automata, $(o,C,r)$ and $(s,B,t)$ such that $(o,C,r)$ accepts a sum of atoms and $(s,B,t)$ accepts no words of length less than two. The machine first outputs proofs that $$o^{\text{T}}C^*r \leq 1$$ $$s^{\text{T}}B^*BBt = s^{\text{T}}Bt.$$
These facts follow from the Boolean axioms and the equation $s^{\text{T}}t + s^{\text{T}}Bt = 0$.
The machine can now output $$\left[
\begin{array}{l}
1 \\ \hline
s \end{array}
\right]^{\text{T}}
\left[
\begin{array}{l|r}
1 & 0\\ \hline
0 & B+Bts^{\text{T}}B \\ \end{array}
\right]^*
\left[
\begin{array}{l}
1 \\ \hline
t \\ \end{array}
\right]
= \gamma^*,$$
which follows from the KAT asterate lemma. Finally, the machine can apply the Boolean axioms to each entry of $$\left[
\begin{array}{l|r}
1 & 0\\ \hline
0 & B+Bts^{\text{T}}B \\ \end{array}
\right]$$
to produce an equivalent simple, $\epsilon$-free transition matrix $D$ (1 becomes the sum of all atoms). It can then output a proof of $$\left[
\begin{array}{l}
1 \\ \hline
s \end{array}
\right]^{\text{T}}
\text{{\Large D}}^*
\left[
\begin{array}{l}
1 \\ \hline
t \\ \end{array}
\right]
= \gamma^*.$$
The proof that the automaton constructed for a term $t$ accepts precisely the guarded strings denoted by $t$ is a straightforward induction.
Reducing the Hoare Theory of KA(T) to the Equational Theory of KA
=================================================================
Finally, we make the simple observation that the reductions in [@bib:co93] and [@bib:ko96] don’t significantly increase the size of the terms.
[*Proofs of equational implications in the Hoare Theory of KA(T) can be produced by a PSPACE transducer.*]{}
Given an alphabet $\Sigma = \{a_1,a_2,...,a_n\}$, let $u = a_1 + a_2 + \cdots + a_n$. In [@bib:co93], it is shown that
$$s \equiv t \Leftrightarrow s + uru = t + uru$$
is a Kleene algebra congruence, therefore $(r = 0 \rightarrow p = q) \leftrightarrow (p + uru = q + uru)$. The same reduction works for KAT, as is shown in [@bib:ko96] - in this case $u$ is only defined to be the sum of all of the atomic programs, not the atomic tests. The transformation from $r = 0 \rightarrow p = q$ to $p + uru = q + uru$ involves only a constant increase in size.
Deterministic [while]{} Programs
================================
Let $P$ be a set of atomic programs, and $B$ be a set of atomic tests. In [@bib:ko97], it is shown how to encode deterministic [while]{} programs as KAT terms: $$p;q = pq$$ $${\bf if}~b~ {\bf then}~p ~{\bf else}~q = bp + \overline{b}q$$ $${\bf if}~b~ {\bf then}~p = bp + \overline{b}$$ $${\bf while}~b~{\bf do}~p = (bp)^*\overline{b}.$$ Let $t$ be a KAT term which is an encoding of a deterministic ${\tt while}$ program. Let $g$ be a guarded string over $(P \cup A_B)$. It is easy to see that the automaton $(u,A,v)$ constructed from $t$ in section 4 satisfies the following:
- There is only one start state $s$ of $(u,A,v)$ with an outgoing transition labeled by an atom $x$ such that first$(g) =x$.
- $|\hat{\delta}_A(s,g)| \leq 1$.
Therefore, when considering the deterministic automaton $(s,D,t)$ obtained from $(u,A,v)$ by the standard subset construction, all states of $(s,D,t)$ corresponding to more than one state of $(u,A,v)$ are inaccessible. This implies that, given two KAT terms $t_1$ and $t_2$, using the above procedure to construct automata for each term and then using the procedure in [@bib:wo08] to generate proofs of equivalence of the automata yields proofs which are only polynomial in $|t_1| + |t_2|$.
[99]{}
Berghammer, R., Möller, B. and Struth, G. (Eds.) *Relational and Kleene-Algebraic Methods in Computer Science*, May 2003.
Cohen, Ernie.
Cohen, E and Kozen, D. and Frederick, S. *Technical Report 96-1598, Computer Science Department, Cornell University.* July 1996.
Kozen, D. . *Infor. and Comput*, 110(2):366-390. May 1994.
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Kozen, D. . *Transactions on Programming Languages and Systems* 19:427-443. May 1997.
Kozen, D. . *Technical Report 98-1669, Computer Science Department, Cornell University*. March 1998.
Kozen, D. . *Trans. Computational Logic*, 1(1):60-76, July 2000.
Kozen, D. and Tiuryn, Jerzy. . *Information Sciences*, 139:187-195, 2001.
Selman, A. . *Algebra Universalis*, 2:20-32, 1972.
Worthington, J. *Proceedings of RelMics10/AKA5*, April 2008 (to appear).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Myers–Perry–AdS–dS black hole exhibits $SO(2,1)\times U(n)$ symmetry in the near horizon limit in the special case that all rotation parameters are equal. We consider a massive relativistic particle propagating on such a background and reduce it to superintegrable spherical mechanics with $U(n)$ symmetry. A complete set of functionally independent $u(n)$ generators realized in the model is given.'
---
LMP-TPU-15/13\
[**Integrable models associated with**]{} 0.3 cm [**Myers–Perry–AdS–dS black hole in diverse**]{} 0.3 cm [**dimensions**]{}\
$
\textrm{\Large Kirill Orekhov\ }
$ 0.7cm [*Laboratory of Mathematical Physics, Tomsk Polytechnic University,\
634050 Tomsk, Lenin Ave. 30, Russian Federation*]{}\
[Email: [email protected]]{}
PACS numbers: 04.70.Bw; 11.30.-j; 02.30.Ik
0.5cm
Keywords: Myers–Perry–AdS–dS black hole, conformal mechanics, integrable models
0
[**1. Introduction**]{}\
In the last decade there has been a surge of interest in the Myers-Perry black hole in arbitrary dimension and especially in its near horizon limit (see, e.g., a recent review [@kl] and references therein). The first reason to be concerned about the near horizon geometries is the duality between the near horizon Kerr black hole and a conformal field theory suggested in [@Strom] (for a review see [@com]). As was proved in [@LuMeiPope], the duality holds true also in higher dimensions and in the presence of a cosmological constant. The second reason is the possibility to build various conformal mechanics models starting from a massive relativistic particle propagating on such backgrounds [@a1]–[@GNS]. In this regard the Myers–Perry black hole with all rotation parameters being equal to each other is of particular interest because its symmetry is enlarged to the unitary algebra (in direct sum with extra $so(2,1)$ algebra in the near horizon case) which is the largest finite-dimensional symmetry algebra possible. In particular, this gives a clue to the construction of new maximally superintegrable models in [@Gal; @GalNers; @GNS].
Note that for a generic conformal mechanics one can always split the radial canonical pair from the angular sector by applying a suitable canonical transformation [@HakLecht1; @HakLecht2]. The dynamics of the angular variables is governed by the Casimir element in the conformal algebra $so(2,1)$. The latter can be viewed as the Hamiltonian of a reduced spherical mechanics which retains symmetries pertaining to the angular sector of the parent conformal mechanics.
A natural one–parameter extension of the Myers–Perry solution can be obtained by including a cosmological constant into the consideration [@GibbLuPagePope]. It is noteworthy that for the special case that all rotation parameters are equal to each other the configuration retains the unitary symmetry and therefore hints at a possibility to construct new superintegrable models associated with it. The goal of this work is to construct such superintegrable models which provide a one–parameter deformation of those built recently in [@GalNers]. The similarities and differences between the two cases are discussed in detail.
The paper is organized as follows. In Sect. 2 a short overview of the extremal Myers–Perry–AdS–dS black hole in arbitrary dimension is given. In Sect. 3 we consider such a metric in $D=2n+1$ dimensions for the special case that all rotation parameters are equal. The near horizon limit is implemented and the associated conformal mechanics is constructed. Next we perform the reduction of the conformal mechanics to its spherical sector. Sect. 4 contains a similar analysis for $D=2n$. In Sect. 5 we discuss the unitary symmetries of the Hamiltonians constructed earlier and show that in odd dimensions the reduced Hamiltonian can be expressed in terms of the first and the second order Casimir invariants of the unitary algebra. Finally, we give a complete set of functionally independent $u(n)$ generators realized in the spherical mechanics and prove their superintegrability. In the concluding Sect. 6 we summarize our results and discuss possible further developments.
[**2. Myers–Perry–AdS–dS metric in arbitrary dimension**]{}\
The Myers–Perry–AdS–dS metric is a solution of Einstein equations in $D$ dimensions with a cosmological constant $\la$ \[einstein\] R\_[ij]{} + (D-1)g\_[ij]{} = 0, which describes a black hole rotating in ($n-\e_D$) spatial two–planes, where $\e_D=0$ for odd dimensions ($D=2n+1$) and $\e_D=1$ for even dimensions ($D=2n$). In Boyer-Lindquist coordinates it reads: $$\label{gen_KadS}
\begin{array}{rl}
ds^2 = & {\displaystyle}{W(1-\la r^2)dt^2 - \frac{U}{\D}dr^2 - \frac{2M}{U}\left(dt-\sum_{i=1}^{n-\e_D}\frac{a_i \m_i^2 d\vf_i}{1+\la a_i^2}\right)^2 - \sum_{i=1}^{n}\frac{r^2+a_i^2}{1+\la a_i^2}d\m_i^2 -}\\
-& {\displaystyle}{\sum_{i=1}^{n-\e_D}\frac{r^2+a_i^2}{1+\la a_i^2}\m_i^2(d\vf_i-\la a_i dt)^2 - \frac{\la}{W(1-\la r^2)}\left(\sum_{i=1}^n\frac{r^2+a_i^2}{1+\la a_i^2}\m_i d\m_i\right),}
\end{array}$$ where
[c]{} [= r\^[\_D-2]{}(1-r\^2)\_[i=1]{}\^[n-\_D]{}(r\^2+a\_i\^2),U = r\^[\_D]{}\_[i=1]{}\^[n]{}\_[j=1]{}\^[n-\_D]{}(r\^2+a\_j\^2),]{}\
[W = \_[i=1]{}\^n.]{}
Above $M$ is the black hole mass, $a_i$ are the rotation parameters, $\vf_i$ are azimuthal angles. It is assumed that the latitudinal angular variables $\m_i$ parameterize the unit sphere \[constraint\] \_[i=1]{}\^n\_i\^2 = 1. In even-dimensional case the $n$–th rotation parameter is set to zero a\_n = 0.
In what follows we shall be mainly concerned with the special case for which all the rotation parameters are equal a\_i = a, where $i = 1,\dots, n-\e_D$. In particular, this greatly simplifies the metric (\[gen\_KadS\]). Below we shall treat the even–, and odd–dimensional cases separately.
The black hole solution with equal rotation parameters has a larger symmetry group as one can rotate various spatial two–planes one into another. In odd dimensions, where the metric takes the form $$\label{KadS_odd}
\begin{array}{rl}
ds^2 = & {\displaystyle}{W(1-\la r^2)dt^2 - \frac{U}{\D}dr^2 - \frac{2M}{U}\left(dt-\frac{a}{1+\la a^2}\sum_{i=1}^n\m_i^2 d\vf_i\right)^2 -}\\
-& {\displaystyle}{\frac{r^2+a^2}{1+\la a^2}\sum_{i=1}^n\m_i^2(\la adt-d\vf_i)^2 - \frac{r^2+a^2}{1+\la a^2}\sum_{i=1}^{n}d\m_i^2,}
\end{array}$$ the vector fields generating these rotations can be written as [@VasuStevensPage2] \[un\_gen\] \_[ij]{} = x\_i\_[y\_j]{} - y\_j\_[x\_i]{} + x\_j\_[y\_i]{} - y\_i\_[x\_j]{}, \_[ij]{} = x\_i\_[x\_j]{} - x\_j\_[x\_i]{} + y\_i[y\_j]{} - y\_j[y\_i]{}, Here we introduced coordinates \[xy\] x\_i = \_i\_i, y\_i = \_i\_i ; [\_i = , \_i = ]{}, which lead also to an equivalent realization: \[ij\]
[c]{} [\_[ij]{} = \_[ij]{}(\_j\_[\_i]{} - \_i\_[\_j]{}) + \_[ij]{}(\_[\_i]{} + \_[\_j]{}),]{}\
[\_[ij]{} = -\_[ij]{}(\_j\_[\_i]{} - \_i\_[\_j]{}) + \_[ij]{}(\_[\_i]{} + \_[\_j]{})]{},
where we denoted $\vf_{ij} = \vf_i - \vf_j$. ${\displaystyle}{\frac{n(n+1)}{2}}$ generators $\r_{ij}$ and ${\displaystyle}{\frac{n(n-1)}{2}}$ generators $\x_{ij}$ all together form the unitary algebra $u(n)$. Note that the existence of the unitary symmetry can be revealed by introducing the complex coordinates z\_k = \_k e\^[i\_k]{} = x\_k + i y\_k.
In even dimensions the metric is
[rl]{} ds\^2 =& [W(1-r\^2)dt\^2 - dr\^2 - (dt-\_[i=1]{}\^[n-1]{}\_i\^2 d\_i)\^2 -]{}\
-& [\_[i=1]{}\^[n-1]{}\_i\^2(d\_i-adt)\^2 - \_[i=1]{}\^[n-1]{}d\_i\^2 - r\^2 d\_n\^2 -]{}\
-&[(\_n d\_n)\^2]{},
and the angular sector splits into $(\m_i, \vf_i)$, $i=1,\dots ,n-1$ part and $\m_n$ part. After passing to $n-1$ latitudinal angles $\n_i$ (see Sect. 4) the unitary symmetry of $(\n_i, \vf_i)$ sector can be described in exactly the same way as in the odd dimensional case. Therefore in even dimensions metric is invariant under the unitary group $u(n-1)$.
We will also need the expression for the inverse metric which was obtained in [@VasuStevensPage2]: \[gen\_inverse\]
[rl]{} g\^ = & [(Q+)\_t\^2 - \_r\^2 +]{}\
+& [\_[i=1]{}\^[n-\_D]{}(aQ + +)\_t\_[\_i]{} -]{}\
-&[\_[i,j=1]{}\^[n-\_D]{}(-\^2 a\^2 Q+-R)\_[\_i]{}\_[\_j]{}+…]{}
where the dots denote terms in the $\m_i$–sector which has to be calculated separately for the even–, and odd–dimensional cases. $Q$ and $R$ in (\[gen\_inverse\]) are defined as follows: $$\label{qr}
\begin{array}{rl}
Q = & {\displaystyle}{\frac{1}{W(1-\la r^2)}+\frac{2M}{U}\frac{1}{(1-\la r^2)^2},}\\
R = & {\displaystyle}{\frac{(2M)^2}{U\D}\frac{2\la a^2(1+\la a^2)}{(1-\la r^2)^2(r^2+a^2)} + \frac{2M}{U}\frac{a}{(r^2+a^2)^2} +}\\
+&{\displaystyle}{\frac{2M}{U}\frac{2\la a^2}{(1-\la r^2)(r^2+a^2)} + \frac{(2M)^2}{U\D}\frac{2a^2(1+\la a^2)}{(1-\la r^2)(r^2+a^2)^2}.}
\end{array}$$ The inverse metric allows one to construct the Hamiltonian of a massive relativistic particle moving on the Myers–Perry–AdS–dS background as a solution $p_0$ of the mass–shell equation $g^{\m\n}p_{\m}p_{\n} = m^2$.
[**3. Odd–dimensional case**]{}\
[*3.1 $D=2n+1$ extremal Myers–Perry–AdS–dS black hole near the horizon*]{}\
For $D = 2n+1$ and equal rotation parameters the metric (\[gen\_KadS\]) can be brought to the form $$\label{KadS1}
\begin{array}{rl}
ds^2 = & {\displaystyle}{\frac{\D}{U}\left(dt-\frac{a}{1+\la a^2}\sum_{i=1}^n\m_i d\vf_i\right)^2 - \frac{U}{\D}dr^2 - \frac{r^2+a^2}{1+\la a^2}\sum_{i=1}^n d\m_i^2 -}\\
-& {\displaystyle}{\frac{1}{r^2}\sum_{i=1}^n\m_i^2\left(adt-\frac{r^2+a^2}{1+\la a^2}d\vf_i\right)^2 + \frac{a^2(1-\la r^2)(r^2+a^2)}{r^2(1+\la a^2)^2}\sum_{i<j}^n\m_i^2\m_j^2(d\vf_i-d\vf_j)^2,}
\end{array}$$ where U = (r\^2+a\^2)\^[n-1]{}, =(1-r\^2)(r\^2+a\^2)\^n-2M.
In the extremal case $\D$ has double zero at the horizon radius $r_0$, i.e.: $$\D(r_0) = \D^{\prime}(r_0) = 0.$$ Solving this equations one can relate the mass and the rotation parameter to the horizon radius $r_0$ and a cosmological constant $$a^2 = (n(1-\vk)-1)r_0^2, \quad 2M = \frac{(n r_0^2)^n(1-\vk)^{n+1}}{r_0^2}; \quad \vk := \la r_0^2.$$ If one approaches the horizon, i.e. $r \rightarrow r_0 + \ve r_0 r$ followed by $\ve \to 0$, the relations $$\D \rightarrow \ve^2 r_0^2 r^2 V, \quad V := \frac{2(nr_0^2)^{n-1}(1-\vk)^{n-1}(n(1-2\vk)-1)}{r_0^2}$$ hold.
In order to describe the near horizon geometry, we follow the procedure in [@LuMeiPope]. First one makes the coordinate transformation: $$\label{transf1}
r \rightarrow r_0 + \ve r_0 r, \quad t \rightarrow \frac{\a t}{\ve}, \quad \vf_i \rightarrow \vf_i + \frac{\b_i t}{\ve},$$ and then takes the limit $\ve \rightarrow 0$. The number coefficients $\a$ and $\b_i$ above are fixed from the condition that the first two terms in (\[KadS1\]) produce the $AdS_2$ metric up to a factor, while the rest is nonsingular $$\label{ab1}
\a = \frac{r_0^2+a^2}{2r_0(n(1-2\vk)-1)}, \quad \b_i = \frac{a(1+\la a^2)}{2r_0(n(1-2\vk)-1)}.$$
The near horizon extremal metric reads \[ex\_KadS1\]
[rl]{} ds\^2 = & [(r\^2 dt\^2-) -\_[i=1]{}\^n\_i\^2(rdt+d\_i)\^2-]{}\
-&[\_[i=1]{}\^n d\_i\^2 + \_[i<j]{}\^n\_i\^2\_j\^2(d\_i-d\_j)\^2,]{}
where we rescaled the azimuthal angular variables as follows: \[transf11\] \_i \_i. It is readily verified that (\[ex\_KadS1\]) is a vacuum solution of the Einstein equations with a cosmological constant (\[einstein\]). It is an extension of the metric constructed in [@Gal] which now includes a cosmological constant $\la$.
A salient feature of the near horizon metric (\[ex\_KadS1\]) is that it exhibits extra symmetries generated by the Killing vectors \[killing1\] D = t\_t-r\_r, K = (t\^2+)\_t - 2tr\_r -\_[i=1]{}\^n\_[\_i]{}, which along with the time translation $H = {\partial}_t$ form the conformal algebra $so(2,1)$.
[*3.2 Conformal mechanics near the horizon of the extremal Myers–Perry–AdS–dS black hole in $D=2n+1$*]{}\
In order to construct the Hamiltonian of a massive relativistic particle moving on the curved background (\[ex\_KadS1\]), we first invert the metric [^1] \[inverse1\]
[rl]{} g\^\_\_ &= [ - r\^2\_r\^2 + \_[i,j=1]{}\^[n-1]{}(\_i\_j-\_[ij]{})\_[\_i]{}\_[\_j]{}-]{}\
&+ [\_[i,j=1]{}\^n( + 1 -)\_[\_i]{}\_[\_j]{}-]{}\
&-[\_[i=1]{}\^n\_t\_[\_i]{}]{},
and then solve the mass–shell condition $g^{\m\n}p_{\m}p_{\n} = m^2$ for the energy \[ham1\]
[c]{} [H = r(-\_[i=1]{}\^n p\_[\_i]{}),]{}\
[Ø= m\^2 + (rp\_r)\^2 +\_[i,j=1]{}\^[n-1]{}(\_[ij]{}-\_i\_j)p\_[\_i]{}p\_[\_j]{} +\_[i,j=1]{}\^n(-)p\_[\_i]{}p\_[\_j]{}]{},\
[=, =,]{}\
[=.]{}
Associated with the Killing vectors (\[killing1\]) are the integrals of motion \[integrals1\]
[c]{} [H = r(-\_[i=1]{}\^n p\_[\_i]{}), D = tH+rp\_r, K = t\^2 H + 2trp\_r + (+\_[i=1]{}\^n p\_[\_i]{}),]{}
which form $so(2,1)$ algebra under the Poisson bracket: {H,D} = H, {H,K} = 2D, {D,K} = K, Computing the Casimir invariant of the $so(2,1)$ algebra \[cas1\] C = HK-D\^2 + P\^2= m\^2 + \_[i,j=1]{}\^[n-1]{}(\_[ij]{}-\_i\_j)p\_[\_i]{}p\_[\_j]{} +\_[i,j=1]{}\^n(-)p\_[\_i]{}p\_[\_j]{}, where we added integral of motion $P^2 = \sum_{i=1}^n(p_{\vf_i})^2$ for convenience, one finds a function on the phase space which depends only on the angular variables and is quadratic in the momenta. Following the ideology in [@HakLecht1; @HakLecht2], it can be considered to be the Hamiltonian of a reduced spherical mechanics. By construction, it inherits the $U(n)$–symmetry of the background, while the decoupling of the radial coordinate is achieved at the expanse of missing $SO(2,1)$. The system (\[cas1\]) is a one–parameter deformation of the model studied in [@Gal; @GalNers]. The detailed discussion of its unitary symmetry and integrability is given below in Sect. 5.
Note that since the Hamiltonian (\[ham1\]) does not depend on the azimuthal angular variables $\vf_i$, the momenta $p_{\vf_i}$ are conserved in time. Setting them to be coupling constants yields a further reduction which, up to a redefinition of the coupling constants, coincides with the maximally superintegrable model analyzed in [@GalNers].
[**4. Even-dimensional case**]{}\
[*4.1 $D=2n$ extremal Myers–Perry–AdS–dS black hole near the horizon*]{}\
For $D=2n$ and equal rotation parameters the metric (\[gen\_KadS\]) can be brought to the form \[KadS2\]
[rl]{} ds\^2=& [(dt-\_[i=1]{}\^[n-1]{}\_i\^2 d\_i)\^2 - dr\^2 - \^2\_[i=1]{}\^[n-1]{} d\_i\^2 - d\^2 -]{}\
-& [\_[i=1]{}\^[n-1]{}\_i\^2(adt-d\_i)\^2 + \_[i<j]{}\^[n-1]{}\_i\^2\_j\^2(d\_i-d\_j)\^2]{},
where we introduced one spherical angle $\theta$ \[sph\_coor\] \_i = \_i \_n = , \_[i=1]{}\^[n-1]{}\_i\^2 = 1 and denoted
[c]{} [U = (r\^2+a\^2\_n\^2)(r\^2+a\^2)\^[n-2]{}, = (r\^2+a\^2)\^[n-1]{}-2M,]{}\
[= r\^2+a\^2\^2, \_ = 1+a\^2\^2.]{}
Imposing the extremality condition (r\_0) = \^(r\_0) = 0, one can link the black hole mass and a cosmological constant to the horizon radius and the rotation parameter = , M = .
In order to construct the near-horizon limit, it suffices to change the coordinates \[transf2\] r r\_0 + r\_0 r, t , \_i \_i + , take $\a$ and $\b_i$ in the form = , \_i = , V = . and finally send $\ve$ to zero. This yields \[ex\_KadS2\]
[rl]{} ds\^2 =& [(r\^2 dt\^2-) - \^2d\_i\^2 - d\^2 -]{}\
-& [\_i\^2(rdt+d\_i)\^2 + \_[i<j]{}\_i\^2\_j\^2(d\_i-d\_j)\^2]{},\
\_0\^2 = & r\_0\^2 + a\^2\^2,
which is a vacuum solution of the Einstein equations in the presence of a cosmological constant. Note that, when deriving the last formula, we rescaled the azimuthal angular variables \[transf21\] \_i \_i and have taken into account the following relations: \^2 r\_0\^2 r\^2, := which hold true in the near horizon limit.
[*4.2 Conformal mechanics near the horizon of the extremal Myers–Perry–AdS–dS black hole in $D=2n$*]{}\
Like in the preceding section, we shall construct the Hamiltonian of a conformal mechanics associated with the near horizon geometry of the extremal Myers–Perry–AdS–dS black hole in $D=2n$ by first inverting the metric \[inverse2\]
[rl]{} g\^\_\_ =& [(-r\^2\_r\^2) - \_[i,j=1]{}\^[n-2]{}(\_[ij]{}-\_i\_j)\_[\_i]{}\_[\_j]{} - \_\^2 -]{}\
-& [\_[i,j=1]{}\^[n-1]{}( - - )\_[\_i]{}\_[\_j]{}-]{}\
-&[2\_[i=1]{}\^[n-1]{}\_t\_[\_i]{}]{}
and then solving the mass–shell condition for the energy \[ham2\]
[rl]{} H = & [r( - \_[i=1]{}\^[n-1]{}p\_[\_i]{})]{},\
Ø= & [+ (rp\_r)\^2 +\_[i=1]{}\^[n-1]{}(p\_[\_i]{})\^2+ \_[i,j=1]{}\^[n-2]{}(\_[ij]{}-\_[i]{}\_[j]{})p\_[\_i]{}p\_[\_j]{} + p\_\^2]{}\
+ & [\_[i,j=1]{}\^[n-1]{}( - -1)p\_[\_i]{}p\_[\_j]{}]{}.
This Hamiltonian possesses conformal symmetry generated by the Killing vectors (\[killing1\]) which gives rise to the integrals of motion realized as in (\[integrals1\]) with $\O$ now taken from the previous line.
Computing the Casimir element in the conformal algebra, one gets the Hamiltonian of the spherical mechanics related to the near horizon geometry of the extremal Myers–Perry–AdS–dS black hole in $D=2n$ \[cas2\]
[rl]{} C = & [HK-D\^2 + P\^2=]{}\
=& [+ \_[i,j=1]{}\^[n-2]{}(\_[ij]{}-\_[i]{}\_[j]{})p\_[\_i]{}p\_[\_j]{} + p\_\^2 + \_0\^2\_[i,j=1]{}\^[n-1]{}( - )p\_[\_i]{}p\_[\_j]{}]{},\
= & [, = , = ]{},
where integral of motion $P^2 = \sum_{i=1}^{n-1}(p_{\vf_i})^2$ was added for convenience. As compared to the model constructed in [@Gal; @GalNers], the Hamiltonian (\[cas2\]) is deformed by the terms which depend on a cosmological constant and, as thus, it provides a one–parameter continuous deformation of the former. The detailed discussion of its unitary symmetry and integrability is given in the next section.
Because the azimuthal angular variables are cyclic, one can consider a further reduction of (\[cas2\]) which is obtained by setting the angular momenta $p_{\vf_i}$ to be coupling constants. This gives the Hamiltonian \[sp\_ham2\] [ = + p\_\^2 - \^+ \^(\_[i,j=1]{}\^[n-2]{}(\_[ij]{}-\_[i]{}\_[j]{})p\_[\_i]{}p\_[\_j]{} + \_[i,j=1]{}\^[n-1]{})]{}, $m^2$, $\s^{\prime}$, $\h^{\prime}$ ang $\gamma_i$ are the coupling constants. This model is a one–parameter deformation of that in [@GalNers].
The proof of superintegrability of (\[sp\_ham2\]) is not affected by the presence of a cosmological constant and proceeds along the same lines as in [@GalNers]. The expression in braces is the maximally superintegrable model studied in [@GalNers]. In this sector one can realize $2(n-2)-1$ functionally independent integrals of motion. The full system (\[sp\_ham2\]) involves one more canonical pair and only one extra integral of motion (the Hamiltonian (\[sp\_ham2\]) itself). The model thus lacks for one integral of motion to be maximally superintegrable.
[**5. Unitary symmetry and superintegrability**]{}\
Let us discuss symmetries and superintegrability of the spherical mechanics constructed above in more detail. Consider first the odd-dimensional case for which the dynamics is governed by the Hamiltonian (\[cas1\]). By construction, it inherits from the parent Hamiltonian (\[ham1\]) the $U(n)$–symmetry realized in the angular sector. The corresponding Killing vector fields are given in (\[un\_gen\]). In particular, one can verify that the Hamiltonian can be expressed via the linear and the quadratic Casimir invariants of $u(n)$ \_1 = \_[i=1]{}\^n\_[ii]{}, \_2 = \_[i,j=1]{}\^n(\_[ij]{}\^2 + \_[ij]{}\^2) as follows \[sph\] H\^[sph]{}\_n = \_2 - \_1\^2= \_[i,j=1]{}\^[n-1]{}(\_[ij]{}-\_i\_j)p\_[\_i]{}p\_[\_j]{} +\_[i=1]{}\^n. For later convenience we invert the transformation (\[transf11\]) and drop the arising constant multiple and a constant term in (\[cas1\]) casting the Hamiltonian into the form \[cc\]
[rl]{} C H\_n = & [H\^[sph]{}\_n - \_1\^2 =]{}\
= & \_2 - (+ )\_1\^2.
This formula shows that $u(n)$ is the spectrum generating algebra of the system. This property is particularly useful in quantum mechanics because a well developed group theoretical framework is available to construct its eigenstates and eigenvalues (see e.g. [@Iachello]).
Let us discuss integrability of the system governed by the Hamiltonian $H_n$ which involves ${2n-1}$ configuration space degrees of freedom. There are $n$ first order Casimir invariants ${{\mathcal{C}}}_1(u(1)),\dots ,{{\mathcal{C}}}_1(u(n))$ which together with $n-1$ second order ones ${{\mathcal{C}}}_2(u(2)),\dots ,{{\mathcal{C}}}_2(u(n))$ form a set of $2n - 1$ functionally independent integrals of motion in involution. Therefore this system is Liouville integrable. The issue of superintegrability is more involved because one needs to count the number of functionally independent integrals of motion among $n^2$ generators $\r_{ij}, \x_{ij}$ of $u(n)$.
Let us use the coordinates $(x_i, y_i)$ (\[xy\]), in which $\r_{ij}$ and $\x_{ij}$ read \[un\_gen1\] \_[ij]{} = x\_i p\_[y\_j]{} - y\_j p\_[x\_i]{} + x\_j p\_[y\_i]{} - y\_i p\_[x\_j]{}, \_[ij]{} = x\_i p\_[x\_j]{} - x\_j p\_[x\_i]{} + y\_i p\_[y\_j]{} - y\_j p\_[y\_i]{}. These expressions provide a canonical realization of $u(n)$.
The number of functionally independent integrals of motion is equal to the rank of the matrix ${\partial}_{\z_a}I_b$, $\z_a$ denote all the phase space coordinates and $I_b$ designates the generators. The case $n = 1$ is trivial. There is one configuration space degree of freedom and one integral of motion. For $n = 2$ there are eight coordinates $\z_a$ and four integrals of motion $I_b$. One can verify that $\text{rank}({\partial}_{\z_a}I_b) = 4$ meaning that all $I_b$ are independent. For $n = 3$ there are twelve coordinates and nine integrals of motion. However, in this case $\text{rank}({\partial}_{\z_a}I_b) = 8$ which implies that one integral is a function of the others. A relation between them can be written explicitly \[rel\] (\_[11]{}(\_[23]{}\^2 + \_[23]{}\^2) + \_[22]{}(\_[13]{}\^2 + \_[13]{}\^2) + \_[33]{}(\_[12]{}\^2 + \_[12]{}\^2)) = \_[23]{}(\_[12]{}\_[13]{} + \_[12]{}\_[13]{}) + \_[23]{}(\_[12]{}\_[13]{} - \_[12]{}\_[13]{}). Note that this relation is of the third order in generators and it does not occur in the completely reduced case because it can not be expressed in terms of $I_{ij} = \r_{ij}^2 + \x_{ij}^2, i<j,$ only (cf. [@GalNers]).
We see that for $n = 2$ and $n = 3$ the number of functionally independent integrals of motion is $4n - 4$. This holds true for all $n \geq 2$ which can be proved by induction. Let us assume that for some $N = n - 1$ there are $4 (n-1) - 4$ functionally independent integrals of motion, which one can choose as follows: \[set\] \_[11]{}, \_[12]{}, \_[12]{}, \_[22]{}, \_[1i]{}, \_[1i]{}, \_[2i]{}, \_[2i]{}, where $i=3,\dots ,n-1$. Then for $N = n$ one adds $2n - 1$ integrals $\r_{in}$ and $\x_{in}$ with $i = 1,\dots ,n-1$ as well as $\r_{nn}$. For each pair of the integrals $\r_{in}$ and $\x_{in}$, where $i = 3,\dots ,n-1$, let us consider the following columns in the matrix ${\partial}_{\z_a}I_b$: \_[\_a]{}{\_[11]{},\_[1i]{},\_[1i]{},\_[ii]{},\_[1n]{},\_[1n]{},\_[in]{},\_[in]{},\_[nn]{}}, \_[\_a]{}{\_[2]{},\_[2i]{},\_[2i]{},\_[ii]{},\_[2n]{},\_[2n]{},\_[in]{},\_[in]{},\_[nn]{}} These columns have exactly the same structure as for $n = 3$, provided one makes the substitutions of indices $(123) \rightarrow (1in)$ and $(123) \rightarrow (2in)$. Therefore they lead to the same relations between the generators as in (\[rel\]) \[rel2\] \_[kk]{}(\_[in]{}\^2 + \_[in]{}\^2) + \_[ii]{}(\_[kn]{}\^2 + \_[kn]{}\^2) + \_[nn]{}(\_[ki]{}\^2 + \_[ki]{}\^2) = \_[in]{}(\_[ki]{}\_[kn]{} + \_[ki]{}\_[kn]{}) + \_[in]{}(\_[ki]{}\_[kn]{} - \_[ki]{}\_[kn]{}), where $k = 1,2$. In order to determine $\r_{nn}$ as a function of other generators, we consider another set of columns \_[\_a]{}{\_[11]{},\_[12]{},\_[12]{},\_[22]{},\_[1n]{},\_[1n]{},\_[2n]{},\_[2n]{},\_[nn]{}}, which leads to the same relation as in (\[rel2\]) with $k = 1, i = 2$. We thus conclude that the generators $\r_{1n}, \x_{1n}, \r_{2n}, \x_{2n}$ are functionally independent. Together with (\[set\]) they form a complete set of $4n - 4$ functionally independent integrals of motion which completes the induction. It follows from the previous discussion that the spherical mechanics in odd dimensions lacks for only one integral of motion to be maximally superintegrable.
The analysis of the even dimensional case with the dynamics governed by the Hamiltonian (\[cas2\]) proceeds along the same lines. First one inverts the transformation (\[transf21\]) which brings the Hamiltonian to the form \[h2\] C \_n = + p\_\^2 + \_0\^2 H\^[sph]{}\_[n-1]{} - (\_[i=1]{}\^[n-1]{}p\_[\_i]{})\^2, where = , = , and $H^{sph}_{n-1}$ is defined in (\[sph\]). This system has $2n - 2$ configuration space degrees of freedom and its Liouville integrability is ensured by the existence of $2n - 2$ commuting independent integrals of motion $H^{sph}_2,\dots ,H^{sph}_{n-1}, p_{\vf_1},\dots ,p_{\vf_{n-1}}, \tilde{H}_n$. It has the same symmetry algebra as $H^{sph}_{n-1}$, i.e. $u(n-1)$. The complete set of $4n - 7$ functionally independent integrals of motion reads \_n, \_[11]{}, \_[12]{}, \_[12]{}, \_[22]{}, \_[1i]{}, \_[1i]{}, \_[2i]{}, \_[2i]{}, where $i = 3,\dots , n - 1$. Therefore the system lacks for two independent integrals of motion to be maximally superintegrable.
[**6. Conclusion**]{}\
To summarize, in this work we have constructed mechanical systems with the conformal and unitary symmetry which result from the near horizon Myers–Perry–AdS–dS black hole geometry in arbitrary dimension. We presented both the Hamiltonians and the integrals of motion as well as performed a reduction to a spherical mechanics which is governed by the Casimir invariant of the conformal group $SO(2,1)$. These models provide one-parameter deformations of the systems constructed recently in [@Gal; @GalNers; @GNS]. It was demonstrated that they are superintegrable but not maximally superintegrable, lacking one integral of motion in the odd–dimensional case and two integrals of motion in the even–dimensional case. A canonical realization of the unitary algebra (\[un\_gen1\]) was studied and the functionally independent generators were identified.
A further reduction of these models was attained by setting momenta canonically conjugate to the azimuthal angular variables to be coupling constants. It was shown that, up to a redefinition of constants, the resulting Hamiltonian in odd dimensions is the same as in the case of a vanishing cosmological constant [@GalNers]. In even dimensions, however, there are extra terms in the reduced Hamiltonian but their presence does not alter the number of functionally independent integrals of motions.
There are several possible developments of this work. A generalization of the present consideration to the case of nonequal rotational parameters is of considerable interest. The case of non–vanishing electromagnetic field is worthy studying as well. And also, new models can be obtained using contractions of $u(n)$ algebra *a-lá* Smorodinsky-Winternitz (see e. g. [@w] and the references therein).
0.5cm [**Acknowledgements**]{}\
This work was supported by the Dynasty Foundation and RFBR grant 13-02-90602-Arm.
[99]{}
[^1]: The constant factor of ${\displaystyle}{\frac{2(n(1-2\vk)-1)}{r_0^2}}$ has been removed by redefining $m^2$. Since the $\m_i$ sector in (\[ex\_KadS1\]) does not mix with other coordinates, the corresponding piece in the metric can be inverted separately.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the two-proton capture reaction of the prominent rapid proton capture waiting point nucleus, $^{68}$Se, that produces the borromean nucleus $^{70}$Kr ($^{68}$Se$+p+p$). We apply a recently formulated general model where the core nucleus, $^{68}$Se, is treated in the mean-field approximation and the three-body problem of the two valence protons and the core is solved exactly. The same Skyrme interaction is used to find core-nucleon and core valence-proton interactions. We calculate $E2$ electromagnetic two-proton dissociation and capture cross sections, and derive the temperature dependent capture rates. We vary the unknown $2^+$ resonance energy without changing any of the structures computed self-consistently for both core and valence particles. We find rates increasing quickly with temperature below $2-4$ GK after which we find rates varying by less than a factor of two independent of $2^+$ resonance energy. The capture mechanism is sequential through the $f_{5/2}$ proton-core resonance, but the continuum background contributes significantly.'
address:
- 'Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark'
- 'Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28042 Madrid, Spain'
author:
- 'D. Hove'
- 'E. Garrido'
- 'A.S. Jensen'
- 'P. Sarriguren'
- 'H.O.U. Fynbo'
- 'D.V. Fedorov'
- 'N.T. Zinner'
title: 'Two-proton capture on the $^{68}$Se nucleus with a new self-consistent cluster model'
---
rp-process ,capture rate ,electric dissociation ,three-body ,mean-field
25.40.Lw ,26.30.Hj ,21.45.-v ,21.60.Jz
Introduction
============
The abundance of most stable nuclei above iron in the universe can be understood as produced by various types of neutron capture [@mat85; @bar06]. However, production of about 40 stable isotopes cannot be explained in this way, but only through similar proton capture processes [@bur57; @arn03; @rau13; @rei14; @pal14]. The basic ignition fuel is a large proton flux arising from a stellar explosion. The sequence of these reactions are then one proton capture after another until the proton dripline is reached and further captured protons are immediately emitted. This dripline nucleus usually must wait to beta-decay to a more stable nucleus which in turn can capture protons anew. This is the “rp-process” [@sch98; @bro02]. These p-nuclei are also believed to be produced by other methods: gamma-proton [@arn03; @woo78] and neutrino-proton processes [@fro06; @fro06b].
The beta-decay waiting time is large for some of these nuclei along the dripline, which for that reason are denoted waiting points [@oin00]. However, another path is possible to follow for borromean proton dripline nuclei where two protons, in contrast to one, are necessary to produce a bound nucleus. Then two protons can be captured before beta-decay occurs [@gri05; @gri01]. The capture time and the corresponding mechanism are therefore important for the description of the outcome of these astrophysical processes [@gor95; @sch06].
We focus in this letter on one of these two-proton capture reactions leading from a prominent waiting point nucleus, $^{68}$Se [@sch07; @tu11], to formation of the borromean proton dripline nucleus, $^{70}$Kr ($^{68}$Se$+p+p$). The specific experimental reaction information is not available, and theoretical estimates are, at least at the moment, unavoidable [@tho04; @erl12; @pfu12]. Traditionally, the reactions have been described as sequential one-proton capture by tunneling through the combined Coulomb plus centrifugal barrier. The tunneling capture mechanisms have been discussed as direct, sequential and virtual sequential decay [@jen10; @gar05; @che07; @rod08]. They are all accounted for in the present formulation. The capture rate depends on temperature through the assumed Maxwell-Boltzmann energy distribution. It is then important to know the energy dependence of the capture cross section for given resonance energies, and especially in the Gamow window [@chr15].
Clearly the desired detailed description requires a three-body model which is available and even applied to the present processes [@hov16; @gar04]. However, the crucial proton-core potentials have so-far been chosen phenomenologically to produce the essentially unknown, but crucial, single-particle energies. A new model involving both core and valence degrees of freedom was recently constructed to provide mean-field proton potentials derived from the effective nucleon-nucleon mean-field interaction [@hov17; @hov17b; @hov17c]. In turn these potentials produce new and different effective three-body potentials, which in the present letter is exploited to investigate the two-proton capture rates.
The techniques are in place all the way from the solution of the coupled core and valence proton system [@hov16; @hov17b; @hov17c], over the self-consistent three-body input and subsequent calculations [@nie01; @jen04], to the capture cross sections and rates [@gar15; @die10; @die11; @die10b]. We shall first sketch the steps in the procedure used in the calculations. Then we shall discuss in more details the numerical results of interest for the astrophysical network computations, which calculates the abundances of the isotopes in the Universe.
Theoretical description
=======================
The basic formulation and the procedure are described in [@hov16; @gar15; @gar15b]. The framework is the three-body technique but based on the proton-core potential derived through the self-consistently solved coupled core-plus-valence-protons equations [@hov17; @hov17b]. The procedure is first to select the three-body method, second to formulate how to calculate the capture rate, and finally to choose the numerical parameters to be used in the computations.
Wave functions
--------------
First, the many-body problem is solved for a mean-field treated core interacting with two surrounding valence protons [@hov16]. The details of this recent model are very elaborate, but already applied on two different neutron dripline nuclei [@hov16; @hov17]. It then suffices to sketch the corresponding details for the present application. Briefly, the novel features are to find the mean-field solution for the core-nucleons in the presence of the external field from the two valence protons. In turn, folding the basic nucleon-nucleon interaction and the core wave function provides the interaction between each valence proton and the core nucleus. The interactions between core and valence nucleons then depend on their respective wave functions, which are found self-consistently by iteration. We emphasize that the same nucleon-nucleon interaction is used both in the core and for this valence-proton core calculation. The crucial main ingredient in the three-body solution is this interaction, which then is provided by the procedure and determined independent of subsequent applications.
The present application exploits the properties of the derived three-body solution. The three-body problem is solved by adiabatic expansion of the Faddeev equations [@gar04] in hyperspherical coordinates. When necessary the continuum is discretized by a large box confinement [@gar15a]. The main coordinate is the hyperradius, $\rho$, defined as the mean radial coordinate in the three-body system [@gar15b; @hov14]. More specifically we have $$\begin{aligned}
(2m_n+m_c) \rho^2
= m_n(\bm{r}_{v_1} - \bm{r}_{v_2})^2
+ m_{c} \sum_{i=1}^2 (\bm{r}_{v_i} - \bm{R}_{c})^2
\label{rhodef}\end{aligned}$$ where $m_n$, $m_{c}$, $\bm{r}_{v_1}$, $\bm{r}_{v_2}$ and $\bm{R}_{c}$ are neutron mass, core mass, valence-proton coordinates and core center-of-mass coordinate, respectively. The three-body wave function is found through this procedure for given angular momenta as functions of the hyperspherical coordinates for the required ground state ($\Psi_{J}$). When necessary the continuum is discretized by a large box confinement and the discretized continuum states ($\psi_{j}^{(i)}$) are calculated.
Reactions
---------
The two-proton capture reaction $p+p+c \leftrightarrow A + \gamma$ cross section $\sigma_{ppc}$ and the photodissociation cross section $\sigma^{\lambda}_{\gamma}$ of order ${\lambda}$ are related [@gar15b]. The three-body energy, $E$, and the ground state energy, $E_{gr}$, determine the photon energy by $E_{\gamma} = E + |E_{gr}|$. The dissociation cross section is then given by $$\begin{aligned}
\sigma^{\lambda}_{\gamma}(E_{\gamma})
=& \frac{(2 \pi)^3 (\lambda +1)}{\lambda ((2\lambda +1)!!)^2} \left( \frac{E_{\gamma}}{\hbar c}\right)^{2\lambda-1} \frac{d}{d E}\mathcal{B}({E}\lambda, 0 \rightarrow \lambda), \label{eq siggam}\end{aligned}$$ where the strength function for the ${E}\lambda$ transition, $$\begin{aligned}
\frac{d}{d E} \mathcal{B}({E}\lambda, 0 \rightarrow \lambda) =
\sum_i \left| \braket{\psi_{\lambda}^{(i)} | | \hat{\Theta}_{\lambda} | |
\Psi_{0}} \right|^2 \delta(E-E_i), \label{eq tran}\end{aligned}$$ is given by the reduced matrix elements, $\braket{\psi_{\lambda}^{(i)} | | \hat{\Theta}_{\lambda} | | \Psi_{0}}$, where $\hat{\Theta}_{\lambda}$ is the electric multipole operator, $\psi_{\lambda}^{(i)}$ is the wave function of energy, $E_i$, for all bound and (discretized) three-body continuum states in the summation. The capture reaction rate, $R_{ppc}$, is given by Ref. [@die11] $$\begin{aligned}
R_{ppc}(E) = & \frac{8 \pi}{(\mu_{cp} \mu_{cp,p})^{3/2}} \frac{\hbar^3}{c^2}
\left( \frac{E_{\gamma}}{E} \right)^2 \sigma^{\lambda}_{\gamma}(E_{\gamma}),
\label{eq rate}\end{aligned}$$ where $\mu_{cp}$ and $\mu_{cp,p}$ are reduced masses of proton and core and proton-plus-core and proton, respectively. For the astrophysical processes in a gas of temperature, $T$, we have to average the rate in Eq. (\[eq rate\]) over the corresponding Maxwell-Boltzmann distribution, $B(E,T) = \frac{1}{2}
E^2 \exp(-E/T)/T^3$, $$\begin{aligned}
\braket{R_{ppc}(E)} = \frac{1}{2T^3} \int E^2 R_{ppc}(E) \exp(-E/T)
\, dE, \label{eq ave rate}\end{aligned}$$ where the temperature is in units of energy.
Interactions
------------
The decisive interaction is first of all related to the mean-field calculation of the core. We use the Skyrme interaction SLy4 [@cha98] with acceptable global average properties. The application on one specific nuclear system requires some adjustment to provide the known borromean character, that is unbound proton-core $f_{5/2}$ resonance at $0.6$ MeV [@san14] and two protons bound to the core. With a minimum of changes we achieve this by shifting all energies while leaving the established structure almost completely unaltered. The simplest consistent such modification is by scaling all the main Skyrme strength parameters, $t_i$, by the same factor, $0.9515$.
The density dependence of the Skyrme interaction can be viewed as a parametrized three-body potential. To simulate that effect we employ a short-range Gaussian, $S_0\exp(-\rho^2/b^2)$, which depends on the three-body hyperradial coordinate $\rho$. We choose $b=6$ fm and leave $S_0$ to fine-tune each of the $0^+$ and $2^+$ three-body energies. This is necessary since the keV-scale of binding is crucial for tunneling through single MeV height barriers. This level of accuracy is beyond the present capability of many-body model calculations. To reproduce the predicted $0^+$ energy of $-1.34$ MeV [@wan12] a three-body strength $S_0 = -17.5$ MeV is needed. The unknown $2^+$ energy is varied from almost bound, zero energy, to the top of the barrier by $S_0$ changing from $-35.05$ MeV to $-26.22$ MeV.
Effective three-body potentials \[sec:3b\]
==========================================
The elaborate numerical calculations produce the sets of coupled “one-body” effective potentials depending on hyperradius as shown in Fig. \[fig:potsw\] for both $0^{+}$ and $2^{+}$. The continuations beyond the $20$ fm in the figure are almost quantitatively Coulomb plus centrifugal behavior and as such reveal no surprises. The kinks and fast bends reflect avoided crossings and related structure changes. They are especially abundant at small distances and around the barriers. The diagonal non-adiabatic coupling terms as well as the diagonal structure-less three-body Gaussian potentials are included in the calculations, but not in the figure. They both leave the structure essentially unaltered although they may change the energies of the solutions rather substantially.
![The effective, adiabatic potentials for the $0^{+}$ (red, solid), and the $2^{+}$ (light-blue, dashed) configuration in $^{68}\text{Se}+p+p$ using the SLy4 Skyrme interaction between core and valence protons, scaled to reproduce the experimental $f_{5/2}$ resonance energy of $0.6$ MeV in $^{68}\text{Se}+p$ [@san14]. The dotted horizontal line is the $0^+$ energy at $-1.34$ MeV from $S_0= -17.5$ MeV. \[fig:potsw\]](CompEffPots.pdf){width="1\linewidth"}
The $0^{+}$ ground-state at $-1.34$ MeV, predicted from systematics [@aud12], is reproduced with the chosen parameters. The structure corresponds to the configuration of the pronounced minimum in the lowest $0^{+}$ potential. No $0^{+}$ resonance are produced by the potentials in Fig. \[fig:potsw\]. The ground state is the final state in the capture process independent of the specific mechanism.
However, the decisive capture process proceeds within the $2^{+}$ continuum from the large to the short-distance attractive region of the potentials shown in Fig. \[fig:potsw\]. This lowest minimum is rather similar to the $0^{+}$ minimum but the non-adiabatic repulsive terms increase the energy substantially. Unfortunately nothing is known about a $2^{+}$ resonance which would strongly influence the capture rate. Consequently the strength, $S_0$, is used to vary the position of the $2^{+}$ resonance from almost bound to disappearance above the barriers. Both the resonance energy, the height, and the rather broad Coulomb shape of the barrier strongly influence the capture process.
The structure of these potentials is substantially simpler than those obtained in [@hov16] where low-lying single-proton states $p_{3/2}$ and $f_{5/2}$ both appeared. The present simplification is an automatic result of the procedure using the nucleon-nucleon mean-field effective interaction to calculate the proton-core potential. This is not an ad hoc assumption, but arises naturally due to identical interactions for both core and valence particles. As such it is a novel deduction embedded in the design of our model. The lack of single-particle states of different parity implies that no $1^-$ three-body resonance states appear in the low-energy region. The transition is then necessarily an $E2$ transition, which contributes to the longer effective lifetime of the system, and could very well be part of the reason this system is a critical waiting point.
Quantitative results
====================
The all-important core-valence proton potential is derived naturally and unambiguously by our mean-field core treatment, as discussed in the previous sections. As a result the two-proton capture cross section follows directly, only depending on the three-body resonance level. This is discussed in the following section, after which the resulting temperature averaged reaction rates are presented. This is supplemented by a discussion of the reaction mechanism and its implication for the possible reactions.
Cross section
-------------
The incident flux of low-energy protons on the core nucleus may result in capture. The corresponding cross section is most easily obtained from calculation of the inverse reaction, that is photodissociation of the $0^+$ ground state, $\Psi_{0}$, of $^{70}$Kr. The discretized continuum states, $\psi_{\lambda}^{(i)}$, are computed and the cross section is obtained from Eqs. (\[eq tran\]) and (\[eq siggam\]) with $\lambda =2$. The two-proton capture cross section of $^{68}$Se, obtained from Eq. (\[eq siggam\]), is shown in Fig. \[fig:cross\] as function of the three-body energy.
![The electromagnetic $E2$ dissociation cross section, $\sigma^{(\lambda=2)}_{\gamma}(E_{\gamma})$, for the proces, $^{70}\text{Kr} + \gamma \rightarrow ^{68}\text{Se}+p+p$, as a function of photon energy. The $0^+$ final state energy is $-1.34$ MeV and the $2^+$ resonance energies are $E=0.5, \, 1.0, \,
2.0,$ and $4.0$ MeV, respectively. The discretized continuum states are obtained using box sizes of $\rho_{max} = 150, 200$ fm. \[fig:cross\]](cross_gam.pdf){width="1\linewidth"}
The peaks in the capture cross section occur at experimentally unknown resonance energies where the tunneling probability is large. We therefore vary the energy from $0.5$ MeV to $4.0$ MeV where the widths of the peaks in the cross section increase with energy as the top of the barrier is approached. We emphasize that the crucial quantity is the resonance energy. This can be tested by varying the number of adiabatic potentials used in the calculation. This results in somewhat different resonance energy which however can be compensated for by use of the three-body potential, which in turn recover the cross section in Fig. \[fig:cross\].
These features are simply understood as enhanced spatial overlaps between the $2^+$ continuum states in the resonance region and the ground state wave function, expressed through Eq. (\[eq tran\]). Beside the resonance contributions we also find significant, although several orders of magnitude smaller, background contributions, which incidentally is independent of the size of the discretization box, as long as it is sufficiently large [@gar15a].
Capture rates
-------------
The capture cross sections are the main ingredient in the calculation of the two-proton absorption rate appropriate for the temperature dependent astrophysical network computation. The average rate in Eq. (\[eq ave rate\]) are shown in Fig. \[fig:rate\] as function of temperature. The Boltzmann smearing factor produces very smooth curves of the same qualitative behavior. They are zero at zero temperature and energy, because the barrier is infinitely thick. All rates then increase to a maximum at the Gamow peak where the best compromise is reached between the decreasing temperature distribution and the increasing tunneling probability.
![The reaction rate for the radiative capture process $^{68}\text{Se}+p+p \rightarrow ^{70}\text{Kr} + \gamma$, as function of temperature for the different $2^+$ resonance energies in Fig. \[fig:cross\]. The black dashed curve is the background contribution. \[fig:rate\]](rate.pdf){width="1\linewidth"}
The peak contribution moves to higher energy and becomes smoother with increasing resonance energy. Above temperatures of a few GK the average rate variation is moderate and the size roughly of order $\simeq 6 \times 10^{-11}$ cm$^{6}[N_{A} mol]^{-2} s^{-1}$. A low-lying resonance energy corresponds to low-lying peak position of larger height. We emphasize that the background without resonance contribution obviously is smaller but only by roughly a factor of two as soon as the temperature exceeds about $4$ GK ($\sim0.34$ MeV). In other words, if temperatures are in the astrophysically interesting range below about $1$ GK, the size variations are substantial, and vice versa above a few GK the details from the microscopic origin are smeared out.
The actual size of the rate may reveal deceivingly little variation at the relatively high temperatures. However, the barrier height and width are all-decisive and both may easily be different for other systems where the single-particle structure at the Fermi energy is different and perhaps more complicated as studied in [@hov16]. The relatively large $2^+$ background contribution might suggest significant corresponding $0^+$ continuum contributions. However, the $0^+$ barriers in Fig. \[fig:potsw\] are larger and the $0^+
\rightarrow 0^+$ transition as well require processes involving atomic electrons. It is again worth emphasizing that a superficially more complete calculation with for example many coupled potentials would provide the same rates after adjusting to the same resonance energy.
![The probability of the three-body, $^{68}\text{Se}+p+p$, wave function for the lowest allowed potential, integrated over directional angles $\left( \sin^2 (\alpha) \cos^2 (\alpha) \int
|\Phi_n(\alpha, \rho, \Omega_x, \Omega_y)|^2 d\Omega_x
d\Omega_y\right)$, as a function of hyperradius, $\rho$, and hyperangle, $\alpha$, related to the Jacobi coordinate system where “x” is between core and proton. \[fig:angDist\]](AngDist.pdf){width="1\linewidth"}
Reaction mechanism
------------------
The rate depends on the capture mechanism. We are here only concerned with three-body capture, but a dense environment would enhance four-body capture processes as discussed in [@die10]. The overall three-body process is tunneling through a barrier of particles in a temperature distribution of given density. Once inside the relatively thick barrier they have essentially only the option of emitting photons to reach the bound ground-state. However, the first of this two-step process can occur through different mechanisms, where the most obvious possibility is to be captured in different angular momentum states. The conservation of angular momentum and parity quantum numbers are crucial in connection with resonance positions. If low-lying $1^-$ continuum states are allowed they would be preferred, and vice versa if prohibited $2^+$ continuum states would be preferred. Low-lying resonances enhance the contributions substantially. This selection depends strongly on the nucleus under investigation.
For a given angular momentum of the three-body continuum states, we still may encounter several qualitatively different ways of absorbing two protons from the continuum [@gar11]. These mechanisms were discussed in [@jen10] for the inverse process of dissociation, that is direct, sequential and virtual sequential decay. They are all accounted for in the present formulation. In [@hov16] we concluded that the direct process is most probable for very low three-body energy when two-body subsystems are unbound. If the energy is larger than stable two-body substructures such intermediate vehicles enhance the rates and the mechanism is sequential.
Even when it is energetically forbidden to populate two-body resonance states it may be advantageous to exploit these structures virtually while tunneling through an also energetically forbidden barrier. This is appropriately named the virtual sequential two-body mechanism. It may be appropriate to emphasize that a similar three-body virtual mechanism is forbidden because the three-body energy is conserved in contrast to the energy of any two-body subsystem.
The mechanism for the present capture process is revealed in Fig. \[fig:angDist\] where the $2^+$ probability integrated over the directional angles is shown for the lowest potential as function of hyperradius and one of the Jacobi angles. It is a strikingly simple structure for hyperradii larger than about $15$ fm, which for these coordinates is equivalent to one proton at that distance from the center of mass of the combined proton-core system. Since the Jacobi angle, $\alpha$, is either close to zero or $\pi/2$, this simply means that one proton is staying very close to the core for all these hyperrradii. Eventually also this proton has to move away from the core since no bound state exist. But the process is sequential through this substructure which can be determined to be the proton-core $f_{5/2}$ resonance.
The higher-lying configurations corresponding to the three following potentials also show precisely the same $f_{5/2}$ structure. This is explained by combining the compact proton-core $f_{5/2}$ resonance with one non-interacting (apart from Coulomb and centrifugal) distant proton in any angular momentum configuration consistent with a $2^+$ structure. The angular momenta capable of combining with $f_{5/2}$ to produce $2^+$ are $p_{1/2}, \, p_{3/2}, \, f_{5/2}, \, f_{7/2},$ and $h_{9/2}$. This also implies that for temperatures much smaller than the $f_{5/2}$ resonance energy it would be energetically advantageous to start the capture process in a configuration corresponding to direct three-body capture. The change of structure, around avoided level crossings, to two-body resonance configurations would then greatly reduce the barrier and substantially enhance the capture rate.
Conclusion
==========
The new model that treats the core and the two valence particles self-consistently and simultaneously is applied on the waiting point nucleus ($^{68}$Se) for the astrophysical rp-process. This is done essentially without any free parameters or phenomenological fitting, which makes the results much less arbitrary than usual three-body calculations. Adding two protons, but not one, produces a bound system, $^{70}$Kr, which is then a borromean nucleus. A moderate overall scaling of the Skyrme interaction SLy4 reproduces the scarcely known properties of these dripline nuclei. Other Skyrme interactions provide very similar results.
We calculate the radiative two-proton capture rate as function of temperature for different resonance energies. We investigate the mechanism and find that sequential capture of one proton after the other by far is dominating. The first available single-particle resonance state, $f_{5/2}$, is the vehicle, whereas the other proton can approach in continuum states of even higher angular momentum. In practice, after tunneling through the barrier into the $2^+$ resonance state, in practice only $E2$ electric transition to the ground state is allowed. Background capture through non-resonance continuum states also contributes significantly to the capture process. The sequential $2^+$ capture mechanism might for other nuclei be replaced by for example the normally larger $1^-$ capture.
In conclusion, the two-proton capture rates at a waiting point at the dripline are successfully calculated with a conceptually relatively simple, but technically advanced, new model. The same effective nucleon-nucleon interaction is used for both the nuclear mean-field and the proton-core calculations. The temperature dependent rate and the corresponding capture mechanism are calculated with less ambiguity than in previous calculations. A number of applications are now feasible.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was funded by the Danish Council for Independent Research DFF Natural Science and the DFF Sapere Aude program. This work has been partially supported by the Spanish Ministerio de Economia y Competitividad under Project FIS2014-51971-P.
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---
abstract: 'Given an elliptic curve $E$ in Legendre form $y^2 = x(x - 1)(x - \lambda)$ over the fraction field of a Henselian ring $R$ of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of $E$ over $R$ which depends only on the valuation of $\lambda$. We provide several examples along with an easy corollary concerning $2$-torsion.'
author:
- Jeffrey Yelton
bibliography:
- 'bibfile.bib'
title: Semistable models of elliptic curves over residue characteristic 2
---
Let $R$ be a Henselian ring of mixed characteristic $(0, 2)$ with a discrete valuation $v : K^{\times} \to {\mathbb{Q}}$ normalized so that $v(2) = 1$, and let $K$ be its fraction field. Let $E$ be the elliptic curve over $K$ defined by an equation of the form $y^2 = f(x)$ for some separable polynomial $f(x) \in K[x]$. After replacing $K$ by a suitable extension and possibly scaling $y$ by an element of $K$ to get an isomorphic elliptic curve, we assume that $\alpha_1, \alpha_2, \alpha_3 \in R$ with $\alpha_2 - \alpha_1 \in R^{\times}$. After possibly applying another isomorphism which translates $x$ by $-\alpha_1$ and then scales it by $(\alpha_2 - \alpha_1)^{-1}$, we further assume that $E$ is in *Legendre form*; that is, $E$ is a smooth projective model of an affine curve given by an equation of the form $$y^2 = f(x) := x(x - 1)(x - \lambda)$$ with $\lambda \in R \smallsetminus \{0, 1\}$ (we denote the point at infinity by $\mathcal{O} \in E(K)$). The purpose of this note is to explicitly find a semistable model of $E$ over a finite algebraic extension of $R$. More precisely, we will find a finite algebraic extension $K' / K$ with ring of integers $R'$ such that $E$ has a model $E^{{\mathrm{ss}}}$ over $R'$ given by explicit formulas and which has either good or (split) multiplicative reduction.
It is well known (see for instance [@serre1989abelian §IV.1.2]) that any elliptic curve over a discrete valuation field has good (resp. multiplicative) reduction over some finite algebraic extension of that field if and only if the valuation of its $j$-invariant is nonnegative (resp. negative). The formula for the $j$-invariant of the Legendre curve $E$ is given as in [@silverman2009arithmetic Proposition III.1.7] by $$\label{eq j}
j(E) = 2^8 \frac{(\lambda^2 - \lambda + 1)^3}{\lambda^2 (\lambda - 1)^2}.$$ For simplicity, we assume throughout this paper that $v(\lambda - 1) = 0$, noting that if $v(\lambda - 1) > 0$, then we have $v(\lambda) = 0$ and the assumption becomes true after replacing $\lambda$ by $1 - \lambda$ and applying the isomorphism given by $(x, y) \mapsto (1 - x, y)$. It follows from this assumption and the formula in (\[eq j\]) that $v(j(E)) = 8 - 2v(\lambda)$ and that therefore any semistable model $E^{{\mathrm{ss}}}$ has good (resp. multiplicative) reduction if and only if $v(\lambda) \leq 4$ (resp. $v(\lambda) > 4$). This explains “why" the formula for the $j$-invariant includes an “extra" factor of $2^8$. The equivalence between potential good reduction and integrality of the $j$-invariant over residue characteristic $2$ is derived by Silverman as [@silverman2009arithmetic Corollary A.1.4] by converting $E$ to its Deuring normal form and arguing via manipulations involving the $j$-invariant. In the course of constructing a semistable model of $E$, we will show essentially the same result more directly and without invoking the $j$-invariant. To the best of the author’s knowledge, such an explicit method of computing semistable models of elliptic curves in Weierstrass form over mixed characteristic $(0, 2)$ is not present in the literature outside of particular examples such as those in [@bouw2015semistable §4.1] (indeed, some of the ideas and notation used in this note were inspired by [@bouw2015semistable]). We believe that the strategy presented here is also applicable to determining semistable models and reduction types for hyperelliptic curves over mixed characteristic $(0, 2)$.
Our general set-up {#S1}
------------------
Any model $E^{{\mathrm{ss}}}$ of an elliptic curve with semistable reduction over residue characteristic $2$ must be determined by an equation of the form $$\label{eq model} y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6,$$ with all $a_i \in R'$ (where $R' / R$ is some finite extension of Henselian rings with fraction fields $K' / K$) and $a_1$ and $a_3$ not both lying in the maximal ideal of $R'$. We write $\bar{E}^{{\mathrm{ss}}}$ for the reduction of $E^{{\mathrm{ss}}}$; it is a projective curve over the residue field which is either smooth or has a single node. We note that at least one of $a_1$ and $a_3$ must be a unit in $R$; that $v(a_1) > 0$ is then sufficient to ensure smoothness of $\bar{E}^{{\mathrm{ss}}}$; and that $v(a_3) > 0$ on the other hand implies that $E^{{\mathrm{ss}}}$ has a node at $(0, 0)$.
An equation of the form given in (\[eq model\]) can be converted to an equation of the form $y^2 = F(x) \in K'[x]$ for some finite extension $K' / K$ by completing the square: we replace $y$ by $y - \frac{1}{2}(a_1 x + a_3)$. Then an isomorphism from the curve $E$ given by $y^2 = f(x)$ to the curve given by $y^2 = F(x)$ must be of the form $(x, y) \mapsto (\alpha + \beta x, \beta^{3/2} y)$ for some $\alpha, \beta \in K'$ (in fact, $K'$ will just be the extension given by adjoining the elements $\alpha$ and $\beta^{1/2}$ to $K$). Given such elements $\alpha, \beta$, we first observe that this isomorphism maps $E$ to the curve defined by $$\label{eq model F} y^2 = F(x) = F_{\alpha, \beta}(X) := (x + \alpha\beta^{-1})(x + \alpha\beta^{-1} - \beta^{-1})(x + \alpha\beta^{-1} - \lambda\beta^{-1}).$$ Now we want to find polynomials $G(x) = x^3 + a_2 x^2 + a_4 x + a_6 \in R'[x]$ and $H(x) = a_1 x + a_3 \in R'[x]$ such that $F = G + \frac{1}{4}H^2$; then the isomorphism $(x, y) \mapsto (x, y + H(x))$ maps the curve given by (\[eq model F\]) to the one given by (\[eq model\]).
For each integer $n \geq 1$, we write $F^{(n)}$ for the $n$th derivative of $F$ divided by $n!$, so that $F^{(n)}(0)$ equals the coefficient of the $x^n$-term of $F$. We compute formulas for the elements $a_1, a_2, a_3$, using the fact that $$G(x) + \frac{1}{4}H(x)^2 = F(x) = x^3 + F^{(2)}(0) x^2 + F^{(1)}(0) x + F(0).$$ Our formulas are as follows: $$\label{eq formulas} \begin{split} a_3 &= 2\sqrt{F(0) - a_6} \\ a_1 &= \frac{2F^{(1)}(0) - a_4}{a_3} = \frac{F^{(1)}(0) - a_4}{\sqrt{F(0) - a_6}} \\ a_2 &= F^{(2)}(0) - \frac{1}{4}a_1^2 = F^{(2)}(0) - \frac{(F^{(1)}(0) - a_4)^2}{4(F(0) - a_6)} \end{split}$$ It will be convenient to fix $a_4 = a_6 = 0$ so that the elements $a_1, a_2, a_3$ are completely determined (up to choosing a sign for $a_3$) by our choice of $\alpha$ and $\beta$ and are given by the slightly simpler formulas $$\label{eq formulas2} a_1 = \frac{F^{(1)}(0)}{\sqrt{F(0)}}; \ a_2 = F^{(2)}(0) - \frac{1}{4}a_1^2 = F^{(2)}(0) - \frac{F^{(1)}(0)^2}{4F(0)}; \ a_3 = 2\sqrt{F(0)}.$$
The $v(\lambda) < 4$ case {#S2}
-------------------------
In this section we assume that $v(j(E)) > 0$. We then see from the formula in (\[eq j\]) that we have $0 \leq m := v(\lambda) < 4$. Since $j(E)$ is integral, the desired model $E^{{\mathrm{ss}}}$ should have good reduction. For any $\lambda$ with $0 \leq v(\lambda) < 4$, we now show how to find algebraic elements $\alpha, \beta \in \bar{K}$ such that we get $v(a_1) > 0$, $v(a_3) = 0$ and even allow $a_2$ to be any integral element that we choose.
\[thm m<4\]
Assume that $0 \leq m < 4$. Choose any $\beta \in \bar{K}$ such that $v(\beta) = \frac{1}{3}m + \frac{2}{3}$ (where $v$ is extended uniquely to a discrete valuation on $K(\beta)$). Let $\alpha$ be a root of the polynomial $$\label{eq val<4} P(X) := 3X^4 - 4(1 + \lambda) X^3 + 6\lambda X^2 - \lambda^2 - \delta,$$ where $\delta \in K(\beta^{1/2})$ satisfies $v(\delta) \geq \frac{4}{3}m + \frac{8}{3}$ (e.g. $\delta = 0$). Then $E$ is isomorphic over $K' := K(\beta^{1/2}, \alpha, F(0)^{1/2})$ to the elliptic curve $E^{{\mathrm{ss}}}$ given by the equation in (\[eq model\]), where $a_4 = a_6 = 0$ and the other coefficients $a_i$ are given by the formulas in (\[eq formulas2\]). The isomorphism $\varphi : E \stackrel{\sim}{\to} E^{{\mathrm{ss}}}$ is given by composing the map $(x, y) \mapsto (\alpha + \beta x, \beta^{3/2}y)$ with the map $(x, y) \mapsto (x, y + \frac{1}{2}(a_1 x + a_3))$.
We have $v(a_1) > 0$, $v(a_2) \geq 0$, and $v(a_3) = 0$ (which implies that $E^{{\mathrm{ss}}}$ has good reduction). Moreover, we have $$a_2 = \frac{\delta}{4\beta \alpha(\alpha - 1)(\alpha - \lambda)}.$$
Assume that we have chosen an algebraic element $\beta$ with $v(\beta) = \frac{1}{3}m + \frac{2}{3}$, an element $\delta \in K(\beta^{1/2})$ satisfying $v(\delta) = \frac{1}{3}m + \frac{2}{3}$, and a root $\alpha$ of the polynomial $P(X)$. The first statement just reaffirms what was shown in the above discussion where the formulas for $a_1, a_2, a_3 \in K'$ were derived, so our main task is to demonstrate the desired bounds for the valuations of these elements.
We note first that $v(\lambda^2) = 2m < \frac{4}{3}m + \frac{8}{3}$, so that the constant coefficient of the polynomial $P$ has valuation equal to $2m$ regardless of our choice of $\delta$. Then since the coefficient of $X^4$ is a unit and the coefficient of $X^3$ (resp. $X^2$) has valuation at least $2 \geq \frac{1}{2}m$ (resp. equal to $1 + m > m$), the Newton polygon of this polynomial consists of a single line segment with slope $\frac{1}{2}m$. It follows that $v(\alpha) = \frac{1}{2}m$. We clearly have $v(\alpha - 1) = 0$ and $v(\alpha - \lambda) = v(\alpha) = \frac{1}{2}m$ as long as $m > 0$. If $m = 0$, we claim that these equalities still hold so that $v(\alpha - 1) = v(\alpha - \lambda) = 0$. To see this, assume that $m = 0$ and consider the polynomials $P(X + 1)$ and $P(X + \lambda)$; it is straightforward to calculate (using the fact that $v(\lambda) = v(\lambda - 1) = 0$) that the Newton polygons of these shifted polynomials both coincide with the Newton polygon of $P$, and the claim follows from the fact that $\alpha - 1$ and $\alpha - \lambda$ are roots of the respective polynomials. We now have $$v(F(0)) = v(\alpha) + v(\alpha - 1) + v(\alpha - \lambda) - 3v(\beta) = 2v(\alpha) - 3v(\beta) = m - m - 2 = -2.$$ The desired equality $v(a_3) = v(2\sqrt{F(0)}) = 0$ immediately follows.
Now we treat the requirement that $v(a_2) \geq 0$, using the formula for $b_2$ given in (\[eq formulas2\]). We use the formulas $$\begin{split} \label{eq formulas3} &\beta^3 F(0) = \alpha(\alpha - 1)(\alpha - \lambda); \ \ \beta^2 F^{(1)}(0) = \alpha(\alpha - 1) + \alpha(\alpha - \lambda) + (\alpha - 1)(\alpha - \lambda); \\
&\beta F^{(2)}(0) = \alpha + (\alpha - 1) + (\alpha - \lambda) \end{split}$$ to expand $4\beta^4 F(0)a_2 = 4(\beta F^{(2)}(0))(\beta^3 F(0)) - (\beta^2 F^{(1)}(0))^2$ as $$\begin{split} 4\alpha^2(\alpha - 1)(\alpha - \lambda) + &4\alpha(\alpha - 1)^2(\alpha - \lambda) + 4\alpha(\alpha - 1)(\alpha - \lambda)^2 \\
- &(\alpha^2(\alpha - 1)^2 + \alpha^2(\alpha - \lambda)^2 + (\alpha - 1)^2(\alpha - \lambda)^2 \\
&\hspace{2em} + 2\alpha^2(\alpha - 1)(\alpha - \lambda) + 2\alpha(\alpha - 1)^2(\alpha - \lambda) + 2\alpha(\alpha - 1)(\alpha - \lambda)^2) \end{split}$$ $$\begin{split} &= 2\alpha^2(\alpha - 1)(\alpha - \lambda) + 2\alpha(\alpha - 1)^2(\alpha - \lambda) + 2\alpha(\alpha - 1)(\alpha - \lambda)^2 - \alpha^2(\alpha - 1)^2 - \alpha^2(\alpha - \lambda)^2 - (\alpha - 1)^2(\alpha - \lambda)^2 \\
&= 2(2\alpha - \lambda)(\alpha)(\alpha - 1)(\alpha - \lambda) - \alpha^2(\alpha - \lambda)^2 + (\alpha - 1)^2 (2\alpha(\alpha - \lambda) - \alpha^2 - (\alpha - \lambda)^2) \\
&= 2(2\alpha - \lambda)(\alpha)(\alpha - 1)(\alpha - \lambda) - \alpha^2(\alpha - \lambda)^2 - (\alpha - 1)^2\lambda^2 \\
&= [4\alpha^4 - 2(2 + 3\lambda)\alpha^3 + 2(3\lambda + \lambda^2)\alpha^2 - 2\lambda^2\alpha] - [\alpha^4 - 2\lambda\alpha^3 + \lambda^2\alpha^2] - [\lambda^2\alpha^2 - 2\lambda^2\alpha + \lambda^2] \end{split}$$ $$\label{eq a_2} = 3\alpha^4 - 4(1 + \lambda)\alpha^3 + 6\lambda \alpha^2 - \lambda^2.$$ Thus, since $P(\alpha) = 0$ can be written as the above expression minus the element $\delta$, we have $\delta = 4\beta^4 F(0)a_2$ (implying the claimed formula for $a_2$). Now the fact that $v(a_2) \geq 0$ is equivalent to saying that $v(\delta) \geq 2 + 4v(\beta) + v(F(0)) = \frac{1}{3}m + \frac{2}{3}$, which was indeed our condition for $\delta$.
It remains only to check that $v(a_1) > 0$. Note that $v(\beta F^{(2)}(0)) \geq \min\{v(\alpha), v(1)\} = 0$. It follows from the formula for $a_2$ in (\[eq formulas2\]) that $v(\frac{1}{4}a_1) \geq \min\{v(F^{(2)}(0)), v(a_2)\} = v(\beta^{-1}) > -2$. Therefore, $v(a_1) - 2 > -2$, hence the desired inequality.
\[ex m=0\]
Suppose we want to find a semistable model $E^{{\mathrm{ss}}}$ of the elliptic curve $E / {\mathbb{Q}}_2$ given by $y^2 = x^3 - 1$ at the prime $(2)$. This elliptic curve is well known to be CM, and so any semistable model $E^{{\mathrm{ss}}}$ should have good reduction; we can also see this by noting that $j(E) = 0$. In fact, $E$ is isomorphic (over $K := {\mathbb{Q}}_2(\omega)$) to the Legendre curve with $\lambda = -\omega^2$, where $\omega := \frac{1}{2}(-1 + \sqrt{-3})$ is a primitive cube root of unity; since $m = v(\lambda) = 0 < 4$, we may apply Theorem \[thm m<4\].
We have $$P(X) = 3X^4 - 4(1 - \omega^2) X^3 - 6\omega^2 X^2 - \omega - \delta.$$ By an easy computation, plugging in $X = \omega$ to the above polynomial yields $8\omega^2 - \delta$, so we may take $\delta = 8\omega^2$ (noting that $v(\delta) = 3 \geq \frac{4}{3}m + \frac{8}{3}$) and $\alpha = \omega$. Then we may choose $\beta$ to be any element with valuation $\frac{1}{3}m + \frac{2}{3} = \frac{2}{3}$, say $\beta = 2^{2/3}$. Now evaluating the formulas in (\[eq formulas2\]) yields the following equation for $E^{{\mathrm{ss}}}$ over the (abelian) extension $K' := K((-3)^{1/4}, 2^{1/3})$. $$y^2 - \omega(-3)^{-1/4}2^{5/3}xy + (-3)^{1/4}y = x^3 + \omega^2(-3)^{-1/2}2^{1/3} x^2$$ We see that $E$ and $E^{{\mathrm{ss}}}$ are isomorphic over $K'$ and that the reduction $\bar{E}^{{\mathrm{ss}}}$ is the nonsingular curve given by $y^2 + y = x^3$.
\[ex m=1\]
Suppose we want to find a semistable model $E^{{\mathrm{ss}}}$ of the elliptic curve $E / {\mathbb{Q}}_2$ given by $y^2 = x^3 - x$ at the prime $(2)$. Just as in the previous example, this elliptic curve is CM, and so any semistable model $E^{{\mathrm{ss}}}$ should again have good reduction. Moreover, $E$ is isomorphic over ${\mathbb{Q}}_2$ to the Legendre curve with $\lambda = 2$, and since $m = v(\lambda) = 1 < 4$, we may apply Theorem \[thm m<4\].
We let $\beta = 2$, noting that this choice of $\beta$ satisfies the requirement that $v(\beta) = \frac{1}{3}m + \frac{2}{3} = 1$. Then we have $$P(X) = 3X^4 - 12X^3 + 12X^2 - 4 - \delta.$$ One can readily check that if we set $\delta = 0$, the roots of this polynomial are $1 \pm \sqrt{1 \pm \frac{2}{\sqrt{3}}}$, where the choices of sign are independent. We take $\alpha = 1 + \sqrt{1 + \frac{2}{\sqrt{3}}}$. Now evaluating the formulas in (\[eq formulas2\]) yields the following equation for $E^{{\mathrm{ss}}}$, over the extension $K' := {\mathbb{Q}}_2(2^{1/2}, 3^{1/4}, \sqrt{\sqrt{3} + 2})$ (which is abelian over ${\mathbb{Q}}_2(i)$ as it is contained in ${\mathbb{Q}}_2(\zeta_{24}, 3^{1/4})$, where $\zeta_{24}$ is a primitive $24$th root of unity). $$y^2 + (3^{1/4} + 3^{3/4})(1 + \frac{2}{\sqrt{3}})^{-1/4} xy + 3^{-1/4}(1 + \frac{2}{\sqrt{3}})^{1/4} y = x^3$$ We see that $E$ and $E^{{\mathrm{ss}}}$ are isomorphic over $K'$ and that the reduction $\bar{E}^{{\mathrm{ss}}}$ is again the nonsingular curve given by $y^2 + y = x^3$.
The $v(\lambda) \geq 4$ case {#S3}
----------------------------
For this section, we adopt exactly the same set-up but treat the complimentary case where $v(j(E)) \leq 0$. In this case, we see from the formula in (\[eq j\]) that we have $m := v(\lambda) \geq 4$. Therefore, under this assumption, any semistable model $E^{{\mathrm{ss}}}$ should have good reduction if and only if $m = 4$; otherwise $E^{{\mathrm{ss}}}$ has multiplicative reduction. As in §\[S2\], we will show how to find algebraic elements $\alpha, \beta \in \bar{K}$ such that evaluating $a_1, a_2, a_3 \in K' := K(\beta^{1/2}, \alpha, F(0)^{1/2})$ using the formulas in (\[eq formulas2\]) yields an equation of the form in (\[eq model\]) (with $a_4 = a_6 = 0$) for an elliptic curve with semistable reduction.
\[thm m>4\]
Assume that $m \geq 4$. Let $\beta \in (K^{\times})^2$ be any element such that $v(\beta) = 2$ (e.g. $\beta = 4$), and choose an element $\alpha \in K$ such that $2 \leq v(\alpha) \leq m - 2$. Then $E$ is isomorphic over $K' := K(F(0)^{1/2})$ to the elliptic curve $E^{{\mathrm{ss}}}$ given by the equation in (\[eq model\]), where $a_4 = a_6 = 0$ and the other coefficients $a_i$ are given by the formulas in (\[eq formulas2\]).
We have $v(a_1) = 0$, $v(a_2) \geq 0$, and $v(a_3) = v(\alpha) - 2$ (when $v(\alpha) > 2$, this directly implies that $E^{{\mathrm{ss}}}$ has multiplicative reduction). The curve $E^{{\mathrm{ss}}}$ has good reduction if $m = 4$ and has multiplicative reduction otherwise.
First of all, we note that $v(\alpha - 1) = 0$. The condition that $v(\alpha) \leq m - 2$ ensures that $v(\lambda) > v(\alpha)$, so $v(\alpha - \lambda) = v(\alpha)$. Therefore, we have $$v(F(0)) = v(\alpha) + v(\alpha - 1) + v(\alpha - \lambda) - 3v(\beta) = 2v(\alpha) - 6;$$ and $$v(F^{(1)}(0)) = v(2\alpha(\alpha - 1) - \lambda(\alpha - 1) + \alpha(\alpha - \lambda)) - 2v(\beta) = \min\{v(\alpha) +1, m, 2v(\alpha)\} - 4 = v(\alpha) - 3.$$ It follows that $v(a_3) = v(\alpha) - 2 \geq 0$ and $v(a_1) = 0$; in particular, $v(a_3) = 0$ if and only if $m = 4$.
We next check that $v(a_2) \geq 0$. In order to do so, we recall the formula in (\[eq a\_2\]) which we derived earlier: $$\label{eq a_2 m>4} 4\beta^4 F(0)a_2 = 3\alpha^4 - 4(1 + \lambda)\alpha^3 + 6\lambda \alpha^2 - \lambda^2.$$ Since $\min\{v(3\alpha^4), v(4(1 + \lambda)\alpha^3), v(6\lambda\alpha^2), v(\lambda^2)\} = \min\{4v(\alpha), 3v(\alpha)+ 2, 2v(\alpha) + m + 1, 2m\} \geq 2v(\alpha) + 4$, we have $v(a_2) = v(4F(0)\beta^4 a_2) - v(4F(0)) - 4v(\beta) \geq 2v(\alpha) + 4 - (2v(\alpha) - 4) - 8 = 0$, as desired.
Finally we assume that $v(\alpha) = 2$ and set out to show that the curve $E^{{\mathrm{ss}}}$ has good reduction if and only if $m = 4$. Any singular point $(x, y)$ on $\bar{E}^{{\mathrm{ss}}}$ satisfies the following set of equations. $$\begin{split} y^2 + \bar{a}_1 xy + \bar{a}_3 y &= x^3 + \bar{a}_2 x^2 \\
\bar{a}_1 y &= x^2 \\
\bar{a}_1 x + \bar{a}_3 &= 0 \end{split}$$ By solving for $x$ and $y$ in the bottom two equations and plugging the results in the top equation, we see that if such a point $(x, y)$ exists, we must have $$\label{eq nonsingularity} \frac{\bar{a}_3^4}{\bar{a}_1^6} + \frac{\bar{a}_3^3}{\bar{a}_1^3} - \frac{\bar{a}_2\bar{a}_3^2}{\bar{a}_1^2} = 0 \implies \frac{\bar{a}_3}{\bar{a}_1}\Big(\frac{\bar{a}_3}{\bar{a}_1^3} + 1 \Big) - \bar{a}_2 = 0.$$ We now show that this is the case if and only if $m > 4$. We compute the following equivalences modulo the prime ideal of $R'$, using the formulas in (\[eq formulas3\]). $$\label{eq nonsingularity2} \begin{split} \frac{a_3}{a_1} = \frac{4F(0)}{2F^{(1)}(0)} &\equiv \frac{-4\alpha^2\beta^{-3}}{-4\alpha\beta^{-2}} = \frac{\alpha}{\beta} \\
\frac{a_3}{a_1^3} = \Big(\frac{a_3}{a_1}\Big)\Big(\frac{4F(0)}{4F^{(1)}(0)^2}\Big) &\equiv \Big(\frac{\alpha}{\beta}\Big) \frac{-4\alpha^2\beta^{-3}}{16\alpha^2\beta^{-4}} = -\frac{\alpha}{4} \end{split}$$ Meanwhile, using what we know from (\[eq a\_2 m>4\]), we compute the equivalence $$\label{eq nonsingularity3} a_2 = \frac{\beta^{-4}(3\alpha^4 - 4(1 + \lambda)\alpha^3 + 6\lambda \alpha^2 - \lambda^2)}{4F(0)} \equiv \frac{\beta^{-4}(\alpha^4 - 4\alpha^3 - \lambda^2)}{-4\alpha^2\beta^{-3}} = -\frac{\alpha^2}{4\beta} + \frac{\alpha}{\beta} + \frac{\lambda^2}{4\alpha^2\beta}.$$ Putting (\[eq nonsingularity2\]) and (\[eq nonsingularity3\]) together, we get $$\frac{a_3}{a_1}\Big(\frac{a_3}{a_1^3} + 1\Big) - a_2 \equiv \frac{\alpha}{\beta} \Big(-\frac{\alpha}{4} + 1\Big) + \frac{\alpha^2}{4\beta} - \frac{\alpha}{\beta} - \frac{\lambda^2}{4\alpha^2\beta} = -\frac{\lambda^2}{4\alpha^2\beta}.$$ Since the valuation of the right-hand term is $2m - 2 - 4 - 2 = 2m - 8$, the above expression is equivalent to $0$ if and only if $m > 4$, and we are done.
It was pointed out to the author by Leonardo Fiore that in the situation of Theorem \[thm m>4\], a semistable model can be obtained by choosing $\alpha$ to be any element satisfying $v(\alpha) \geq 2$ (e.g. $\alpha = 0$), as long as we allow the possibility that $a_4 \neq 0$ or $a_6 \neq 0$. Indeed, there is an isomorphism (defined over $R$) between any two such models induced by translating $x$ by the integral element $\beta^{-1}(\alpha_1 - \alpha_2) \in R$, where $\alpha_1$ and $\alpha_2$ are the choices of $\alpha$ determining the models.
We now recall that the $2$-torsion subgroup $E[2] \subset E(\bar{K})$ is given by $\{\mathcal{O}, (0, 0), (1, 0), (\lambda, 0)\}$.
\[cor 2-torsion\]
Assume that $m > 4$ and construct the semistable model $E^{{\mathrm{ss}}}$ of $E$ as in the statement of Theorem \[thm m>4\]. The reduction of the $2$-torsion subgroup $E^{{\mathrm{ss}}}[2]$ coincides with the subset consisting of the infinity point $\bar{\mathcal{O}}$ and the cusp $P$ of $\bar{E}^{{\mathrm{ss}}}$; the inverse images of $\{\bar{\mathcal{O}}\}$ and $\{P\}$ correspond to the subgroup $\{\mathcal{O}, (1, 0)\} \subset E[2]$ and its coset $\{(0, 0), (\lambda, 0)\} \subset E[2]$ respectively.
It is clear that the infinity point $\mathcal{O}$ of $E$ gets sent to $\bar{\mathcal{O}} \in \bar{E}^{{\mathrm{ss}}}$. Now since $\varphi : E \stackrel{\sim}{\to} E^{{\mathrm{ss}}}$ sends the first coordinate of any point $(x, y) \in E(K') \smallsetminus \{\mathcal{O}\}$ to $\beta^{-1}(x - \alpha)$, we see that the first coordinate of the image $\varphi((1, 0)) \in E^{{\mathrm{ss}}}(K')$ (resp. of each image $\varphi((0, 0)), \varphi((\lambda, 0)) \in E^{{\mathrm{ss}}}(K')$) reduces to $\infty$ (resp. $-\frac{\bar{\alpha}}{\bar{\beta}}$). As in the proof of Theorem \[thm m>4\], the cusp $P$ has $x$-coordinate $-\frac{\bar{a}_3}{\bar{a}_1} = -\frac{\bar{\alpha}}{\bar{\beta}}$. Since $\bar{\mathcal{O}}$ (resp. $P$) is the only point of $\bar{E}^{{\mathrm{ss}}}$ whose first coordinate is $\infty$ (resp. $-\frac{\bar{\alpha}}{\bar{\beta}}$), we are done.
In a similar fashion to how we proved the above corollary, it is straightforward to show directly from Theorem \[thm m>4\] (resp. Theorem \[thm m<4\]) that in the case that $v(\lambda) = 4$ (resp. $0 \leq v(\lambda) < 4$), the elements $\mathcal{O}, (1, 0) \in E[2]$ are mapped via $\varphi$ composed with reduction to the infinity point $\bar{\mathcal{O}}$ of $\bar{E}^{{\mathrm{ss}}}$ and the elements $(0, 0), (\lambda, 0) \in E[2]$ map to another point of $\bar{E}^{{\mathrm{ss}}}$ (resp. the elements of $E[2]$ are all mapped to the infinity point $\bar{\mathcal{O}}$ of $\bar{E}^{{\mathrm{ss}}}$). Since the image of $E[2]$ under $\varphi$ composed with reduction must be contained in the $2$-torsion subgroup $\bar{E}^{{\mathrm{ss}}}[2]$, we see in this way that when $v(\lambda) = 4$ (or equivalently, when $j(\bar{E}^{{\mathrm{ss}}}) \neq 0$), the reduced curve $\bar{E}^{{\mathrm{ss}}}$ is ordinary. This is one direction of the equivalence given in [@silverman2009arithmetic Exercise 5.7], which states that an elliptic curve over a field of characteristic $2$ is supersingular if and only if its $j$-invariant is $0$. Since the other direction of that equivalence implies that $\bar{E}^{{\mathrm{ss}}}$ is supersingular in the $v(\lambda) < 4$ case, we see that $\bar{E}^{{\mathrm{ss}}}[2] = \{\bar{\mathcal{O}}\}$ coincides with the reduction of $E[2] \cong E^{{\mathrm{ss}}}[2]$.
\[ex m=4\]
Consider the elliptic curve $E / {\mathbb{Q}}_2$ given by $y^2 = x(x - 1)(x - 16)$. This curve is already in Legendre form with $\lambda = 16$. Since $m = v(\lambda) = 4$, any semistable model $E^{{\mathrm{ss}}}$ will have good reduction, and we may apply Theorem \[thm m>4\].
We set $\alpha = \beta = 4$, noting that $v(\alpha) = 2 = m - 2$. Now evaluating the formulas in (\[eq formulas2\]) yields the following equation for $E^{{\mathrm{ss}}}$, over the extension $K' := {\mathbb{Q}}_2(i)$. $$y^2 + 3ixy + 3iy = x^3 + x^2$$ It is easy to check directly that the reduction $\bar{E}^{{\mathrm{ss}}}$, given by $y^2 + xy + y = x^3 + x^2$, is nonsingular.
\[ex m=6\]
Consider the elliptic curve $E / {\mathbb{Q}}_2$ given by $y^2 = x(x - 1)(x - 64)$. The curve is again already in Legendre form, this time with $\lambda = 64$ so $m = v(\lambda) = 6$. Again, we may apply Theorem \[thm m>4\], but in this case, the semistable model $E^{{\mathrm{ss}}}$ we arrive at will have multiplicative reduction.
As before, we set $\beta = 4$, but this time, we let $\alpha = 8$, noting that $2 < v(\alpha) = 3 \leq m - 4$. Now evaluating the formulas in (\[eq formulas2\]) yields the following equation for $E^{{\mathrm{ss}}}$, over the extension $K' := {\mathbb{Q}}_2(i)$. $$y^2 + 7ixy + 14iy = x^3 + 2x^2$$ The reduction $\bar{E}^{{\mathrm{ss}}}$ is $y^2 + xy = x^3$, which visibly has a node at the point $(0, 0)$; hence, $E^{{\mathrm{ss}}}$ has (split) multiplicative reduction, as expected.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We give an overview over recent calculations of baryonic correlator functions with finite mass quarks in view on their applicability for QCD sum rules. The QCD sum rule method is then demonstrated within the Heavy Quark Effective Theory.'
---
MZ-TH/00-24\
hep-ph/0006072\
June 2000
[**QCD sum rules as applied to heavy baryons[^1]**]{}\
[Stefan Groote]{}\
[Institut für Physik der\
Johannes-Gutenberg-Universität,\
55099 Mainz, Germany]{}
Mesonic and baryonic correlators
================================
Data sets which are produced to an huge amount especially by the so-called $B$ factories like Barbar, Belle, Cleo and Hera B allow for the discovery of excited states of hadrons containing the $b$ quark and other heavy quarks. In order to follow this development on the experimental side, theorists are asked to develop and use methods to analyze such excited states. There is one main obstacle on this way which consists of the scale differences that occur. While perturbative calculations including a heavy quark can effectively be done only in the high energy range, there is an extrapolation of these results down to the energy range needed where we expect excited states are to be located.
One of the most powerful methods which was developed long ago are the QCD sum rules. In this talk we will only concentrate on the QCD sum rule approach developed by Shifman, Vainshtein and Zakharov, called SVZ approach [@ShifmanVainshteinZakharov]. This approach makes special assumptions about the spectral density of the correlator function related to the hadron and uses the Borel transform for the extrapolation. There are a lot of calculations for the correlator function of hadrons, especially for those containing a heavy quark. The calculations were performed within perturbative QCD as well as HQET and the massless limit. In the first part of this talk I will concentrate on calculations using perturbative QCD. In all cases the baryons are a kind of “stepchild” of the theorists, so we work hard to fill this gap.
Baryonic correlators in QCD
---------------------------
The mesonic correlator function for two vector currents has already been calculated ten years ago by Generalis [@Generalis] even in the case of two different and non-vanishing masses. Our aim is therefore to extend these calculations to the baryonic case. A first step has already been done by a recent publication [@GrooteKoernerPivovarov]. In this publication the spectral density is calculated for a three-quark current (two massless and one of finite mass) of the form $$J_B=[u^{iT}Cd^j]\Psi^k{\varepsilon}_{ijk}.$$ The result presented in [@GrooteKoernerPivovarov] is the one for the mass part $\Pi_m(-q^2)$ of the correlator $$\Pi(-q^2)={q\kern-5.5pt/}\Pi_q(-q^2)+m\Pi_m(-q^2).$$ This part is independently interesting because it can directly be compared with the result obtained within HQET, as we will see later. The momentum part $\Pi_q(-q^2)$ is still under construction, we expect a publication in the next few months. As mentioned before, we can easily reconstruct the correlator function $$\Pi_m(-q^2)=\int_{m^2}^\infty\frac{\rho_m(s)ds}{s-q^2}$$ if we know the spectral density. This is given by $$\rho_m(s)=\frac1{128\pi^4}\rho(s),\qquad
\rho=s^2\left\{\rho_0\left(1+\frac{\alpha_s}\pi\ln{\left(\frac{\mu^2}{m^2}\right)}
\right)+\frac{\alpha_s}\pi\rho_1\right\}$$ where $$\rho_0(s)=1+9z-9z^3-z^3+6z(1+z)\ln z$$ and $$\begin{aligned}
\label{corr1}
\lefteqn{\rho_1(s)\ =\ 9+\frac{665}9z-\frac{665}9z^2-9z^3
-\left(\frac{58}9+42z-42z^2-\frac{58}9z^3\right)\ln(1-z)\,+}\nonumber\\&&
+\left(2+\frac{154}3z-\frac{22}3z^2-\frac{58}9z^3\right)\ln z
+4\left(\frac13+3z-3z^2-\frac13z^3\right)\ln(1-z)\ln z\,+\nonumber\\&&
+12z\left(2+3z+\frac19z^2\right)\left(\frac12\ln^2z-\zeta(2)\right)
+4\left(\frac23+12z+3z^2-\frac13z^3\right){{\rm Li}}_2(z)\,+\nonumber\\&&
+24z(1+z)\left({{\rm Li}}_3(z)-\zeta(3)-\frac13{{\rm Li}}_2(z)\ln z\right).\end{aligned}$$ This result is obtained by using basic integrals of the kind $$V(\alpha,\beta;q^2/m^2)=\int{\frac{d^Dp}{(2\pi)^D}}\frac1{(p^2+m^2)^\alpha(q-p)^{2\beta}}$$ which are a generalization of the standard object $G(\alpha,\beta)$ of the massless calculation.
Comparison with the limits
--------------------------
Here we only want to stress that we were able to compare this result with the two limits, i.e. the massless limit and the heavy quark or near-threshold limit. For the massless limit we obtain $$m\rho(s)=m_{\overline{\rm MS}}(\mu)s^2\left\{1+\frac{\alpha_s}\pi
\left(2\ln{\left(\frac{\mu^2}{s}\right)}+\frac{31}3\right)\right\}$$ where the relation between the pole (or invariant) mass parameter $m$ and the $\overline{\rm MS}$ mass $m_{\overline{\rm MS}}$ reads $$m=m_{\overline{\rm MS}}\left\{1+\frac{\alpha_s}\pi
\left(\ln{\left(\frac{\mu^2}{m^2}\right)}+\frac43\right)\right\}.$$ With this explicit finite mass representation we can even obtain terms like $m^2\ln(\mu^2/m^2)$ which are absent in the effective theory of massless quarks but result from perturbative contributions of heavy quark condensates $\langle\bar\Psi\Psi\rangle$. For the other limit, i.e. the near-threshold limit $E\rightarrow 0$ with $s=(m+E)^2$ we obtain $$\label{hqet}
\rho^{\rm thr}(m,E)=\frac{16E^5}{5m}\left\{1+\frac{\alpha_s}\pi
\ln{\left(\frac{\mu^2}{m^2}\right)}+\frac{\alpha_s}\pi\left(\frac{54}5+\frac{4\pi^2}9
+4\ln{\left(\frac{m}{2E}\right)}\right)\right\}+O(E^6).$$ The invariant function $\rho_m$ suffices to determine the complete leading HQET behaviour since one has ${q\kern-5.5pt/}\rho_q+m\rho_m\rightarrow m({v\kern-5pt\raise1pt\hbox{$\scriptstyle/$}\kern1pt}+1)\rho$ for the leading term. In this region the appropriate device to compute the limit of the correlator is HQET (see e.g. [@Georgi; @Neubert]). Writing $$m\rho^{\rm thr}(m,E)=C(m/\mu,\alpha_s)^2\rho^{\rm HQET}(E,\mu)$$ we obtain the known result for $\rho^{\rm HQET}(E,\mu)$ [@GrooteKoernerYakovlev1] with matching coefficient $C(m/\mu,\alpha_s)$ [@GrozinYakovlev]. In this case the matching procedure allows one to restore the near-threshold limit of the full correlator starting from the simpler effective theory near threshold [@EichtenHill] (see also [@thresh]). Note that the higher order corrections in $E/m$ to Eq. (\[hqet\]) can be easily obtained from the explicit result given in Eq. (\[corr1\]). Indeed, the next-to-leading order correction in low energy expansion reads $$\Delta\rho^{\rm thr}(m,E)=-\frac{88E^6}{5m^2}
\left\{1+\frac{\alpha_s}\pi\left(\ln{\left(\frac{\mu^2}{m^2}\right)}
+\frac{376}{33}+\frac{4\pi^2}9+\frac{140}{33}\ln{\left(\frac{m}{2E}\right)}\right)\right\}.$$ It is a much more difficult task to obtain this result starting from HQET. In Fig. \[fig1\] we compare components of the baryonic spectral function in leading and next-to-leading order. Shown is the ratio $\rho_1(s)/\rho_0(s)$ where we put $\mu=m$ for simplicity. One can see that a simple interpolation between the two limits can give a rather good approximation for the next-to-leading order correction in the complete region of $s$.
A specific feature of the calculation
-------------------------------------
The two-loop and three-loop Feynman diagrams which had to be calculated for the determination of the spectral function are shown in Fig. \[fig2\]. For our calculation we extensively used the fact that the diagrams can be composed by glueing together subdiagrams (see also Ref. [@melone]). The so-called scalar spectacle diagram in Fig. \[fig2a\] is calculated as the convolution of two heavy-light spectral densities, $$\rho(s)=\int\lambda(s,s_1,s_2)\rho_1(s_1)\rho_2(s_2)ds_1ds_2$$ where the convolution function is given by the remaining line.
QCD sum rules for the HQET
==========================
The QCD sum rules for the correlator calculated above have not yet been constructed. Instead we will present the principles of the SVZ approach to QCD sum rules as applied to leading order HQET with results for heavy baryons taken from previous publications [@GrooteKoernerYakovlev1; @GrooteKoernerYakovlev2].
Construction of QCD sum rules
-----------------------------
The QCD sum rules can be constructed by taking care of two possible expressions for the two-point correlator which in this case reduces to a scalar correlator function $P(\omega)$, $$\Pi(\omega=p\cdot v)=i\int e^{ip\cdot x}
\langle T\{J(x),\bar J(0)\}\rangle d^4x
=\Gamma'\frac{1+{v\kern-5pt\raise1pt\hbox{$\scriptstyle/$}\kern1pt}}2\bar\Gamma'\frac12{\rm Tr}(\Gamma\bar\Gamma)
2{\rm\,Tr}(\tau\tau^\dagger)P(\omega).$$ On the one hand side, the function $P(\omega)$ satisfies the dispersion relation $$P(\omega)=\int_0^\infty\frac{\rho(\omega')d\omega'}{\omega'-\omega-i0}
+\mbox{subtraction},$$ where $\rho(\omega)$ is the spectral density in HQET. On the phenomenological side the two-point correlator is represented by the spectral representation $$P(\omega)=\frac{\frac12|F_B|^2}{\bar\Lambda-\omega-i0}
+\sum_{X\ne B}\frac{\frac12|F_X|^2}{\omega_X-\omega-i0}
+\mbox{subtr.}$$ where $\bar\Lambda=m_B-m_Q$ is the ground state energy of the baryon and $F_B$ the residue. The main assumption of the SVZ approach is that the remaining sum can be approximated by the integral of the spectral density given by the dispersion relation and starting from some threshold energy $E_C$. The combination of the phenomenological and the theoretical identity for the correlator function then leads to $$\frac{\frac12|F_B|^2}{\bar\Lambda-\omega-i0}=\int_0^{E_C}
\frac{\rho(\omega')d\omega'}{\omega'-\omega-i0}+\mbox{subtr.}$$
The Borel transformation
------------------------
This formula is not useful since the spectral density as calculated in the Euklidean domain is reliable only for negative values of $\omega$, while the integral is to be calculated mainly at $\omega=\bar\Lambda$. This region of integration can be reached by an extrapolation using higher and higher derivatives when $\omega$ goes to $-\infty$. This extrapolation is expressed by the Borel transformation (cf. e.g. Ref. [@Neubert]) $$\hat B_T^{(\omega)}(f(\omega))=\lim_{-\omega,n\rightarrow\infty}
\frac{(-\omega)^{n+1}}{n!}\frac{d^n}{d\omega^n}f(\omega),\qquad
T=\frac{-\omega}n\quad\mbox{fixed}.$$ The Borel transformation is by construction a derivative and therefore also cancels the (constant) subtraction terms. The Borel parameter $T$ is an unphysical quantity in units of an energy, and the obtained values should be mostly independent on this parameter. This will be the main criterion in analyzing the sum rules. The Borel transformation leads to the final form of the QCD sum rule, $$\label{sumrule1}
\frac12|F_B|^2e^{-\bar\Lambda/T}=K(E_C,T),\qquad K(E_C,T)=\int_0^{E_C}
\rho(\omega)e^{-\omega/T}d\omega+\sum_i\frac{A_i}{T^i}.$$ At the end we can take the derivative with respect to the inverse Borel parameter and obtain a second sum rule, $$\label{sumrule2}
\bar\Lambda=-\frac12\frac\partial{\partial(T^{-1})}\ln K(E_C,T).$$
QCD sum rule analysis
---------------------
The procedure of the sum rule analysis is as follows: First we calculate the theoretical expression for $K(E_C,T)$ by calculating the spectral density within perturbation theory. This has been done for the next-to-leading order of the $d=0$ term as well as the $d=3$ term in the Operator Product Expansion. For brevity I will only mention the results of the analysis for the $d=0$ radiative corrections. For the analysis we have to select a “sum rule window”, i.e. a range for the Borel parameter $T$ in which the sum rule analysis is performed. The boundaries of this window is given by heuristic arguments. If the Borel parameter becomes too small, the nonperturbative contributions which are badly known blow up. If on the other hand the Borel parameter becomes too large, the fact that the Borel parameter appears as “temperature” in the sum rule causes the problem that higher and higher excitations contribute to the ground state. The region of reliability and so the sum rule window is therefore roughly given by $\Lambda_{\rm QCD}<T<2\bar\Lambda$.
Now we use the second sum rule (\[sumrule2\]) to determine the ground state energy parameter $\bar\Lambda$. This is done by varying the continuum threshold parameter $E_C$ in order to obtain a rather stable value for the quantities with respect to the unphysical parameter $T$ within the sum rule window. The value obtained for the ground state energy can then be used in the first sum rule (\[sumrule1\]) to determine the absolute value $|F_B|$ of the residue. More detailed considerations are found in Ref. [@GrooteKoernerYakovlev1]. Our results read $$\begin{aligned}
E_C(\Lambda_Q)&=&1.2\pm 0.1{{\rm\,GeV}},\nonumber\\
\bar\Lambda(\Lambda_Q)&=&0.77\pm 0.05{{\rm\,GeV}},\\
|F(\Lambda_Q)|&=&0.022\pm 0.001{{\rm\,GeV}}^3\nonumber\end{aligned}$$ while the $O(\alpha_s)$ results are given by $$\begin{aligned}
E_C(\Lambda_Q)&=&1.1\pm 0.1{{\rm\,GeV}},\nonumber\\
\bar\Lambda(\Lambda_Q)&=&0.77\pm 0.05{{\rm\,GeV}},\\
|F(\Lambda_Q)|&=&0.027\pm 0.001{{\rm\,GeV}}^3\nonumber\end{aligned}$$ where the errors are estimated by looking at the stability with respect to different values for $E_C$. Taking the experimental results for the masses of the baryons, namely $m(\Lambda_c)=2284.9\pm 0.6{{\rm\,MeV}}$ and $m(\Lambda_b)=5642\pm 50{{\rm\,MeV}}$ [@PDG], our central value $\bar\Lambda(\Lambda_Q)$ for the bound state energy suggests pole masses of $m_c=1520\pm 100{{\rm\,MeV}}$ and $m_b=4880\pm 100{{\rm\,MeV}}$.
Conclusion and Outlook
======================
The QCD sum rule analysis for heavy baryons requires the calculation of the spectral density related to the correlator function of baryonic currents. If this spectral density is calculated, the sum rule analysis can be used as powerful tool to extract phenomenological non-perturbative quantities like bound state energies and form factors. I have presented the sum rule analysis for an example within HQET. In order to perform a sum rule analysis for baryons containing finite mass quarks the following steps are still missing and will be done in the near future:
- The calculation of the momentum part of the spectral density (nearly finished)
- The matching procedure for the spectral density
- The calculation of spectral densities for baryons with different quantum numbers
Moreover it is planned to develop light-cone sum rules for the three-point function of heavy baryons in order to determine the Isgur-Wise function.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I want to thank my collaborators J.G. Körner, A.A. Pivovarov and O.I. Yakovlev for the continuing and fruitful collaboration. I would like to thank the organizers of this conference for their hospitality. This work is supported by a grant given by the DFG.
[99]{} M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. [**B147**]{} (1979) 385; [**B147**]{} (1979) 448 S.C. Generalis, J. Phys. [**G16**]{} (1990) 785 S. Groote, J.G. Körner and A.A. Pivovarov, Phys. Rev. [**D61**]{} (2000) 071501 H. Georgi, Nucl. Phys. [**B363**]{} (1991) 301 M. Neubert, Phys. Rep. [**245**]{} (1994) 259 S. Groote, J.G. Körner and O.I. Yakovlev, Phys. Rev. [**D55**]{} (1997) 3016 A.G. Grozin and O.I. Yakovlev, Phys. Lett. [**285 B**]{} (1992) 254 E. Eichten and B. Hill, Phys. Lett. [**234 B**]{} (1990) 511 S. Groote and A.A. Pivovarov, “Threshold expansion of Feynman diagrams within a configuration space technique”, Report No. MZ-TH/00-07, hep-ph/0003115, to be published in Nucl. Phys. B S. Groote, J.G. Körner and A.A. Pivovarov, Nucl. Phys. [**B542**]{} (1999) 515; Eur. Phys. J. [**C11**]{} (1999) 279; Phys. Lett. [**443 B**]{} (1998) 269 S. Groote, J.G. Körner and O.I. Yakovlev, Phys. Rev. [**D56**]{} (1997) 3943 Particle Data Group, “Review of Particle Properties”, Eur. Phys. J. [**C3**]{} (1998) 1
[^1]: Invited talk given at the conference “Heavy Quark Physics 5”, Dubna, Russia, 6–8 April 2000, to appear in the proceedings
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{
"pile_set_name": "ArXiv"
}
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---
abstract: |
We consider the subgroup ${\it lp}G_{k,1}$ of length preserving elements of the Thompson-Higman group $G_{k,1}$ and we show that all elements of $G_{k,1}$ have a unique ${\it lp}G_{k,1} \cdot F_{k,1}$ factorization. This applies to the Thompson-Higman group $T_{k,1}$ as well. We show that ${\it lp}G_{k,1}$ is a “diagonal” direct limit of finite symmetric groups, and that ${\it lp}T_{k,1}$ is a $k^{\infty}$ Prüfer group. We find an infinite generating set of ${\it lp}G_{k,1}$ which is related to reversible boolean circuits.
We further investigate connections between the Thompson-Higman groups, circuits, and complexity. We show that elements of $F_{k,1}$ cannot be one-way functions. We show that describing an element of $G_{k,1}$ by a generalized bijective circuit is equivalent to describing the element by a word over a certain infinite generating set of $G_{k,1}$; word length over these generators is equivalent to generalized bijective circuit size.
We give some coNP-completeness results for $G_{k,1}$ (e.g., the word problem when elements are given by circuits), and $\#{\mathcal P}$-completeness results (e.g., finding the ${\it lp}G_{k,1} \cdot F_{k,1}$ factorization of an element of $G_{k,1}$ given by a circuit).
author:
- 'Jean-Camille Birget [^1]'
title: 'Factorizations of the Thompson-Higman groups, and circuit complexity '
---
Introduction
============
The Thompson groups, introduced by Richard J. Thompson [@Th0; @McKTh; @Th], and their generalization by Graham Higman [@Hig74], are well known for their amazing properties and their importance in combinatorial group theory and topology. In this paper we focus on the computational role of these groups, continuing the work started in [@BiThomps; @BiCoNP], and we study some subgroups of the Thompson-Higman groups that are motivated by circuit complexity. We emphasize that we view the Thompson groups as [*a model of computation*]{} and not just a source of algorithmic problems. Indeed, since the Thompson group $V$ has a faithful partial action on the set $\{0,1\}^*$ of all bitstrings, it is natural to consider combinational circuits for computing elements of $V$; i.e., we can view every element of $V$ as the input-output function of an acyclic digital circuit (see [@BiCoNP]). The elements of $V$ are bijections, hence we will see connections between $V$ and [*reversible computing*]{}. More precisely, words over certain generating sets of $V$ will be seen to be equivalent to circuits made of (generalized) bijective gates.
Combinational circuits have fixed-length inputs and fixed-length outputs, which is not the case for elements of $V$; but the notion of a circuit can be adapted in order to be applied to the computation of elements of $V$. Moreover, for bijective circuits the fixed length of inputs and outputs implies that the circuit is length preserving. This leads to the question what the [*length preserving*]{} elements of $V$ are, and how arbitrary elements of $V$ are related to length preserving elements of $V$. The length preserving elements of $V$ turn out to form an interesting subgroup, called ${\it lp}V$, and we will show that every element of $V$ can be factored in a unique way as a product of an element of ${\it lp}V$ and an element of the Thompson group $F$. This factorization carries over to the Thompson group $T$, where we have a unique ${\it lp}T \cdot F$ factorization. All of this generalizes to the Thompson-Higman groups $G_{k,1}$ and $T_{k,1}$.
The group ${\it lp}G_{k,1}$ is locally finite; it is a “diagonal” direct limit of finite symmetric groups, and it is simple when $k$ is even; ${\it lp}T_{k,1}$ is a $k^{\infty}$ Prüfer group. The connection with bijective (a.k.a. “reversible”) circuits leads to an interesting infinite generating set of ${\it lp}V$. We show that a description of an element of $G_{k,1}$ by a bijective circuit is equivalent to a description by a word over a certain infinite generating set of $G_{k,1}$; bijective circuit size is closely related to the word size over a certain infinite generating set of $G_{k,1}$. This shows that $G_{k,1}$ (and especially $V$) can serve as a model for bijective computing, with equivalent complexity.
We also investigate the computational complexity of some problems in $G_{k,1}$. We show that when an element $\varphi \in G_{k,1}$ is given by a bijective circuit (or by a general non-bijective circuit), the question whether $\varphi$ is the identity, and the question whether $\varphi$ is maximally extended, are coNP-complete problems; this is an application of [@BiCoNP] where $G_{k,1}$ and its connection with circuits was used to construct a finitely presented group with coNP-complete word problem. We show that elements of $F_{k,1}$ cannot be one-way functions, i.e., from a circuit for $f \in F_{k,1}$ one can easily find a circuit for $f^{-1}$. And we show that when $\varphi \in G_{k,1}$ is given by a bijective circuit, the problem of finding the ${\it lp}G_{k,1} \cdot F_{k,1}$ factorization of $\varphi$ is $\#{\mathcal P}$-complete in general (and also under some restrictions).
[**Definition of the Thompson-Higman group**]{}
The rest of this Introduction consists of a brief, but complete, definition of the Thompson-Higman groups $G_{k,1}$, $T_{k,1}$, and $F_{k,1}$. We follow the exposition of [@BiThomps; @BiCoNP], based on partial actions on finite words, which simplifies the connections with circuits. Compare with the definition in [@Th] (based on infinite sequences), [@Hig74] (based on automorphisms of certain algebras), [@Scott] (based on words and similar to this paper, but with different terminology), [@CFP] (based on finite trees), [@BrinSqu] (based on piecewise linear maps between real numbers, see also [@CFP]), [@Dehornoy] (related to associativity or commutativity in term rewriting, which was Thompson’s original view).
To define the Thompson-Higman group $G_{k,1}$ we fix an alphabet $A$ of cardinality $|A| = k$. Let $A^*$ denote the set of all finite [*words*]{} over $A$ (i.e., all finite sequences of elements of $A$); this includes the empty word $\varepsilon$. The [*length*]{} of $w \in A^*$ is denoted by $|w|$; let $A^n$ denote the set of words of length $n$. For two words $u,v \in A^*$ we denote their [*concatenation*]{} by $uv$ or by $u \cdot v$; for sets $B, C \subseteq A^*$ the concatenation is $BC = \{uv : u \in B, v \in C\}$. A [*right ideal*]{} of $A^*$ is a subset $R \subseteq A^*$ such that $RA^* \subseteq R$. A generating set of a right ideal $R$ is a set $C$ such that $R$ is the intersection of all right ideals that contain $C$. A right ideal $R$ is called [*essential*]{} iff $R$ has a non-empty intersections with every right ideal of $A^*$. For words $u,v \in A^*$, we say that $u$ is a [*prefix*]{} of $v$ iff there exists $z \in A^*$ such that $uz = v$. A [*prefix code*]{} is a subset $C \subseteq A^*$ such that no element of $C$ is a prefix of another element of $C$. A prefix code is [*maximal*]{} iff it is not a strict subset of another prefix code. One can prove that a right ideal $R$ has a unique minimal (under inclusion) generating set, and that this minimal generating set is a prefix code; this prefix code is maximal iff $R$ is an essential right ideal.
Partial functions on $A^*$ will play a big role. For $f: A^* \to A^*$, let Dom$(f)$ denote the domain and let Im$(f)$ denote the image (range) of $f$. A [*restriction*]{} of $f$ is any function $f_1: A^* \to A^*$ such that Dom$(f_1) \subseteq$ Dom$(f)$, and such that $f_1(x) = f(x)$ for all $x \in {\rm Dom}(f_1)$. An [*extension*]{} of $f$ is any function on $A^*$ of which $f$ is a restriction.
An [*isomorphism*]{} between right ideals $R_1, R_2$ of $A^*$ is a bijection $\varphi: R_1 \to R_2$ such that for all $r_1 \in R_1$ and all $z \in A^*$: $\varphi(r_1z) = \varphi(r_1) \cdot z$; the isomorphism $\varphi$ can be described by a bijection between the prefix codes that minimally generate $R_1$, respectively $R_2$.
One can prove that an isomorphism $\varphi$ between essential right ideals has a unique maximal extension (as an isomorphism between essential right ideals), denoted max $\varphi$. So, max $\varphi$ has no extension (other than itself) to an isomorphism between essential right ideal.
Finally, the Thompson-Higman group $G_{k,1}$ is defined to consist of all maximally extended isomorphisms between finitely generated essential right ideals of $A^*$. The multiplication consists of composition followed by maximal extension: $\varphi \cdot \psi = $ max$(\varphi \circ \psi)$. Note that we let $G_{k,1}$ act partially and faithfully on $A^*$ on the [*left*]{}.
Thompson and Higman proved that $G_{k,1}$ is finitely presented. Also, when $k$ is even $G_{k,1}$ is simple, and when $k$ is odd $G_{k,1}$ has a simple normal subgroup of index 2.
Every element $\varphi \in G_{k,1}$ can be described by a bijection between two finite maximal prefix codes; this bijection can be described concretely by a finite function [*table*]{}. When $\varphi$ is described by a maximally extended isomorphism between essential right ideals, $\varphi: R_1 \to R_2$, we call the minimum generating set of $R_1$ the [*domain code*]{} of $\varphi$, denoted domC$(\varphi)$, and we call the minimum generating set of $R_2$ the [*image code*]{} of $\varphi$, denoted imC$(\varphi)$; because of the uniqueness of maximal extension, domC$(\varphi)$ and imC$(\varphi)$ are uniquely determined by $\varphi$. We call the cardinality $|{\rm domC}(\varphi)| = $ $|{\rm imC}(\varphi)|$ the [*table size*]{} of $\varphi$, denoted $\|\varphi\|$. In [@BiThomps] it was proved that for all $\varphi, \psi \in G_{k,1}$: $\|\varphi \psi\| \leq \|\varphi\| + \|\psi\|$. The concepts of domC$(\varphi)$, imC$(\varphi)$, table, and $\|\varphi\|$, can also be used when $\varphi$ is not maximally extended.
For any finite generating set $\Gamma$ of $G_{k,1}$ and any $\varphi \in G_{k,1}$, we define the [*word length*]{} of $\varphi$ over $\Gamma$ as the length of a shortest word over $\Gamma \cup \Gamma^{-1}$ that represents $\varphi$; it is denoted by $|\varphi|_{\Gamma}$. In [@BiThomps] it was proved that for any finite generating set $\Gamma$ of $G_{k,1}$, the word length and the table size are closely related; for all $\varphi \in G_{k,1}$: $c' \ \|\varphi\| \ \leq \ |\varphi|_{\Gamma} \ \leq \ $ $c \ \|\varphi\| \ \log_2 \|\varphi\|$ (for some constants $c, c' >0$ depending on $\Gamma$ but not on $\varphi$). Asymptotically, for most $\varphi \in G_{k,1}$ we also have $|\varphi|_{\Gamma} \ \geq \ $ $c'' \ \|\varphi\| \ \log_2 \|\varphi\|$ (for some constant depending on $\Gamma$, $0 < c'' < c$). However, for $\varphi \in F_{k,1}$ it was proved in [@CFP] that $c' \ \|\varphi\| \ \leq \ |\varphi|_{\Gamma} \ $ $ \leq \ c \ \|\varphi\|$.
We will use the well-known finitely presented subgroups $F_{k,1}$ and $T_{k,1}$ of $G_{k,1}$, introduced in [@Th0] and [@Hig74]. The groups $F_{2,1}$ (also called $F$) and $T_{2,1}$ (also called $T$) have a large literature; a few examples are [@McKTh], [@McKTh; @Th], [@BrinSqu], [@CFP], [@GhysSergiescu], [@BrownGeo], [@ClearyTaback], [@Brin97], [@Brin99], [@GubaSapir], [@BCST]. Below we will introduce the subgroups ${\it lp}G_{k,1}$ and ${\it lp}T_{k,1}$ of length preserving elements of $G_{k,1}$, respectively $T_{k,1}$.
We will need the exact definition of $F_{k,1}$ and $T_{k,1}$ in the setting of partial actions on words (in $A^*$), and to do so we need some preliminary definitions. Assuming that a linear order has been chosen for the alphabet $A$, we can consider the dictionary order on $A^*$, denoted $\leq_{\rm d}$, and defined as follows. For any $x_1, x_2 \in A^*$ we say that $x_1 \leq_{\rm d} x_2$ (i.e., $x_1$ precedes $x_2$ in the dictionary order) iff either (1) $x_1$ is a prefix of $x_2$, or, (2) letting $p$ denote the longest common prefix of $x_1$ and $x_2$, we have: $x_1 = pa_1v_1$, $x_2 = pa_2v_2$, with $a_1 < a_2$ (for some letters $a_1, a_2 \in A$, and words $v_1, v_2 \in A^*$, where $<$ is the strict order in $A$).
A partial map $f: A^* \to A^*$ is said to [*preserve the dictionary order*]{} iff for all $x_1, x_2 \in {\rm Dom}(f)$ we have: $x_1 \leq_{\rm d} x_2$ iff $f(x_1) \leq_{\rm d} f(x_2)$.
We also want to define “cyclical preservation” of the dictionary order. Here we will simply write $<$ for $<_{\rm d}$ (strict dictionary order). A cyclical order of a finite maximal prefix code $P \subset A^*$ is a listing $(x_0, x_1, \ldots, x_{|P|-1})$ of all the elements of $P$ such that for some integer $s$: $(x_s, \ldots, x_{|P|-1}, x_0, \ldots, x_{s-1})$ is the listing of $P$ in dictionary order. In other words, a cyclical order of $P$ is a cyclic permutation of the dictionary order on $P$.
We say that a partial map $f: A^* \to A^*$ [*cyclically preserves the dictionary order*]{} iff for all finite sequences $(x_0, x_1, \ldots, x_{n-1})$ we have: $(x_0, x_1, \ldots, x_{n-1})$ is a cyclical order of some finite maximal prefix code iff $(f(x_0), f(x_1), \ldots, f(x_{n-1}))$ is a cyclical order of some finite maximal prefix code.
The groups $F_{k,1}$ and $T_{k,1}$ can be defined as follows, from the point of view of partial actions on finite words (see [@BiThomps]).
\[F\_and\_T\] Assume that a linear order has been chosen for the alphabet $A$, where $|A| = k$. Then $F_{k,1}$ consists of the elements of $G_{k,1}$ that preserve the dictionary order of $A^*$, and $T_{k,1}$ consists of the elements of $G_{k,1}$ that cyclically preserve the dictionary order of $A^*$.
[**Another view of $F_{k,1}$:**]{} The elements of $F_{k,1}$ can be given the following interpretation. First we define the concept of a rank function on a (partial) order structure $(S,\leq)$. The [*rank*]{} of an element $t \in S$ is
${\rm rank}_S(t) = |\{ x \in S : x < t \}|$,
i.e., the number of elements that strictly precede $t$. Every element of $\varphi \in G_{k,1}$ can be represented (after appropriate restriction) by a bijective partial function $\varphi: A^* \to A^*$ such that ${\rm imC}(\varphi) = A^n$ for some $n>0$, and ${\rm domC}(\varphi)$ is some finite maximal prefix code of cardinality $k^n$ (where $|A| = k$). If we view the elements of $A^n$ as the integers $\{0,1, \ldots, k^n -1 \}$ in base-$k$ representation we have:
$F_{k,1}$ [*consists of all elements of $G_{k,1}$ that can be represented by rank functions*]{}
${\rm rank}_P(.): P \to \{0,1, \ldots, k^n -1 \}$,
where $n$ ranges over the positive integers, and $P \subset A^*$ ranges over all maximal prefix codes of cardinality $k^n$. This point of view will help us later in proving that elements of $F_{k,1}$ can have high computational complexity, even when their domain code ${\rm domC}(\varphi)$ has an easy membership problem (see Theorem \[numberP\_fact2\]).
[**Overview:**]{} This paper consists of the following parts:
Part 1 consists of sections 2, 3, and 4. We introduce the subgroup ${\it lp}G_{k,1}$ of length preserving elements of the Thompson-Higman group $G_{k,1}$, and we give the ${\it lp}G_{k,1} \cdot F_{k,1}$ factorization of $G_{k,1}$; we generalize this unique factorization to other subgroups of $G_{k,1}$.
Section 5 makes the transition from part 1 to part 2, by giving a connection between circuits and some properties of ${\it lp}G_{k,1}$.
Part 2 consists of sections 6 and 7. We study $V$ as a model for reversible circuits. We also investigate the complexity of some problems: We show that elements of $F$ cannot be one-way functions, and we show that finding the ${\it lp}V \cdot F$ factorization of an element of $V$ given by a circuit is $\#{\mathcal P}$-complete.
The subgroups ${\it lp}G_{k,1}$ and ${\it lp}T_{k,1}$
======================================================
The Thompson-Higman group $G_{k,1}$ contains all finite symmetric groups, and this inspires the definition of the subgroup ${\it lp}G_{k,1}$ of all length-preserving elements of $G_{k,1}$. We will denote ${\it lp}G_{2,1}$ also by ${\it lp}V$. Another motivation of ${\it lp}V$, which we will develop more later, is the computation of elements of $V$ and ${\it lp}V$ by digital circuits. Indeed, circuits traditionally have a fixed length for inputs and a fixed length for outputs (corresponding to fixed numbers of wires); for bijective functions this means length preservation.
The subgroup of length-preserving elements of the Thompson-Higman group $G_{k,1}$ is lp$G_{k,1} = \{ \varphi \in G_{k,1} : $ $\forall x \in$ [Dom]{}$(\varphi), \, |x| = |\varphi(x)| \}$. Similarly we define lp$T_{k,1} = T_{k,1} \, \cap \, {\it lp}G_{k,1}$.
Restriction or extension of a length-preserving partial function $A^* \to A^*$, representing an element of $G_{k,1}$, is again length preserving, so [*lp*]{}$G_{k,1}$ is well-defined as a subset of the group $G_{k,1}$. The inverse of a length-preserving partial function is also length-preserving. After a restriction, if necessary, any finite set of elements of [*lp*]{}$G_{k,1}$ can be represented by permutations of the same set $A^m$, for any large enough $m$. Hence [*lp*]{}$G_{k,1}$ is closed under composition. It follows that [*lp*]{}$G_{k,1}$ is a subgroup of $G_{k,1}$. The group [*lp*]{}$G_{k,1}$ is [*locally finite*]{} (i.e., every finitely generated subgroup is finite), and [*lp*]{}$G_{k,1}$ contains all the finite symmetric groups ${\mathfrak S}_{A^n}$, for all $n \geq 1$.
Assume we restrict an element $\varphi \in {\it lp}G_{k,1}$ so that its domain and image codes are both $A^m$ for some $m$. Then the additional overall restriction operation (which replaces each $\varphi(x) = y$ by the $k$-tuple $\varphi(x \, a) = y \, a$, where $a$ ranges over $A$) leads to the following embeddings:
$\otimes {\rm id}_A: \pi \in {\mathfrak S}_{A^n} \ \hookrightarrow \ $ $\pi \otimes {\rm id}_A \in {\mathfrak S}_{A^{n+1}}$,
where for all $x \in A^n$ and $a \in A$ we define $(\pi \otimes {\rm id}_A)(x a) = \pi(x) \cdot a$ (where $\cdot$ denotes concatenation). This type of embedding of symmetric groups is called [*diagonal*]{} [@Zalesskii], [@Hartley]. Moreover, when $|A| = k$ is [*even*]{} then the above embedding factors through the alternating group
${\mathfrak S}_{A^n} \ \hookrightarrow \ $ ${\mathfrak A}_{A^{n+1}} \ \subset {\mathfrak S}_{A^{n+1}}$.
Indeed we have the following generalization of an observation of [@Shende] (see also Section 5 below): For any positive integer $n$ and any $\pi \in {\mathfrak S}_{A^n}$, the permutation $\pi \otimes {\rm id}_A$ is [*even*]{}. Indeed, the transformation $\pi \to \pi \otimes {\rm id}_A$ replaces one transposition $(u|v)$ of $\pi$ (with $u,v \in A^n$) by the sequence of transpositions $(ua_1|va_1) \ldots (ua_k|va_k)$, i.e., $k$ transpositions with $k$ even; here $A = \{a_1, \ldots, a_k\}$.
The above embeddings yield the following.
\[lpV\_infinite\_alt\] The group ${\it lp}G_{k,1}$ is isomorphic to the direct limit of the sequence of diagonal embeddings $\otimes {\rm id}_A: {\mathfrak S}_{A^n} \ \hookrightarrow$ $ \ {\mathfrak S}_{A^{n+1}}$.
When $|A| = k$ is even, ${\it lp}G_{k,1}$ is isomorphic to the direct limit of the sequence of embeddings ${\mathfrak S}_{A^n} \ \hookrightarrow \ $ ${\mathfrak A}_{A^{n+1}} \ \subset {\mathfrak S}_{A^{n+1}}$.
These are examples of the direct limits of symmetric groups considered in [@KegelWehrfritz], chapter 6, and in [@Hartley], section 1.5. The embedding maps are of “diagonal” type, in the terminology of these references. By these references we also conclude that when $k$ is even, ${\it lp}G_{k,1}$ is a simple group, and when $k$ is odd, ${\it lp}G_{k,1}$ has a simple subgroup of index 2 (via the parity map). In any case, it also follows from [@KegelWehrfritz] and [@Hartley] that ${\it lp}G_{k,1}$ is different from the finitary symmetric group and the finitary alternating group; indeed, the finitary symmetric group does not contain any Prüfer groups, whereas ${\it lp}G_{k,1}$ contains ${\it lp}T_{k,1}$ which is a group of Prüfer type (as we shall see next). However, ${\it lp}G_{k,1}$ also contains many copies of the finitary symmetric group (as was mentioned in [@Th]).
The observations above apply also to the Thompson group $T_{k,1}$. Let us denote by ${\mathbb Z}_{A^n}$ the cyclic subgroup of ${\mathfrak S}_{A^n}$ generated by the permutation $w_i \mapsto w_{(i+1) \, {\rm mod} \, k^n}$, where $(w_i : i = 0,1, \ldots, k^n -1)$ is the listing of $A^n$ in dictionary order. ${\mathbb Z}_{A^n}$ consists of the elements of ${\mathfrak S}_{A^n}$ that cyclically preserve the dictionary order. Just as for the symmetric groups on $A^n$, the restriction operation of $G_{k,1}$ gives an embedding of ${\mathbb Z}_{A^n}$ into ${\mathbb Z}_{A^{n+1}}$, by the transformation $\otimes {\rm id}_A$ which sends the generator $(w_i \mapsto w_{(i+1) \, {\rm mod} \, k^n})$ of ${\mathbb Z}_{A^n}$ to the element $(v_j \mapsto v_{(j+k) \, {\rm mod} \, k^{n+1}})$ of ${\mathbb Z}_{A^{n+1}}$. Here, $(v_j : j = 0,1, \ldots, k^{n+1} -1)$ is the listing of $A^{n+1}$ in dictionary order. Thus we have:
\[lpT\_Prufer\] The group ${\it lp}T_{k,1}$ is isomorphic to the $k^{\infty}$ Prüfer group, given by the direct limit of the sequence of embeddings ${\mathbb Z}_{A^n} \hookrightarrow {\mathbb Z}_{A^{n+1}}$, where the embeddings are determined by the restriction operation of $G_{k,1}$.
The $k^{\infty}$ Prüfer group is isomorphic to the multiplicative group of the complex $k^n$th roots of unity (for all $n>0$), or the additive group of $k$-ary rationals modulo 1, i.e., $\{ \frac{m}{k^n} \ {\rm mod} \ 1 \ : \ n, m \in {\mathbb N} \}$.
Length-preserving order-preserving factorization of $G_{k,1}$ and $T_{k,1}$
============================================================================
Let [**1**]{} denote the identity of $G_{k,1}$.
\[lp\_cap\_F\] If an element of $F_{k,1}$ has a representation $f: A^* \to A^*$ such that ${\rm domC}(f) = {\rm imC}(f)$ then $f$ represents the identity. Hence, $F_{k,1} \ \cap \ {\it lp}G_{k,1} \ = \ \{ {\bf 1} \}.$
[**Proof.**]{} Since ${\rm domC}(f) = {\rm imC}(f)$, $f$ is a permutation of ${\rm domC}(f)$. For any finite set of words, the only permutation that preserves the dictionary order is the identity.
We saw already that every element $\varphi \in {\it lp}G_{k,1}$ can be represented by a permutation of $A^m$ for some $m>0$, so ${\rm domC}(\varphi) = {\rm imC}(\varphi)$ for every $\varphi \in {\it lp}G_{k,1}$. $\Box$
\[factorization\_lpV\_F\].\
$\bullet$ We have $G_{k,1} \ = \ {\it lp}G_{k,1} \cdot F_{k,1}$ where every element $\varphi$ of $G_{k,1}$ has a [*unique*]{} factorization $\varphi = \pi \cdot f$ with $\pi \in {\it lp}G_{k,1}$ and $f \in F_{k,1}$.
$\bullet$ Symmetrically there is a unique factorization $G_{k,1} \ = \ F_{k,1} \cdot {\it lp}G_{k,1}$.
$\bullet$ For $T_{k,1}$ there are unique factorizations $T_{k,1} \ = \ {\it lp}T_{k,1} \cdot F_{k,1} $ $ \ = \ $ $F_{k,1} \cdot {\it lp}T_{k,1}$.
[**Proof.**]{} Uniqueness of the factorization follows immediately from Lemma \[lp\_cap\_F\]: If $\pi_1 f_1 = \pi_2 f_2$ then $\pi_2^{-1} \pi_1 = f_2 f_1^{-1} \in \, F_{k,1} \cap {\it lp}G_{k,1} = $ $\{ {\bf 1} \}$, hence $\pi_2^{-1} \pi_1 = {\bf 1} = f_2 f_1^{-1}$. Existence follows from the following factorization algorithm, whose input is any $\varphi \in G_{k,1}$.
[*Factorization algorithm*]{}:
\(1) Restrict $\varphi$ so that its image code becomes $A^n$ for some $n>0$. Let $P$ be the corresponding domain code (of cardinality $k^n$). So now $\varphi$ is represented by a bijection $P \to A^n$.
\(2) Let $f: P \to A^n$ be the unique element of $F_{k,1}$ determined by the finite maximal prefix codes $P$ and $A^n$.
\(3) Let $\pi(.) = \varphi \ f^{-1}(.)$; then $\varphi = \pi f$. \[End of algorithm.\]
We claim that $\pi \in {\it lp}G_{k,1}$. Indeed, the domain code and the image code of $\pi$ are both $A^n$; hence $\pi$ preserves length.
In the case of $T_{k,1}$ we observe that if $\varphi \in T_{k,1}$ then the unique factorization $\varphi = \pi \, f$ yields $\pi = \varphi \, f^{-1} \in T_{k,1}$ (since $\varphi \in T_{k,1}$ and $F_{k,1} \subset T_{k,1}$). $\Box$
Observe that $f \in F_{k,1}$, produced by the factorization algorithm, is the ranking function of $P$, when we view $A^n$ as the integers $\{0,1, \ldots, k^n -1\}$ in base-$k$ notation.
We will examine how the table sizes of $\pi \in {\it lp}G_{k,1}$ and $f \in F_{k,1}$ are related to the table size of $\varphi$ when $\varphi = \pi \, f$. It turns out that $\pi$ and $f$ can have exponentially larger size than $\varphi$. In a later section we’ll consider other complexity measures for $\pi$ and $f$.
\[table\_size\_pi\_f\] For all $n > 2$ there are elements $\varphi_n \in T_{2,1}$ whose factorization $\varphi_n = \pi_n f_n$ leads to an exponential increase in table size. More precisely, $\varphi_n$ can be found so that $\|\varphi_n\| = n$, and $\|\pi_n\| = \|f_n\| = 2^{n-1}$.
[**Proof.**]{} Let us pick $\varphi_n \in T_{2,1}$ given by the following table, over the alphabet $A = \{a,b\}$:
$$\varphi_n \ = \ \left[
\begin{array}{ccc cc ccc}
a^{n-1} \ & \ a^{n-2}b \ & \ \ldots \ & \ a^ib \ & \ \ldots \ & \ a^2b \ & \
ab \ & \ b \\
a^{n-2}b \ & \ a^{n-3}b \ & \ \ldots \ & \ a^{i-1}b \ & \ \ldots \ & \ ab \ &
\ b \ & \ a^{n-1}
\end{array} \right]$$
So, $\varphi_n$ is a cyclic permutation of the finite maximal prefix code $\{a^{n-1}\} \cup \{a^i b : i = n-2, \ldots , 1, 0 \}$. One observes that $\varphi_n$ is reduced (unextendable) as given by the table, hence $\|\varphi_n\| = n$.
The longest words in the image code of $\varphi_n$ in the above table have length $n-1$. When we restrict $\varphi_n$ and let its image code become $\{a,b\}^{n-1}$ we obtain the following table of size $2^{n-1}$ for $\varphi_n$, where $x_j$ ranges over $\{a,b\}^j$ (for $j=1, \ldots, n-2$):
$$\varphi_n \ = \ \left[
\begin{array}{ccc cc ccc}
a^{n-1} \ & \ a^{n-2}b x_1 \ & \ \ldots \ & \ a^ib x_{n-i-1} \ & \ \ldots \
& \ a^2b x_{n-3} \ & \ ab x_{n-2} \ & \ b \\
a^{n-2}b \ & \ a^{n-3}b x_1 \ & \ \ldots \ & \ a^{i-1}b x_{n-i-1} \ & \ \ldots
\ & \ ab x_{n-3} \ & \ b x_{n-2} \ & \ a^{n-1}
\end{array} \right]$$
Then in the factorization $\varphi_n(.) = \pi_n f_n(.)$ we have:
$$f_n \ = \ \left[
\begin{array}{ccc ccc cc}
a^{n-1} \ & \ a^{n-2}b \, a \ & \ a^{n-2}b \, b \ & \ \ldots
\ & \ a^ib \, a^{n-i-1} \ & \ a^ib \, s(x_{n-i-1}) \ & \ a^{i-1}b a^{n-i}
\ & \ \ldots \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a^{n-1} \ & \ a^{n-2}b \ & \ a^{n-3}b \, a \ & \ \ldots
\ & \ a^ib \, b^{n-i-2} \ & \ a^{i-1}b x_{n-i-1} \ & \ a^{i-1}b \, b^{n-i-1}
\ & \ \ldots \
\end{array} \right.$$
$$\left.
\begin{array}{ccc ccc}
\ \ldots \ & \ a^2b \, a^{n-3} \ & \ a^2b \, s(x_{n-3})
\ & \ ab \, a^{n-2} \ & \ ab \, s(x_{n-2}) \ & \ b \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ \ldots \ & \ a^2b \, b^{n-4} \ & \ ab x_{n-3}
\ & \ b \, b^{n-3} \ & \ bx_{n-2} \ & \ b^{n-1}
\end{array} \right]$$
where each $x_j$ ranges over $\{a,b\}^j - \{b^j\}$, and where $s(x_j)$ denotes the [*successor*]{} of $x_j$ in the dictionary order; hence, $s(x_j)$ ranges over $\{a,b\}^j - \{a^j\}$. In the table, the strings $x_j$ and the strings $s(x_j)$ appear in dictionary order. We also have
$$\pi_n \ = \ \left[
\begin{array}{ccc ccc cc}
b^{n-1} \ & \ a^{n-1} \ & \ a^{n-2}b \ & \ a^{n-3}b \, a \ & \ \ldots
\ & \ a^ib \, b^{n-i-2} \ & \ a^{i-1}b \, x_{n-i-1} \ & \ \ldots \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a^{n-1} \ & \ a^{n-2}b \ & \ a^{n-3}b \, a \ & \ a^{n-3}b \, b \ & \ \ldots
\ & \ a^{i-1}b \, a^{n-i-1} \ & \ a^{i-1}b \, s(x_{n-i-1}) \ & \ \ldots
\end{array} \right.$$
$$\left.
\begin{array}{ccc ccc}
\ldots \ & \ a^2b \, b^{n-4} \ & \ a^2b \, x_{n-3}
\ & \ ab \, b^{n-3} \ & \ b \, x_{n-2} \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ldots \ & \ ab \, a^{n-3} \ & \ ab \, s(x_{n-3})
\ & \ b \, a^{n-2} \ & \ b \, s(x_{n-2})
\end{array} \right]$$
where the words $x_j$ and $s(x_j)$ range over the same values and have the same meaning as for $f_n$.
One sees in the table of $\pi_n$ that for every argument $x$, $\pi_n(x)$ differs from $x$ in the right-most letter: whenever $x$ ends in $a$, $\pi_n(x)$ ends in $b$, and vice versa. Hence, $\pi_n$ as given by the table, is reduced (cannot be extended). Hence, $\|\pi_n\|$ is the size of the above table, i.e., $2^{n-1}$. Similarly, in the table for $f_n$, $x$ and $f_n(x)$ differ in the right-most letter, except when $x = a^{n-1}$ or $x = b$. Hence $f_n$ as given by the table is reduced, and $\|f_n\| = 2^{n-1}$. $\Box$
Other factorizations of $G_{k,1}$ and $T_{k,1}$
===============================================
We will give an infinite collection of torsion subgroups $S$ of $G_{k,1}$ that can be used for factoring $G_{k,1}$ as $S \cdot F_{k,1}$.
If $P \subset A^*$ is a finite maximal prefix code then for every $n \geq 0$, the overall restriction operation in $G_{k,1}$ determines a diagonal embedding
$\otimes {\rm id}_A: \pi \in {\mathfrak S}_{PA^n} \ \hookrightarrow$ $ \ \pi \otimes {\rm id}_A \in {\mathfrak S}_{PA^{n+1}}$
where $(\pi \otimes {\rm id}_A)(xa) = \pi(x) \cdot a$, for all $x \in PA^n$, $a \in A$. This is a generalization of the embedding ${\mathfrak S}_{A^n}$ $\hookrightarrow {\mathfrak S}_{A^{n+1}}$ that we saw earlier (which was the special case when $P$ consists of just the empty word). We then take the direct limit of this sequence of symmetric groups and obtain a subgroup of $G_{k,1}$, denoted by
$\bigcup_{n \geq 1} {\mathfrak S}_{P A^n}$.
Just as for ${\it lp}G_{k,1}$, when $k = |A|$ is even the group $\bigcup_{n \geq 1} {\mathfrak S}_{P A^n}$ is simple, and when $k$ is odd the group has a simple subgroup of index 2 (via the parity map).
\[PAm\_factorization\] If $P \subset A^*$ is a finite maximal prefix code then $S \ = \ \bigcup_{n \geq 1} {\mathfrak S}_{P A^n}$ is a subgroup of $G_{k,1}$, and we have $G_{k,1} = S \cdot F_{k,1}$. Moreover, $S \cap F_{k,1} = \{ {\bf 1} \}$, hence we have a unique factorization.
The group $T_{k,1}$ has the subgroup $Z \ = \ \bigcup_{n \geq 1} {\mathbb Z}_{P A^n}$ and we have $T_{k,1} = Z \cdot F_{k,1}$, with unique factorization for every element of $T_{k,1}$.
[**Proof.**]{} Every element $\varphi$ of ${\mathfrak S}_{P A^n}$, as an element of $G_{k,1}$, has finite domain and image codes that are the same: ${\rm domC}(\varphi) = {\rm imC}(\varphi)$. Hence by Lemma \[lp\_cap\_F\], if $\varphi \in F_{k,1}$ then $\varphi = {\bf 1}$. Hence, $S \cap F_{k,1} = \{ {\bf 1} \}$, which implies uniqueness of the factorization as we saw in the beginning of the proof of Theorem \[factorization\_lpV\_F\].
To prove existence of the factorization we use the same factorization algorithm as in the proof of Theorem \[factorization\_lpV\_F\]. Let $\varphi': P_1 \to Q_1$ represent any element of $G_{k,1}$, where $P_1$ and $Q_1$ are finite maximal prefix codes. Then by restriction we obtain a representation of the same element of the form $\varphi: P_2 \to P A^n$, where it suffices to choose $n$ such that $P A^n A^* \subseteq Q_1A^*$; since $P$ and $Q_1$ are finite maximal prefix codes, such an $n$ exists. The remainder of the proof follows from the same idea as for Theorem \[factorization\_lpV\_F\]. We let $f: P_2 \to P A^n$ be the (unique) element of $F_{k,1}$ determined by the finite maximal prefix codes $P_2$ and $P A^n$, and let $\pi = \varphi \, f^{-1}$; then ${\rm domC}(\pi) = {\rm imC}(\pi) = P A^n$, hence $\pi \in {\mathfrak S}_{P A^n}$.
When $\varphi \in T_{k,1}$ the unique factorization $\varphi = \pi \, f$ satisfies $\pi = \varphi \, f^{-1} \in T_{k,1}$ (since $F_{k,1} \subset T_{k,1}$), hence $\pi \in T_{k,1} \cap S = Z$. $\Box$
Higman (in [@Hig74], Section 6) shows that the question whether a given element of $G_{k,1}$ has finite order, is decidable. The following theorem shows that every element of finite order of $G_{k,1}$ belongs to some subgroup ${\mathfrak S}_P$, for some finite maximal prefix code $P$. Note that ${\rm domC}(\varphi) = {\rm imC}(\varphi) = P$ iff $\varphi \in {\mathfrak S}_P$.
\[torsion\_elements\] Let $\Phi \in G_{k,1}$. Then $\Phi$ has finite order iff for some restriction $\varphi$ of $\Phi$ we have ${\rm domC}(\varphi) = {\rm imC}(\varphi)$.
[**Proof.**]{} If ${\rm domC}(\varphi) = {\rm imC}(\varphi) = P$ then $\varphi \in {\mathfrak S}_P$, hence $\varphi$ has finite order.
Conversely, suppose that $\Phi$ is of finite order $r$, i.e., $\Phi^r(.) = {\rm id}(.)$ with $r > 0$, and $\Phi^i(.) \neq {\rm id}(.)$ for $0 \leq i < r$. By sufficiently restricting $\Phi$ we obtain maximal finite prefix codes $P_0, P_1, \ldots, P_r \subset A^*$ such that for some restriction $\varphi: A^* \to A^*$ of $\Phi$ we have $P_0 \stackrel{\varphi}{\longrightarrow} P_1 $ $\stackrel{\varphi}{\longrightarrow} \ \ldots \ $ $ \stackrel{\varphi}{\longrightarrow} \ P_{r-1} $ $\stackrel{\varphi}{\longrightarrow} P_r$, and $\varphi(P_i) = P_{i+1}$ for $i=0,1, \ldots, r-1$. Since $\varphi^r(.) = {\rm id}(.)$ it follows that $P_0 = P_r$.
[**Claim:**]{} For every $x \in P_0$, $C_x = \{ \varphi^i(x) : i = 0, 1, \ldots, r-1\}$ is a prefix code.\
(Note: We only claim that no two $\varphi^i(x)$ are strict prefixes of each other; we do not rule out that $\varphi^i(x) = \varphi^j(x)$ for some $0 \leq i \neq j < r$.)
Proof of the Claim. If, by contradiction, we have $\varphi^{\ell}(x) = x \, z$, for a non-empty word $z \in A^*$ and $x \in P_0$, then for all $m \geq 0$ we have: $\varphi^{m \ell}(x) = x \, z^m$. This implies that $\bigcup_{\ell=0}^{r-1} P_{\ell}$ contains words of arbitrarily large length, which contradicts the fact that the prefix codes $P_{\ell}$ are finite.
It follows that when $i>j$ then $\varphi^j(x) \in P_j$ cannot be a strict prefix of $\varphi^i(x)$, since applying $\varphi^{-j}$ to $\varphi^i(x) = \varphi^j(x) \, z$ yields $\varphi^{i-j}(x) = x \, z$.
Similarly, if we have $\varphi^i(x) = \varphi^j(x) \, z$ for a non-empty word $z \in A^*$ and $x \in P_0$ and if $i<j$ then, applying $\varphi^{r-j}$ yields $\varphi^{r+i-j}(x) = x \, z$, and the reasoning in the first paragraph (with $\ell = r+i-j$) again yields a contradiction. We conclude that $\varphi^i(x)$ and $\varphi^j(x)$ cannot be strict prefixes of each other. \[End, Proof of Claim.\]
The Claim implies that $\varphi(C_x) = C_x$ and that $C_x$ is a cycle of $\varphi$. For each $x \in P_0$ we have a cycle $C_x$ as above. For different $x \in P_0$ the corresponding cycles yield either the same set or disjoint sets, i.e., for each $x, y \in P_0$, either $C_x = C_y$ or $C_x \cap C_y = \emptyset$. So, $P_0$ is partitioned into cycles of $\varphi$, hence $\varphi(P_0) = P_0$. $\Box$
As a consequence of Theorem \[torsion\_elements\] and Lemma \[lp\_cap\_F\] we recover a result of Brin and Squier [@BrinSqu]:
The group $F_{k,1}$ is torsion-free.
\[PAm\_equal\_diff\] If $P_1, P_2 \subset A^*$ are finite maximal prefix codes let $S_i = \ \bigcup_{n \geq 0} {\mathfrak S}_{P_i A^n}$ for $i = 1$ or $2$. We have:
$S_1 = S_2$ iff $\{ P_1A^n : n \geq 0\} \cap \{ P_2A^m : m \geq 0\} \neq \emptyset$,
When $S_1 \neq S_2$, the subgroup generated by $S_1 \cup S_2$ contains infinitely many elements of $F_{k,1}$.
[**Proof.**]{} If $P_1A^N = P_2A^M$ for some $M,N \geq 0$ then $S_1 = $ $\bigcup_{n \geq 0} {\mathfrak S}_{P_1 A^n} = $ $\bigcup_{n \geq 0} {\mathfrak S}_{P_1 A^N A^n}$, since ${\mathfrak S}_{P_1 A^i} \hookrightarrow {\mathfrak S}_{P_1 A^N}$ when $i \leq N$. Similarly, $\bigcup_{m \geq 0} {\mathfrak S}_{P_2A^M A^n} = S_2$. Now, since $P_1A^N = P_2A^M$ we have $\bigcup_{n \geq 0} {\mathfrak S}_{P_1 A^N A^n} = $ $\bigcup_{m \geq 0} {\mathfrak S}_{P_2A^M A^n}$, hence $S_1 = S_2$.
In the other direction, under the condition $\{ P_1A^n : n \geq 0\} \cap \{ P_2A^m : m \geq 0\} = \emptyset$ we will prove that the subgroup of $G_{k,1}$ generated by $S_1$ and $S_2$ together contains some non-identity elements of $F_{k,1}$. Since $S_1$ and $S_2$ are torsion groups whereas $F_{k,1}$ is torsion-free, this implies that $S_1 \neq S_2$.
[**Claim.**]{} There exist $n_0, m_0 \geq 0$ such that $P_1A^{n_0} \cap P_2A^{m_0} \neq \emptyset$, and $P_1A^{n_0} \neq P_2A^{m_0}$. Moreover, there are $v_1 \in P_1A^{n_0} - P_2A^{m_0}$ and $v_2 \in P_2A^{m_0} - P_1A^{n_0}$ such that $v_1$ is a strict prefix of $v_2$.
Proof of the Claim: First, since each $P_1$ is a finite maximal prefix code, every long enough word belongs to $P_1A^*$; e.g., every word $w \in A^*$ of length $\geq {\rm max}\{ |p| : p \in P_1\}$ belongs to $P_1A^*$. Therefore, for all $m$ large enough (e.g., all $m \geq {\rm max}\{ |p| : p \in P_1\}$) we have $P_2A^m \subseteq P_1A^*$. Let $m_0 \geq 0$ be such that $P_2A^{m_0} \subseteq P_1A^*$, and let us consider the possible $n \geq 0$ such that $P_2A^{m_0} \subseteq P_1A^nA^*$. For every $p_2u \in P_2A^{m_0}$ there exists exactly one $p_1 v \in P_1A^n$ such that $p_1 v$ is a prefix of $p_2u$. If $p_1 v \neq p_2u$, we can increase the length of $v$ (i.e., increase $n$) to move $p_1 v$ closer to $p_2u$, until $p_1 v = p_2u$. Thus, there exists $n_0$ such that $P_1A^{n_0} \cap P_2A^{m_0} \neq \emptyset$.
Finally, $P_1A^{n_0} \neq P_2A^{m_0}$ by the hypothesis that $\{ P_1A^n : n \geq 0\} \cap \{ P_2A^m : m \geq 0\} = \emptyset$. Since $P_1A^{n_0} \neq P_2A^{m_0}$ and since $P_1A^{n_0}$ and $P_2A^{m_0}$ are finite maximal prefix codes, $P_1A^{n_0}$ and $P_2A^{m_0}$ are not strict subsets of each other. Hence there exist $w_1 \in P_1A^{n_0} - P_2A^{m_0}$ and $v_2 \in P_2A^{m_0} - P_1A^{n_0}$. Moreover, since $P_2A^{m_0} \subset P_1A^{n_0} A^*$ we have: For any $v_2 \in P_2A^{m_0}$ there exists $v_1 \in P_1A^{n_0}$ such that $v_1$ is a prefix of $v_2$. Since $v_2 \notin P_1A^{n_0}$, $v_1$ is a strict prefix of $v_2$. \[This proves the Claim.\]
We will now construct an element $\gamma_1 \in S_1$ whose $S_2 \cdot F_{k,1}$ factorization is of the form $\gamma_1 = \pi_2 \, f$ with $f \neq {\bf 1}$. From this we obtain two elements $\gamma_1 \in S_1$ and $\pi_2 \in S_2$ such that $\pi_2^{-1} \gamma_1 = f \in F_{k,1}$ with $f \neq {\bf 1}$.
Since $P_1A^{n_0} \cap P_2A^{m_0} \neq \emptyset$, there is $u_1 \in P_1A^{n_0} \cap P_2A^{m_0}$. Using $u_1$ and the words $v_1$ and $v_2$ from the Claim, we now define $\gamma_1 = (u_1|v_1)$; i.e., $\gamma_1$ is the permutation of $P_1A^{n_0}$ that transposes the two words $u_1$ and $v_1$, and fixes the rest of $P_1A^{n_0}$.
By Theorem \[PAm\_factorization\], $\gamma_1 = \pi_2 \, f$ for a unique $\pi_2 \in S_2$ and $f \in F_{k,1}$. The factorization algorithm given in the proof of Theorem \[PAm\_factorization\] finds $\pi_2$ and $f$ by restricting $\gamma_1$ so that its image code becomes ${\rm imC}(\gamma_1) = P_2A^{m_0}$; the image codes of $f$ and of $\pi_2$ (not necessarily maximally extended), as well as the domain code of $\pi_2$, will also be $P_2A^{m_0}$. The table of $\gamma_1$ is
$ \gamma_1 \ = \ \left[ \begin{array}{cc l}
u_1 & v_1 \ & \ \ \ {\rm identity \ on} \\
v_1 & u_1 \ & \ \ \ P_1A^{n_0} - \{u_1,v_1\}
\end{array} \right]. $
By restricting so as to make ${\rm imC}(\gamma_1) = P_2A^{m_0}$ we obtain a table of the form
$ \gamma_1 \ = \ \left[ \begin{array}{lll ll}
\ldots \ & \ u_1 z \ & \ \ldots \ & \ v_1 \ & \ \ldots \\
\ldots \ & \ v_1 z \ (= v_2) \ & \ \ldots \ & \ u_1 \ & \ \ldots
\end{array} \right]. $
Here $z \in A^*$ is such that $v_2 = v_1 z$ and $z$ is non-empty (recall that $v_1$ is a strict prefix of $v_2$). Hence for this restriction of $\gamma_1, \pi_2$ and $f$ we have: ${\rm domC}(\gamma_1) = {\rm domC}(f)$ contains $u_1 z$. But since $u_1 \in P_2 A^{m_0}$ and since $P_2 A^{m_0}$ is a prefix code we find that $u_1 z \notin P_2 A^{m_0} = {\rm imC}(f)$. Hence, ${\rm domC}(f) \neq {\rm imC}(f)$, therefore $f$ is not the identity. Since $F_{k,1}$ is torsion-free, the conclusion follows. $\Box$
\[PAm\_conjugate\] If $P_1, P_2 \subset A^*$ are finite maximal prefix codes let $S_i = \ \bigcup_{n \geq 0} {\mathfrak S}_{P_i A^n}$ for $i = 1$ or $2$. If $|P_1A^N| = |P_2A^M|$ for some $N, M \geq 0$ then as subgroups of $G_{k,1}$, $S_1 = \theta^{-1} \ S_2 \ \theta$ for some $\theta \in F_{k,1}$.
[**Proof.**]{} Let $\theta$ be the element of $F_{k,1}$ such that $\theta: P_1A^N \to P_2A^M$; then $\theta$ can be restricted such that $\theta: P_1A^NA^n \to P_2A^MA^n$ for all $n \geq 0$. Then as subgroups of $G_{k,1}$, $S_1 = \theta^{-1} \ S_2 \ \theta$. $\Box$
[**Element-specific factorizations**]{}
For any element $\varphi \in G_{k,1}$ with ${\rm domC}(\varphi) = P$ and ${\rm imC}(\varphi) = Q$ we can apply Theorem \[PAm\_factorization\] to obtain the factorizations $\varphi(.) = \pi_Q \, f(.) = f \, \pi_P(.)$, where $f: P \to Q$ belongs to $F_{k,1}$, $\pi_Q \in {\mathfrak S}_Q$ and $\pi_P \in {\mathfrak S}_P$. Moreover, ${\mathfrak S}_P = f^{-1} \, {\mathfrak S}_Q \, f$. Note that in this factorization, $\|f\|, \|\pi_Q\|, \|\pi_P\| \leq \|\varphi\|$.
If $\varphi, \psi \in G_{k,1}$ are such that ${\rm domC}(\varphi) = P$, ${\rm imC}(\varphi) = Q = {\rm domC}(\psi)$, and ${\rm imC}(\psi) = R$, then (since domain and ranges match) we have the following multiplication formula for the factorization of $\psi \, \varphi(.)$. If $\varphi(.) = \pi_Q^{\varphi}\, f^{\varphi}$ and $\psi(.) = \pi_R^{\psi} \, f^{\psi}$ then $\psi \, \varphi(.) = \pi \, f(.)$, where $\pi \ = \ \pi_R^{\psi} \, f^{\psi} \pi_Q (f^{\psi})^{-1} \ \in $ ${\mathfrak S}_R$, and $f \ = \ f^{\psi} \, f^{\varphi} \ \in F_{k,1}$.
[**Questions left open:**]{} What are all the torsion subgroups of $G_{k,1}$? What are all the torsion, non-torsion, or torsion-free subgroups $S$ of $G_{k,1}$ for which there is a unique factorization $G_{k,1} = S \cdot F_{k,1}$? Are the groups $\bigcup_{n \geq 0} {\mathfrak S}_{P_1 A^n}$ and $\bigcup_{n \geq 0} {\mathfrak S}_{P_2 A^n}$ non-isomorphic if they do not obey the conditions of Theorem \[PAm\_conjugate\]?
Generators of [*lp*]{}$V$ and reversible computing
==================================================
We are interested in the computation of elements of $V$ and of [*lp*]{}$V$ by circuits. For general information on circuits see [@Wegener; @Handb]; good references on reversible circuits are [@FredToff; @Toff80Memo; @Toff80Conf; @Shende]. We will use the following fundamental results from the field of [*reversible computing*]{}:
$\bullet$ (V. Shende, A. Prasad, I. Markov, J. Hayes [@Shende]) Every [*even*]{} permutation of the set $\{0,1\}^n$ can be computed by a circuit constructed only from bijective gates of type [not, c-not, cc-not]{}.
The gates [not, c-not, cc-not]{} are well known in the field of reversible computing, and are defined as follows:
[not]{}: $x \in \{0,1\} \longmapsto \overline{x} \in \{0,1\}$ is the usual negation operation;
[c-not]{}: $(x,y) \in \{0,1\}^2 \longmapsto (x, y \oplus x) \in $ $\{0,1\}^2$ is the [*controlled*]{} [not]{}, also called the “Feynman gate”;
[cc-not]{}: $(x, y, z) \in \{0,1\}^3 \longmapsto $ $(x, y, z \oplus (x \& y))$ $\in \{0,1\}^3$ is the [*doubly controlled*]{} [not]{}, with $\oplus$ denoting the usual exclusive [or]{} (i.e., addition modulo 2), and $\&$ denoting the logical [and]{} (i.e., multiplication modulo 2). The doubly controlled [not]{} is usually called the “Toffoli gate” [@FredToff; @Toff80Memo; @Toff80Conf].
$\bullet$ [@Shende] For any positive integer $n$ and any permutation $\pi \in {\mathfrak S}_{2^n}$, the permutation
$(x_1, \ldots, x_n, x_{n+1})$ $\in$ $\{0,1\}^{n+1}$ $ \ \longmapsto \ \pi(x_1, \ldots, x_n) \cdot x_{n+1} \in \{0,1\}^{n+1}$
is [*even*]{}. Here, “$\cdot$” denotes concatenation. Indeed, one transposition $(u|v)$ (with $u,v \in \{0,1\}^n$) is now replaced by $(u0|v0) \, (u1|v1)$ (i.e., two transpositions).
As a consequence, every odd permutation of $\{0,1\}^n$ can be computed by a circuit that only makes use of bijective gates of type [not, c-not, cc-not]{}, and that uses an extra identity wire $x_{n+1} \mapsto x_{n+1}$.
$\bullet$ (T. Toffoli [@Toff80Memo; @Toff80Conf]) An odd permutation of $\{0,1\}^n$ cannot be computed by any circuit containing only bijective gates with fewer than $n$ input-output wires. Hence for odd permutations, the extra identity wire is necessary for bijective computing with a finite collection of gate types.
The above results have some interesting consequences for the group [*lp*]{}$V$:
First, the overall restriction operation for elements of the Thompson group $V$ (which replaces each $\varphi(x) = y$ by the pair $\varphi(x0) = y0$, $\varphi(x1) = y1$ for all $x$ in the domain of $\varphi$) now receives a very concrete interpretation for elements of [*lp*]{}$V$: For an element $\varphi$ of [*lp*]{}$V$, the overall restriction is equivalent to adding an [*identity wire*]{} at the “bottom” of the circuit (i.e., at the right-most position for boolean variables).
Second, another consequence of the above concerns the generators of [*lp*]{}$V$. Let $N, C, T$ be the partial maps $\{0,1\}^* \to \{0,1\}^*$ defined as follows, where $w \in \{0,1\}^*$ is any bitstring; $N: x_1w \mapsto \overline{x}_1w$, $C: x_1x_2w \mapsto x_1 \, (x_2 \oplus x_1) \, w$, and $T: x_1x_2x_3w \mapsto x_1x_2 \, (x_3 \oplus (x_2 \& x_1)) \, w$. These maps are just the [not, c-not, cc-not]{} gates, [*applied only to the first (left-most) bits*]{} of a binary string. We leave $N, C, T$ undefined on bit strings that are too short.
Note that the engineering convention consists of using the same name (e.g., “not”, “c-not”, etc.) for the same operation on different variables in an sequence of variables. But this convention would not be correct in our setting; e.g., negating the first bit in a string is different from negating the second bit. In order to implement the operations $N, C, T$ on all bit-positions, i.e., in order to obtain the gate types [not, c-not, cc-not]{} in the engineering sense of the word, we also introduce the position transpositions $\tau_{i,j}: \{0,1\}^* \to \{0,1\}^*$ (where $1 \leq i<j$), defined by
$\tau_{i,j}: \ $ $u \, x_i \, v \, x_j \, w \ \longmapsto \ u \, x_j \, v \, x_i \, w$,
where $u, v, w \in \{0,1\}^*$, $|u| = i-1$, $|v| = j-1-i$; and we leave $\tau_{i,j}(s)$ undefined when $|s| < j$. Note that $\tau_{i,j}$ does not transpose a pair of words $\in \{0,1\}^*$, but boolean variables (or positions within words).
\[nct\_generators\_lpV\] The group [*lp*]{}$V$ is generated by the set $\{N, C, T\} \, \cup \, \{\tau_{i,i+1} : 1 \leq i \}$. More generally, [*lp*]{}$G_{k,1}$ is generated by $\Gamma_k \, \cup $ $\{\tau_{i,i+1} : 1 \leq i \}$ for some finite set $\Gamma_k$.
The finite alternating group ${\mathfrak A}_{2^n}$ (acting on $\{0,1\}^n$) is generated by the set $\{N, C, T\} \cup \{\tau_{i,i+1} : 1 \leq i < n\}$.
[**Proof.**]{} The Proposition is immediate from the above observations, and in particular the work [@Shende]. $\Box$
[**Application: An intuitive generating set for $V$**]{}
The ${\it lp}V \cdot F$ factorization, together with the nice generating set given above for ${\it lp}V$ enables us to find a finite generating set for $V$ with a nice “physical” interpretation. It follows from the ${\it lp}V \cdot F$ factorization that $\{N, C, T\} \, \cup \, \{\tau_{i,i+1} : 1 \leq i \}$ $ \, \cup \ \{\sigma, \sigma_1\}$ is a generating set of $V$, where $\{\sigma, \sigma_1\}$ is the generating set of $F$ given in [@CFP] with tables
$\sigma \ = \ \left[ \begin{array}{lll}
00 & 01 & 1 \\
0 & 10 & 11
\end{array} \right] $, $\sigma_1 \ = \ \left[ \begin{array}{llll}
0 & 100 & 101 & 11 \\
0 & 10 & 110 & 111
\end{array} \right]. $
We will see that $\sigma_1$ is a “controlled lowering” of $\sigma$ (defined below). In [@BiThomps] we saw that $\sigma$ can be viewed as the ${\mathbb Z}$-shift $0^n 1 \mapsto 0^{n-1}1$, $01 \mapsto 10$, $1^n 0 \mapsto 1^{n+1} 0$, on the maximal prefix code $0^*01 \cup 1^*10$.
Since $V$ is finitely generated, only a finite subset of $\{\tau_{i,i+1} : 1 \leq i \}$ will be needed for generating $V$. Surprisingly, it turns out that in the presence of the other generators, the Toffoli gate $T$ will not be needed for $V$. In detail we have:
\[generators\_V\] The Thompson group $V$ is generated by the finite set $\{N,\ C, \ \tau_{1,2}, \ \sigma, \ \sigma_1 \}$, where $N$ is the [not]{} gate applied to the first wire, $C$ is [c-not]{} (controlled [not]{}, a.k.a. Feynman gate) applied to the first two wires, $\tau_{1,2}$ is the transposition of the first two wires, and $\sigma, \ \sigma_1$ generate the Thompson group $F$.
[**Proof.**]{} We start out with the Higman generators $\kappa, \lambda, \mu, \nu$ of $V$ (see [@Hig74]), whose tables are
$\kappa \ = \ \left[ \begin{array}{ll}
0 & 1 \\
1 & 0
\end{array} \right] $, $\lambda \ = \ \left[ \begin{array}{lll}
00 & 01 & 1 \\
00 & 1 & 01
\end{array} \right] $, $\mu \ = \ \left[ \begin{array}{lll}
0 & 10 & 11 \\
10 & 0 & 11
\end{array} \right] $, $\nu \ = \ \left[ \begin{array}{llll}
00 & 01 & 10 & 11 \\
00 & 10 & 01 & 11
\end{array} \right] $.
We see that $\kappa = N$ and $\nu = \tau_{1,2}$. For $\lambda$ and $\mu$ we apply the ${\it lp}V \cdot F$ factorization algorithm, which leads to
$\lambda(.) \ = \ \left[ \begin{array}{llll}
00 & 01 & 10 & 11 \\
00 & 10 & 11 & 01
\end{array} \right] \ \cdot \
\left[ \begin{array}{llll}
00 & 010 & 011 & 1 \\
00 & 01 & 10 & 11
\end{array} \right](.) $
The right factor belongs to $F$, hence it is generated by $\{ \sigma, \sigma_1\}$. It is easy to check that the first factor is equal to $\tau_{1,2} \cdot C(.)$; recall that $C$ has the table $\left[ \begin{array}{llll}
00 & 01 & 10 & 11 \\
00 & 01 & 11 & 10
\end{array} \right] \ = \
\left[ \begin{array}{lll}
0 & 10 & 11 \\
0 & 11 & 10
\end{array} \right] $.
A similar calculation leads to a factorization $\mu(.) \ = \ $ $\tau_{1,2} \cdot C \cdot \tau_{1,2} \cdot N \cdot \tau_{1,2} \cdot f(.)$, for some $f \in F$. $\Box$
[**The lowering operation**]{}
The following operation, inspired by circuits, gives further insight into ${\it lp}V$ and $F$. For any integer $d>0$ we define
$\varphi \in G_{k,1} \ \longmapsto \ (\varphi)_d \in G_{k,1}$ by
$(\varphi)_d(zx) = z \ \varphi(x)$ for all $z \in A^d$, $x \in {\rm Dom}(\varphi)$.
Recall that $A^d$ is the set of all words of length $d$ over $A$. It is easy to see that for each $d>0$, the operation $\varphi \to (\varphi)_d$ is an [*endomorphism*]{} of $G_{k,1}$, which is injective but not surjective; it is also an endomorphism of ${\it lp}G_{k,1}$, of $F_{k,1}$, of $T_{k,1}$, and of ${\it lp}T_{k,1}$.
The circuit interpretation of the operation $\varphi \to (\varphi)_d$ is that the “gate” $\varphi$ is lowered by $d$ positions in the circuit through the introduction of $d$ identity wires on top of the “gate” $\varphi$ (i.e., at the left end of the list of input variables). While $\varphi$ is applied to the boolean variables $x_1, x_2, \ldots$, the lowered gate will be applied to the variables $x_{d+1}, x_{d+2}, \ldots$. This is commonly done in circuits, as it allows the designer to place gates at any place in the circuit. In electrical engineering, traditionally no distinction is made between a gate; e.g., the [c-not]{} operation and its lowerings are all just called “[c-not]{} gates”. The lowering operation is an important link between circuits and their representation by groups or monoids of functions.
The lowering operation can be expressed in terms of the transpositions $\tau_{i,i+1}$, although the formula depends on the length $\ell$ of the longest word $\in A^*$ appearing in the table of $\varphi$. We have: $(\varphi)_d(.) \ = \ \pi^{-1} \ \varphi \ \pi(.)$, where $\pi$ is the following permutation of bit positions:
If if $d+1 > \ell$ then $ \ \ \pi(.) \ = \ $ $\left( \begin{array}{cccc cccc}
\hspace{-.1in} 1 & 2 & \ldots & \ell & d+1 & d+2 & \ldots & d+\ell \\
\hspace{-.1in} d+1 & d+2 & \ldots & d+\ell & 1 & 2 & \ldots & \ell
\end{array} \hspace{-.09in} \right)$.
If $d+1 \leq \ell$ then $ \ \ \pi(.) \ = \ $ $\left( \begin{array}{cccc cccc}
\hspace{-.1in} 1 & 2 & \ldots & d & d+1 & d+2 & \ldots & d+\ell \\
\hspace{-.1in} 1+\ell & 2+\ell & \ldots & d+\ell & 1 & 2 & \ldots & \ell
\end{array} \hspace{-.09in} \right)$.
When we write elements of $G_{k,1}$ as elements of the Cuntz algebra ${\mathcal O}_k$ (according to [@BiThomps] and [@Nekrash]), we see that the lowering operation is an endomorphism of ${\mathcal O}_k$ given by the formula
$\gamma \in {\mathcal O}_k \ \longmapsto \ (\gamma)_d = $ $\sum_{z \in A^d} z \, \gamma \, {\overline z} \ \in \ {\mathcal O}_k$.
Note that all transpositions of variables (or wires) $\tau_{i,i+1}$ are obtained from the transposition of variables $\tau_{1,2}$ by $\tau_{i,i+1} = (\tau_{1,2})_{i-1}$. So, the lowering operations, together with a finite set of elements of ${\it lp}G_{k,1}$, yields a generating set of ${\it lp}G_{k,1}$. For $G_{k,1}$ we already saw that the transpositions of variables are redundant as generators (since $G_{k,1}$ is finitely generated), but that the use of the transpositions of variables shortens the word length; we will see (Theorem \[VwordsCircuits\]) that when the transpositions of variables are added to a finite generating set of $G_{k,1}$ then the word length becomes approximately the same as the bijective-circuit complexity. For $F_{k,1}$ it will be interesting to consider generating sets of the form $\Gamma \cup \ \bigcup_{d \geq 1} (\Gamma)_d$, where $\Gamma$ is any finite generating set of $F_{k,1}$, and where $(\Gamma)_d = \{ (f)_d : f \in \Gamma \}$ (see the open problems at the end of Section 6).
We can define the [**controlled lowering operation**]{}; we fix any string $c \in A^*$, called the “control string” and define
$\varphi \in G_{k,1} \ \longmapsto \ (\varphi)_c \in G_{k,1}$ by
$(\varphi)_c(cx) \ = \ c \ \varphi(x)$ for all $x \in {\rm Dom}(\varphi)$, and
$(\varphi)_c(p \alpha) \ = \ p \alpha$ where $p <_{{\rm pref}} c$, $\alpha \in A$, $p \alpha \not\leq_{{\rm pref}} c$;
so here $p$ is any strict prefix of $c$ (i.e., $p \neq c$), and $\alpha \in A$ is such that $p \alpha$ is [*not*]{} a prefix of $c$. So, domC$((\varphi)_c) \ = \ c \cdot {\rm domC}(\varphi) \ \cup \ $ $\{p \alpha : p <_{{\rm pref}} c, \ \alpha \in A, \ $ $p \alpha \not\leq_{{\rm pref}} c\}$, and imC$((\varphi)_c) \ = \ $ $c \cdot {\rm imC}(\varphi) \ \cup \ $ $\{p \alpha : p <_{{\rm pref}} c, \ \alpha \in A, \ $ $p \alpha \not\leq_{{\rm pref}} c\}$.
It is easy to see that for each $c \in A^*$, the operation $\varphi \to (\varphi)_c$ is an [*endomorphism*]{} of $G_{k,1}$, which is injective but not surjective; it is also an endomorphism of ${\it lp}G_{k,1}$, of $F_{k,1}$, of $T_{k,1}$, and of ${\it lp}T_{k,1}$. In Cuntz algebra notation, the operation takes the form
$\gamma \in {\mathcal O}_k \ \longmapsto \ (\gamma)_c \ = \ $ $c \, \gamma \, {\overline c} \ + \ $ $\sum_{p, \alpha } p \alpha \, {\overline {p \alpha}} \ \in \ {\mathcal O}_k$.
where $p$ ranges over the strict prefixes of $c$ and $\alpha$ ranges over the letters of $A$ such that $p \alpha$ is not a prefix of $c$. In ${\mathcal O}_k$ the controlled lowering operation is a multiplicative endomorphism, but it is not additive.
Observe that for the generators $\{ \sigma, \sigma_1 \}$ of $F$ seen before, the second generator is the controlled lowering of the first with control string 1; this explains our notation for $\sigma_1$.
Generalized word problem, distortion of $F_{k,1}$ and ${\it lp}G_{k,1}$ in $G_{k,1}$
=====================================================================================
Over any finite generating set of $G_{k,1}$ the generalized word problem of $F_{k,1}$ in $G_{k,1}$ can be decided in [*cubic*]{} deterministic time.
Similarly, over any finite generating set of $G_{k,1}$ the generalized word problem of ${\it lp}G_{k,1}$ in $G_{k,1}$, and more generally, the generalized word problem of $\bigcup_m {\mathfrak S}_{PA^m}$ in $G_{k,1}$ (for any finite maximal prefix code $P$) can be decided in [*cubic*]{} deterministic time.
[**Proof.**]{} By Proposition 4.2 of [@BiThomps], if $\varphi$ is given by a word of length $n$ over a finite generating set of $G_{k,1}$ then a table for $\varphi$ (not necessarily maximally extended) can be computed in time $O(n^3)$. By Proposition 3.5 of [@BiThomps], the length $n$ provides a linear upper bound on the size of this table. Also, every table entry has length $\leq c \ n$. More precisely, the table has the form $((x_1,y_1), \dots, (x_N,y_N))$, where $|x_i|, |y_i|, N \leq c \ n$ (for some constant $c \geq 1$). The sets $\{x_1, \dots, x_N\}$ and $\{y_1, \dots, y_N\}$ are maximal prefix codes, and $\varphi(x_i) = y_i$ for $i = 1, \dots, N$.
To check whether $\varphi$ belongs to $F_{k,1}$ we first sort the table according to the input entries, with respect to dictionary order; more precisely, we sort the pairs of the table $((x_1,y_1), \dots, (x_N,y_N))$ according the $x$-coordinates, in time $\leq O(n^2 \, {\rm log} \, n)$; indeed, there are $O(n \, {\rm log} \, n)$ sorting steps, and since each word has length $\leq c \ n$, each word comparison takes time $O(n)$. Then we check whether the resulting $x$-sorted table is now also in sorted form regarding the $y$-coordinates; this takes quadratic time, as there are $O(n)$ words of length $O(n)$.
To check whether $\varphi$ belongs to ${\it lp}G_{k,1}$ we check, in time $\leq O(n^2)$, that $|x_i| = |y_i|$ for $i = 1, \ldots, N$. And to check whether $\varphi$ belongs to $\bigcup_m {\mathfrak S}_{PA^m}$ we note first that $P$ is a fixed finite maximal prefix code, independent of $\varphi$. We restrict $\varphi$ so that every table entry receives length $\geq {\rm max}\{|p| : p \in P\}$. This multiplies the table size of $\varphi$ by a constant, at most (since $P$ and ${\rm max}\{|p| : p \in P\}$ are fixed). So we can assume that each $x_i$ and $y_i$ in the table of $\varphi$ has a prefix in $P$. Now for $x_1$, find the prefix $p_1 \in P$ of $x_1$, so $x_1 = p_1s_1$ for some $s_1 \in A^*$, and let $m_0 = |s_1|$. Thus, $\varphi$ belongs to $\bigcup_m {\mathfrak S}_{PA^m}$ iff $\varphi \in {\mathfrak S}_{PA^{m_0}}$. So, we now write each $x_i$ and each $y_i$ in the form $p \, s$ with $p \in P$ and check that $|s| = m_0$; this holds (for all $s$ obtained) iff $\varphi \in$ ${\mathfrak S}_{PA^{m_0}}$. Checking this takes time $\leq O(n^2)$. $\Box$
Since ${\it lp}G_{k,1} \cap F_{k,1} = \{ {\bf 1} \}$, we have the following equivalence: $w = {\bf 1}$ (as elements of $G_{k,1}$) iff $w \in {\it lp}G_{k,1}$ and $w \in F_{k,1}$. Thus, the word problem of $G_{k,1}$ reduces (by a one-to-one linear-time reduction) to the conjunction of the generalized word problem of ${\it lp}G_{k,1}$ in $G_{k,1}$ and the generalized word problem of $F_{k,1}$ in $G_{k,1}$. (Here, the reduction function is just the identity map.) The same is true with ${\it lp}G_{k,1}$ replaced by $\bigcup_m {\mathfrak S}_{PA^m}$ (for any chosen finite maximal prefix code $P$).
Hence, the deterministic (or nondeterministic, or co-nondeterministic) time complexity of the word problem of $G_{k,1}$ is a lower bound for the deterministic (respectively nondeterministic, or co-nondeterministic) time complexity of the generalized word problem of $\bigcup_n {\mathfrak S}_{PA^m}$ in $G_{k,1}$ or the generalized word problem of $F_{k,1}$ in $G_{k,1}$, or both. More formally, we have the following:
We say that a language (or decision problem) $L$ is [*as hard as*]{} coNP iff there is a coNP-complete problem $L_0$ such that for every function $t(.)$ that is a deterministic time complexity lower bound for infinitely many instances of $L_0$ we have: Some function $\geq c \cdot t(.))$ is a deterministic time complexity lower bound for infinitely many instances of $L$ (for some constant $c > 0$).
\[defn\_wordlength\] Let $G$ be a group with generating set $A$. Suppose every generator $\alpha \in A$ has been assigned a “length” $|\alpha| \in {\mathbb N}$. Typically, if $A$ is finite then $|\alpha| =1$ for all $\alpha \in A$. For the position transpositions $\tau_{i,j}$ ($1 \leq i < j$) we take $|\tau_{i,j}| = j$.
The length of a word $w = a_1 \ldots a_n$ over $A$ is defined by $|w| = \sum_{j=1}^n |a_j|$.
The [*word length*]{} $|g|_A$ of $g \in G$ over $A$ is defined to be the shortest length of any word (over $A$) that represents $g$.
For a group with generating set $A$ we often say “a word over $A$” when we actually mean “a word over $A \cup A^{-1}$”; we will also use the notation $A^{\pm 1}$ for $A \cup A^{-1}$.
\[as\_hard\_as\_coNP\] Let $\Gamma_{k,1}$ be a finite generating set of $G_{k,1}$ but suppose that elements of $G_{k,1}$ are given over the infinite generating set $\Gamma_{k,1} \cup \{\tau_{i,j}: 1 \leq i<j \}$. Then the generalized word problem, in $G_{k,1}$, of either $F_{k,1}$ or ${\it lp}G_{k,1}$, or both, is as hard as coNP.
Similarly, if $P \in A^*$ is a finite maximal prefix code then the generalized word problem, in $G_{k,1}$, of either $F_{k,1}$ or $\bigcup_n {\mathfrak S}_{PA^m}$, or both, is as hard as coNP. Moreover, these problems are in coNP.
[**Proof.**]{} The word problem of $G_{k,1}$ over the generating set $\Gamma_{k,1} \cup \{\tau_{i,j}: 1 \leq i<j \}$ is coNP-complete [@BiCoNP]. The hardness then follows from the above conjunctive reduction. $\Box$
\[defn\_distortion\] Let $G_1$ be a group with generating set $A_1$, and let $G_2$ be a subgroup of $G_1$ with generating set $A_2$. A function $f: {\mathbb N} \to {\mathbb N}$ is called a [**distortion**]{} function for $G_2$ within $G_1$, with respect to the generators $A_1$, respectively $A_2$, iff for all $g_2 \in G_2$: $|g_2|_{A_2} \leq f(|g_2|_{A_1})$.
[*The*]{} distortion function of $G_2$ within $G_1$, with respect to the generators $A_1$, respectively $A_2$, is the smallest distortion function.
If we use [*finite*]{} generating sets for both $G_{k,1}$ and $F_{k,1}$ then $F_{k,1}$ has linear distortion in $G_{k,1}$.
[**Proof.**]{} For any element $g \in G_{k,1}$ we have $\|g\| \leq c_1 \, |g|_G$, by Proposition 3.5 of [@BiThomps]; here, $\|g\|$ is the table size of the element $g$, $c_1$ is a positive constant, and $|g|_G$ is the word length of $g$ over some chosen, fixed finite generating set of $G_{k,1}$. By Theorem 2.5 of [@CFP], $|g|_F \leq c_2 \, \|g\|$, where $c_2$ is a positive constant, and $|g|_F$ is the word length of $g$ over some chosen, fixed finite generating set of $F_{k,1}$. Hence, $|g|_F \leq c_1c_2 \, |g|_G$, so the distortion of $F_{k,1}$ in $G_{k,1}$ is linear. $\Box$
[**Problems left open:**]{}
1\. Over the generating set $\Gamma_{k,1} \cup \{\tau_{i,j}: 1 \leq i<j \}$ of $G_{k,1}$, are the generalized word problems of the subgroups $F_{k,1}$, ${\it lp}G_{k,1}$, and $\bigcup_n {\mathfrak S}_{PA^n}$ each coNP-complete?
2\. We saw that ${\it lp}V$ is generated by $\{ N, C, T\} \ \cup \ \{\tau_{i,i+1}: 1 \leq i \}$. What is the [*distortion*]{} of ${\it lp}V$ (over this generating set) within the Thompson group $V$ (with $V$ over the generating set $\Gamma_V \ \cup \ \{\tau_{i,i+1}: 1 \leq i \}$, where $\Gamma_V$ is any finite generating set of $V$)? We will see in the next Section that this distortion has a close connection to the relation between different kinds of bijective circuits.
3\. We saw that $F$ is generated by a two-element set $\{ \sigma, \sigma_1\}$, and hence also by $\{ (\sigma)_d, (\sigma_1)_d : d > 0\}$. What is the [*distortion*]{} of $F$ within $V$, when $F$ is taken over the generating set $\{ (\sigma)_d, (\sigma_1)_d : d > 0\}$, and $V$ is taken over the generating set $\Gamma_V \, \cup \{\tau_{i,i+1}: 1 \leq i \}$?
Complexity of $F$ and of the factorization of $V$
=================================================
We saw that in the factorization $\varphi = \pi \, f$ with $\pi \in {\it lp}V$ and $f \in F$, the table sizes of $\pi$ and of $f$ can be exponentially larger than the table size of $\varphi$. We will now investigate the circuit complexity of $\pi$ and $f$, compared to that of $\varphi$. We will also show that if $f \in F$ then the circuit complexity of $f^{-1}$ is not much higher than the circuit complexity of $f$; in other words, the elements of the Thompson group $F$ do not have much computational asymmetry (and in particular, they cannot be one-way functions). And we will show that some problems in $V$ are coNP-complete or $\#{\mathcal P}$-complete; in particular, the problem of finding the ${\it lp}V \cdot F$ factorization is $\#{\mathcal P}$-complete. In this section we focus on the Thompson groups $V$ and $F$, but the results could easily be extended to $G_{k,1}$ and $F_{k,1}$.
Circuit complexity and Thompson groups
--------------------------------------
Since an element $\varphi \in V$ is a partial function mapping bitstrings to bitstrings, it is natural to view $\varphi$ as a boolean function, to be computed by a boolean circuit. However, unless $\varphi \in {\it lp}V$, the inputs and the outputs of $\varphi$ do not have a fixed length. So the traditional concept of a combinational boolean circuit cannot be applied directly to elements of $V$.
Let $\varphi: P \to Q$ be a bijection between finite maximal prefix codes $P,Q \subset \{0,1\}^*$, representing an element of $V$. We will use ternary logic over the alphabet $\{0,1,\bot\}$, where $\bot$ is a new letter used for padding bitstrings. Let $m$ is the length of the longest bitstring in $P$, and let $n$ is the length of the longest bitstring in $Q$. We define $\varphi^{\bot}: \{0,1,\bot\}^m \to \{0,1,\bot\}^n$ as follows:
For $p \in P$, $\varphi^{\bot}(p \, \bot^{m -|p|}) \ = \ $ $ \varphi(p) \ \bot^{n - |\varphi(p)|}$.
For $x \in \{0,1,\bot\}^m - \{ p \, \bot^{m -|p|} : p \in P\}$ we let $\varphi^{\bot}(x) \ = \ \bot^n$.
We will use the notation
$P^{\bot} \ = \ \{ p \, \bot^{m -|p|} : p \in P\}$ $ \ = \ P\bot^* \ \cap \ \{0,1,\bot\}^m$, where $m = {\rm max}\{ |p| : p \in P\}$;
$Q^{\bot} \ = \ \{ q \, \bot^{n -|q|} : q \in Q\}$ $ \ = \ Q\bot^* \ \cap \ \{0,1,\bot\}^n$, where $n = {\rm max}\{ |q| : q \in Q\}$.
We call $P^{\bot}$, $Q^{\bot}$, and $\varphi^{\bot}$ the [**padding**]{} of $P$, $Q$, respectively $\varphi$. Note: For $\varphi \in V = G_{2,1}$, the padding $\varphi^{\bot}$ is [*not*]{} to be viewed as an element of [*lp*]{}$G_{3,1}$.
We observe that for the restrictions to $P^{\bot}$ or to $Q^{\bot}$ we have $(\varphi^{\bot}|_{P^{\bot}})^{-1} = (\varphi^{-1})^{\bot}|_{Q^{\bot}}$; the restriction $\varphi^{\bot}|_{P^{\bot}}$ is bijective (but $\varphi^{\bot}$ is not bijective in general). When ${\rm imC}(\varphi) = {\rm domC}(\psi)$ we also have $(\psi \varphi)^{\bot} = \psi^{\bot}\varphi^{\bot}$.
To compute the function $\varphi^{\bot}: \{0,1,\bot\}^m \to \{0,1,\bot\}^n$ we consider ternary-logic combinational circuits with gates over the alphabet $\{0,1,\bot\}$. We assume that a finite, computationally universal set of ternary logic gates has been chosen; we ignore the details since they only affect the circuit complexity by a constant multiple. We also use the (unbounded) set of wire-swap operations $\tau_{i,i+1}$.
For such a circuit, the [**size of the circuit**]{} is defined to be the number of gates together with the number of wires (links between gates or between gates and inputs or outputs). Note that a “lowered gate” $(\gamma)_d$ (i.e., the gate $\gamma$ applied to the wires $d+1, d+2$, etc., as defined at the end of Section 5) is counted as one gate (independently of $d$). Also, in a circuit each wire-crossing $\tau_{i,i+1}$ will be counted as one gate (independently of $i$). Note that here we are talking about circuit size, not about word length.
[**Remarks**]{}:\
(1) The idea of padding with $\bot$ works for the Thompson-Higman group $G_{k,1}$ in general, by using $(A \cup \{\bot\})$-valued logic. The gates that we use include the wire-crossings $\tau_{i,i+1}$.\
(2) In [@BiDistor] we will follow another, more algebraic, approach for defining circuit complexity of elements of $V$. We embed $V$ into a certain finitely generated partial transformation [*monoid*]{} $M$ acting on $\{0,1,\bot\}^*$, and we take the word-length of $\varphi$ in $M$ as the circuit complexity of $\varphi$. We will prove in [@BiDistor] that there are monoids $M$ that, over certain generators, can “simulate” logic gates, and that in such monoids word-length is closely related to circuit complexity.
Since the functions $\varphi: P \to Q$ considered here are elements of $V$, hence bijective, it is natural to also introduce [**bijective $\{0,1,\bot\}$-valued circuits**]{}. A $\{0,1,\bot\}$-valued circuit is said to be bijective iff the gates that make up the circuit are the wire-swap operations $\tau_{i,i+1}$ ($i \geq 1$), and a set of gates derived from the elements of some fixed finite generating set $\Gamma_V$ of $V$. The latter means, more precisely, that the gates derived from $\Gamma_V$ are of the form $((\gamma)_d)^{\bot}$ where each $\gamma$ is a restriction of an element of $\Gamma_V$. Recall that $(\gamma)_d$ (for $d \geq 0$) is the lowering of $\gamma$ (defined at the end of Section 5).
In this paper, unless we specifically mention “bijective” or “$\{0,1,\bot\}$-valued”, the word “circuit” will refer to a general boolean circuit (not necessarily bijective).
[*Comparison between $\{0,1,\bot\}$-valued circuits and boolean (i.e., $\{0,1\}$-valued) circuits:*]{}
In the general (not necessarily bijective) case, a $\{0,1,\bot\}$-valued circuit can be simulated by a traditional binary-logic circuit (e.g., by encoding the ternary values $0, 1, \bot$ by the binary strings $00, 11, 01$ respectively, with $10$ also serving as a code for $\bot$). Thus, there is no essential difference between general $\{0,1,\bot\}$-valued circuits and general boolean circuits.
However, bijective $\{0,1,\bot\}$-valued circuits have greater generality than bijective boolean circuits. First, bijective boolean circuits have input-output functions belonging to ${\it lp}V$ only; on the other hand, input-output functions of bijective $\{0,1,\bot\}$-valued circuits are the paddings of all the elements of $V$. Also, the input-output function of a bijective $\{0,1,\bot\}$-valued circuit is only bijective as a function $P^{\bot} \to Q^{\bot}$, not as a function $\{0,1,\bot\}^m \to \{0,1,\bot\}^n$, whereas the input-output function of a bijective boolean circuit is a permutation of $\{0,1\}^m$ for some $m$.
Moreover, even for $\varphi \in {\it lp}V$ the smallest size of a bijective $\{0,1,\bot\}$-valued circuit computing $\varphi$ is the word length of $\varphi$ over the generators of $V$ (as we shall show in Theorem \[VwordsCircuits\] below), whereas the smallest size of a bijective boolean circuit computing $\varphi$ is the word length of $\varphi$ over the generators of ${\it lp}V$. Thus, we will see that the relation between the two bijective circuit sizes is approximately the [*distortion*]{} of ${\it lp}V$ within $V$. Here the generating set of $V$ is $\Gamma_V \cup \{ \tau_{i,i+1} : 1 \leq i\}$ for any finite generating set $\Gamma_V$ of $V$, and ${\it lp}V$ is generated by $\{N,C,T\} \cup \{ \tau_{i,i+1} : 1 \leq i\}$ as seen before. Finding the distortion of ${\it lp}V$ within $V$ is one of our open problems mentioned at the end of Section 6. Theorem \[VwordsCircuits\] below will give a precise connection between the two kinds of bijective circuit sizes, word lengths in $V$ or in ${\it lp}V$, and the distortion of ${\it lp}V$ in $V$.
Let $\Gamma_{k,1}$ be a finite generating set of $G_{k,1}$, and let $w$ be a word over the generating set $\Gamma_{k,1} \, \cup $ $ \{\tau_{i,i+1} : 1 \leq i \}$ of $G_{k,1}$. The [**length**]{} of $w = a_1 \ldots a_n$ is $|w| \ = \ \sum_{j=1}^n |a_j|$, where $|a_j| = 1$ if $a_j \in \Gamma_{k,1}$ and $|a_j| = i+1$ if $a_j = \tau_{i,i+1}$.
For $\varphi \in G_{k,1}$, the [**word length**]{} of $\varphi$ over $\Gamma_{k,1} \, \cup \, \{\tau_{i,i+1} : 1 \leq i \}$ is the shortest length of any word (over the above generators) that represents $\varphi$.
Observe that $\tau_{i,i+1}$ is counted differently for circuit size than for word length ($\tau_{i,i+1}$ is counted as 1 in circuit size but as $i+1$ in the word length).
The following definition compares bijective padded circuits for elements of ${\it lp}V$ with boolean bijective circuits. Note that in this definition we only consider circuits for computing elements of ${\it lp}V$.
An [*unpadding cost function*]{} from bijective $\{0,1,\bot\}$-valued circuits to bijective binary circuits is any function $U: {\mathbb N} \to {\mathbb N}$ such that the following holds: For all $\varphi \in {\it lp}V$ and any bijective $\{0,1,\bot\}$-valued circuit of size $m$ for $\varphi$ there exists a bijective binary circuit of size $\leq U(m)$ for $\varphi$.
[*The*]{} unpadding cost function $u(.)$ is the minimum unpadding cost function.
Two functions $f_1, f_2: {\mathbb N} \to {\mathbb N}$ are said to be [ **linearly related**]{} iff there are constants $c_0, c_1, c_2$, all $\geq 1$, such that for all $n \geq c_0$: $f_1(n) \leq c_1 \, f_2(c_1 n)$ and $f_2(n) \leq c_2 \, f_1(c_2 n)$.
The functions $f_1, f_2$ are said to be [**polynomially related**]{} iff there are constants $c_0, c_1, c_2$, all $\geq 1$, such that for all $n \geq c_0$: $f_1(n) \leq c_1 \, f_2(c_1 n^{c_1})^{c_1}$ and $f_2(n) \leq c_2 \, f_1(c_2 n^{c_2})^{c_2}$.
The following theorem motivates $\{0,1,\bot\}$-valued circuits, as well as the concept of word length over the generating set $\Gamma_V \, \cup $ $ \{\tau_{i,i+1} : 1 \leq i \}$ for $V$. It again motivates our use of the infinite set $ \{\tau_{i,i+1} : 1 \leq i \}$ for generating $V$, inspite of the fact that $V$ is finitely generated. It also reinforces the connection between Thompson groups and bijective (“reversible”) computing, seen before. In [@BiDistor] we will generalize Theorem \[VwordsCircuits\] to a connection between general circuit size and the word size in the “Thompson monoids” (the latter being a generalization of the Thompson-Higman groups to monoids). We will only state the Theorem for $V$ and for binary or $\{0,1,\bot\}$-valued circuit, although it could easily be generalized to $G_{k,1}$. First a lemma:
\[length\_in\_table\] Let $a_n, \ldots, a_1 \in V$ be given by table and let $\ell$ be the length of the longest words in the tables of $a_n, \ldots, a_1$. Then $a_n, \ldots, a_1$ have restrictions $\alpha_n, \ldots, \alpha_1$, respectively, such that
$\bullet$ ${\rm domC}(\alpha_{j+1}) = {\rm imC}(\alpha_j)$ for $n >j \geq 1$, and
$\bullet$ all words in the tables of $\alpha_j$ ($n \geq j \geq 1$) have lengths $\leq n \, \ell$.
[**Proof.**]{} For $\varphi \in V$ we will describe the table of $\varphi$ as a set of input-output pairs, of the form $\{(u_i, v_i) : i = 1, \ldots, I\}$. We also use tables to represent elements of $V$ in non-maximally extended form; we will mention explicitly when we assume maximal extension.
[**Claim.**]{}
* Let $a_i \in V$ (for $i = 1, \ldots, n$) be given by tables $\{(x_j^{(i)},y_j^{(i)}) : j = 1, \ldots, r\}$, not necessarily in reduced form. Thus ${\rm domC}(a_i) = \{ x_j^{(i)} : j = 1, \ldots, r\}$, and ${\rm imC}(a_i) = \{ y_j^{(i)} : j = 1, \ldots, r\}$. We assume that ${\rm domC}(a_{i+1}) = {\rm imC}(a_i)$ for $i = 1, \ldots, n-1$ Let $\ell$ be an upper bound on the length of all the words in $\bigcup_{i=1}^n {\rm domC}(a_i) \cup {\rm imC}(a_i)$. For $j = 1, \ldots, r$, let $S_j = \{ s_{j,k} : 1 \leq k \leq |S_j|\}$ be a finite maximal prefix code over $\{0,1\}$.*
Then ${\alpha}_i$ (for $i = 1, \ldots, n$), defined by the table $\{(x_j^{(i)}s_{j,k}, \ y_j^{(i)}s_{j,k}) : 1 \leq j \leq r, $ $1 \leq k \leq |S_j|\}$, is a restriction of $a_i$ satisfying:
$\bullet$ ${\rm domC}(\alpha_{i+1}) = {\rm imC}(\alpha_i)$ for $n > i \geq 1$, and
$\bullet$ all words in the tables of $\alpha_i$ ($n \geq i \geq 1$) have lengths $\leq $ ${\rm max}\{|s_i| : 1 \leq i \leq r\} + \ell$.
Proof of the Claim: Since $\{ x_j^{(i+1)} : j = 1, \ldots, r\} = $ ${\rm domC}(a_{i+1}) = {\rm imC}(a_i) = \{ y_j^{(i)} : j = 1, \ldots, r\}$, it follows immediately that $\{ x_j^{(i+1)} s_{j,k} : 1 \leq j \leq r, 1 \leq k \leq |S_j|\} = $ $\{ y_j^{(i)} s_{j,k} : 1 \leq j \leq r, 1 \leq k \leq |S_j|\}$. Hence ${\rm domC}(\alpha_{i+1}) = {\rm imC}(\alpha_i)$.
Also, $|x_j^{(i)} s_{j,k}| \leq \ell + {\rm max}\{|s_i| : 1 \leq i \leq r\}$ (and similarly for $|y_j^{(i)} s_{j,k}|$), hence we have the claimed length bound. This proves the Claim.
Let us now prove Lemma \[length\_in\_table\] by induction on $n$. The Lemma is obvious when $n=1$. Given $a_i \in V$ (for $i = n, \ldots, 1$, with $n \geq 2$), we use the Lemma by induction for $a_{n-1}, \ldots, a_1$. So we can assume that ${\rm domC}(a_{i+1}) = {\rm imC}(a_i)$ for $n-1 > i \geq 1$, and all words in the tables of $a_i$ ($n-1 \geq i \geq 1$) have lengths $\leq (n-1) \, \ell$. Let us denote ${\rm domC}(a_{i+1}) = {\rm imC}(a_i)$ by $P_i$; so we have $P_0 \stackrel{a_1}{\to} P_1 \stackrel{a_2}{\to} \ \ldots \ $ $ \stackrel{a_{n-2}}{\longrightarrow} P_{n-2}$ $ \stackrel{a_{n-1}}{\longrightarrow} P_{n-1}$ .
We consider the product $a_n \cdot (a_{n-1} \ldots a_1)$. We will find a restriction $\alpha_n$ of $a_n$, and a restriction $\overline{a_{n-1} \ldots a_1}$ of $a_{n-1} \ldots a_1$, such that ${\rm domC}(\alpha_n) = {\rm imC}(\overline{a_{n-1} \ldots a_1})$. We also want to restrict $a_{n-1}, \ldots, a_1$ to functions $\alpha_{n-1}, \ldots, \alpha_1$ such that ${\rm domC}(\alpha_{i+1}) = {\rm imC}(\alpha_i)$ for $n-1 > i \geq 1$. Two cases arise:
Case 1: Every word in $P_{n-1}$ has length $\geq \ell$.
By the assumptions of Lemma \[length\_in\_table\], every word in the table of $a_n$ has length $\leq \ell$. Therefore, all we need to do to obtain $\alpha_n$ and $\alpha_{n-1}, \ldots, \alpha_1$ is to restrict $a_n$ so that ${\rm domC}(\alpha_n)$ becomes $P_{n-1}$. No restriction of $a_{n-1} \ldots a_1$ is needed, i.e., $\alpha_i = a_i$ for $i = n-1, \ldots, 1$, and $\overline{a_{n-1} \ldots a_1} = a_{n-1} \ldots a_1$. Hence the longest word in $P_0, P_1, \ldots, P_{n-1}$ has length $\leq (n-1) \, \ell$.
The longest word in the table of $\alpha_n$ has length $\leq {\rm max}\{|p| : p \in P_{n-1}\} + \ell$, by the Claim (applied to $\alpha_n$ and $a_{n-1} \ldots a_1$). Hence the longest word in the table of $\alpha_n$ has length $ \leq (n-1) \, \ell + \ell = n \, \ell$.
Case 2: Some word in $P_{n-1}$ has length $< \ell$.
We restrict $a_{n-1}$ so as to make all words in ${\rm imC}(\alpha_{n-1})$ have length $\geq \ell$, as follows. For any $y_j \in P_{n-1}$ with $|y_j| < \ell$ we consider the finite maximal prefix code $S_j = \{0,1\}^{\ell - |y_j|}$. We restrict $a_{n-1}$ in such a way that $y_j$ is replaced by $y_j \cdot S_j$, i.e., $P_{n-1}$ becomes $(P_{n-1} - \{y_j\}) \, \cup \, y_j \cdot S_j$. Note that all words in $y_j \cdot S_j$ have length $\ell$. After every word in $P_{n-1}$ of length $< \ell$ has been replaced, let $\overline{P}_{n-1}$ be the resulting finite maximal prefix code. Now we apply the Claim in order to restrict all of $a_{n-1}, \ldots, a_1$. As a result, each $\alpha_i$ (for $i = n-1, \ldots, 1$) receives a table with words of length $\leq (n-1) \, \ell + {\rm max}\{|s| : s \in \bigcup_j S_j\}$ $\leq (n-1) \, \ell + \ell = n \, \ell$. Note that in these restrictions, the length of the words in $P_{n-1}$ only increases for the very short words (namely, words of length $< \ell$ are replaced by words of length $\ell$). Hence, after restriction, the words in $\overline{P}_{n-1}$ still have length $\leq (n-1) \, \ell$.
Next we restrict $a_n$, as in case 1. Since after restriction, the words in $\overline{P}_{n-1}$ have length $\leq (n-1) \, \ell$, the longest word in the table of $\alpha_n$ has length $ \leq (n-1) \, \ell + \ell = n \, \ell$. $\Box$
\[VwordsCircuits\] [**(1)**]{} For the elements $\varphi \in V$, the minimum [*size*]{} of bijective $\{0,1,\bot\}$-valued circuits that compute $\varphi$ is polynomially related to the [*word length*]{} of $\varphi$ in $V$ (over $\Gamma_V \, \cup $ $ \{\tau_{i,i+1} : 1 \leq i \}$). More precisely, there are constants $c_1, c_2 >0$ such that $s_{\varphi} \leq c_1 \, |\varphi|_{_V}^2$, and $|\varphi|_{_V} \leq c_2 \, s_{\varphi}$, where $|\varphi|_{_V}$ is word length of $\varphi \in V$ over $\Gamma_V \, \cup $ $ \{\tau_{i,i+1} : 1 \leq i \}$, and $s_{\varphi}$ is the $\{0,1,\bot\}$-valued bijective circuit size of $\varphi$.
[**(2)**]{} For the elements $\varphi \in {\it lp}V$, the minimum size of bijective binary circuits is polynomially related to the word length of $\varphi$ in ${\it lp}V$ (over $\{ N,C,T\} \, \cup \, \{\tau_{i,i+1} : 1 \leq i \}$). More precisely, the word length $|\varphi|_{_{{\it lp}V}}$ over $\{ N,C,T\} \, \cup \, \{\tau_{i,i+1} : 1 \leq i \}$, and the binary circuit size $b_{\varphi}$ of $\varphi$ satisfy: $b_{\varphi} \leq c_1 \, |\varphi|_{_{{\it lp}V}}^2$, and $|\varphi|_{_{{\it lp}V}} \leq c_2 \, b_{\varphi}$ (for some constants $c_1, c_2 >0$).
[**(3)**]{} The [**distortion**]{} function $d(.)$ of ${\it lp}V$ in $V$ (over the generators mentioned above), and the [**unpadding cost**]{} function $u(.)$ for bijective $\{0,1,\bot\}$-valued circuits, are polynomially related. More precisely, for some constants $c, c'>0$ and for all $x > 0$ we have $u(x) \leq c \ d(c \, x)^2$ and $d(x) \leq c' \ u(c' \, x^2)$.
[**Proof.**]{} (1) For $\varphi \in V$ let $s_{\varphi}$ be the circuit size of $\varphi$ over $\Gamma_V^{\pm 1} \, \cup \, \{\tau_{i,i+1}: 1 \leq i \}$, let $|\varphi|$ be the word length over $\Gamma_V^{\pm 1} \, \cup \, \{\tau_{i,i+1}: 1 \leq i \}$.
$\bullet$ Proof that $s_{\varphi} \leq c_1 \, |\varphi|^2$ (for some constant $c_1>0$): Let $w = a_1 \ldots a_n$ be a shortest word that represents $\varphi$, where $a_j \in \Gamma_V^{\pm 1} \, \cup \, \{\tau_{i,i+1}: 1 \leq i \}$ for $1 \leq j \leq n = |w| = |\varphi|$. We restrict the generators $a_j$ as in Lemma \[length\_in\_table\] so that they can be composed, and $a_j$ will only have bitstrings of length $\leq c_1 \, |\varphi|$ in its table. Thus, the word $w$ becomes a $\{0,1,\bot\}$-valued circuit consisting of the $n = |\varphi|$ operations $a_j$ ($1 \leq j \leq n$), and each $a_j$ has $\leq c_1 \, |\varphi|$ wires; so the circuit for $\varphi$ has size $\leq c_1 \, |\varphi| \, |\varphi|$.
$\bullet$ Proof that $|\varphi| \leq c_2 \, s_{\varphi}$ (for some constant $c_2 >0$): Consider any smallest bijective circuit $C$ over $\Gamma_V^{\pm 1} \, \cup \, \{\tau_{i,i+1}: 1 \leq i \}$, of size $s_{\varphi}$, computing $\varphi$. This circuit is a sequence $(a_1, \ldots, a_n)$ where $n = |C| = s_{\varphi}$. Each $a_j$ is either of the form $\tau_{i,i+1}$, or $a_j$ is the padding of a restriction of $(\gamma)_d$ with $d \geq 0$ and $\gamma \in \Gamma_V^{\pm 1}$. Hence, the sequence $(a_1, \ldots, a_n)$ is a word of length $s_{\varphi}$ representing $\varphi$. To obtain a word over $\Gamma_V^{\pm 1} \, \cup \, \{\tau_{i,i+1}: 1 \leq i \}$ we express the lowering operation in terms of position transpositions, as at the end of Section 5; then $(\gamma)_d$ becomes $\pi^{-1} \, \gamma \, \pi$, where $\pi$ is the composition of $\leq 2 \, m$ position transpositions of the form $\tau_{i,d+i}$ or $\tau_{i,m+i}$. Here $m$ is the length of the longest word in any of the tables for the elements $\gamma \in \Gamma_V$; since $\Gamma_V$ is fixed and finite, $m$ is a constant. A transposition $\tau_{i,d+i}$ can be written as the composition of $< 2d$ transpositions of the form $\tau_{j,j+1}$. Let $((\gamma_j)_{d_j} : j = 1, \ldots, J)$ be the list of all the lowered gates that occur in the circuit $C$; then $\sum_{j=1}^J d_j < |C|$, since for each $(\gamma_j)_{d_j}$ there are $d_j$ wires in $C$ that are output wires of other gates (or that are inputs of $C$), and that are counted as part of the size of $C$. Thus we obtain a word of length $< c_2 \, s_{\varphi}$ (for some constant $c_2 > 0$), representing $\varphi$.
\(2) The proof is very similar to the proof of (1).
\(3) By (1) and (2) and by the definition of distortion we have: $c \ \sqrt{b_{\varphi}} \ \leq \ |\varphi|_{_{{\it lp}V}} \ \leq \ $ $d(|\varphi|_{_V}) \ \leq \ d(c' \ s_{\varphi})$, hence $b_{\varphi} \ \leq \ c'' \ d(c' \ s_{\varphi})^2$. Here, $c, c', c''$ are constants. By the definition of the unpadding cost function it follows that $u(x) \leq c'' \ d(c' \ x)^2$.
Also by (1) and (2) and by the definition of the unpadding cost function we have: $c \, |\varphi|_{_{{\it lp}V}} \ \leq \ b_{\varphi} \ \leq \ $ $u(s_{\varphi}) \ \leq \ u(c' \ |\varphi|_{_V}^2)$, hence $|\varphi|_{_{{\it lp}V}} \ \leq \ c'' \ u(c' \ |\varphi|_{_V}^2)$. Here, $c, c', c''$ are constants. By the definition of distortion it follows that $d(x) \leq c'' \ u(c' \, x^2)$. $\Box$
\[inverting\_f\] Consider any element of the Thompson group $F$, represented by a bijection $f: P \to Q$ that preserves the dictionary order, where $P$ and $Q$ are finite maximal prefix codes. If $f$ can be computed by a $\{0,1,\bot\}$-valued circuit of size $s$ then $f^{-1}: Q \to P$ can be computed by a combinational circuit size $m(m+1) \, s + O(m^2 n)$, where $m = {\rm max}\{|p| : p \in P\}$ and $n = {\rm max}\{|q| : q \in Q\}$.
Moreover, a circuit for $f^{-1}$ can be found from a circuit for $f$ deterministically in polynomial time in terms of $s, m, n$.
[**Proof.**]{} Suppose $f^{\bot}(x \bot^{m-|x|}) = y \bot^{n-|y|}$, and $y \bot^{n-|y|}$ is given. The idea for inverting $f^{\bot}$ is simple: Since $f$ preserves the dictionary order we can find $x \bot^{m-|x|}$ by adapting the classical [*binary search*]{} algorithm. This algorithm is usually used for searching in a sorted array; but it works in a similar way for inverting any order-preserving map.
A few technical details have to be discussed before we give an algorithm for inverting $f^{\bot}$. For many strings $z \in \{0,1,\bot\}^n$ there is no inverse image under $f^{\bot}$; in that case our inversion algorithm will output $\bot^m$. For example, $z \not \in \{0,1\}^*\bot^*$ has no inverse. Also, since the range of $f^{\bot}$ is $Q^{\bot} \cup \{\bot^n\}$ where $Q$ is a finite maximal prefix code we have: If $y \bot^{n-|y|}$ has an inverse then there is no inverse for any strict prefix of $y$; i.e., if $w \in \{0,1\}^*$ is a strict prefix of $y$ then $w \bot^{n-|w|}$ has no inverse. On the other hand, for every $v \in \{0,1\}^n$, there exists exactly one prefix $y$ of $v$ such that $y \bot^{n-|y|}$ has an inverse. Similarly, for every $u \in \{0,1\}^m$, there exists exactly one prefix $x$ of $u$ such that $f^{\bot}(x \bot^{m-|x|})$ $\neq \bot^n$.
The binary search can be pictured on the complete binary tree with vertex set $\{0,1\}^{\leq m}$, with root $\varepsilon$ (the empty word), and leaves $\{0,1\}^m$; the children of a vertex $v \in \{0,1\}^{< m}$ are $v0$ and $v1$. The search uses a variable vertex $v$, initialized to $\varepsilon$, and proceeds from $v$ to $v0$ or $v1$, until success, or until $v$ becomes a leaf.
[Algorithm]{} (for inverting $f^{\bot}$)
For any input $z \in \{0,1,\bot\}^n$ it is easy to check whether $z \not\in \{0,1\}^*\bot^*$; in that case the output is $\bot^m$. Assume from now on that the input is of the form $y \bot^{n-|y|}$, with $y \in \{0,1\}^*$, $|y| \leq n$. Let $v \in \{0,1\}^{\leq m}$ be a bitstring, initialized to $v = \varepsilon$ (the empty string). Below, $<_{_d}$ and $>_{_d}$ refer to the dictionary order.
[repeat the following until the exit command]{}\
[begin]{}
find the prefix $x$ of $v \, 1 \, 0^{m-1-|v|}$ (if $|v| < m$) or the prefix $x$ of $v$ (if $|v| = m$), such that
$f^{\bot}(x \bot^{m-|x|}) \neq \bot^n$ (this is done by trying all prefixes of $v \, 1 \, 0^{m-1-|v|}$, respectively of $v$);
[if]{} $f^{\bot}(x \bot^{m-|x|}) = y \bot^{n-|y|}$ [then]{} return $x \bot^{m-|x|}$ as output, and [exit]{};
[if]{} $|v| =m$ [then]{} return $\bot^m$ as output, and [exit]{};
[if]{} $f^{\bot}(x \bot^{m-|x|}) <_{_d} \ y \bot^{n-|y|}$ and $|v|<m$ [then]{} replace $v$ by $v1$;
[else]{} (i.e., when $f^{\bot}(x \bot^{m-|x|}) >_{_d} \ y \bot^{n-|y|}$ and $|v|<m$) replace $v$ by $v0$;
[end (of repeat loop)]{};
[end (of algorithm).]{}
Let us show that this program can be implemented by an acyclic circuit of size $m(m+1) s + O(m^2 n)$. The loop of the program is executed at most $m$ times, and each execution of the loop will be implemented as one of $m$ stages of the complete circuit.
Each execution of the loop takes at most $(m+1)s + O(mn)$ gates.
The first part of the loop (namely, to “find the prefix $x$”) requires $m$ copies of the circuit of $f^{\bot}$, each of which is followed by $O(n)$ gates to check equality with $\bot^n$, followed by a tree of $n$ [and]{}-gates. Moreover, recall that when we want to apply the same type of gate to different variables (wires) we need to permute wires (using bit position transpositions $\tau_{i,j}$). Similarly, permutations may need to be applied to the output wires of a gate. This adds at most a constant number of operations for each gate. So the first part of the loop uses $ms+ O(mn)$ gates.
The first [if]{} condition requires another copy of the circuit of $f^{\bot}$, followed by $O(n)$ gates to compare the result with $y \bot^{n-|y|}$ for equality and to check for $<_{_d}$ or $>_{_d}$ in the dictionary order. The $<_{_d}$- or $>_{_d}$-comparison of two strings of the same length can be done by a finite automaton, reading both strings in parallel from left to right; if the inputs are restricted to strings of length $n$, this automaton can then be turned into a [*prefix circuit*]{} (of Ladner and Fischer [@LadnerFischer]). The Ladner-Fischer circuit consists of $\leq 4n$ copies of a gate that implements the (fixed) transition function of the finite automaton. The prefix circuit uses fan-out $< n$; however, there is also a bounded-fan-out design for the prefix circuit, using just $< 9n$ gates (see p. 205 of [@LakshDhall]). Moreover, applying the same gate to different variables first requires some permutations of wires; this introduces a constant factor (since gates have a fixed number of input-output wires, so only a fixed number of wires are permuted back and forth). Checking whether $|v| = m$ is equivalent to checking absence of $\bot$, which requires $O(n)$ gates. So overall the if-statements require $s + O(n)$ gates.
Finally, the above description amounts to a polynomial-time procedure for producing the circuit that implements $(f^{\bot})^{-1}$. $\Box$
As a consequence, $(f^{\bot})^{-1}$ is not much harder to compute than $f^{\bot}$ itself. So, without need to define the concept of a one-way function in detail we can conclude that for any reasonable definition of “one-way function” we have:
The Thompson group $F$ does not contain any one-way functions.
Recall that in our algorithm for finding the ${\it lp}V \cdot F$ factorization of $\varphi \in V$, the element $\varphi$ is first restricted so as to make ${\rm imC}(\varphi) = \{0,1\}^n$. We will show next that this restriction does not increase circuit complexity much, and that we can find a circuit for certain restrictions. On the other hand, we saw in Theorem \[coNP\_circuits\](3) that the opposite operation, namely finding the maximal extension, is hard.
\[makingImC\_n\] Every $\varphi \in V$ has a restriction $\Phi$ such that ${\rm imC}(\Phi) = \{0,1\}^n$, and such that the circuit size of $\Phi$ is only polynomially larger than the circuit size of $\varphi$.
More precisely, assume $\varphi^{\bot}$ has a circuit of size $s$, with $m$ input variables and $n$ output variables (over $\{0,1,\bot\}$). Then the restriction $\Phi$ with ${\rm imC}(\Phi) = \{0,1\}^n$ has a circuit of size $\leq 4s \, (n+m+1)$, with $n+m$ input variables and $n$ output variables. Moreover, such a circuit for $\Phi^{\bot}$ can be found from the given circuit for $\varphi^{\bot}$ deterministically in polynomial time (as a function of $s, m, n$); i.e., there is a polynomial-time reduction from the problem of finding a circuit for $\Phi$ to the problem of finding a circuit for $\varphi$ (for $\{0,1,\bot\}$-valued circuits).
[**Proof.**]{} We can view $\varphi$ as a bijection $P \to Q$ where $P,Q \subset \{0,1\}^*$ are finite maximal prefix codes. Since the circuit for $\varphi^{\bot}$ has $n$ output variables, we have $n = {\rm max}\{|y| : y \in Q\}$. Let $\Phi: P_1 \to \{0,1\}^n$ be the restriction of $\varphi$ with image code $\{0,1\}^n$, where $P_1$ is the finite maximal prefix code obtained when $\varphi$ is restricted to make the image code $\{0,1\}^n$. Let $m = {\rm max}\{|x| : x \in P\}$. Thus, all words in the finite maximal prefix code $P_1$ have length $\leq n+m$. We now construct a $\{0,1, \bot\}$-valued circuit for $\Phi^{\bot}$, with $n+m$ input variables and $n$ output variables. On an input $x \in \{0,1, \bot\}^{n+m}$ the circuit behaves as follows:
$\bullet$ If $x \notin \{0,1\}^*\bot^*$, the output is $\Phi(x) = \bot^n$.
To check whether $x \notin \{0,1\}^*\bot^*$ we consider all $n+m-1$ pairs $(x_i,x_{i+1})$ of neighboring input variables (for $i=1, \ldots, n+m-1$) and check whether any of them have values $(x_i,x_{i+1}) = (\bot,0)$ or $= (\bot,1)$, using $n+m-1$ gates. To produce the output $\bot^n$ in that case, the $n+m-1$ gates above feed into a tree of [or]{} gates whose output is 1 iff $(\bot,0)$ or $(\bot,1)$ occurs anywhere in input pairs. The [or]{}-tree and the output $\bot^n$ require $< 2(n+m)$ gates. Thus so far we have $< 3(n+m)$ gates in total.
$\bullet$ If $x \in \{0,1\}^*\bot^*$, since $\Phi^{\bot}$ has $m+n$ input wires, we write $x = u \, \bot^i$ with $u \in \{0,1\}^{n+m-i}$. We look at each prefix $p$ of $u = pz$, in order of increasing length $|p| = 0, 1, \ldots, m+n-i$, and feed $p \, \bot^{m-|p|}$ into $\varphi^{\bot}$.
- If $\varphi^{\bot}(p \, \bot^{m-|p|}) = \bot^n$, we ignore $p$ and look at the next prefix of $u$.
- If $\varphi^{\bot}(p \, \bot^{m-|p|}) = q \, \bot^{m-|q|}$ for some $q \in \{0,1\}^*$, we conclude that $\varphi(p) = q$ and $\varphi(u) = \varphi(pz) = qz$. Hence, if $|z| = n - |q|$ we produce the output $\Phi(u \, \bot^i) = qz \in \{0,1\}^n$; so $\Phi$ agrees with $\varphi$ and has imC$(\Phi) = \{0,1\}^n$. If $|z| \neq n - |q|$ we produce the output $\Phi(u \, \bot^i) = \bot^n$. (No new prefixes of $u$ will be considered.)
In the above construction, the circuit of $\varphi^{\bot}$ is repeated $m+n+1$ times, since an input of length $m+n$ has $\leq m+n+1$ prefixes. So this part of the circuit has size $s \, (m+n+1)$. We need another $3n \, (m+n+1)$ gates to combine the outputs of the $(m+n+1)$ copies of the $\varphi^{\bot}$-circuit: If one of the $\varphi^{\bot}$-circuits produces an output in $\{0,1\}^n$, that output has to be the final output; if all the copies of the $\varphi^{\bot}$-circuit produce $\bot^n$, then $\bot^n$ should be the final output.
Finally, the total circuit for $\Phi^{\bot}$ has size $\leq 3(n+m) + s \, (m+n+1) + 3n \, (m+n+1)$ $ \leq (s+3n+3)(m+n+1)$ $ \leq 4s \, (m+n+1)$ (the last “$\leq$” holds since $s \geq m+n$ and $m,n \geq 1$). The above description of the construction of a circuit for $\Phi^{\bot}$ is a deterministic algorithm whose running time is a polynomial in $s$. $\Box$
An immediate consequence if Lemma \[makingImC\_n\] is the following:
\[f\_imC\_01m\] Assume $f \in F$ has a $\{0,1,\bot\}$-valued circuit of size $\leq s$, with $m$ input variables and $n$ output variables. Then the restriction of $f$ with ${\rm imC}(f) = \{0,1\}^n$, i.e., the restriction of $f$ that makes $f$ a rank function ${\rm rank}_P(.)$, has a $\{0,1,\bot\}$-valued circuit of size $\leq 4s \, (m+n+1)$.
In other words, representing elements of $F$ by rank functions does not lead to a large increase in circuit complexity.
\[complexity\_of\_factors\] Let $\varphi$ be an element of the Thompson group $V$, and let $\varphi = \pi \cdot f$ be its ${\it lp}V \cdot F$ factorization. Let $\varphi: P \to \{0,1\}^n$ be a representation of $\varphi$ by a bijection from a finite maximal prefix code $P \subseteq \{0,1\}^{\leq m}$ onto $\{0,1\}^n$. Suppose that the rank function of $P$ can be computed by a circuit of size $\leq s$.
Then $f$ has circuit complexity $\leq s$, and the circuit complexities of $\varphi$ and of $\pi$ differ by at most $m(m+1) \, s + O(m^2 n)$. The circuit complexities of $\varphi^{-1}$ and $\pi^{-1}$ differ by at most $s$.
Moreover, the circuits for $f$ and $\pi$ can be found in deterministic polynomial time.
[**Proof.**]{} We apply our ${\it lp}V \cdot F$ factorization algorithm. Since $\varphi$ already has imC$(\varphi) = \{0,1\}^n$, we have $f = {\rm rank}_P(.)$, and hence by assumption, $f$ has circuit complexity $\leq s$.
To obtain a circuit for $\pi = \varphi \, f^{-1}$ we use Theorem \[inverting\_f\] to obtain a circuit for $f^{-1}$ of size $\leq m(m+1) \, s + O(m^2 n)$, where $m = {\rm max}\{|p| : p \in P\}$; then we compose the circuit for $f^{-1}$ with the circuit for $\varphi$.
To obtain a circuit for $\varphi = \pi \cdot f$ from a circuit for $\pi$ we just compose the circuit for $f$ and the circuit for $\pi$. $\Box$
A consequence of Proposition \[complexity\_of\_factors\] is the following. If an element of $V$ has a representation $\varphi: P \to \{0,1\}^n$ (for some $n>0$), and if $P$ is a finite maximal prefix code with easy rank function, then $\varphi$ and of $\pi$ have similar circuit complexities; $\varphi^{-1}$ and $\pi^{-1}$ also have similar circuit complexities. Thus we have:
If there exists a one-way bijection $\varphi: P \to \{0,1\}^n$ (for some $n>0$), where $P$ is a finite maximal prefix code with easy rank function, then there exists a one-way permutation $\pi$ of $\{0,1\}^n$.
coNP-complete and $\#{\mathcal P}$-complete problems in the Thompson groups
---------------------------------------------------------------------------
The following coNP-completeness results are similar to the well-known coNP-completeness of questions about circuits, except that here we deal with circuits that compute bijections, in the sense defined at the beginning of this section.
\[coNP\_circuits\] The following decision problems are [**coNP-complete**]{}:\
(1) Given two $\{0,1,\bot\}$-valued bijective circuits, do they compute the same element of $V$?\
(2) Given two $\{0,1,\bot\}$-valued bijective circuits for computing elements $\psi, \varphi \in V$, is $\psi$ the maximal extension of $\varphi$?\
(3) Given a $\{0,1,\bot\}$-valued bijective circuit, does it compute the identity element of $V$?
The problems remain coNP-complete when the given $\{0,1,\bot\}$-valued circuits are general (not necessarily bijective).
[**Proof.**]{} Let us first check that these problems are in coNP. Problems (1), (3) and (4) are variants of the classical circuit equivalence problem. For problem (2), we can check in coNP whether $\psi$ and $\varphi$ represent the same element of $V$. To check in NP whether $\psi$ is [*not*]{} maximally extended, guess entries $(x0,y0), (x1,y1)$ in the table of $\psi$; the lengths of $x$ and $y$ are no larger than the size of the given circuit for $\psi$, and the fact that $(x0,y0)$ and $(x1,y1)$ are in the table of $\psi$ can be checked rapidly using the circuit for $\psi$.
Hardness: Problem (3) is a special case of (2) and of (1) (letting $\psi$ be the identity map with domain and image codes consisting of just the empty word), so (2) and (1) are at least as hard as (3). The hardness of (3) is a consequence of the fact that the word problem of $G_{3,1}$ over the generating set $\Gamma_{3,1} \cup $ $\{\tau_{i,j} : 1 \leq i < j\}$ is coNP-complete (proved in [@BiCoNP]), and the fact (proved in Theorem \[VwordsCircuits\]) that every word over $\Gamma_{3,1} \cup $ $\{\tau_{i,j} : 1 \leq i < j\}$ has a $\{0,1,\bot\}$-valued circuit whose size is linearly bounded by the size of the word. $\Box$
Proposition \[complexity\_of\_factors\] shows that under certain conditions the ${\it lp}V \cdot F$ factorization is easy to find. The next Theorems show that in general, finding the ${\it lp}V \cdot F$ factorization is $\#{\mathcal P}$-hard, even when circuits for the rank functions of the domain code and image code are given.
To define the class $\#{\mathcal P}$ we consider functions of the form $f: A^* \to \{0,1\}^*$, where $A$ is a finite alphabet, and elements of $\{0,1\}^*$ are interpreted as non-negative integers in binary representation. Intuitively, for a function $f$ in $\#{\mathcal P}$ and for $x \in A^*$, $f(x)$ is the number of ways a relation that is parameterized by $x$ can be satisfied. More precisely we will use the following definition of the $\#{\mathcal P}$; see e.g. [@Handb].
\[numberP\] A function $f: A^* \to \{0,1\}^*$ is in $\#{\mathcal P}$ iff there is a relation $R \subseteq A^* \times B^*$ (where $B$ is a finite alphabet) such that\
(1) for all $x \in A^*$: $f(x) \ = \ |\{ w \in B^* : (x,w) \in R\}|$, with $f(x) \in \{0,1\}^*$ interpreted as an integer;\
(2) $R$ is in ${\mathcal P}$ (deterministic polynomial time);\
(3) $R$ is polynomially balanced (also called “polynomially honest”); i.e., there is a polynomial $p(.)$ such that for all $(x,w) \in R$, $|x| \leq p(|w|)$ and $|w| \leq p(|x|)$.
\[numberP\_rank\] [**(Ranking problem for finite maximal prefix codes)**]{}\
The following problem is $\#{\mathcal P}$-complete.\
[Input:]{} A $\{0,1,\bot\}$-valued circuit that accepts a finite maximal prefix code $P \subset \{0,1\}^*$, and $x \in P$.\
[Output:]{} The rank of $x$ in $P$ according to dictionary order.
[**Proof.**]{} The problem is clearly in $\#{\mathcal P}$ since ${\rm rank}_P(x)$ is the number of words $w \in B^*$ (here $B = \{0,1\}$) satisfying the relation “$w \in P$ and $w <_d x$”. Moreover, the prefix code $P$ is given by a circuit, whose size is counted as part of the input size of the problem, so the relation “$w \in P$ and $w <_d x$” can be verified in deterministic polynomial time.
Next, we will reduce the $\#{\mathcal P}$-complete problem \#SAT to our problem. For a boolean formula $\beta(x_1, \ldots, x_n)$ with $n$ boolean variables, let $T \subseteq \{0,1\}^n$ be the set of truth-value assignments that make $\beta$ true. Although $T$ is a finite prefix code, $T$ is not maximal, and the cardinality $|T|$ is not necessarily a power of 2; however, finding $|T|$, given $\beta$, is precisely the $\#{\mathcal P}$-complete problem \#SAT. We will use $T$ to construct a finite maximal prefix code $P$ (with $|P|$ a power of 2), whose ranking function determines $|T|$. We use the notation ${\overline T} = \{0,1\}^n - T$. Let
$P_T \ = \ 00{\overline T} \ \cup \ 00T0 \ \cup \ 00T1 \ \cup \ $ $01\, \{0,1\}^n \ \cup \ 1T \ \cup \ $ $ 1 {\overline T}0 \ \cup \ 1 {\overline T}1$.
Then $P_T$ is a finite maximal prefix code of cardinality $|P_T| = 2^{n+2}$. Membership in $P_T$ is easily decided by the formula $\beta$.
Finally, $|T|$ is easily derived from the rank of $00 1^n$ or of $001^n1$ in $P_T$. Indeed, if $1^n \in {\overline T}$ then ${\rm rank}_{P_T}(001^n) +1 = |T| \cdot 2 + |{\overline T}| = |T| + 2^n$; if $1^n \in T$ then ${\rm rank}_{P_T}(001^n1) +1 = |T| + 2^n$. Hence, $|T|$ can easily be obtained from ${\rm rank}_{P_T}(01^n)$ or ${\rm rank}_{P_T}(01^n1)$; the numbers are written in binary, so the representation of $2^n$ is not large. $\Box$
\[numberP\_fact1\] [**(${\it lp}V \cdot F$ factorization problem, given $\Phi: P \to \{0,1\}^n$)**]{}\
The following problem is $\#{\mathcal P}$-complete.\
[Input:]{} A $\{0,1,\bot\}$-valued circuit that computes a bijection $\Phi: P \to \{0,1\}^n$ (where $P$ is a finite maximal prefix code over $\{0,1\}$), and $x \in P$.\
[Output:]{} The rank of $x$ in $P$ according to dictionary order. (Recall that ${\rm rank}_P(.)$ is the $F$-part in the ${\it lp}V \cdot F$ factorization of $\Phi$.)
The problem remains $\#{\mathcal P}$-complete if we assume that circuits for both $\Phi$ and $\Phi^{-1}$ are given. Also, evaluating $\pi$ or $\pi^{-1}$ is $\#{\mathcal P}$-complete (where $\Phi(.) = \pi \, f(.)$ is the ${\it lp}V \cdot F$ factorization).
[**Proof:**]{} The problem is in $\#{\mathcal P}$ because the circuit for $\Phi$ can also be used to test membership in $P$. To show $\#{\mathcal P}$-hardness, let $P_T$ be as in Theorem \[numberP\_rank\] above, where $T \subseteq \{0,1\}^n$ is the set of truth value assignments that make a given boolean formula $\beta$ true; again, ${\overline T}$ denotes $\{0,1\}^n - T$.
$P_T \ = \ 00{\overline T} \ \cup \ 00T0 \ \cup \ 00T1 \ \cup \ $ $01\, \{0,1\}^n \ \cup \ 1T \ \cup \ $ $ 1 {\overline T}0 \ \cup \ 1 {\overline T}1$.
Let $\Phi: P_T \to \{0,1\}^{n+2}$ be the bijection defined as follows for all $x \in \{0,1\}^n$:
$00x \in 00{\overline T} \ \longmapsto \ 11x \in 11{\overline T}$
$00x0 \in 00T0 \ \longmapsto \ 0x0 \in 0T0$
$00x1 \in 00T1 \ \longmapsto \ 0x1 \in 0T1$
$01x \in 01 \, \{0,1\}^n \ \longmapsto \ 10x \in 10 \, \{0,1\}^n$
$1x0 \in 1 {\overline T}0 \ \longmapsto \ 0x0 \in 0 {\overline T}0$
$1x1 \in 1 {\overline T}1 \ \longmapsto \ 0x1 \in 0{\overline T}1$
$1x \in 1 T \ \longmapsto \ 11x \in 11 T$.
Clearly $\Phi$ and $\Phi^{-1}$ can easily be computed from the boolean formula $\beta$, and they have small circuits that can be derived from the boolean formula $\beta$.
Let $\Phi = \pi \, f$ be the ${\it lp}V \cdot F$ factorization of $\Phi$; then $f = {\rm rank}_{P_T}(.)$. We saw in Theorem \[numberP\_rank\] above that evaluating ${\rm rank}_{P_T}(.)$ is a $\#{\mathcal P}$-complete problem. Thus by the reduction of $f$ to $f^{-1}$ in Theorem \[inverting\_f\], the problem of computing $f^{-1}$ is also $\#{\mathcal P}$-complete.
To show that the evaluations of $\pi$ and $\pi^{-1}$ are $\#{\mathcal P}$-hard, note that $f = \pi^{-1} \Phi$ and $f^{-1} = \Phi^{-1} \pi$; since $\Phi$ and $\Phi^{-1}$ are easy to evaluate, this reduces the $\#{\mathcal P}$-complete evaluation problems for $f$ and $f^{-1}$ to the evaluation of $\pi^{-1}$, respectively $\pi$. $\Box$
The above Theorem means that ranking in $P_T$ according to the dictionary order is hard, but there may exist another bijection $P_T \to \{0,1\}^n$, namely $\Phi$, which provides an easy ranking in $P_T$.
\[numberP\_fact2\] [**(${\it lp}V \cdot F$ factorization, given $\varphi: P_0 \to Q_0$, rank$_{P_0}$(.) and rank$_{Q_0}$(.))**]{}\
The following problem is $\#{\mathcal P}$-complete.\
[Input]{}, consisting of three parts:\
$\bullet$ A $\{0,1,\bot\}$-valued circuit that computes a bijection $\varphi: P_0 \to Q_0$ (where $P_0$ and $Q_0$ are finite maximal prefix codes over $\{0,1\}$),\
$\bullet$ two $\{0,1,\bot\}$-valued circuits that compute the rank functions of $P_0$, respectively $Q_0$,\
$\bullet$ and $x \in P_1$ (where $P_1$ is the domain code that $\varphi$ receives when it is restricted so as to have [*imC*]{}$(\varphi)$ $= \{0,1\}^n$, where $n = {\rm max}\{ |q| : q \in Q_0\}$).\
[Output:]{} The rank of $x$ in $P_1$ according to dictionary order.
The problems remains $\#{\mathcal P}$-complete if we assume that circuits for both $\varphi$ and $\varphi^{-1}$ are given.
Also, evaluating $\pi$ or $\pi^{-1}$ is $\#{\mathcal P}$-complete (where $\varphi = \pi \, f$ be the ${\it lp}V \cdot F$ factorization).
[**Proof:**]{} The problem is in $\#{\mathcal P}$ because the circuit for $\varphi$ can also be used to obtain a circuit for the restriction $P_1 \to \{0,1\}^n$ of $\varphi$, by Lemma \[makingImC\_n\]; this circuit can then be used to test membership in $P_1$.
To show $\#{\mathcal P}$-hardness, let $T$ and $P_T$ be as in the proofs of Theorems \[numberP\_rank\] and \[numberP\_fact2\]. Let
$P_0 \ = \ \{00,01,1\} \cdot \{0,1\}^n$ $ \ = \ 00T \ \cup \ 00{\overline T} \ \cup \ 01 \, \{0,1\}^n $ $ \ \cup \ 1T \ \cup \ 1{\overline T}$,
$Q_0 \ = \ \{0,10,11\} \cdot \{0,1\}^n$ $ \ = \ 0T \ \cup \ 0{\overline T} \ \cup \ 10 \, \{0,1\}^n $ $ \ \cup \ 11T \ \cup \ 11{\overline T}$,
and define $\varphi: P_0 \to Q_0$ by
$00x \in 00{\overline T} \ \longmapsto \ 11x \in 11{\overline T}$
$00x \in 00T \ \longmapsto \ 0x \in 0T$
$01x \in 01\, \{0,1\}^n \ \longmapsto \ 10x \in 10 \, \{0,1\}^n$
$1x \in 1{\overline T} \ \longmapsto \ 0x \in 0{\overline T}$.
$1x \in 1T \ \longmapsto \ 11x \in 11T$
Then $\varphi$, ${\rm rank}_{P_0}(.)$, and ${\rm rank}_{Q_0}(.)$, and their inverses have small circuits, that are easily derived from the boolean formula $\beta$.
Next, we restrict $\varphi$ in such a way that its image code becomes $\{0,1\}^{n+2}$. The resulting bijection is exactly the bijection $\Phi: P_T \to \{0,1\}^{n+2}$ of the proof of Theorem \[numberP\_fact1\]. All the claimed conclusions of Theorem \[numberP\_fact2\] now follow from Theorem \[numberP\_fact1\]. $\Box$
The above $\#{\mathcal P}$-completeness results imply that finding circuits for the ${\it lp}V \cdot F$ factors $\pi, f$ of $\varphi \in V$ is difficult (if ${\mathcal P} \neq NP$, etc.). However, whether this implies that the factors require large circuits remains a very difficult open problem.
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[**Jean-Camille Birget**]{}\
Dept. of Computer Science\
Rutgers University at Camden\
Camden, NJ 08102, USA\
[[email protected]]{}
[^1]: Supported by NSF grant CCR-0310793.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For any object $A$ in a simplicial model category ${\mathcal{M}}$, we construct a topological space ${\widehat{A}}$ which classifies linear functors whose value on an open ball is equivalent to $A$. More precisely for a manifold $M$, and ${\mathcal{O}(M)}$ its poset category of open sets, weak equivalence classes of such functors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ are shown to be in bijection with homotopy classes of maps $[M,{\widehat{A}}]$. The result extends to higher degree homogeneous functors. At the end we explain a connection to a classification result of Weiss.'
author:
- |
Paul Arnaud Songhafouo Tsopméné\
Donald Stanley
title: ' **Classification of homogeneous functors in manifold calculus**'
---
Introduction
============
Let $M$ be a manifold and let ${\mathcal{O}(M)}$ be the poset of open subsets of $M$. In order to study the space of smooth embeddings of $M$ inside another manifold, Goodwillie and Weiss [@good_weiss99; @wei99] introduced the theory of manifold calculus, which is one incarnation of calculus of functors. One can define manifold calculus as the study of contravariant functors from ${\mathcal{O}(M)}$ to Top, the category of spaces. Being a calculus of functors, its philosophy is to take a functor $F$ and replace it by its Taylor tower $\{T_k(F) {\longrightarrow}T_{k-1}(F)\}_{k \geq 1}$, which converges to the original functor in good cases, very much like the approximation of a function by its Taylor series. The functor $T_kF$ is the *polynomial approximation* to $F$ of degree $\leq k$. The difference between $T_kF$ and $T_{k-1}F$, or more precisely the homotopy fiber of the canonical map $T_kF {\longrightarrow}T_{k-1}F$, belongs to a class of objects called *homogeneous functors* of degree $k$. In [@wei99 Theorem 8.5], Weiss proves a deep result about the classification of homogeneous functors of degree $k$. Specifically, he shows that any such functor is equivalent to a functor constructed out of a fibration $p \colon Z \to F_k(M)$ over the unordered configuration space of $k$ points in $M$, with a preferred section (germ) near the fat diagonal of $M$.
In this paper we classify homogeneous functors of degree $k$ from ${\mathcal{O}(M)}$ into any simplicial model category ${\mathcal{M}}$. Such functors are determined by their values on disjoint unions of $k$ balls [@paul_don17-2 Lemma 6.5]. Let ${\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})$ denote the category of homogeneous functors $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ such that $F(U) \simeq A$ for any $U$ diffeomorphic to the disjoint union of exactly $k$ open balls (see [@paul_don17-2 Definition 6.2]). Let ${\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})\slash we$ denote the collection of weak equivalence classes of such functors. For spaces $X$ and $Y$, we let $[X, Y]$ be the standard notation for the set of homotopy classes of maps from $X$ to $Y$. We classify objects of ${\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})$ not through fibrations, but instead by maps from $F_k(M)$ to a certain topological space. Specifically, we have the following, which is the main result of this paper.
\[main\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let $A \in {\mathcal{M}}$. Then there is a topological space ${\widehat{A}}$ such that for any manifold $M$,
1. if $k = 1$, there is a bijection $${\mathcal{F}}_{1A} ({\mathcal{O}(M)}; {\mathcal{M}})\slash we \ \ \cong \ \ \left[M, {\widehat{A}}\right].$$
2. If $k \geq 2$ and ${\mathcal{M}}$ has a zero object, there is a bijection $${\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})\slash we \ \ \cong \ \ \left[F_k(M), {\widehat{A}}\right].$$
One may ask the following natural questions.
1. How is ${\widehat{A}}$ constructed?
2. What do we know about ${\widehat{A}}$?
3. How is our classification related to that of Weiss?
To answer the first question, let ${\widetilde{\Delta}^n}, n \geq 0$, denote the poset whose objects are nonempty subsets of $\{0, \cdots, n\}$, and whose morphisms are inclusions. We construct ${{\mathcal{C}}_A}\subseteq {\mathcal{M}}$, a small subcategory consisting of a certain collection of fibrant-cofibrant objects of ${\mathcal{M}}$ that are weakly equivalent to $A$. The morphisms of ${{\mathcal{C}}_A}$ are the weak equivalences between its objects (see Definition \[ca\_defn\]). Define ${\widehat{A}_{\bullet}}$ as the simplicial set whose $n$-simplices are contravariant functors ${\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ that are required to be fibrant with respect to the injective model structure on ${{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}}$. Face maps are defined in the standard way, while degeneracies are more intricate (see Definition \[disj\_defn\]). It turns out that ${\widehat{A}_{\bullet}}$ is a Kan complex. We define ${\widehat{A}}$ as the geometric realization of ${\widehat{A}_{\bullet}}$.
For the second question, we do not know that much about ${\widehat{A}}$. By definition it is connected, and we believe its fundamental group is the group of (derived) homotopy automorphisms of $A$. Further computations seem hard. Regarding the third question, let us consider Weiss’ result as mentioned above. In addition to this, he proves that the fiber $p^{-1}(S)$ of the fibration $p \colon Z \to F_k(M)$ that classifies a homogeneous functor $E \colon {\mathcal{O}(M)}\to \text{Top}$ of degree $k$ is homotopy equivalent to $E(U_S)$, where $U_S$ is a tubular neighborhood of $S$ so that $U_S$ is diffeomorphic to a disjoint union of $k$ open balls [@wei99]. So the classification of objects of ${\mathcal{F}}_{kA}({\mathcal{O}(M)}; \text{Top})$ amounts to the classification of fibrations over $F_k(M)$ with a section near the fat diagonal and whose fiber is $A$. In the case $k=1$, the fat diagonal is empty and we are just looking at fibrations over $M$ whose fiber is $A$. It is well known that there is a classifying space for such fibrations, namely $B{\text{haut}}A$ where ${\text{haut}}A$ denotes the topological/simplicial monoid of (derived) homotopy automorphisms of $A$. If $k > 1$ and ${\mathcal{M}}= \text{Top}_*$, the category of pointed spaces, one has a similar classifying space for the objects of ${\mathcal{F}}_{kA}({\mathcal{O}(M)}; \text{Top}_*)$. For a general simplicial model category ${\mathcal{M}}$ we believe our classifying space, ${\widehat{A}}$, is homotopy equivalent to $B{\text{haut}}A$, but we do not know how to prove this. We also believe there should be another approach (which does not involve ${\widehat{A}}$) to try to show that ${\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})\slash we $ is in one-to-one correspondence with homotopy classes of maps $F_k(M) \to B{\text{haut}}A$. We will say more about all this in Section \[comparison\_section\].
One may use Theorem \[main\_thm\] to set up the concept of *characteristic classes* or *invariants* of homogeneous functors though we do not know whether ${\widehat{A}}$ is homotopy equivalent to $B{\text{haut}}A$. Let $F \in {{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}$ and let $f \colon F_k(M) {\longrightarrow}{\widehat{A}}$ denote the classifying map of $F$. One can define the *characteristic classes* or *invariants* of $F$ as the cohomology classes $f^*(H^*({\widehat{A}})) \subseteq H^*(F_k(M))$. If two functors are weakly equivalent, then by Theorem \[main\_thm\] they are homotopic and therefore they are equal in cohomology. It would be interesting to see what kind of characteristic classes one could recover using our approach, or if other more traditional classifying spaces can be seen as special cases of our construction.
**Strategy of the proof of Theorem \[main\_thm\].** To prove the first part, we need four intermediate results. Let ${\mathcal{T}^M}$ be a triangulation of $M$, that is, a simplicial complex homeomorphic to $M$ together with a homeomorphism ${\mathcal{T}^M}{\longrightarrow}M$. There is no need for the *link condition* (which says that the link of any simplex is a piecewise-linear sphere). Associated with ${\mathcal{T}^M}$ is the poset ${\mathcal{U}(\mathcal{T}^M)}\subseteq {\mathcal{O}(M)}$, which was introduced in [@paul_don17 Section 4.1] and recalled in Definition \[utm\_defn\]. That poset is one of the key objects of this paper as it enables us to connect many categories. Associated with ${\mathcal{U}(\mathcal{T}^M)}$ is the poset ${\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}\subseteq {\mathcal{O}(M)}$ defined as $${\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}= \{\text{$B$ diffeomorphic to an open ball such that $B \subseteq U_{\sigma}$ for some $\sigma$} \}.$$ It turns out that ${\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}$ is a basis for the topology of $M$. For a subposet ${\mathcal{S}}\subseteq {\mathcal{O}(M)}$, we denote by ${\mathcal{F}}_A({\mathcal{S}}; {\mathcal{M}})$ the category of isotopy functors $F \colon {\mathcal{S}}{\longrightarrow}{\mathcal{M}}$ such that $F(U)$ is weakly equivalent to $A$ for any $U$ diffeomorphic to an open ball. The first result we need is Lemma 6.5 from [@paul_don17-2] which says that the categories ${\mathcal{F}}_{1A} ({\mathcal{O}}(M); {\mathcal{M}})$ and ${\mathcal{F}}_A({\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}; {\mathcal{M}})$ are weakly equivalent (in the sense of [@paul_don17-2 Definition 6.3]), that is, $$\begin{aligned}
\label{res_eqn2}
{\mathcal{F}}_{1A} ({\mathcal{O}}(M); {\mathcal{M}}) {\simeq}{\mathcal{F}}_A({\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}; {\mathcal{M}}). \end{aligned}$$ Looking closer at the proof of Lemma 6.5 from [@paul_don17-2], one can see that the hypothesis that ${\mathcal{M}}$ has a zero object is not needed when $k=1$. Since ${\mathcal{U}(\mathcal{T}^M)}$ is a very good cover (in the sense of [@paul_don17 Definition 4.1]) of $M$, we have the following which can be proved along the lines of [@paul_don17 Proposition 4.7]. $$\begin{aligned}
\label{res_eqn3}
{\mathcal{F}}_A({\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}; {\mathcal{M}}) {\simeq}{{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}. \end{aligned}$$ (See also Proposition \[fum\_butm\_prop\].) As we can see, (\[res\_eqn2\]) and (\[res\_eqn3\]) are just local versions of some results from [@paul_don17; @paul_don17-2]. The following two technical results are new and proved using model category techniques. To state them, let ${{\mathcal{C}}_A}$ as above. Let ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ denote the category of isotopy functors from ${\mathcal{U}(\mathcal{T}^M)}$ to ${{\mathcal{C}}_A}$. By definition, there is an obvious functor $\phi \colon {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}{\hookrightarrow}{{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$. Though we do not define a functor in the other direction, we succeed to prove that the localization of $\phi$ is an equivalence of categories. That is, $$\begin{aligned}
\label{res_eqn4}
{{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}[W^{-1}] \simeq {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}[W^{-1}].\end{aligned}$$ (See Proposition \[fuca\_prop\].) To get (\[res\_eqn4\]), we show that the localization of $\phi$ is essentially surjective and fully faithful, the essentially surjectivity being the most difficult part. The final result we need is stated as follows. Let ${\widehat{A}_{\bullet}}$ as above. One can associate to ${\mathcal{T}^M}$ a canonical simplicial set denoted ${\mathcal{T}_{\bullet}^{M}}$. We have the bijection $$\begin{aligned}
\label{res_eqn5}
{{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}\slash we \ \ \cong \ \ \left[{\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}}\right].\end{aligned}$$ (See Proposition \[fuca\_iso\_prop\].) To get (\[res\_eqn5\]) we construct explicit maps between the sets involved. The hardest part is to show that those maps are well defined. Defining ${\widehat{A}}$ as above, and noticing that the geometric realization of ${\mathcal{T}_{\bullet}^{M}}$ is $M$, one deduces Theorem \[main\_thm\] -(i) from (\[res\_eqn2\])-(\[res\_eqn5\]).
The second part of Theorem \[main\_thm\] is an immediate consequence of the first part and the following weak equivalence, which is [@paul_don17-2 Theorem 1.3]. $$\begin{aligned}
\label{res_eqn1}
{{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}{\simeq}{\mathcal{F}}_{1A} ({\mathcal{O}}(F_k(M)); {\mathcal{M}}). \end{aligned}$$ Theorem \[main\_thm\] has many hypotheses including the following: ${\mathcal{M}}$ is a model category and . Note that the two underlined terms are not needed to prove (\[res\_eqn3\])-(\[res\_eqn5\]).
**Outline** The plan of the paper is as follows (see also the Table of Contents at the beginning of the paper). In Section \[dtn\_section\] we prove basic results we will use later. Section \[ahd\_section\] defines the simplicial set ${\widehat{A}_{\bullet}}$ and proves Proposition \[ahd\_prop\], which says that ${\widehat{A}_{\bullet}}$ is a Kan complex. First we construct a small category ${{\mathcal{C}}_A}\subseteq {\mathcal{M}}$ out of a model category ${\mathcal{M}}$ and an object $A \in {\mathcal{M}}$. Next we construct a specific fibrant replacement functor ${\mathcal{R}}\colon {{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}} {\longrightarrow}{{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}}$, which is essentially used to define degeneracy maps of ${\widehat{A}_{\bullet}}$. In Section \[fautm\_futma\_section\] we prove Proposition \[fuca\_prop\] or (\[res\_eqn4\]). In Sections \[lambda\_section\], \[theta\_section\] we prove Proposition \[fuca\_iso\_prop\] or (\[res\_eqn5\]). Section \[pmr\_section\] is dedicated to the proof of the main result of this paper: Theorem \[main\_thm\]. Finally, in Section \[comparison\_section\], we state a conjecture saying how our classification is related to that of Weiss.
**Convention, notation, etc.** These will be as in [@paul_don17 Section 2], with the following additions. Throughout this paper the letter $M$ stands for a second-countable smooth manifold. The only place we need $M$ to be second-countable is Section \[infinite\_case\_subsection\]. We write ${\mathcal{M}}$ for a *model category* [@hovey99 Definition 1.1.4], while $A$ is an object of ${\mathcal{M}}$. As part of the definition, the factorizations in ${\mathcal{M}}$ are functorial. Wherever necessary, additional conditions on ${\mathcal{M}}$ will be imposed. In the sequel, the term simplicial complex means geometric simplicial complex. We write $[n]$ for the set $\{0, \cdots, n\}$, $[n]_i$ for $[n] \backslash \{i\}$, and $[n]_{ij}$ for the set $[n] \backslash \{i, j\}$. Also we let $\{a_0, \cdots, \widehat{a}_i, \cdots, a_s\}$ denote the set $\{a_0, \cdots, a_s\} \backslash \{a_i\}$. If ${\mathcal{S}}$ is a small category and ${\mathcal{C}}$ is a subcategory of ${\mathcal{M}}$, we write ${\mathcal{C}}^{{\mathcal{S}}}$ for the category of contravariant functors from ${\mathcal{S}}$ to ${\mathcal{C}}$. An object of that category is called *${\mathcal{S}}$-diagram* or just *diagram* in ${\mathcal{C}}$. As usual, weak equivalences in ${\mathcal{C}}^{{\mathcal{S}}}$ are natural transformations which are objectwise weak equivalences.
**Acknowledgments** This work has been supported by Pacific Institute for the Mathematical Sciences (PIMS), the University of Regina and the Natural Sciences and Engineering Research Council of Canada (NSERC), that the authors acknowledge. We would also like to thank Jim Davis for helping us with PL topology.
Basic results and recollections {#dtn_section}
===============================
As we said in the introduction, this section presents some basic results we will use later. It is organized as follows. In Section \[dtn\_subsection\] we define the posets ${\widetilde{\Delta}^n}$, ${\partial^i}{\widetilde{\Delta}^n}$, and ${\partial {\widetilde{\Delta}^n}}$, and endow the collection $\{{\widetilde{\Delta}}^n\}_{n \geq 0}$ with a natural cosimplicial structure. In Section \[model\_category\_subsection\] we endow the category of ${\widetilde{\Delta}^n}$-diagrams with the injective model structure and discuss some basic results about fibrant diagrams. In Section \[local\_category\_subsection\], we recall some classical facts about the localization of categories.
The posets ${\widetilde{\Delta}^n}$, ${\partial^i}{\widetilde{\Delta}^n}$ and ${\partial {\widetilde{\Delta}^n}}$ {#dtn_subsection}
-----------------------------------------------------------------------------------------------------------------
\[dtn\_defn\] For $n \geq 0$, define ${\widetilde{\Delta}}^n$ to be the poset whose objects are nonempty ordered subsets $\alpha = \{a_0, \cdots, a_s\}$ of $\{0, \cdots, n\}$. Of course the order on $\alpha$ is induced by the natural order of the set $\{0, \cdots, n\}$. Morphisms of ${\widetilde{\Delta}^n}$ are inclusions.
Anytime we write $\alpha = \{a_0, \cdots, a_s\}$, it will always mean $a_0 \leq \cdots \leq a_s$.
\[dtn\_rmk\] The poset ${\widetilde{\Delta}}^n$ has distinguished morphisms, namely $$d^i \colon \{a_0, \cdots, \widehat{a}_i, \cdots, a_s\} {\longrightarrow}\{a_0, \cdots, a_s\}, \quad 0 \leq i \leq s.$$ One can check that every morphism of ${\widetilde{\Delta}}^n$ can be written as a composition of $d^i$’s.
\[dt2\_expl\] The following diagram is the poset ${\widetilde{\Delta}}^2$. $${\widetilde{\Delta}}^2 = \xymatrix{ & & & \{0\} \ar[ld]_-{d^1} \ar[rd]^-{d^1} & & & \\
& & \{0, 1\} \ar[rd]^-{d^2} & & \{0, 2\} \ar[ld]_-{d^1} & & \\
& & & \{0, 1, 2\} & & & \\
\{1\} \ar[rruu]^-{d^0} \ar[rrr]_-{d^1} & & & \{1, 2\} \ar[u]^-{d^0} & & & \{2\} \ar[lll]^-{d^0} \ar[lluu]_-{d^0} }$$
Varying $n$ we get the collection ${\widetilde{\Delta}^{\bullet}}= \{{\widetilde{\Delta}}^n\}_{n \geq 0}$, which turns out to be endowed with a natural cosimplicial structure defined as follows. Let $\Delta$ be the category whose objects are sets of the form $[n] =\{0, \cdots, n\}, n \geq 0$, endowed with the natural order, and whose morphisms are non-decreasing maps. Let $d^i \colon [n] {\longrightarrow}[n+1]$ and $s^k \colon [n+1] {\longrightarrow}[n]$ be the special morphisms of $\Delta$ (see [@goe_jar09 Section I.1]). It is well known that $d^i$ and $s^k$ satisfy the cosimplicial identities.
\[di\_sk\_defn\]
1. Define a functor $d^i \colon {\widetilde{\Delta}}^n {\longrightarrow}{\widetilde{\Delta}}^{n+1}, 0 \leq i \leq n+1$, as $
d^i(\{a_0, \cdots, a_s\}) := \{d^i(a_0), \cdots, d^i(a_s) \}.
$
2. Define a functor $s^k \colon {\widetilde{\Delta}}^{n+1} {\longrightarrow}{\widetilde{\Delta}}^n, 0 \leq k \leq n$, as $
s^k(\{a_0, \cdots, a_s\}) := \{s^k(a_0), \cdots, s^k(a_s)\}.
$
On morphisms, define $d^i$ and $s^k$ in the obvious way.
The following proposition is straightforward.
\[cosimpl\_rel\_prop\] The functors $d^i$ and $s^k$ we just defined satisfy the cosimplicial identities. That is, ${\widetilde{\Delta}^{\bullet}}= \{{\widetilde{\Delta}}^n\}_{n \geq 0}$ is a cosimplicial category.
\[pdtn\_defn\]
1. For $i \in \{0, \cdots, n\}$, define ${\partial^i}{\widetilde{\Delta}^n}\subseteq {\widetilde{\Delta}^n}$ to be the full subposet whose objects are nonempty subsets of $[n]_i$.
2. Define $
{\partial {\widetilde{\Delta}^n}}:= \bigcup_{i=0}^n {\partial^i}{\widetilde{\Delta}^n}.
$ Equivalently, ${\partial {\widetilde{\Delta}^n}}$ is the full subposet of ${\widetilde{\Delta}^n}$ whose objects are nonempty proper subsets of $\{0, \cdots, n\}$.
By the definitions, one can easily see that there is a canonical isomorphism between ${\partial^i}{\widetilde{\Delta}^n}$ and ${\widetilde{\Delta}}^{n-1}$. That is, ${\partial^i}{\widetilde{\Delta}^n}\cong {\widetilde{\Delta}}^{n-1}$. Similarly, one has ${\partial^i}{\widetilde{\Delta}^n}\cap {\partial^j}{\widetilde{\Delta}^n}\cong {\widetilde{\Delta}}^{n-2}$. We will make these identifications throughout the paper.
Model category structure on ${\mathcal{M}}^{{\widetilde{\Delta}^n}}$ {#model_category_subsection}
--------------------------------------------------------------------
\[pt\_defn\] Let ${\mathcal{T}}$ be a simplicial complex. Define ${P({\mathcal{T}})}$ to be the poset whose objects are (non-degenerate) simplices of ${\mathcal{T}}$. Given two objects $\alpha, \sigma \in {P({\mathcal{T}})}$, there is a morphism $d^{\alpha \sigma} \colon \alpha {\longrightarrow}\sigma$ if and only if $\alpha$ is a face [^1] of $\sigma$.
\[pt\_rmk\] From Definition \[pt\_defn\], it is clear that ${P({\mathcal{T}})}$ is isomorphic to ${\widetilde{\Delta}^n}$ when ${\mathcal{T}}$ is the standard geometric $n$-simplex. That is, $P(\Delta^n) \cong {\widetilde{\Delta}^n}$. The same remark holds for the poset ${\partial {\widetilde{\Delta}^n}}$ introduced in Definition \[pdtn\_defn\]. That is, $P(\partial \Delta^n) \cong {\partial {\widetilde{\Delta}^n}}$.
\[matching\_defn\] Let ${\mathcal{T}}$ and ${P({\mathcal{T}})}$ be as in Definition \[pt\_defn\].
1. For an object $\sigma \in {P({\mathcal{T}})}$, define $\partial\sigma$ as the simplicial complex whose simplices are nonempty proper faces of $\sigma$.
2. Let $F \in {\mathcal{M}}^{{P({\mathcal{T}})}}$. The *matching object* of $F$ at $\sigma \in {P({\mathcal{T}})}$, denoted $M_{\sigma}(F)$, is defined as $$M_{\sigma} (F) := \underset{\alpha \in P(\partial\sigma)}{\text{lim}} \; F(\alpha).$$
3. For $F \in {\mathcal{M}}^{{P({\mathcal{T}})}}$ and $\sigma \in {P({\mathcal{T}})}$, the canonical map $
F(\sigma) {\longrightarrow}M_{\sigma} (F),
$ provided by the universal property of limit, is called the *matching map* of $F$ at $\sigma$.
\[model\_prop\] Let $T$ and ${P({\mathcal{T}})}$ as above. There exists a model structure on the category ${\mathcal{M}}^{{P({\mathcal{T}})}}$ of ${P({\mathcal{T}})}$-diagrams in ${\mathcal{M}}$ such that weak equivalences and cofibrations are objectwise. Furthermore, a map $F {\longrightarrow}G$ is a (trivial) fibration if and only if the induced map $F(\sigma) {\longrightarrow}G(\sigma) \times_{M_{\sigma}(G)} M_{\sigma} (F)$ is a (trivial) fibration for all $\sigma \in {P({\mathcal{T}})}$.
This follows from two facts. The first one is the fact that the poset ${P({\mathcal{T}})}$ is clearly a *direct category* in the sense of [@hovey99 Definition 5.1.1]. So any $F \in {\mathcal{M}}^{{P({\mathcal{T}})}}$ is an *inverse diagram* [^2] (remember that for us diagram means contravariant functor). The second fact is [@hovey99 Theorem 5.1.3].
The category of diagrams, ${\mathcal{M}}^{{P({\mathcal{T}})}}$, will be always endowed with this model structure.
It is not difficult to see that a diagram $F \colon P({\Delta^n}) {\longrightarrow}{\mathcal{M}}$ is fibrant if and only if all of its $k$-faces, $0 \leq k \leq n$, are fibrant $P(\Delta^k)$-diagrams. By *$k$-face* of $F$, we mean the diagram $F\circ P(\tau)$, where $\tau \colon \Delta^k \hookrightarrow \Delta^n$ is the canonical inclusion map.
\[fibrant\_prop\] Let ${\mathcal{T}}$ be a finite simplicial complex, and let ${P({\mathcal{T}})}$ be as in Definition \[pt\_defn\]. Let $F \colon {P({\mathcal{T}})}{\longrightarrow}{\mathcal{M}}$ be a fibrant diagram. Then the limit $\underset{\alpha \in {P({\mathcal{T}})}}{\lim} F(\alpha)$ is a fibrant object of ${\mathcal{M}}$.
Since $F$ is fibrant, it follows that for every $\sigma \in {P({\mathcal{T}})}$, $F(\sigma)$ is fibrant. Furthermore the limit of the diagram $F$ is the same as its homotopy limit again because $F$ is fibrant. Applying now [@hir03 Theorem 18.5.2], we get the desired result.
\[lnk\_defn\] Recall the poset ${\partial {\widetilde{\Delta}^n}}$ from Definition \[pdtn\_defn\]. For $n \geq 1$ and $0 \leq k \leq n$, define ${\widetilde{\Lambda}^n_k}\subseteq {\widetilde{\Delta}^n}$ as the full subposet whose objects are nonempty proper subsets of $[n]$ except $[n]_k$. That is, $
\text{ob} ({\widetilde{\Lambda}^n_k}) = \text{ob} ({\partial {\widetilde{\Delta}^n}}) \backslash [n]_k.
$
\[we\_prop\] Let $F \colon {\widetilde{\Lambda}^n_k}{\longrightarrow}{\mathcal{M}}$ be a fibrant diagram in which every morphism is a weak equivalence. Then for any $\alpha' \in {\widetilde{\Lambda}^n_k}$, the canonical projection $p_{\alpha'} \colon \underset{\alpha \in {\widetilde{\Lambda}^n_k}}{\lim} F(\alpha) {\longrightarrow}F(\alpha')$ is a weak equivalence.
By inspection, one can see that the indexing category ${\widetilde{\Lambda}^n_k}$ is contractible. One can then apply the dual of [@cis09 Corollary 1.18] to get the desired result.
Localization of categories {#local_category_subsection}
--------------------------
The aim of this section is to recall some classical results about the localization of categories we need. Our main references are [@dwyer_spa95] and [@hovey99]. The material of this section will be used in Section \[fautm\_futma\_subsection\].
Let ${\mathcal{E}}$ be a model category and let ${\mathcal{D}}\subseteq {\mathcal{E}}$ be a full subcategory (not necessarily a model subcategory) of ${\mathcal{E}}$. One should think of ${\mathcal{E}}$ as the diagram category ${\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}$ and ${\mathcal{D}}$ as the category ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ that will be introduced in Definition \[fuca\_defn\]. We denote by ${\mathcal{W}}_{{\mathcal{D}}}$ the class of weak equivalences of ${\mathcal{E}}$ that lie in ${\mathcal{D}}$. We will use the standard notation ${\mathcal{D}}[{\mathcal{W}}_{{\mathcal{D}}}^{-1}]$ for the localization of ${\mathcal{D}}$ with respect to ${\mathcal{W}}_{{\mathcal{D}}}$. Roughly speaking, ${\mathcal{D}}[{\mathcal{W}}_{{\mathcal{D}}}^{-1}]$ has the same objects as ${\mathcal{D}}$, and morphisms of ${\mathcal{D}}[{\mathcal{W}}_{{\mathcal{D}}}^{-1}]$ are strings $(f_1, \cdots, f_r)$ of composable arrows where $f_i$ is either an arrow of ${\mathcal{D}}$ or the formal inverse $w^{-1}$ of an arrow $w$ of ${\mathcal{W}}_{{\mathcal{D}}}$.
Recall the notion of cylinder object for $X \in {\mathcal{E}}$, denoted $X \times I$, from [@hovey99 Definition 1.2.4], and let $i_0$ and $i_1$ be the canonical maps from $X$ to $X \times I$. The following result is straightforward.
\[izio\_prop\] Assume that for any $X \in {\mathcal{D}}$ there is a cylinder object $X \times I$ for $X$ in ${\mathcal{D}}$. Then one has $i_0 = i_1$ in the category ${\mathcal{D}}[{\mathcal{W}}_{{\mathcal{D}}}^{-1}]$.
For $f, g \colon X {\longrightarrow}Y$ in ${\mathcal{D}}$, if $f$ is homotopic to $g$ (see [@hovey99 Definition 1.2.4]), we will write $f \sim g$.
[@hovey99 Corollary 1.2.6] \[htpy\_prop\] Let $f, g \colon X {\longrightarrow}Y$ be morphisms of ${\mathcal{E}}$. Assume $X$ cofibrant and $Y$ fibrant. If $f \sim g$ then there is a left homotopy $H \colon X \times I {\longrightarrow}Y$ from $f$ to $g$ using any cylinder object $X \times I$.
The following proposition is also straightforward.
\[ho\_prop\] Assume that ${\mathcal{D}}$ is closed under taking fibrant and cofibrant replacements, and let $X, Y \in {\mathcal{D}}$.
1. Then there is a natural isomorphism $$\xymatrix{\underset{{\mathcal{D}}}{\text{Hom}} (QX, RY)\slash \sim \; \ar[rr]^-{\cong}_-{\theta'} & & \underset{{\mathcal{D}}[{\mathcal{W}}_{{\mathcal{D}}}^{-1}]}{\text{Hom}} (X, Y)}.$$
2. If $X$ and $Y$ are both fibrant and cofibrant, the map $\underset{{\mathcal{D}}}{\text{Hom}} \; (X, Y) \stackrel{\varphi}{{\longrightarrow}} \underset{{\mathcal{D}}[{\mathcal{W}}_{{\mathcal{D}}}^{-1}]}{\text{Hom}} (X, Y)$ induced by the canonical functor ${\mathcal{D}}{\longrightarrow}{\mathcal{D}}[{\mathcal{W}}_{{\mathcal{D}}}^{-1}]$ is surjective.
Part (i) follows from [@hovey99 Theorem 1.2.10]. Part (ii) comes from the fact that the following canonical triangle commutes and the map $\pi$ is surjective. $$\xymatrix{\underset{{\mathcal{D}}}{\text{Hom}} (X, Y) \ar[rr]^-{\varphi} \ar[rrd]_-{\pi} & & \underset{{\mathcal{D}}[{\mathcal{W}}_{{\mathcal{D}}}^{-1}]}{\text{Hom}} (X, Y) \\
& & \underset{{\mathcal{D}}}{\text{Hom}} (X, Y) \slash \sim. \ar[u]_-{\cong}^-{\theta'}}$$
The simplicial set ${\widehat{A}_{\bullet}}$ {#ahd_section}
============================================
In this section ${\mathcal{M}}$ is a model category, and $A$ is an object of ${\mathcal{M}}$. The goal here is to construct a simplicial set ${\widehat{A}_{\bullet}}$, which classifies homogeneous functors in manifold calculus, out of ${\mathcal{M}}$ and $A$. We also show that ${\widehat{A}_{\bullet}}$ is a Kan complex (see Proposition \[ahd\_prop\]). We need ${\widehat{A}_{\bullet}}$ to be a Kan complex because of (\[res\_eqn5\]), which involves homotopy classes [^3] of simplicial maps into ${\widehat{A}_{\bullet}}$. This section is organized as follows. In Section \[ca\_subsection\] we define a small category ${{\mathcal{C}}_A}$, which will play the role of the target category for many functors including $n$-simplices of ${\widehat{A}_{\bullet}}$. In Section \[mcalr\_subsection\] we define an explicit fibrant replacement functor ${\mathcal{R}}\colon {{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}} {\longrightarrow}{{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}}$, which will be essentially used to define degeneracy maps of ${\widehat{A}_{\bullet}}$. Lastly, in Section \[ahd\_subsection\], we define ${\widehat{A}_{\bullet}}$ and prove Proposition \[ahd\_prop\].
The category ${{\mathcal{C}}_A}$ {#ca_subsection}
--------------------------------
In Section \[ahd\_subsection\], we will define an $n$-simplex of ${\widehat{A}_{\bullet}}$ as a contravariant functor from ${\widetilde{\Delta}^n}$ to a subcategory ${\mathcal{C}}$ of ${\mathcal{M}}$ that satisfies certain conditions. To guarantee that the collection of all $n$-simplices is actually a set, we need ${\mathcal{C}}$ to be small. This section defines a small category ${{\mathcal{C}}_A}$, which will play the role of ${\mathcal{C}}$.
For each morphism $f \colon X {\longrightarrow}Y$ of ${\mathcal{M}}$, choose two functorial factorizations: $$\xymatrix{X \ \ \ar[rr]^-{f} \ar@{>->}[rd]_-{\sim} & & Y \\
& V_f \ar@{->>}[ru] & } \qquad \xymatrix{X \ \ \ar[rr]^-{f} \ar@{>->}[rd] & & Y \\
& W_f \ar@{->>}[ru]_-{\sim} & }$$ We construct ${{\mathcal{C}}_A}$ by induction. Let $QRA$ be a fibrant-cofibrant replacement of $A$. Define ${{\mathcal{C}}_A}^0 := \{QRA\},$ the full subcategory of ${\mathcal{M}}$ with a single object. Assume we have defined a full subcategory ${{\mathcal{C}}_A}^{i-1} \subseteq {\mathcal{M}}, i \geq 1,$ and recall the poset ${\partial {\widetilde{\Delta}^n}}$ from Definition \[pdtn\_defn\]. Let $X \in {{\mathcal{C}}_A}^{i-1}$ and let $F \colon \partial {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}^{i-1}$ be a fibrant diagram with respect to the injective model structure we described in Proposition \[model\_prop\]. Let $\phi \colon X {\longrightarrow}\underset{\alpha \in \partial {\widetilde{\Delta}^n}}{\text{lim}} \; F(\alpha)$ be a morphism of ${\mathcal{M}}$, and consider the functorial factorization $$\begin{aligned}
\label{mcale_eqn}
\xymatrix{X \ar[rr]^-{\phi} \ \ \ar@{>->}[rd]_-{\sim}^-{\tau} & & \underset{\alpha \in \partial {\widetilde{\Delta}^n}}{\text{lim}}F(\alpha) \\
& Z_{(X, F, \phi)} \ar@{->>}[ru]_-{{\overline{p}}}}\end{aligned}$$ where $Z_{(X, F, \phi)}:= V_{\phi}$. Also consider the functorial factorization to the canonical map $g \colon X \coprod X {\longrightarrow}X$: $$\begin{aligned}
\label{mcalh_eqn}
\xymatrix{X \coprod X \ar[rr]^-{g} \ \ \ar@{>->}[rd] & & X \\
& Z_X \ar@{->>}[ru]_-{\sim}^-{\pi}}\end{aligned}$$ where $Z_X:= W_g$. Now recall the poset ${\widetilde{\Lambda}^n_k}$ from Definition \[lnk\_defn\]. Let $R\underset{\text{Cat}}{\text{Hom}} (\partial {\widetilde{\Delta}^n}; {{\mathcal{C}}_A}^{i-1})$ denote the set of fibrant objects of $({{\mathcal{C}}_A}^{i-1})^{\partial {\widetilde{\Delta}^n}}$, and let $\widetilde{R}\underset{\text{Cat}}{\text{Hom}} ({\widetilde{\Lambda}^n_k}, {{\mathcal{C}}_A}^{i-1})$ denote the set of fibrant diagrams $F \colon {\widetilde{\Lambda}^n_i}{\longrightarrow}{{\mathcal{C}}_A}^{i-1}$ sending every morphism to a weak equivalence. Define three full subcategories ${\mathcal{E}}_A^i$, ${\mathcal{H}}^i_A $ and ${\mathcal{D}}^i_A$ of ${\mathcal{M}}$ as $$\text{ob}({\mathcal{E}}^i_A) := \left\{ Z_{(X, F, \phi)} \ | \; X \in {{\mathcal{C}}_A}^{i-1},\ F \in R\underset{\text{Cat}}{\text{Hom}} (\partial {\widetilde{\Delta}^n}; {{\mathcal{C}}_A}^{i-1}), \phi \in \underset{{\mathcal{M}}}{\text{Hom}} (X, \underset{\alpha \in \partial {\widetilde{\Delta}^n}}{\text{lim}}F(\alpha)) \right\},$$
$$\text{ob}({\mathcal{H}}^i_A) := \left\{Z_X \ | \; X \in {{\mathcal{C}}_A}^{i-1} \right\},$$ and $$\text{ob} ({\mathcal{D}}^i_A) = \left\{Q \underset{{\widetilde{\Lambda}^n_k}}{\lim} \, F \ | \; F \in \widetilde{R}\underset{\text{Cat}}{\text{Hom}} ({\widetilde{\Lambda}^n_k}, {{\mathcal{C}}_A}^{i-1}), 0 \leq k \leq n \right\},$$ where $Q(-)$ stands of course for the cofibrant replacement functor. Also define the full subcategory ${\mathcal{A}}_A^i \subseteq {\mathcal{M}}$ as $$\textup{ob}({\mathcal{A}}_A^i) := \left\{ R \underset{\Uppsi}{\textup{colim}}\, F| \ F \in \widetilde{Q}\underset{\text{Cat}}{\text{Hom}} (\Uppsi, {{\mathcal{C}}_A}^{i-1}) \right\},$$ where
1. $R(-)$ is the fibrant replacement functor.
2. $\Uppsi$ is the poset whose objects are $\emptyset, \{1\}, \cdots, \{n\}$ for some positive integer $n$ and whose morphisms are inclusions.
3. $\widetilde{Q}\underset{\text{Cat}}{\text{Hom}} (\Uppsi, {{\mathcal{C}}_A}^{i-1})$ is the set of covariant functors $F \colon \Uppsi {\longrightarrow}{{\mathcal{C}}_A}^{i-1}$ such that $F(f)$ is an acyclic cofibration for every morphism $f$ of $\Uppsi$.
[@lam_stan05 Section 3] \[meno\_defn\] A diagram of shape $\Uppsi$, denoted by a collection of maps $\{f_1, \cdots, f_n\}$ in ${\mathcal{M}}$ with same domain, is called *menorah*.
\[ca\_defn\] Let ${\mathcal{M}}$ be a model category, and let $A \in {\mathcal{M}}$. Define ${{\mathcal{C}}_A}\subseteq {\mathcal{M}}$ as the full subcategory $${{\mathcal{C}}_A}:= \bigcup_{i \geq 0} {{\mathcal{C}}_A}^i, \quad \text{where} \quad {{\mathcal{C}}_A}^i := \left\{ \begin{array}{ccc}
\{QRA\} & \text{if} & i =0 \\
{\mathcal{E}}_A^i \cup {\mathcal{H}}_A^i \cup {\mathcal{D}}^i_A \cup {\mathcal{A}}_A^i \cup {{\mathcal{C}}_A}^{i-1} & \text{if} & i \geq 1
\end{array} \right.$$
From the definition, there are five kinds of objects in ${{\mathcal{C}}_A}$.
1. The distinguished object, $QRA$, will be used in Lemma \[esur\_lem\] to prove that a certain functor is essentially surjective.
2. Objects of the form $Z_{(X, F, \phi)}$ will be used in Section \[mcalr\_subsection\], Lemmas \[esur\_lem\], \[htpy1\_lem\], and other places.
3. Objects of the form $Z_X$ will be used in Lemma \[ffull\_lem\] to prove that a certain functor is faithful.
4. Objects of the form $Q \underset{{\widetilde{\Lambda}^n_k}}{\lim} \, F$ will be used in Proposition \[ahd\_prop\] to prove that ${\widehat{A}_{\bullet}}$ is a Kan complex.
5. Lastly, objects of the form $R \underset{\Uppsi}{\textup{colim}}\,F$ will be used in Section \[ahd\_subsection\] to define the degeneracy maps of ${\widehat{A}_{\bullet}}$.
The category ${{\mathcal{C}}_A}$ has nice features given by the following two propositions.
\[ca\_prop0\] The collection of objects of ${{\mathcal{C}}_A}$ is a set. That is, the category ${{\mathcal{C}}_A}$ is small.
By induction, it is easy to show that for all $i$ the collection of objects of ${{\mathcal{C}}_A}^i$ is a set. This implies the proposition.
\[ca\_prop\] Every object of ${{\mathcal{C}}_A}$ is fibrant, cofibrant, and weakly equivalent to $A$.
1. To see that every object of ${{\mathcal{C}}_A}$ is fibrant, we will proceed by induction on $i$. If $i =0$ then the statement is clearly true since $QRA$ is fibrant. Assume that every object of ${{\mathcal{C}}_A}^{i-1}$ is fibrant, and let $Z \in {{\mathcal{C}}_A}^i$. If $Z$ is equal to $Z_{(X, F, \phi)}$ as in (\[mcale\_eqn\]) then $Z$ is fibrant since the map ${\overline{p}}$ is a fibration and its target is fibrant (by Proposition \[fibrant\_prop\]). If $Z$ is equal to $Z_X$ as in (\[mcalh\_eqn\]), the same argument applies. If $Z$ is equal to $Q\underset{{\widetilde{\Lambda}^n_k}}{\lim}\, F$ for some $F \in R\underset{\text{Cat}}{\text{Hom}} ({\widetilde{\Lambda}^n_k}, {{\mathcal{C}}_A}^{i-1})$, then $Z$ is fibrant since $\underset{{\widetilde{\Lambda}^n_k}}{\lim}\, F$ is fibrant (by Proposition \[fibrant\_prop\]). Objects of the form $Z=R \underset{\Uppsi}{\textup{colim}}\,F$ are obviously fibrant.
2. Similarly, one can show by induction that every object of ${{\mathcal{C}}_A}$ is cofibrant. Objects of the form $R \underset{\Uppsi}{\textup{colim}}\,F$ deserve a special attention. They are cofibrant basically because every morphism of the diagram $F$ is an acyclic cofibration, and thanks to the shape of $F$, one can compute its colimit by taking “successive pushouts”. In fact, one can easily show that the map from $F(\emptyset)$ to $\underset{\Uppsi}{\textup{colim}} \, F$ is an acyclic cofibration (by using the fact that the pushout of a cofibration is again a cofibration, and the pushout of a weak equivalence along a cofibration is a weak equivalence). This implies that $\underset{\Uppsi}{\textup{colim}} \, F$ is cofibrant since $F(\emptyset)$ is cofibrant by the induction hypothesis.
3. We proceed again by induction to prove that every object of ${{\mathcal{C}}_A}$ is weakly equivalent to $A$. The base case is obvious. Assume that the statement holds for $i-1$. Let $Z \in {{\mathcal{C}}_A}^i$. If $Z = Z_{(X, F, \phi)}$ or $Z = Z_X$, then $Z$ is weakly equivalent to $A$ by the induction hypothesis and the fact that the maps $\tau$ and $\pi$ from (\[mcale\_eqn\]) and (\[mcalh\_eqn\]) respectively are both weak equivalences. Now assume that $Z = Q\underset{{\widetilde{\Lambda}^n_k}}{\lim}\, F$. Since $Q(-)$ is the cofibrant replacement functor, we have the weak equivalence $Z \stackrel{\sim}{{\longrightarrow}} \underset{{\widetilde{\Lambda}^n_k}}{\lim}\, F$. Furthermore the limit $\underset{{\widetilde{\Lambda}^n_k}}{\lim} F$ is weakly equivalent to $A$ by the induction hypothesis and the fact that the map from the limit of $F$ to each piece of the diagram is a weak equivalence. This latter fact comes from Proposition \[we\_prop\]. Lastly, objects of the form $Z=\underset{\Uppsi}{\textup{colim}} \, F$ are weakly equivalent to $A$ since the map $F(\emptyset) {\longrightarrow}\underset{\Uppsi}{\textup{colim}} \, F$ is a weak equivalence as explained in the previous part.
This ends the proof.
Specific fibrant replacement functor ${\mathcal{R}}\colon {{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}} {\longrightarrow}{{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}}$ {#mcalr_subsection}
-----------------------------------------------------------------------------------------------------------------------------------------------------------------
On the category ${{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}}$, consider the model structure described in Proposition \[model\_prop\]. For our purposes, we need to construct a specific fibrant replacement functor ${\mathcal{R}}\colon {{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}} {\longrightarrow}{{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}}$ that has nice properties (see Proposition \[mcalr\_prop\] below). The idea is to first take the fibrant replacement of $0$-simplices, then $1$-simplices, and so on. So we need to proceed by induction on $n$.
$\bullet$ For $n=0$, define ${\mathcal{R}}\colon {{\mathcal{C}}_A}^{{\widetilde{\Delta}}^0} {\longrightarrow}{{\mathcal{C}}_A}^{{\widetilde{\Delta}}^0}$ as ${\mathcal{R}}(F):= F$. This makes sense since a diagram $F \colon {\widetilde{\Delta}}^0 {\longrightarrow}{{\mathcal{C}}_A}$ consists of a single object, $F(\{0\})$, which is fibrant thanks to Proposition \[ca\_prop\].
$\bullet$ Let $n=1$, and let $F \colon {\widetilde{\Delta}}^1 {\longrightarrow}{{\mathcal{C}}_A}$ be an object of ${{\mathcal{C}}_A}^{{\widetilde{\Delta}}^1}$. We want to define a fibrant diagram ${\mathcal{R}}(F) \colon {\widetilde{\Delta}}^1 {\longrightarrow}{{\mathcal{C}}_A}$. If $F$ is fibrant, define ${\mathcal{R}}(F):=F$. If $F$ is not fibrant, define ${\mathcal{R}}(F)$ as follows. Set $F=\{X_0 \leftarrow X_{01} \rightarrow X_1\}$, where $X_i:= F(\{i\})$ and $X_{01}:= f(\{0, 1\})$. Consider the map $\phi \colon X_{01} {\longrightarrow}\underset{\alpha \in {\partial}{\widetilde{\Delta}}^1}{\lim} F(\alpha)$ provided by the universal property of limit. Also consider the projection $p_{\alpha'} \colon \underset{\alpha \in {\partial}{\widetilde{\Delta}}^1}{\lim} F(\alpha) {\longrightarrow}F(\alpha'), \alpha' \in {\partial}{\widetilde{\Delta}}^1$. Replacing $X$ by $X_{01}$ and $n$ by $1$ in (\[mcale\_eqn\]), we get $\overline{X}_{01}:=Z_{(X_{01}, F, \phi)} \in {{\mathcal{C}}_A}$ together with the map $\overline{p} \colon \overline{X}_{01} {\longrightarrow}\underset{\alpha \in {\partial}{\widetilde{\Delta}}^1}{\lim} F(\alpha)$. Define ${\mathcal{R}}(F)$ as $${\mathcal{R}}(F):= \left\{\xymatrix{X_0 & & \overline{X}_{01} \ar[ll]_-{p_{\alpha_0}{\overline{p}}} \ar[rr]^-{p_{\alpha_1}{\overline{p}}} & & X_1}\right\},$$ where $\alpha_0 = \{0\}$ and $\alpha_1 = \{1\}$. Note that by definition the functor ${\mathcal{R}}$ has the following property: $${\mathcal{R}}(Fd^i) = {\mathcal{R}}(F)d^i, \quad \text{for all} \quad d^i \colon {\widetilde{\Delta}}^0 {\longrightarrow}{\widetilde{\Delta}}^1.$$ $\bullet$ Assume we have defined ${\mathcal{R}}\colon {{\mathcal{C}}_A}^{{\widetilde{\Delta}}^{k}} {\longrightarrow}{{\mathcal{C}}_A}^{{\widetilde{\Delta}}^{k}}$ for all $k \leq n-1$. Also assume that ${\mathcal{R}}$ has the following property: $$\begin{aligned}
\label{mcalr_eqn}
{\mathcal{R}}(Fd^i) = {\mathcal{R}}(F)d^i \quad \text{for all} \quad d^i \colon {\widetilde{\Delta}}^{n-2} {\longrightarrow}{\widetilde{\Delta}}^{n-1}.\end{aligned}$$ Let $F \in {{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}}$. We need to define ${\mathcal{R}}(F) \colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$. If $F$ is fibrant, define ${\mathcal{R}}(F):= F$. Assume $F$ is not fibrant, and define ${\mathcal{R}}(F)$ as follows. The idea is to first take the fibrant replacement of all $(n-1)$-faces of $F$ using the induction hypothesis (this will produce a new functor ${\overline{F}}\colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$), and then substitute ${\overline{F}}([n])$ by the appropriate $Z$ in ${{\mathcal{C}}_A}$. So we will proceed in two steps.
1. Construction of ${\overline{F}}$. For $0 \leq j \leq n$, define ${\overline{F}}^j \colon {\partial}^j {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ as $
{\overline{F}}^j := {\mathcal{R}}(Fd^j). $ By using (\[mcalr\_eqn\]), and the definitions it is straightforward to see that ${\overline{F}}^j$ agrees with ${\overline{F}}^i$ on the intersection ${\partial}^i {\widetilde{\Delta}^n}\cap {\partial}^j {\widetilde{\Delta}^n}$. This allows us to define ${\overline{F}}\colon {\partial}{\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ on the boundary of ${\widetilde{\Delta}^n}$ as ${\overline{F}}(\alpha):= {\overline{F}}^i (\alpha)$ if $\alpha \in {\partial}^i {\widetilde{\Delta}^n}$. Since the functor ${\overline{F}}$ is a fibrant replacement of $F|{\partial}{\widetilde{\Delta}^n}$, there is a weak equivalence $\eta$ from the latter to the former. Define ${\overline{F}}([n]):= F([n])$. For $\alpha \in {\partial}{\widetilde{\Delta}^n}$, $d^{\alpha [n]} \colon \alpha \hookrightarrow [n]$ a morphism of ${\widetilde{\Delta}^n}$, define ${\overline{F}}(d^{\alpha [n]}$) as the composition $$\xymatrix{F([n]) \ar[rr]^-{F(d^{\alpha [n]})} & & F(\alpha) \ar[rr]^-{\eta[\alpha]} & & {\overline{F}}(\alpha)},$$ where $\eta[\alpha]$ is the component of the natural transformation $\eta$ at $\alpha$. This completes the definition of ${\overline{F}}\colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$.
2. The boundary of ${\overline{F}}$, that is ${\overline{F}}|{\partial}{\widetilde{\Delta}^n}$, is certainly fibrant, but ${\overline{F}}$ itself might not be fibrant as the matching map $\phi \colon {\overline{F}}([n]) {\longrightarrow}\underset{\alpha \in {\partial}{\widetilde{\Delta}^n}}{\lim} {\overline{F}}(\alpha)$ might not be a fibration. To fix this, consider the object $Z_{({\overline{F}}([n]), {\overline{F}}, \phi)} \in {{\mathcal{C}}_A}$, which comes equipped with ${\overline{p}}\colon Z_{({\overline{F}}([n]), {\overline{F}}, \phi)} {\longrightarrow}\underset{\alpha \in {\partial}{\widetilde{\Delta}^n}}{\lim} {\overline{F}}(\alpha)$ (see (\[mcale\_eqn\])). Define $${\mathcal{R}}(F)|{\partial}{\widetilde{\Delta}^n}:= {\overline{F}}, \qquad \text{and} \qquad {\mathcal{R}}(F) ([n]):= Z_{({\overline{F}}([n]), {\overline{F}}, \phi)}.$$ For $\alpha' \in {\partial}{\widetilde{\Delta}^n}, d^{\alpha'[n]} \colon \alpha' \hookrightarrow [n],$ define ${\mathcal{R}}(F)(d^{\alpha'[n]}):= p_{\alpha'}{\overline{p}}$, where $p_{\alpha'} \colon \underset{\alpha \in {\partial}{\widetilde{\Delta}^n}}{\lim} {\overline{F}}(\alpha) {\longrightarrow}{\overline{F}}(\alpha')$ is the canonical projection as usual. This completes the definition of ${\mathcal{R}}(F)$, which is indeed fibrant and has the required property by construction.
Since the factorization (\[mcale\_eqn\]) is functorial, a simple induction argument on $n$ shows that the assignment $F \mapsto {\mathcal{R}}(F)$ is functorial. That functor has nice properties given by the following.
\[mcalr\_prop\] Let $F \colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ be a contravariant functor.
1. If $F$ is fibrant then ${\mathcal{R}}(F) = F$.
2. If $\tau \colon {\widetilde{\Delta}}^k {\longrightarrow}{\widetilde{\Delta}}^n$ is injective, then ${\mathcal{R}}(F\tau) = {\mathcal{R}}(F)\tau$.
3. If $\tau \colon {\widetilde{\Delta}}^k {\longrightarrow}{\widetilde{\Delta}}^n$ is injective and $F\tau$ is fibrant, then ${\mathcal{R}}(F)\tau = F\tau$, equivalently ${\mathcal{R}}(F)|{\widetilde{\Delta}}^k = F|{\widetilde{\Delta}}^k$. In other words, if any face of $F$ is fibrant, it appears in ${\mathcal{R}}(F)$.
This follows immediately from the construction of ${\mathcal{R}}$.
The simplicial set ${\widehat{A}_{\bullet}}$ {#ahd_subsection}
--------------------------------------------
To prove the main result of this paper (that is, Theorem \[main\_thm\]), we need to construct a simplicial set $X_{\bullet}$ with the following properties.
1. $X_{\bullet}$ is a Kan complex.
2. There is a pair $$\xymatrix{{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A}) \ar@<1ex>[rr]^-{\Lambda} & & \underset{\textup{sSet}}{\textup{Hom}}({\mathcal{T}_{\bullet}^{M}}, X_{\bullet}) \ar@<1ex>[ll]^-{\Theta} }$$ of maps (for the meaning of “${\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})$”, see Definition \[fuca\_defn\]) that satisfies the following four conditions:
1. $\Lambda (F)$ is homotopic to $\Lambda(F')$ whenever $F$ is weakly equivalent to $F'$;
2. $\Theta(f)$ is weakly equivalent to $\Theta(f')$ whenever $f$ is homotopic to $f'$;
3. $\Theta \Lambda (F)$ is weakly equivalent to $F$ for any $F$;
4. $\Lambda \Theta (f)$ is homotopic to $f$ for any $f$.
The natural candidate for $X_{\bullet}$ is the simplicial set ${\widehat{A}_{\bullet}}'$ defined as follows. An $n$-simplex of ${\widehat{A}_{\bullet}}'$ is just a contravariant functor $\sigma \colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$. The simplicial structure of ${\widehat{A}_{\bullet}}'$ is the one induced by the cosimplicial structure of $\widetilde{\Delta}^{\bullet}$ (see Proposition \[cosimpl\_rel\_prop\]). The issue with the simplicial ${\widehat{A}_{\bullet}}'$ is that it might not satisfy (A), and whether the other conditions are satisfied can depend on how $\Lambda$ is defined. We can show that ${\widehat{A}_{\bullet}}'$ satisfies (B)- (i) for some $\Lambda$, and (B)- (iv) for another $\Lambda$, but we do not know how to prove that these two conditions are met for the same $\Lambda$. For the natural $\Lambda$ (with $X_{\bullet}={\widehat{A}_{\bullet}}'$), we can easily prove that $\Lambda \Theta = id$ and $\Theta \Lambda = id$, but we do not know how to prove that it satisfies (B) -(i).
We now define ${\widehat{A}_{\bullet}}$. In proposition \[ahd\_prop\] below we will prove that ${\widehat{A}_{\bullet}}$ is a Kan complex. In the upcoming sections, we will show that ${\widehat{A}_{\bullet}}$ meets the remaining conditions.
\[ahn\_defn\] Let ${{\mathcal{C}}_A}$ be the category from Definition \[ca\_defn\], and let ${\widetilde{\Delta}}^n$ be the poset from Definition \[dtn\_defn\]. Define ${\widehat{A}}_n$ as the collection of contravariant functors $\sigma \colon {\widetilde{\Delta}}^n {\longrightarrow}{{\mathcal{C}}_A}$ satisfy the following two conditions:
1. $\sigma$ sends every morphism to a weak equivalence;
2. $\sigma$ is a fibrant ${\widetilde{\Delta}^n}$-diagram.
Note that by Proposition \[ca\_prop\], for all $x \in {\widetilde{\Delta}}^n$, $\sigma(x)$ is weakly equivalent to $A$. Also note that each ${\widehat{A}_n}$ is a set since the category ${\widetilde{\Delta}^n}$ is small by definition, and ${{\mathcal{C}}_A}$ is small as well (by Proposition \[ca\_prop0\]).
\[disj\_defn\] Recall the functor $d^i \colon {\widetilde{\Delta}}^{n-1} {\longrightarrow}{\widetilde{\Delta}}^n$ from Definition \[di\_sk\_defn\].
1. Define the face map $d_i \colon {\widehat{A}}_n {\longrightarrow}{\widehat{A}}_{n-1}, 0 \leq i \leq n$, as $d_i (\sigma) := \sigma d^i$.
2. The degeneracy maps are defined in Section \[sj\_subsubsection\] below.
Intuitively, $d_i(\sigma)$ can be defined “geometrically” as follows. Thinking of $\sigma \in {\widehat{A}}_n$ as $\sigma({\widetilde{\Delta}}^n)$, the object $d_i (\sigma)$ is nothing but the $(n-1)$-face of $\sigma$ opposite to the vertex $\sigma(\{i\})$. For instance, for $n=2$, consider the diagram obtained by applying $\sigma$ to the diagram from Example \[dt2\_expl\]. For $i = \{0, 1\}$, one has $$d_0 (\sigma) = \left\{\xymatrix{\sigma(\{1\}) & & \sigma(\{1, 2\}) \ar[ll]_-{\sigma(d^1)} \ar[rr]^-{\sigma(d^0)} & & \sigma(\{2\}) } \right\},$$ $$d_1 (\sigma) = \left\{\xymatrix{\sigma(\{0\}) & & \sigma(\{0, 2\}) \ar[ll]_-{\sigma(d^1)} \ar[rr]^-{\sigma(d^0)} & & \sigma(\{2\}) } \right\}.$$
### Definition of the degeneracies $s_j \colon {\widehat{A}_n}{\longrightarrow}{\widehat{A}}_{n+1}$ {#sj_subsubsection}
First we need the following.
Let $k \geq 1$. Consider the set of sequences $(i_1, \cdots, i_k)$ of length $k$ where $i_p$ is an non-negative integer for every $p$. Define on that set the equivalence relation $\sim$ generated by $$\begin{aligned}
\label{seq_eqn}
(i_1, \cdots, i_{p-1}, i_p, \cdots, i_k) \sim (i_1, \cdots, i_p+1, i_{p-1}, \cdots, i_k), \ \textup{$i_{p-1} \leq i_p$}
\end{aligned}$$ This means that when $i_{p-1} \leq i_p$, we switch $i_{p-1}$ and $i_p$, then we add $1$ to $i_p$, the other numbers remaining unchanged.
So two sequences $(i_1, \cdots, i_k)$ and $(r_1, \cdots, r_k)$ are in relation if one can be obtained from the other by using (\[seq\_eqn\]) as many times as needed. Given a simplicial set $X_{\bullet}$ and two sets $X_{m_0}$ and $X_{m_k}$, we assign to each sequence $(i_1, \cdots, i_k)$, with $0 \leq i_{k-p+1} \leq m_{p-1}$ for all $p$, a sequence of degeneracy maps $(s_{i_k}, \cdots, s_{i_1})$ between $X_{m_0}$ and $X_{m_k}$: $$\xymatrix{X_{m_0} \ar[r]^-{s_{i_k}} & \cdots \ar[r]^-{s_{i_1}} & X_{m_k}}$$ So $(i_1, \cdots, i_k) \sim (r_1, \cdots, r_k)$ amounts to saying that the composite $s_{i_1} \cdots s_{i_k}$ can be obtained from $s_{r_1} \cdots s_{r_k}$ by using the simplicial identity $s_is_j = s_{j+1}s_i, i \leq j,$ as many times as needed.
We now define $s_j \colon {\widehat{A}}_n {\longrightarrow}{\widehat{A}}_{n+1}, 0 \leq j \leq n$ by induction on $n$ as follows. Recall the functor ${\mathcal{R}}$ from Section \[mcalr\_subsection\]. Also recall the functor $s^j \colon {\widetilde{\Delta}}^{n+1} {\longrightarrow}{\widetilde{\Delta}^n}$ from Definition \[di\_sk\_defn\]. Given a simplex $\sigma \in {\widehat{A}_n}$, the composite $\sigma s^j$ is not a priori an element of ${\widehat{A}}_{n+1}$ as it might fail to be fibrant. Actually, $\sigma s^j$ is an $(n+1)$-simplex of ${\widehat{A}_{\bullet}}'$, the simplicial set we defined at the beginning of Section \[ahd\_subsection\]. We still denote the face maps of ${\widehat{A}_{\bullet}}'$ by $d_i$.
For $n = 0$, $s_0 \colon {\widehat{A}}_0 {\longrightarrow}{\widehat{A}}_1$ is defined as $s_0(\sigma):= {\mathcal{R}}(\sigma s^0)$, where $s^0 \colon {\widetilde{\Delta}}^1 {\longrightarrow}{\widetilde{\Delta}}^0$.
Let $n \geq 1$.
1. Suppose that for every $m \leq n$ we have defined the degeneracy maps $s_j \colon {\widehat{A}}_{m-1} {\longrightarrow}{\widehat{A}}_m, 0 \leq j \leq m-1$.
2. Assume that the simplicial identities
$d_is_j = s_{j-1}d_i$ if $i < j$ $d_js_j = id = d_{j+1}s_j$
------------------------------------ -------------------------------------
$d_is_j = s_jd_{i-1}$ if $i > j+1$ $s_is_j = s_{j+1}s_i$ if $i \leq j$
are satisfied for all $s_j \colon {\widehat{A}}_{m-1} {\longrightarrow}{\widehat{A}}_m, 0 \leq j \leq m-1, m \leq n$.
3. Assume that for all $m \leq n$, for all $\sigma \in {\widehat{A}}_m$, there are weak equivalences
1. $d_i (\sigma s^j) \stackrel{\sim}{{\longrightarrow}} s_{j-1} (d_i\sigma)$, if $i < j$, and
2. $d_i (\sigma s^j) \stackrel{\sim}{{\longrightarrow}} s_{j} (d_{i-1}\sigma)$, if $i > j+1$.
4. Suppose that for every $m_0 \leq \cdots \leq m_{k+1} \leq n+1$, for any sequences $(i_0, \cdots, i_k)$ and $(r_0, \cdots, r_k)$, with $0 \leq i_{k-p}, r_{k-p} \leq m_p$ for all $p$, such that $(i_0, \cdots, i_k)$ $\sim$ $(r_0, \cdots, r_k)$, there exists an acyclic cofibration $$\begin{aligned}
\label{acy_eqn}
\phi_{r_0\cdots r_k}^{m_{k+1}} \colon \xymatrix{\lambda s^{i_k} \cdots s^{i_1}s^{i_0} ([m_{k+1}]) \ \ \ar@{>->}[r]^-{\sim} & (s_{r_{1}} \cdots s_{r_k} \lambda)s^{r_0} ([m_{k+1}])}
\end{aligned}$$ for every non-degenerate simplex $\lambda \in {\widehat{A}}_{m_0}$.
Define $s_j \colon {\widehat{A}_n}{\longrightarrow}{\widehat{A}}_{n+1}$ as follows. Let $\sigma \in {\widehat{A}_n}$. Then there exists a unique non-degenerate simplex $\lambda \in {\widehat{A}}_{m_0}$ and a sequence $(s_{i_k}, \cdots, s_{i_1})$ between ${\widehat{A}}_{m_0}$ and ${\widehat{A}_n}$ such that $\sigma = s_{i_1} \cdots s_{i_k} \lambda$. (Note that if $\sigma$ is itself non-degenerate, then $m_0=n$, $s_{i_1}=\cdots=s_{i_k} = id$, and $\lambda = \sigma$.) Consider the composite $\sigma s^j$.
$\bullet$ If $i< j$, by (IH3), there exists a weak equivalence $d_i (\sigma s^j) \stackrel{\sim}{{\longrightarrow}} s_{j-1} (d_i\sigma)$. This allows us to replace the face $d_i (\sigma s^j)$ by $s_{j-1} (d_i \sigma)$.
$\bullet$ Similarly, if $i > j+1$, replace the face $d_i (\sigma s^j)$ by $s_j (d_{i-1} \sigma)$.
$\bullet$ This defines a contravariant functor $\widehat{\sigma s^j} \colon {\widetilde{\Delta}}^{n+1} {\longrightarrow}{{\mathcal{C}}_A}$ as $\widehat{\sigma s^j}([n+1]):= \sigma s^j ([n+1])$, and $$\widehat{\sigma s^j}|{\partial}^i {\widetilde{\Delta}}^{n+1} := \left\{\begin{array}{ccc}
s_{j-1}(d_i \sigma) & \textup{if} & i < j \\
s_j (d_{i-1} \sigma) & \textup{if} & i > j+1 \\
\sigma & \textup{if} & i \in \{j, j+1\}.
\end{array} \right.$$ On morphisms of ${\widetilde{\Delta}}^{n+1}$, define $\widehat{\sigma s^j}$ in the obvious way.
$\bullet$ We now replace $\widehat{\sigma s^j}([n+1])$ by an object $Y \in {{\mathcal{C}}_A}$ defined as follows. Let $i_0:=j$. Recalling Definition \[meno\_defn\], define the menorah $\mathbb{D} \colon \Uppsi {\longrightarrow}{{\mathcal{C}}_A}$ as the collection of morphisms from (\[acy\_eqn\]) with $m_k+1 = n+1$. That is, $$\begin{aligned}
\label{meno_eqn}
\mathbb{D}:= \left\{\phi_{r_0\cdots r_k}^{n+1}| \ (r_0, \cdots, r_k) \sim (i_0, \cdots, i_k)\right\}. \end{aligned}$$ Thanks to Lemma \[comp\_lem\] below, there exists a natural transformation $\underset{\Uppsi}{\textup{colim}}\, \mathbb{D} \stackrel{\sim}{{\longrightarrow}} \widehat{\sigma s^j}|{\partial}{\widetilde{\Delta}}^{n+1}$ given by the universal property of colimit (here $\underset{\Uppsi}{\textup{colim}}\, \mathbb{D}$ is viewed as the constant diagram). Taking the fibrant replacement (objectwise) of that morphism, we get a weak equivalence $$\varphi: R\underset{\Uppsi}{\textup{colim}}\, \mathbb{D} \stackrel{\sim}{{\longrightarrow}} R\widehat{\sigma s^j}|{\partial}{\widetilde{\Delta}}^{n+1}= \widehat{\sigma s^j}|{\partial}{\widetilde{\Delta}}^{n+1}$$ in ${{\mathcal{C}}_A}$. (Remember $R\underset{\Uppsi}{\textup{colim}}\, \mathbb{D}$ is an object of ${{\mathcal{C}}_A}$ by Definition \[ca\_defn\].) Define $Y:= R\underset{\Uppsi}{\textup{colim}}\, \mathbb{D}$. The map $\varphi$ gives rise to a contravariant functor $\overline{\sigma s^j} \colon {\widetilde{\Delta}}^{n+1} {\longrightarrow}{{\mathcal{C}}_A}$ defined as $\overline{\sigma s^j}([n+1]):= Y$ and $\overline{\sigma s^j}| {\partial}{\widetilde{\Delta}}^{n+1} := \widehat{\sigma s^j}|{\partial}{\widetilde{\Delta}}^{n+1}$.
The $(n+1)$-simplex $s_j(\sigma)$ is defined as $s_j(\sigma):= {\mathcal{R}}(\overline{\sigma s^j})$.
By construction, it is straightforward to show that the hypothesis (IH2), (IH3), and (IH4) are verified if one replaces $n$ by $n+1$.
\[comp\_lem\] Consider the menorah $\mathbb{D}$ above (\[meno\_eqn\]).
1. For any sequence $(r_0, \cdots, r_k)$ such that $(r_0, \cdots, r_k)$ $\sim$ $(i_0, \cdots, i_k)$, there exists a natural transformation $$\psi_r \colon (s_{r_1} \cdots s_{r_k} \lambda) s^{r_0} ([n+1]) {\longrightarrow}\widehat{\sigma s^{i_0}}|{\partial}{\widetilde{\Delta}}^{n+1},$$ which is a weak equivalence.
2. In other to lighten the notation, we denote $\phi_r:= \phi_{r_0\cdots r_k}^{n+1}$. For any other sequence $(t_0, \cdots, t_k) \sim (i_0, \cdots, i_k)$, the following square commutes for every $x \in {\partial}{\widetilde{\Delta}}^{n+1}$. $$\xymatrix{\lambda s^{i_k} \cdots s^{i_0} ([n+1]) \ \ \ar@{>->}[r]^-{\phi_r}_-{\sim} \ar@{ >->}[d]_-{\phi_t}^-{\sim} & (s_{r_1}\cdots s_{r_k} \lambda)s^{r_0} ([n+1]) \ar[d]^-{\psi_r}_-{\sim} \\
(s_{t_1} \cdots s_{t_k} \lambda) s^{t_0} ([n+1]) \ar[r]_-{\psi_t}^-{\sim} & {\gamma_{\sigma}}(x)}$$
We begin with the first part. Let $0 \leq l \leq n+1$. We need to define $$\psi_r \colon (s_{r_1} \cdots s_{r_k} \lambda) s^{r_0} ([n+1]) {\longrightarrow}(d_l \widehat{\sigma s^{i_0}}) (x)$$ for every $x \in {\partial}^l {\widetilde{\Delta}}^{n+1}$. We define $\psi_r$ when $x = [n+1]_l \cong [n]$. Then for the other values of $x$, $\psi_r$ is defined as the obvious composition. We will leave to the reader to check that the map $\psi_r$ is indeed a natural transformation. Let us treat only the case $l < j=i_0$, the cases $l > j+1$ and $l \in \{j, j+1\}$ being similar.
First of all, define $d_l s_{i_0}s_{i_1} \cdots s_{i_k} \lambda([n]):= s_{i_0-1}d_ls_{i_1} \cdots s_{i_k} \lambda ([n])$ and $d_l s_{r_0}s_{r_1} \cdots s_{r_k} \lambda ([n]):=$ $s_{r_0'}d_{l'} s_{r_1} \cdots s_{r_k} \lambda ([n])$, where $(r_0', l', r_1, \cdots, r_k) \sim (r_0, r_1, \cdots, r_k)$. On the one side we have
$$\begin{aligned}
d_l \widehat{\sigma s^{i_0}} ([n]) & = & s_{i_0-1}d_ls_{i_1} \cdots s_{i_k} \lambda ([n]) \textup{ by definition} \\
& = & d_l s_{i_0}s_{i_1} \cdots s_{i_k}\lambda ([n]) \textup{ by definition} \\
& = & d_l s_{r_0}s_{r_1} \cdots s_{r_k} \lambda ([n]) \textup{ by hypothesis and (IH2)} \\
& = & s_{r_0'} \cdots s_{r_k'}d_{l''} \lambda ([n]) \textup{ by definition and (IH2)}
\end{aligned}$$
On the other side, we have the weak equivalences $$(s_{r_1} \cdots s_{r_k} \lambda) s^{r_0} ([n+1]) \stackrel{\sim}{{\longrightarrow}} d_l ((s_{r_1} \cdots s_{r_k} \lambda)s^{r_0})([n]) \stackrel{\sim}{{\longrightarrow}} s_{r_0'}d_{l'} s_{r_1} \cdots s_{r_k} \lambda ([n])$$ The first one is obvious, while the second is nothing but (IH3). Using (IH2), we get that $s_{r_0'}d_{l'} s_{r_1} \cdots s_{r_k} \lambda ([n])$ $=$ $s_{r_0'}s_{r_1'} \cdots s_{r_k'} d_{l''} \lambda ([n])$. We thus obtain the required map $$\psi_r \colon (s_{r_1} \cdots s_{r_k} \lambda) s^{r_0} ([n+1]) \stackrel{\sim}{{\longrightarrow}} s_{r_0'}s_{r_1'} \cdots s_{r_k'} d_{l''} \lambda ([n]) = d_l \widehat{\sigma s^{i_0}} ([n]).$$ If $d_l$ happens to disappear when using (IH2) or $d_{l''}\lambda$ happens to be a degenerate simplex, we define $\psi_r$ is in a similar fashion.
The second part is straightforward. This ends the proof.
1. From the definition, it is straightforward to see that $s_j(\sigma)={\mathcal{R}}(\sigma s^j)$ whenever $\sigma$ and all of its faces are non-degenerate.
2. A natural question one may ask is to know why we do not define all $s_j$’s by the simple formula ${\mathcal{R}}(\sigma s^j)$. The reason is the fact that if we do so, the simplicial identity $s_is_j = s_{j+1}s_i, i \leq j$, won’t hold when $i< j$.
Consider the $1$-simplex $$\sigma = \left\{\xymatrix{X_0 & & X_{01} \ar[ll]_-{d_1} \ar[rr]^-{d_0} & & X_1 } \right\},$$ where $X_i:= \sigma(\{i\}), X_{01}:= \sigma(\{0, 1\}),$ and $d_i:= \sigma(d^i)$. We have
$$\sigma s^0 = \xymatrix{ & & & X_0 & & & \\
& & X_0 \ar[ru]^-{id} \ar[lldd]_-{id} & & X_{01} \ar[lu]_-{d_1} \ar[rrdd]^-{d_0} & & \\
& & & X_{01} \ar[lu]_-{d_1} \ar[ru]^-{id} \ar[d]_-{id} & & & \\
X_0 & & & X_{01} \ar[rrr]_-{d_0} \ar[lll]^-{d_1} & & & X_1 }$$ One can notice that the $1$-face $\{X_0 \stackrel{id}{\leftarrow} X_0 \stackrel{id}{\rightarrow} X_0\}$ is not fibrant as the matching map $(id, id) \colon X_0 {\longrightarrow}X_0 \times X_0$ is the diagonal map, which is not a fibration. The two other $1$-faces are both equal to $\sigma$, which is fibrant. Since one face is not fibrant, it follows that the whole diagram $\sigma s^0$ is not fibrant as well. If $\sigma$ is non-degenerate, by applying ${\mathcal{R}}$ to $\sigma s^0$ we get $s_0\sigma$, which looks like $$s_0\sigma = {\mathcal{R}}(\sigma s^0) = \xymatrix{ & & & X_0 & & & \\
& & \widetilde{X}_0 \ar[ru] \ar[lldd] & & X_{01} \ar[lu]_-{d_1} \ar[rrdd]^-{d_0} & & \\
& & & \widetilde{X}_{01} \ar[lu] \ar[ru] \ar[d] & & & \\
X_0 & & & X_{01} \ar[rrr]_-{d_0} \ar[lll]^-{d_1} & & & X_1 }$$
\[di-sk\_prop\] The collection ${\widehat{A}_{\bullet}}= \{{\widehat{A}}_n\}_{n\geq 0}$ endowed with $d_i \colon {\widehat{A}}_n {\longrightarrow}{\widehat{A}}_{n-1}$ and $s_j \colon {\widehat{A}}_n {\longrightarrow}{\widehat{A}}_{n+1}$ forms a simplicial set.
The simplicial identity $d_id_j = d_{j-1}d_i, i < j,$ follows from the definition and Proposition \[cosimpl\_rel\_prop\], while the other four simplicial identities follow immediately from the construction of $s_j$.
\[dibeta\_rmk\] For every $n \geq 0$, for every $\sigma, \sigma' \in {\widehat{A}_n}$, for any natural transformation $\beta \colon \sigma {\longrightarrow}\sigma'$, there exist two natural transformations $d_i \beta \colon d_i (\sigma) {\longrightarrow}d_i(\sigma)$ and $s_j\beta \colon s_j(\sigma) {\longrightarrow}s_j(\sigma')$ for all $i, j$. This is straightforward to prove by induction on $n$.
### Proving that ${\widehat{A}_{\bullet}}$ is a Kan complex
\[ahd\_prop\] The simplicial set ${\widehat{A}_{\bullet}}$ is a Kan complex.
Let $n \geq 0$, and let $k \in \{0, \cdots, n\}$. Consider $n$ $(n-1)$-simplices $\sigma_0, \cdots, \widehat{\sigma}_k, \cdots, \sigma_n$ of ${\widehat{A}_{\bullet}}$ such that $
d_i (\sigma_j) = d_{j-1} (\sigma_i), i < j,$ and $i, j \neq k.
$ Our goal is to construct an $n$-simplex $\sigma \in {\widehat{A}}_n$ such that $d_i\sigma = \sigma_i$ for all $i \neq k$. We first need to construct an intermediate functor $\overline{\sigma} \colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$. Recall the poset ${\widetilde{\Lambda}^n_k}$ from Definition \[lnk\_defn\]. For $i \in \{0, \cdots, n\}, i \neq k,$ define $\overline{\sigma} \colon {\widetilde{\Lambda}^n_k}{\longrightarrow}{{\mathcal{C}}_A}$ as $\overline{\sigma} (\alpha) := \sigma_i (\alpha)$ for $\alpha \in {\partial}^i {\widetilde{\Delta}^n}\cong {\widetilde{\Delta}}^{n-1}$. It is straightforward to see that this is well defined on the intersection ${\partial^i}{\widetilde{\Delta}^n}\cap {\partial^j}{\widetilde{\Delta}^n}$. Now define $$\overline{\sigma}([n]) := Q \underset{\alpha \in {\widetilde{\Lambda}^n_k}}{\lim} \overline{\sigma}(\alpha), \quad \overline{\sigma}([n]_k):= \overline{\sigma}([n]), \quad \text{and} \quad \overline{\sigma}(d^k):= id,$$ where $d^k \colon [n]_k {\longrightarrow}[n]$ is a morphism of ${\widetilde{\Delta}^n}$. For $\alpha' \in {\widetilde{\Lambda}^n_k}$, and $d^{\alpha'[n]} \colon \alpha' {\longrightarrow}[n]$ a morphism of ${\widetilde{\Delta}^n}$, define $\overline{\sigma}(d^{\alpha'[n]})$ as the composition $$\xymatrix{Q \underset{\alpha \in {\widetilde{\Lambda}^n_k}}{\lim} \overline{\sigma}(\alpha) \ar[rr]_-{\sim} & & \underset{\alpha \in {\widetilde{\Lambda}^n_k}}{\lim} \overline{\sigma}(\alpha) \ar[rr]^-{p_{\alpha'}} & & \overline{\sigma}(\alpha')},$$ where $p_{\alpha'}$ is the canonical projection as usual. Define $\sigma:= {\mathcal{R}}(\overline{\sigma})$. By construction and Proposition \[we\_prop\], the functor $\overline{\sigma} \colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ thus defined belongs to ${\widehat{A}_n}$. Moreover, using the properties of ${\mathcal{R}}$ from Proposition \[mcalr\_prop\], one has $d_i \sigma = \sigma_i$ for all $i \neq k$. This proves the proposition.
The functor categories ${{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$ and ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ {#fautm_futma_section}
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From now on, we let ${\mathcal{T}^M}$ denote a triangulation of $M$. Let ${{\mathcal{C}}_A}\subseteq {\mathcal{M}}$ be the small subcategory constructed in Section \[ca\_subsection\]. In this section we recall an important poset ${\mathcal{U}(\mathcal{T}^M)}$ and introduce two categories: ${{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$ and ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$. The first one is the category of isotopy functors ${\mathcal{U}(\mathcal{T}^M)}{\longrightarrow}{\mathcal{M}}$ that send every object to an object weakly equivalent to $A$, while the second is the category of isotopy functors from ${\mathcal{U}(\mathcal{T}^M)}$ to ${{\mathcal{C}}_A}$. By the definitions there is an inclusion functor $\phi \colon {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}{\hookrightarrow}{{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$. The goal of this section is to prove (\[res\_eqn4\]) or Proposition \[fuca\_prop\] below, which says that the localization of $\phi$ with respect to weak equivalences is an equivalence of categories. We begin with the definition of ${\mathcal{U}(\mathcal{T}^M)}$.
The poset ${\mathcal{U}(\mathcal{T}^M)}$
----------------------------------------
The poset ${\mathcal{U}(\mathcal{T}^M)}$ was introduced by the authors in [@paul_don17 Section 4.1]. We recall its definition, and refer the reader to [@paul_don17 Section 4.1] for more explanation.
\[utm\_defn\] First take two barycentric subdivisions of ${\mathcal{T}^M}$, and then define $U_{\sigma}$ as the interior of the star of $\sigma$. An object of ${\mathcal{U}(\mathcal{T}^M)}$ is defined to be $U_{\sigma}, \sigma \in {\mathcal{T}^M}$. There is a morphism $U_{\sigma} {\longrightarrow}U_{\sigma'}$ if and only if $\sigma$ is a face of $\sigma'$. In other words, morphisms of ${\mathcal{U}(\mathcal{T}^M)}$ are just inclusions.
The poset ${\mathcal{U}(\mathcal{T}^M)}$ is related to the poset $P({\mathcal{T}^M})$ we introduced in Definition \[pt\_defn\] as follows.
\[utm\_pt\_rmk\] From the definitions, one has a canonical isomorphism ${\mathcal{U}(\mathcal{T}^M)}\stackrel{\cong}{{\longrightarrow}} P({\mathcal{T}^M}), U_{\sigma} \mapsto \sigma$. When ${\mathcal{T}^M}= {\Delta^n}$, there is an isomorphism ${\mathcal{U}}({\Delta^n}) \cong {\widetilde{\Delta}^n}$, where ${\widetilde{\Delta}^n}$ is the poset from Definition \[dtn\_defn\].
\[utm\_rmk\]
1. It is clear that every morphism of ${\mathcal{U}(\mathcal{T}^M)}$ is a composition of $d^i$’s, where $$\begin{aligned}
\label{di_eqn}
d^i \colon U_{\langle v_{a_0}, \cdots, \widehat{v}_{a_i}, \cdots, v_{a_n}\rangle} {\hookrightarrow}U_{\langle v_{a_0}, \cdots, v_{a_n}\rangle}
\end{aligned}$$
2. It is also clear that every morphism of ${\mathcal{U}(\mathcal{T}^M)}$ is an isotopy equivalence since the inclusion of one open ball of $M$ inside another one is always an isotopy equivalence [@hirsch76 Chapter 8].
The functor categories ${{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$ and ${{\mathcal{F}}_A({\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}; {\mathcal{M}})}$
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Here we prove (\[res\_eqn3\]) or Proposition \[fum\_butm\_prop\] below. We begin with a couple of definitions.
\[butm\_defn\] Define ${\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}$ to be the collection of all subsets $B$ of $M$ diffeomorphic to an open ball such that $B$ is contained in some $U_{\sigma} \in {\mathcal{U}(\mathcal{T}^M)}$.
Certainly ${\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}$ is a basis for the topology of $M$ that contains ${\mathcal{U}(\mathcal{T}^M)}$. Often ${\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}$ will be thought of as the poset whose objects are $B \in {\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}$, and whose morphisms are inclusions.
\[fum\_defn\] Let ${\mathcal{S}}\subseteq {\mathcal{O}(M)}$ be a subcategory. Define ${\mathcal{F}}_A({\mathcal{S}}; {\mathcal{M}})$ to be the category whose objects are isotopy functors (see [@paul_don17-2 Definition 4.4]) $F \colon {\mathcal{S}}{\longrightarrow}{\mathcal{M}}$ such that for every $U \in {\mathcal{S}}$, $F(U)$ is weakly equivalent to $A$. Of course morphisms are natural transformations.
We are mostly interested in the case when ${\mathcal{S}}= {\mathcal{U}(\mathcal{T}^M)}$ or ${\mathcal{S}}={\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}$.
If two categories ${\mathcal{C}}$ and ${\mathcal{D}}$ are weakly equivalent in the sense of [@paul_don17-2 Definition 6.3], we write ${\mathcal{C}}\simeq {\mathcal{D}}$.
\[fum\_butm\_prop\] The category ${{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$ is weakly equivalent to the category ${{\mathcal{F}}_A({\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}; {\mathcal{M}})}$. That is, $${{\mathcal{F}}_A({\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}; {\mathcal{M}})}\simeq {{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}.$$
Define $\varphi \colon {{\mathcal{F}}_A({\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}; {\mathcal{M}})}{\longrightarrow}{{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$ as the restriction to ${\mathcal{U}(\mathcal{T}^M)}$. That is, $\varphi(G):= G|{\mathcal{U}(\mathcal{T}^M)}$. To define a functor $\psi$ in the other way, let $F \colon {\mathcal{U}(\mathcal{T}^M)}{\longrightarrow}{\mathcal{M}}$ be an object of ${{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$. For $B \in {\mathcal{B}_{\mathcal{U}(\mathcal{T}^M)}}$ define $\psi(F)(B):= F(U_{\sigma_B})$, where $U_{\sigma_B}$ is provided by the axiom $(C_3)$ from [@paul_don17 Definition 4.1]. From the same axiom, one can define $\psi(F)$ on morphisms in the standard way. If $\beta \colon F {\longrightarrow}F'$ is a morphism of ${{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$, define $\psi(\eta)(B):= \beta[U_{\sigma_B}]$. It is straightforward to see that $\varphi$ and $\psi$ preserve weak equivalences. It is also straightforward to check that $\phi \psi = id$ and $\psi \phi \simeq id$. This proves the proposition.
Proving that ${{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}\ \simeq \ {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ {#fautm_futma_subsection}
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The goal here is to prove (\[res\_eqn4\]) or Proposition \[fuca\_prop\] below, which says that the localization of ${{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$ is equivalent to the localization of a certain small category ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ that we now define.
\[fuca\_defn\]
1. Define ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ as the category of isotopy functors (see [@paul_don17-2 Definition 4.4]) from ${\mathcal{U}(\mathcal{T}^M)}$ to ${{\mathcal{C}}_A}$. To simplify the notation, we will often write ${\mathcal{K}}$ for ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ in this section. That is, ${\mathcal{K}}:= {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}.$ Weak equivalences of ${\mathcal{K}}$ are natural transformations which are objectwise weak equivalences. We denote the class of weak equivalences of ${\mathcal{K}}$ by ${\mathcal{W}}_{{\mathcal{K}}}$.
2. Define ${\mathcal{L}}:= {{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})}$ (see Definition \[fum\_defn\]). We denote the class of weak equivalences of ${\mathcal{L}}$ by ${\mathcal{W}}_{{\mathcal{L}}}$ .
By Definition \[fuca\_defn\] and Propsotion \[ca\_prop\], one has ${\mathcal{K}}\subseteq {\mathcal{L}}$.
The subcategory ${\mathcal{K}}\subseteq {\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}$ might not be a model category on its own right as it might not be closed under factorizations or under taking small limits. The same remark applies to ${\mathcal{L}}$.
Nevertheless ${\mathcal{K}}$ and ${\mathcal{L}}$ are closed under certain things described in Proposition \[mcalrb\_prop\] below. Before we state it, we need to introduce a functor. Let $n$ be the dimension of $M$. Consider the fibrant replacement functor ${\mathcal{R}}\colon {{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}} {\longrightarrow}{{\mathcal{C}}_A}^{{\widetilde{\Delta}^n}}$ from Section \[mcalr\_subsection\]. According to Remark \[utm\_pt\_rmk\], one has an isomorphism ${\mathcal{U}}({\Delta^n}) \cong {\widetilde{\Delta}^n}$. This enables us to regard ${\mathcal{R}}$ as a functor ${\mathcal{R}}\colon {{\mathcal{C}}_A}^{{\mathcal{U}}({\Delta^n})} {\longrightarrow}{{\mathcal{C}}_A}^{{\mathcal{U}}({\Delta^n})}$. From the definition of ${\mathcal{R}}$, and the fact that the simplicial complex ${\mathcal{T}^M}$ can be built up by gluing together the $\Delta^n$’s, the functor ${\mathcal{R}}$ can be extended in the obvious way to a functor that we denote $$\begin{aligned}
\label{mcalrb_eqn}
{\overline{{\mathcal{R}}}}\colon {{\mathcal{C}}_A}^{{\mathcal{U}(\mathcal{T}^M)}} {\longrightarrow}{\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}. \end{aligned}$$
\[mcalrb\_prop\]
1. For every $F \in {\mathcal{K}}$, ${\overline{{\mathcal{R}}}}F$ belongs to ${\mathcal{K}}$.
2. The category ${\mathcal{L}}$ is closed under taking cofibrant and fibrant replacements.
3. For every $F \in {\mathcal{K}}$ (respectively $G \in {\mathcal{L}}$) there exists a cylinder object $F \times I$ in ${\mathcal{K}}$ (respectively $G \times I$ in ${\mathcal{L}}$)
So it makes sense to talk about homotopy in ${\mathcal{K}}$ and ${\mathcal{L}}$ provided that the source is cofibrant and the target is fibrant.
Parts (i) and (ii) follow from the definitions. Regarding (iii), let $F \in {\mathcal{K}}$ and let $U \in {\mathcal{U}(\mathcal{T}^M)}$. Consider (\[mcalh\_eqn\]) with $F(U)$ in place of $X$, and define $F \times I \colon {\mathcal{U}(\mathcal{T}^M)}{\longrightarrow}{{\mathcal{C}}_A}$ on objects as $(F \times I)(U):= Z_{F(U)}$, and in the obvious way on morphisms. Since weak equivalences and cofibrations are both objectwise, it follows that $F \times I$ is a cylinder object for $F$. Certainly $F \times I$ belongs to ${\mathcal{K}}$. A similar construction can be performed in ${\mathcal{L}}$.
Before we state and prove the main result (Proposition \[fuca\_prop\]) of this section, we need three preparatory lemmas. We will use the notation and terminology from Section \[local\_category\_subsection\].
\[esur\_lem\] Consider the categories ${\mathcal{K}}$ and ${\mathcal{L}}$ from Definition \[fuca\_defn\]. Then the functor $$\begin{aligned}
\label{phi_eqn}
\phi \colon {\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}] {\longrightarrow}{\mathcal{L}}[{\mathcal{W}}_{{\mathcal{L}}}^{-1}], \quad F \mapsto F,
\end{aligned}$$ induced by the inclusion functor ${\mathcal{K}}{\hookrightarrow}{\mathcal{L}}$ is essentially surjective.
Let $n=\text{dim} M$ as above. Since the simplicial complex ${\mathcal{T}^M}$ can be built up by gluing together the ${\Delta^n}$’s, it is enough to prove the lemma when ${\mathcal{T}^M}= \Delta^n$. Set ${\mathcal{K}}_n = {\mathcal{F}}({\mathcal{U}}(\Delta^n); {{\mathcal{C}}_A})$ and ${\mathcal{L}}_n = {\mathcal{F}}_A({\mathcal{U}}(\Delta^n); {\mathcal{M}})$. The idea of the proof is to proceed by induction on $n$ by showing that for all $n \geq 0$, for all $F \in {\mathcal{L}}_n[{\mathcal{W}}_{{\mathcal{L}}_n}^{-1}]$, there exist ${\overline{G}}\in {\mathcal{K}}_n[{\mathcal{W}}_{{\mathcal{K}}_n}^{-1}]$ and a zigzag of weak equivalences $\xymatrix{F \ar[r]^-{\sim} & RF & QRF \ar[l]_-{\sim} \ar[r]^-{\sim}_-{{\overline{\beta}}} & {\overline{G}}}.$
For $n=0$, the standard geometric simplex $\Delta^n$ has only one vertex, say $v$. So ${\mathcal{U}}(\Delta^0) = \{U_v\}$. Let $F \colon {\mathcal{U}}(\Delta^0) {\longrightarrow}{\mathcal{M}}$ be an object of ${\mathcal{L}}_0[{\mathcal{W}}_{{\mathcal{L}}_0}^{-1}]$. Define ${\overline{G}}\colon {\mathcal{U}}(\Delta^0) {\longrightarrow}{{\mathcal{C}}_A}$ as ${\overline{G}}(U_v):= QRA$. By the definition of ${\mathcal{L}}_0$, the object $F(U_v)$ is weakly equivalent to $QRA$. So $QRF(U_v)$ is also weakly equivalent to $QRA$, that is, there exists a zigzag of weak equivalences $$\begin{aligned}
\label{zz_eqn}
\xymatrix{QRF(U_v) & A_1 \ar[l]_-{\sim}^-{f_0} \ar[r]^-{\sim}_-{f_1} & A_2 & \cdots \ar[l]_-{\sim} \ar[r]^-{\sim} & A_{s-1} & A_s \ar[l]_-{\sim}^-{f_{s-1}} \ar[r]^-{\sim}_-{f_s} & QRA.}
\end{aligned}$$ Using standard techniques from model categories and the fact that the objects $QRF(U_v)$ and $QRA$ are both fibrant and cofibrant, one can replace (\[zz\_eqn\]) by a direct morphism ${\overline{\beta}}[U_v] \colon QRF(U_v) \stackrel{\sim}{{\longrightarrow}} QRA= {\overline{G}}(U_v)$. This proves the base case. Assume that the statement is true for all $k \leq n-1$, and let $F \in {\mathcal{L}}_n[{\mathcal{W}}_{{\mathcal{L}}_n}^{-1}]$. We need to find ${\overline{G}}\in {\mathcal{K}}_n[{\mathcal{W}}_{{\mathcal{K}}_n}^{-1}]$ and a weak equivalence ${\overline{\beta}}\colon QRF \stackrel{\sim}{{\longrightarrow}} {\overline{G}}$. First of all let $\Delta^n = {\langle v_0, \cdots, v_n\rangle}$, and define $\partial {\mathcal{U}}(\Delta^n) \subseteq {\mathcal{U}}(\Delta^n)$ as the full subposet whose objects are $U_{\sigma}$’s with $\sigma$ be a simplex of the boundary of $\Delta^n$. By the induction hypothesis there exist an isotopy functor $G \colon \partial {\mathcal{U}}(\Delta^n) {\longrightarrow}{{\mathcal{C}}_A}$ and a natural transformation $\beta \colon QRF|\partial {\mathcal{U}}(\Delta^n) \stackrel{\sim}{{\longrightarrow}} G$. From the base case, there is a weak equivalence $g \colon QRF(U_{{\langle v_0, \cdots, v_n\rangle}}) \stackrel{\sim}{{\longrightarrow}} QRA$. Since $QRA$ and $QRF(U_{{\langle v_0, \cdots, v_n\rangle}})$ are both fibrant and cofibrant, it follows by Proposition \[ho\_prop\] that $g$ admits a homotopy inverse, say $f$. Let $
H \colon QRF(U_{{\langle v_0, \cdots, v_n\rangle}}) \times I {\longrightarrow}QRF(U_{{\langle v_0, \cdots, v_n\rangle}})
$ denote a homotopy from $fg$ to $id$. Now define $G' \colon {\mathcal{U}}(\Delta^n) {\longrightarrow}{{\mathcal{C}}_A}$ as $$G'|\partial {\mathcal{U}}(\Delta^n) := G, G'(U_{{\langle v_0, \cdots, v_n\rangle}}) := QRA,
\text{ and }
G'(d^i) := \beta[U_{{\langle v_0, \cdots, \widehat{v}_i, \cdots, v_n\rangle}}] \circ F(d^i) \circ f,$$ where $d^i$ is the map from (\[di\_eqn\]). On the compositions we define $G'$ in the obvious way (that is, $G'(a \circ b) := G'(b) \circ G'(a)$). It is straightforward to see that $G'$ is a contravariant functor. One may take ${\overline{G}}$ to be $G'$ and ${\overline{\beta}}$ to be $\beta'$, where $\beta' \colon F {\longrightarrow}G'$ is defined as $\beta$ on $\partial {\mathcal{U}}(\Delta^n)$ and $g$ on $U_{{\langle v_0, \cdots, v_n\rangle}}$. The issue with that definition is the fact that the square involving $G'(d^i)$, $F(d^i)$, $g$ and $\beta[U_{{\langle v_0, \cdots, \widehat{v}_i, \cdots, v_n\rangle}}]$ is only commutative up to homotopy. To fix this, consider the following commutative diagram. $$\xymatrix{QRF(U_{{\langle v_0, \cdots, v_n\rangle}}) \ar[rr]^-{g}_-{\sim} \ar@{>->}[dd]_-{i_0}^-{\sim} & & G'(U_{{\langle v_0, \cdots, v_n\rangle}}) \ar[dd]_-{p} \ar@{>->}[rd]^-{\sim}_{\tau} & \\
& & & Z_{(A,G',p)} \ar@{->>}[ld]^-{{\overline{p}}} \\
QRF(U_{{\langle v_0, \cdots, v_n\rangle}}) \times I \ar[rr]_-{{\mathcal{H}}} \ar@{.>}[rrru]^-{\overline{{\mathcal{H}}}} & & \underset{U_{\sigma} \in \partial {\mathcal{U}}(\Delta^n)}{\text{lim}} G'(U_{\sigma}) & }$$ In that diagram the square involving $g, {\mathcal{H}}, i_0$ and $p$ is induced by the commutative square $$\xymatrix{QRF(U_{{\langle v_0, \cdots, v_n\rangle}}) \ar[rrrr]^-{g} \ar@{>->}[d]_-{i_0} & & & & G'(U_{{\langle v_0, \cdots, v_n\rangle}}) \ar[d]_-{\sim}^-{G'(d^i)} \\
QRF(U_{{\langle v_0, \cdots, v_n\rangle}}) \times I \ar[rrrr]_-{\beta[U_{{\langle v_0, \cdots, \widehat{v}_i, \cdots, v_n\rangle}}]F(d^i)H} & & & & G'(U_{{\langle v_0, \cdots, \widehat{v}_i, \cdots, v_n\rangle}}),}$$ while ${\overline{p}}\tau$ is the factorization of $p$ such that $Z_{(QRA, G', p)}$ belongs to ${{\mathcal{C}}_A}$, and $\overline{{\mathcal{H}}}$ is given by the lifting axiom. Now define ${\overline{G}}\colon {\mathcal{U}}(\Delta^n) {\longrightarrow}{{\mathcal{C}}_A}$ as $${\overline{G}}|\partial {\mathcal{U}}(\Delta^n):= G', \quad {\overline{G}}(U_{{\langle v_0, \cdots, v_n\rangle}}):= Z_{(QRA, G', p)} \quad \text{and} \quad {\overline{G}}(d^i):= P_{U_{\sigma_i}} \circ {\overline{p}},$$ where $
\sigma_i:= {\langle v_0, \cdots, \widehat{v}_i, \cdots, v_n\rangle}$ and $P_{U_{\sigma_i}} \colon \underset{U_{\sigma} \in \partial {\mathcal{U}}(\Delta^n)}{\text{lim}} G'(U_{\sigma}) {\longrightarrow}G'(U_{\sigma_i})
$ is the canonical projection. Also define $${\overline{\beta}}|\partial {\mathcal{U}}(\Delta^n) := \beta \qquad \text{and} \qquad {\overline{\beta}}[U_{{\langle v_0, \cdots, v_n\rangle}}] := \overline{{\mathcal{H}}} \circ i_1,$$ where $i_1 \colon QRF(U_{{\langle v_0, \cdots, v_n\rangle}}) {\longrightarrow}QRF(U_{{\langle v_0, \cdots, v_n\rangle}}) \times I$ is the canonical inclusion. It is straightforward to see that ${\overline{\beta}}$ is a weak equivalence. This proves the lemma.
\[full\_lem\] Let ${\mathcal{K}}$ and ${\mathcal{L}}$ be as in Lemma \[esur\_lem\]. Then the functor $\phi$ from (\[phi\_eqn\]) is full.
Let $F_1, F_2 \in {\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}]$. We need to show that the canonical map $$\begin{aligned}
\label{phii_eqn}
\phi_{F_1F_2} \colon \underset{{\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}]}{\text{Hom}} (F_1, F_2) {\longrightarrow}\underset{{\mathcal{L}}[{\mathcal{W}}_{{\mathcal{L}}}^{-1}]}{\text{Hom}} (\phi(F_1), \phi(F_2))
\end{aligned}$$ induced by $\phi$ is surjective. To do this, let $f_1 \colon QF_1 \stackrel{\sim}{{\longrightarrow}} F_1$ be a cofibrant replacement of $F_1$. One can take $QF_1 = F_1$ and $f_1 =id$ by Propositions \[model\_prop\], \[ca\_prop\]. Also let $f_2 \colon F_2 \stackrel{\sim}{{\longrightarrow}} {\overline{{\mathcal{R}}}}F_2$ be a fibrant replacement of $F_2$, which lies in ${\mathcal{K}}$ thanks to Proposition \[mcalrb\_prop\]. Consider the following commutative square. $$\begin{aligned}
\label{full_eqn}
\xymatrix{\underset{{\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}]}{\text{Hom}} (F_1, F_2) \ar[rr]^-{\phi_{F_1F_2}} & & \underset{{\mathcal{L}}[{\mathcal{W}}_{{\mathcal{L}}}^{-1}]}{\text{Hom}} (\phi(F_1),\phi(F_2)) \\
\underset{{\mathcal{K}}}{\text{Hom}} (F_1, {\overline{{\mathcal{R}}}}F_2) \ar@{=}[r] \ar[u]^-{\theta} & \underset{{\mathcal{L}}}{\text{Hom}} (F_1, {\overline{{\mathcal{R}}}}F_2) \ar[r]_-{\pi} & \underset{{\mathcal{L}}}{\text{Hom}} (F_1, {\overline{{\mathcal{R}}}}F_2) \slash \sim \ar[u]_-{\cong}^-{\theta'} }
\end{aligned}$$ In that square $\theta$ is defined as the string $\theta(f):= (f, f_2^{-1})$, $\pi$ is the canonical surjection, and the isomorphism $\theta'$ comes from Proposition \[ho\_prop\]. The equality comes from the fact that ${\mathcal{K}}$ is a full subcategory of ${\mathcal{L}}$ by definition. Since the composition $\theta' \circ \pi$ is surjective, and since the square commutes, it follows that $\phi_{F_1F_2}$ is a surjective map. This ends the proof.
\[ffull\_lem\] Let ${\mathcal{K}}$ and ${\mathcal{L}}$ be as in Lemma \[esur\_lem\]. Then the functor $\phi$ from (\[phi\_eqn\]) is faithful.
Let $F_1, F_2 \in {\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}]$. As we mentioned in the proof of Lemma \[full\_lem\], $QF_1 = F_1$ and ${\overline{{\mathcal{R}}}}F_2$ belongs to ${\mathcal{K}}$. We need to show that the map $\phi_{F_1F_2}$ from (\[phii\_eqn\]) is injective. Consider the diagram (\[phii\_eqn\]), and let $\eta_0, \eta_1 \in \underset{{\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}]}{\text{Hom}} (F_1, F_2)$ such that $\phi_{F_1F_2}(\eta_0) = \phi_{F_1F_2}(\eta_1)$. This latter equality implies that $\eta'_0$ is homotopic to $\eta'_1$, where $\eta'_i := \theta'^{-1} (\phi_{F_1F_2}(\eta_i))$. By Propositions \[htpy\_prop\], \[mcalrb\_prop\], there is a left homotopy $H \colon F_1 \times I {\longrightarrow}{\overline{{\mathcal{R}}}}F_2$ for some cylinder object $F_1 \times I \in {\mathcal{L}}$ for $F_1$. The key point of the proof is the fact that one can always choose $F_1 \times I$ in ${\mathcal{K}}$ thanks to Proposition \[mcalrb\_prop\]. Now consider the following commutative diagram in ${\mathcal{K}}$. $$\xymatrix{F_1 \ar[rrd]^-{\eta'_0} \ar[d]_-{i_0} & & \\
F_1 \times I \ar[rr]^-{H} & & {\overline{{\mathcal{R}}}}F_2 \\
F_1 \ar[rru]_-{\eta'_1} \ar[u]^-{i_1}}$$ Since $i_0=i_1$ in ${\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}]$ by Proposition \[izio\_prop\], it follows that $\eta'_0 = \eta'_1$ in ${\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}]$, which implies $\eta_0 = \eta_1$. This proves the lemma.
\[fuca\_prop\] Let ${\mathcal{K}}$ and ${\mathcal{L}}$ be as in Definition \[fuca\_defn\]. Then the inclusion ${\mathcal{K}}{\hookrightarrow}{\mathcal{L}}$ induces an equivalence of categories between the localizations ${\mathcal{K}}[{\mathcal{W}}_{{\mathcal{K}}}^{-1}]$ and ${\mathcal{L}}[{\mathcal{W}}_{{\mathcal{L}}}^{-1}]$.
This follows immediately from Lemmas \[esur\_lem\], \[full\_lem\] and \[ffull\_lem\] as it is well known that a fully faithful and essentially surjective functor is an equivalence of categories.
The map ${\overline{\Lambda}}\colon {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}\slash we {\longrightarrow}[{\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}}]$ {#lambda_section}
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Recall the category ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ (which was called ${\mathcal{K}}$ for convenience in Section \[fautm\_futma\_subsection\]) from Definition \[fuca\_defn\]. It turns out that it is a small category since ${\mathcal{U}(\mathcal{T}^M)}$ and ${{\mathcal{C}}_A}$ are both small. In this section and the next one, we will view ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ as a set. Choose a well-order on the set of vertices of ${\mathcal{T}^M}$, and let ${\mathcal{T}_{\bullet}^{M}}$ denote the canonical associated simplicial set. Recall the simplicial set ${\widehat{A}_{\bullet}}$ from Section \[ahd\_subsection\], and let ${\underset{\text{sSet}}{\text{Hom}}({\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}})}$ denote the set of simplicial maps ${\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$. The goal of this section is to define a map $\Lambda \colon {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}{\longrightarrow}{\underset{\text{sSet}}{\text{Hom}}({\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}})}$ and prove Proposition \[lambda\_prop\] below, which says that $\Lambda(F)$ is homotopic to $\Lambda(F')$ whenever $F$ is weakly equivalent to $F'$. The notion of homotopy between two simplicial maps we use is the one from [@goe_jar09 Section I.6].
We begin with the construction of $\Lambda$. Define $\Lambda(F):= f \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}, \sigma \mapsto f_{\sigma}$ as follows. First consider the fibrant replacement functor ${\overline{{\mathcal{R}}}}$ from (\[mcalrb\_eqn\]). Let $n \geq 0,$ and let $\sigma = \langle v_0, \cdots, v_n \rangle \in {\mathcal{T}^M}_n$. Depending on the fact that $\sigma$ is degenerate or not we need to deal with two cases.
1. If $\sigma$ is non-degenerate, define $f_{\sigma} \colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ on objects as $
f_{\sigma}(\{a_0, \cdots, a_s\}) := {\overline{{\mathcal{R}}}}(F)(U_{\langle v_{a_0}, \cdots, v_{a_s}\rangle}).
$ On morphisms, it is enough to define $f_{\sigma}$ on the generators $d^i$’s from Remark \[dtn\_rmk\]. For a morphism $
\theta^i \colon U_{\langle v_{a_0}, \cdots, \widehat{v}_{a_i}, \cdots, v_{a_s}\rangle} {\hookrightarrow}U_{\langle v_{a_0}, \cdots, v_{a_s}\rangle}
$ of ${\mathcal{U}(\mathcal{T}^M)}$, define $f_{\sigma}(d^i) :={\overline{{\mathcal{R}}}}(F)(\theta^i)$.
2. If $\sigma$ is degenerate, a classical result on simplicial sets claims the existence of a unique non-degenerate simplex $\lambda$ of some degree $p \leq n$ and a unique non-decreasing surjection $s \colon [n] {\longrightarrow}[p]$ such that $\sigma = {\mathcal{T}_{\bullet}^{M}}(s)(\lambda)$. Define $f_{\sigma} := {\widehat{A}_{\bullet}}(s)(f_{\lambda})$. Here ${\mathcal{T}_{\bullet}^{M}}$ and ${\widehat{A}_{\bullet}}$ are indeed viewed as contravariant functors from the standard simpicial category $\Delta$ to sets.
Certainly $f_{\sigma}$ belongs to ${\widehat{A}_n}$ and $f \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ is a simplicial map. This completes the definition of $\Lambda$.
Now on the set ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ define the equivalence relation $we$ as $F \ we \ F'$ if and only if $F$ is weakly equivalent to $F'$, that is, there is a zigzag of weak equivalences between $F$ and $F'$. Let $[{\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}}]$ be the set of homotopy classes of simplicial maps from ${\mathcal{T}_{\bullet}^{M}}$ to ${\widehat{A}_{\bullet}}$.
\[lamb\_defn\] Consider the map $\Lambda \colon {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}{\longrightarrow}{\underset{\text{sSet}}{\text{Hom}}({\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}})}$ and the equivalence relation $we$ we just defined. Thanks to Proposition \[lambda\_prop\] below $\Lambda$ passes to the quotient. Define ${\overline{\Lambda}}\colon {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}\slash we {\longrightarrow}[{\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}}]$ to be the resulting quotient map.
\[lambda\_prop\] Let $F, F' \in {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$. Assume that there is a zigzag of weak equivalences between $F$ and $F'$. Then $\Lambda(F)$ is homotopic to $\Lambda(F')$.
\[we\_defn\] Let $f, f' \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ be simplicial maps.
1. A morphism $\beta \colon f {\longrightarrow}f'$ consists of a collection $
\beta = \{\beta_{\sigma} \colon f_{\sigma} {\longrightarrow}f'_{\sigma}\}_{n \geq 0, \sigma \in {\mathcal{T}^M}_n}
$ of natural transformations such that $$\begin{aligned}
\label{cc_disk_eqn}
d_i \beta_{\sigma} = \beta_{d_i\sigma} \quad \text{and} \quad s_k\beta_{\sigma} = \beta_{s_k\sigma} \quad \text{for all $\sigma$, $i, k$}.
\end{aligned}$$ (see Remark \[dibeta\_rmk\].)
2. We say that $\beta$ is a *weak equivalence* if for every $n \geq 0$, for every $\sigma \in {\mathcal{T}_{\bullet}^{M}}$, $\beta_{\sigma}$ is a weak equivalence of ${\widetilde{\Delta}^n}$-diagrams.
\[htpy1\_lem\] Let $f, f' \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ be simplicial maps. Assume that there is a morphism $\beta \colon f {\longrightarrow}f'$ which is a weak equivalence (see Definition \[we\_defn\]). Then $f$ is homotopic to $f'$.
Idea of proof. For $n \geq 0$, we let $\Delta[n]$ be the standard simplicial model for $\Delta^n$. To prove Lemma \[htpy1\_lem\] we need to construct a homotopy $H \colon {\mathcal{T}_{\bullet}^{M}}\times \Delta[1] {\longrightarrow}{\widehat{A}_{\bullet}}$ between $f$ and $f'$. So, for all $n \geq 0, \sigma \in {\mathcal{T}^M}_n, t \in \Delta[1]_n$, we need to define $H(\sigma, t) \in {\widehat{A}_n}$. To get a better idea of the construction of $H(\sigma, t)$, let us treat a specific example. Let $n=2, \sigma = v_{012}:= \langle v_0, v_1, v_2\rangle$ and $t= 011$. We need the following equation, which comes from the fact that $f$ is a simplicial map and the definition of $d_i \colon {\widehat{A}}_n {\longrightarrow}{\widehat{A}}_{n-1}$. $$\begin{aligned}
\label{fdi_eqn}
f_{d_i \sigma}([n-1]) = f_{\sigma} ([n]_i), \quad \sigma \in {\mathcal{T}^M}_n. \end{aligned}$$ Recalling the functor ${\mathcal{R}}$ from Section \[mcalr\_subsection\], define $H(v_{012}, 011):= {\mathcal{R}}({\overline{H}})$, where ${\overline{H}}$ is the diagram from (\[hb\_eqn\]) defined as follows.
$$\begin{aligned}
\label{hb_eqn}
\xymatrix{ & & & {\overline{H}}(v_0, 0) & & & \\
& & {\overline{H}}(v_{01}, 01) \ar[ru]^-{\sim}_-{d_1} \ar[lldd]_-{\sim}^-{d_0} & & {\overline{H}}(v_{02}, 01) \ar[lu]_-{\sim}^-{d_1} \ar[rrdd]^-{\sim}_-{d_0} & & \\
& & & {\overline{H}}(v_{012}, 011) \ar[lu]_-{\sim}^-{d_2} \ar[ru]^-{\sim}_-{d_1} \ar[d]_-{\sim}^-{d_0} & & & \\
{\overline{H}}(v_1, 1) & & & {\overline{H}}(v_{12}, 11) \ar[rrr]_-{\sim}^-{d_0} \ar[lll]^-{\sim}_-{d_1} & & & {\overline{H}}(v_2, 1) }
\end{aligned}$$
${\overline{H}}(v_i, 0):= f_{v_i}([0])$ ${\overline{H}}(v_i, 1):= f'_{v_i}([0])$ ${\overline{H}}(v_{012}, 011):=f_{v_{012}}([2])$
------------------------------------------------ ------------------------------------------------ --------------------------------------------------
${\overline{H}}(v_{01}, 01):= f_{v_{01}}([1])$ ${\overline{H}}(v_{02}, 01):= f_{v_{02}}([1])$ ${\overline{H}}(v_{12}, 11):= f'_{v_{12}}([1])$
The morphism $d_2 \colon {\overline{H}}(v_{012}, 011) {\rightarrow}{\overline{H}}(v_{01}, 01)$ is defined as the composition $$\xymatrix{f_{v_{012}}([2]) \ar[rr]^-{f_{v_{012}}(d^2)} & & f_{v_{012}}([1]) = f_{v_{01}}([1])},$$ where the equality comes from (\[fdi\_eqn\]). The map $d_0 \colon {\overline{H}}(v_{012}, 011) {\longrightarrow}{\overline{H}}(v_{12}, 11)$ is defined as the composition $$\xymatrix{f_{v_{012}}([2]) \ar[rr]^-{f_{v_{012}}(d^0)} & & f_{v_{012}} (\{1, 2\}) = f_{v_{12}}([1]) \ar[rr]^-{\beta_{v_{12}}[1]} & & f'_{v_{12}}([1]).}$$ The other morphisms are defined in a similar fashion. The diagram ${\overline{H}}$ thus obtained commutes thanks to (\[cc\_disk\_eqn\]) and the fact that for every $\sigma$, $\beta_{\sigma}$ is a natural transformation. Note that there is a weak equivalence from $f_{v_{012}}([2])$ to the fibrant replacement of the boundary of $(\ref{hb_eqn})$, where $f_{v_{012}}([2])$ is viewed as the constant diagram at $f_{v_{012}}([2])$.
First of all recall the posets ${\partial {\widetilde{\Delta}^n}}$ and ${\partial^i}{\widetilde{\Delta}^n}$ from Definition \[pdtn\_defn\]. Our goal is to define a simplicial map $H \colon {\mathcal{T}_{\bullet}^{M}}\times \Delta[1] {\longrightarrow}{\widehat{A}_{\bullet}}$ making (\[htpy\_eqn\]) commute. We will proceed by induction on the skeletons ${\text{sk}}_n({\mathcal{T}_{\bullet}^{M}}\times \Delta[1])$. More precisely, we will show that for all $n \geq 0$, there exist a simplicial map $H_n \colon {\text{sk}}_n({\mathcal{T}_{\bullet}^{M}}\times \Delta[1]) {\longrightarrow}{\widehat{A}_{\bullet}}$ and a weak equivalence $$\begin{aligned}
\label{eta_eqn}
\eta_{n+1} \colon {f_{\sigma}}([n+1]) \stackrel{\sim}{{\longrightarrow}} {\mathcal{H}}_{(n+1)t}^{\sigma} \quad \forall \sigma \in {\mathcal{T}^M}_{n+1}, \forall t \in \Delta[1]_{n+1},
\end{aligned}$$ where ${\mathcal{H}}_{(n+1)t}^{\sigma} \colon \partial \widetilde{\Delta}^{n+1} {\longrightarrow}{{\mathcal{C}}_A}$, defined as $$\begin{aligned}
\label{mcalh0_eqn}
{\mathcal{H}}_{(n+1)t}^{\sigma}|{\partial^i}\widetilde{\Delta}^{n+1} := H_n(d_i\sigma, d_it),
\end{aligned}$$ is a fibrant diagram and ${f_{\sigma}}([n+1])$ is viewed as the constant functor at ${f_{\sigma}}([n+1])$.
$\bullet$ On the $0$-skeleton, define $H_0 \colon {\text{sk}}_0({\mathcal{T}_{\bullet}^{M}}\times \Delta[1]) {\longrightarrow}{\widehat{A}_{\bullet}}$ as $H_0(v, 0) := f_{v}$ and $H_0(v, 1):= f'_{v}$.
Let $\sigma = \langle v_0, v_1\rangle \in {\mathcal{T}^M}_1$ and let $t \in \Delta[1]_1$. Consider the poset $$\widetilde{\Delta}^1 = \left\{ \xymatrix{ \{0\} \ar[rr]^-{d^1} & & \{0, 1\} & & \{1\} \ar[ll]_-{d^0} } \right\},$$ and the functor ${\mathcal{H}}_{1t}^{\sigma} \colon \partial \widetilde{\Delta}^1 {\longrightarrow}{{\mathcal{C}}_A}$ from (\[mcalh0\_eqn\]). We need to define a natural transformation $\eta_1 \colon {f_{\sigma}}([1]) \stackrel{\sim}{{\longrightarrow}} {\mathcal{H}}_{1t}^{\sigma}$. Depending on the value of $t \in \{00, 11, 01\}$, we need to deal with three cases.
1. If $t=00$ then ${\mathcal{H}}_{1t}^{\sigma} \colon \partial \widetilde{\Delta}^1 {\longrightarrow}{{\mathcal{C}}_A}$ is given by ${\mathcal{H}}_{1t}^{\sigma}(\{0\}) = f_{v_0}(\{0\})$ and ${\mathcal{H}}_{1t}^{\sigma}(\{1\}) = f_{v_1}(\{0\})$. Note that by (\[fdi\_eqn\]) one has $f_{v_0}(\{0\}) = {f_{\sigma}}(\{0\})$ and $f_{v_1}(\{0\}) = {f_{\sigma}}(\{1\})$. Now define $$\eta_1[\{0\}] \colon {f_{\sigma}}(\{0, 1\}) {\longrightarrow}f_{v_0}(\{0\}) \quad \text{and} \quad \eta_1[\{1\}] \colon {f_{\sigma}}(\{0, 1\}) {\longrightarrow}f_{v_1}(\{0\})$$ as $\eta_1[\{0\}] := {f_{\sigma}}(d^1)$ and $\eta_1[\{1\}] := {f_{\sigma}}(d^0)$.
2. If $t = 11$ then ${\mathcal{H}}_{1t}^{\sigma}$ is given by ${\mathcal{H}}_{1t}^{\sigma}(\{0\}) = f'_{v_0}(\{0\})$ and ${\mathcal{H}}_{1t}^{\sigma}(\{1\}) = f'_{v_1}(\{0\})$. By hypothesis there is a weak equivalence $\beta_{\sigma} \colon {f_{\sigma}}\stackrel{\sim}{{\longrightarrow}} {f'_{\sigma}}$. Define $\eta_1[\{0\}]$ and $\eta_1[\{1\}]$ as $$\eta_1[\{0\}] := {\beta_{\sigma}}[\{0\}] \circ {f_{\sigma}}(d^1) \quad \text{and} \quad \eta_1[\{1\}] := {\beta_{\sigma}}[\{1\}] \circ {f_{\sigma}}(d^0).$$
3. If $t=01$ then ${\mathcal{H}}_{1t}^{\sigma}(\{0\}) = f_{v_0}(\{0\})$ and ${\mathcal{H}}_{1t}^{\sigma}(\{1\}) = f'_{v_1}(\{0\})$. Define $$\eta_1[\{0\}] := {f_{\sigma}}(d^1) \quad \text{and} \quad \eta_1[\{1\}] := {\beta_{\sigma}}[\{1\}] \circ {f_{\sigma}}(d^0).$$
Clearly, in each of the above cases, the natural transformation $\eta_1$ is a weak equivalence and the diagram ${\mathcal{H}}_{1t}^{\sigma}$ is fibrant.
$\bullet$ Assume that the statement is true for all $k \leq n-1$. To prove it for $k=n$, there are two things to do. The first one is to define $H_n \colon {\text{sk}}_n({\mathcal{T}_{\bullet}^{M}}\times \Delta[1]) {\longrightarrow}{\widehat{A}_{\bullet}}$, and the second is to get (\[eta\_eqn\]). To define $H_n$, let $\sigma \in {\mathcal{T}^M}_n$ and let $t \in \Delta[1]_n$. Define $H_n(\sigma, t) \colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ as $$H_n(\sigma, t) := \left\{ \begin{array}{ccc}
f_{\sigma} & \text{if } t = \underbrace{0 \cdots 0}_{n+1} =:t_0^{n+1} \\
f'_{\sigma} & \text{if } t = \underbrace{1 \cdots 1}_{n+1} =: t_1^{n+1} \\
\psi_n & \text{otherwise},
\end{array} \right.$$ where $\psi_n \colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ is defined as follows. By the induction hypothesis, there is a weak equivalence $\eta_n \colon {f_{\sigma}}([n]) \stackrel{\sim}{{\longrightarrow}} {\mathcal{H}}_{nt}^{\sigma}$ with ${\mathcal{H}}_{nt}^{\sigma}$ fibrant. Consider the following factorization of the limit of $\eta_n$ such that $Z \in {{\mathcal{C}}_A}$. $$\begin{aligned}
\label{fsz_diag}
\xymatrix{{f_{\sigma}}([n]) \ar@{ >->}[rd]_{\theta_{\sigma}}^-{\sim} \ar[rr]^-{\lim (\eta_n)} & & \underset{{\partial {\widetilde{\Delta}^n}}}{\text{lim}} {\mathcal{H}}_{nt}^{\sigma}. \\
& Z \ar@{->>}[ru]_-{{\overline{p}}} & }
\end{aligned}$$ Now define $$\psi_n|{\partial^i}{\widetilde{\Delta}^n}:= H_{n-1}(d_i\sigma, d_it), \quad \psi_n ([n]) := Z \quad \text{and} \quad \psi_n (d^i):= p_i \circ {\overline{p}},$$ where $
p_i \colon \underset{{\partial {\widetilde{\Delta}^n}}}{\text{lim}} {\mathcal{H}}_{nt}^{\sigma} {\longrightarrow}{\mathcal{H}}_{nt}^{\sigma} ([n]_i)
$ is the canonical projection as usual. It is straightforward to check that $\psi_n$ belongs to ${\widehat{A}}_n$. To get (\[eta\_eqn\]), let $\alpha \in {\mathcal{T}^M}_{n+1}$ and let $t \in \Delta[1]_{n+1}$. We need to deal with three cases.
1. If $t=t_0^{n+2}$, define $\eta_{n+1}[[n+1]_i]:= f_{\alpha}(d^i)$, where $d^i \colon [n+1]_i {\hookrightarrow}[n+1]$ is the inclusion map.
2. If $t = t_1^{n+2}$ we define $\eta_{n+1}[[n+1]_i]:= \beta_{\alpha}[[n+1]_i] \circ f_{\alpha}(d^i)$.
3. If $t \notin \{t_0^{n+2}, t_1^{n+2}\}$, one has $\eta_{n+1}[[n+1]_i]:= \theta_{d_i\alpha} \circ f_{\alpha}(d^i)$, where $\theta_{d_i\alpha}$ is the map from (\[fsz\_diag\]).
On the other objects of $\partial \widetilde{\Delta}^{n+1}$, define $\eta_{n+1}$ as the obvious compositions. Clearly ${\mathcal{H}}^{\alpha}_{(n+1)t}$ is fibrant. It is straightforward to check that $\eta_{n+1}$ is a natural transformation which is a weak equivalence. It is also straightforward to check that the map $H$ thus defined is a homotopy from $f$ to $f'$. This proves the lemma.
Now we can prove the main result of this section.
Let $\eta \colon F \stackrel{\sim}{{\longrightarrow}} F'$ be a weak equivalence of ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$. Set $\Lambda(F) = f$ and $\Lambda(F') = f'$. For $n \geq 0$ and $\sigma = \langle v_0, \cdots, v_n \rangle \in {\mathcal{T}^M}_n$, if $\sigma$ is non-degenerate, define $\beta_{\sigma} \colon f_{\sigma} {\longrightarrow}f'_{\sigma}$ as $
\beta_{\sigma}[\{a_0, \cdots, a_s\}] := ({\overline{{\mathcal{R}}}}\eta) [U_{\langle v_{a_0}, \cdots, v_{a_s}\rangle}].
$ If $\sigma$ is degenerate, there exist a unique non-degenerate simplex $\lambda$ and a sequence of degeneracy maps $s_{i_1}, \cdots, s_{i_k}$ such that $\sigma = s_{i_1} \cdots s_{i_k} (\lambda)$. By Remark \[dibeta\_rmk\], we have a map $\varphi \colon s_{i_1} \cdots s_{i_k}(f_{\lambda}) {\longrightarrow}s_{i_1} \cdots s_{i_k}(f'_{\lambda})$ induced by $\beta_{\lambda} \colon f_{\lambda} {\longrightarrow}f'_{\lambda}$. Define $\beta_{\sigma}:= \varphi$. By definition $\beta_{\sigma}$ is a weak equivalence for all $\sigma$. It is straightforward to check that the collection $\{\beta_{\sigma}\}_{\sigma}$ satisfies (\[cc\_disk\_eqn\]).
Applying Lemma \[htpy1\_lem\], we have that $f$ is homotopic to $f'$. Now assume that there is a zigzag $\xymatrix{F & \bullet \cdots \bullet \ar[l]_-{\sim} \ar[r]^-{\sim} & F'}$. Applying the first part to each map of that zigzag, and taking the inverse homotopy associated to each backward arrow, we have a homotopy between $\Lambda(F)$ and $\Lambda(F')$. This proves the proposition.
The map ${\overline{\Theta}}\colon [{\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}}] {\longrightarrow}{{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}\slash we$ {#theta_section}
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In Section \[lambda\_section\], or more precisely in Definition \[lamb\_defn\], we defined a map ${\overline{\Lambda}}\colon {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}\slash we {\longrightarrow}[{\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}}]$. The goal of this section is to construct its inverse, and thus get (\[res\_eqn5\]). We continue to use the well-order on the set of vertices of ${\mathcal{T}^M}$ we chose in the previous section.
\[tt\_defn\] Define a map $\Theta \colon {\underset{\text{sSet}}{\text{Hom}}({\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}})}{\longrightarrow}{{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$ as $\Theta(f):= F$, where $F \colon {\mathcal{U}(\mathcal{T}^M)}{\longrightarrow}{{\mathcal{C}}_A}$ is defined on objects as $F(U_{\sigma}):= {f_{\sigma}}([n])$ for $\sigma = \langle v_{a_0}, \cdots, v_{a_n}\rangle$. On morphisms, it is enough to define $F$ only on $d^i$’s from Remark \[utm\_rmk\]. Define $
F(d^i) := f_{\sigma}(d^i) \colon f_{\sigma}([n]) {\longrightarrow}f_{\sigma}([n]_i).
$
\[ttb\_defn\] Thanks to Proposition \[theta\_prop\] below, the map $\Theta$ passes to the quotient. Define ${\overline{\Theta}}\colon [{\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}}] {\longrightarrow}{{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}\slash we$ to be the resulting quotient map.
\[theta\_prop\] Let $f, f' \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ be two simplicial maps such that $f$ is homotopic to $f'$. Then there exists a zigzag of weak equivalences $\xymatrix{\Theta(f) & \bullet \cdots \bullet \ar[l]_-{\sim} \ar[r]^-{\sim} & \Theta(f')}$ in ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$.
The proof of Proposition \[theta\_prop\] occupies the section and will be given at the end. Our strategy goes through two big steps. In the first one we prove the result when the poset ${\mathcal{U}(\mathcal{T}^M)}$ is finite. In the case where ${\mathcal{U}(\mathcal{T}^M)}$ is infinite, the idea of the proof is to write $F=\Theta(f)$ as the homotopy limit of a certain diagram $E_1F_1 {\longleftarrow}E_2F_2 {\longleftarrow}\cdots$, where $F_i$ is the restriction of $F$ to a finite subposet ${{\mathcal{U}}({\mathcal{T}}^{M_i})}\subseteq {\mathcal{U}(\mathcal{T}^M)}$ and $E_iF_i$ is the right Kan extension of $F_i$ along the inclusion ${{\mathcal{U}}({\mathcal{T}}^{M_i})}{\hookrightarrow}{\mathcal{U}(\mathcal{T}^M)}$. Using the fact that $F_i$ is weakly equivalent to $F'_i$ by the first step, one can deduce the proposition. This section is organized as follows. In Section \[finite\_case\_subsection\] we prove Lemma \[htpy2\_lem\] below, which says that if $f, f' \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ are homotopic, then $\Theta(f)$ is weakly equivalent to $\Theta(f')$ provided that ${\mathcal{T}^M}$ has finite number of simplices. Since the proof of that lemma is technical, we begin with a special case: ${\mathcal{T}^M}= \Delta^1$. Section \[infinite\_case\_subsection\] deals with the case where ${\mathcal{T}^M}$ has infinite number of simplices.
Case where ${\mathcal{T}^M}$ has finite number of simplices {#finite_case_subsection}
-----------------------------------------------------------
\[htpy1-1\_lem\] Let ${\mathcal{T}^M}= \Delta^1$, and let $f, f' \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ be simplicial maps that are homotopic. Then there exists a zigzag of weak equivalences $\xymatrix{\Theta(f) & \bullet \cdots \bullet \ar[l]_-{\sim} \ar[r]^-{\sim} & \Theta(f')}$ in ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$.
Let $H$ be a homotopy between $f$ and $f'$. For the sake of simplicity, we will often use the notation $v_{0 \cdots n} := \langle v_0, \cdots, v_n \rangle$ in this proof and the next one. Consider the poset ${{\mathcal{U}}(\Delta^1)}= \{ \xymatrix{ U_{v_0} \ar[r]^-{d^1} & U_{v_{01}} & U_{v_1} \ar[l]_-{d^0}}\}$. Also consider Figure \[sdt\_splx\], which is a subdivision of $\Delta^1 \times [0, 1]$ into two $2$-simplices, namely $$\langle (v_0, 0), (v_0, 1), (v_1, 1)\rangle \quad \textup{and} \quad \langle (v_0, 0), (v_1, 0), (v_1, 1)\rangle$$
We now explain the algorithm that produces functors out of $H$ and Figure \[sdt\_splx\]. Define the *barycenter* of $\sigma_0:= \langle (v_0, 0), (v_1, 0), (v_1, 1)\rangle$ as the pair $(v_{011}, 001)$, and that of $\sigma_1:=\langle (v_0, 0), (v_0, 1), (v_1, 1)\rangle$ as the pair $(v_{001}, 011)$. In general, the barycenter of $\langle (v_{i_0}, j_0), (v_{i_1}, j_1), (v_{i_2}, j_2)\rangle$ is defined to be $(v_{i_0i_1i_2}, j_0j_1j_2)$. Applying the homotopy $H$ to those barycenters, we get the commutative diagram (\[big\_eqn\]) below in which we make the following simplifications at the level of notation. We write $H(v_{011}, 001)$ for $H(v_{011}, 001)(\{0, 1,2\})$, $H(v_{01}, 01)$ for $H(v_{01}, 01)(\{0,1\})$, and so on. Also we write $d_i$ for $H(-, -)(d^i)$.
$$\begin{aligned}
\label{big_eqn}
\xymatrix{H(v_0, 1) & & H(v_{01}, 11) \ar[ll]_-{d_1}^-{\sim} \ar[rr]^-{d_0}_-{\sim} & & H(v_1, 1) \\
& & H(v_{001}, 011) \ar[u]^-{\sim}_-{d_0} \ar[lld]_-{d_2}^-{\sim} \ar[d]^-{d_1}_-{\sim} & & \\
H(v_{00}, 01) \ar[uu]^-{d_0}_-{\sim} \ar[dd]^-{\sim}_-{d_1} & & H(v_{01}, 01) \ar[rruu]^-{d_0}_-{\sim} \ar[lldd]^-{\sim}_-{d_1} & & H(v_{11}, 01) \ar[uu]^-{\sim}_-{d_0} \ar[dd]^-{d_1}_-{\sim} \\
& & H(v_{011}, 001) \ar[u]^-{\sim}_-{d_1} \ar[d]^-{d_2}_-{\sim} \ar[rru]^-{\sim}_-{d_0} & & \\
H(v_0, 0) & & H(v_{01}, 00) \ar[ll]_-{\sim}^-{d_1} \ar[rr]^-{\sim}_-{d_0} & & H(v_1, 0) }
\end{aligned}$$
Now define the bottom of (\[big\_eqn\]) as a functor $F_0^L \colon {{\mathcal{U}}(\Delta^1)}{\longrightarrow}{{\mathcal{C}}_A}$, where the letter $L$ stands for lower. Specifically, we have $$F_0^L (U_{v_0}) := H(v_0, 0), \ F_0^L (U_{v_{01}}) := H(v_{01}, 00), \ F_0^L (U_{v_1}) := H(v_1, 0), \ \text{and} \ F_0^L (d^i):= d_i.$$ Certainly that functor is the same as $F$. The functor associated to the barycenter of the simplex $\sigma_0$ is defined as (B stands for barycenter) $$F_0^B := \left\{ \xymatrix{H(v_{0}, 0) & & H(v_{011}, 001) \ar[ll]_-{d_1d_2}^-{\sim} \ar[rr]^-{d_0}_-{\sim} & & H(v_{11}, 01) } \right\},$$ while the functor corresponding to its upper face (which is the same as the functor associated to the lower face of $\sigma_1$) is defined as $$F_0^U = F_1^L := \left\{ \xymatrix{H(v_{0}, 0) & & H(v_{01}, 01) \ar[ll]_-{d_1}^-{\sim} \ar[rr]^-{d_0}_-{\sim} & & H(v_1, 1) } \right\}.$$ Here U stands for upper of course. Lastly, the functor corresponding to the barycenter of the $2$-simplex $\sigma_1$ is defined as $$F_1^B := \left\{ \xymatrix{H(v_{00}, 01) & & H(v_{001}, 011) \ar[ll]_-{d_2}^-{\sim} \ar[rr]^-{d_0d_0}_-{\sim} & & H(v_1, 1) } \right\},$$ and that associated to its upper face, denoted $F_1^U$, is defined as the top of (\[big\_eqn\]). Clearly one has the following zigzag of weak equivalences, which are all natural since (\[big\_eqn\]) is commutative. $$\xymatrix{F= F_0^L & & F_0^B \ar[ll]_-{(id, d_2, d_1)}^-{\sim} \ar[rr]^-{(id, d_1, d_0)}_-{\sim} & & F_0^U=F_1^L & & F_1^B \ar[ll]_-{(d_1, d_1, id)}^-{\sim} \ar[rr]^-{(d_0, d_0, id)}_-{\sim} & & F_1^U =F'.}$$ This ends the proof.
\[htpy2\_lem\] Let $f, f' \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ be simplicial maps that are homotopic. Assume that ${\mathcal{T}^M}$ has finite number of simplices. Then there exists a zigzag of weak equivalences $\xymatrix{\Theta(f) & \bullet \cdots \bullet \ar[l]_-{\sim} \ar[r]^-{\sim} & \Theta(f')}$ in ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$.
Since the simplicial complex ${\mathcal{T}^M}$ has finite number of simplices, one can assume without loss of generality that it is a subcomplex of $\Delta^n$ for some $n$. Set $F = \Theta(f)$ and $F' = \Theta(f')$, and let $H \colon {\mathcal{T}_{\bullet}^{M}}\times \Delta[1] {\longrightarrow}{\widehat{A}_{\bullet}}$ be a homotopy from $f$ to $f'$. We need to construct a zigzag of weak equivalences between the functors $F, F' \colon {\mathcal{U}(\mathcal{T}^M)}{\longrightarrow}{{\mathcal{C}}_A}$. Following the special case above, the idea is to first subdivide ${\Delta^n}\times [0, 1]$ into $(n+1)$ $(n+1)$-simplices in a suitable way. Each $(n+1)$-simplex will produce three functors (one for the upper face, one for the lower face, and one for the barycenter), and two natural transformations like $\xymatrix{\bullet & \bullet \ar[l]_-{\sim} \ar[r]^-{\sim} & \bullet}$.
Let us consider the prism ${\Delta^n}\times I$, I = \[0, 1\], and set $${\Delta^n}\times \{t\} = \langle (v_0, t), \cdots, (v_n, t) \rangle, \quad t \in \{0, 1\}.$$ We will sometimes write $v_j$ for $(v_j, 0)$. For $0 \leq i \leq n$, define an $(n+1)$-simplex $$\sigma_i := \big\langle (v_0, 0), \cdots, (v_{n-i}, 0), (v_{n-i}, 1), \cdots, (v_n, 1)\big\rangle.$$ Certainly ${\Delta^n}\times [0, 1]$ is the union of simplices $\sigma_i, 0 \leq i \leq n$, each intersecting the next in an $n$-simplex face. As we said earlier, each $\sigma_i$ produces three functors $F_i^L, F_i^B, F_i^U \colon {\mathcal{U}(\mathcal{T}^M)}{\longrightarrow}{{\mathcal{C}}_A}$ that we now define. Consider the diagram $$\xymatrix{{\mathcal{U}(\mathcal{T}^M)}\ar[r]^-{\Psi_{{\mathcal{T}^M}}}_-{\cong} & P({\mathcal{T}^M}) \ar@<3ex>[r]^-{\phi_i^U} \ar@<0ex>[r]^-{\phi_i^B} \ar@<-3ex>[r]_-{\phi_i^L} & P(\sigma_i) \cap P({\mathcal{T}^M}\times I) \ \ar@{^{(}->}[r]^-{\iota} & P({\mathcal{T}^M}\times I) \ar[r]^-{\Phi}_-{\cong} & {\mathcal{U}}({\mathcal{T}^M}\times I) \ar[d]^-{\overline{H}} \\
& & & & {{\mathcal{C}}_A}}$$ defined as follows. $\bullet$ $P(-)$ is the construction from Definition \[pt\_defn\].
$\bullet$ The functor $\Psi_{{\mathcal{T}^M}}$ is defined as $\Psi_{{\mathcal{T}^M}}(U_{\lambda}):= \lambda$. Clearly this is an isomorphism of categories. The functor $\Phi$ is defined as $\Phi:= \Psi_{{\mathcal{T}^M}\times I}^{-1}$.
$\bullet$ $\iota$ is just the inclusion functor.
$\bullet$ Defining $\overline{H}$. Recall the map $\Theta$ from Definition \[tt\_defn\], and let us denote it by $\Theta_{{\mathcal{T}^M}}$. Then one has the map $$\Theta_{{\mathcal{T}^M}\times I} \colon \underset{\textup{sSet}}{\textup{Hom}}(({\mathcal{T}^M}\times I)_{\bullet}, {\widehat{A}_{\bullet}}) {\longrightarrow}{\mathcal{F}}({\mathcal{U}}({\mathcal{T}^M}\times I); {{\mathcal{C}}_A}).$$ The functor $\overline{H}$ is defined as $\overline{H}:= \Theta_{{\mathcal{T}^M}\times I} (H)$.
$\bullet$ We now define the functor $\phi^B_i \colon P({\mathcal{T}^M}) {\longrightarrow}P(\sigma_i) \times P({\mathcal{T}^M}\times I)$. Let $\lambda=\langle v_{a_0}, \cdots, v_{a_s}\rangle$ be an object of $P({\mathcal{T}^M})$. We need to deal with four cases.
1. If $n-i = a_j$ for some $j$, define $$\phi^B_i(\lambda):= \big\langle (v_{a_0}, 0), \cdots, (v_{a_j}, 0), (v_{a_j}, 1), (v_{a_{j+1}}, 1), \cdots, (v_{a_s}, 1)\big\rangle.$$
2. If $a_j < n-i < a_{j+1}$ for some $j$, define $$\phi^B_i(\lambda):= \big\langle (v_{a_0}, 0), \cdots, (v_{a_j}, 0), (v_{a_{j+1}}, 1), \cdots, (v_{a_s}, 1)\big\rangle.$$
3. If $n-i < a_j$ for all $j$, define $$\phi^B_i(\lambda):= \big\langle (v_{a_0}, 1), \cdots, (v_{a_s}, 1)\big\rangle.$$
4. If $n-i > a_j$ for all $j$, define $$\phi^B_i(\lambda):= \big\langle (v_{a_0}, 0), \cdots, (v_{a_s}, 0)\big\rangle.$$
$\bullet$ Define $\phi^U_i$ as $$\phi^U_i(\lambda):= \big\langle (v_{a_0}, 0), \cdots, \widehat{(v_{a_j}, 0)}, (v_{a_j}, 1), (v_{a_{j+1}}, 1), \cdots, (v_{a_s}, 1)\big\rangle$$ if $n-i=a_j$ for some $j$, and $\phi^U_i (\lambda):= \phi^B_i(\lambda)$ otherwise. $\bullet$ Define $\phi^L_i$ as $$\phi^L_i(\lambda):= \big\langle (v_{a_0}, 0), \cdots, (v_{a_j}, 0), \widehat{(v_{a_j}, 1)}, (v_{a_{j+1}}, 1), \cdots, (v_{a_s}, 1)\big\rangle$$ if $n-i=a_j$ for some $j$, and $\phi^L_i (\lambda):= \phi^B_i(\lambda)$ otherwise.
For $X \in \{B, L, U\}$, define $F^X_i$ as the composite $$F^X_i:= \overline{H} \circ \iota \circ \phi^X_i \circ \psi.$$ From the definitions, there are two natural transformations $\alpha_i \colon \phi^L_i {\longrightarrow}\phi^B_i$ and $\beta_i \colon \phi^U_i {\longrightarrow}\phi^B_i$ which are nothing but the inclusions (that is, for every $\lambda \in P({\mathcal{T}^M})$, the components of $\alpha_i$ and $\beta_i$ at $\lambda$ are the obvious inclusions). These give rise to a zigzag $F^L_i {\longleftarrow}F^B_i {\longrightarrow}F^U_i$ (remember $\overline{H}$ is contravariant). Each map from that zigzag is a weak equivalence since the functor $\overline{H}$ sends every morphism to a weak equivalence by the definitions. Clearly, one has $F^L_0 = F, F^U_n = F'$, and $F^U_i = F^L_{i+1}$ for all $i$. This ends the proof.
Case where ${\mathcal{T}^M}$ has infinite number of simplices {#infinite_case_subsection}
-------------------------------------------------------------
In this section we assume that ${\mathcal{T}^M}$ has infinitely many simplices. Since $M$ is second-countable by assumption (see the introduction), it follows that ${\mathcal{T}^M}$ has countably many simplices. This enables us to choose a family $${\mathcal{T}}^{M_1} \subseteq \cdots \subseteq {\mathcal{T}}^{M_i} \subseteq {\mathcal{T}}^{M_{i+1}} \subseteq \cdots \subseteq {\mathcal{T}^M}$$ of subcomplexes of ${\mathcal{T}^M}$ satisfying the following two conditions:
1. Each ${\mathcal{T}}^{M_i}$ has finitely many simplices;
2. $P({\mathcal{T}}^{M_{i+1}}) = P({\mathcal{T}}^{M_i}) \cup \{z\}$, where $z \notin P({\mathcal{T}}^{M_i})$ and for every simplex $x$ of ${\mathcal{T}}^{M_i}$, $z$ is not a face of $x$.
Let $\{{{\mathcal{U}}({\mathcal{T}}^{M_i})}\}_{i \geq 1}$ be the corresponding family of subposets of ${\mathcal{U}(\mathcal{T}^M)}$. (Certainly, one has $\bigcup_i {{\mathcal{U}}({\mathcal{T}}^{M_i})}= {\mathcal{U}(\mathcal{T}^M)}$.) Let $R_i \colon {{\mathcal{U}}({\mathcal{T}}^{M_i})}{\hookrightarrow}{\mathcal{U}(\mathcal{T}^M)}$ and $R_{ij} \colon {{\mathcal{U}}({\mathcal{T}}^{M_j})}{\hookrightarrow}{{\mathcal{U}}({\mathcal{T}}^{M_i})}$ denote the inclusion functors. Consider the pair $$R_i^* \colon \xymatrix{{\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}} \ar@<1ex>[rr] & & {\mathcal{M}}^{{{\mathcal{U}}({\mathcal{T}}^{M_i})}} \ar@<1ex>[ll]}: E_i,$$ where $R_i^*$ is the restriction functor (that is, $R_i^* (F):= F|{{\mathcal{U}}({\mathcal{T}}^{M_i})}$), and $E_i$ is nothing but the right Kan extension functor along $R_i$, that is, $$\begin{aligned}
\label{ei_eqn}
E_i(F)(x):= \underset{y {\rightarrow}x, y \in {{\mathcal{U}}({\mathcal{T}}^{M_i})}}{\lim} F(y) = \underset{{{\mathcal{U}}({\mathcal{T}}^{M_i})}\downarrow x}{\lim} \mathbb{D}_i.\end{aligned}$$ Here $\mathbb{D}_i \colon {{\mathcal{U}}({\mathcal{T}}^{M_i})}\downarrow x {\longrightarrow}{\mathcal{M}}$ is the functor that sends $(y {\longrightarrow}x)$ to $F(y)$. Clearly $E_i$ is right adjoint to $R_i^*$. Similarly, we have a diagram $\mathbb{D}_{ij}$, and a pair of adjoint functors $$R_{ij}^* \colon \xymatrix{{\mathcal{M}}^{{{\mathcal{U}}({\mathcal{T}}^{M_i})}} \ar@<1ex>[rr] & & {\mathcal{M}}^{{{\mathcal{U}}({\mathcal{T}}^{M_j})}} \ar@<1ex>[ll]}: E_{ij},$$ where $E_{ij}(F)(x)$ is of course the limit of $\mathbb{D}_{ij}$.
Before we prove Proposition \[theta\_prop\], we need five lemmas.
\[facts\_lem\] Let $j \leq i$. Then
1. For every $F \in {\mathcal{M}}^{{{\mathcal{U}}({\mathcal{T}}^{M_j})}}$, $E_i (E_{ij} (F))$ is naturally isomorphic to $E_j(F)$. That is, $E_i (E_{ij} (F)) \cong E_j(F)$.
2. Let $F \in {\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}$ be a fibrant diagram. Then the diagram $\mathbb{D}_i$ above as well as $\mathbb{D}_{ij}$ is fibrant for any $x \in {\mathcal{U}(\mathcal{T}^M)}$.
Part (i) follows immediately from the definitions. To see part (ii), let $x \in {\mathcal{U}(\mathcal{T}^M)}$. Then, by Definition \[utm\_defn\], there exists a unique simplex $\sigma_x \in {\mathcal{T}^M}$ such that $x = U_{\sigma_x}$. Associated with $\sigma_x$ is the poset ${\mathcal{U}}(\sigma_x) = \{U_{\lambda}| \ \lambda \subseteq \sigma_x\}$. Clearly we have ${{\mathcal{U}}({\mathcal{T}}^{M_i})}\downarrow x = {{\mathcal{U}}({\mathcal{T}}^{M_i})}\cap {\mathcal{U}}(\sigma_x)$. So $\mathbb{D}_i$ is nothing but the restriction of $F$ to ${{\mathcal{U}}({\mathcal{T}}^{M_i})}\cap {\mathcal{U}}(\sigma_x)$, which is definitely fibrant since $F$ is fibrant by assumption. Likewise, one can show that $\mathbb{D}_{ij}$ is fibrant.
Let $\Theta$ be the map that appears in Proposition \[theta\_prop\], and let $f, f' \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ be simplicial maps that are homotopic. Consider the categories ${\mathcal{M}}$ and ${{\mathcal{C}}_A}$ as in Definition \[ca\_defn\], and set $F = \Theta(f)$ and $F'= \Theta (f')$. By the definitions, the functors $F, F' \in {{\mathcal{C}}_A}^{{\mathcal{U}(\mathcal{T}^M)}} \subseteq {\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}$ are both fibrant and cofibrant. For convenience, we will regard these as functors into ${\mathcal{M}}$. Define $F_i := F|{{\mathcal{U}}({\mathcal{T}}^{M_i})}$ and $F'_i := F'|{{\mathcal{U}}({\mathcal{T}}^{M_i})}$.
\[fib\_lem\] For every $i$ there exist ${\overline{F}_i}\in {\mathcal{M}}^{{{\mathcal{U}}({\mathcal{T}}^{M_i})}}$ and two weak equivalences $\xymatrix{F_i & {\overline{F}_i}\ar[l]_-{\alpha_i}^-{\sim} \ar[r]^-{\alpha'_i}_-{\sim} & F'_i}$ satisfying the following two conditions: (a) ${\overline{F}_i}$ is cofibrant. (b) The map $(\alpha_i, \alpha'_i) \colon {\overline{F}_i}{\longrightarrow}F_i \times F'_i$ is a fibration.
Consider the map $\Theta_i \colon \underset{\text{sSet}}{\text{Hom}} ({\mathcal{T}}^{M_i}_{\bullet}, {\widehat{A}_{\bullet}}) {\longrightarrow}{\mathcal{F}}({{\mathcal{U}}({\mathcal{T}}^{M_i})}; {{\mathcal{C}}_A})$ defined in the same way as $\Theta$, and let $f_i = f|{\mathcal{T}}^{M_i}_{\bullet}$ and $f'_i = f'|{\mathcal{T}}^{M_i}_{\bullet}$. Then it is clear that $f_i$ is homotopic to $f'_i$, $\Theta_i(f_i) = F_i$ and $\Theta_i(f'_i) = F'_i$. Applying Lemma \[htpy2\_lem\], we get a zigzag $$\begin{aligned}
\label{fi_zigzag}
\xymatrix{F_i & \bullet \ar[l]_-{\sim} \ar[r]^-{\sim} & \cdots & \bullet \ar[l]_-{\sim} \ar[r]^-{\sim} & F_i'}
\end{aligned}$$ Using now the fact that $F_i$ and $F_i'$ are both fibrant and cofibrant, and some standard techniques from model categories, one can construct out of (\[fi\_zigzag\]) a short zigzag (of the form indicated in the lemma) which has the required properties.
Let $j \leq i$, and let $\alpha_i$ and $\alpha'_i$ be as in Lemma \[fib\_lem\]. Then there exists a map ${\overline{F}_j}{\longleftarrow}R_{ij}^* {\overline{F}_i}$ making the following diagram commute. $$\begin{aligned}
\label{fi_diag}
\xymatrix{R^*_{ij}F_i \ar[d]_-{id} & & R^*_{ij} {\overline{F}_i}\ar[ll]_-{\sim}^-{R^*_{ij}(\alpha_i)} \ar[rr]^-{\sim}_-{R^*_{ij}(\alpha'_i)} \ar[d]^-{\sim} & & R^*_{ij} F'_i \ar[d]^-{id} \\
F_j & & {\overline{F}_j}\ar[ll]_-{\sim}^-{\alpha_j} \ar[rr]^-{\sim}_-{\alpha'_j} & & F'_j.}
\end{aligned}$$
Since $\alpha_j$ is a fibration by Lemma \[fib\_lem\], and since $F_j$ is cofibrant, by the lifting axiom, there exists a map $\alpha_j^{-1} \colon F_j {\longrightarrow}{\overline{F}_j}$ making the obvious square commute. From the construction of the zigzag (\[fi\_zigzag\]) (look closer at the proof of Lemma \[htpy2\_lem\]), the following square commutes up to homotopy. $$\begin{aligned}
\label{fi_sq1}
\xymatrix{R^*_{ij} F_i \ar[rrr]^-{R^*_{ij} (\alpha'_i)R^*_{ij}(\alpha_i^{-1})} \ar[d]_-{id} & & & R^*_{ij} F'_i \ar[d]^-{id} \\
F_j \ar[rrr]_-{\alpha'_j \alpha_j^{-1}} & & & F'_j.}
\end{aligned}$$ Since $\alpha'_j$ is a fibration, and since $R^*_{ij} {\overline{F}_i}$ is cofibrant (this is because ${\overline{F}_i}$ is cofibrant by Lemma \[fib\_lem\], and cofibrations are objectwise), the lifting axiom guarantees the existence of a map $R_{ij}^* {\overline{F}_i}\stackrel{\phi}{{\longrightarrow}} {\overline{F}_j}$ that makes the righthand square of the following diagram commute. $$\begin{aligned}
\label{fi_sq2}
\xymatrix{R^*_{ij}F_i \ar[d]_-{id} & & R^*_{ij} {\overline{F}_i}\ar[ll]_-{\sim}^-{R^*_{ij}(\alpha_i)} \ar[rr]^-{\sim}_-{R^*_{ij}(\alpha'_i)} \ar[d]^-{\sim}_-{\phi} & & R^*_{ij} F'_i \ar[d]^-{id} \\
F_j & & {\overline{F}_j}\ar[ll]_-{\sim}^-{\alpha_j} \ar[rr]^-{\sim}_-{\alpha'_j} & & F'_j.}
\end{aligned}$$ Combining this with the fact that the square (\[fi\_sq1\]) commutes up to homotopy, we deduce that the lefthand square commutes up to homotopy as well. By Lemma \[reg\_lem\] below, one can then replace $\phi$ by a map $R_{ij}^* {\overline{F}_i}\stackrel{{\overline{\phi}}}{{\longrightarrow}} {\overline{F}_j}$ that makes the whole diagram commute.
\[reg\_lem\] Consider the following diagram in a model category ${\mathcal{M}}$. $$\xymatrix{A_0 \ar[d]_-{g_0} & B \ar[l]_-{f_0} \ar[r]^-{f_1} \ar[d]^-{g} & A_1 \ar[d]^-{g_1} \\
D_0 & C \ar[l]^-{f'_0} \ar[r]_-{f'_1} & D_1.}$$ Assume that each square commutes up to homotopy. Also assume that $B$ is cofibrant. If the map $(f'_0, f'_1) \colon C {\longrightarrow}D_0 \times D_1$ is a fibration, then there exists $\overline{g}$ homotopic to $g$ that makes the whole diagram commute.
Since each square commutes up to homotopy, there exists a homotopy $H \colon B \times I {\longrightarrow}D_0 \times D_1$ from $(g_0f_0, g_1f_1)$ to $(f'_0, f'_1) \circ g$, which fits into the following commutative diagram. $$\xymatrix{ & B \ar[r]^-{g} \ar@{>->}[d]_-{i_1}^-{\sim} & C \ar@{->>}[d]^-{(f'_0, f'_1)} \\
B \ \ar@{>->}[r]_-{i_0} & B \times I \ar[r]_-{H} \ar[ru]^-{\psi} & D_0 \times D_1.}$$ The canonical inclusion $i_1$ is an acyclic cofibration since $B$ is cofibrant [@dwyer_spa95 Lemma 4.4]. The map $(f'_0, f'_1)$ is a fibration by assumption, and the map $\psi$ is provided by the lifting axiom. Now define $\overline{g} = \psi \circ i_0$. It is straightforward to check that $\overline{g}$ does the work.
We still need an important lemma. From the definition of the right Kan extension (\[ei\_eqn\]), there is a canonical map $\eta_i \colon E_i F_i {\longrightarrow}E_{i-1}F_{i-1}$ induced by the universal property of limit. These maps fit into the covariant diagram ${\mathbb{E}}\colon {\mathbb{N}}{\longrightarrow}{\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}$ defined by ${\mathbb{E}}(i) = E_iF_i$ and ${\mathbb{E}}(i {\rightarrow}(i-1)) = \eta_i$. Here ${\mathbb{N}}$ is viewed as the poset $\{1 {\leftarrow}2 {\leftarrow}3 {\leftarrow}\cdots\}$.
\[kappa\_lem\] The canonical map $ \kappa \colon F {\longrightarrow}\underset{{\mathbb{N}}}{\text{holim}} \ {\mathbb{E}}$ is a weak equivalence.
We begin by claiming that for every $x \in {\mathcal{U}(\mathcal{T}^M)}$, there exists $s \in {\mathbb{N}}$ such that $F(x) \cong E_rF_r (x)$ for all $r \geq s$. To see this, let $x \in {\mathcal{U}(\mathcal{T}^M)}$. Since $\bigcup_i {{\mathcal{U}}({\mathcal{T}}^{M_i})}= {\mathcal{U}(\mathcal{T}^M)}$, there exists $s \in {\mathbb{N}}$ such that $x \in {{\mathcal{U}}({\mathcal{T}}^{M_s})}$. Let $r \geq s$. Since the sequence $\{{{\mathcal{U}}({\mathcal{T}}^{M_i})}\}_i$ is increasing, the indexing category ${{\mathcal{U}}({\mathcal{T}}^{M_r})}\downarrow x$ has a terminal object, namely $x \stackrel{id}{{\longrightarrow}} x$. This implies (by the definition (\[ei\_eqn\]) of $E_r$) that the canonical map $F(x) {\longrightarrow}E_rF_r (x)$ is an isomorphism. Thanks to that isomorphism, we have $F \cong \underset{i}{\lim} E_i F_i$. To end the proof, it suffices to show that the diagram $\mathbb{E} \colon{\mathbb{N}}{\longrightarrow}{\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}$ is fibrant. By Proposition \[model\_prop\], we have to show that the matching map of ${\mathbb{E}}$ at $i$ is a fibration for any $i$. This is the case for $i =1$ since $E_1F_1$ is fibrant (because of the fact that $F$ is fibrant). Let $i \geq 2$. Then the matching map at $i$ is nothing but the canonical map $\eta_i \colon E_i F_i {\longrightarrow}E_{i-1} F_{i-1}$. Looking at the definition of a fibration (see Proposition \[model\_prop\]) in ${\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}$, we need to show that the map $p_x$ from the following commutative diagram is a fibration for every $x \in {\mathcal{U}(\mathcal{T}^M)}$ to conclude that $\eta_i$ is a fibration. $$\xymatrix{E_iF_i (x) \ar[rd]^-{p_x} \ar[rrd]^-{(\eta_i)_x} \ar[rdd]_-{\theta_{xi}} & & \\
& \text{PB} \ar[d]^-{\lambda_x} \ar[r]_-{\beta_x} & E_{i-1}F_{i-1} (x) \ar[d]^-{\theta_{x(i-1)}}\\
& \underset{y {\rightarrow}x, y \neq x}{\lim} E_iF_i (y) \ar[r]^-{M_x(\eta_i)} & \underset{y {\rightarrow}x, y \neq x}{\lim} E_{i-1}F_{i-1} (y) }$$ So let $x \in {\mathcal{U}(\mathcal{T}^M)}$. Let $z \in {\mathcal{U}(\mathcal{T}^M)}$ such that ${{\mathcal{U}}({\mathcal{T}}^{M_i})}= {{\mathcal{U}}({\mathcal{T}}^{M_{i-1}})}\cup \{z\}$. We need to deal with two cases depending on the fact that $x = z$ or $x \neq z$.
1. If $x = z$, then $\theta_{x(i-1)}$ is an isomorphism. Since the pullback of an isomorphism is an isomorphism, it follows that $\lambda_x$ is an isomorphism. Since $\theta_{xi}$ is exactly the matching map of $E_iF_i$ at $z =x$, the map $\theta_{xi}$ is a fibration. Thus $p_x$ is a fibration.
2. Assume that $x \neq z$. We have two cases. If there is no map from $z$ to $x$, then by the definitions, we have $E_{i-1}F_{i-1} (y) = E_iF_i (y)$ for all $y {\rightarrow}x$. This implies that $M_x(\eta_i), \beta_x, (\eta_i)_x,$ and $p_x$ are isomorphisms. If there is a map $z {\rightarrow}x$, one can see that $\theta_{x(i-1)}$ and $\theta_{xi}$ are both isomorphisms since $x \notin {{\mathcal{U}}({\mathcal{T}}^{M_{i-1}})}$ and $x \notin {{\mathcal{U}}({\mathcal{T}}^{M_i})}$. This implies that $\lambda_x$ and $p_x$ are also isomorphisms.
We thus obtain the desired result.
We are now ready to prove the main result of this section: Proposition \[theta\_prop\].
We need to show that $\Theta (f) = F$ and $\Theta (f') = F'$ are weakly equivalent in ${{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$. By Proposition \[fuca\_prop\], it is enough to show that $\phi F \simeq \phi F'$ in ${\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})$ (see Definition \[fuca\_defn\]). Here $\phi \colon {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}{\longrightarrow}{\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})$ is the obvious functor defined by $\phi(G) = G$. From now on, we will regard $F$ and $F'$ as objects of ${\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})$. As before, we let $F_i$ (respectively $F'_i$) denote the restriction of $F$ (respectively $F'$) to ${{\mathcal{U}}({\mathcal{T}}^{M_i})}$. Taking the adjoint to (\[fi\_diag\]), we get (\[fi2\_diag\]), which is of course a commutative diagram. $$\begin{aligned}
\label{fi2_diag}
\xymatrix{F_i \ar[d] & & {\overline{F}_i}\ar[ll]_-{\sim}^-{\alpha_i} \ar[rr]^-{\sim}_-{\alpha'_i} \ar[d] & & F'_i \ar[d] \\
E_{ij} F_j & & E_{ij} {\overline{F}_j}\ar[ll]_-{\sim}^-{E_{ij} (\alpha_j)} \ar[rr]^-{\sim}_-{E_{ij} (\alpha'_j)} & & E_{ij} F'_j.}
\end{aligned}$$ From Lemma \[facts\_lem\] -(ii), it is easy to see why the maps $E_{ij}(\alpha_j)$ and $E_{ij}(\alpha'_j)$ are both weak equivalences. Applying now the functor $E_i$ to (\[fi2\_diag\]), and using Lemma \[facts\_lem\] -(i), we get $$\begin{aligned}
\label{fi3_diag}
\xymatrix{E_iF_i \ar[d] & E_i{\overline{F}_i}\ar[l]_-{\sim}^-{\beta_i} \ar[r]^-{\sim}_-{\beta'_i} \ar[d] & E_iF'_i \ar[d] \\
E_j F_j & E_j{\overline{F}_j}\ar[l]_-{\sim}^-{\beta_j} \ar[r]^-{\sim}_-{\beta'_j} & E_j F'_j,}
\end{aligned}$$ where $\beta_i:= E_i(\alpha_i)$. This gives rise to two weak equivalences: $\xymatrix{{\mathbb{E}}& \overline{{\mathbb{E}}} \ar[l]_-{\sim}^-{\beta} \ar[r]^-{\sim}_-{\beta'} & {\mathbb{E}}'}$, where $\overline{{\mathbb{E}}} \colon {\mathbb{N}}{\longrightarrow}{\mathcal{M}}^{{\mathcal{U}(\mathcal{T}^M)}}$ is the obvious functor defined by $\overline{{\mathbb{E}}}(i) = E_i {\overline{F}_i}$. Recalling the map $\kappa$ from Lemma \[kappa\_lem\], we have the zigzag $$\begin{aligned}
\label{fi4_diag}
\xymatrix{F \ar[r]^-{\sim}_-{\kappa} & \underset{{\mathbb{N}}}{\text{holim}} \ {\mathbb{E}}& & \underset{{\mathbb{N}}}{\text{holim}} \ \overline{{\mathbb{E}}} \ar[ll]_-{\sim}^-{\text{holim}(\beta)} \ar[rr]^-{\sim}_-{\text{holim}(\beta')} & & \underset{{\mathbb{N}}}{\text{holim}} \ {\mathbb{E}}' & F', \ar[l]_-{\sim}^-{\kappa'}}
\end{aligned}$$ which completes the proof [^4].
We close this section with the following result.
\[fuca\_iso\_prop\] The map ${\overline{\Lambda}}$ from Definition \[lamb\_defn\] is a bijection. That is, $$\xymatrix{{{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}/we \ar[r]_-{\cong}^-{{\overline{\Lambda}}} & [{\mathcal{T}_{\bullet}^{M}}, {\widehat{A}_{\bullet}}]}.$$ In fact the inverse of ${\overline{\Lambda}}$ is the map ${\overline{\Theta}}$ from Definition \[ttb\_defn\].
From the definitions, it is easy to see that $\Theta\Lambda (F) \simeq F$ for every $F \in {{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})}$. So ${\overline{\Theta}}\circ {\overline{\Lambda}}= id$. On the other hand, let $f \colon {\mathcal{T}_{\bullet}^{M}}{\longrightarrow}{\widehat{A}_{\bullet}}$ be a simplicial map. By construction $\Theta(f)$ is fibrant since ${f_{\sigma}}\colon {\widetilde{\Delta}^n}{\longrightarrow}{{\mathcal{C}}_A}$ is fibrant for any $n \geq 0, \sigma \in {\mathcal{T}^M}_n$. Therefore ${\overline{{\mathcal{R}}}}\Theta (f) = \Theta (f)$, where ${\overline{{\mathcal{R}}}}$ is the fibrant replacement funtor from (\[mcalrb\_eqn\]). This implies that $\Lambda \Theta (f) = f$, and therefore ${\overline{\Lambda}}\circ {\overline{\Theta}}= id$, which completes the proof.
Proof of the main result {#pmr_section}
========================
We now have all ingredients to prove Theorem \[main\_thm\], which is the main result of this paper. Roughly speaking it classifies a class of functors called *homogeneous* (see [@paul_don17-2 Definition 6.2]).
\[defn:fkom\] Define ${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}})$ as the category of homogeneous functors $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ such that $F(U) \simeq A$ for every $U$ diffeomorphic to the disjoint union of exactly $k$ open balls.
\[fkfu\_lem\] Let ${\mathcal{M}}$ be a simplicial model category that has a zero object, and let ${{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}$ be as in Definition \[defn:fkom\]. Let ${\mathcal{T}}^{F_k(M)}$ be a triangulation of $F_k(M)$, and let ${\mathcal{F}}_A({{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}; {\mathcal{M}})$ be the category from Definition \[fum\_defn\]. Then the categories ${{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}$ and ${\mathcal{F}}_A({{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}; {\mathcal{M}})$ are weakly equivalent.
By using the same approach as that we used to prove [@paul_don17-2 Theorem 1.3 & Lemma 6.5], one has $$\begin{aligned}
\label{we_eq0}
{{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}\simeq {{\mathcal{F}}_{1A} ({\mathcal{O}}(F_k(M)); {\mathcal{M}})},
\end{aligned}$$ and $$\begin{aligned}
\label{we_eq1}
{{\mathcal{F}}_{1A} ({\mathcal{O}}(F_k(M)); {\mathcal{M}})}\simeq {\mathcal{F}}_A ({\mathcal{B}}_{{{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}}; {\mathcal{M}}),
\end{aligned}$$ where ${\mathcal{F}}_A ({\mathcal{B}}_{{{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}}; {\mathcal{M}})$ is the category from Definition \[fum\_defn\]. Furthermore, by Proposition \[fum\_butm\_prop\], we have $$\begin{aligned}
\label{we_eq2}
{\mathcal{F}}_A ({\mathcal{B}}_{{{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}}; {\mathcal{M}}) \simeq {\mathcal{F}}_A ({{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}; {\mathcal{M}}).
\end{aligned}$$ Combining (\[we\_eq0\]), (\[we\_eq1\]), and (\[we\_eq2\]), we get the desired result.
We are now ready to prove Theorem \[main\_thm\].
For the first part, we refer the reader to the introduction. The second part is proved as follows. Let ${\mathcal{T}}^{F_k(M)}$ as above. From Lemma \[fkfu\_lem\] and [@paul_don17-2 Remark 6.4], we deduce that the localizations of ${{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}$ and ${\mathcal{F}}_A ({{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}; {\mathcal{M}})$ are equivalent in the classical sense. Furthermore, by Proposition \[fuca\_prop\], we have that the localization of the latter category is equivalent to the localization of ${\mathcal{F}}({{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}; {{\mathcal{C}}_A})$. This implies that weak equivalence classes of ${{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}$ are in one-to-one correspondence with weak equivalence classes of ${\mathcal{F}}({{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}; {{\mathcal{C}}_A})$. That is, $$\begin{aligned}
\label{pthm_eqn}
{{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}\slash we \ \cong \ {\mathcal{F}}({{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}; {{\mathcal{C}}_A}) \slash we.
\end{aligned}$$ Applying Proposition \[fuca\_iso\_prop\], we get $$\begin{aligned}
\label{pthm_eqn2}
{\mathcal{F}}({{\mathcal{U}}({\mathcal{T}}^{F_k(M)})}; {{\mathcal{C}}_A}) \slash we \ \cong \ \left[{\mathcal{T}}^{F_k(M)}_{\bullet}, {\widehat{A}_{\bullet}}\right].
\end{aligned}$$ Define ${\widehat{A}}$ to be the geometric realization of ${\widehat{A}_{\bullet}}$. That is, ${\widehat{A}}:= |{\widehat{A}_{\bullet}}|$. Since $|{\mathcal{T}}^{F_k(M)}_{\bullet}| \cong F_k(M)$, it follows that $
\left[{\mathcal{T}}^{F_k(M)}_{\bullet}, {\widehat{A}_{\bullet}}\right] \cong \left[F_k(M), {\widehat{A}}\right].
$ This proves the theorem.
How our classification is related to that of Weiss {#comparison_section}
==================================================
In this section we briefly recall the Weiss classification of homogeneous functors, and we explain a connection to our classification result (Theorem \[main\_thm\]). As usual, we let (respectively $\text{Top}_*$) denote the category of spaces (respectively pointed spaces).
We begin with Weiss’ result about the classification of homogeneous functors. Let $p \colon Z {\longrightarrow}F_k(M)$ be a fibration. Define $F \colon {\mathcal{O}(M)}{\longrightarrow}\text{Top}$ as $F(U) = \Gamma(p; F_k(U))$, the space of sections of $p$ over $F_k(U)$. It turns out that $F$ is polynomial of degree $\leq k$ (see [@wei99 Example 7.1]). Define another functor $G \colon {\mathcal{O}(M)}{\longrightarrow}\text{Top}$ as follows. Let $M^k \slash \Sigma_k$ denote the orbit space of the action of the symmetric group $\Sigma_k$ on the $k$-fold product $M^k$. Let $\Delta_k M$ be the complement of $F_k(M)$ in $M^k \slash \Sigma_k$. (The space $\Delta_k M$ is the so-called *fat diagonal* of $M$.) Define $
G(U) := \underset{N \in {\mathcal{N}}}{\text{hocolim }} \Gamma(p; F_k(U) \cap N),
$ where ${\mathcal{N}}$ stands for the poset of neighborhoods of $\Delta_k M$. The space $G(M)$ should be thought of as the space of sections near the fat diagonal of $M$. Let $\eta \colon F {\longrightarrow}G$ be the canonical map induced by the inclusions $F_k(U) \cap N \subseteq F_k(U)$. It turns out that $\eta$ is nothing but the canonical map $T_k F {\longrightarrow}T_{k-1}F$ since $T_{k-1}F$ is equivalent to $G$ (see [@wei99 Propositions 7.5 and 7.6 ]). Selecting a point $s$ in $G(M)$, if one exists, we define $E_{p,s} \colon {\mathcal{O}(M)}{\longrightarrow}\textup{Top}$ as the homotopy fiber of $\eta$ over $s$, that is, $E_{p, s} := \textup{hofiber} (\eta)$. It follows from [@wei99 Example 8.2] that $E_{p, s}$ is homogeneous of degree $k$. Weiss’ classification says that every homogeneous functor can be constructed in this way. Specifically, we have the following result.
[@wei99 Theorem 8.5] \[thm:wchf\] Let $E \colon {\mathcal{O}(M)}{\longrightarrow}\textup{Top}$ be homogeneous of degree $k$. Then there is a (levelwise) homotopy equivalence $E {\longrightarrow}E_{p, s}$ for a fibration $p \colon Z \to F_k(M)$ with a section $s$ near the fat diagonal of $M$.
Weiss also describes the fiber of $p$ in terms of $E$.
[@wei99 Proposition 8.4 and Theorem 8.5] \[prop:fiber\] Let $E$ be as in Theorem \[thm:wchf\], and suppose $E$ is classified by a fibration $p \colon Z \to F_k(M)$. Then the fiber $p^{-1}(S)$ over $S \in F_k(M)$ is homotpy equivalent to $E(U_S)$, where $U_S$ is a tubular neighborhood of $S$, so that $U_S$ is diffeomorphic to the dijoint union of $k$ open balls.
Combining Theorem \[thm:wchf\] and Proposition \[prop:fiber\], we get that the classification of the objects of ${\mathcal{F}}_{kA}({\mathcal{O}(M)}; \text{Top})$ (see Definition \[defn:fkom\]) amounts to the classification of fibrations over $F_k(M)$ with a section $s$ near the fat diagonal, and whose fiber is homotopy equivalent to $A$. Note that in the case $k=1$ the fat diagonal is empty, and we just look at fibrations over $M$ with fiber $A$.
We now explain a connection to our classification result. As usual, ${\mathcal{M}}$ is an arbitrary simplicial model category, and $A$ is an object of ${\mathcal{M}}$. Assume that $A$ is both fibrant and cofibrant. Let ${\text{haut}}A$ be the simplicial monoid of self weak equivalences $A \stackrel{\sim}{\to} A$. Define B${\text{haut}}A:= \overline{W}{\text{haut}}A$, where $\overline{W}$ is the functor from [@may92 p. 87] or [@goe_jar09 p. 269]. (Note that $\overline{W}$ lands in the category of simplicial sets.) We still denote by B${\text{haut}}A$ the geometric realization of B${\text{haut}}A$. So depending on the context, $\text{Bhaut}A$ should be interpreted as either a simplicial set or a topological space. We have the following two conjectures.
\[conj:ah\_BhautA\] Let ${\mathcal{M}}$ be a simplicial model category, and let $A \in {\mathcal{M}}$ be an object which is both fibrant and cofibrant. Let ${\widehat{A}_{\bullet}}$ be the simplicial set from Section \[ahd\_subsection\] constructed out of $A$, and let ${\widehat{A}}$ be its geometric realization. Then ${\widehat{A}}$ is weakly equivalent to $\emph{Bhaut} A$. That is, ${\widehat{A}}\simeq \emph{Bhaut} A$.
\[conj:thm\] Let ${\mathcal{M}}$ and $A$ be as in Conjecture \[conj:ah\_BhautA\]. Then for any manifold $M$,
1. if $k = 1$, there is a bijection ${\mathcal{F}}_{1A}({\mathcal{O}(M)}; {\mathcal{M}})/we \ \cong \big[M, \emph{Bhaut} A\big]$,
2. if $k \geq 2$ and ${\mathcal{M}}$ has a zero object, there is a bijection ${{\mathcal{F}}_{kA} ({\mathcal{O}(M)}; {\mathcal{M}})}/we \ \cong \big[F_k(M), \emph{Bhaut} A\big]$.
One way to get the latter is to prove Conjecture \[conj:ah\_BhautA\] and then use our Theorem \[main\_thm\]. When ${\mathcal{M}}$ is the category of spaces, we have that B${\text{haut}}A$ classifies fibrations with fiber homotopy equivalent to $A$ (see [@may75 Corollary 9.5]). We wanted to prove Conjecture \[conj:thm\] and state it as the main result of this paper, but we couldn’t find a way to do it. We tried another interesting approach, which does not involve ${\widehat{A}}$ at all, that we now explain.
Another attempt to demonstrate Conjecture \[conj:thm\]
------------------------------------------------------
We discuss the case $k=1$; the cases $k \geq 2$ can be treated similarly. In the introduction we provided the proof of the first part of Theorem \[main\_thm\] that can be summarized as $${\mathcal{F}}_{1A}({\mathcal{O}(M)}; {\mathcal{M}})/we \ \cong \ {\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A})/we \ \cong \big[ M, {\widehat{A}}\big].$$ For the purposes of the new approach, we need to replace ${{\mathcal{C}}_A}$ with a larger category ${{\mathcal{C}}_A}'$ defined as follows. The objects of ${{\mathcal{C}}_A}'$ are the objects of ${\mathcal{M}}$ which are connected to $A$ by a zigzag of weak equivalences. The morphisms of ${{\mathcal{C}}_A}'$ are the weak equivalences between its objects. By definition ${{\mathcal{C}}_A}'$ is connected. Note that the category ${{\mathcal{C}}_A}'$ is nothing but what Dwyer and Kan call *special classification complex* of $A$ (see [@dwyer_kan84 Section 2.2] where they use the notation $scA$ instead). The usefulness of ${{\mathcal{C}}_A}'$ is due to the fact that its (enriched) nerve (which is denoted below by $d{\widetilde{N}}{{\mathcal{C}}_A}'$) has the homotopy type of $\text{Bhaut}A$ [@dwyer_kan84 Proposition 2.3], and depends only on the weak equivalence class of $A$.
Using the same approach as that we used to prove Theorem \[main\_thm\], one can show that there is a bijection $${\mathcal{F}}_{1A}({\mathcal{O}(M)}; {\mathcal{M}})/we \ \cong \ {\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A}')/we.$$ This makes sense if and only if ${\mathcal{F}}_{1A}({\mathcal{O}(M)}; {\mathcal{M}})$ is viewed as the category of homogeneous functors of degree $1$ whose morphisms are natural transformations which are (objectwise) weak equivalences. The next step is to try to see whether ${\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A}')/we$ is in bijection with $[M, \text{Bhaut}A]$. For this, consider the diagram
$$\xymatrix{{\mathcal{F}}({\mathcal{U}(\mathcal{T}^M)}; {{\mathcal{C}}_A}')/we \ar[r]^-{\widetilde{N}}_-{\cong} & \underset{\text{s}^2\text{Set}}{\text{Hom}} (\widetilde{N}{\mathcal{U}(\mathcal{T}^M)}, \widetilde{N}{{\mathcal{C}}_A}')/\simeq & \underset{\text{sSet}}{\text{Hom}} (N{\mathcal{U}(\mathcal{T}^M)}, N{{\mathcal{C}}_A}')/\simeq \ar[l]_-{\alpha}^-{\cong} \ar[d]^-{\psi_*} \\
\big[ |N{\mathcal{U}(\mathcal{T}^M)}|, |{\mathcal{R}}d\widetilde{N}{{\mathcal{C}}_A}'|\big] & \underset{\text{sSet}}{\text{Hom}} (N{\mathcal{U}(\mathcal{T}^M)}, {\mathcal{R}}d\widetilde{N}{{\mathcal{C}}_A}')/\sim \ar[l]_-{|-|}^-{\cong} & \underset{\text{sSet}}{\text{Hom}} (d\widetilde{N}{\mathcal{U}(\mathcal{T}^M)}, d\widetilde{N}{{\mathcal{C}}_A}')/\simeq \ar[l]_-{\varphi_*} \\
\big[ M, |d\widetilde{N}{{\mathcal{C}}_A}'|\big] \ar[u]^-{|\varphi|_*}_-{\cong} \ \ \ \ \ \ \ \ \cong & \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \big[ M, \text{Bhaut} A\big], }$$
where
1. ${\mathcal{U}(\mathcal{T}^M)}$ is viewed as a simplicially enriched category with the constant simplicial set at $\underset{{\mathcal{U}(\mathcal{T}^M)}}{\text{Hom}} (u, v)$ for every $u, v \in {\mathcal{U}(\mathcal{T}^M)}$. The simplicial enrichment of ${{\mathcal{C}}_A}' \subseteq {\mathcal{M}}$ is of course induced by that of ${\mathcal{M}}$. Note that since the source, ${\mathcal{U}(\mathcal{T}^M)}$, is discrete, we have that every functor $F \colon {\mathcal{U}(\mathcal{T}^M)}\to {{\mathcal{C}}_A}'$ is a simplicial functor in the sense that $F$ respects the simplicial enrichment;
2. $\text{s}^2\text{Set}$ is the category of bisimplicial sets, and $\widetilde{N}$ is the bisimplicial nerve functor defined as follows. For a simplicially enriched category ${\mathcal{A}}$, $\widetilde{N} {\mathcal{A}}$ is the simplicial object in simplicial sets whose $k$-simplices are given by $$(\widetilde{N} {\mathcal{A}})_k = \coprod_{(a_0, \cdots, a_k) \in {\mathcal{A}}^{k+1}} \prod_{i=0}^{k-1} \underset{{\mathcal{A}}}{\textbf{Hom}} (a_i, a_{i+1}),$$ where $\underset{{\mathcal{A}}}{\textbf{Hom}} (-, -)$ stands for the simplicial hom-set functor. It turns out that ${\widetilde{N}}$ is fully faithful. The equivalence relation “$\simeq$” that appears in $\underset{\text{s}^2\text{Set}}{\text{Hom}} (\widetilde{N}{\mathcal{U}(\mathcal{T}^M)}, \widetilde{N}{{\mathcal{C}}_A}')/\simeq$ is the one generated by homotopies. That is, two bisimplicial maps $f, g \colon {\widetilde{N}}{\mathcal{U}(\mathcal{T}^M)}\to {\widetilde{N}}{{\mathcal{C}}_A}'$ are in relation with respect to $\simeq$ if they are connected by a zigzag $f \leftarrow f_1 \rightarrow \cdots \leftarrow f_n \rightarrow g$ where $f_i \to f_j$ means that there is a homotopy from $f_i \to f_j$;
3. $N$ is the ordinary/discrete nerve functor, sSet is the category of simplicial sets as usual, and the equivalence relation “$\simeq$” that appears in $\underset{\text{sSet}}{\text{Hom}}(N{\mathcal{U}(\mathcal{T}^M)}, N{{\mathcal{C}}_A}')/\simeq$ is generated by homotopies in the same sense as before. The isomorphism $\alpha$ is defined in the standard way.
4. $d$ is the diagonal functor $d \colon \text{s}^2\text{Set} \to \text{sSet}$, and $\psi_*$ is the map induced by the canonical map $\psi \colon N{{\mathcal{C}}_A}' \to d{\widetilde{N}}{{\mathcal{C}}_A}'$. Note that by definition $d{\widetilde{N}}{\mathcal{U}(\mathcal{T}^M)}= N{\mathcal{U}(\mathcal{T}^M)}$.
5. ${\mathcal{R}}$ is a fibrant replacement functor ${\mathcal{R}}\colon \text{sSet} \to \text{sSet}$, $\sim$ is the usual homotopy relation, and $\varphi_*$ is the map induced by the fibrant replacement $\varphi: \xymatrix{d{\widetilde{N}}{{\mathcal{C}}_A}' \ \ar@{>->}[r]^-{\sim} & {\mathcal{R}}d{\widetilde{N}}{{\mathcal{C}}_A}'}$;
6. $|-|$ is of course the geometric realization functor, and $|\varphi|_*$ is the map induced by $|\varphi|$. Note that $|N{\mathcal{U}(\mathcal{T}^M)}|$ is homeomorphic to $M$;
7. The final bijection comes from the fact that $|d{\widetilde{N}}{{\mathcal{C}}_A}'| \simeq \text{Bhaut}A$ as mentioned above.
This is another candidate for a possible proof of Conjecture \[conj:thm\], but the problem here is that we do not know how to show that $\psi_*$ and $\varphi_*$ are both isomorphisms.
[99]{} D.C. Cisinski, *Locally constant functors*, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 3, 593–614. W. G. Dwyer, D. M. Kan, *A classification theorem for diagrams of simplicial sets*, Topology 23 (1984), no. 2, 139–155. W. G. Dwyer, J. Spalinski, *Homotopy theories and model categories*, Handbook of algebraic topology, 73–126, North-Holland, Amsterdam, 1995. P. G. Goerss and J. F. Jardine, *Simplicial Homotopy Theory*, Springer, vol. 174, 2009. T. G. Goodwillie and M. Weiss, *Embeddings from the point of view of immersion theory: Part II*, Geom. Topol. 3 (1999) 103–118. M. W. Hirsch, *Differential Topology*, Springer-Verlag, New York Heidelberg, Berlin, 1976. P. Hirschhorn, *Models categories and their localizations*, Mathematical Surveys and Monographs, 99. American Mathematical Society, Providence, RI, 2003. M. Hovey, *Model categories*, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. P. Lambrechts, D. Stanley, *Algebraic models of Poincaré embeddings*, Algebr. Geom. Topol. 5 (2005), 135–182. P. May, *Classifying spaces and fibrations*, Mem. Amer. Math. Soc. 1 (1975), no. 155. J. P. May, *Simplicial objects in algebraic topology*, University of Chicago Press, Chicago, IL, 1992. D. Pryor, *Special open sets in manifold calculus*, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 1, 89–103. P. A. Songhafouo Tsopméné, D. Stanley, *Very good homogeneous functors in manifold calculus*, Colloquium Mathematicum 158 (2019), no. 2, 265–297. P. A. Songhafouo Tsopméné, D. Stanley, *Polynomial functors in manifold calculus*, Topology and its Applications 248 (2018) 75–116. M. Weiss, *Embedding from the point of view of immersion theory I*, Geom. Topol. 3 (1999), 67–101.
*E-mail address: [email protected]*
*E-mail address: [email protected]*
[^1]: face will always mean face of any dimension. For instance one possible choice of $\alpha$ is $\sigma$ itself.
[^2]: An *inverse diagram* is either a covariant functor out of a small inverse category [@hovey99 Definition 5.1.1] or a contravariant functor out of a small direct category.
[^3]: If $X_{\bullet}$ and $Y_{\bullet}$ are two simplicial sets, one can consider the homotopy relation (see [@goe_jar09 Section I.6]) in the set of simplicial maps from $X_{\bullet}$ to $Y_{\bullet}$. If one wants that relation to be an equivalence relation, one has to require $Y_{\bullet}$ to be a Kan complex [@goe_jar09 Corollary 6.2].
[^4]: One may ask the question to know whether the objects that appear in diagrams (\[fi2\_diag\]), (\[fi3\_diag\]), and (\[fi4\_diag\]) belong to ${\mathcal{F}}_A({\mathcal{U}(\mathcal{T}^M)}; {\mathcal{M}})$. The answer is yes. This is straightforward by using Lemma \[facts\_lem\] -(ii) and some classical properties of homotopy limit.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is important to understand the strong external magnetic field generated at the very beginning of high energy nuclear collisions. We study the effect of the magnetic field on the charmonium yield and anisotropic distribution in Pb+Pb collisions at the LHC energy. The time dependent Schrödinger equation is employed to describe the motion of $c\bar c$ pairs. We compare our model prediction of non-collective anisotropic parameter $v_2$ of $J/\psi$s with CMS data at high transverse momentum. This is the first attempt to measure the magnetic field in high energy nuclear collisions.'
author:
- 'Xingyu Guo$^1$, Shuzhe Shi$^1$, Nu Xu$^{2,3}$, Zhe Xu$^1$, Pengfei Zhuang$^1$'
title: Magnetic Field Effect on Charmonium Production in High Energy Nuclear Collisions
---
It is widely accepted that the most strong magnetic field in nature can be generated in the early stage of relativistic heavy ion collisions. The peaked magnitude of the field can reach $eB \sim 5 m_\pi^2$ for semi-central Au+Au collisions at the Relativistic Heavy Ion Collider (RHIC) and $70 m_\pi^2$ for semi-central Pb+Pb collisions at the Large Hadron Collider (LHC) [@rafelski; @skokov; @asakawa; @voronyuk; @ou; @deng; @liu], where $e$ is the electron charge and $m_\pi$ the pion mass. Considering the interaction between the field and the hot medium formed in the later stage of the collisions, one could study fundamental QCD topological structures [@gursoy] in hot and dense nuclear matter such as chiral magnetic effect [@vilenkin; @metlitshi; @newman; @kharzeev1], chiral magnetic wave [@kharzeev2; @gorbar; @burnier], chiral separation effect [@kharzeev3; @kharzeev4; @fukushima], chiral vortical effect [@kharzeev5; @kharzeev6], chiral electric separation effect [@huang], and enhancement of elliptic flow of charged particles [@tuchin; @basar1; @basar2]. Recent experimental results on the study of the QCD intrinsic properties at RHIC and LHC can be found in Ref. [@xuzhaoshou].
Although we have crude estimation on the strength of the initial magnetic field, it may decay very fast and survive only in the very beginning of heavy ion collisions. So far, no experimental determination on the magnetic field has been made. In this Letter, we propose using the anisotropic production of high momentum charmonia to probe the initial magnetic field in high energy nuclear collisions. Due to their large masses, heavy quarks are produced in the early stage through hard scatterings. For low momentum charmonia, they can be produced in the initial stage and regenerated [@pbm; @rapp; @thews; @yan] in the hot medium, and both will be suppressed through Debye screening [@matsui]. High momentum charmonia, on the other hand, are purely formed before the formation of the hot medium. Except interaction with the magnetic field, which will be discussed here, the high momentum charmonia are almost not affected by the hot medium and therefore can be used as an ideal tool to probe the initial magnetic field in high energy nuclear collisions. Note, in this work we only focus on the spectators induced filed. The effects of the delayed decay of the field [@gursoy] caused by the conducting medium will not change our main conclusions.
Since the life time of the magnetic field $t_B \sim 0.1$ fm/c [@deng] is much shorter than the charmorium formation time $t_f \sim 0.5$ fm/c [@gerschel] from a $c\bar c$ pair to a state $\Psi=J/\psi,\psi',\chi_c,\cdots$, the interaction between the $c\bar c$ pair and the field could cause some profound effects on (i) Charmonium yields: the magnetic field induced force will change the charmonium fractions $|C_\Psi|^2$ in the $c\bar c$ pair $|c\bar c\rangle =C_\Psi|\Psi\rangle$ and thus alters the relative yields among different charmonium states, and (ii) Charmonium distribution: the specific direction along the magnetic field breaks down the rotational symmetry of the system, which leads to an anisotropic charmonium production in the transverse plane. Both effects can be experimentally checked.
As charm quarks are heavy enough in comparison with the inner movement inside a charmonium bound state which is determined by the $J/\psi$ radius due to the uncertainty principle, $m_c\sim 1.3$ GeV $>p\sim 1/r=1/(0.5\text {fm})=0.4$ GeV, we can ignore the relativistic effect in considering the inner structure of a charmonium. We employ the time dependent Schrödinger equation to describe the evolution of a $c\bar c$ pair wave function $\Phi(t,{\bf r}_c,{\bf r}_{\bar c})$, $$\label{schroedinger}
i{\partial\over \partial t}\Phi=\hat H\Phi$$ with the Hamiltonian operator $$\label{hamiltonia}
\hat H={\left(\hat {\bf p}_c-q{\bf A}_c\right)^2\over 2m_c}+ {\left(\hat {\bf p}_{\bar c}+q{\bf A}_{\bar c}\right)^2\over 2m_c}+V,$$ where ${\bf A}$ is the magnetic potential, $q=2e/3$ the charm quark electron charge, and $V$ the potential between the quark and antiquark. We take the Cornell potential together with the spin-spin interaction [@lattice], $$V(r) = -\frac{\alpha}{r} + \sigma r + \beta e^{-\gamma r} {\bf s}_c\cdot{\bf s}_{\bar c}.$$ By fitting the charmonium spectrum in vacuum, one can fix the potential parameters as $m_c=1.29$ GeV, $\sigma=0.174$ GeV$^2$, $\alpha=0.312$, $\beta=1.982$ GeV and $\gamma=2.06$ GeV.
By separating the $c\bar c$ wave function into a center-of-mass part and a relative part $\Phi=\Phi_R\Phi_r$ with the center-of-mass coordinator ${\bf R}=({\bf r}_c+{\bf r}_{\bar c})/2$ and relative coordinator ${\bf r}={\bf r}_c-{\bf r}_{\bar c}$, and further expanding the relative part in terms of the charmonium wave functions $\Psi({\bf r})$, $$\label{expansion}
\Phi = {1\over \sqrt{2\pi}}e^{i{\bf P}_k\cdot{\bf R}-i{{\bf P}_p^2\over 4m_c}t}\sum_\Psi C_\Psi e^{-iE_\Psi t} \Psi,$$ the probability $|C_\psi(t)|^2$ for the $c\bar c$ pair to be in the charmonium state $\Psi$ satisfies the evolution equation $$\label{cpsi}
{d\over dt}C_\Psi=\sum_{\Psi'}e^{i(E_\psi-E_{\psi'})t}C_{\Psi'}\int d^3{\bf r}\Psi^*({\bf r})\hat H_B\Psi'({\bf r}),$$ where we have separated the Hamiltonian into a vacuum part and a magnetic field dependent part, $$\label{h}
\hat H=\hat H_0 + \hat H_B,$$ with $$\label{hb}
\hat H_B= {-\boldsymbol \mu}\cdot {\bf B}-{q\over 2m_c}(\hat {\bf P}_p\times {\bf B})\cdot{\bf r}+{q^2\over 4m_c}({\bf B}\times {\bf r})^2.$$ The charmonium energy $E_\psi$ and wave function $\Psi({\bf r})$ in Eqs.(\[expansion\]) and (\[cpsi\]) are determined by $\hat H_0$, $$\label{h0}
\hat H_0\Psi=E_\Psi \Psi.$$
Since we have now a special direction, the direction of ${\bf B}$, the kinetic momentum ${\bf P}_k={\bf P}-q({\bf A}_c-{\bf A}_{\bar c})$ with the total momentum ${\bf P}={\bf p}_c+{\bf p}_{\bar c}$ is no longer conserved, and the conserved momentum is the pseudo momentum ${\bf P}_p = {\bf P}+q({\bf A}_c-{\bf A}_{\bar c})$ [@alford]. In Eq.(\[hb\]) the $c\bar c$ pair interaction with the magnetic field is manifested by three terms: the first term is the spin-field interaction with the spin magnetic moment ${\boldsymbol \mu}=q/m_c({\bf s}_c-{\bf s}_{\bar c})$, the second term comes from the Lorentz force which is proportional to the $c\bar c$ momentum, and the third term is the harmonic potential which is quadratic in $q{\bf B}$ and therefore its effect is much smaller in comparison with the first and second terms which are linear in $q{\bf B}$. At high momentum, the Lorentz force is the dominant magnetic field effect. Note that the spin-field interaction makes the spin angular momentum no longer a conserved quantity, the spin singlet $\eta_c$ will couple with one of the triplet $J/\psi$ which carries zero spin component along the magnetic field $s_B=0$ [@alford; @yang]. On the other hand, the Lorentz force and the harmonic potential breaks the rotational symmetry in the coordinate space.
Let us take the nuclear colliding direction as the $z$-axis and the impact parameter ${\bf b}$ parallel to the $x$-axis. While the created magnetic field in heavy ion collisions depends on the events, and the magnitude and direction of the field fluctuate in space and time [@rafelski; @skokov; @asakawa; @voronyuk; @ou; @deng; @liu], we consider here an averaged magnetic field ${\bf B}$ along the $y-$axis in the space-time region determined by the colliding energy and nuclear geometry, $$\label{b}
{\bf B} = \left\{ \begin{array}{ll}
B{\bf e}_y, & 0<t<t_B, \frac{x^2}{(R_A-b/2)^2}+\frac{y^2}{(b/2)^2}+\frac{\gamma_c^2z^2}{(b/2)^2} < 1, \\
0, & \text {others}. \\
\end{array}\right.$$ which, from the relation ${\bf B}=\nabla \times {\bf A}$, leads to $A_x=Bz$ and $A_y=A_z=0$. For Pb+Pb collisions with centrality $40\%$ and at LHC energy $\sqrt{s_{NN}}=2.76$ TeV, the geometry parameters and the Lorentz factor are fixed to be $R_A=6.6$ fm, $b=8$ fm and $\gamma_c=1400$. From the MC simulation [@deng], we take the magnitude and the life time of the magnetic field $eB=25 m_\pi^2$ and $t_B=0.2$ fm/c. At RHIC energy, the initially created magnetic field is much weaker, $eB\simeq m_\pi^2$ [@deng]. If the field decays very fast before the formation of the hot medium where the Faraday induction may prolong the field [@gursoy], the magnetic field effect on the charmonium production can be safely neglected. In the following we will focus on the magnetic field effect at LHC energy.
To solve the dynamical equation (\[cpsi\]) for the probability magnitude $C_\Psi(t)$, we need the initial value $C_\Psi(0)$ or the initial wave function $\Phi_r(0)$. Suppose the relative motion of the $c\bar c$ pair is initially described by a compact Gaussian wave package $$\label{initial}
\Phi_r(0)\sim e^{-{({\bf r}-{\bf r}_0)^2\over \sigma_0^2}}$$ with averaged relative coordinate ${\bf r}_0$ and distribution width $\sigma_0$. Since there is no reason to choose a special direction before the magnetic field is introduced, the azimuth angles of ${\bf r}_0=r_0(\sin\theta_0\cos\phi_0,\sin\theta_0\sin\phi_0,\cos\theta_0)$ are randomly distributed, and we do ensemble average over $\theta_0$ and $\phi_0$ in the final state calculations. The remaining two parameters $r_0$ and $\sigma_0$ can be determined by fitting the charmonium fractions in p+p collisions. Suppose an initial point-like wave function $\delta({\bf r}-{\bf r}')$ develops as $e^{({\bf r}-{\bf r}')^2/(v^2t^2)}$ with a constant expansion velocity $v$, the initial Gaussian wave package evolves as $$\label{phit}
\Phi_r(t) \sim \int d^3{\bf r}' e^{-{({\bf r}'-{\bf r}_0)^2\over \sigma_0^2}}e^{-{({\bf r}-{\bf r}')^2\over v^2t^2}} \sim e^{-{({\bf r}-{\bf r}_0)^2\over \sigma_t^2}}$$ in p+p collisions. It is always a Gaussian wave package but with a time dependent width $\sigma_t^2=\sigma_0^2+v^2t^2$. Calculating the probability $|C_\Psi(t_f)|^2 = |\langle\Psi|\Phi_r(t_f)\rangle|^2$ in vacuum with $B=0$ at the charmonium formation time $t_f$, and taking the experimentally observed decay branch ratios ${\cal B}(\Psi\to J/\psi)$ [@pdg], we obtain the fractions of the direct $J/\psi$ production and feed down from the excited states in the finally observed prompt $J/\psi$s, $$\label{ratio}
R_\Psi(t)={|C_\Psi(t)|^2{\cal B}(\Psi\to J/\psi)\over \sum_\Psi |C_\Psi(t)|^2{\cal B}(\Psi\to J/\psi)}$$ with the definition of ${\cal B}(J/\psi\to J/\psi)=1$. Taking $R(J/\psi)=60\%$, $R(\psi')=10\%$ and $R(\chi_c)=30\%$ from the recent p+p data at LHC energy [@atlas; @cms1; @lhcb] and the charmonium formation time $t_f=0.5$ fm/c [@gerschel], the remaining two parameters in the initial wave package $\Phi_r(0)$ are fixed to be $r_0=0.68$ fm and $\sigma_0=0.02$ fm, which correspond to the expansion velocity $v=0.72$c and the width $\sigma_{t_f}=0.38$ fm.
With the known initial wave function $\Phi_r(0)$ or the initial probability magnitudes $C_{J/\psi}(0):C_{\psi'}(0):C_{\chi_c}(0)=1:0.5:1.2$, we can solve the evolution equation (\[cpsi\]) for $C_\Psi(t)$ in the time region $0<t<t_B$ when the magnetic field exists. After that the wave function evolves according to the expansion rule (\[phit\]) with $\Phi_r(t_B,{\bf r})$ as the initial condition, $$\label{phit2}
\Phi_r(t,{\bf r}) \sim \int d^3{\bf r}' \Phi_{r'}(t_B,{\bf r}')e^{-{({\bf r}-{\bf r}')^2\over v^2(t-t_B)^2}}\ \ \ \ \ \ \text {for}\ \ t>t_B.$$
We now show our numerical results about the magnetic field effect on the charmonium production in heavy ion collisions. We focus on the central rapidity region in Pb+Pb collisions with impact parameter $b=8$ fm and at LHC energy $\sqrt {s_{NN}}=2.76$ TeV. In our calculation, the initial $c\bar c$ pairs which are determined by the colliding energy and the nuclear geometry are assumed to be distributed homogeneously in the overlapped region of the two colliding nuclei. To have a good precision in solving the coupled equations (\[cpsi\]) for $C_\Psi(t)$, we take a cutoff in the sum over the charmonium eigenstate $\Psi$: the main quantum number $n\leq 6$ and the orbital angular momentum number $l\leq 7$.
In Fig.\[fig1\], the $J/\psi$ fractions $R_\Psi(t)$ from different channels are shown as functions of $c\bar c$ evolution time. The results with and without the external magnetic field are displayed by thick and thin lines, respectively. These fractions are extracted at fixed transverse momentum $p_T=10$ GeV/c from Pb+Pb collisions at impact parameter $b=8$ fm and LHC energy $\sqrt s_{NN}=2.76$ TeV. In case there is no magnetic field, the direct $J/\psi$ production and feed down from $\psi'$ are slightly enhanced, while the contribution from $\chi_c$ is weakly suppressed, indicating a $\chi_c$ decay to $J/\psi$ and $\psi'$ with quantum selection rule $\delta l=1$. When the magnetic field is turned on, the converting from the high lying state $\chi_c$ to $J/\psi$ and $\psi'$ becomes much more dramatic, as one can see the thick lines in Fig.\[fig1\]. Since the magnetic field introduces a special direction, the $c\bar c$ motion becomes anisotropic. Here we fix the $c\bar c$ azimuthal angle in the transverse plane $\varphi=\arctan (p_y/p_x)=0$ where the magnetic field effect is expected to be the strongest. In the time period $t<t_B$, the external magnetic field serves as a stimulator that enhances the quantum mechanics allowed transition from $\chi_c$ to $J/\psi$ and $\psi'$. After the field is off at $t>t_B$, the variation of all the fractions with time becomes mild again. At the formation time $t_f=0.5$ fm/c, the relative enhancement for both direct $J/\psi$ production and feed down from $\psi'$ are found to be $10\%$, and the contribution from $\chi_c$ decay is relatively suppressed by $23\%$.
![(Color online) The time evolution of $J/\psi$s from direct production (solid lines) and feed down from $\psi'$ (dashed lines) and $\chi_c$ (dot-dashed lines). The thick and thin lines represent the results from the calculations with and without the external magnetic filed, respectively. As indicated by the vertical short-dashed line, the magnetic field only lasts during the time $t<t_B= 0.2$ fm/c. []{data-label="fig1"}](fig1.eps){width="48.00000%"}
![(Color online) The magnetic field induced anisotropic behavior of the fractions $R_\Psi$ in the $J/\psi$ yield at the formation time $t_f$. []{data-label="fig2"}](fig2.eps){width="48.00000%"}
Having discussed the different contributions to $J/\psi$ production, we now look at the yields for $J/\psi$, $\psi'$ and $\chi_c$. For $J/\psi$, the ratio between the yields $N_{J/\psi}^{(B)}$ and $N_{J/\psi}^{(0)}$ with and without the magnetic field can be expressed in terms of the probabilities $|C_\Psi^{(B)}(t_f)|^2$ and $|C_\Psi^{(0)}(t_f)|^2$ and the fractions $R_\Psi^{(0)}(t_f)$ at the formation time $t_f$, $$\begin{aligned}
\label{ratio2}
{N_{J/\psi}^{(B)}\over N_{J/\psi}^{(0)}}&=&{\sum_\Psi |C_\Psi^{(B)}(t_f)|^2{\cal B}(\Psi\to J/\psi)\over \sum_\Psi |C_\Psi^{(0)}(t_f)|^2{\cal B}(\Psi\to J/\psi)}\nonumber\\
&=&\sum_\Psi {|C_\Psi^{(B)}(t_f)|^2\over |C_\Psi^{(0)}(t_f)|^2}R_\Psi^{(0)}(t_f).\end{aligned}$$ Using the above calculated probabilities and fractions at LHC energy, the maximum $J/\psi$ enhancement at $\varphi=0$ and $p_T=10$ GeV/c is $13\%$. For the excited states, neglecting the feed down from the higher eigen states of $\hat H_0$, we have the yield ratios $N_{\psi'}^{(B)}/N_{\psi'}^{(0)}=|C_{\psi'}^{(B)}(t_f)|^2/|C_{\psi'}^{(0)}(t_f)|^2=1.29$ for $\psi'$ and $N_{\chi_c}^{(B)}/N_{\chi_c}^{(0)}=|C_{\chi_c}^{(B)}(t_f)|^2/|C_{\chi_c}^{(0)}(t_f)|^2=0.84$ for $\chi_c$, which mean, due to the magnetic field, a relative enhancement of $29\%$ and suppression of $16\%$ for $\psi'$ and $\chi_c$ productions in Pb+Pb collisions at the LHC energy.
Fig.\[fig2\] shows the magnetic field induced anisotropic property of the fractions $R_\Psi(t_f)$ in the transverse plane at charmonium formation time $t_f$. The strength of the Lorentz force acting on the charmonia is most strong at $\varphi=0$, and decreases monotonously with increasing azimuthal angle $\varphi$. Finally at $\varphi=\pi/2$, the force disappears and only the weak harmonic potential exists, the fractions approach to their vacuum values.
![(Color online) The magnetic field induced transverse momentum dependence of the fractions $R_\Psi$ in the $J/\psi$ yield at the formation time $t_f$. []{data-label="fig3"}](fig3.eps){width="48.00000%"}
![(Color online) The transverse momentum dependence of $J/\psi$ $v_2$. The solid and dashed lines are the initially produced non-collective $J/\psi$ $v_2$ with and without magnetic field, and the data at high $p_t$ are from the CMS collaboration [@cms2]. As a comparison, the collective $J/\psi$ $v_2$ from a transport model [@zhou] is shown as the dot-dashed line.[]{data-label="fig4"}](fig4.eps){width="50.00000%"}
We now show in Fig.\[fig3\] the magnetic field induced transverse momentum dependence of the fractions $R_\Psi(t_f)$ at charmonium formation time $t_f$. We again fix the angle $\varphi=0$ to see the maximum magnetic field effect. At $p_T=0$ the Lorentz force disappears, the slight deviation of the fractions from the vacuum values comes from the weak harmonic potential. The strength of the Lorentz force increases with $p_T$, and the change of the fractions for all channels becomes larger till $p_T \sim 3$ GeV/c. In higher $p_T$ region, the change stops and the transverse momentum dependence seems flat. Although the force is increasing with $p_T$, the higher the $p_T$ the shorter the time the $c\bar c$ pair experiences the field. The combination of the momentum dependence of the Lorentz force and the finite time of the external magnetic field leads to the observed saturation in Fig.\[fig3\].
Usually, the observed event anisotropy $v_2$ is discussed in the framework of hydrodynamics, representing the collective motion of the medium created in high energy nuclear collisions [@zhou; @rapp2; @gossiaux]. At the LHC energy, $J/\psi$ $v_2$ has been reported in the region of $p_T<10$ GeV/c [@alice]. Since the collective motion is dominated by the bulk interactions in relatively low momentum region, those high $p_T$ charmonia generated in the initial stage are not expected to be sensitive to the nature of the hot medium. However, the Lorentz force induced anisotropic production in the transverse plane, shown in Fig.\[fig2\], may result in a non-collective $J/\psi$ $v_2$ at high $p_T$. The $J/\psi$ $v_2$ as a function of rapidity and transverse momentum is defined as $$\label{v2}
v_2(\eta,p_t) = \frac{\int_0^{2\pi} N_{J/\psi}(\eta,p_t,\varphi)\cos(2\varphi) d\varphi}{\int_0^{2\pi} N_{J/\psi}(\eta,p_t,\varphi) d\varphi}.$$ The numerical result at central rapidity is shown in Fig.\[fig4\]. In normal calculations without considering the magnetic field [@zhou; @rapp2; @gossiaux], the initial $J/\psi$s are isotropically produced with $v_2=0$ (dashed line). However, due to the magnetic field induced Lorentz force, the high $p_T$ $J/\psi$s acquire sizable non-collective $v_2$ (solid line), which explains reasonably well the CMS data [@cms2] for prompt $J/\psi$s in the high $p_T$ region. As a comparison, We show in Fig.\[fig4\] also the collective $v_2$ (dot-dashed line) calculated from a dynamical transport model [@zhou]. Different from the collective flow which comes from the $J/\psi$ regeneration at low and intermediate $p_T$, the non-collective $v_2$ induced by the magnetic field is mainly in high $p_T$ region. As one can see in Figs.\[fig1\], \[fig2\] and \[fig3\], $\chi_c$ is suppressed, corresponding to the $J/\psi$ and $\psi'$ enhancement. Hence it is interesting to point out that if the observed non-collective $v_2$ for $J/\psi$ is positive, the high $p_T$ $v_2$ for $\chi_c$ should be negative.
In summary, by solving the time dependent Schrödinger equation for the $c\bar c$ pairs, we discussed the effects of the initially created magnetic field on the relative yields and anisotropic distributions of the charmonium states in high energy nuclear collisions. For Pb+Pb collisions at LHC, we found (i) the directly produced and $\psi'$ decayed $J/\psi$s are enhanced, while the $J/\psi$s from $\chi_c$ decay are suppressed, and (ii) the non-collective $J/\psi$ $v_2$ at large transverse momentum, $p_T \geq 8 $ GeV/c, is as large as $4\%$ and comparable with the CMS data. We wish to point out that in high energy nuclear collisions the initial magnetic field is related to several important measurements reflecting the intrinsic structure of the QCD. The measurement of the strong $J/\psi$ anisotropy at large $p_T$ can be used to diagnose the magnetic filed in such collisions.
[**Acknowledgement:**]{} The work is supported by the NSFC, MOST and DOE grant Nos. 11275103, 11335005, 2013CB922000, 2014CB845400, 2015CB856900 and DE-AC03-76SF00098.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cavalcanti, Fatori, and Ma, *Attractors for wave equations with degenerate memory*, J. Differential Equations (2016). Under certain extra assumptions (namely on the nonlinear term), we show the existence of a compact attracting set which provides further regularity for the global attractor and show that it consists of regular solutions.'
address: 'Joseph L. Shomberg, Department of Mathematics and Computer Science, Providence College, Providence, RI 02918, USA'
author:
- 'Joseph L. Shomberg'
title: Regular global attractors for wave equations with degenerate memory
---
Introduction
============
An elastic body perturbed from equilibrium may undergo a restoring force subject to both frictional and viscoelastic dissipation mechanisms. The problem under consideration is the wave equation with degenerate viscoelastic dissipation in the unknown $u=u(x,t)$ $$\begin{aligned}
u_{tt} - \Delta u + \int_0^\infty g(s){{\rm div}}[a(x)\nabla u(t-s)]ds + b(x)u_t + f(u) = h(x) \quad \text{in} \ \Omega\times\mathbb{R}^+, \label{tpde}\end{aligned}$$ defined on a bounded domain $\Omega$ in $\mathbb{R}^3$ with smooth (at least class $\mathcal{C}^2$) boundary $\Gamma$. The equation is subject to Dirichlet boundary conditions $$\begin{aligned}
u(x,t) = 0 \quad \text{on} \ \Gamma\times\mathbb{R}^+, \label{tbc}\end{aligned}$$ and the initial conditions $$\begin{aligned}
u(x,0) = u_0(x) \quad \text{and} \quad u_t(x,0) = u_1(x) \quad \text{at} \ \Omega\times\{0\}. \label{tic}\end{aligned}$$
This problem was recently treated, to the extent of global well-posedness and global attractors, in [@CFM-16]. The novelty here being the degenerate nature of the viscoelasticity. Similar problems have yielded several important results as well. We mention some other works concerning semilinear wave equations with memory. On the asymptotic behavior of solutions (in the sense of global attractors) see , and on rates of decay of solutions one can also see [@LiZh2011; @santos07; @Tahamtani_799].
To the problem under consideration here, the well-posedness was carried out under the guise of semigroup methods. Here, local mild solutions and regular (or “strong” solutions) are obtained using the fact that the underlying operator is the infinitesimal generator of a strongly continuous semigroup of contractions on the Hilbertian phase space $\mathcal{H}$, and the other condition naturally being that the nonlinear term defines a locally Lipschitz continuous functional also on $\mathcal{H}$.
The main result concerning the asymptotic behavior of - in [@CFM-16] consists in demonstrating the existence of a finite dimensional global attractor for the semidynamical system $(\mathcal{H},S(t)).$ For this, the authors of [@CFM-16] rely on [@Ch-La-10 Proposition 7.9.4 and Theorem 7.9.6]. That is, the problem is of the asymptotically smooth gradient system class where the set of stationary points is bounded. The so-called quasi-stability of the dynamical system $(\mathcal{H},S(t))$ involves finding a suitable (relatively) compact seminorm on $\mathcal{H}$ (i.e., the approach is similar to finding a global attractor via an $\alpha$-contraction method). Instead of characterizing the global attractor as the omega-limit set of some bounded absorbing set $\mathcal{B}$ in $\mathcal{H},$ i.e. $\mathcal{A}=\omega(\mathcal{B}),$ the global attractor in this work is characterized with properties from the gradient system so that the global attractor is described by the union of unstable manifolds connecting the set of stationary points $\mathcal{N}$, i.e. $\mathcal{A}=\mathbb{M}^u(\mathcal{N})$. Unlike the methods used to prove the existence of a global attractor by virtue of the former characterization, in the latter no (explicit) bounded absorbing set $\mathcal{B}$ nor any (explicit) uniform bound on solutions is used to prove the existence of the global attractor. Finally, it seems that an explicit bound in terms of some of the parameters of the problem (Lipschitz constant, etc.) can be given to the fractal dimension of the global attractor (indeed, see [@Chueshov-15 Theorem 3.4.5]). These results are obtained without assuming the two damping terms satisfy a geometric control condition (cf. e.g. [@Joly-Camille-13]).
To treat the memory term, we define a past history variable using the [*[relative displacement history]{}*]{}, for all $x\in\Omega\subset \mathbb{R}^3$ and $s,t\in\mathbb{R}^+$, $$\eta^t(x,s) := u(x,t) - u(x,t-s). \label{memory-defn}$$ In order for this formulation to make sense, we also need to prescribe the past history of $u(x,t)$, $t<0$. Observe, from we readily find the useful identity $$\int_0^\infty g(s){{\rm div}}[a(x)\nabla u(t-s)]ds = - \int_0^\infty g(s){{\rm div}}[a(x)\nabla\eta^t(s)]ds + k_0{{\rm div}}[a(x)\nabla u(s)],$$ where $k_0:=\int_0^\infty g(s) ds$ assumed to be sufficiently small below (see ). Thus, equations - have an equivalent form in the unknowns $u=u(x,t)$ and $\eta^t=\eta^t(x,s),$ for all $x\in\Omega$ and $s,t\in\mathbb{R}^+$, $$\begin{aligned}
u_{tt} & - {{\rm div}}[(1-k_0a(x))\nabla u]-\int_0^\infty g(s){{\rm div}}[a(x)\nabla\eta^t(s)]ds + b(x)u_t + f(u) = h(x), \label{pde-1} \\
\eta_t & = -\eta_s + u_t, \label{pde-2}\end{aligned}$$ with boundary conditions, for all $(x,t)\in\Gamma\times\mathbb{R}^+$, $$\begin{aligned}
u(x,t) = 0 \quad \text{and} \quad \eta^t(x,s) = 0, \label{bc}\end{aligned}$$ and the following initial conditions at $t=0,$ $$\begin{aligned}
u(x,0) = u_0(x), \quad u_t(x,0) = u_1(x) \quad \text{and} \quad \eta^t(x,0) = 0, \quad \eta^0(x,s) = \eta_0(x,s). \label{ic}\end{aligned}$$
In this article, we aim to provide a regularity result to the global attractors found in [@CFM-16] for the problem -.
Preliminaries
=============
This section contains a summary of the assumptions and main results of [@CFM-16].
[*[A word about notation:]{}*]{} we will often drop the dependence on $x$ and even $t$ or $s$ from the unknowns $u(x,t)$ and $\eta^t(x,s)$ writing only $u$ and $\eta^t$ instead. The norm in the space $L^p(\Omega)$ is denoted $\|\cdot\|_p$ except in the common occurrence when $p=2$ where we simply write the $L^2(\Omega)$ norm as $\|\cdot\|$. The $L^2(\Omega)$ product is simply denoted $(\cdot,\cdot).$ Other Sobolev norms are denoted by occurrence; in particular, since we are working with the homogeneous Dirichlet boundary conditions , in $H^1_0(\Omega)$, we will use the equivalent norm $$\|u\|_{H^1_0(\Omega)}=\|\nabla u\|,$$ and in particular, $$\begin{aligned}
\|u\| \le \frac{1}{\sqrt{\lambda_1}}\|\nabla u\|, \label{Poincare}\end{aligned}$$ where $\lambda_1>0$ denotes the first eigenvalue of the Dirichlet–Laplacian. With $D(-\Delta)=H^2(\Omega)\cap H^1_0(\Omega),$ we are able to define, for any $s\ge0,$ $$H^s:=D((-\Delta)^{s/2}).$$ Given a subset $B$ of a Banach space $X$, denote by $\|B\|_X$ the quantity $\sup_{x\in B}\|x\|_X$. Finally, in many calculations $C$ denotes a [*[generic]{}*]{} positive constant which may or may not depend on several of the parameters involved in the formulation of the problem, and $Q(\cdot)$ will denote a [*[generic]{}*]{} positive nondecreasing function.
Concerning the model problem, we make the following assumptions.
(H1)
: Let $a\in C^1({\overline{\Omega}})$ be such that $${\mathrm{meas}}\{ x\in\Gamma:a(x)>0 \} >0,$$ and $$\mathcal{V}^1_a:=\left\{\psi\in L^2(\Omega):\int_\Omega a(x)|\nabla\psi(x)|^2 dx < \infty,\ \psi_{\mid\Gamma}=0 \right\},$$ is a Hilbert space endowed with the product $$(\chi,\psi)_{\mathcal{V}^1_a}:=\int_\Omega a(x)\nabla\chi(x) \cdot \nabla\psi(x) dx.$$ (Two examples are given in [@CFM-16].) Above $\psi_{\mid\Gamma}=0$ is meant in the sense of trace which is well-defined when $\mathcal{V}^1_a\hookrightarrow W^{1,1}(\Omega)$. In addition, we also assume the continuous embeddings hold $$H^{1}_0(\Omega)\hookrightarrow \mathcal{V}^1_a\hookrightarrow L^2(\Omega),$$ and also that $Au:={{\rm div}}(a(x)\nabla u)$ is a self-adjoint non-positive operator.
(H2)
: Assume $b\in L^\infty(\Omega)$ is a non-negative function and $c_0$ is a constant satisfying, for all $x\in\Omega,$ $$\inf_{x\in\Omega}\{a(x)+b(x)\} \ge c_0 > 0. \label{away}$$
(H3)
: Assume $g\in C^1(\mathbb{R}^+)\cap L^1(\mathbb{R}^+)$ satisfies, for all $s\ge0,$ $$g(s) \ge 0 \quad \text{and} \quad g'(s) \le -\delta g(s). \label{kernel}$$ We also impose on $g$ the smallness condition $$k_0:=\int_0^\infty g(s) ds < \|a\|^{-1}_{\infty}. \label{small-g}$$
Assumption (H1) allows us to set the space for the past history function $\eta^t.$ Indeed, define $$\label{past-space}
\mathcal{M}^{0}:=L^2_g(\mathbb{R}^+;\mathcal{V}^{1}_a)=\left\{ \eta(x,s):\int_0^\infty g(s)\|\eta(x,s)\|^2_{\mathcal{V}^{1}_a}ds <\infty \right\}$$ which is Hilbert with the product $$(\eta,\zeta)_{\mathcal{M}^{0}}:=\int_0^\infty g(s) \left( \int_\Omega a(x)\nabla\eta(x,s)\cdot \nabla\zeta(x,s) dx \right) ds.$$
\[r:degenerate\] It should be noted that in [@CFM-16], the assumption (H2) allows one to view the role of the frictional damping coefficient $b$ as an arbitrarily small complementary damping in the following sense: if $\omega_0:=\{x\in\mathbb{R}^3:a(x)=0\}$, then what is only required is $b(x)>0$ on any neighborhood of $\omega_0$.
\[r:a-bnd\] Equation of assumption (H3) implies $g$ decays to zero exponentially. Moreover, by , we have that, for all $x\in{\overline{\Omega}}$, $$\begin{aligned}
0<\ell_0 \le 1-k_0a(x) \label{ell-ineq}\end{aligned}$$ where $$\ell_0:=1-k_0\|a\|_\infty.$$
Now we make our final assumptions.
(H4)
: Let $f\in C^2(\Omega)$ and assume there exists $C_f>0$ such that, for all $s\in\mathbb{R},$ $$|f''(s)| \le C_f (1+|s|). \label{assm-f-1}$$ (Hence, the nonlinear term is allowed to attain critical growth.) We also assume that $$\liminf_{|s|\rightarrow\infty} \frac{f(s)}{s} > -\ell_0\lambda_1 \label{assm-f-2}$$ cf. .
The two conditions and are used in [@GGPS05] which treats the asymptotic behavior of a phase-field equation with memory. The assumption implies there is a constant $C>0$ such that for all $r,s\in\mathbb{R}$ $$|f(r)-f(s)| \le C |r-s|(1+|r|^2+|s|^2). \label{cons-f-1}$$ The condition appears in many recent works on semilinear wave equations with memory (e.g. [@FePeAn2016]) and the strongly damped wave equation (this condition refers to the [*sub*]{}critical setting of those problems), see for example . By we find that for some $\alpha\in(0,\lambda_1)$, there exists $\rho_f>0$ so that, for all $s\in\mathbb{R},$ there hold $$f(s)s \ge -\ell_0\alpha s^2 - \rho_f \label{cons-f-2}$$ and, for $F(s):=\int_0^s f(\sigma)d\sigma$, $$F(s) \ge -\frac{\ell_0\alpha}{2}s^2 - \rho_f. \label{cons-f-3}$$ Observe though both and follow when is replaced by the less general assumption, $$\label{cond-99}
\liminf_{|s|\rightarrow\infty} f'(s) \ge -\ell_0\lambda_1.$$ Assumption and condition appear in equations with memory terms [@CGG11; @Conti-Mola-08; @CPS06; @PPZ08].
Concerning the new regularity results described in section \[s:reg\], we additionally assume the following assumptions hold along with (H1)-(H4).
(H1r)
: Suppose $a\in C^1({\overline{\Omega}})$ is such that $$\mathcal{V}^2_a:=\left\{\psi\in L^2(\Omega):\int_\Omega a(x) \left( |\Delta\psi(x)|^2 + |\psi(x)|^2 \right) dx < \infty,\ \psi_{\mid\Gamma}=0 \right\},$$ is a Hilbert space endowed with the product $$(\chi,\psi)_{\mathcal{V}^2_a}:=\int_\Omega a(x) \left( \Delta\chi(x) \Delta\psi(x) + \chi(x) \psi(x) \right) dx.$$ Also, assume the continuous embedding holds $$\mathcal{V}^2_a\hookrightarrow H^1_0(\Omega).$$
It should be noted that the embedding $D(-\Delta)\hookrightarrow \mathcal{V}^2_a$, where $D(-\Delta):=H^2(\Omega)\cap H^1_0(\Omega)$, does not hold. The interested reader should see [@CRV-08 Section 3] where it is shown $H^2(\Omega)\not\subseteq\mathcal{V}^2_a$.
(H4r)
: Assume that there exists $\vartheta>0$ such that, for all $s\in\mathbb{R},$ $$f'(s)\ge-\vartheta. \label{assm-f-3}$$
The last assumption appears in . Such a bound is commonly utilized to obtain the precompactness property for the semiflow associated with evolution equations where the use of fractional powers of the Laplace operator present a difficulty, if they are even well-defined.
Throughout the remainder of this article, we simply denote - under assumptions (H1)-(H4) and (H1r) and (H4r) as problem P.
The finite energy phase-spaces we study problem P in involve the following Hilbert spaces. First, $$\mathcal{H}^{0}:= H^{1}(\Omega)\times L^2(\Omega)\times \mathcal{M}^{0},$$ endowed with the norm whose square is given by, for $U=(u,v,\eta)\in\mathcal{H}^{0},$ $$\|U\|^2_{\mathcal{H}^{0}}:=\|\nabla u\|^2 + \|v\|^2 +\|\eta\|^2_{\mathcal{M}^{0}}.$$ Later we also require $$\mathcal{M}^1:=L^2_g(\mathbb{R}^+;\mathcal{V}^2_a)=\left\{ \eta:\int_0^\infty g(s)\|\eta(s)\|^2_{\mathcal{V}^2_a}ds <\infty \right\}$$ and $$\mathcal{H}^1:= H^2(\Omega)\times H^1(\Omega)\times \mathcal{M}^1,$$ with the norm whose square is given by, for $U=(u,v,\eta)\in\mathcal{H}^1,$ $$\|U\|^2_{\mathcal{H}^1}:=\|u\|^2_{H^{2}(\Omega)} + \|v\|^2_{H^1(\Omega)} +\|\eta\|^2_{\mathcal{M}^1}.$$ Here $H^1(\Omega)$ is normed with $$\|\psi\|_{H^1(\Omega)} = \left( \|\nabla \psi\| + \|\psi\| \right)^{1/2},$$ and concerning the $H^2(\Omega)$ norm above, we know by $H^2$-elliptic regularity theory (cf. e.g. [@GT83 section 8.4]), $$\begin{aligned}
\|\psi\|_{H^2(\Omega)} \le C\left( \|\Delta \psi\| + \|\psi\| \right), \label{reg-est}\end{aligned}$$ for some constant $C>0.$
So that we may write problem P in an operator formulation, we also define the following spaces, $$\begin{aligned}
D(T):=\{ \eta\in\mathcal{M}^0:\eta_s\in\mathcal{M}^0,\ \eta(0)=0 \}, \notag \end{aligned}$$ where $\eta_s$ denotes the distributional derivative of $\eta$ and the equality $\eta(0)=0$ is meant as $$\lim_{s\rightarrow0}\|\eta(s)\| = 0,$$ and $$\begin{aligned}
D(\mathcal{L}):=\left\{ U=(u,v,\eta)\in\mathcal{H}^0 \left| \begin{array}{l} v\in H^1_0(\Omega),\ \eta\in D(T), \\ {{\rm div}}[(1-k_0a(x))\nabla u] + \displaystyle\int_0^\infty g(s) {{\rm div}}[a(x)\nabla\eta(s)] ds \in L^2(\Omega) \end{array} \right. \right\}, \notag\end{aligned}$$ to which we observe that there holds $D(\mathcal{L})\subset\mathcal{H}^1.$ On these spaces we defined the associated operators $$\begin{aligned}
T\eta:=-\eta_s, \quad \text{for}\ \eta\in D(T), \notag\end{aligned}$$ and $$\begin{aligned}
\mathcal{L}U:= \begin{pmatrix} v \\ {{\rm div}}[(1-k_0a(x))\nabla u] + \displaystyle\int_0^\infty g(s){{\rm div}}[a(x)\nabla\eta(s)] ds - b(x)v \\ v+T\eta \end{pmatrix}, \quad \text{for}\ U\in D(\mathcal{L}). \notag\end{aligned}$$ For each $t\in[0,T]$, the equation $$\eta^t_t = T\eta^t + v(t) \label{memory-2}$$ holds as an ODE in $\mathcal{M}^0$ subject to the initial condition $$\eta^0=\eta_0\in\mathcal{M}^0. \label{memory-3}$$ Concerning the IVP (\[memory-2\])-(\[memory-3\]), we have the following proposition (cf. [@Pata-Zucchi-2001]).
\[t:gen-T\] The operator $T$ with domain $D(T)$ the generator of the right-translation semigroup. Moreover, $\eta^t$ can be explicitly represented by $$\eta^t(s) = \left\{ \begin{array}{ll} u(t)-u(t-s) & \text{if}\ 0\le s\le t \\
\eta_0(s-t) + u(t) - u(0) & \text{if}\ s>t. \end{array} \right. \label{eta}$$
Next we define the nonlinear functional by $$\begin{aligned}
\mathcal{F}(U):=(0,-f(u)+h,0). \notag\end{aligned}$$ Problem P can now be written as the abstract Cauchy problem on $\mathcal{H}^0$, $$\label{acp}
\left\{ \begin{array}{ll} \displaystyle\frac{d}{dt}U = \mathcal{L}U+\mathcal{F}(U), & t>0, \\ U(0)=U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0. \end{array} \right.$$ Later, when we are concerned with the regularity properties of problem P, we will also be interested in a more regular subspace of $\mathcal{H}^0$ (this is discussed further below).
Concerning the spaces $\mathcal{V}^1_a$ and $\mathcal{V}^2_a$ from above, it is important to note that although the injection $\mathcal{V}^1_a \hookleftarrow \mathcal{V}^2_a$ is compact, it does not follow that the injection $\mathcal{M}^0 \hookleftarrow \mathcal{M}^1$ is. Indeed, see [@Pata-Zucchi-2001] for a counterexample. Moreover, this means the embedding $\mathcal{H}^1\hookrightarrow\mathcal{H}^0$ is not compact. Such compactness between the “natural phase spaces” is essential to obtaining further regularity for the global attractors and even for the construction of finite dimensional exponential attractors. To alleviate this issue we follow [@GRP01; @Pata-Zucchi-2001] (also see [@CPS06; @GMPZ10]) and define the so-called [*[tail function]{}*]{} of $\eta\in\mathcal{M}^{0}$ by, for all $\tau\ge0,$ $$\mathbb{T}(\tau;\eta) := \int\limits_{(0,1/\tau)\cup(\tau,\infty)} g(s) \|\nabla\eta(s)\|^2 ds.$$ With this we set, $$\mathcal{T}^1 := \left\{ \eta\in\mathcal{M}^1 : \eta_s\in\mathcal{M}^{0},\ \eta(0)=0,\ \sup_{\tau\ge1} \tau\mathbb{T}(\tau;\eta)<\infty \right\}.$$ The space $\mathcal{T}^1$ is Banach with the norm whose square is defined by $$\|\eta\|^2_{\mathcal{T}^1} := \|\eta\|^2_{\mathcal{M}^1} + \|\eta_s\|^2_{\mathcal{M}^{0}} + \sup_{\tau\ge1} \tau\mathbb{T}(\tau;\eta). \label{new-norm}$$ Importantly, the embedding $\mathcal{T}^1\hookrightarrow\mathcal{M}^0$ is compact. (We should mention that although the works [@CPS06; @GMPZ10] treat PDE with an [*[integrated past history]{}*]{} variable, the compactness issue still applies to models with a relative displacement history variable, such as here. In fact, the compactness issue is more delicate in this setting; one must introduce so-called “tail functions,” cf. [@CPS06 Lemma 3.1] or [@GMPZ10 Proposition 5.4]). Hence, let us now also define the space $$\begin{aligned}
\mathcal{K}^1 := H^{2}(\Omega)\times H^{1}(\Omega)\times \mathcal{T}^1, \label{reg-compact}\end{aligned}$$ and the desired [*[compact]{}*]{} embedding $\mathcal{K}^1\hookrightarrow\mathcal{H}^0$ holds. Again, each space is equipped with the corresponding graph norm whose square is defined by, for all $U=(u,v,\eta)\in\mathcal{K}^1$, $$\|U\|^2_{\mathcal{K}^1} := \|u\|^2_{H^2(\Omega)} + \|v\|^2_{H^1(\Omega)} + \|\eta\|^2_{\mathcal{T}^1}. \label{reg-norm}$$
Concerning the IVP (\[memory-2\])-(\[memory-3\]), we will also call upon the following (cf. [@CPS06 Lemmas 3.6]).
\[what-2\] Let $\eta_0\in D(T)$. Assume there is $\rho>0$ such that, for all $t\ge0$, $\|\nabla u(t)\|\le\rho$. Then there is a constant $C>0$ such that, for all $t\ge0$, $$\begin{aligned}
\sup_{\tau\ge1} \tau\mathbb{T}(\tau;\eta^t) \le 2 \left( t+2 \right)e^{-\delta t} \sup_{\tau\ge1} \tau\mathbb{T}(\tau;\eta_0) + C\rho^2. \notag\end{aligned}$$
We now report some results from [@CFM-16] who only need to assume (H1)-(H4) hold. The following result is from [@CFM-16 Theorem 2.1]. The proof follows by relying on classical semigroup theory; namely, the operator $\mathcal{L}$ is the infinitesimal generator of a $C^0$-semigroup of contractions $e^{\mathcal{L}t}$ in $\mathcal{H}^0$ (cf. [@CFM-16 Lemma 3.1]) and the local Lipschitz continuity of $\mathcal{F}:\mathcal{H}^0\rightarrow\mathcal{H}^0$.
\[t:well-posedness\] Given $h\in L^2(\Omega)$ and $U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0$, problem P possesses a unique global mild solution satisfying the regularity $$\label{mild-reg}
u\in C([0,\infty);H^1_0(\Omega)), \quad u_t\in C([0,\infty);L^2(\Omega)) \quad \text{and} \quad \eta^t\in C([0,\infty);\mathcal{M}^0).$$ If $U_0=(u_0,u_1,\eta_0)\in D(\mathcal{L})$, the solution is regular and satisfies $$\label{str-reg}
U\in C([0,\infty);D(\mathcal{L})).$$ In addition, if $Z^i(t)=(u^i(t),u^i_t(t),\eta^{i,t})$, $i=1,2$, are any two mild solutions to problem P corresponding to the initial data $Z^1_0,Z^2_0\in\mathcal{H}^0$, respectively, where $\|Z^1_0\|_{\mathcal{H}^0}\le R$ and $\|Z^2_0\|_{\mathcal{H}^0}\le R$ for some $R>0$, then for any $T>0$ and for all $t\in[0,T],$ $$\label{cont-dep}
\|Z^1(t)-Z^2(t)\|_{\mathcal{H}^0} \le e^{Q(R)T}\|Z^1(0)-Z^2(0)\|_{\mathcal{H}^0}$$ for some positive nondecreasing function $Q(\cdot)$.
The next result depends on [@CFM-16 Lemma 3.3]. For this we define the “energy functional” which is used to extend local solutions to global ones, as well as demonstrate the gradient structure of problem P. $$\label{energy}
E(t) := \|u_t(t)\|^2 + \int_\Omega (1-k_0a(x))|\nabla u(t)|^2 dx + \|\eta^t\|^2_{\mathcal{M}^0} + 2\int_\Omega \left( F(u(t))-h(x)u(t) \right)dx.$$
The energy $E(t)$ is non-increasing along any solution $U(t)=(u(t),u_t(t),\eta^t)$. In addition, there exists $\delta_0,C_{fh}>0$, independent of $U$, such that for all $t\ge0,$ $$\label{lower-e}
E(t) \le \delta_0\|(u(t),u_t(t),\eta^t)\|^2_{\mathcal{H}^0} - C_{fh}.$$
The following is [@CFM-16 Theorem 2.2].
\[t:global-attr\] Let $h\in L^2(\Omega)$ and $U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0$. The dynamical system $(\mathcal{H}^0,S(t))$ generated by the mild solutions of Problem P is gradient and possesses a global attractor $\mathcal{A}$ which has finite (fractal) dimension and coincides with the unstable manifold $\mathbb{M}^n(\mathcal{N})$ of stationary solutions of problem P.
The final two results here will be useful in the next section. Each result follows from the existence of a (bounded) attractor in $\mathcal{H}^0$. The first result provides a uniform bound on the mild solutions of problem P and some extremely important dissipation integrals, and the second provides the existence of an absorbing set in a natural way.
\[t:unif-bnd\] For each $R>0$ and every $U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0$ such that $\|U_0\|_{\mathcal{H}^0}\le R$, there holds, for all $t\ge0,$ $$\label{unif-bnd}
\|S(t)U_0\|_{\mathcal{H}^0}\le Q(R)$$ for some positive nondecreasing function $Q(\cdot).$ In addition, there exists a function $Q(\cdot)$ such that $$\label{unif-bnd-2}
\int_0^\infty \left( \|\sqrt{b(x)}u_t(\tau)\|^2 + \delta\|\eta^\tau\|^2_{\mathcal{M}^0} \right) d\tau \le Q(R).$$ Consequently, there also holds $$\label{unif-bnd-3}
\int_0^\infty \|u_t(\tau)\|^2 d\tau \le Q(R).$$
The first result is a consequence of the existence of a global/universal attractor.
To show , let $R>0$ be given and $U_0\in\mathcal{H}^0$ be such that $\|U_0\|_{\mathcal{H}^0}\le R.$ Next we formally derive the “energy identity” associated with problem P by multiplying by $2u_t$ to then integrate over $\Omega$; this yields (cf. [@CFM-16 Equation (3.7)]), $$\begin{aligned}
\frac{d}{dt} E + 2\int_0^\infty g(s) \int_\Omega a(x)\nabla\eta^t(s)\cdot \nabla u_t dx ds + 2\|\sqrt{b(x)}u_t\|^2 = 0. \label{ued-1}\end{aligned}$$ where $E$ is the energy functional . Observe, thanks to , and , we readily find $C(R)>0$ such that, for all $t\ge0,$ $$\label{ued-4}
|E(t)| \le C(R).$$ Next we note that with $_2$ there holds, $$\begin{aligned}
2\int_0^\infty g(s) \int_\Omega a(x)\nabla\eta^t(s)\cdot \nabla u_t dx ds & = \frac{d}{dt}\|\eta^t\|^2_{\mathcal{M}^0} + \int_0^\infty g(s) \frac{d}{ds}\|\eta^t\|^2_{\mathcal{V}^1_a} ds, \notag\end{aligned}$$ and applying yields, $$\begin{aligned}
\int_0^\infty g(s)\frac{d}{ds}\|\eta^t(s)\|^2_{\mathcal{V}^1_a} ds & = - \int_0^\infty g'(s)\|\eta^t(s)\|^2_{\mathcal{V}^1_a} ds \notag \\
& \ge \delta\int_0^\infty g(s)\|\eta^t(s)\|^2_{\mathcal{V}^1_a} ds. \label{ued-5}\end{aligned}$$ Hence, we have $$\begin{aligned}
\frac{d}{dt} E + \delta\|\eta^t\|^2_{\mathcal{M}^0} + 2\|\sqrt{b(x)}u_t\|^2 \le 0. \label{ued-7}\end{aligned}$$ Thus, integrating over $(0,t)$ produces .
Now we show easily follows from . Indeed, using the Mean Value Theorem for Definite Integrals, for each $\tau\ge0$, there is $\xi_\tau\in\Omega$ so that $$\|\sqrt{b(x)}u_t\|^2 = \int_\Omega b(x)|u_t(\tau)|^2 dx = b(\xi_\tau)\|u_t(\tau)\|^2.$$ Now consider $$\int_0^\infty b(\xi_\tau)\|u_t(\tau)\|^2 d\tau = \int_0^\infty \|\sqrt{b(x)}u_t(\tau)\|^2 d\tau,$$ and $b(x)\not\equiv0$ on $\Omega$, then $b(\xi_\tau)>0$ for each $\tau\ge0$. Define $b_*:=\inf_{\tau\ge0}b(\xi_\tau)>0$. So with we find $$\int_0^\infty \|u_t(\tau)\|^2 d\tau \le \frac{1}{b_*}Q(R).$$ The thesis follows with hypotheses (H5). The proof is complete.
\[t:abs-set\] The semigroup of solution operators $S(t)$ admits a bounded absorbing set $\mathcal{B}$ in $\mathcal{H}^0$; that is, for any subset $B\subset \mathcal{H}^0$, there exists $t_B\ge0$ (depending on $B$) such that for all $t\ge t_B$, $S(t)B\subset\mathcal{B}$.
The proof follows directly from the fact that the attractor $\mathcal{A}$ is bounded in $\mathcal{H}^0$; e.g., a ball in $\mathcal{H}^0$ of radius $\|\mathcal{A}\|_{\mathcal{H}^0}+1$ is an absorbing set in $\mathcal{H}^0$.
\[r:slow\] Unfortunately we do not know the rate of convergence of any bounded subset in $\mathcal{H}^0$ to the global attractor $\mathcal{A}.$ Moreover, there are several applications in the literature (not containing equations with degeneracies in crucial diffusion or damping terms) in which the rate of convergence of any bonded subset $B$ of $\mathcal{H}^0$ [*[is]{}*]{} exponential in the sense that there is a constant $\varpi>0$ such that for any nonempty bounded subset $B\subset\mathcal{H}^0$ and for all $t\ge0$ there holds, $${\mathrm{dist}}_{\mathcal{H}^0}(S(t)B,\mathcal{B}) \le Q(R)e^{-\varpi t}.$$ Here, given two subsets $U$ and $V$ of a Banach space $X$, the [*[Hausdorff semidistance]{}*]{} between them is $${\mathrm{dist}}_{X}(U,V):=\sup_{u\in U}\inf_{v\in V}\|u-v\|_X.$$
Regularity {#s:reg}
==========
The aim of this section, and indeed the aim of this article, is to show the existence of a smooth compact subset of $\mathcal{H}^0$ containing the global attractor $\mathcal{A}.$ This is achieved by finding a suitable subset $\mathcal{C}$ of $\mathcal{K}^1\hookrightarrow\mathcal{H}^0$; hence, $\mathcal{C}$ is compact in $\mathcal{H}^0.$ To this end we decompose the semigroup of solution operators by showing it splits into uniformly decaying to zero and uniformly compact parts. With this we obtain asymptotic compactness for the associated semigroup of solution operators. The procedure requires some technical lemmas and a suitable Grönwall type inequality; the presentation follows [@Frigeri10; @Gal-Shomberg15]. The argument developed here will also be relied on to establish the existence of a compact attracting set. As a reminder to the reader, throughout this section (and the next) we assume the hypotheses (H1r), (H3r) and (H4r) hold in addition to (H1)-(H4).
The main result in this section is the following.
\[t:main-reg\] There exists a closed and bounded subset $\mathcal{C}\subset \mathcal{K}^1$ and a conatant $\omega>0$ such that for every nonempty bounded subset $B\subset \mathcal{H}^0$ and for all $t\ge0$, there holds $$\begin{aligned}
{\mathrm{dist}}_{\mathcal{H}^0}(S(t)B,\mathcal{C}) \le Q(\|B\|_{\mathcal{H}^0})e^{-\omega t}. \label{trans-1}\end{aligned}$$ Consequently, the global attractor $\mathcal{A}$ (cf. Theorem \[t:global-attr\]) is bounded in $\mathcal{K}^1$ and trajectories on $\mathcal{A}$ are regular solutions of the form $$\label{regular}
u\in C([0,\infty);H^2(\Omega)), \quad u_t\in C([0,\infty);H^1(\Omega)) \quad \text{and} \quad \eta^t\in C([0,\infty);\mathcal{T}^1).$$
The proof first requires several lemmas.
Set $$\label{psi}
\psi(s):=f(s)+\beta s \quad \text{with} \quad \beta \ge \vartheta \quad \text{so that} \quad \psi'(s) \ge 0$$ and set $\Psi(s):=\int_0^s\psi(\sigma)d\sigma.$ (We remind the reader of .) Let $U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0$. Decompose - into the functions $v$, $w$, $\xi$ and $\zeta$ where $v+w=u$ and $\xi+\zeta=\eta$ satisfy, respectively, problem V and problem W which are given by $$\left\{ \begin{array}{ll} v_{tt} - {{\rm div}}[(1-k_0a(x))\nabla v] - \displaystyle\int_0^\infty g(s){{\rm div}}[a(x)\nabla\xi^t(s)]ds + b(x)v_t + \psi(u) - \psi(w) = 0 & \text{in}\ \Omega\times\mathbb{R}^+ \\
\xi^t_t = -\xi^t_s + v_t & \text{in}\ \Omega\times\mathbb{R}^+ \\
v(x,t) = 0, \quad \xi^t(x,s) = 0 & \text{on}\ \Gamma\times\mathbb{R}^+ \\
v(x,0) = u_0(x), \quad v_t(x,0) = u_1(x), \quad \xi^t(x,0) = 0, \quad \xi^0(x,s) = \eta_0(x,s) & \text{at}\ \Omega\times\{0\}
\end{array}\right. \label{problem-v}$$ and $$\left\{ \begin{array}{ll} w_{tt} - {{\rm div}}[(1-k_0a(x))\nabla w] - \displaystyle\int_0^\infty g(s){{\rm div}}[a(x)\nabla\zeta^t(s)]ds + b(x)w_t + \psi(w) = h(x) + \beta u & \text{in}\ \Omega\times\mathbb{R}^+ \\
\zeta^t_t = -\zeta^t_s + w_t & \text{in}\ \Omega\times\mathbb{R}^+ \\
w(x,t) = 0, \quad \zeta^t(x,s) = 0 & \text{on}\ \Gamma\times\mathbb{R}^+ \\
w(x,0) = 0, \quad w_t(x,0) = 0, \quad \zeta^t(x,0) = 0, \quad \zeta^0(x,s) = 0 & \text{at}\ \Omega\times\{0\}.
\end{array}\right. \label{problem-w}$$ We now define the operators $K(t)U_0:=(w(t),w_t(t),\zeta^t)$ and $Z(t)U_0:=(v(t),v_t(t),\xi^t)$ using the associated global mild solutions to problem V and problem W (the existence of such solutions follows in a similar manor to the semigroup methods used to establish the well-posedness for problem P; cf. Theorem \[t:well-posedness\] and the regularity described in ).
The first of the subsequent lemmas shows that the operators $K(t)$ are bounded bounded on $\mathcal{H}^0.$
The following lemma provides an estimate that will be extremely important later in this section.
\[t:ride\] For each $U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0$ there exists a unique global weak solution $$W:=(w,w_t,\zeta^t)\in C([0,\infty);\mathcal{H}^0) \label{wer-0}$$ to problem W. Moreover, for each $R>0$ and for all $U_0\in\mathcal{H}^0$ with $\|U_0\|_{\mathcal{H}^0}\le R$, there holds, for all $t\ge0,$ $$\label{wer-1}
\|K(t)U_0\|_{\mathcal{H}^0}\le Q(R)$$ for some nonnegative increasing function $Q(\cdot).$ There also holds $$\label{wer-2}
\int_0^\infty \|w_t(\tau)\|^2 d\tau \le Q(R).$$ In addition, for every ${\varepsilon}>0$ there exists a function $Q_{\varepsilon}(\cdot)\sim{\varepsilon}^{-1}$ such that for every $0\le s\le t$, $R>0$ and $U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0$ with $\|U_0\|_{\mathcal{H}^0}\le R,$ there holds $$\begin{aligned}
\int_s^t & \left( \|u_t(\tau)\|^2 + \|\sqrt{b(x)}u_t(\tau)\|^2 + \delta\|\eta^\tau\|^2_{\mathcal{M}^0} + \|w_t(\tau)\|^2 + \|\sqrt{b(x)}w_t(\tau)\|^2 + \delta\|\zeta^\tau\|^2_{\mathcal{M}^0} \right) d\tau \notag \\
& \le \frac{{\varepsilon}}{2}(t-s) + Q_{\varepsilon}(R). \label{wed-0}\end{aligned}$$ Finally, there holds $$\begin{aligned}
\int_t^{t+1} & \left( \|u_t(\tau)\|^2 + \delta\|\eta^\tau\|^2_{\mathcal{M}^0} + \|\sqrt{b(x)}w_t(\tau)\|^2 + \|w_t(\tau)\|^2 + \delta\|\zeta^\tau\|^2_{\mathcal{M}^0} \right) d\tau \le Q(R). \label{star}\end{aligned}$$
As we have already stated above, the existence of global mild solutions satisfying follows by arguing as in the proof of Theorem \[t:well-posedness\]. The bound essentially follows from the existence of a global attractor for problem P (cf. Corollary \[t:unif-bnd\]). The dissipation property follows by arguing exactly as in the proof of Corollary \[t:unif-bnd\] keeping in mind both $u^{(1)}$ and $u^{(b)}$ make sense, and that we are able to utilize the bound for either one.
We are now interested in establishing . Indeed, multiplying $_1$ by $2w_t$ and integrating over $\Omega$, applying $_2$ and applying an estimate like , all with $w$ and $\zeta$ in place of $u$ and $\eta$, respectively, and $E_w$ denoting the corresponding functional $E$, produces (in place of ) $$\begin{aligned}
\frac{d}{dt} & E_w + \delta\|\zeta\|^2_{\mathcal{M}^0} + 2\|\sqrt{b(x)}w_t\|^2 \le 2\beta(u,w_t). \label{wed-1}\end{aligned}$$ Since $$\begin{aligned}
2\beta(u,w_t) = 2\beta(u_t,w) + 2\beta\frac{d}{dt}(u,w) \notag\end{aligned}$$ and by $$\begin{aligned}
2\beta(u_t,w) & \le \beta^2C(R)\|u_t\| \notag \\
& \le {\varepsilon}+ C_{\varepsilon}\|u_t\|^2, \notag\end{aligned}$$ so the differential inequality becomes $$\begin{aligned}
\frac{d}{dt} & \{ E_w - 2\beta(u,w) \} + \delta\|\zeta^\tau\|^2_{\mathcal{M}^0} + 2\|\sqrt{b(x)}w_t\|^2 \le {\varepsilon}+ C_{\varepsilon}\|u_t\|^2. \label{wed-3}\end{aligned}$$ In light of and , adding $\|u_t\|^2+\|\sqrt{b(x)}u_t(\tau)\|^2 + \delta\|\eta^\tau\|^2_{\mathcal{M}^0}$ to both sides of and integrating the result over $(s,t)$ then applying , and for problem W produces the desired estimate .
To show , we now add in the bound $\|u_t\|^2+\delta\|\eta\|^2_{\mathcal{M}^0}+2\|w_t\|^2\le C(R)$ into , and this time estimate the right-hand side with $C(R)+\|w_t\|^2$ to obtain $$\begin{aligned}
\frac{d}{dt} & E_w + \|u_t(\tau)\|^2 + \delta\|\eta^\tau\|^2_{\mathcal{M}^0} + \|\sqrt{b(x)}w_t(\tau)\|^2 + \|w_t(\tau)\|^2 + \delta\|\zeta^\tau\|^2_{\mathcal{M}^0} \le C(R). \label{wed-99}\end{aligned}$$ Integrating over $(t,t+1)$ and applying for problem W yields .
\[t:uniform-decay\] For each $U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0$ there exists a unique global weak solution $$V:=(v,v_t,\xi^t)\in C([0,\infty);\mathcal{H}^0) \label{opp-0}$$ to problem V. Moreover, for each $R>0$ and for all $U_0\in\mathcal{H}^0$ with $\|U_0\|_{\mathcal{H}^0}\le R$, there exists $\omega_1>0$ such that, for all $t\geq 0$, $$\|Z(t)U_{0}\|_{\mathcal{H}^0}\le Q(R)e^{-\omega_1 t} \label{opp-1}$$ for some positive nondecreasing function $Q(\cdot).$ Thus, the operators $Z(t)$ are uniformly decaying to zero in $\mathcal{H}^0$.
As we have already stated above, the existence of global mild solutions satisfying follows by arguing as in the proof of Theorem \[t:well-posedness\]. It suffices to show .
Let $R>0$ and $U_0=(u_0,u_1,\eta_0)\in\mathcal{H}^0$ be such that $\|U_0\|_{\mathcal{H}^0}\le R.$ Next we rewrite the term $b(x)v_t$ in equation $_1$ as $(b(x)+1)v_t- v_t$. Then multiply the result in $L^2(\Omega)$ by $v_t+{\varepsilon}v$, where ${\varepsilon}>0$ will be chosen below. When we include the basic identity $$\begin{aligned}
(\psi(u)-\psi(w),v_t) & = \frac{d}{dt}\left\{(\psi(u)-\psi(w),v)-\frac{1}{2}(\psi'(u)v,v) \right\} \notag \\
& - ((\psi'(u)-\psi'(w))w_t,v) + \frac{1}{2}(\psi''(u)u_t,v^2) \notag\end{aligned}$$ to the result and use $_2$, we find that there holds, for almost all $t\ge0,$ $$\begin{aligned}
\frac{d}{dt} & \left\{ \|v_t\|^2 + 2{\varepsilon}(v_t,v) + \int_\Omega (1-k_0a(x))|\nabla v|^2 dx + \|\xi^t\|^2_{\mathcal{M}^0} + {\varepsilon}\|\sqrt{b(x)}v\|^2 \right. \notag \\
& \left. + 2(\psi(u)-\psi(w),v) - (\psi'(u)v,v) \right\} \notag \\
& - 2{\varepsilon}\|v_t\|^2 + 2{\varepsilon}\int_\Omega (1-k_0a(x)) |\nabla v|^2 dx - \int_0^\infty g'(s)\|\xi^t(s)\|^2_{\mathcal{V}^1_a} ds \notag \\
& + 2{\varepsilon}\int_0^\infty g(s) \int_\Omega a(x) \nabla\xi^t(s)\cdot\nabla v dx ds + 2\|\sqrt{b(x)}v_t\|^2 \notag \\
& - 2(\psi'(u)-\psi'(w))w_t,v) + (\psi''(u)u_t,v^2) + 2{\varepsilon}(\psi(u)-\psi(w),v) \notag \\
& = 0. \label{opp-2} \end{aligned}$$
We now consider the functional defined by $$\begin{aligned}
\mathbb{V}(t) & := \|v_t(t)\|^2 + 2{\varepsilon}(v_t(t),v(t)) + \int_\Omega (1-k_0a(x))|\nabla v(t)|^2 dx + \|\xi^t\|^2_{\mathcal{M}^0} + {\varepsilon}\|\sqrt{b(x)}v(t)\|^2 \notag \\
& + 2(\psi(u(t))-\psi(w(t)),v(t)) - (\psi'(u(t))v(t),v(t)) \label{opp-3}\end{aligned}$$ We now will show that, given $U(t)=(u(t),u_{t}(t),\eta^t), W(t)=(w(t),w_{t}(t),\zeta^t)\in \mathcal{H}^0$ are uniformly bounded with respect to $t\ge0$ by some $R>0$, there are constants $C_1,C_2>0$, independent of $t$, in which for all $V(t)=(v(t),v_t(t),\xi^t)\in \mathcal{H}^0$, $$C_1\|V(t)\|_{\mathcal{H}^0}^{2} \le \mathbb{V}(t) \le C_2\|V(t)\|_{\mathcal{H}^0}^{2}. \label{opp-4}$$ To this end we begin by estimating the following product with , $$\begin{aligned}
2{\varepsilon}|(v_t,v)| & \le {\varepsilon}\|v_t\|^2 + {\varepsilon}\|v\|^2 \notag \\
& \le {\varepsilon}\|v_t\|^2 + \frac{{\varepsilon}}{\lambda_1}\|\nabla v\|^2, \label{opp-4.5}\end{aligned}$$ and $$\begin{aligned}
{\varepsilon}\|\sqrt{b(x)}v\|^2 & \le {\varepsilon}\|\sqrt{b}\|^2_\infty\|v\|^2 \notag \\
& \le \frac{{\varepsilon}}{\lambda_1}\|b\|_\infty\|\nabla v\|^2. \label{opp-4.6}\end{aligned}$$ Concerning the terms in the functional $\mathbb{V}$ that involve the nonlinear term $\psi$, using , , and the embedding $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$, and also , there holds $$\begin{aligned}
|(\psi ^{\prime }(u)v,v) | & \leq C\left( 1+\|\nabla u\|^{2}\right) \|\nabla v\|\|v\| \notag \\
& \le {\varepsilon}\|\nabla v\|^{2} + C_{\varepsilon}(R)\|v\|^{2}, \label{opp-5} \end{aligned}$$ where the constant $0<C_{\varepsilon}\sim{\varepsilon}^{-1}.$ From assumption and $$2(\psi(u)-\psi(w),v) \geq 2(\beta -\vartheta )\|v\|^{2}. \label{opp-6}$$Hence, for $\beta=\beta({\varepsilon})$ sufficiently large, the combination of (\[opp-5\]) and (\[opp-6\]) produces, $$\begin{aligned}
2(\psi(u)-\psi(w),v) - (\psi'(u)v,v) & \ge 2(\beta-\vartheta)\|v\|^2 - {\varepsilon}\|\nabla v\|^2 - C_{\varepsilon}(R)\|v\|^2 \notag \\
& \geq - {\varepsilon}\|\nabla v\|^2. \label{opp-7}\end{aligned}$$ With , and we attain the lower bound for the functional $\mathbb{V}$, $$\mathbb{V} \ge \left( \ell_0 - \frac{{\varepsilon}}{\lambda_1}(2+\|b\|_\infty) - {\varepsilon}\right) \|\nabla v\|^2 + \left( 1-{\varepsilon}\right)\|v_t\|^2 + \|\xi^t\|^2_{\mathcal{M}^0}. \notag$$ So for a sufficiently small ${\varepsilon}>0$ fixed (which also fixes the choice of $\beta$), there is $m_0>0$ in which, for all $t\geq 0$, we have that $$\mathbb{V}(t) \ge m_0\|(v(t),v_{t}(t),\xi^t)\|_{\mathcal{H}^0}^{2}. \label{opp-8}$$ Now by the (local) Lipschitz continuity of $f$, the embedding $H^1_0(\Omega)\hookrightarrow L^2(\Omega)$, the uniform bounds on $u$ and $w$, and the Poincaré inequality , it is easy to check that with there holds $$\begin{aligned}
2(\psi(u)-\psi(w),v) & \le 2\|\psi(u)-\psi(w)\| \|v\| \notag \\
& \le C(R)\|\nabla v\|^{2}. \label{opp-8.5}\end{aligned}$$ Also, using , (\[assm-f-1\]), (\[assm-f-2\]) and the bound (\[unif-bnd\]), there also holds $$|(\psi'(u)v,v)| \leq C(R)\|\nabla v\|^{2}. \label{opp-9}$$ Thus, with , and referring to some of the above estimates, the right-hand side of (\[opp-4\]) also follows.
Moving forward, we now work on . In light of the estimates $$\begin{aligned}
2|((\psi'(u)-\psi'(w))w_{t},v)| & \le C( 1 + \|\nabla u\| + \|\nabla w\|) \|w_{t}\| \|v\|^2 \notag \\
& \le \frac{1}{2\beta}\|v\|^{2} + C(R)\|w_{t}\|^{2}\mathbb{V}, \label{opp-11}\end{aligned}$$ and $$\begin{aligned}
|(\psi''(u)u_{t},v^2)| & \le C ( 1 + \|\nabla u\| ) \|u_{t}\| \|v\|^{2} \notag \\
& \le \frac{1}{2\beta}\|v\|^{2} + C(R)\|u_{t}\|^{2} \mathbb{V}, \label{opp-12}\end{aligned}$$ (here the constants $C(R)>0$ also depend on $\beta>0$) we see that with , , as well as , and , the differential identity (\[opp-2\]) becomes $$\begin{aligned}
\frac{d}{dt} & \mathbb{V} + {\varepsilon}\|v_t\|^2 + 2{\varepsilon}\ell_0 \|\nabla v\|^2 + \delta \|\xi^t\|^2_{\mathcal{M}^0} \notag \\
& + 2{\varepsilon}\int_0^\infty g(s) \int_\Omega a(x) \nabla \xi^t(s)\cdot\nabla v dx ds + 2\|\sqrt{b(x)}v_t\|^2 + \left( 2{\varepsilon}(\beta-\vartheta) - \frac{1}{\beta} \right)\|v\|^2 \notag \\
& \le C(R) \left( \|u_t\|^2 + \|w_t\|^2 \right) \mathbb{V} + 3{\varepsilon}\mathbb{V}, \label{opp-14} \end{aligned}$$ where we also added $3{\varepsilon}\|v_t\|^2$ to both sides (observe, $3{\varepsilon}\|v_t\|^2\le 3{\varepsilon}\mathbb{V}$). We now seek a suitable control on the product $$\begin{aligned}
\left| 2{\varepsilon}\int_0^\infty g(s) \int_\Omega a(x) \nabla \xi^t(s)\cdot\nabla v dx ds \right| & \le 2{\varepsilon}\int_0^\infty g(s) \left| \int_\Omega a(x) \nabla \xi^t(s)\cdot\nabla v dx \right| ds \notag \\
& = 2{\varepsilon}\int_0^\infty g(s) \left| (\xi^t(s),v)_{\mathcal{V}^1_a} \right| ds \notag \\
& \le 2{\varepsilon}\|\xi^t\|_{\mathcal{M}^0} \|\nabla v\| \notag \\
& \le 2\sqrt{{\varepsilon}}\|\xi^t\|^2_{\mathcal{M}^0} + \frac{{\varepsilon}\sqrt{{\varepsilon}}}{2}\|\nabla v\|^2. \label{opp-14.3}\end{aligned}$$ For sufficiently large $\beta>0$, we may omit the positive terms $2\|\sqrt{b(x)}v_t\|^2+(2{\varepsilon}(\beta-\vartheta)-\frac{1}{\beta})\|v\|^2$ from the left-hand side of so that it becomes, with , $$\begin{aligned}
\frac{d}{dt} & \mathbb{V} + {\varepsilon}\|v_t\|^2 + {\varepsilon}\left( 2\ell_0-\frac{\sqrt{{\varepsilon}}}{2} \right) \|\nabla v\|^2 + \left( \delta - 2\sqrt{{\varepsilon}} \right) \|\xi^t\|^2_{\mathcal{M}^0} \notag \\
& \le C(R) \left( \|u_t\|^2 + \|w_t\|^2 + 3{\varepsilon}\right) \mathbb{V}. \label{opp-14.5} \end{aligned}$$ For any ${\varepsilon}>0$ sufficiently small so that $$2\ell_0-\frac{\sqrt{{\varepsilon}}}{2} >0 \quad \text{and} \quad \delta - 2\sqrt{{\varepsilon}}>0,$$ we can find a constant $m_1>0$, thanks to , such that can be written as the following differential inequality, to hold for almost all $t\ge0,$ $$\begin{aligned}
\frac{d}{dt} & \mathbb{V} + {\varepsilon}m_1\mathbb{V} \le C(R) \left( \|u_t\|^2 + \|w_t\|^2 + 3{\varepsilon}\right) \mathbb{V}. \label{opp-15} \end{aligned}$$ Here we recall Proposition \[GL\] and Lemma \[t:ride\]. Applying these to (\[opp-15\]) yields, for all $t\ge0,$ $$\mathbb{V}(t) \le \mathbb{V}(0) e^{Q(R)}e^{-m_1 t/2}, \label{opp-16}$$ for some positive nondecreasing function $Q(\cdot).$ By virtue of (\[opp-4\]) and the initial conditions provided in , $$\begin{aligned}
\mathbb{V}(0) & \le C_{2}(R) \|(v(0),v_t(0),\xi^0)\|_{\mathcal{H}^0}^{2} \notag \\
& \le C_{2}(R) \left( \|\nabla u_0\|^2 + \|u_1\|^2 + \|\eta_0\|^2_{\mathcal{M}^0} \right) \notag \\
& \le Q(R). \notag\end{aligned}$$ Therefore (\[opp-16\]) shows that the operators $Z(t)$ are uniformly decaying to zero. The proof is finished. [**$<<<<<$**]{}
The remaining lemmas will show that the operators $K(t)$ are asymptotically compact on $\mathcal{H}_0$. In order to establish this, we prove that the operators $K(t)$ are uniformly bounded in $\mathcal{K}^1\hookrightarrow\mathcal{H}^0.$
Due to the nature of the proof of the following lemma, we also need to assign the past history for the term $w_t$. Indeed, from below we need to consider the initial condition $$\zeta^0_t(x,s) = -\zeta^0_s(x,s) = -w_t(x,0-s).$$ However, since $u=v+w$, we can write $$-u_t(x,0-s) = -v_t(x,0-s) - w_t(x,0-s)$$ and hence assume that $$\begin{aligned}
v_t(x,0-s) = u_t(x,0-s) = -\eta^0_t(x,s) \quad \text{and} \quad w_t(x,0-s) = 0. \label{initial}\end{aligned}$$
\[t:diff-bnd\] For each $R>0$ and for all $U_{0}=(u_{0},u_{1},\eta_0)\in \mathcal{H}^0$ such that $\|U_{0}\|_{\mathcal{H}^0} \le R$, there holds for all $t\ge0$ $$\begin{aligned}
\|\partial_tK(t)U_{0}\|^2_{\mathcal{H}^0} = \|\nabla w_t(t)\|^2 + \|w_{tt}(t)\|^2 + \|\zeta^t_t\|^2_{\mathcal{M}^0} \le Q(R) \label{lpp-12}\end{aligned}$$ for some positive nondecreasing function $Q(\cdot)$.
For all $x\in\Omega$ and $t,s\in\mathbb{R}^+$, set $H(x,t):=w_t(x,t)$ and $X^t:=\zeta^t_t(s).$ Differentiating problem W with respect to $t$ yields the system $$\label{problem-H}
\left\{ \begin{array}{ll} H_{tt} - {{\rm div}}[(1-k_0a(x))\nabla H] - \displaystyle\int_0^\infty g(s){{\rm div}}[a(x)\nabla X^t(s)]ds + b(x)H_t + \psi'(w)H = \beta u_t & \text{in}\ \Omega\times\mathbb{R}^+ \\
X^t_t = -X^t_s + H_t & \text{in}\ \Omega\times\mathbb{R}^+ \\
H(x,t) = w_t(x,t) = 0, \quad X^t(x,s) = \zeta^t_t(x,s) & \text{on}\ \Gamma\times\mathbb{R}^+ \\
H(x,0) = w_t(x,0) = 0, \quad H_t(x,0) = w_{tt}(x,0) = -f(0)-u_1 \quad \text{(from \eqref{problem-w})} & \text{at}\ \Omega\times\{0\} \\
X^t(x,0) = w_t(x,t) - w_t(x,t-0) = 0, \quad X^0(x,s) = 0 \quad \text{(see \eqref{initial})} & \text{at}\ \Omega\times\{0\}.
\end{array}\right.$$ Multiply equation $_1$ by $H_t+{\varepsilon}H$ for some ${\varepsilon}>0$ to be chosen below. To this result we apply the identities $$(\psi'(w)H,H_t) = \frac{1}{2}\frac{d}{dt}(\psi'(w)H,H) - \frac{1}{2}(\psi''(w)w_t,H^2),$$ and (here we rely on $_2$) $$\begin{aligned}
\int_0^\infty g(s) & \int_\Omega a(x)\nabla X^t(s)\nabla H_t(t) dxds \notag \\
& = \frac{1}{2}\frac{d}{dt}\|X^t\|^2_{\mathcal{M}^0} + \int_0^\infty g(s)\frac{d}{ds}\|X^t(s)\|^2_{\mathcal{V}^1_a}ds \notag \\
& = \frac{1}{2}\frac{d}{dt}\|X^t\|^2_{\mathcal{M}^0} - \int_0^\infty g'(s)\|X^t(s)\|^2_{\mathcal{V}^1_a}ds \notag\end{aligned}$$ so that together we find $$\begin{aligned}
& \frac{d}{dt} \left\{ \|H_t\|^2 + 2{\varepsilon}(H_t,H) + \int_\Omega (1-k_0a(x))|\nabla H|^2dx + \|X^t\|^2_{\mathcal{M}^0} + (\psi'(w)H,H) \right\} \notag \\
& - 2{\varepsilon}\|H_t\|^2 + 2{\varepsilon}\|\sqrt{b(x)}H_t\|^2 + 2{\varepsilon}(b(x)H_t,H) + 2{\varepsilon}\int_\Omega (1-k_0a(x))|\nabla H|^2dx + 2{\varepsilon}(\psi'(w)H,H) \notag \\
& - 2\int_0^\infty g'(s)\|X^t(s)\|^2_{\mathcal{V}^1_a}ds + 2{\varepsilon}\int_0^\infty g(s)\int_\Omega a(x)\nabla X^t(s)\cdot \nabla H(t) dx ds \notag \\
& = (\psi''(w)w_t,H^2) + 2\beta(u_t,H_t) + 2\beta{\varepsilon}(u_t,H). \label{lpp-1}\end{aligned}$$ Next we recall and find $$\begin{aligned}
-2\int_0^\infty g'(s)\|X^t(s)\|^2_{\mathcal{V}^1_a} ds & \ge 2\delta\|X^t\|^2_{\mathcal{M}^0}, \label{lpp-2}\end{aligned}$$ and $$\begin{aligned}
2{\varepsilon}\int_0^\infty g(s)\int_\Omega a(x)\nabla X^t(s)\cdot \nabla H(t) dx ds & \ge -\delta\|X^t\|^2_{\mathcal{M}^0} - \frac{{\varepsilon}^2}{\delta}\|\nabla H\|^2, \label{lpp-3}\end{aligned}$$ where the last inequality follows from . For all ${\varepsilon}>0$ and $t\ge0$, define the functional $$\begin{aligned}
\mathbb{I}(t):=\|H_t(t)\|^2 + 2{\varepsilon}(H_t(t),H(t)) + \int_\Omega (1-k_0a(x))|\nabla H(t)|^2dx + \|X^t\|^2_{\mathcal{M}^0} + (\psi'(w)H(t),H(t)). \label{lpp-4}\end{aligned}$$ Thanks to and since $\psi'>0$, there is a constant $C>0$, sufficiently small, so that $$\begin{aligned}
C\left( \|H_t(t)\|^2 + \ell_0\|\nabla H(t)\|^2 + \|X^t\|^2_{\mathcal{M}^0} \right) \le \mathbb{I}(t). \label{lpp-4.5}\end{aligned}$$ At this point we can write - with as $$\begin{aligned}
& \frac{d}{dt}\mathbb{I} - 2{\varepsilon}\|H_t\|^2 + 2{\varepsilon}\|\sqrt{b(x)}H_t\|^2 + 2{\varepsilon}(b(x)H_t,H) + \left( 2{\varepsilon}\ell_0 - \frac{{\varepsilon}^2}{\delta} \right)\|\nabla H\|^2 \notag \\
& + \delta\|X^t\|^2_{\mathcal{M}^0} + 2{\varepsilon}(\psi'(w)H,H) \notag \\
& \le 2(\psi''(w)w_t,H^2) + 2\beta(u_t,H_t) + 2\beta{\varepsilon}(u_t,H). \label{lpp-5}\end{aligned}$$ Next, let us rely on the uniform bounds and to estimate the products on the right-hand side $$\begin{aligned}
2|(\psi''(w)w_t,H^2)| & \le 2\|\psi''(w)w_t H^2\|_{1} \notag \\
& \le 2\|\psi''(w)w_t\|_{3/2}\|H\|^2_6 \notag \\
& \le 2\|\psi''(w)\|_{6}\|w_t\|\|H\|^2_6 \notag \\
& \le C(R)\|w_t\|\|\nabla H\|^2 \notag \\
& \le C(R)\|w_t\|\mathbb{I}, \label{lpp-7}\end{aligned}$$ $$\begin{aligned}
2\beta|(u_t,H_t)+{\varepsilon}(u_t,H)| & \le C(R)\|H_t\| + C(R)\|\nabla H\| \notag \\
& \le C_{\varepsilon}(R) + {\varepsilon}\|H_t\|^2 + {\varepsilon}^2\|\nabla H\|^2, \label{lpp-8}\end{aligned}$$ where $C_{\varepsilon}\sim{\varepsilon}^{-1}\wedge{\varepsilon}^{-2}$. Also, we know $$\begin{aligned}
2{\varepsilon}(\psi'(w)H,H) & \ge 2{\varepsilon}^2(\beta-\vartheta)\|H\|^2 >0. \label{lpp-8.5}\end{aligned}$$ Thus, combining - yields $$\begin{aligned}
\frac{d}{dt}\mathbb{I} & - 3{\varepsilon}\|H_t\|^2 + 2{\varepsilon}\|\sqrt{b(x)}H_t\|^2 + {\varepsilon}\left( 2\ell_0-{\varepsilon}\left(\frac{1}{\delta}+1\right)\right)\|\nabla H\|^2 + \delta\|X^t\|^2_{\mathcal{M}^0} \notag \\
& \le C(R)\|w_t\|\mathbb{I} + C_{\varepsilon}(R). \label{lpp-10}\end{aligned}$$ Since $4{\varepsilon}\|H_t\|^2\le4{\varepsilon}\mathbb{I}$, adding this to makes the differential inequality (we also omit $2{\varepsilon}\|\sqrt{b(x)}H_t\|^2$) $$\begin{aligned}
\frac{d}{dt}\mathbb{I} & + {\varepsilon}\|H_t\|^2 + {\varepsilon}\left( 2\ell_0-{\varepsilon}\left(\frac{1}{\delta}+1\right)\right)\|\nabla H\|^2 + \delta\|X^t\|^2_{\mathcal{M}^0} \notag \\
& \le C(R)\left( \|w_t\| + {\varepsilon}\right)\mathbb{I} + C_{\varepsilon}(R). \label{lpp-10.5}\end{aligned}$$ We now find that for any ${\varepsilon}>0$ small so that $$2\ell_0-{\varepsilon}\left(\frac{1}{\delta}+1\right) >0,$$ then $$\begin{aligned}
& \frac{d}{dt}\mathbb{I} + {\varepsilon}\mathbb{I} \le C(R)\left( \|w_t\| + {\varepsilon}\right)\mathbb{I} + C_{\varepsilon}(R) \label{lpp-11}\end{aligned}$$ to which we now apply Proposition \[GL2\] and the bounds and to conclude that, for all $t\ge0$, there holds $$\begin{aligned}
\mathbb{I}(t) \le C(R) \mathbb{I}(0)e^{-{\varepsilon}t/2} + C_{\varepsilon}(R). \label{lpp-11.5} \end{aligned}$$ Moreover, with and the initial conditions in we find that there is a constant $C>0$ (with ${\varepsilon}>0$ now fixed) in which $$\begin{aligned}
\|H_t(t)\|^2 + \|\nabla H(t)\|^2 + \|X^t\|^2_{\mathcal{M}^0} & \le C(R). \notag\end{aligned}$$ This establishes and completes the proof.
We derive the immediate consequence of and .
\[t:part-z\] Under the assumptions of Lemma \[t:diff-bnd\], there holds for all $t\ge0,$ $$\begin{aligned}
\|\zeta^t_s\|_{\mathcal{M}^0} \le Q(R). \label{part-z}\end{aligned}$$
Before we continue, we derive a further estimate for $\zeta^t$.
Under the assumptions of Lemma \[t:diff-bnd\], there holds for all $t\ge0$, $$\begin{aligned}
\|\nabla\zeta^t\|_{L^2_g(\mathbb{R}^+;L^2(\Omega))} \le C_\delta. \label{ccc-2}\end{aligned}$$
Formally multiplying $_2$ in $L^2_g(\mathbb{R}^+;L^2(\Omega))$ by $-\Delta \zeta^t(s)$ and estimating the result yields the differential inequality $$\begin{aligned}
\frac{d}{dt}\|\nabla\zeta^t\|^2_{L^2_g(\mathbb{R}^+;L^2(\Omega))} & = -\int_0^\infty g(s) \frac{d}{ds} \|\nabla\zeta^t(s)\|^2 ds + (\nabla w_t,\nabla\zeta^t)_{L^2_g(\mathbb{R}^+;L^2(\Omega))} \notag \\
& = \int_0^\infty g'(s) \|\nabla\zeta^t(s)\|^2 ds + (\nabla w_t,\nabla\zeta^t)_{L^2_g(\mathbb{R}^+;L^2(\Omega))} \notag \\
& \le -\delta\int_0^\infty g(s) \|\nabla\zeta^t(s)\|^2 ds + \frac{2}{\delta}\|\nabla w_t\|^2 + \frac{\delta}{2}\|\nabla\zeta^t\|^2_{L^2_g(\mathbb{R}^+;L^2(\Omega))} \notag \\
& = -\frac{\delta}{2}\|\nabla\zeta^t\|^2_{L^2_g(\mathbb{R}^+;L^2(\Omega))} + \frac{2}{\delta}\|\nabla w_t\|^2. \label{ccc-1}\end{aligned}$$ Hence, applying the bound to , we find the differential inequality which holds for almost all $t\ge0$ $$\begin{aligned}
\frac{d}{dt}\|\nabla\zeta^t\|^2_{L^2_g(\mathbb{R}^+;L^2(\Omega))} + \frac{\delta}{2}\|\nabla\zeta^t\|^2_{L^2_g(\mathbb{R}^+;L^2(\Omega))} \le C_\delta \notag\end{aligned}$$ where $0<C_\delta\sim\delta^{-1}.$ Applying a straight-forward Grönwall inequality and the initial conditions in produces the desired bound . This concludes the proof.
\[t:uniform-compactness\] Under the assumptions of Lemma \[t:diff-bnd\], the following holds for all $t>0$, $$\|K(t)U_{0}\|_{\mathcal{K}^1} \le Q(R), \label{compact}$$ for some positive nondecreasing function $Q(\cdot)$. Furthermore, the operators $K(t)$ are uniformly compact in $\mathcal{H}^0$.
The proof consists of several parts. In the first part, we derive further bounds for some higher order terms. We begin by rewriting/expanding as $$\begin{aligned}
& w_{tt} +k_0\nabla a(x)\cdot\nabla w +(1-k_0a(x))(-\Delta)w \notag \\
& - \displaystyle\int_0^\infty g(s)\nabla a(x)\cdot\nabla\zeta^t(s)ds + \displaystyle\int_0^\infty g(s)a(x)(-\Delta)\zeta^t(s)ds + b(x)w_t + \psi(w) = \beta u. \label{qww-0.1}\end{aligned}$$ Next, Using the relative displacement history definition of the memory space term $$\label{memory-w}
\zeta^t(s) := w(x,t) - w(x,t-s),$$ we rewrite the integral $$\begin{aligned}
\displaystyle\int_0^\infty g(s)a(x)(-\Delta)\zeta^t(s)ds = k_0 a(x)(-\Delta)w - \int_0^\infty g(s)a(x)(-\Delta)w(t-s)ds. \label{qww-0.2}\end{aligned}$$ Combining and shows takes the useful alternate form $$\begin{aligned}
& w_{tt} - \Delta w - \int_0^\infty g(s) a(x) (-\Delta) w(t-s) ds + b(x)w_t + \psi(w) \notag \\
& + k_0 \nabla a(x)\cdot\nabla w - \int_0^\infty g(s) \nabla a(x)\cdot \nabla\zeta^t(s) ds = \beta u. \label{qww-1}\end{aligned}$$We now report six identities that will be used below: $$\begin{aligned}
(w_{tt},(-\Delta)w) = \frac{d}{dt}(\nabla w_t,\nabla w) - \|\nabla w_t\|^2, \label{qww-1.05}\end{aligned}$$ $$\begin{aligned}
-\int_0^\infty & g(s)(a(x)(-\Delta)\underbrace{w(t-s)}_{=w(t)-\zeta^t(s)},(-\Delta)w_t(t)) ds \notag \\
& = -\int_0^\infty g(s)(a(x)(-\Delta)w(t),(-\Delta)w_t(t)) ds + \int_0^\infty g(s)(a(x)(-\Delta)\zeta^t(s),(-\Delta)\underbrace{w_t(t)}_{=\zeta^t_t(s)+\zeta^t_s(s)})ds \notag \\
& = -\frac{k_0}{2}\frac{d}{dt}\|w\|^2_{\mathcal{V}^2_a} + \frac{1}{2}\frac{d}{dt}\|\zeta^t\|^2_{\mathcal{M}^1} + \frac{1}{2}\int_0^\infty g(s)\frac{d}{ds}\|\zeta^t(s)\|^2_{\mathcal{V}^2_a} ds, \label{qww-1.1}\end{aligned}$$ $$\begin{aligned}
-\int_0^\infty & g(s)(a(x)(-\Delta)\underbrace{w(t-s)}_{=w(t)-\zeta^t(s)},(-\Delta)w(t)) ds = -k_0\|w\|^2_{\mathcal{V}^2_a} + \int_0^\infty g(s)(a(x)(-\Delta)\zeta^t(s),(-\Delta)w(t)) ds, \label{qww-1.2}\end{aligned}$$ $$\begin{aligned}
(b(x)w_t,(-\Delta)w_t) = \frac{d}{dt}(b(x)w_t,(-\Delta)w) - (b(x)w_{tt},(-\Delta)w), \label{qww-1.3}\end{aligned}$$ $$\begin{aligned}
k_0(\nabla a(x)\cdot \nabla w,(-\Delta)w_t) = \frac{d}{dt}k_0(\nabla a(x)\cdot \nabla w,(-\Delta)w) - k_0(\nabla a(x)\cdot \nabla w_{t},(-\Delta)w), \label{qww-1.4}\end{aligned}$$ and $$\begin{aligned}
-\int_0^\infty & g(s)(\nabla a(x)\cdot \nabla\zeta^t(s),(-\Delta)w_t(t)) ds \notag \\
& = -\frac{d}{dt} \int_0^\infty g(s)(\nabla a(x)\cdot \nabla\zeta^t(s),(-\Delta)w(t)) ds + \int_0^\infty g(s)(\nabla a(x)\cdot \nabla\zeta^t_t(s),(-\Delta)w(t)) ds. \label{qww-1.5}\end{aligned}$$ Next we multiply in $L^2(\Omega)$ by $(-\Delta)w_t+(-\Delta)w$ to obtain, in light of -, the differential identity $$\begin{aligned}
& \frac{d}{dt}\left\{ \|\nabla w_t\|^2 + 2(\nabla w_t,\nabla w) + \|\Delta w\|^2 - k_0\|w\|^2_{\mathcal{V}^2_a} + \|\zeta^t\|^2_{\mathcal{M}^1} \right. \notag \\
& \left. + 2(b(x)w_t,(-\Delta)w) + 2k_0(\nabla a(x)\cdot\nabla w,(-\Delta)w) - 2\int_0^\infty g(s)(\nabla a(x)\cdot\nabla\zeta^t(s),(-\Delta)w(t)) ds \right\} \notag \\
& - 2\|\nabla w_t\|^2 + 2\|\Delta w\|^2 + \int_0^\infty g(s) \frac{d}{ds}\|\zeta^t(s)\|^2_{\mathcal{V}^2_a} ds - 2k_0\|w\|^2_{\mathcal{V}^2_a} \notag \\
& + 2\int_0^\infty g(s)(a(x)(-\Delta)\zeta^t(s),(-\Delta)w(t)) ds - 2(b(x)w_{tt},(-\Delta)w) + 2(b(x)w_t,(-\Delta)w) \notag \\
& + 2(\psi'(w)\nabla w,\nabla w_t) + 2(\psi(w),(-\Delta)w) - 2k_0(\nabla a(x)\cdot\nabla w_t,(-\Delta)w) + 2k_0(\nabla a(x)\cdot\nabla w,(-\Delta)w) \notag \\
& + 2\int_0^\infty g(s)(\nabla a(x)\cdot\nabla\zeta^t_t(s),(-\Delta)w(t)) ds - 2\int_0^\infty g(s)(\nabla a(x)\cdot\nabla\zeta^t(s),(-\Delta)w(t)) ds \notag \\
& = 2\beta(\nabla u,\nabla w_t) + 2\beta(u,(-\Delta)w). \label{qww-2}\end{aligned}$$
We now seek a constant $m_2>0$ sufficiently small so that we can write the above differential identity in the following form $$\label{qww-3}
\frac{d}{dt}\Phi + c m_2\Phi \le Q(R)$$ where $$\begin{aligned}
& \Phi(t) := \|\nabla w_t(t)\|^2 + 2(\nabla w_t(t),\nabla w(t)) + \|\Delta w(t)\|^2 - k_0\|w(t)\|^2_{\mathcal{V}^2_a} + \|\zeta^t\|^2_{\mathcal{M}^1} \notag \\
& + 2(b(x)w_t(t),(-\Delta)w(t)) + 2k_0(\nabla a(x)\cdot\nabla w(t),(-\Delta)w(t)) - 2\int_0^\infty g(s)(\nabla a(x)\cdot\nabla\zeta^t(s),(-\Delta)w(t)) ds. \label{qww-3.2}\end{aligned}$$ The important lower bound holds $$\begin{aligned}
\Phi \ge C_1(\|\Delta w\|^2 + \|\nabla w_t\|^2 + \|\zeta^t\|^2_{\mathcal{M}^1}) - C_2(R) \label{qww-3.5}\end{aligned}$$ for some constants $C_1,C_2(R)>0,$ and essentially follows from some basic estimates, the bounds , , , , the Poincaré inequality and with the assumptions on the functions $a$ and $b$. Indeed, we estimate, for all ${\varepsilon}>0,$ $$\begin{aligned}
2|(\nabla w_t,\nabla w)| & \le {\varepsilon}\|\nabla w_t\|^2 + \frac{1}{{\varepsilon}}\|\nabla w\|^2 \notag \\
& \le {\varepsilon}\|\nabla w_t\|^2 + C_{\varepsilon}(R), \label{qww-3.51}\end{aligned}$$ $$\begin{aligned}
- k_0\|w\|^2_{\mathcal{V}^2_a} & = -k_0\int_\Omega a(x)|\Delta w|^2 dx \notag \\
& \ge -k_0 \|a\|_\infty\|\Delta w\|^2, \label{qww-3.52}\end{aligned}$$ $$\begin{aligned}
2|(b(x)w_t,(-\Delta)w)| & \le \frac{1}{{\varepsilon}}\|b(x)w_t\| + {\varepsilon}\|\Delta w\|^2 \notag \\
& \le C_{\varepsilon}(R) + {\varepsilon}\|\Delta w\|^2, \label{qww-3.53}\end{aligned}$$ $$\begin{aligned}
2k_0|(\nabla a(x)\cdot\nabla w,(-\Delta)w)| & \le \frac{k_0^2}{{\varepsilon}}\|\nabla a(x)\cdot\nabla w\|^2 + {\varepsilon}\|\Delta w\|^2 \notag \\
& \le C_{\varepsilon}(R) + {\varepsilon}\|\Delta w\|^2, \label{qww-3.54}\end{aligned}$$ and $$\begin{aligned}
2\int_0^\infty g(s)|(\nabla a(x)\cdot\nabla\zeta^t(s),(-\Delta)w(t))| ds & \le \int_0^\infty g(s)\left( \frac{1}{{\varepsilon}}\|\nabla a(x)\cdot\nabla\zeta^t(s)\|^2 + {\varepsilon}\|\Delta w(t)\|^2 \right) ds \notag \\
& \le \frac{1}{{\varepsilon}}\int_0^\infty g(s) \|\nabla a\|^2_\infty\|\nabla\zeta^t(s)\|^2 ds + {\varepsilon}\int_0^\infty g(s)\|\Delta w(t)\|^2 ds \notag \\
& \le \frac{1}{{\varepsilon}}\|\nabla a\|^2_\infty\|\nabla\zeta^t\|^2_{L^2_g(\mathbb{R}^+;L^2(\Omega))} + {\varepsilon}k_0\|\Delta w\|^2 \notag \\
& \le C_{\varepsilon}(R) + {\varepsilon}k_0\|\Delta w\|^2. \label{qww-3.55}\end{aligned}$$ Applying - to gives us the lower bound for all ${\varepsilon}>0$, $$\begin{aligned}
\Phi \ge (1-{\varepsilon})\|\nabla w_t\|^2 + (\ell_0-(2+k_0){\varepsilon}) \|\Delta w\|^2 + \|\zeta^t\|^2_{\mathcal{M}^1} - C_{\varepsilon}(R).\end{aligned}$$ For any fixed $0<{\varepsilon}<\min\{1,\ell_0/(2+k_0)\}$, we obtain .
Returning to the aim of , we first add $$3\|\nabla w_t\|^2 + 2(\nabla w_t,\nabla w)$$ to both sides of , and also insert $$\begin{aligned}
\int_0^\infty g(s) \frac{d}{ds}\|\zeta^t(s)\|^2_{\mathcal{V}^2_a} ds & = -\int_0^\infty g'(s) \|\zeta^t(s)\|^2_{\mathcal{V}^2_a} ds \notag \\
& \ge \delta\int_0^\infty g(s) \|\zeta^t(s)\|^2_{\mathcal{V}^2_a} ds \notag \\
& = \delta\|\zeta^t\|^2_{\mathcal{M}^1}. \notag\end{aligned}$$ Putting these together and using the second inequality in , becomes the differential inequality $$\begin{aligned}
& \frac{d}{dt}\Phi + \|\nabla w_t\|^2 + 2(\nabla w_t,\nabla w) + 2\|\Delta w\|^2 - 2k_0\|w\|^2_{\mathcal{V}^2_a} + \delta\|\zeta^t\|^2_{\mathcal{M}^1} \notag \\
& + 2(b(x)w_t,(-\Delta)w) + 2k_0(\nabla a(x)\cdot\nabla w,(-\Delta)w) - 2\int_0^\infty g(s)(\nabla a(x)\cdot\nabla\zeta^t(s),(-\Delta)w(t)) ds \notag \\
& \le 3\|\nabla w_t\|^2 + 2(\nabla w_t,\nabla w) + 2(b(x) w_{tt}, (-\Delta)w) + 2k_0(\nabla a(x)\cdot\nabla w_t,(-\Delta)w) \notag \\
& - 2\int_0^\infty g(s)(a(x)(-\Delta)\zeta^t(s),(-\Delta)w(t)) ds - 2\int_0^\infty g(s)(\nabla a(x)\cdot \nabla \zeta^t_t(s),(-\Delta)w(t)) ds \notag \\
& - 2(\psi'(w)\nabla w,\nabla w_t) - 2(\psi(w),(-\Delta)w) + 2\beta(\nabla u,\nabla w_t) + 2\beta(u,(-\Delta)w). \label{qww-4}\end{aligned}$$ (We should mention that the final bound of is now realized to control the $\nabla\zeta^t_t$ term appearing on the right-hand side.) Next we employ some basic inequalities, the assumptions on $a$ and $b$, the assumptions , the bounds , and , and finally even the continuous embedding $\mathcal{V}^2_a\hookrightarrow H^1_0(\Omega)$ of (H1r) to control the right-hand side of with the estimates $$\begin{aligned}
3\|\nabla w_t\|^2 + 2(\nabla w_t,\nabla w) - 2(\psi'(w)\nabla w,\nabla w_t) + 2\beta(\nabla u,\nabla w_t) \le C(R), \label{qww-5}\end{aligned}$$ $$\begin{aligned}
2(b(x) w_{tt},(-\Delta)w) & \le C(R) + \frac{1}{4}\|\Delta w\|^2, \label{qww-6}\end{aligned}$$ $$\begin{aligned}
2k_0(\nabla a(x)\cdot\nabla w_t,(-\Delta)w) & \le C(R) + \frac{1}{4}\|\Delta w\|^2, \label{qww-7}\end{aligned}$$ $$\begin{aligned}
-2\int_0^\infty g(s)(a(x)(-\Delta)\zeta^t(s),(-\Delta)w(t)) ds & = -2\int_0^\infty g(s)(\zeta^t(s),w(t))_{\mathcal{V}^2_a} ds \notag \\
& \le 2\int_0^\infty g(s)\|\zeta^t(s)\|_{\mathcal{V}^2_a}\|w(t)\|_{\mathcal{V}^2_a} ds \notag \\
& \le {\varepsilon}\|\zeta^t\|^2_{\mathcal{M}^1} + \frac{2}{{\varepsilon}k_0}k_0\|w\|^2_{\mathcal{V}^2_a}, \label{qww-8} \end{aligned}$$ $$\begin{aligned}
-2\int_0^\infty g(s)(\nabla a(x)\cdot \nabla \zeta^t_t(s),(-\Delta)w(t)) ds & \le 2\int_0^\infty g(s)\|\nabla a(x)\cdot \nabla \zeta^t_t(s)\| \|\Delta w(t)\| ds \notag \\
& \le 2\int_0^\infty g(s)\|\nabla a\|_\infty \|\nabla \zeta^t_t(s)\| \|\Delta w(t)\| ds \notag \\
& \le \frac{1}{{\varepsilon}}\|\nabla a\|^2_\infty \int_0^\infty g(s) \|\nabla \zeta^t_t(s)\|^2 ds + {\varepsilon}\int_0^\infty g(s) \|\Delta w(t)\|^2 ds \notag \\
& = \frac{1}{{\varepsilon}}\|\nabla a\|^2_\infty \|\zeta^t_t\|^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))} + {\varepsilon}k_0\|\Delta w\|^2 \notag \\
& \le C_{\varepsilon}(R) \|\zeta^t_t\|^2_{\mathcal{M}^0} + {\varepsilon}k_0\|\Delta w\|^2 \notag \\
& \le C_{\varepsilon}(R) + {\varepsilon}k_0\|\Delta w\|^2, \label{qww-9} \end{aligned}$$ $$\begin{aligned}
-2(\psi(w),(-\Delta)w) & \le C(R) + \frac{1}{4}\|\Delta w\|^2, \label{qww-10}\end{aligned}$$ and $$\begin{aligned}
2\beta(u,(-\Delta)w) & \le C(R) + \frac{1}{4}\|\Delta w\|^2. \label{qww-11}\end{aligned}$$ Hence, - show the right-hand side of is controlled with, for all ${\varepsilon}>0,$ $$C_{\varepsilon}(R) + (1+{\varepsilon}k_0)\|\Delta w\|^2 + \frac{2}{{\varepsilon}k_0}k_0\|w\|^2_{\mathcal{V}^2_a} + {\varepsilon}\|\zeta^t\|^2_{\mathcal{M}^1}.$$ Now fixing $0<{\varepsilon}<\min\{1/k_0,\delta\}$ and setting $$m_2=m_2(k_0,\delta):=\min\{1-{\varepsilon}k_0,\delta-{\varepsilon}\}>0 \quad \text{and} \quad c=c(k_0):=2\left( 1+\frac{1}{{\varepsilon}k_0} \right)$$ we arrive at the desired estimate .
So now we integrate the linear differential inequality and apply $\Phi(0)=0$. Thus, $$\begin{aligned}
& \|\Delta w(t)\|^2 + \|\nabla w_t(t)\|^2 + \|\zeta^t\|^2_{\mathcal{M}^1} \le Q_{\delta}(R), \label{qww-12}\end{aligned}$$ for some positive nondecreasing function $Q_{\delta}(\cdot) \sim \delta^{-1}.$ By combining , and the Poincaré inequality , we see that, with the $H^2$-elliptic regularity estimate , we have with uniform bounds $$w(t)\in H^2(\Omega) \quad \text{and} \quad w_t(t)\in H^1(\Omega) \quad \forall\ t>0.$$ Additionally, collecting the bounds and establishes that, for all $t\ge0$, $$\begin{aligned}
\|\zeta^t\|^2_{\mathcal{M}^1} + \|\zeta^t_s\|^2_{\mathcal{M}^0} \le Q_{\delta}(R). \label{qww-99}\end{aligned}$$ Lastly, to show holds we need to control the last term of the norm . With the bound , we apply the conclusion of Lemma \[what-2\] here in the form $$\begin{aligned}
\sup_{\tau\ge1} \tau\mathbb{T}(\tau;\zeta^t) & \le 2 \left( t+2 \right)e^{-\delta t} \sup_{\tau\ge1} \tau\mathbb{T}(\tau;\zeta_0) + C(R) \notag \\
& \le C(R). \label{strong-bound-1} \end{aligned}$$ where the last inequality follows from the null initial condition given in $_4$. Together, the estimates - show that holds. This completes the proof.
We now prove the main theorem.
Define the subset $\mathcal{C}$ of $\mathcal{K}^1$ by $$\mathcal{C} := \{ U=(u,v,\eta) \in \mathcal{K}^1: \|U\|_{\mathcal{K}^1} \le Q(R) \},$$ where $Q(R)>0$ is the function from Lemma \[t:uniform-compactness\], and $R>0$ is such that $\|U_{0}\|_{\mathcal{H}^0} \le R.$ Let now $U_0=(u_{0},u_{1},\eta_0) \in \mathcal{B}$ (the bounded absorbing set of Corollary \[t:abs-set\] endowed with the topology of $\mathcal{H}^0$). Then, for all $t\ge0$ and for all $U_{0}\in \mathcal{B}$, $S(t)U_{0} = Z(t)U_{0} + K(t)U_{0}$, where $Z(t)$ is uniformly and exponentially decaying to zero by Lemma \[t:uniform-decay\], and, by Lemma \[t:uniform-compactness\], $K(t)$ is uniformly bounded in $\mathcal{K}^1.$ In particular, there holds $$\mathrm{dist}_{\mathcal{H}^0}(S(t)\mathcal{B},\mathcal{C}) \le Q(R)e^{-\omega t}.$$ The proof is finished.
Conclusions
===========
We have show that the global attractors associated with a wave equation with degenerate viscoelastic dissipation in the form of degenerate memory possesses more regularity than previously obtained in [@CFM-16]. This is established under reasonable assumptions by showing the existence of a compact attracting set to which global attractor resides. Moreover, the global attractor consists of regular solutions. The main difficulties encountered here are due to the degeneracy of the dissipation term as well as obtaining compactness for the memory term.
\[appendix\]
We include two frequently used Grönwall-type inequalities that are important to this paper. The first can be found in [@Pata-Zelik-06 Lemma 5]; the second in .
\[GL\] Let $\Lambda :\mathbb{R}^+\rightarrow \mathbb{R}^+$ be an absolutely continuous function satisfying $$\frac{d}{dt}\Lambda(t) + 2\eta \Lambda(t) \le h(t)\Lambda(t)+k,$$ where $\eta>0$, $k\ge0$ and $\int_s^t h(\tau)d\tau \le \eta(t-s)+m$, for all $t\ge s\ge0$ and some $m\ge 0$. Then, for all $t\ge0$, $$\Lambda(t) \le \Lambda(0)e^{m}e^{-\eta t}+\frac{ke^m}{\eta}.$$
\[GL2\] Let $\Phi:[0,\infty)\rightarrow[0,\infty)$ be an absolutely continuous function such that, for some ${\varepsilon}>0$, $$\frac{d}{dt}\Phi(t)+2{\varepsilon}\Phi(t)\le f(t)\Phi(t)+h(t)$$ for almost every $t\in[0,\infty)$, where $f$ and $h$ are functions on $[0,\infty)$ such that $$\int_s^t|f(\tau)|d\tau \le \alpha(1+(t-s)^\lambda), \quad \sup_{t\ge0}\int_t^{t+1}|h(\tau)|d\tau\le\beta$$ for some $\alpha,\beta\ge0$ and $\lambda\in[0,1)$. Then $$\Phi(t)\le\gamma\Phi(0)e^{-{\varepsilon}t}+K$$ for every $t\in[0,\infty)$, for some $\gamma=\gamma(f,{\varepsilon},\lambda)\ge1$ and $K=K({\varepsilon},\lambda,f,h)\ge0.$
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is indebted to the anonymous referees for their careful reading of the manuscript and for their helpful comments and suggestions—in particular, for the reference [@CRV-08].
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{
"pile_set_name": "ArXiv"
}
|
TECHNION-PH-2017-4\
April 2017\
**Reexamining the photon polarization in $B \to K\pi\pi\gamma$**
Michael Gronau
*Physics Department, Technion, Haifa 32000, Israel*
Dan Pirjol
National Institute of Physics and Nuclear Engineering
Bucharest, Romania
> We reexamine, update and extend a suggestion we made fifteen years ago for measuring the photon polarization in $b \to s\gamma$ by observing in $B \to K\pi\pi\gamma$ an asymmetry of the photon with respect to the $K\pi\pi$ plane. Asymmetries are calculated for different charged final states due to intermediate $K_1(1400)$ and $K_1(1270)$ resonant states. Three distinct interference mechanisms are identified contributing to asymmetries at different levels for these two kaon resonances. For $K_1(1400)$ decays including a final state $\pi^0$ an asymmetry around $+30\%$ is calculated, dominated by interference of two intermediate $K^*\pi$ states, while an asymmetry around $+10\%$ in decays including final $\pi^+\pi^-$ is dominated by interference of $S$ and $D$ wave $K^*\pi$ amplitudes. In decays via $K_1(1270)$ to final states including a $\pi^0$ a negative asymmetry is favored up to $-10\%$ if one assumes $S$ wave dominance in decays to $K^*\pi$ and $K\rho$, while in decays involving $\pi^+\pi^-$ the asymmetry can vary anywhere in the range $-13\%$ to $+24\%$ depending on unknown phases. For more precise asymmetry predictions in the latter decays we propose studying phases in $K_1 \to K^*\pi, K\rho$ by performing dedicated amplitude analyses of $B\to J/\psi(\psi') K\pi\pi$. In order to increase statistics in studies of $B\to K\pi\pi\gamma$ we suggest using isospin symmetry to combine in the same analysis samples of charged and neutral $B$ decays.
Introduction
============
Flavor-changing radiative $B$ meson decays provide important tests for the standard model. A crucial feature, which has not yet been tested experimentally in these processes, is the dominantly left-handed polarization of the photon in $b \to s\gamma$. In several extensions of the standard model the photon in $b\to s\gamma$ acquire a sizable right-handed component due to chirality flip along a heavy fermion line in the electroweak loop process [@RH]. A very early test for probing the dominantly left-handed photon polarization through time-dependent CP asymmetries, induced by interference of a large left-handed $b$ amplitude and a small left-handed $\bar b$ amplitude, was suggested in Ref. [@Atwood:1997zr] and pursued experimentally by the Babar [@Aubert:2004pe] and Belle [@Ushiroda:2006fi] Collaborations. Several years later a second test, reminiscent of a method measuring the tau neutrino helicity in $\tau \to a_1\nu_\tau, a_1 \to \rho\pi$ [@Kuhn:1982di; @Albrecht:1990jf], was proposed based on measuring final particle momenta in $B^{0,+} \to K \pi\pi\gamma$ [@Gronau:2001ng; @Gronau:2002rz]. The photon polarization, a parity-odd quantity, was shown to be related to an asymmetry between the number of photons emitted in the two sides of the plane defined by $K\pi\pi$ in their center-of-mass frame. Since this asymmetry is odd also under time-reversal, a potentially large asymmetry requires that the decay amplitude acquires a nontrivial sizable phase due to final state interactions. Such a large calculable phase was shown to be produced in $B^+\to K^0\pi^+\pi^0\gamma$ and $B^0 \to K^+\pi^-\pi^0\gamma$ by two interfering amplitudes involving $K^{*+}$ and $K^{*0}$ intermediate resonances [@Gronau:2001ng; @Gronau:2002rz].
A calculation of the decay $B \to K_1(1400)\gamma \to K\pi\pi^0\gamma$, through interfering amplitudes for intermediate $K^{*0}\pi$ and $K^{*+}\pi$ states, was shown to lead to a sizable integrated asymmetry around $34\%$ [@Gronau:2001ng; @Gronau:2002rz]. The feasibility of observing such a large asymmetry in future experiments has been discussed in this work, assuming a branching ratio ${\cal B}(B \to K_1(1400)\gamma) = 0.7\times 10^{-5}$ as estimated in some models [@Veseli:1995bt]. The process $B \to K_1(1270)\gamma$, observed a few years later with a considerably larger branching ratio \[see Eqs. (\[BRK1(1270)\]) and (\[BRup(1400)\]) below\], was studied subsequently [@Kou:2010kn] under model-dependent assumptions about the strong decay $K_1(1270) \to K \pi\pi$, thereby introducing a considerable uncertainty in the polarization analysis [@Tayduganov:2011ui]. Quite recently the same authors proposed an alternative approach for obtaining this hadronic information by studying the process $B \to J/\psi K_1 \to J/\psi K\pi\pi$ in parallel with $B \to K_1\gamma \to K\pi\pi\gamma$ [@Kou:2016iau]. A photon polarization analysis combining contributions from several kaon resonances with $J^P = 1^+, 1^-, 2^+$ has been outlined in Ref. [@Gronau:2002rz], but would have to be treated further by experimental methods due to its complexity.
The purpose of this paper is to reexamine the situation in $B \to K_1(1270) \gamma \to K \pi\pi\gamma$ while drawing a comparison with $B \to K_1(1400)\gamma \to K\pi\pi\gamma$ which we studied only partially in Refs. [@Gronau:2001ng; @Gronau:2002rz]. In contrast to Ref.[@Kou:2010kn] which applied a quark pair creation model for describing the strong decay $K_1(1270) \to K\pi\pi$, our approach will be purely phenomenological using as much information as possible from experiments. We will discuss a few sources for the photon up-down asymmetry with respect to the $K_1$ decay plane, that are related to different types of interference occurring in $K_1$ decays.
In Section 2 we summarize the current relevant experimental data, including branching ratios and certain final state interaction phases for $K_1$ decays to $K^*\pi$ and $\rho K$ leading to $K\pi\pi$ final states. A detailed derivation of relations between covariant and partial wave amplitudes describing the latter processes is presented in Section 3 in order to resolve a discrepancy between relations used in Refs. [@Gronau:2001ng; @Gronau:2002rz] and Refs. [@Kou:2010kn; @Tayduganov:2011ui]. General expressions for decay amplitudes of $K_1 \to K\pi\pi$ are obtained in Section 4, distinguishing between hadronic final states involving $\pi^+\pi^-$ and $\pi^{\pm}\pi^0$. The photon up-down asymmetry in $B \to K\pi\pi\gamma$ with respect to the $K\pi\pi$ plane is calculated in Section 5 for these final states, separately for intermediate $K_1(1400)$ and $K_1(1270)$ resonant states. We discuss the role of three potential sources for an asymmetry. Section 6 uses approximate isospin symmetry in radiative $B$ decays to suggest combining charged and neutral $B\to K\pi\pi\gamma$ decays in order to increase statistics in studies of the photon polarization. Finally we conclude in Section 7.
Experimental situation
======================
$B\to K\pi\pi\gamma$
--------------------
Following the suggestions made in Refs. [@Gronau:2001ng; @Gronau:2002rz] for measuring the photon polarization in $B \to K\pi\pi\gamma$ several experiments reported measuring these processes. Inclusive branching ratios were measured in four charged modes, $B^+ \to K^+\pi^-\pi^+\gamma$, $B^0 \to K^0\pi^+\pi^-\gamma$, $B^+ \to K^0\pi^+\pi^0\gamma$ and $B^0 \to K^+\pi^-\pi^0\gamma$, for an hadronic invariant mass $m(K\pi\pi)$ in a range between 1 GeV$/c^2$ and 1.8 or 2 Gev$/c^2$. Both the Belle [@Nishida:2002me; @Yang:2004as] and Babar [@Aubert:2005xk] collaborations have observed the first two charged and neutral $B$ decay modes involving a pair of charged pions resulting in the following averaged branching ratios [@Olive:2016xmw]: \[BR1\] [B]{}(B\^+ K\^+\^-\^+) & = & (2.76 0.22)10\^[-5]{} ,\
[B]{}(B\^0 K\^0\^+\^-) & = & (1.95 0.22)10\^[-5]{} . Babar has also measured branching ratios for decay modes involving a neutral pion [@Aubert:2005xk]: \[BR2\] [B]{}(B\^+ K\^0\^+\^0) & = & (4.6 0.5)10\^[-5]{} ,\
[B]{}(B\^0 K\^+\^-\^0) & = & (4.1 0. 4)10\^[-5]{} .
Exclusive radiative $B^+$ decays involving the charged kaon resonance $K^+_1(1270)$ decaying to $K^+\pi^-\pi^+$ have been reported by Belle [@Yang:2004as], \[BRK1(1270)\] [B]{}(B\^+ K\_1\^+(1270) ) = (4.3 1.3)10\^[-5]{} . Radiative $B$ decays to $K^*_2(1430)$, first reported by the CLEO collaboration [@Coan:1999kh], (B K\^\*\_2(1430)) = (1.7 0.6)10\^[-5]{} , were observed subsequently by Babar at a similar rate [@Aubert:2003zs], \[BRK\*2\] [B]{}(B\^+ K\^[\*+]{}\_2(1430) ) & = & (1.4 0.4)10\^[-5]{} ,\
[B]{}(B\^0 K\^[\*0]{}\_2(1430) ) & = & (1.24 0.24)10\^[-5]{} . We note that so far none of the $K\pi\pi\gamma$ modes observed by Belle included a $\pi^0$ in the final state, in contrast to several of the above measurements by Babar. Belle also obtained upper bounds at $90\%$ confidence level for decays involving $K_1(1400)$ to final states including $\pi^+\pi^-$, using only about $18\%$ of their final data set [@Yang:2004as], \[BRup(1400)\] [B]{}(B\^+ K\_1\^+(1400) ) & < & 1.510\^[-5]{} ,\
[B]{}(B\^0 K\_1\^0(1400) ) & < & 1.210\^[-5]{} . These upper bounds are a factor of two larger than the branching ratio assumed in Ref. [@Gronau:2001ng].
A first attempt for measuring the photon polarization in $B \to K\pi\pi\gamma$ was made by the LHCb collaboration [@Aaij:2014wgo; @Veneziano:2015ggl]. Nearly 14,000 signal events were reconstructed in the all charged mode $B^+ \to K^+\pi^-\pi^+\gamma$. The formalism developed in Refs. [@Gronau:2001ng; @Gronau:2002rz], extended to include interference of a few kaon resonances, was applied to decay distributions for four $K\pi\pi$ mass intervals in the overall range $1.1 - 1.9$ GeV$/c^2$. The final result, a nonzero up-down asymmetry at $5.2\sigma$, was insufficient for providing a significantly quantitative measurement of the photon polarization.
$K_1 \to K\pi\pi$
-----------------
An analysis of the photon polarization in $B \to K\pi\pi\gamma$ via intermediate $K_1(1400)$ and $K_1(1270)$ resonances requires knowledge of branching ratios for these kaon resonances decaying into $K^*\pi$ and $\rho K$ states, and of magnitudes and relative phases between corresponding partial wave decay amplitudes. The situation in decays of $K_1(1400)$ is described in Table \[tab.K1400\]. This information is based solely on a thirty-six-year-old experiment [@Daum:1981hb] performing a partial wave analysis for $J^P=1^+$ $K\pi\pi$ states produced by $K^-p$ diffractive scattering with couplings to $K^*\pi$ and $\rho K$ in both S and D waves. In addition to measuring the ratio of $S$ and $D$ wave $K_1(1400)$ branching ratios into $K^*\pi$, some tantalizing information, $\delta_{DS}^{(K^*\pi)}\sim 260^\circ$, $\alpha_S \sim 40^\circ$, has been obtained for two relevant phases, between $K^*\pi$ $S$ and $D$ partial wave amplitudes and between $S$ wave amplitudes for $K^*\pi$ and $\rho K$, respectively.
Mode ${\cal B}$ $\Gamma_D/\Gamma_S$ $ \delta_{DS}$ $|\vec p|$(MeV)
---------- -------------- --------------------- ---------------- -----------------
$K^*\pi$ $(94\pm 6)$% $0.04 \pm 0.01$ $-$ $401$
$\rho K$ $(3 \pm 3)$% $-$ $-$ $291$
: Branching fractions and particle momenta for the main decay modes of $K_1(1400)$ [@Olive:2016xmw].
\[tab.K1400\]
Mode ${\cal B}$ [@Olive:2016xmw] $\Gamma_D/\Gamma_S$ [@Olive:2016xmw] $\delta_{DS}$ $|\vec p|$(MeV) ${\cal B}$ Fit 1 [@Guler:2010if] ${\cal B}$ Fit 2 [@Guler:2010if] Average
---------- ----------------------------- -------------------------------------- --------------- ----------------- ---------------------------------- ---------------------------------- ---------
$\rho K$ $(42\pm 6)$% $-$ $-$ 46 $(57.3\pm 3.5)$% $(58.4\pm 4.3)$% 57.9%
$K^*\pi$ $(16\pm 5)$% $1.0 \pm 0.7$ $-$ 302 $(26.0\pm 2.1)$% $(17.1\pm 2.3)$% 21.6%
: Branching fractions and particle momenta for the main decay modes of $K_1(1270)$ [@Olive:2016xmw; @Guler:2010if].
\[tab.K1270\]
The situation in decays of $K_1(1270)$ is displayed in Table \[tab.K1270\]. The left-hand side is based on the same $K^- p$ scattering experiment [@Daum:1981hb], while the right-hand side quotes results obtained much more recently by the Belle Collaboration through an amplitude analysis determining the resonant structure of the $K^+\pi^-\pi^+$ final state in $B^+ \to J/\psi K^+\pi^-\pi^+$ [@Guler:2010if]. The difference between the $K_1(1270)$ decay branching ratios obtained in these two different methods seems to be associated with a third decay channel of $K_1(1270)$ involving $K^*_0(1430)\pi$, for which a sizable branching ratio of $(28 \pm 4)\%$ was claimed in [@Daum:1981hb] in contrast to a negligible branching ratio around two percent reported in [@Guler:2010if]. A rather crude measurement exists for the ratio of $S$ and $D$ wave branching ratios into $K^*\pi$ [@Daum:1981hb]. However no direct information exists on two relevant phases, between $K^*\pi$ partial wave amplitudes and between $S$ wave amplitudes for $K^*\pi$ and $\rho K$. A relative phase around $\phi(\rho K)-\phi(K^*\pi) \sim -40^\circ$ has been measured between total $\rho K$ and $K^*\pi$ decay amplitudes [@Guler:2010if]. Assuming that these two amplitudes are dominated by an $S$ wave, this would imply $\alpha_S \sim -40^\circ$.
Covariant and partial wave $K_1\to K^*\pi,\rho K$ amplitudes
============================================================
The amplitude for an axial-vector meson decaying to a vector meson and a pseudoscalar meson has two equivalent descriptions, in terms of two covariant amplitudes and in terms of S and D partial wave amplitudes. The polarization analysis for $B \to K\pi\pi\gamma$ is based on covariant amplitudes [@Gronau:2001ng; @Gronau:2002rz] while data are given in terms of partial wave amplitudes. In this section we will prove relations between these two descriptions which will be used in our forthcoming analysis. While these relations were given briefly in Refs. [@Gronau:2001ng; @Gronau:2002rz], different relations have been used by the authors of [@Kou:2010kn; @Tayduganov:2011ui] quoting Ref. [@Chung:1971ri] with no detail. Here we wish to settle this discrepancy by proving these relations in some detail.
Consider, for instance $K_1 \to K^* \pi$. The covariant amplitude for $K_1^+(p,\ep) \to K^{*0}(p',\ep')\pi^+(p_\pi)$, involving particles with four-momenta $p, p', p_{\pi}$ and polarization vectors $\ep, \ep'$, is given by: \[inv\] [M]{}\^1 = A\_[K\^\*]{}(’\^\*) + B\_[K\^\*]{}(p\_)(’\^\*p\_) . In the $K_1$ rest frame ($\vec p=0$) we define $z$ as the direction of the $K^*$ momentum, while the pion moves in the direction $-z$. The three possible initial spin-one $K_1$ states involving spin projection $\lambda=+1, 0, -1$ along $z$ are denoted $|1, \lambda\rangle$. The three polarization vectors $\ep$ and $\ep'$ for these three states $\lambda=\pm 1,0$ are: \[lam+1\] =1 &:& = ’ = (0, -(e\_1 + ie\_2)) ,\
\[lam-1\] =-1 &:& = ’ = (0, (e\_1 - ie\_2)) ,\
=0 &:& = (0, e\_3), ’ = (|p\_|/m\_[K\^\*]{}, (E\_[K\^\*]{}/m\_[K\^\*]{})e\_3) . For $\lambda=1$ this is the form of $\ep$ in the $K_1$ rest frame. The same form in this frame, identical to its form in the $K^*$ rest frame, applies to $\ep'$ because a Lorentz transformation along $z$ does not change the $x, y$ components, mixing only the $t, z$ components. For $\lambda=0$ $\ep'$ is obtained from $\ep'(K^*) = (0, \vec e_3)$ in the rest frame of $K^*$ by a Lorentz transformation to the rest frame of $K_1$ using $\gamma = E_{K^*}/m_{K^*}, \gamma\beta = -|p_\pi|/m_{K^*}$, ’\_0(K\_1) & = & = |p\_|/m\_[K\^\*]{} ,\
’\_3(K\_1) & = & = E\_[K\^\*]{}/m\_[K\^\*]{} . We note that the transversity condition $\ep' \cdot p'=0$ is satisfied for all polarization states $\lambda$ of the $K^*$ meson. In particular, for $\lambda=0$ we have, using $p'(K^*) = (E_{K^*}, |\vec p_\pi|\vec e_3)$, p’(K\^\*)’(K\^\*) -1mm= -1mm E\_[K\^\*]{}|p\_|/m\_[K\^\*]{} - |p\_|E\_[K\^\*]{}/m\_[K\^\*]{} =0 .
The covariant decay amplitude (\[inv\]) can now be calculated for these three polarization states: \[Mpm1\] & = & 1: (’\^\*) = -(e\_1 ie\_2)(e\_1 ie\_2) = -1; p\_= 0\
&& \^1\_[=1]{} = -A\_[K\^\*]{} . \[M0\] && =0: ’\^\* = -; p\_ = |p\_|; ’\^\*p\_= + =\
&& \^1\_[= 0]{} = - A\_[K\^\*]{} + B\_[K\^\*]{} .
Let us now write decay amplitudes for the three polarization states $|1, \lambda\rangle$ in terms of amplitudes for S and D waves, $L=0, 2$, noting that the angular momentum states carry $L_z =0$ ($m=0$) for $K^*$ and $\pi$ moving in $\pm z$ directions. Using SU(2) Clebsch-Gordan coefficients $(l~0; 1~\lambda | 1~\lambda)$ and absorbing a factor $1/\sqrt{5}$ in the definition of the D-wave amplitude, we have: \[SDpm1\] [M]{}\^1\_[= 1]{} = (0 0 ; 1 1 | 1 1)C\^[(K\^\*)]{}\_S + (2 0; 1 1|1 1)C\^[(K\^\*)]{}\_D = C\^[(K\^\*)]{}\_S + C\^[(K\^\*)]{}\_D , \[SD0\] [M]{}\^1\_[= 0]{} = (0 0 ; 1 0 | 1 0)C\^[(K\^\*)]{}\_S + (2 0; 1 0|1 0)C\^[(K\^\*)]{}\_D = C\^[(K\^\*)]{}\_S - C\^[(K\^\*)]{}\_D . Squaring magnitudes of these amplitudes and averaging over the three polarizations states of the $K_1$ meson, one obtains \_[=0,1]{} |[M]{}\^1\_|\^2 = |C\^[(K\^\*)]{}\_S|\^2 + |C\^[(K\^\*)]{}\_D|\^2 , implying a decay rate \[rate\] (K\_1 K\^\*) = (|C\^[(K\^\*)]{}\_S|\^2 + |C\^[(K\^\*)]{}\_D|\^2) |p\_|.
Comparing Eqs. (\[Mpm1\]) and (\[M0\]) with (\[SDpm1\]) and (\[SD0\]) one obtains \[ASD\] -A\_[K\^\*]{} & = & C\^[(K\^\*)]{}\_S + C\^[(K\^\*)]{}\_D ,\
-B\_[K\^\*]{} & = & C\^[(K\^\*)]{}\_S( - 1) + C\^[(K\^\*)]{}\_D( + 2) , or \[BSD\] -B\_[K\^\*]{}|p\_|\^2 = C\^[(K\^\*)]{}\_S() + C\^[(K\^\*)]{}\_D( ) . These relations agree with those applied in Refs. [@Gronau:2001ng; @Gronau:2002rz] using a different convention for partial wave amplitudes. \[The amplitudes $C_{S,D}^{(K^*\pi)}$ are related to $c_{S,D}$ occurring in Eq. (20) of [@Gronau:2002rz] by $C_S^{(K^*\pi)} = - c_S$ and $C_D ^{(K^*\pi)} = - \sqrt2 \frac{m_{K_1} |\vec p_\pi|^2}{2m_{K^*}+E_{K^*}}c_D$.\]
While the expression for $A_{K^*\pi}$ agrees with the one quoted by the authors of Ref. [@Kou:2010kn], these authors used a different relation for $B_{K^*\pi}$. Their Eq. (27) reads in our notation [@eq27], \[Bkou\] -B\_[K\^\*]{}|p\_|\^2 = . We find that this relation is in disagreement with (\[BSD\]), and is therefore incorrect.
Relations similar to (\[ASD\]) and (\[BSD\]) apply to $K_1 \to K\rho$: \[ASDr\] -A\_[K]{} & = & C\^[(K)]{}\_S + C\^[(K)]{}\_D ,\
\[BSDr\] -B\_[K]{}|p\_K|\^2 & = & C\^[(K)]{}\_S() + C\^[(K)]{}\_D( ) .
Decay amplitudes for $K_1 \to K\pi\pi$
======================================
The two pairs of processes in Eqs. (\[BR1\]) and (\[BR2\]) obtain contributions from $K_1(1270)$ and $K_1(1400)$ resonances. The decays of these resonances to different $K\pi\pi$ charged modes may be divided into two distinct pairs distinguished by their intermediate resonant decay channels [@Gronau:2002rz]. The first pair involves two single decay channels into $K^*\pi$ and $K\rho$, K\^+\_1 {
[c ]{} K\^[\*0]{}\^+ K\^+ \^0
} K\^+\^-\^+ , K\^0\_1 {
[c ]{} K\^[\*+]{}\^- K\^0 \^0
} K\^0\^+\^- , while the second pair obtains contributions from two interfering $K^*\pi$ decay channels in addition to a single $K\rho$ channel, K\^+\_1 {
[c ]{} K\^[\*+]{}\^0 K\^[\*0]{} \^+ K\^0 \^+
} K\^0\^+\^0 , K\^0\_1 {
[c ]{} K\^[\*+]{}\^- K\^[\*0]{}\^0 K\^+\^-
} K\^+\^-\^0 .
The decays of $K_1^+$ and $K_1^0$ within each pair are related to each other by isospin reflection $u \leftrightarrow d$, implying equal amplitudes in the isospin symmetry limit: \[type3\] A(K\^+\_1 K\^+\^-\^+) & = & A(K\^0\_1 K\^0\^+\^-) ,\
\[type2\] A(K\^+\_1 K\^0\^+\^0) & = & A(K\^0\_1 K\^+\^-\^0) . These two amplitudes, for final states characterized by two charged pions in one case and by a pair of charged and neutral pions in the other, will be studied separately. Only the second pair of amplitudes has been analyzed for $K_1(1400)$ in Refs. [@Gronau:2001ng; @Gronau:2002rz].
Decays involving two charged pions $K_1^+ \to K^+\pi^+\pi^-,
K_1^0\to K^0\pi^+\pi^-$
------------------------------------------------------------
The $K^{*0}\pi^+$ contribution to the decay amplitude for $K_1^+(p,\ep) \to K^+(p_3)\pi^+(p_1)\pi^-(p_2)$ is obtained by convoluting the amplitude (\[inv\]) with the amplitude for $K^{*0} \to K^+\pi^-$, A(K\^[\*0]{} K\^+ \^-) = g\_[\_[K\^\*K]{}]{}’(p\_2 - p\_3) , including a Breit-Wigner propagator for the $K^*$, && \^[(K\^\*)]{}\_[23]{} ,\
&& [B]{}\^[(K\^\*)]{}\_[23]{} (s\_[23]{} - m\^2\_[K\^\*]{} - im\_[K\^\*]{}\_[K\^\*]{})\^[-1]{} , s\_[23]{} (p\_2 + p\_3)\^2 . A similar contribution due to $K_1^+ \to K^+\rho^0, \rho^0\to \pi^+\pi^-$ involves invariant amplitudes $A_{\rho K}, B_{\rho K}$ describing $K_1 \to K\rho$, the strong coupling $g_{\rho\pi\pi}$ and a Breit-Wigner propagator for the $\rho$, ${\cal B}^{(\rho)}_{12} \equiv (s_{12} - m^2_\rho - im_\rho\Gamma_\rho)^{-1}$. Specifically, we define the amplitude A(\^0 \^+ \^-) = g\_[\_]{}’(p\_1 - p\_2) .
Adding these two contributions and neglecting a non-resonant term (which is justified in $K_1(1400)$ more than in $K_1(1270)$ - see Tables \[tab.K1400\] and \[tab.K1270\]), the total covariant amplitude for $K_1^+(p,\ep) \to K^+(p_3)\pi^+(p_1)\pi^-(p_2)$ is given by \[calM\] [M]{} = C\_1(p\_1) - C\_2(p\_2) , where C\_i = C\_i\^[(K\^\*)]{} + C\_i\^[(K)]{} (i=1, 2) , \[Ci\] & C\_1\^[(K\^\*)]{} = g\_[\_[K\^\*K]{}]{}[B]{}\^[(K\^\*)]{}\_[23]{}{A\_[K\^\*]{} + B\_[K\^\*]{}p\_1(p\_2-p\_3) - \[A\_[K\^\*]{} - B\_[K\^\*]{}((pp\_1) - m\^2\_)\]} ,\
& C\_1\^[(K)]{} = g\_[B]{}\^[()]{}\_[12]{}\[A\_[K]{} - B\_[K]{}(pp\_1 - pp\_2)\] ,\
& C\_2\^[(K\^\*)]{} = -g\_[\_[K\^\*K]{}]{}[B]{}\^[(K\^\*)]{}\_[23]{}(2A\_[K\^\*]{}) ,\
& C\_2\^[(K)]{} = - g\_[B]{}\^[()]{}\_[12]{}\[A\_[K]{} + B\_[K]{}(pp\_1 - pp\_2)\] .
The four scalar products of two momenta in (\[Ci\]), $p_1\cdot p_2$, $p_1\cdot p_3$, $p \cdot p_1$ and $p\cdot p_2$, may all be written in terms of $s_{13}$ and $s_{23}$. That is, $C_i$ are functions of these two variables and the decay amplitude has the explicitly covariant form = C\_1(s\_[13]{}, s\_[23]{})(p\_1) - C\_2(s\_[13]{}, s\_[23]{})(p\_2) .
Decays involving a neutral pion $K_1^+ \to K^0\pi^+\pi^0,
K_1^0\to K^+\pi^-\pi^0$
---------------------------------------------------------
In these decays the amplitude has the same structure as (\[calM\]), \[M’\] [M’]{} = C’\_1(s\_[13]{},s\_[23]{})(p\_1) - C’\_2(s\_[13]{}, S\_[23]{})(p\_2) , with two contributions to $C'_{1,2}$ from $K^{*0}\pi$ and $K^{*+}\pi$ and one contribution from $K\rho^{\pm}$. The overall contribution from $K^*\pi$ is antisymmetric under the exchange of the two pion momenta and, using isospin, is expressed in terms of the same quantities $C_i^{(K^*\pi)}$ given in (\[Ci\]): \[C’\] C\_i\^[(K\^\*)]{} = \[C\^[(K\^\*)]{}\_i - C\^[(K\^\*)]{}\_i (p\_1 p\_2)\] . The single contribution from $K\rho$ is C\_i\^[(K)]{} = C\_i\^[(K)]{} .
Experimental information on ratios of amplitude
-----------------------------------------------
In the next section studying the photon polarization in $B \to K_1\gamma,
K_1 \to K\pi\pi$, which depends on interference of amplitudes, we will need ratios of certain quantities which we calculate now.
The strong couplings $g_{_{K^*K\pi}}$ and $g_{\rho\pi\pi}$ occurring in (\[Ci\]) (for which we used a slightly different convention in [@Gronau:2002rz]) are obtained from the $K^*$ and $\rho$ widths. Using && (K\^[\*0]{}K\^+\^-) = 23 \_[K\^\*]{}(K\^\*K) = |g\_[K\^\* K]{}|\^2 |p\_|\^3 ,\
&& (\^0 \^+\^-) = \_() = |g\_|\^2|p\_|\^3 , where $\Gamma_{K^*} = 51$ MeV, $\Gamma_{\rho} = 150$ MeV, $|\vec p_\pi|_{K^*\to K\pi} = 289$ MeV, $|\vec p_\pi|_{\rho\to \pi\pi} = 364$ MeV [@Olive:2016xmw], we calculate \[gratio\] = 1.29 . This compares well with an SU(3) prediction = - 2 .
The quantities $A_{K^*\pi}, B_{K^*\pi}, A_{K\rho}, B_{K\rho}$ in Eqs. (\[Ci\]) may be obtained from $S$ and $D$ wave amplitudes measured in $K_1^+ \to K^{*0}\pi^+$ and $K_1^+\to K^+\rho^0$ decays, denoted $C^{(K^*\pi)}_{S,D}$ and $C^{(K\rho)}_{S,D}$, using Eqs. (\[ASD\]) (\[BSD\]) for the first process and (\[ASDr\]) (\[BSDr\]) for the second. Branching ratios for $K_1 \to K^*\pi$ and $K_1 \to K\rho$ summed over all charged modes and corresponding ratios of decay rates for $S$ and $D$ waves waves were given in Tables \[tab.K1400\] and \[tab.K1270\] for $K_1(1400)$ and $K_1(1270)$, respectively. We will denote by $\delta_{DS}^{(K^*\pi)}$ and $\delta_{DS}^{(K\rho)}$ relative phases between $S$ and $D$ wave amplitudes in $K_1^+ \to K^{*0}\pi^+$ and $K_1^+ \to K^+\rho^0$, respectively, and by $\kappa_S$ and $\alpha_S$ the magnitude and phase of the ratio of $S$ wave amplitudes for these decays, \_[DS]{}\^[(K\^\*)]{} (C\_D\^[(K\^\*)]{}/C\_S\^[(K\^\*)]{}), \_[DS]{}\^[(K)]{} (C\_D\^[(K)]{}/C\_S\^[(K)]{}), \_S e\^[i\_S]{} C\_S\^[(K)]{}/C\_S\^[(K\^\*)]{}.
Ratios of amplitude will now be calculated separately for $K_1(1400)$ and $K_1(1270)$ applying Eqs. (\[ASD\]) (\[BSD\]) to $K_1^+ \to K^{*0}\pi^+$ and (\[ASDr\]) (\[BSDr\]) to $K_1^+ \to K^+ \rho^0$. Meson masses will be taken from [@Olive:2016xmw].
- $K_1(1400)$ Using $|C_D^{(K^*\pi)}|^2/|C_S^{(K^*\pi)}|^2= 0.04 \pm 0.01$ and since the branching ratio $K_1 \to \rho K$ is very small, we calculate: \[ratio1\] & = & ,\
\[ratio2\] & = & \~0.45 , The ratio $|C_S^{(K\rho)}|/|C_S^{(K^*\pi)}|$ may be obtained from \[45\] = , implying together with (\[gratio\]) and assuming a central value ${\cal B}(K_1 \to K\rho) = 3\%$, \[ratio3\] \_S & = &0.19 e\^[i\_S]{} . The factor of 2 on the right-hand side of (\[45\]) is due to the specific choice of the modes $K_1^+\to K^{*0}\pi^+$ and $K_1^+\to \rho^0 K^+$ used to define the couplings $C_S^{(K^*\pi)}$ and $C_S^{(\rho K)}$, while the branching ratios on the left-hand side are for final states summed over all charges.
- $K_1(1270)$ Taking the central value in $|C_D^{(K^*\pi)}|^2/|C_S^{(K^*\pi)}|^2= 1.0 \pm 0.7$ and assuming that $S$ wave dominates $K_1 \to \rho K$ because of an extremely small available phase space, we find: & = & ,\
& = & 0.51 . Using for branching ratios of $K_1 \to K^*\pi$ and $K_1\to K\rho$ averages of the two Belle fits in Table \[tab.K1270\], we calculate \_S & = & 5.42 e\^[i\_S]{} . A relative phase $\phi(\rho K)- \phi(K^*\pi) \sim -40^\circ$ between total amplitudes has been measured by the Belle collaboration [@Guler:2010if]. However, translating this into a constraint on $\alpha_S$ requires information about the partial wave amplitudes which is not available.
Photon polarization and asymmetry in $B\to K\pi\pi\gamma$
=========================================================
We have shown that the decay amplitude for $K_1(\vec p=0,\vec\ep) \to \pi(p_1)\pi(p_2)K(p_3)$ in the $K_1$ rest frame has the general structure \_[rest]{} = C\_1(s\_[13]{}, s\_[23]{})p\_1- C\_2(s\_[13]{}, s\_[23]{})p\_2= J , where \[J\] J C\_1(s\_[13]{}, s\_[23]{})p\_1 - C\_2(s\_[13]{}, s\_[23]{})p\_2 . Considering now $B\to K_1\gamma$ followed by $K_1 \to K\pi\pi$ we wish to study the angular distribution of the photon with respect to the $K_1$ decay plane as function of the photon polarization. For completeness we will derive this relation, although some parts of the derivation can be found in Refs. [@Gronau:2001ng; @Gronau:2002rz]. One reason for presenting this complete analysis is correcting a sign error in defining a specific direction in this previous work.
Working in the rest frame of $K_1$, we take the photon momentum $\vec p_\gamma$ along the $-z$ direction, and the $B$ meson momentum along the $+z$ direction. There are two amplitudes for $\bar B\to K_1\gamma$ decays, corresponding to left- and right-handed photons $$\begin{aligned}
\mathcal{M}_L \equiv \mathcal{A}(\bar B \to K_1 \gamma_L) \,,\quad
\mathcal{M}_R \equiv \mathcal{A}(\bar B \to K_1 \gamma_R)~.\end{aligned}$$ Defining the photon polarization parameter, $$\begin{aligned}
\lambda_\gamma \equiv \frac{|\mathcal{M}_R|^2 -|\mathcal{M}_L|^2}
{|\mathcal{M}_R|^2 +|\mathcal{M}_L|^2}\,,\end{aligned}$$ we would like to determine $\lambda_\gamma$ through the angular distribution of the decay products of the $K_1$ meson.
The amplitude for $\bar B \to K\pi\pi\gamma_{L,R}$ is proportional to the decay amplitude $K_1(p,\epsilon)\to K\pi\pi$ with $\epsilon=\epsilon_{L,R}$ corresponding to the transverse polarization states $|\lambda = \mp 1\rangle$ of the $K_1$ meson in its rest frame \[see Eqs.(\[lam+1\]) (\[lam-1\])\], (|B K\_[L,R]{}) = \_[L,R]{} (\_[1]{}J) = \_[L,R]{} (J\_x i J\_y) . Squaring the amplitude, |(|B K\_[L,R]{})|\^2 &=& 12|\_[L,R]{}|\^2 (J\_x i J\_y)(J\_x\^\* i J\_y\^\*)\
&=& 12|\_[L,R]{}|\^2 { |J\_x|\^2 + |J\_y|\^2 2(J\_x J\_y\^\*) }, and summing over the two photon polarization states, one obtains \[avsquare\] && \_[=L,R]{}|(|B K\_)|\^2\
&& = 12 { |\_L|\^2 + |\_R|\^2 } .
We denote by $\hat n = (\vec p_1\times \vec p_2)/|\vec p_1\times \vec p_2|$ the normal to the $K_1$ decay plane defined by the two pions momenta. The orientation of the $K\pi\pi$ plane with respect to the $(\hat e_x,\hat e_y,
\hat e_z)$ axes is determined by three Euler-like angles $(\theta,\phi,\psi)$. The polar angles $(\theta, \psi)$ define the orientation of $\hat n$ with respect to $\hat e_z$ such that $\cos\theta = \hat e_z \cdot \hat n$, and the third angle $\phi$ parameterizes rotations of the $K\pi\pi$ plane around $\hat n$. The intersection of the $K\pi\pi$ plane with the $(\hat e_x, \hat e_y)$ plane is the nodal line, and its angle with respect to $\hat e_x$ is $\psi$. We denote unit vectors in the $K\pi\pi$ plane by $(\hat e_1,\hat e_2)$ such that $\hat e_3 \equiv \hat n$, and define $\phi$ as the angle between the nodal line and $\hat e_1$.
The vector $\vec J$ lies in the $(\hat e_1, \hat e_2)$ plane. Its components in the $(\hat e_x,\hat e_y,\hat e_z)$ coordinates can be expressed in terms of the angles introduced above: $$\begin{aligned}
\label{Jxyz}
&& J_x = (J_1 \cos\phi + J_2 \sin\phi) \cos\psi
- (- J_1 \sin\phi + J_2 \cos\phi) \sin\psi\cos\theta~, \nn \\
&& J_y = (J_1 \cos\phi + J_2 \sin\phi) \sin\psi
+ (- J_1 \sin\phi + J_2 \cos\phi) \cos\psi\cos\theta~, \nn \\
&& J_z = (-J_1 \sin\phi + J_2 \cos\phi) \sin\theta~.\end{aligned}$$ These equations are obtained by noting that the projections of $\vec J$ in the $K\pi\pi$ plane, $J_\parallel$ along the nodal line and $J_\perp$ perpendicular to it, are $$\begin{aligned}
\label{Jpp}
J_\parallel = J_1 \cos\phi + J_2 \sin\phi \,, \quad
J_\perp = - J_1 \sin\phi + J_2 \cos\phi\,.\end{aligned}$$ The components along the $(\hat e_x,\hat e_y)$ directions are $$\begin{aligned}
&& J_x = J_\parallel \cos\psi - J_\perp \sin \psi \cos\theta~, \\
&& J_y = J_\parallel \sin\psi + J_\perp \cos \psi \cos\theta~.\end{aligned}$$ Substituting (\[Jpp\]) in these relations leads to (\[Jxyz\]).
Using |J\_x|\^2 + |J\_y|\^2 = |J|\^2 - |J\_z|\^2 , and averaging over $\phi$ implies for the first term in (\[avsquare\]), $$\begin{aligned}
\frac{1}{2\pi} \int_0^{2\pi} d\phi (|\vec J_x|^2 + |J_y|^2) =
|\vec J|^2 ( 1 - \frac12 \sin^2\theta) = \frac12 |\vec J|^2 (1 + \cos^2\theta)~.\end{aligned}$$ The second term multiplying $\lambda_\gamma$ is $$\begin{aligned}
\mbox{Im}(J_x J_y^*) &=& \mbox{Im}(J_\parallel J_\perp^*) \cos^2\psi \cos\theta
- \mbox{Im}(J_\parallel^* J_\perp) \sin^2\psi \cos \theta \\
&=& 2 \mbox{Im}(J_\parallel J_\perp^*) \cos\theta =
2 \mbox{Im}(J_1 J_2^*) \cos\theta =
\mbox{Im}[\hat n \cdot (\vec J \times \vec J^*)] \cos\theta
\nn \,.\end{aligned}$$ Thus, after averaging over rotations in the $K\pi\pi$ decay plane (angle $\phi$) and around the $\hat e_z$ axis (angle $\psi$), the decay distribution in the angle $\theta$ is given by $$\begin{aligned}
\frac{d\Gamma}{ds_{13} ds_{23} d\cos\theta} = C(s_{13}, s_{23}) \left\{
|\vec J|^2 (1 + \cos^2\theta) + \lambda_\gamma 2 \mbox{Im}[\hat n \cdot
(\vec J \times \vec J^*)] \cos\theta \right\}\,.\end{aligned}$$
The second term in this decay distribution is sensitive to the photon polarization parameter $\lambda_\gamma$. Its contribution can be isolated by forming an up-down asymmetry with respect to the angle $\theta$. At each point $(s_{13},s_{23})$ in the Dalitz plot one may define an up-down asymmetry with respect to the $\hat e_z$ axis (s\_[13]{},s\_[23]{}) (\_0\^[/2]{} d - \_[/2]{}\^d ) .
We have seen in (\[C’\]) that for $K\pi\pi$ final states including a $\pi^0$ the overall contribution from the two $K^*\pi$ intermediate states to $C_{1,2}$, which enters the definition of $\vec J$ in (\[J\]), is antisymmetric under an exchange of the two pion momenta. Consequently the interference of the two $K^*\pi$ contributions, which for an intermediate $K_1(1400)$ is a dominant source for a photon up-down asymmetry in $B \to K\pi\pi^0\gamma$ (see next subsection), is antisymmetric under $s_{13} \leftrightarrow s_{23}$, and thus vanishes when being integrated over the entire Dalitz plot. For this reason one redefines a slightly modified integrated up-down asymmetry by multiplying the numerator with $\mbox{sgn}(s_{13}-s_{23})$ which is also antisymmetric in $(s_{13},s_{23})$, $$\begin{aligned}
\tilde \mathcal{A} \equiv \frac{1}{\frac83 \langle |\vec J|^2 \rangle}
2\lambda_\gamma \langle \mbox{sgn}(s_{13}-s_{23}) \mbox{Im}(\hat n \cdot
(\vec J \times \vec J^*))\rangle =
\frac34 \frac{\langle \mbox{sgn}(s_{13}-s_{23}) \mbox{Im}(\hat n \cdot
(\vec J \times \vec J^*))\rangle}{\langle |\vec J|^2\rangle} \lambda_\gamma~.
\nonumber\\\end{aligned}$$ The angular brackets denote integration over the Dalitz plot, $\langle \cdots \rangle = \int\cdots ds_{13} ds_{23}$. This asymmetry may be formulated also as an up-down asymmetry with respect to an angle $\tilde \theta$ defined by $\cos\tilde\theta \equiv \mbox{sgn}(s_{13}-s_{23}) \cos\theta$, where $\tilde \theta $ is the angle between $\hat e_z$ and the normal to the plane determined by $\vec p_{fast} \times \vec p_{slow}$ [@sign].
Three mechanisms for a photon asymmetry
---------------------------------------
Given the expressions of $C_i(s_{13},s_{23})$ occurring in amplitudes for $K_1\to K\pi\pi$ decays and the experimental information about these amplitudes as described in Sec.4, we are now ready to calculate the photon up-down asymmetry with respect to the $K\pi\pi$ decay plane.
As mentioned in the introduction, a nonzero up-down asymmetry which is odd under time-reversal requires two interfering amplitudes with a nonzero relative phase due to final state interactions. We identify three types of interference which involve such potential phases:
- \(a) Interference of amplitudes for two $K^*\pi$ intermediate states. Such interference, involving $K^{*0}\pi^+, K^{*+}\pi^0$ and $K^{*0}\pi^0,
K^{*+}\pi^-$ in $K_1^+ \to K^0\pi^+\pi^0$ and $K_1^0\to K^+\pi^-\pi^o$ respectively, occurs only in decays involving a final neutral pion. The amplitude for these $K_1^{+,0} \to K\pi\pi^0$ decays is given in Sec.4.2. The relevant strong phase originates in an overlap of two isospin-related Breit-Wigner $K^{*0}$ and $K^{*+}$ resonance bands in the Dalitz plot. The contribution of this interference to the asymmetry includes also interference of $S$ and $D$ wave $K^*\pi$ amplitudes which depends on $\delta_{DS}^{(K^*\pi)}$ and vanishes for $\delta_{DS}^{(K^*\pi)}=0$. We denote an asymmetry from interference of this kind by $\tilde{\cal A}_a$.
- \(b) Interference between $K^*\pi$ and $K\rho$ amplitudes. Such interference occurs in all $K_1 \to K\pi\pi$ decays including both $K_1^+ \to K^+\pi^+\pi^-,
K_1^0\to K^0\pi^+\pi^-$ and $K_1^+ \to K^0\pi^+\pi^0,
K_1^0\to K^+\pi^-\pi^0$. This contribution to an asymmetry is affected by an overlap in the Dalitz plot of the $K^*$ and $\rho$ bands and depends on the two relative phases $\delta_{DS}^{(K^*\pi)}$ and $\alpha_S$. We denote this contribution to an asymmetry by $\tilde{\cal A}_b$.
- \(c) Interference of $S$ and $D$ wave amplitudes in $K_1 \to K^*\pi$. This kind of interference occurs in all four $K_1 \to K\pi\pi$ charged modes. Because of an assumed negligible $D$ wave amplitude in $K_1 \to K\rho$ due to very limited available phase space (in particular in $K_1(1270) \to K\rho$), we neglect a similar interference in these decays. The interference between $S$ and $D$ wave $K^*\pi$ amplitudes does not depend on overlapping bands in the Dalitz plot and on $\alpha_S$. The resulting asymmetry depends on $\alpha_S$ (through the asymmtry denominator) and on $\delta_{DS}^{(K^*\pi)}$ and vanishes for $\delta^{(K^*\pi)}_{DS}=0$. This contribution to an asymmetry will be denoted $\tilde{\cal A}_c$.
Results will now be presented for up-down photon asymmetries with respect to the $K\pi\pi$ decay plane, which we calculate separately for decays involving $K_1(1400)$ and $K_1(1270)$ resonant states. In addition to total asymmetries we will present asymmetries due to interference of type (a) in decays involving a final neutral pion, and due to interference of types (b) and (c) for decays involving a $\pi^+\pi^-$ pair. We point out that the total asymmetry in the latter decays is the sum $\tilde{\cal A}_{\rm total} = \tilde{\cal A}_b + \tilde{\cal A}_c$, in which $\tilde A_b$ and $\tilde A_c$ depend on both $\delta_{DS}^{(K^*\pi)}$ and $\alpha_S$.
Photon asymmetry due to $B \to K_1(1400)\gamma$
-----------------------------------------------
### $B^+\to K^0\pi^+\pi^0\gamma$ and $B^0\to K^+\pi^-\pi^0\gamma$
$\delta^{(K^*\pi)}_{DS}$(degrees) 0 45 90 135 180 225 270 315
----------------------------------- ------ ------ ------ ------ ------ ------ ------ ------
$\tilde{\cal A}_a$ 0.30 0.21 0.14 0.14 0.19 0.28 0.34 0.35
$\tilde{\cal A}_{\rm total}$ 0.30 0.21 0.15 0.14 0.20 0.29 0.35 0.36
: Up-down photon asymmetry $\tilde{\cal A}$ in $B^+\to K^0\pi^+\pi^0\gamma$ from intermediate $K_1(1400)$. The asymmetry $\tilde{\cal A}_a$ neglects a contribution of a $\rho K$ amplitude as described in the text. For the total asymmetry we use $\alpha_S=40^\circ$, a value favored by the analysis of [@Daum:1981hb].
\[tab.asym1400(2)\]
Table \[tab.asym1400(2)\] shows total asymmetries and asymmetries of type (a) calculated for a large range of phases $\delta_{DS}^{(K^*\pi)}$, assuming for the total asymmetry a value $\alpha_S=40^\circ$ favored by [@Daum:1981hb]. We note that in decays involving a final state $\pi^0$ the total asymmetry is completely dominated by interference of type (a) of two amplitudes for two $K^*\pi$ intermediate states and is therefore practically independent on $\alpha_S$. This follows from the dominance of the $K^*\pi$ mode and the negligible $K_1(1400)$ decay branching ratio into $K\rho$. The asymmetry $\tilde A_{\rm total}=0.30$ at $\delta_{DS}^{(K^*\pi)}=0$ is purely due to to an overlap of two equal strength (by isospin) Breit-Wigner $K^{*0}$ and $K^{*+}$ bands in the Dalitz plot. Using a value of $\delta_{DS}^{(K^*\pi)}$ around $260^\circ$, as indicated by the partial wave analysis performed in Ref. [@Daum:1981hb], one expects a slightly larger asymmetry of $34\%$ [@Gronau:2001ng; @Gronau:2002rz].
### $B^+ \to K^+\pi^+\pi^-\gamma$ and $B^0 \to K^0\pi^+\pi^-\gamma$.
$\delta^{(K^*\pi)}_{DS}$ (degrees) 0 45 90 135 180 225 270 315
------------------------------------ ------ ------- ------- ------- ------ ------ ------ ------
$\tilde{\cal A}_b$ 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.00
$\tilde{\cal A}_c$ 0 -0.07 -0.10 -0.07 0.0 0.07 0.10 0.07
$\tilde{\cal A}_{\rm total}$ 0 -0.07 -0.10 -0.07 0.01 0.08 0.11 0.07
: Up-down photon asymmetry $\tilde{\cal A}$ in $B^+\to K^+\pi^+\pi^-\gamma$ from intermediate $K_1(1400)$. Asymmetries $\tilde{\cal A}_b$ and $\tilde{\cal A}_c$ are defined in the text. The asymmetries are calculated for $\alpha_S=40^\circ$, a value favored by the analysis in [@Daum:1981hb].
\[tab.asym1400(3)\]
Table \[tab.asym1400(3)\] presents asymmetries of types (b) and (c) and total asymmetries for the same range of values of $\delta_{DS}^{(K^*\pi)}$ as in Table \[tab.asym1400(2)\], assuming $\alpha_S= 40^\circ$ as mentioned above and $|C_D^{(K^*\pi)}|/|C_S^{(K^*\pi)}| = 0.2$ (See Table \[tab.K1400\].) The total asymmetry is seen to be dominated by terms of type (c) due to interference of $S$ and $D$ wave amplitudes in $K_1(1400) \to K^*\pi$, while terms of type (b) originating in interference between $K^*\pi$ and $K\rho$ amplitudes are negligible. This can be traced back to the very small $K\rho$ branching ratio of $K_1$ decay which is completely dominated by $K^*\pi$. (See Table \[tab.K1400\].) While for arbitrary $\delta_{DS}^{(K^*\pi)}$ the total asymmetry may be positive or negative, it is predicted to be about $+10\%$ for $\delta_{DS}^{(K^*\pi)} \sim 260^\circ$ which is favored by the analysis in Ref. [@Daum:1981hb].
Photon asymmetry due to $B \to K_1(1270)\gamma$
-----------------------------------------------
### $B^+\to K^0\pi^+\pi^0\gamma$ and $B^0\to K^+\pi^-\pi^0\gamma$
$\delta^{(K^*\pi)}_{DS}$(degrees) 0 45 90 135 180 225 270 315
----------------------------------- ------- ------- ------- ------- ------- ------- ------- -------
$\tilde{\cal A}_a$ 0.02 -0.00 -0.04 -0.10 -0.05 0.08 0.06 0.04
$\tilde{\cal A}_{\rm total}$ -0.09 -0.10 -0.10 -0.10 -0.07 -0.07 -0.08 -0.09
: Up-down photon asymmetry in $B^+\to K^0\pi^+\pi^0\gamma$ from intermediate $K_1(1270)$. The asymmetry $\tilde{\cal A}_a$ neglects a contribution of a $\rho K$ amplitude as described in the text. For the total asymmetry we assume $\alpha_S=-40^\circ$, an approximate value obtained from Ref. [@Guler:2010if] by assuming $S$ wave dominance of $K_1(1270)$ decays to $K^*\pi$ and $K\rho$.
\[tab.asym1270(2)\]
Table \[tab.asym1270(2)\] shows total asymmetries and asymmetries of type (a) due to interference of amplitudes for two $K^*\pi$ intermediate states for $B^+\to K^0\pi^+\pi^0\gamma$ decays via $K_1(1270)$ as functions of $\delta_{DS}^{(K^*\pi)}$. The total asymmetry is predicted to lie in a narrow range between $-7\%$ and $-10\%$, considerably smaller than the corresponding asymmetry via $K_1(1400)$ given in Table \[tab.asym1400(2)\]. While the latter was shown to be positive the former is negative. Unlike the situation we encountered with $K_1(1400)$, the total asymmetry via $K_1(1270)$ is not dominated by interference of type (a). This can be traced back to the small branching ratio of $K_1(1270)$ decay into $K^*\pi$ relative to its considerably larger decay rate into $\rho K$.
### $B^+ \to K^+\pi^+\pi^-\gamma$ and $B^0 \to K^0\pi^+\pi^-\gamma$
$(\delta^{(K^*\pi)}_{DS}, \alpha_S)$ (degrees) (90,0) (270,270) (225,135) (30,30)
------------------------------------------------ -------- ----------- ----------- --------- -- -- -- --
$\tilde{\cal A}_b$ -0.05 -0.08 +0.12 -0.05
$\tilde{\cal A}_c$ -0.08 +0.08 +0.12 -0.02
$\tilde{\cal A}_{\rm total}$ -0.13 +0.00 +0.24 -0.07
: Up-down photon asymmetry in $B^+\to K^+\pi^+\pi^-\gamma$ from intermediate $K_1(1270)$ assuming $|C_D^{(K^*\pi)}|/|C_S^{(K^*\pi)}| = 1$. The asymmetries $\tilde{\cal A}_b$ and $\tilde{\cal A}_c$ are defined in the text. A full range of values for $\tilde{\cal A}_{\rm total}$ is calculated for selected values of $\delta_{DS}^{(K^*\pi)}$ and $\alpha_S$.
\[tab.asym1270(3)\]
$(\delta^{(K^*\pi)}_{DS}, \alpha_S)$ (degrees) (90,0) (270,270) (225,135) (30,30)
------------------------------------------------ -------- ----------- ----------- --------- -- -- -- --
$\tilde{\cal A}_b$ -0.05 -0.08 +0.09 -0.04
$\tilde{\cal A}_c$ -0.05 +0.05 +0.05 -0.02
$\tilde{\cal A}_{\rm total}$ -0.11 -0.03 +0.14 -0.06
: Up-down photon asymmetry in $B^+\to K^+\pi^+\pi^-\gamma$ from intermediate $K_1(1270)$ assuming $|C_D^{(K^*\pi)}|/|C_S^{(K^*\pi)}| = 0.2$. The asymmetries $\tilde{\cal A}_b$ and $\tilde{\cal A}_c$ are defined in the text. A full range of values for $\tilde{\cal A}_{\rm total}$ is calculated for selected values of $\delta_{DS}^{(K^*\pi)}$ and $\alpha_S$.
\[tab.asym1270(3b)\]
Table \[tab.asym1270(3)\] shows photon asymmetries calculated for $B^+\to K^+\pi^+\pi^-\gamma$ from intermediate $K_1(1270)$ assuming $|C_D^{(K^*\pi)}|/|C_S^{(K^*\pi)}| = 1$. In the absence of experimental information on $\delta_{DS}^{(K^*\pi)}$ and $\alpha_S$ we varied these two phases over their entire range of $0^\circ - 360^\circ$ searching for an overall range of $\tilde{\cal A}_{\rm total}$. The asymmetries presented in the table correspond to four cases: The largest positive and negative total asymmetries, $+24\%$ and $-13\%$, a vanishing total asymmetry (obtained also for other values of the two phases) and a fourth case involving arbitrarily chosen two phases of $30^\circ$ each. In Table \[tab.asym1270(3b)\] we present asymmetries assuming $|C_D^{(K^*\pi)}|/|C_S^{(K^*\pi)}| = 0.2$, calculated for the same four pairs of phases ($\delta_{DS}^{(K^*\pi)},\alpha_S)$ as in Table \[tab.asym1270(3)\]. We also found two extreme values of the total asymmetry, $+15\%$ and $-13\%$, obtained for phases $(270^\circ,135^\circ)$ and $(90^\circ,315^\circ)$, respectively, and asymmetries $\le 1\%$ obtained for other values including a continuum range $(90^\circ-135^\circ,90^\circ-135^\circ)$.
We conclude that without further phase information about $\alpha_S$ and $\delta_{DS}^{(K^*\pi)}$ the total asymmetry can have any value ranging from $-13\%$ to $+24\%$. Typical contributions of asymmetries of types (b) and (c) have comparable magnitudes which may enhance or cancel each other in the total asymmetry. Comparing the entries in Tables \[tab.asym1270(3)\] and \[tab.asym1270(3b)\] shows that for certain phases the value of $|C_D^{(K^*\pi)}|/|C_S^{(K^*\pi)}|$ may have a significant effect on the photon asymmetry.
Isospin symmetry in $B\to K\pi\pi\gamma$
========================================
We have pointed out that the two pairs of strong decay amplitudes of $K_1^+ \to K\pi\pi$ and $K_1^0\to K\pi\pi$ in Eqs.(\[type3\]) and (\[type2\]), for final states related by isospin reflection $u \leftrightarrow d$, are equal in the isospin symmetry limit. Does a similar relation hold approximately for corresponding weak decay amplitudes $B^+ \to K\pi\pi\gamma$ and $B^0\to K\pi\pi\gamma$?
Isospin breaking in radiative $B$ meson decays has been studied in the literature and found to be small. For two recent brief reviews including theoretical and experimental references see Ref. [@Jung:2015yma; @Paz:2017wow]. Isospin asymmetry at a level of $5\%$, consistent with zero at $2\sigma$, was measured by the Belle and Babar collaborations for $B\to K^*\gamma$ [@Olive:2016xmw; @Nakao:2004th; @Aubert:2009ak], A\_I(BK\^\*) = 0.052 0.026 . Isospin breaking in inclusive radiative decays $B \to X_s\gamma$ is expected to be further suppressed and has been measured at this level by Babar [@Aubert:2005cua], A\_I(B X\_s) = -0.006 0.058 0.009 0.024 .
Thus one may assume that the following two approximate isospin equalities hold at a few percent level also for radiative decays to the $K_1$ resonances: \[B+B0\] A(B\^+ K\_1\^+K\^+\^+\^-) & & A(B\^0 K\_1\^0K\^0\^-\^+) ,\
A(B\^+ K\_1\^+K\^0\^+\^0) & & A(B\^0 K\_1\^0K\^+\^-\^0) . At this level of approximation one may therefore study the photon polarization by combining data for charged and neutral $B$ decays. This should double the statistics. One must pay some attention to the definition of an up-down asymmetry for these two pairs of processes by considering the isospin reflection, $u \leftrightarrow d$, which relates the final kaon and two pions in $B^+$ decays to corresponding final mesons in $B^0$ decays.
Conclusion
==========
In this paper we reexamined, updated and extended a suggestion made fifteen years ago to measure the photon polarization in $b \to s\gamma$ by observing in $B \to K\pi\pi\gamma$ an asymmetry of the photon with respect to the $K\pi\pi$ plane. Asymmetries were calculated for different charged final states due to intermediate $K_1(1400)$ and $K_1(1270)$ resonant states. Three interference mechanisms were identified playing different roles in decays involving these two kaon resonances.
- The situation is quite simple in decays via $K_1(1400)$, for which an upper bound ${\cal B}(B \to K_1(1400)\gamma) < (1.2-1.5)\times 10^{-5}$ has been measured using less than $20\%$ of the Belle total data sample involving $\pi^+\pi^-$. As $K_1(1400)$ is dominated by $K^*\pi$ decays, the total symmetry in decays involving a final state $\pi^0$ is large and positive favoring values around $30\%$ from an overlap of two Breit-Wigner $K^*$ bands of equal strength. The asymmetry in decays involving a final state $\pi^+\pi^-$ pair, dominated by interference of $S$ and $D$ wave amplitudes in $K_1(1400) \to K^*\pi$, is considerably smaller favoring a value around $10\%$. As these asymmetries show some dependence on the phase $\delta^{(K*\pi)}_{DS}$ between $S$ and $D$ wave amplitudes in $K_1(1400) \to K^*\pi$ which has only been measured in [@Daum:1981hb], an independent measurement of this phase in dedicated amplitude analyses of $B\to J/\psi(\psi') K\pi\pi$ decays would be useful.
- The situation is considerably more involved in decays via $K_1(1270)$, for which a branching ration ${\cal B}(B \to K_1(1270)\gamma) \sim 4\times 10^{-5}$ has been measured. There are two reasons for this situation. First, the $K_1(1270)$ decays more frequently to $K\rho$ than to $K^*\pi$, for which the branching ratio is only around $20\%$. Consequently, the total asymmetry in decays involving a final state $\pi^0$ is not dominated by interference of two intermediate $K^*\pi$ states. A second reason for being unable to predict an asymmetry in decays involving an intermediate $K_1(1270)$ is lack of information about final state interaction phases in its decays to $K^*\pi$ and $K\rho$. Assuming $S$ wave dominance of $K_1(1270)$ decays to $K^*\pi$ and $K\rho$ an analysis in [@Guler:2010if] implies a value $\alpha_S \sim -40^\circ$ for the relative phase between these two amplitudes. Using this value of $\alpha_S$ the asymmetry is predicted to be negative and at most $-10\%$ for a final state involving a $\pi^0$.
The situation in decays via $K_1(1270)$ involving a final state $\pi^+\pi^-$ pair is more uncertain because there is no information about the two relevant phases, $\alpha_S$ and $\delta_{DS}^{(K^*\pi)}$, and there exists only a crude measurement of $|C_D^{(K^*\pi)}|/|C_S^{(K^*\pi)}|$. Varying these phases over their entire range of $0^\circ - 360^\circ$ we calculated total asymmetries between $-13\%$ and $+24\%$, depending to some extent on the ratio of $D$ and $S$ amplitudes. The asymmetry obtains comparable contributions from interference of $K^*\pi$ and $K\rho$ amplitudes and interference of $S$ and $D$ wave $K^*\pi$ amplitudes, which may act constructively or destructively with respect to one another. Major progress in predicting these asymmetries would be achieved by measuring the phases $\alpha_S$ and $\delta_{DS}^{(K^*\pi)}$ and improving the current measurement of the $D$ to $S$ ratio in $K_1(1270)\to K^*\pi$. This could be achieved in dedicated amplitude analyses of $B \to J/\psi(\psi') K\pi\pi$ decays to be performed in the future by the Belle II Collaboration at SuperKEKB [@Abe:2010gxa; @Aushev:2010bq].
Finally, in order to increase statistics in studies of the photon polarization, we suggest using approximate isospin symmetry (\[B+B0\]) for combining in the same analysis $B\to K\pi\pi\gamma$ decays for charged and neutral $B$ mesons. So far the Belle collaboration used less than $20\%$ of their total data sample to obtain the branching ratio (\[BRK1(1270)\]) for $B^+ \to K^+_1(1270)\gamma$ and the separate upper bounds (\[BRup(1400)\]) on $B^+ \to K^+_1(1400)\gamma$ and $B^0\to K^0_1(1400)\gamma$ for final states involving $\pi^+\pi^-$ [@Yang:2004as]. We urge the Belle collaboration to combine $B^+$ and $B^0$ decays when analyzing their full data sample for these decays, and to study also final states including a $\pi^0$ in combined $B^+$ and $B^0$ decay samples.
We wish to thank Karim Trabelsi for asking very useful questions which motivated this work and Jonathan Rosner for helpful correspondence.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Gabriele Croci (University of Milano-Bicocca, INFN & CNR),\
Fabrizio Murtas (INFN & CERN),\
Filippo Resnati (CERN)\
\
Organising committee of the Academia-Industry Matching Event\
A. Breskin (Weizmann Institute), A. Delbart (CEA),\
S. Duarte Pinto (CERN), I. Giomataris (CEA),\
B. Guerrard (ILL), R. Hall-Wilton (ESS),\
J. Le Goff (CERN), F. Murtas (INFN & CERN),\
A. Pacheco (CERN), L. Ropelewski (CERN),\
M. Titov (CEA), T. Tsarfati (CERN)
title: |
RD51-NOTE-2015-012\
\
\
Summary of RD-51 Academia-Industry Matching Event\
Second Special Workshop on Neutron Detection with MPGDs
---
Introduction
============
The aim of this document is to summarise the discussion and the contributions from the 2nd Academia-Industry Matching Event on Detecting Neutrons with MPGDs [@neutronEvent2] which took place at CERN on the $16^{th}$ and the $17^{th}$ of March 2015. The first event of this kind [@neutronEvent1], organised in 2013, was summarised in [@summary1]. These events provide a platform for discussing the prospects of Micro-Pattern Gaseous Detectors (MPGDs) [@MPGD] for thermal and fast neutron detection, commercial constraints and possible solutions. The aim is to foster the collaboration between the particle physics community, the neutron detector users, instrument scientists and fabricants. This document is not meant to be a comprehensive review of the neutron detection with gaseous detectors, instead it is an addendum and a continuation of the previous summary.
Very good position resolution, high particle flux capability, radiation tolerance, low material budget, large surfaces and low energy threshold are the key features which make MPGDs flexible and widespread devices in High Energy Physics experiments. These features make them interesting solutions also for the next generation neutron scattering instruments and beam monitors. The development of *non-standard* neutron detectors, possibly based on MPGDs, is important not only because of the $^{3}$He shortage, which forces to find urgently valuable alternatives, but also to extend the capabilities of the actual detectors (e.g. very high particle flux capabilities).
MPGDs are gas-based particle detectors that extend the capabilities of the Multi-Wire Proportional Chambers, even them largely used by the High Energy Physics and neutron scattering communities. In MPGDs the electrodes to shape the electric field in the gas volume are printed with photolithographic techniques on solid substrates. Due to the presence of *cathode* electrodes in the vicinity of the amplification region, MPGDs features a fast evacuation of ions, and therefore very good capabilities to cope with high particle fluxes. The electric field is modestly affected by spurious deposits on the electrodes, reducing very much the *classical* ageing. The readout electrodes, placed on solid and often rigid substrate, allow position resolutions of the order of tens of micrometer. Since 1988, when the first detector of this kind was invented, the Micro-Strip Gas Chamber (MSGC) [@Oed:1988], several MPGDs families have been developed. The two most common are based on MicroMeGas [@Giomataris:1996] and GEM [@Sauli:1997] solutions.
RD51 is the world’s largest collaboration for the development and application of Micro-Pattern Gaseous Detectors with the aim of facilitating the development of advanced gas-avalanche detector technologies and associated electronic readout systems, for applications in basic and applied research. In this context, RD51 organised a series of Academia-Industry matching events, in order to advertise the capabilities of MPGDs, and to bring out the issues and the constraints for possible MPGDs applications outside the fundamental research.
Neutrons are widely used in modern matter physics in a variety of applications. From the very beginning, thermal neutron diffraction has been used to study crystals and their properties, exploiting the fact that neutrons have a higher penetration in matter than X-rays, of which they constitute a natural complementary probe. Neutrons can penetrate samples of large thickness (many cm) to reveal details of the crystal structure of the bulk of the specimen. On the other hand, neutrons are not stopped by the walls of sample containers, thus allowing testing in high-vacuum or high-pressure conditions. Moreover, thermal neutrons are widely used to test molecular dynamics, in a way similar to Raman or IR measurements, with the additional advantage of not being subject to EM selection rules and being more sensitive to light elements. Finally, more recent applications of neutrons as a probe for matter physics are neutron radiography and tomography, neutron Spin-Echo and Neutron Resonance Capture Analysis. Many neutron sources devoted to matter physics are nowadays available throughout the world, both steady-state (i. e. reactor-based) or pulsed (i. e. accelerator driven). Exhaustive and comprehensive documents are [@Squires; @Windsor].
This document targets both instrument scientists of beam lines at neutron sources and researcher that need to develop a high rate, large area, low cost fast neutron diagnostic devices for all neutron-linked physics applications (plasma physics, medical physics, space physics, etc.).
Thermal neutron detectors
=========================
Thermal neutrons are used as a probe to investigate the matter properties because they are non-damaging and highly penetrating. Neutron scattering instruments can measure at the same time both energy and momentum transfer. Unlike the X-rays, the neutrons scatter against the nuclei of the atoms. It happens that the total neutron cross-sections are *complementary* to the one of X-rays: Neutrons interact also with very light atoms, and mainly with hydrogen. Neutron elastic scattering brings information about the position of the atoms, while inelastic scattering gives insights of the atoms dynamics. Though not the only production methods, fission (in a reactor) and spallation (in a proton accelerator) are the two most common, efficient and effective ways to produce high intensity neutron beams. The detail discussion of neutron production, thermalisation and transport is beyond the scope of this document.
![Neutron capture cross-sections as a function of the neutron energy for different nuclei.[]{data-label="fig:CrossSection"}](crossSections.jpg){width="0.8\linewidth"}
Thermal neutrons are not energetic enough to be directly detectable via elastic scattering. The detection principle is based on the absorption and the conversion of the neutron into charged particles via nuclear reactions. The most exploited, for which the cross-sections as a function of the neutron energy are shown in figure \[fig:CrossSection\], reactions are: $^{3}$He(n,p)$^{3}$H, $^{6}$Li(n,$\alpha$)$^{3}$H, $^{10}$B(n,$\alpha$)$^{7}$Li, $^{X}$Gd(n,e$^{-}$)$^{X+1}$Gd$^{+}$, $^{235}$U(n,f). The converter (in the form of fluid or solid) can either be the active material of the detector, as in the case of $^{3}$He tubes, or simply the target from where the secondary charged particles are emitted. In MPGDs the typical converter is a set of thin foils usually evaporated or sputtered with the sensitive atoms onto a thin aluminium substrate. The material, the thickness and the geometry of the converter is optimised for the needed sensitivity to the given energies of the neutrons.
In general within a detector, a neutron may do not interact at all, be scattered and potentially be detected far from the first interaction point, be absorbed, but not detected, or be properly detected. This gives rise to a very complex detector behaviour that can usually be summarised with quantities such as efficiency, spatial resolution, time resolution, high particle flux capability, radiation tolerance, gamma sensitivity, background discrimination, time stability, and ageing.
In the next paragraphs we summarise the contributions concerning the thermal neutron detection according to what in our believe are the most important discussion topics.
Simulations
-----------
Deterministic and Monte Carlo (MC) simulations are valuable and *costless* tools to address the research toward new neutron detectors. MC methods are suited to study stochastic processes, and in particular the transport of radiation through matter. The accuracy and reliability of MC predictions depends on the model and the input data libraries. Therefore, simple experiments must be setup to benchmark their performances.
In the contribution of Lina Quintieri [@Quintieri_talk], the thickness of LiF and $^{10}$B coating on a single layer and a multi layer cathode was optimised using GEANT4 [@GEANT4] and FLUKA [@FLUKA] simulation packages for the GEM *side-on* [@SideOn] detector. The two simulations give similar results, and the $^{10}$B behaviour is well in agreement with the experimental data. The optimal thickness of the $^{10}$B is found to be 1.8 $\mu$m for five layers of $^{10}$B converter.
![Simulated detection efficiency for different Gd-based converters.[]{data-label="fig:GdSim"}](GdSim.png){width="0.8\linewidth"}
$^{6}$Li and $^{10}$B are not the only converter material studied in simulations. Dorothea Pfeiffer [@Pfeiffer_talk] simulated with GEANT4 the performance of cathodes singly and doubly coated with natural Gadolinium, $^{155}$Gd, $^{157}$Gd, Gd$_2$O$_3$ and enriched Gd$_2$O$_3$. As shown in figure \[fig:GdSim\], 9 $\mu$m of $^{155}$Gd has a conversion efficiency up to a very promising 44% for 25 meV neutrons. The simulation considers also the energy distribution of the conversion electrons in several thicknesses of gas. GEANT4 is interfaced with GARFIELD++ [@GARFIELD++] for the simulation of the transport and the amplification of the primary ionisation electrons.
Coating
-------
Solid converters deposited on adequate substrates are the most suitable solutions for MPGD based neutron detectors. B$_4$C is one of the most popular materials, that can be enriched with $^{10}$B. In order to further increase the detection efficiency, efforts are ongoing to deposit pure metal boron films [@Pietropaolo_talk]. A collaboration between ENEA, INFN, Columbus Superconductors SpA [@Columbus] and RHP Technology [@RHP] resulted in tests performed with an evaporation system at ENEA Frascati, a radio-frequency sputtering at ENEA Casaccia, and High Power Impulse Magnetron Sputtering at INFN LNF. The coatings were successfully used in the multi-layer cathode of the GEM *side-on* detector, to reach efficiencies as high as 31% for 5 meV neutrons [@Claps].
Some of the ESS neutron scattering instruments will require very large converter surfaces. The more common $^{10}$B$_4$C, in this case, seems preferable. Nevertheless, the ability to mass-produce with the acceptable quality and speed hundreds of m$^2$ must be proven [@Robinson_talk]. Since the Summer 2014, the ESS detector coatings workshop in Linköping is equipped with a 4 magnetrons sputtering machine from CemeCon [@Cemecon], able to coat at the same time several large ($50\times10$cm$^2$) targets. 27000 blades (more than 100 m$^2$) for a full IN5 sector demonstrator [@Khaplanov] were coated with $^{10}$B$_4$C in 45 days, proving the mass production capabilities of the ESS coatings workshop in Linköping. $10^{14}$ n/cm$^2$ on a 1 $\mu$m thick film are proven not to influence on adhesion, composition, morphology, and structure. B$_4$C can be properly transferred to several substrates like aluminium, stainless steel, alumina, silicon, copper and Kapton. Difficult substrate, where the adhesion is not optimal and more R&D is needed are glass, Teflon, nickel and MgO.
Efficiency
----------
![Comparison of a diffractogram of a bronze sample from a boron-GEM and an $^3$He tube. The total measurement time is about 20 h. The active area of the GEM detector is about $5\times10$ cm$^2$, while the active area of the $^3$He tube is about $2.5\times15$ cm$^2$.[]{data-label="fig:tof"}](tof.png){width="0.8\linewidth"}
The first diffractogram recorded with a GEM based detector was reported [@Croci_talk]. A *low efficiency* B$_4$C coated aluminium cathode was used for this measurement at INES-ISIS. The efficiency was increased of about a factor of three with a $^{10}$B enriched boron carbide cathode. The diffractogram of a bronze sample is shown in figure \[fig:tof\] in comparison to the one recorded with an $^3$He tube. The poorer peak resolution is mainly attributed to the lack of a neutron collimation in front of the detector. The resolution was improved summing the spectra from pads that lie on the same Debey-Scherrer cone. In order to further increase the detection efficiency, new cathode geometries are under study. In particular, the new BAND-GEM detector, with B$_4$C-coated *lamellas* that resample the field shaping electrodes of a TPCs, was tested at the RD2D beam-line at IFE, demonstrating about 15% efficiency for neutrons with energies of 34.5 meV.
![Beam profile from a Gd-GEM at RD2D beam-line at IFE. The band structure on the sides are intentional dis-homogeneities in the cathode.[]{data-label="fig:scatter"}](scatter.png){width="0.8\linewidth"}
The Neutron Macromolecular Crystallography (NMX) instrument at ESS requires detectors with efficiency larger than 20%, spatial resolution of about 200 $\mu$m, capabilities to cope with very high local neutron fluxes and a wide range of incident neutron angles. This seems to imply that the neutron capture must occur in a single conversion layer. Li and B based converters are not enough efficient, instead gadolinium, despite other complications, seems promising. For the first time, a triple GEM was coupled to a 250 $\mu$m natural Gd cathode and exposed to a neutron beam [@Pfeiffer_talk] at RD2D beam-line at IFE. The detection efficiency was estimated to be about 10%, smaller than the simulated one, but with large systematic uncertainties. The profile of a square beam is shown in figure \[fig:scatter\]. The spatial resolution is estimated from the sharp edge marked in red. The TPC analysis described in the next paragraph improves the spatial resolution by a factor of three. The spatial resolution was estimated to be of the order of 1 mm, dominated by the neutron scattering on the materials in front of the detector. In spite of the fact that gadolinium-based detectors are more sensitive to gamma, the signal to background ratio in the RD2D beam-line conditions was 100:1. Boron-based detectors, depending on the gain, can reach gamma rejections of the order of $10^-7$ [@Khaplanov:2013].
Space resolution
----------------
![Two views of a $^7$Li or $\alpha$ track from a neutron conversion in $^{10}$B. The point marked with the circle is the extrapolated point of the neutron capture.[]{data-label="fig:events"}](event.png){width="0.8\linewidth"}
In the case of $^{10}$B and Gd, the typical range of the secondary charged particles in gas is several millimetres, apparently in contrast to some ESS instruments requirements of sub-millimetre position resolution. A single GEM detector with a $^{10}$B$_4$C coated cathode and an anode segmented into two orthogonal sets of 400 $\mu$m wide strips demonstrated position resolutions on the two views better than 200 $\mu$m [@Resnati_talk; @Pfeiffer]. The detector is used as an imaging device, profiting from an analysis based on the concept of the Time Projection Chamber (TPC). The neutron capture point in the converter is extrapolated on an event by event basis from the $\alpha$ and $^7$Li tracks (see figure \[fig:events\]). With a resistive MicroMeGas, similar results are obtained on one strip view. The other one is affected by the electrons evacuation through the resistive layer, and the signal shapes are not suitable for the TPC analysis. With respect to the *center of gravity* approach, the TPC analysis improves the position resolution of a factor of 5. Since the analysis is independent of the detector details, it can be exploited in several occasions. A similar analysis significantly improves the position resolution on Gd-based detectors [@Pfeiffer_talk]. Given the curly trajectories of tens keV electrons from the Gd neutron conversion, a refined analysis can improve even more significantly the Gd-GEM performance.
![Closeup of the holes of the micro-bulk MicroMeGas. The electrode is segmented into strips. In transparency one can notice the orthogonal segmentation of the other electrode.[]{data-label="fig:strips"}](strips.png){width="0.8\linewidth"}
There are circumstances where severe constraints drive the entire design of the detector. Extremely low *material budgets* are necessary when the detector is installed in the neutron beam as profiler [@Diakaki_talk]. Detection efficiency is clearly not an issue, position resolutions of about 1 mm, easily achievable in other conditions, become a challenge when almost no metal is allowed in the detector. A micro-bulk MicroMeGas with orthogonally segmented anode and *mesh* is tested in a neutron beam at GELINA FACILITY IRMM-GEEL [@GEEL]. The 1 mm wide strips on the anode and the mesh provide two dimensional readout, without an addition of a dedicated 2D anode. The picture in figure \[fig:strips\] shows a closeup of the segmented electrode of the MicroMeGas. In transparency the orthogonal strips can be noticed.
Rate capability
---------------
MPGDs were invented to improve the position resolution and the particle flux capability of the MWPCs. In particular, GEMs at gains of few thousands are known to well cope with fluxes of the order of 1 MHz/mm$^2$ of minimum ionising particles crossing 3 mm drift region. The high neutron flux capability of a *standard* triple GEM with the cathode coated with B$_4$C was tested at the G3-2 irradiation station at the ORPHEE reactor (LLB-Saclay). The saturation of the measured rate above 10 MHz/cm$^2$ is due to the electronics dead time, while the detector gain is independent of the particle flux in these ranges [@Croci_talk].
Fast Neutron Detectors
======================
In order to detect fast neutrons (E $>$ 1 keV), different techniques have been used, but most of them are based on the usage of hydrogenated materials that can either moderate fast neutrons down to the thermal levels, or produce charged particles by elastic scattering that can ionise the active gas of different kind of MPGDs. Both GEM and $\mu$Megas detectors have been developed to detect fast neutrons in several applications linked to beam monitoring, neutron cross-section measuring and total dose estimations. Applications related both to spallation sources, fusion reactors and medical field profited from the introduction of Micro-patterned gaseous detector to improve instrumentation performance.
The properties of MPGD based fast neutron beam monitors can be summarised as follows:
- Neutrons are converted through nuclear reactions on
- A hydrogenated material (such as polyethylene or polypropylene),
- A material containing boron or lithium (after thermalisation),
- A uranium ($^{235}$U) sheet,
- A gas containing helium or hydrogen.
- Minimal perturbation of neutron beam,
- Minimal introduction of background,
- Capability of reconstructing the 2D shape of the beam,
- Capability of on-line measuring the neutron flux also up to very high rates,
- Efficiency from 10$^{-2}$ to 10$^{-7}$,
- Space resolution from few mm to cm,
- Time resolution of few ns,
- Gamma ray rejection from 10$^{-6}$ to 10$^{-7}$.
Micromegas detectors coupled with $^{10}$B or $^{235}$U cathodes have been used at nTOF [@nTOF] since 2001 to monitor the beam profile. The device, with an active area of $6\times6$ cm$^2$ is composed by several $\mu$bulk Micromegas inserted in an aluminium gas chamber together with converter samples.
New tracking X-Y Micromegas beam monitors based on the $\mu$bulk technique have also been developed giving the possibility to improve space resolution through the track reconstruction algorithm. These detectors (see Figure \[mm1\]) installed on the nTOF beam pipe are used to measure both the neutron beam profile (see Figure \[mm2\]) and the neutron flux. These detectors have a very low material budget and thus do not introduce any perturbation in the beam itself [@Diakaki_talk].
![Beam profile reconstructed through X-Y $\mu$Megas.[]{data-label="mm2"}](muMeg_nTOF-mod.pdf)
![Beam profile reconstructed through X-Y $\mu$Megas.[]{data-label="mm2"}](muMegas_BeamProf-mod.pdf)
This kind of detector have been used also to measure fission cross-sections (n,f) and (n,charged particles) of materials like $^{235}$U, $^{240}$Pu, $^{242}$Pu and $^{33}$S. The detector has been recently improved with the use of a 4 pads-anode instead of a standard X-Y strips that allows reducing the capacitance and increasing the counting rate capability and the S/N ratio (see Figure \[mm3\]). This kind of chamber has been already used to measure the neutron flux (see Figure \[mm4\]). The active area of the device is still a circle with diameter of 6 cm [@talk_mm2].
![Picture of the 4 pads Micromegas.[]{data-label="mm3"}](muMeg_4Pads_Photo-mod.pdf)
![nTOF neutron flux reconstructed with the 4 pads Micromegas[]{data-label="mm4"}](muMeg_EaR1_4Pads_Flux-mod.pdf)
Other Micromegas devices have been developed as portable and directional fast neutron detectors. This devices are able to detect neutrons from few keV up to several MeV by exploiting He based gas mixture and measuring the recoil atoms produced by a neutron interaction. The active area covered by this portable devices is 100 cm$^2$ [@talk_mm3]
GEM-based fast neutron detectors have also been presented for applications related to spallation sources, to diagnostic of neutral beam injector facilities for future fusion reactors and to medical applications. Several GEM detectors of different sizes (from $10\times10$ cm$^2$ up to $20\times35$ cm$^2$ [@gab1; @gab2] ) have been used to measure the nTOF beam properties (both energy spectrum and profile). These chambers are provided with different kind of cathodes (both polyethylene, polypropylene and borated) and have different sizes. 2D beam images as well as neutron energy spectra have been measured by exploiting the neutron time of flight technique and show that the thermal and fast neutron components of the beam have different shapes. The space resolution achievable with these devices depends upon the pad dimension and ranges from 1 mm up to 10 mm. Recent development is the realisation of a medium size area GEM based spectrometer equipped with a cathode subdivided in different areas that are sensitive to different neutron energies (see Figure \[gem1\]. Areas have all different shapes and some of them are borated and if covered with polyethylene slabs work like a planar Bonner-sphere [@talk_gem1].
![GEM-based neutron spectrometer and its event display.[]{data-label="gem1"}](Spectrometer.pdf){width="0.95\linewidth"}
Another medium-size ($35\times20$ cm$^2$) GEM detector has been developed for applications related to thermonuclear fusion in a facility called SPIDER that represents the prototype of Neutral beam injector for ITER. The neutron production due to fusion reactions between beam deuterons and deuterons implanted in the SPIDER dump will be a few times 10$^{15}$ neutrons per SPIDER pulse. The neutron source intensity is suitable for diagnostic applications. A neutron diagnostic was designed in order to provide the map of the beam intensity with a spatial resolution approaching the size of individual beamlets is described. The proposed detection system is called Close-contact Neutron Emission Surface Mapping (CNESM) and it is placed right behind the beam dump as close as possible to the neutron emitting surface (around 30 cm). The detectors employed in this diagnostic system are nGEM chambers [@nGEM]. This device was tested at the ISIS spallation source and shows same performance as small area ($10\times10$ cm$^2$) detectors [@gab2]. Figure \[gem2\] shows the reconstructed ISIS beam profile. Its uniformity of response all over the active area is higher than 90% proving that larger area GEM based fast neutron detectors can be realised. This kind of chambers can be also used as beam monitors for fast neutron lines at spallation source (like ChipIR at ISIS) or for other irradiation lines [@talk_gem2]..
![Beam Profile reconstruction using the $35\times20$ cm$^2$ nGEM detector for SPIDER. The spatial resolution achieved in this case is about 10 mm due to the pad size (13 mm $\times$ 22 mm).[]{data-label="gem2"}](nGEM-Spider-BP-mod.pdf)
Finally a medical application related to hadron-therapy was presented. The aim is the measurement of the neutron flux due to the proton interaction within the patient body during the treatment: due to spallation reaction in the body, a very high neutron flux: For example for a proton beam of 172 MeV/u, an average differential neutron flux $\Phi_E$ of 10$^4$ $[(MeV/u \times cm^2)^{-1} Gy^{-1}]$ is produced. A GEM based device (MONDO) coupled with plastic scintillators and with a CMOS camera is being designed to measure the neutron flux and the direction of the produced neutrons. This device convert neutrons from 20 to 200 MeV into charged particles that produce scintillation lights in the fibre that is subsequently converted into photo-electrons by CsI coated GEMs. The photo-electrons are then further multiplied and generate secondary light that can be detected by a very high resolution commercial CMOS photon camera. Figure \[gem4\] shows a schematic of the system and gives more details about the camera. One of the most important features is that with such a scheme the electronics is kept far away from the neutron beam thus increasing its survival time. Fluka simulations show that this system gives the possibility to measure the neutron flux and to understand the direction of the neutrons [@talk_gem3].
![Schematic of the MONDO device.[]{data-label="gem4"}](Mondo-Scheme-mod.pdf)
Coupling MPGD detectors to CMOS read-out gives the possibility to realise very high resolution detectors. A quad-Timepix 2 chip was used as anodic readout of a Triple GEM detector and this device was called GEMPIX [@Murtas]. The Timepix 2 chip has an active area of $1.4\times1.4$ cm$^2$ and is composed by $254\times254$ pixels each with an area of 55 $\mu$m $\times$ 55 $\mu$m. The GEMPix detector (picture in figure \[fig:GEMPix\]) has an active area of $28\times28$ mm$^2$ since it is read-out by a quad-Timepix (4 chips). Such a high spatial resolution implies the capability of recognising every single interacted track giving the possibility to operate the particle identification by exploiting the track morphology. This opens the way to a so called microscopic-analysis that improves the reconstruction of macroscopic features like beam profile. Capability of standard GEMPix detector under MIPs irradiation were presented as well as a preliminary measurement in a mixed neutron field. By exploiting either the TOT or the TOA both 2D maps as well as energy spectra were reconstructed. This kind of chip can be also used coupled to standard silicon detectors that proved to be able to reconstruct the nTOF beam profile as well as to measure the neutron flux and the neutron energy spectrum [@talk_gem4]. All the results shown during the workshop prove that micro pattern Gaseous Detectors offer very high performances for fast neutron detection in different applications field giving the possibility to improve already existing technology or to explore new areas.
![Image of a GEMPix detector.[]{data-label="fig:GEMPix"}](GEMPix.jpg)
Electronics
===========
Several front end electronics have been used coupled to MPDG-based neutron detectors but, at the moment, they are all derived or adapted from electronics developed either for silicon or Multi Wire Proportional Chambers, so none of them has been especially developed for MPGDs. Nevertheless, some of them have proven to give reliable results also when applied to neutron detectors based on MPGDs. Table \[table\] shows some of the electronics chips that have been recently used coupled with MPGDs. Standard requirement for electronics to be used for neutron measurement is radiation hardness, since it is known that neutrons can be harmful for chips since they induce Single Event Upsets (SEU) that can lead even to the disruption of the chip itself. The RD51 collaboration [@MPGD] has recently developed a system called SRS (Scalable Readout System) that is meant to be a base DAQ to be interfaced with different kind of Front-End Chips. Up to now the only chip that has been used with the SRS is the APV25 analog chip, that was also used for the read-out of $\mu$TPC thermal neutron chamber and of Gadolinium GEM detectors (both presented during this workshop [@Pfeiffer_talk; @Resnati_talk]. Although APV25 coupled to SRS does not give the possibility to acquire at high rates (it is limited at order of kHz), its feature of being bunched in 25 ns gates can be beneficial for very precise ToF measurements of fast neutrons. For example, if it is triggered by the proton beam T0 (that is when the beam impinges on a spallation target) it can be used to improve the fast neutron energy spectrum measurements that have been shown during the workshop. Other chips that will be coupled to the SRS in the near future are the GEMROK and the VMM2 (see the table for details).
[|l|l|l|l|l|l|l|l|l|l|l|l|]{} Name & Costumer & Maker & Type & Chan & Shaping &
-----
Det
Cap
-----
&
---------
Sens
(mV/fC)
---------
& Noise &
-------
Time
stamp
-------
&
------
Max
Rate
------
&
------
Rad
Hard
------
\
Carioca &
--------
LhCb
Muon
(MWPC)
--------
&
------
CERN
INFN
------
&
--------
Discr.
A-S-D
MWPC
--------
& 8 & 15 ns & <220pF & 10 &
----------
600e
(100 pF)
----------
& N/A &
--------
10
MHz/ch
--------
&
------
Up
to 1
Mrad
------
\
-------------
GEMROK
(MSGCROK
derivative)
-------------
&
----------
DETNI
Neutrons
----------
&
------------
Krakow
Heidelberg
HMI Berlin
------------
&
-------------
A-2S-PD-A-T
MSGC
GEM
Peak
Analog
Pipe
Prompt
disc TS
-------------
& 32 &
----------
Dual
25/85 ns
----------
& aim 25 pF & 2 &
-------
aim
200 e
-------
&
-------
12-14
bit
8 ns
-------
& N/A & N/A\
APV25 &
------------
CMS
Tracker
(Si Strip)
------------
& RAL &
---------
Analog
Pipe
A-S-BUF
---------
& 128 &
---------
50 ns +
fast
decon
---------
& low & 1 mA/fC &
--------
2000 e
430+61
/pF
--------
& LHC &
--------
100 Hz
with
SRS
--------
&\
VFAT & CMS &
------
CEA
INFN
CERN
------
&
-------
Disc
Pipe
A-S-
D-BUF
-------
& 128 & 25-400 ns & 5-80 pF & 1-50 &
--------
1000 +
40/pF
--------
& LHC & N/A & N/A\
CBC & CMS Si Strip & RAL &
--------
A-S-D
Analog
prot
--------
& 128 & 20 ns & 3-6 pF & 50 &
---------
1000 e
at 10pF
---------
&
------
Pipe
Only
------
& N/A & N/A\
VMM &
-----------------
Atlas
Muon Micromegas
Thin
gap
-----------------
& BNL &
--------
A-S-D-
PA-TOT
--------
& 64 & 25-200 ns & wide & 1-9 &
---------
1000 e
at 10pF
---------
& N/A & N/A & N/A\
VMM2 &
-----------------
Atlas
Muon Micromegas
Thin gap
-----------------
& BNL &
---------
+6bit
peak
FADC
10 bit
slowADC
---------
& 64 & 25-200 ns & wide & 1-9 &
---------
1000 e
at 10pF
---------
& N/A & N/A & N/A\
TIMEPIX2 &
------------
GEM
Micromegas
------------
& CERN & & & & & & & & &\
Other chips that have extensively been used both with thermal and fast neutron detectors are the CARIOCA-GEM digital and self-triggered chips [@CARIOCA]. These chips (each with 8 channels) were realised in order to equip the LHCB-GEMs and have recently found several new applications. As presented during the workshop [@Croci_talk], these chips have a maximum count rate capability per channel of about 10 MHz and have been used coupled with a custom made LNF-INFN FPGA motherboard. The system of CARIOCA+FPGA gives the possibility both to reconstruct the 2D map of a padded GEM detector and to perform TOF measurement either by exploiting the internal feature of movable gate of the FPGA. In addition if the Logical Signal coming from the CARIOCA chip is properly adapted it can be interfaced to already existing DAQ system, such as the DAE of the ISIS facility [@DAE]. New chips called GEMINI are being developed by INFN (MIB and LNF sections) in order to substitute and improve the CARIOCA performances. Another very interesting chip that was presented during the workshop was the AGET+ASAD electronics. This chip was tested in combination with MicroMegas detectors and is an analog and auto-triggered chip featuring 64 channels, a sampling rate of 100 MHz and a dynamic range from 120 fC to 10 pC. In addition through this chip it is possible to supply high voltage. This chip was used both to get signals from grounded anode strips and from mesh strips put at high voltages. These gave the possibility to perform time-of-flight measurements at neutron sources.
Conclusions
===========
During this second workshop dedicated to neutron detection with MPGD, we noticed a sharp improvement with respect to the first edition both regarding detector manufacturing techniques and performances.
The construction at Linköping University (Sweden) of a new coating machine for enriched boron carbide deposition over large areas allows to realise new geometrical configurations of the detectors in order to increase detection efficiency.
We strongly think that the next workshop on this topic must involve instrument scientists working at spallation neutron sources in order to better understand their requirements and to realise highly integrated detection systems with performances better than present instrumentation.
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A. Oed, “Position Sensitive Detector with Microstrip Anode for electron Multiplication with Gases,” Nucl. Instr. Meth. A 263 (1988) 351.
Y. Giomataris *et al.*, “MICROMEGAS: A High granularity position sensitive gaseous detector for high particle flux environments,” Nucl. Instr. Meth. A 376 (1996) 29.
F. Sauli, “GEM: A new concept for electron amplification in gas detectors,” Nucl. Instr. Meth. A 386 (1997) 531.
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<http://www.isis.stfc.ac.uk/>
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A. Pietropaolo, “Thin film Boron deposition for GEM-based neutron detectors,” <https://indico.cern.ch/event/365840/contribution/3/attachments/727404/998128/04-_PIETROPAOLO_CERN_Meeting_def.pdf>
<http://www.columbussuperconductors.com/>
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G. Claps *et al.*, “$^3$He-free triple GEM thermal neutron detector,” EPL 105 (2014) 22002.
L. Robinson, “A Deposition Facility for Boron Carbide,” <https://indico.cern.ch/event/365840/contribution/7/attachments/727408/998134/CERN_ESS_Detector_Coatings_WS-150317_ver2.pdf>
<http://www.cemecon.de>
A. Khaplanov, “Design and Produc/on of a 2.4 Square Meter Neutron Detector as a Demonstrator of B10 Layer Technology for Neutron ScaBering Instruments,” <https://indico.esss.lu.se/indico/event/223/material/slides/1.pdf>
G. Croci, “Development of GEM-based thermal neutron detectors,” <https://indico.cern.ch/event/365840/contribution/4/attachments/727407/998132/G.Croci_-_MPGDNeutrons2015_v1.pdf>
A. Khaplanov *et al.*, “Investigation of gamma-ray sensitivity of neutron detectors based on thin converter films,” 2013 JINST 8 P10025.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Equilibrium and motion of a contact line are viewed as analogs of phase equilibrium and motion of an interphase boundary. This point of view makes evident the tendency to minimization of the length of the contact line at equilibrium. The concept of line tension is, however, of limited applicability, in view of a qualitatively different relaxation response of the contact line, compared to a two-dimensional curve. Both the analogy and qualitative distinction extend to a non-equilibrium situation arising due to coupling with reversible substrate modification. Under these conditions, the contact line may suffer a variety of chemo-capillary instabilities (fingering, traveling and oscillatory), similar to those of dissipative structures in nonlinear non-equilibrium systems. The preference order of the various instabilities changes, however, significantly due to a different way the interfacial curvature is relaxed.'
author:
- 'L. M. Pismen'
title: ' Chemo-capillary instabilities of a contact line '
---
Introduction
============
Instabilities of a contact line are commonplace in various settings (for a recent review, see Refs. [@bonn; @matarev]). In many practical applications, instabilities are detrimental when they, for example, impair smooth coating or spotless dewetting. Instabilities may lead, however, to formation of fine patterns or enhanced spreading in microfluidic applications, and cause such fascinating phenomena as spontaneous motion in non-equilibrium systems. Most thoroughly studied instabilities are of hydrodynamic origin, being caused, for example, by enhanced gravitational or Marangoni driving at a perturbed contact line, and are most commonly related to formation of capillary ridges [@matarev]. Quantitative characterization of these instabilities is commonly based on lubrication approximation, and is strongly affected by the notorious contact line singularity resolved on a microscopic scale.
We will concentrate here on a different kind of instability, caused by substrate modification rather than hydrodynamics, and therefore apt to occur at slow velocities and small scales. We start in Sect. \[S1\] we discussing an analogy between equilibrium of a fluid body bounded by the contact line and thermodynamic phase equilibrium in two dimensions, which is made evident by variational representation of lubrication equations accounting for the action of surface tension and disjoining pressure. We will argue, however, that this analogy does not justify extending the notion of two-dimensional line tension to the contact line, in view of a qualitatively different quasi-elastic response of the contact line dependent on three-dimensional effects hidden in the lubrication description.
In the main part of the paper, we will develop the thermodynamic analogy in a different direction by considering chemo-capillary instabilities in a non-equilibrium setting arising due to coupling with reversible substrate modification. As a representative example, we will consider a set-up used in observations of spontaneous motion of droplets induced by surfactant adsorption [@j05l]. This system has been described by a model combining hydrodynamics in lubrication approximation with a linear reaction-diffusion equation for an adsorbed species, and studied both numerically [@th05] and analytically [@p06]. We reconsider it here as an analog of nonlinear models responsible for the formation of non-equilibrium structures [@book]. There are remarkable similarities here, leading to the same variety of instabilities, which include, in addition to traveling instability, fingering instability and oscillations commonly found in other physical settings. The character of transitions is, however, strongly influenced by the specific wavenumber dependence of the contact line relaxation response, different from the elastic response of an interphase boundary in two dimensions.
We will derive equations of motion of a curved contact line in Sect. \[S11\] by combining the effective equation of motion of a rectilinear contact line derived through resolving the contact line singularity [@pe08] with the curvature response derived here in a local approximation and fitting the established theory [@bonn; @deGennes85]. After the diffusive surfactant model is introduced in Sect. \[S12\], the various instabilities of a contact line delimiting a static or moving semi-infinite fluid layer or bounding a circular droplet are analyzed in the following Sections.
Equilibrium {#S1}
===========
The evolution equation of the layer thickness $h$ in lubrication approximation can be written in a variational form [@p02] $$\partial_t h = - \nabla \cdot \kappa(h) \nabla\, \frac{\delta W}{\delta h},
\label{jst2}$$ The driving potential $W$ is of the Cahn–Hilliard type, appropriate for the case when the “order parameter” is conserved: $$W = \int \left[\frac{\gamma}{2} |\nabla h|^2
+ \Phi(h) \right]{{\rm d}}^2 {\mbox{\boldmath $x$}}.
\label{jst3}$$
The two terms in the integrand are the interfacial energy with the surface tension $\gamma$ and the net adhesion energy $\Phi(h)$; $\nabla$ is the two-dimensional gradient operator. Other extrinsic interaction terms, such as gravity, having the same algebraic form, can be added here, but will be further assumed to be negligible, as they operate on far larger scales. The mobility coefficient $\kappa(h)$ depends on $h$ in a strongly nonlinear fashion when it is of hydrodynamic origin, e.g. $\kappa(h) = \mbox{$\frac{1}{3}$} \eta^{-1}h^3$ for the Stokes flow with the dynamic viscosity $\eta$ and no slip on the solid substrate. The validity of the lubrication hydrodynamic approximation used in derivation of Eq. (\[jst2\]), as well as of the concept of shape-independent adhesion energy, is formally restricted to films with a large aspect ratio, implying small contact angles. It is universally used nevertheless for the analysis of both equilibrium and non-equilibrium films leading to qualiatively correct results even outside its formal applicability limits.
It is clear from the variational form of Eq. (\[jst2\]) that the functional $W$ is minimized in a stationary configuration, which is defined by the Euler–Lagrange equation $$\gamma\nabla^2 h - \Pi(h) = 0,
\label{jst4}$$ where $\Pi(h) = \Phi'(h)$ is the disjoining pressure. One can consider two basic configurations. A *macroscopic* system (or a part thereof) can be viewed as an infinite layer asymptotically flat at spatial infinity. Practically, the layer may be shaped at distances far exceeding the range of adhesion forces by gravity or other macroscopic forces. The enormous macroscopic to microscopic scale ratio, which may reach 7–8 orders of magnitude, leaves ample space for an asymptotically flat region at intermediate distances. The asymptotic slope is identified then with the equilibrium contact angle.
Consider, for example, a commonly used expression for disjoining pressure of the type $$\label{af}
\Pi(h) = \frac{A}{h^3}\left[1-\left(\frac{h_f}{h}\right)^{n}\right],$$ characterizing a partially wetting liquid with the net van der Waals interaction parameter $A$ forming a precursor film of the thickness $h_f$. The equilibrium contact angle $\theta_e$ is computed by solving the one-dimensional version of Eq. (\[jst4\]), $$\gamma h''(x) - \Pi(h) = 0,
\quad h(-\infty)=h_f, \quad h'(\infty)=\theta_e,
\label{ppst}$$ where $x$ is the coordinate normal to the contact line. Using the phase plane representation $h'(x)=p(h)$ and integrating yields $$\theta_e = \left[2 \Phi(h_f)/\gamma \right]^{1/2}.
\label{th}$$
A representative example of a disjoining pressure of a more complex form [@sharma89] combines wetting van der Waals and dewetting polar interactions: $$\label{shf}
\Pi(h) = - \frac{A}{h^3} + B e^{-h/\ell}.$$ In a dimensionless form with $h$ scaled by the attenuation scale $\ell$ and $\Pi$ scaled by $A/\ell^3$, this function depends on a single parameter $\chi=B\ell^3/A$: $$\label{sf}
\Pi(h) = - h^{-3} + \chi e^{-h}.$$ The condition $f(h_f)=0$, which has positive roots at $\chi> \chi_f =(e/3)^3 \approx 0.7439$, determines the thickness $h_f$ of a thin film in equilibrium with a flat macroscopic liquid layer. The equilibrium contact angle computed in the same way as (\[th\]) is positive at $\chi > \chi_0 = e^2/8 \approx 0.9236$.
An alternative configuration is a *microdroplet* of a limited volume (or several microdroplets). The volume constraint can be treated by extracting from Eq. (\[jst3\]) the term $\mu V = \mu \int h({\mbox{\boldmath $x$}})\, {{\rm d}}^2 {\mbox{\boldmath $x$}}$, where the Lagrange multiplier $\mu$ plays a role of chemical potential. For the function (\[af\]), the simplest solution of the respective Euler–Lagrange equation, $$\gamma\nabla^2 h - \Pi(h) +\mu = 0,
\label{jstmu}$$ is a parabolic cap of a radius $R$ on the top of a precursor layer with the thickness somewhat exceeding $h_f$. The Lagrange multiplier $\mu$ can be identified then with the chemical potential or the vapor phase in equilibrium with the droplet of curvature radius $R/\theta_e$, i.e. with the vapor concentration just below the dew point when $R \gg h_f\theta_e$. For the function (\[sf\]), the solutions at at $\chi < \chi_0$ and a suitable value of $\mu>0$ can take the form of a straight-line front separating domains with alternative values of the stationary film thickness or of a circular “pancake” or “hole”.
Both macroscopic and microscopic configurations must be stable when the length of the contact line is minimal. Equation (\[ppst\]) or (\[jstmu\]) with a suitable function $\Pi(h)$ can be recognized as equations of phase equilibrium with the layer thickness $h$ playing the role of the order parameter. Respectively, Eq. (\[jst2\]), is recognized as the Cahn–Hilliard equation governing relaxation to the equilibrium state. The only unconventional features are a strongly nonlinear dependence of the mobility on the order parameter and a possibility for the latter to be infinite in a system of infinite extent. This analogy allows us to consider the question of contact line stability in the general framework of stability of phase equilibria.
The question of stability of different configurations of an equilibrium contact line has been discussed in connection with the notion of *line tension* [@Dietrich07]. Unlike a line in two dimensions, deformations of a contact line inadvertently cause deformations of the surface of the enclosed fluid volume, and therefore instability may not arise even when the line tension is formally negative [@Mechkov07]. Moreover, line tension of a contact line, unlike surface tension in three or line tension in two dimensions, is not a fundamental property. If it is defined as a constituent part of the overall energy proportional to the line length, it remains dependent on the shape of the interface in the vicinity of the contact line, and therefore cannot be separated from surface tension and wetting properties. As noted in Ref. [@Mechkov07], stability of an equilibrium contact line is fully determined by the applicable dependence of the disjoining pressure on the local layer thickness $h$, which determines both the equilibrium contact angle and the interface shape in the vicinity of the contact line affecting the apparent line tension. Mechkov *et al* [@Mechkov07] proved stability of a circular contact line and stability of a capillary ridge on short wavelengths. In the lubrication approximation, this result is evident: both the transitional area near the contact line and a front separating domains with different film thickness carry positive energy, and their length should be minimized by the globally stable configuration.
Equation of motion for the contact line {#S11}
=======================================
Instabilities of a contact line or a front between alternative equilibrium thickness levels may arise in a one-component fluid only in a non-equilibrium setting. The key ingredient of the stability analysis in different situations is the equation of motion for the contact line. A naive approach based on the local curvature of the contact line and line tension that would be sufficient if the problem was truly two-dimensional is not applicable here, since the three-dimensional configuration of the interface in the vicinity of the contact line plays a crucial role. Strictly speaking, the problem is nonlocal, since any distortion of the contact line causes pressure changes within the liquid layer or droplet that affect all other locations along the contact line. The problem becomes tractable only when pressure fluctuations are neglected and the analysis is based on a local equation of contact line motion. Such an equation can be extracted from the analysis of slow displacement or spreading of droplets based on the solvability condition for perturbed stationary configurations [@pp04; @p06; @pe08]. The theory is based on matching of perturbations induced by displacement of the contact line in the macroscopic domain and in the contact line vicinity where the dynamic contact line singularity is resolved through either slip or the presence of a precursor film. For a droplet spreading due to a difference between the apparent macroscopic angle $\theta$ and the equilibrium contact angle $\theta_e$, the spreading velocity $U$ is [@pe08] $$U = \frac{1}{3} Q U_0 (\theta^3 - \theta_e^3), \qquad Q = \frac{1}{3 \ln (aR/\ell)},
\label{uth}$$ where $U_0=\gamma/\eta$ is the characteristic viscous velocity and the argument of the logarithm contains the ratio of a macroscopic length $R$ (droplet radius) to a microscopic length $\ell$ (thickness of the precursor layer or slip length); the numerical factor $a$ is dependent on the functional form of the disjoining pressure and/or other microscopic and macroscopic inputs. The linearized form of Eq. (\[uth\]) applicable when the difference $\vartheta=\theta - \theta_e$ is small is $$U = Q \theta_e^2 U_0 \vartheta.
\label{uth0}$$ If this expression is applied to compute the displacement velocity of a droplet under the action of a variable equilibrium contact angle, the result coincides precisely with that obtained earlier in a somewhat different way [@p06]. We will further adopt Eq. (\[uth\]) and its linearization as the local equation of motion of the contact line also for non-circular geometry, neglecting nonlocal effects which are likely to manifest themselves only as weak corrections of the numerical geometric factor in the argument of the logarithm.
Perturbations of the apparent contact angle $\vartheta$ can be computed in the following way. Let the contact line be shifted from an unperturbed position $\Gamma(y)$ along the outer normal by an increment $\xi(y)$, where $y$ is a spanwise coordinate. The displacement is supposed to be small on the macroscopic scale, although it may be large compared to the microscopic scale $\ell$; the essential requirement is that the spanwise derivative $\xi'(y)$ be small, so that the corrugation wavelength be much larger than $\ell$. Projecting the perturbation upon $\Gamma$ implies the perturbation of the local film thickness $\widetilde h(\Gamma,y) = \theta_e \xi(y)$. The global perturbation is obtained by solving the two-dimensional Laplace equation $\nabla^2 \widetilde{h}=0$ in the *unperturbed* geometry, and $\vartheta$ is subsequently computed as the normal derivative ${\mbox{\boldmath $n$}} \cdot \nabla \widetilde{h}=0$ at the contact line.
Consider first relaxation of a perturbed rectilinear contact line bounding the liquid layer at $x<0$ with a constant incline $\theta_e$. The contact line perturbation is presented as a Fourier integral $\xi(y)= \int \zeta_k \cos ky\, dk$. For small deviations of the contact angle, the perturbation of the film thickness at the axis $x=0$ (i.e. at the nominal position of the undisturbed contact line) induced by a Fourier component with $k \ll\ell^{-1}$ is $\widetilde h(0,y)= \theta_e \zeta_k \cos ky$. The perturbation of the film thickness $\widetilde h(x,y)$ in the bulk of the film obtained by solving the Laplace equation with this boundary condition is $$\widetilde h(x,y) = \theta_e \zeta_k\, {{\rm e}}^{|k|x} \cos ky .
\label{hthx}$$ This yields the perturbation of the contact angle $$\vartheta = - \widetilde h_x(0,y) = - |k| \theta_e \zeta_k \cos ky .
\label{hth}$$ Using this in (\[uth0\]) yields the Fourier component of the contact line displacement velocity $$\dot\zeta_k = - Q U_0 \theta_e^3 |k| \zeta_k.
\label{zth}$$ We see that the restoring force is proportional to $|k|$ rather than $k^2$ as it would be for a truly two-dimensional elastic line. This is in agreement with the qualitative argument [@deGennes85; @JdeGennes84; @Raphael01] stipulating that the capillary energy (ostensibly proportional to $k^2$) should be integrated over the penetration length of the order $|k|^{-1}$, thus leading to a reduced power of the wavenumber dependence. This makes the response of the contact line “superdiffusive”, e.g. the curvature radius $r$ of a small bulge grows with time as $r \propto t^{2/3}$ rather than $r \propto t^{1/2}$. The relaxation rate proportional to $|k|$ is also supported by full numerical computations [@Snoeijer07], which have shown that it breaks down only near the Landau–Levich entrainment limit. In this limit, the approximation breaks down, as the contact line perturbation computed as above becomes comparable with the unperturbed contact angle.
Another example to be studied in detail below is a circular droplet of the radius $R$. The unperturbed profile is a parabolic cap $$\label{hcap}
h_0(r) =\frac{\theta_eR}{2} \left[1 - \left(\frac{r}{R}\right)^2\right],$$ where $r$ is the radial coordinate $r$. The perturbation modes dependent on the polar angle $\phi$ are $\xi_n=\zeta_n \cos n\phi$ with an integer $n$. The solution of the Laplace equation for the perturbed interface is $$\widetilde h(r,\phi) = \theta_e\zeta_n \,(r/R)^n \cos n \phi ,
\label{hthr1}$$ Taking also into account the contribution of $h_0'(r)$ yields the perturbation of the apparent contact angle at the contact line $$\vartheta = \theta_e -h_0'(R+\xi_n) - \widetilde h_r(R,\phi)
= - \frac{n-1}{R}\, \theta_e \zeta_n \cos n \phi .
\label{hthr}$$ In a special case of a symmetric perturbation ($n=0$), the perturbation of $\theta$ is related to the perturbation of $R$ by the conservation condition of the droplet volume $V=\frac{4}{3}\theta_eR^3$, which yields $\vartheta =-\zeta_n\theta_e/R$. The expression (\[hthr\]) can be extended also to this case if $n-1$ is replaced by its absolute value $|n-1|$.
Dynamics of a diffusive surfactant {#S12}
==================================
A disturbance displacing the contact line may be caused by chemical inhomogeneities, which can be accounted for by assuming a linear dependence of the equilibrium contact angle on the local concentration of an adsorbed chemical: $$\theta_e = \theta_0 [1 - b (c - c_0)],
\label{cth}$$ where $\theta_0$ is the reference contact angle at $c=c_0$ and $b$ is a proportionality constant. Modification of wetting properties by adsorption or chemical reactions has been used in a number of experiments to induce spontaneous motion of droplets on solid substrate. In the following, we will consider a reversible setup allowing for restoration of substrate properties [@j05l]. Variation of the contact angle is caused in this system by deposition of a surfactant from a bulk phase (assumed to have negligible viscosity) and its dissolution underneath a droplet or liquid layer. The theoretical model describing stationary droplet motion in this setup [@p06] is based on a surface diffusion equation with a constant adsorption rate and linear desorption kinetics. This leads to the surfactant diffusion equation written in the dimensionless form $$c_t = \nabla^2 c - c + H({\mbox{\boldmath $x$}}) .
\label{diff}$$ The surfactant coverage $c$ is scaled here by the coverage in equilibrium with the constant surfactant concentration in the continuous phase, time by the inverse desorption rate constant $k_d$, and length, by $\sqrt{D/k_d}$, where $D$ is the surfactant diffusivity on the substrate; $H({\mbox{\boldmath $x$}})$ is the Heaviside step function equal to 1 outside and 0 inside the footprint of the droplet or liquid layer. For simplicity, it is assumed that the desorption rate is constant all over the substrate and the bulk concentration in the liquid layer is negligible.
We will apply now the same model in conjunction with Eq. (\[uth\]) to explore stability of a contact line. In this formulation, the problem is almost identical to that of stability of non-equilibrium structures in the FitzHugh–Nagumo or similar models [@book]. First, we have to find the stationary solution of Eq. (\[diff\]) for the unperturbed configuration. For a semi-infinte layer at $x<0$, the solution dependent on a single coordinate $x$ is $$c_s(x) = \left\{ \begin{array}{lcc}
\frac{1}{2}\,{{\rm e}}^{x} & \mbox{ at } & x \le 0, \\
1 - \frac{1}{2}\, {{\rm e}}^{-x} & \mbox{ at } & x \ge 0.
\end{array} \right.
\label{3veq1s}$$ The reference concentration $c_0$ can be identified with the stationary concentration at the contact line $c_s(0)=1/2$.
For a circular droplet, the stationary solution $c=c_s(r)$ of Eq. (\[diff\]) is $$c_s= \left\{ \begin{array}{lcc}
R K_1(R)I_0(r) & \mbox{ at } & r\le R \\
1 - R I_1(R)K_0(r)] & \mbox{ at } & r\ge R,
\end{array} \right.
\label{32soln}$$ where $I_n,\:K_n$ are modified Bessel functions. The values of $c_s(R)$ given by both lines of this formula are the same, in view of the identity $K_1(R)I_0(R) + K_0(R)I_1(R)= 1/R$. This common value should be identified with the reference concentration $c_0$.
The concentration perturbation due to a contact line displacement can be computed in a simple way by observing that any shift of the contact line position by an increment $\xi \ll 1$ is equivalent to switching on or off the source term in Eq. (\[diff\]) in a narrow region near the contact line. This contributes to the equation for the perturbation $\widetilde c=c-c_s$ a source localized at the unperturbed contact line position and proportional to its displacement: $$\partial_t \widetilde c = \nabla^2 \widetilde c - \widetilde c - \xi \, \delta (\Gamma).
\label{3sVeq}$$
The dynamic equation for the contact line displacement is obtained by linearizing Eq. (\[uth\]). Using Eq. (\[cth\]) and expanding the surfactant concentration in the vicinity of the unperturbed contact line position yields $\theta_e =\theta_0[1 - b(\widetilde c + j \xi)]$, where $j$ is the stationary surfactant flux into the liquid across the contact line. Combining this with Eq. (\[uth0\]) yields the equation for the local contact line displacement $$\dot\xi = - \chi [\vartheta/\theta_0 - b(\widetilde c + j\,\xi) ].
\label{zth1}$$ We have transformed here to the dimensionless form based on the chemical length and time scales. The remaining dimensionless parameter $\chi =Q U_0 \theta_0^3/\sqrt{Dk_d}$ is proportional to ratio of the characteristic capillary velocity $U_0=\gamma/\eta$ to the characteristic chemical velocity $\sqrt{Dk_d}$. This ratio is typically very large, but is effectively reduced being multiplied by the cube of the contact angle, which must be small when the lubrication approximation is applicable.
Although perturbations are habitually presumed to be arbitrarily small for the purpose of linear stability analysis, the method is actually also applicable to finite perturbations of the contact line position, provided they are small compared to the characteristic surfactant diffusion scale.
Instability of a rectilinear contact line {#S31}
=========================================
For a rectilinear contact line displaced by a harmonic perturbation, $\vartheta$ in Eq. (\[zth1\]) is given by Eq. (\[hth\]) and the stationary flux following from Eq. (\[3veq1s\]) is $j= c_s'(0) = 1/2$. Presenting the concentration perturbation in the spectral form $\widetilde c(x,y,t) = \psi(x,t)\cos ky$ and setting in the perturbation equations (\[3sVeq\]), (\[zth1\]) $\psi, \zeta_k \sim {{\rm e}}^{\lambda t}$ leads to the eigenvalue problem $$\begin{aligned}
\lambda \zeta_k &=& - \chi [ |k| \zeta_k - b(\psi + \zeta_k/2) ] ,
\label{zeq} \\
&& \psi_{xx} - q^2\psi = \zeta_k \delta (x),
\label{3spsieq} \end{aligned}$$ where $q^2=1 +\lambda + k^2$. The solution of Eq. (\[3spsieq\]), presuming Re $q>0$, is $$\psi(x) = -\frac{\zeta_k}{2q} \, {{\rm e}}^{-q|x|}.
\label{3sVsoln}$$ Using this in Eq. (\[zeq\]) yields the dispersion relation $$\lambda = - \chi \left[ |k| - \frac{b}{2} \left(1 - \frac{1}{q}\right) \right] .
\label{3disp0}$$ The stability condition is Re $\lambda<0$. As expected, $\lambda$ vanishes at $k=0$, which reflects the translational symmetry of the rectilinear line. Unlike a similar problem in Ref. [@book] with $k$ replaced by $k^2$, the derivative ${{\rm d}}\lambda/{{\rm d}}k$ at $k=0$ is always negative, so that the contact line is stable to long-scale perturbations. Stability is lost as $\lambda$ vanishes at $$k = k_c = \sqrt{\frac{1+ \sqrt{5}}{2}}, \qquad b = b_c =\frac{(3+ \sqrt{5})^{3/2}}{ \sqrt{1+ \sqrt{5}}}.
\label{bk}$$ A band of unstable modes around $k=k_c$ (which turns out to coincide with the golden ratio) widens at $b>b_c$. In dimensional units, the wavelength of growing perturbations is of the same order of magnitude as the diffusional length $\sqrt{D/k_d}$.
![Monotonic and oscillatory instability regions in the parametric plane $\chi, b$ for a rectilinear contact line.[]{data-label="fstraight"}](fstraight){width="8cm"}
In addition, oscillatory instability is possible when $\chi$ is sufficiently large. It first arises in the long-scale low-frequency mode at $\chi =\chi_c= 4/b$, and can be detected by expanding Eq. (\[3disp0\]) near this point. For $\chi -\chi_c= \epsilon \ll 1$, the suitable scaling is $$\lambda = \epsilon \lambda_r +{{\rm i}}\epsilon^{1/2} \widetilde \omega,
\qquad k = \epsilon \widetilde{k}.
\label{keps}$$ The leading terms of the real and imaginary parts of the expansion are, respectively, of the order 1 and 3/2. This yields two lowest order equations for the real and imaginary parts of $\lambda$ as functions of $k$: $$\widetilde{k} - \frac{3}{16} b\, \widetilde \omega^2=0, \qquad
b \, \widetilde \omega \left( 2b -12 \lambda_r -5\widetilde \omega^2 \right)=0.
\label{keps1}$$ The first of these gives the frequency as the function of the wavenumber: $$\omega= 4\sqrt{ \frac{k}{3b}}.
\label{kom}$$ The second equation gives the $O(\epsilon)$ real part of the eigenvalue $$\lambda_r = \frac{b}{6} - \frac{20}{9} \frac{\widetilde k}{b},
\label{klam}$$ indicating that instability indeed occurs at $\chi >\chi_c$ and disappears as $k$ increases. Higher orders of the expansion give $O(\epsilon^2)$ corrections to the real, and $O(\epsilon^{3/2})$ corrections to the imaginary parts of $\lambda$. Within the instability region, wave modes with finite wavelength also become unstable and are likely to exhibit there the fastest growth rate. The monotonic and oscillatory instability regions in the parametric plane $\chi, b$ are shown in Fig. \[fstraight\].
Instability of a circular droplet {#S32}
=================================
For a circular droplet, $\vartheta$ in Eq. (\[zth1\]) is given by Eq. (\[hthr\]), and the eigenvalue problem for coupled spectral equations of the displacement and concentration perturbation analogous to Eqs. (\[zeq\]), (\[3spsieq\]) has the common form for any integer $n \ge 0$: $$\begin{aligned}
&& \lambda \zeta_n = - \chi [ |n-1| \zeta_n - b(\psi + j\,\zeta_n ) ] ,
\label{zeqr} \\
&& \psi_{rr} + \frac{1}{r}\psi_r - \left(q^2 + \frac{n^2}{r^2}\right)\psi =
\zeta_n \delta (r-R),
\label{3spsieqr} \end{aligned}$$ where $q^2=1 +\lambda$. The solution of Eq. (\[3spsieqr\]) is $$\psi(r)= \left\{ \begin{array}{lcc}
- R \zeta_n K_n(qR)I_n(qr) &\mbox{ at } & r\le R ,\\
- R \zeta_n I_n(qR)K_n(qr) &\mbox{ at } & r\ge R.
\end{array} \right.
\label{3sVsolnd}$$ Using this, together with the expression for the stationary flux $j= c_s'(R) = R K_1(R)I_1(R)$ following from Eq. (\[32soln\]), in (\[zeqr\]) yields the dispersion relation $$\frac{\lambda}{\chi} = - \frac{|n-1|}{R} + b R \left[I_1(R)K_1(R) - I_n(qR)K_n(qR)\right] .
\label{3disp0r}$$
As expected, $\lambda$ vanishes at $n=1$, which corresponds to a shift without deformation. For $n \neq 1$, the critical value $b=b_n$ for a given radius verifies Eq. (\[3disp0r\]) with $\lambda=0$: $$\frac{1}{b_n} = \frac{R^{2}}{|n-1|} \, \left[I_1(R)K_1(R) -I_n(qR)K_n(qR)\right] .
\label{posbc}$$ For $n =0$, the value $b_0$ given by this relation is negative, so that no monotonic instability can arise in the symmetric mode. The most dangerous instability mode is the one with the lowest critical value of $b_\perp(R)=\min_n b_n(R)$. For small droplets, the dipole mode deforming the circular droplet into an ellipse, is the most dangerous one. With the growing radius, the most dangerous monotonic instability mode shifts to larger values of $n$, as shown in Fig. \[fcircle\]. The absolute lower limit of monotonic instability is $b_2 \approx 5.158$ at $R\approx 2.337$, which is below the respective limit $b_c \approx 6.6604$ for the rectilinear contact line given by Eq. (\[bk\]); the latter limit is approached asymptotically at $R \to \infty$, as the envelope of the family of curves in Fig. \[fcircle\] gradually rises. The wavenumber $k=n/R$ rises as well on the average, as seen in the inset in the same Figure, coming gradually closer to the value $k_c \approx 1.272$ of Eq. (\[bk\]).
![Monotonic instability thresholds for a circular droplet of radius $R$. The numbers indicate the value of $n$. Instability occurs above the lower envelope of the family of curves. Inset: the wavenumber $k=n/R$ for the most dangerous mode; the dashed line indicates the respective value for the rectilinear contact line.[]{data-label="fcircle"}](fcircle){width="8cm"}
![The lower limit of the product $b\chi$ at the traveling instability limit as a function of radius $R$. Inset: The lower limit $\chi_t$ at which the traveling instability precedes the monotonic one. []{data-label="fmove"}](fmove){width="8cm"}
Besides the monotonic instability, dynamic instabilities dependent on the parameter $\chi$ are possible. Those are traveling instability setting the droplet into motion and oscillatory or wave instability. The threshold of traveling instability is detected, according to the general algorithm [@book], by differentiating the dispersion relation (\[3disp0r\]) for $n=1$ with respect to $\lambda$ and solving the resulting equation at $\lambda=0$. The droplet sets into spontaneous motion at $$R[I_0(R)K_1(R)- I_1(R)K_2(R)]> 2/b \chi.
\label{3ct2}$$ The lowest value of the product $b \chi$ enabling traveling instability $b \chi \approx 11.46$ is attained at $R \approx 1.5866$ (see Fig. \[fmove\]). The lower limit $\chi=\chi_t$ at which the traveling instability precedes the monotonic one for droplets with a certain radius can be obtained by using in Eq. (\[3ct2\]) the critical value $b_\perp$ for the most dangerous mode given by Eq. (\[posbc\]). The plot $\chi_t(R)$ is shown in the inset of Fig. \[fmove\].
Oscillatory instability can be detected numerically by solving coupled equations for the real and imaginary parts of Eq. (\[3disp0r\]). Setting $\lambda = {{\rm i}}\omega$ we obtain at the instability threshold $$\begin{aligned}
&& \frac{1}{b} = \frac{R^{2}}{|n-1|} \, \left[I_1(R)K_1(R) -\mbox{Re}\,F_n(R,\omega)\right], \label{posb} \\
&&\frac{1}{\chi} = -\frac{bR}{\omega} \,\mbox{Im} \, F_n(R,\omega),
\label{posc} \\
&& F_n(R,\omega) =
I_n \left(R\sqrt{1+{{\rm i}}\omega} \right)
K_n \left( R\sqrt{1+{{\rm i}}\omega}\right).
\label{posf} \end{aligned}$$ Using Eq. (\[posb\]) to eliminate $b$, we are left with a single equation (\[posc\]), which can be solved numerically to compute the frequency $\omega$ at the Hopf bifurcation point for a droplet of a given radius $R$.
For $n \ge 2$, the locus of oscillatory instability branches off the locus of monotonic instability defined by Eq. (\[posbc\]) at the point of double zero eigenvalue where the frequency vanishes. Close to this line, $\omega$ is small, and a Taylor series can be used. The leading term in the expansion of Eq. (\[posb\]) is of the order $O(\omega^2)$, while the leading term in the expansion of Eq. (\[posc\]) is linear in $\omega$. Setting the latter to zero gives the value of $\chi=\chi_n$ required to obtain the double zero as a function of the radius $R$; the same relation can be obtained most readily by differentiating Eq. (\[3disp0r\]) with respect to $\lambda$. The resulting condition is a generalization of Eq. (\[3ct2\]) $$R[I_{n-1}(R)K_n(R)- I_n(R)K_{n+1}(R)] > 2/b \chi_n .
\label{3ct2d}$$ The value of $\chi_n$ given by this relation consistently rises with $n$, and is always higher than the respective value for traveling instability obtained for $n=1$. This proves that low-frequency oscillatory instability never occurs before traveling instability sets in.
The remaining possibility is oscillatory instability with a finite frequency, which is also feasible (and, indeed, is most likely to occur) in a symmetric mode. Rather than solving Eqs. (\[posb\]), (\[posc\]) directly, it is more instructive again to compute the lower limit $\chi=\chi_o$ at which this instability precedes the monotonic one for a droplet with a certain radius. For this purpose, $\omega$ is computed by solving numerically Eq. (\[posb\]) with $b=b_\perp$, and then $\chi_o$ is evaluated using Eq. (\[posc\]). The computation shows that the frequency at the Hopf bifurcation decreases monotonically, going down to zero as the limit of a rectilinear contact line approaches, in agreement with the results of Sect. \[S31\]. The curve $\chi_o(R)$ for the symmetric mode lies, however, slightly higher than the curve $\chi_t(R)$, so that traveling instability always occurs first.
Instability of a moving contact line {#S4}
====================================
The analysis can be extended to the case when the difference between the equilibrium and apparent contact angles is not small and the unperturbed line is moving with a certain velocity $U$. The deviation $\vartheta$ is now redefined as a perturbation of the apparent contact angle at the moving contact line $\theta$, which, in accordance with Eq. (\[uth\]), equals to $( \theta_e^3 + 3U/QU_0)^{1/3}$. In the same approximation as in Sect. \[S11\], $\vartheta$ is given for a rectilinear contact line by Eq. (\[hth\]) with $\theta_e$ replaced by $\theta$. The equation of the local contact line displacement (\[zth1\]) has to be written now for the displacement $\xi$ relative to the unperturbed moving line, and is modified to $$\dot\xi = - \left(\chi+3u \right) \,\vartheta/\theta
+ b\chi(\widetilde c + j\,\xi) ,
\label{zthv}$$ where $u=U/\sqrt{k_dD}$ is the dimensionless velocity.
The flux through the contact line defined by solving Eq. (\[diff\]) in the comoving frame is $j= c_s'(0) = 1/\sqrt{4+u^2}$, and the spectral equations (\[zeq\]), (\[3spsieq\]) are modified to $$\begin{aligned}
\lambda \zeta_k &=& -\left( \chi+ 3u \right) |k| \, \zeta_k
+ b\chi\left(\psi + \frac{\zeta_k}{\sqrt{4+u^2}}\right) ,
\label{zeqv} \\
&& \psi_{xx} - u\psi_x - q^2\psi = \zeta_k \delta (x).
\label{3spsiv} \end{aligned}$$ The solution of the last equation is $$\psi(x) = -\frac{\zeta_k}{\sqrt{4q^2+u^2}} \,
\exp \left[-\frac{x}{2} \left(u \pm \sqrt{4q^2+u^2 }\right) \right] ,
\label{3sVsolnv}$$ where the positive and negative signs apply, respectively, at $x>0$ and $x<0$. Using this in Eq. (\[zeqv\]) yields the dispersion relation $$\lambda = - \left( \chi+ 3u \right)|k|
+ b\chi \left( \frac{1}{\sqrt{4+u^2}}
- \frac{1}{\sqrt{4q^2+u^2 }}\right) .
\label{3dispv}$$ Notably, the coefficient at $|k|$ vanishes at $u=-\chi/3$, which corresponds to the Landau–Levich entrainment limit; the theory becomes inapplicable close to this point [@Snoeijer07]. Besides the coefficient at $|k|$, the dispersion relation is insensitive to reversing the direction of motion, but the reference concentration $c_0$ defined as the surfactant concentration on the unperturbed contact line, is different in the two cases.
![Loci of monotonic (dashed lines) and oscillatory (solid line) instability in the parametric plane $\chi, b$ for the rectilinear contact line moving with the speed $u=\pm0.1$ (as indicated by numbers at the curves). The stable domain is in below both respective curves.[]{data-label="fv"}](fv){width="8cm"}
Monotonic instability is observed at $$\begin{aligned}
k_c &=& \sqrt{\frac{1}{2}\left( 1+ \sqrt{5}\right) \left(1+ \frac{u^2}{4} \right)}, \label{kv} \\
b_c &=& \frac{\left(3+\sqrt{5}\right)^{3/2} \left(1+u^2/4\right) (3 u+\chi)}{\chi
\sqrt{1+\sqrt{5}} }.
\label{bv} \end{aligned}$$ Besides the velocity dependence, a qualitatively significant change, compared to Eq. (\[bk\]), is the dependence of the monotonic instability threshold on the parameter $\chi$ which appears at any non-zero velocity; the marginal wavelength $k_c$ remains, however, independent of $\chi$. At $\chi \to 0$, the velocity dependence becomes singular: a retreating contact line $u<0$ is always unstable under these conditions, while for an advancing line ($u>0$) the threshold rises sharply.
Oscillatory instability emerges, as at $u=0$, in a long-wave low-frequency mode, and can be detected by expanding Eq. (\[3dispv\]) using the scaling (\[keps\]) as before. The $O(\epsilon^{1/2})$ term gives the parametric relation at the instability threshold $$b\chi= 4(1+u^2/4)^{3/2} .
\label{bchiv}$$ The dispersion relation $\omega( k)$ following from the $O(\epsilon)$ term is $$\omega=\sqrt{k \left(u^2+4\right) \left[ b+\frac{\left(u^2+4\right)^{3/2}}{6u}\right]}.
\label{komv}$$ For $u>0$, this relation is qualitatively similar to Eq. (\[kom\]); the function $\lambda_r (\widetilde k)$ defined by the $O(\epsilon^{3/2})$ term is likewise similar to Eq. (\[klam\]). The overall bifurcation diagram changes, however significantly, compared to that in Fig. \[fstraight\]. The loci of monotonic and oscillatory bifurcation fail to intersect already at $u > 0.2012$; at higher velocities, only oscillatory instability is relevant. At $u < 0$, on the contrary, monotonic instability prevails at small $\chi$ (see Fig. \[fv\]).
Discussion {#S5}
==========
We have seen in Sect. \[S1\] a remarkable similarity between the contact line equilibrium (described in lubrication approximation) and equilibrium of an interphase boundary in two dimensions. Likewise, chemo-capillary instabilities of the contact line driven out of equilibrium by coupling with surfactant adsorption (or other non-equilibrium process affecting wetting properties of the substrate) are remarkably similar to instabilities of two-dimensional non-equlibrium structures. A substantial difference stems, however, from relaxation response of a curved contact line being proportional to the absolute value rather than square of the wavenumber $k$. This may be attributed to a “two-and-a-half-dimensional” character of the lubrication approximation. In view of this specific response, the concept of line tension lifted from the two-dimensional world is not duly applicable to the contact line. This is manifested, in particular, in the paradox of stability of the contact line at an apparently negative line tension [@Dietrich07; @Mechkov07].
Out of equilibrium, the modified wavenumber dependence brings about more subtle changes, compared to instabilities of dissipative structures in two dimensions [@book]. The onset of instability of a rectilinear contact line shifts to a finite wavelength, and traveling instability prevails over the oscillatory one for circular droplets. We have seen in Sects. \[S31\], \[S32\] that monotonic instability occurs on the wavelength of the same order of magnitude as the surfactant diffusion length. It should cause fingering in the case of macroscopic liquid volumes of a size far exceeding this length, like it is commonly observed in spreading of surfactant-laden liquid layers [@matar]. It can also explain blebbing and fission of moving droplets observed in the experiment [@Nagai]. Instabilities with lower $n$ may result in splitting of microdroplets with a radius of the same order of magnitude as the diffusion length. Dynamic instabilities are enhanced by the increased ratio of the characteristic hydrodynamic and reaction-diffusion velocities expressed by the parameter $\chi$. One should be aware, however, that the variety of instabilities predicted by theory may be masked in actual experiments by surface roughness arresting contact line displacement.
[9]{} D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, Rev. Mod. Phys. **81**, 739 (2009). R. V. Craster and O. K. Matar, Rev. Mod. Phys. **81**, 1131 (2009). Y. Sumino, N. Magome, T. Hamada, and K. Yoshikawa, Phys. Rev. Lett. **94**, 068301 (2005). K. John, and M. Bär, and U. Thiele, Eur. Phys. J. E **18**, 183 (2005). L. M. Pismen, Phys. Rev. E [**74**]{}, 041605 (2006). L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer, Berlin (2006). L. M. Pismen and J. Eggers, Phys. Rev. E [**78**]{}, 056304 (2008). P. G. de Gennes, Rev. Mod. Phys. **57**, 827 (1985). L. M. Pismen, Colloids and Surfaces A **206**, 11 (2002). A. Sharma, Langmuir [**9**]{}, 3580 (1993). L. Schimmele, M. Napiorkowski, and S. Dietrich, J. Chem. Phys. **127**, 164715 (2007) S. Mechkov, G. Oshanin, M. Rauscher, M. Brinkmann, A.M. Cazabat, and S. Dietrich, EPL **80**, 66002 (2007). L. M. Pismen and Y. Pomeau, Phys. Fluids, [**16**]{} 2604 (2004). J. H. Snoeijer, B. Andreotti, G. Delon, and M. Fermigier, J. Fluid Mech., **579** 63 (2007). J. F. Joanny and P. G. de Gennes, J. Chem. Phys. 81, 552 (1984). R. Golestanian and E. Raphaël, Phys. Rev. E **64**, 031601 (2001). O. K. Matar and R. V. Craster, Soft Matter, **5**, 3801 (2009). K. Nagai, Y. Sumino, H. Kitahata, and K. Yoshikawa, Phys. Rev. E [**71**]{}, 065301 (2005).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study unification in the $SU(5)$ model with an extra Dirac multiplet in the ${\bf 24}$ representation. After spontaneous symmetry breaking we have at low energies a singlet, a colorless triplet and a neutral color-octet. All other particles can be taken at the unification scale. This combination leads to a unification very near the Planck scale. The triplet is light, its neutral component is a dark matter candidate. The model is in agreement with a recently derived anomaly condition, that implies that the number of (Weyl)-fermions has to be a multiple of 16.'
address: |
Institut für Physik,\
Albert-Ludwigs-Universität Freiburg,\
79104 Freiburg, Germany
author:
- 'J. J. van der Bij'
title: 'A simple $SU(5)$ model with unification near the Planck scale.'
---
Introduction
============
The spectrum of known particles leads one to consider the unification of the known gauge forces into a larger group. The simplest one is the gauge group $SU(5)$. Larger groups, like $SO(10)$ or $E(6)$ are also popular. However the known particles do not lead to a unification of the gauge couplings. Therefore one needs to add new particles to the theory. Also dark matter exists, which is most easily explained by the presence of new particles around the electroweak scale. One is therefore interested in finding a suitable set of new particles to get a consistent view of unification. Very popular is the extension towards a supersymmetric model. However this extension is not completely without problems. Let us set out what would be desirable for a successful description of unification. We can consider the following conditions.
1. The gauge coupling constants should unify at a high scale.
2. Preferably the scale should be very near the Planck scale, in order to have a link with gravity.
3. The extra particles should not lead to proton decay.
4. There should be a dark matter candidate.
5. There should not be extra interactions in conflict with phenomenology. In particular flavor changing neutral currents are dangerous.
6. Preferably there should be an independent reason for the choice of gauge group and representations.
The supersymmetric model works very well regarding point one. Doubling the standard model particles with their supersymmetric partners leads quite precisely to a unification within $SU(5)$. On the other points it does not do so well. The unification scale is below the Planck scale, so a unification with gravity is somewhat difficult to imagine. Proton decay has to be prevented by extra symmetries and/or fine-tuning of parameters. A dark matter candidate can be constructed by the imposition of R-parity, which however is an extra principle unrelated to supersymmetry itself. One has to finely tune a large number of parameters to stay out of conflict with experiment. So the balance is not purely positive. Given the fact, that the LHC is starting to constrain the theory quite severely[@ichep], one should keep an open mind and look at possibilities outside supersymmetric unification. In particular one would like to take into account point six in the above list as well. Normally this is ignored and one takes the structure of the known particles as motivation only. However recently some progress in this question has been made[@vdbij1; @vdbij2]. On the basis of a gravitational anomaly within a cosmological context, some constraints on the possible existence of particle multiplets were found. More precisely the indication is that the number of gauge fields has to be a multiple of 8 and the number of (Weyl)-fermions a multiple of 16. This fits in well with the known particle content and with the gauge group $SU(5)$, however without supersymmetry.
The model
=========
Given the above we therefore will consider a $SU(5)$ unification without supersymmetry. A promising model was described in [@perez1]. The main feature of this model for our purpose is the addition of an extra set of fermions in the representation ${\bf 24}$ of $SU(5)$. Under the subgroup $SU(3) \times SU(2) \times U(1)$ this representation is decomposed as: $$\begin{aligned}
{\bf 24} = &(\rho_8) \oplus (\rho_3) \oplus (\rho_{(3,2)}) \oplus (\rho_{(\bar 3, 2)}) \oplus (\rho_0) \\
\nonumber
= &({\bf 8,1,0}) \oplus ({\bf 1,3,0}) \oplus ({\bf 3,2, -5/6})\oplus ({\bf 3,2, 5/6})
\oplus({\bf 1,1,0})\end{aligned}$$ In a first stage of symmetry-breaking these particles receive a mass through a coupling to a ${\bf 24_{H}}$ Higgs field and also have a direct mass term. The Lagrangian density is: $${\cal L}_{mass, Yukawa} = M\, Tr({\bf 24}^2) + \lambda \, Tr({\bf 24}^2 \, {\bf 24_H}) + h.c.$$ The masses derived from this Lagrangian can be written as [@perez1]: $$M_0= \mid M -\Lambda \mid,\,
M_3= \mid M -3 \Lambda \mid,\,
M_8= \mid M + 2\Lambda \mid,\,
M_{32}= \mid M - 1/2 \Lambda \mid$$ Here $\Lambda$ is a complex number coming from the product of the Yukawa coupling and the vacuum expectation value of the ${\bf 24}$ Higgs field. Using also extra Higgses that were introduced in order to have a seesaw mechanism for neutrino masses a unification was possible, however rather complicated, relying strongly on having certain Higgses to be light. Naively one would like to have these Higgs fields to be at the unification scale. Ignoring the presence of the Higgs fields, assuming that they have a mass at the unification scale, the one-loop renormalization group equations read:
$$\begin{aligned}
2\pi(\alpha_1^{-1}-\alpha_U^{-1}) &=&\frac{41}{10} \ln(m_U/m_Z) + \frac{10}{3} \ln(m_U/m_{32}) \\
\nonumber
2\pi(\alpha_2^{-1}-\alpha_U^{-1}) &=&-\frac{19}{6} \ln(m_U/m_Z) + \frac{4}{3} \ln(m_U/m_3) +
2 \ln(m_U/m_{32})\\ \nonumber
2\pi(\alpha_3^{-1}-\alpha_U^{-1}) &=&-7 \ln(m_U/m_Z)
+ 2 \ln(m_U/m_8) + \frac{4}{3} \ln(m_U/m_{32})\end{aligned}$$
In these formulas $m_U$ and $\alpha_U$ are the unification scale and the unified fine structure constant at the unification scale. Given the formula (3) for the masses a sensible unification in this form is not possible, without adding light scalars into the renormalization group running [@perez1].
Actually the model in this form suffers from two problems. First a ${\bf 24}$ representation has 24 fermions, which therefore violates the anomaly[@vdbij1; @vdbij2] condition. Secondly the fermions are Weyl-fermions. Therefore the Yukawa term in eq.(2) is necessarily of the D-type, being symmetric in the interchange of two types of fermions. As a consequence the $\rho_{32}$ fields stay light. This is not conducive to unification, as is easily seen from eq.(4).
The solution to both problems is simplicity itself. One takes the fermions in the ${\bf 24}$ representation to be Dirac fermions, instead of Weyl fermions, thereby doubling the number of fermions. This leads to 48 fields, which is divisible by 16. Morever for Dirac fermions there is a second independent Yukawa interaction of F-type, being antisymmetric in the group indices and proportional to the structure constants of the group. This leads to an extra contribution to the mass of the $\rho_{32}$ fields, which one can move to the unification scale. Furthermore using Dirac fermions one has particle number conservation, which makes the lightest particle within this multiplet a candidate for dark matter.
Going from Weyl-fermions to Dirac-fermions the contribution from the fermions are doubled. The one-loop renormalization equations now read:
$$\begin{aligned}
2\pi(\alpha_1^{-1}-\alpha_U^{-1}) &=& \frac{41}{10} \ln(m_U/m_Z) + \frac{20}{3} \ln(m_U/m_{32})\\
\nonumber
2\pi(\alpha_2^{-1}-\alpha_U^{-1}) &=& -\frac{1}{2} \ln(m_U/m_Z) - \frac{8}{3} \ln(m_3/m_Z) +
4 \ln(m_U/m_{32})\\ \nonumber
2\pi(\alpha_3^{-1}-\alpha_U^{-1}) &=& -3 \ln(m_U/m_Z)
- 4 \ln(m_8/m_Z) + \frac{8}{3} \ln(m_U/m_{32})\end{aligned}$$
In this equation we take the $\rho_{32}$ field to have a mass at the unification scale. We therefore leave out the terms $ \ln(m_U/m_{32})$. If we furthermore assume that the triplet is the only dark matter component of the universe, its mass can be calculated from the dark matter abundance. Following [@cirelli1; @cirelli2] we find $m_3 = 1.9\, TeV$. The difference with [@cirelli1; @cirelli2] is a factor $1/\sqrt{2}$ in the mass because of the Dirac nature of the fields.
Using as input the central values[@pdg] : $$\begin{aligned}
\alpha_1^{-1} &=& 59.000, \\ \nonumber
\alpha_2^{-1} &=& 29.572, \\ \nonumber
\alpha_3^{-1} &=& 8.425, \end{aligned}$$
one can solve solve the equations and finds:
$$\begin{aligned}
m_8 &=& 6.80 \,\,10^6\, GeV, \\ \nonumber
M_U &=& 4.55\,\, 10^{18}\, GeV, \\ \nonumber
a_U &=& 0.029, \end{aligned}$$
Discussion
==========
The results of the model are quite satisfactory and appear to fulfil the conditions set out in the introduction. The spectrum is simple. The unification scale is quite close to the Planck scale and comes out naturally.
The triplet fields are light. The neutral component is $166\, MeV$ lighter than the charged ones, due to radiative corrections. This makes the neutral component an excellent candidate for the dark matter of the universe. If the dark matter would consist of more components, for instance an admixture of scalar singlets, the mass of the triplet fermions would have to be smaller. This would lead to a further increase in the unification scale towards the Planck mass. On the other hand lowering $m_{32}$ leads to a lower unification scale; one can even go down to the limits from proton decay. The phenomenology of the triplets at the LHC is similar to the phenomenology of light triplet scalars. The typical signature of such particles has been studied in [@cirelli1; @cirelli2; @lopez; @perez2].
The values in eq.(7) are somewhat qualitative and cannot be taken as precision predictions. They depend of course on the uncertainties in the determination of the coupling constants, higher order corrections in the renormalization group, contributions of scalars, non-perturbative effects from gravity etc. The main conclusion however, that the triplet can be light, and the unification can take place very close to the Planck scale, will stay true. Therefore this model makes two testable predictions. First one expects a triplet of light Dirac fermions at the LHC. Secondly one expects proton decay to be absent in experiments in the near future, as the unification scale is near the Planck mass, which makes the proton lifetime too large. However the last prediction can be avoided when $m_{32}$ is smaller than the unification scale.
Non-supersymmetric unification with ${\bf 24}$’s has been sparsely considered in the literature. See for instance [@krasnikov:1993], where however more emphasis was put on the Higgs field representations. Some recent models are given in [@frigerio; @kannike]. The precise content with a Dirac ${\bf 24}$, which is needed to fulfil the condition in [@vdbij1; @vdbij2] appears not to have been discussed. The presence of a Dirac ${\bf 24}$ would seem to point to an underlying supersymmetry, possibly even N=2 supersymmetry. Indeed a similar spectrum of fermions was found in the model in [@jones1; @jones2], which was introduced as an example of a finite unified theory. For a review of finite unified theories see [@zoupanos]. This model could satisfy the conditions of[@vdbij1; @vdbij2], since further Higgs representations come in pairs ${\bf 5}$ and ${\bf \bar 5}$. However given the above we would expect supersymmetry to become relevant only near the Planck scale, which is not unreasonable as local supersymmetry implies gravity.\
[**Acknowledgement**]{} I thank Prof. G. Zoupanos for discussions.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present a measurement noise reduction scheme based on information flow of a chaotic system. This scheme operates on conditions of chaoticity and well-defined noise level, not depending on other detailed characteristics of noise. Starting with a simple map and full knowledge of dynamics, we extend the basic idea to general form applicable to higher dimensional systems. Reducing noise in Lorenz system is demonstrated as an example. Inferring dynamics without [*a priori*]{} knowledge is then discussed by proposing an indicator which measures predictability.'
author:
- Seung Ki Baek
title: Nonlinear Noise Reduction Scheme Based on Information Flow
---
It has been of great importance in communication and experimental research how to filter off noisy parts from the signal. As the broad-band spectrum of signals from nonlinear chaotic systems usually makes traditional linear filters unfeasible, many researchers have studied noise reduction methods applicable to nonlinear systems [@Kost1988; @Kost1989; @Hammel1990; @Farmer1991; @Davies1994; @Schreiber1993; @Kost1993; @Davies1998; @Brocker2001]. It is widely known that there are two kinds of noise : [*measurement noise*]{} means corruption of data in observation process without interfering dynamics itself, while [*dynamical noise*]{} denotes the perturbation of the system coupled to dynamics, occurring at each time step. The noise reduction problem is quite different for each case and we treat measurement noise in this paper. There exists a [*true*]{} orbit $\{Y_k\}_{k=1}^{N}$ satisfying certain dynamics $Y_{k+1} = M(Y_k)$ for $1 \le k \le N-1$, while one observes only a [*noisy*]{} orbit $\{X_k\}_{k=1}^{N}$ given by $X_k = Y_k + \eta_k$ for small $|\eta_k|<\delta$, where $\eta_k$ and $\delta$ denotes [*noise*]{} and [*noise level*]{}, respectively. We would like to obtain a less noisy orbit $\{X'_k\}_{k=1}^{N}$, and most approaches take this problem through minimizing a target function with constraints, such as $$\label{Lagrangian}
S = \sum_{k=1}^{N}|X'_k - X_k|^2 + \sum_{k=1}^{N-1}\{M(X'_k)-X'_{k+1}\} \lambda_k,$$ where $\lambda_k$ is a Lagrangian multiplier [@Farmer1991]. Minimizing $S$ corresponds to maximizing [*likelihood*]{} function $\bar{P}$ within a time interval $[t-\alpha, t+\beta]$ : $$\begin{aligned}
\label{likelihood}
&\bar{P}(&M^\alpha(X_{t-\alpha}),\ldots, M^{-\beta} (X_{t+\beta}))\nonumber\\
&\propto& \prod_{j=\alpha}^{j=-\beta} \exp \left( -\frac{1}{2 \sigma^2} \left| \frac{M^j (X_{t-j})-Y_t}{dM^j (Y_{t-j})} \right|^2 \right)\end{aligned}$$ where $dM$ is the derivative of $M$, under the assumption that the sequence $\{\eta_k\}$ is independently Gaussian distributed with standard deviation $\sigma$. Those probability distributions of position at different times are transported to a particular time, distorted by chaotic dynamics $M$, and the true data point is restricted to their intersection. Thus maximum of joint probability function $\bar{P}$ estimates the position of true data point at that time. We shall discuss how this calculation is simplified if we consider information aspects as in communication area.
Studies on communication using chaos [@Hayes; @Bollt; @Cuomo] has been carried out from the understanding of chaos control[@OGY; @Kantz] and chaos synchronization [@Pecora]. The main issues in this field are how to encode information using chaotic signal with dynamics already known to both of transmitter and receiver, and how to build a system persistent from noise occurring in communication channel, which corresponds to measurement noise. Rosa et al. [@Rosa] illustrated a filtering method using $2x \bmod 1$ map. This method, which will be called [*Rosa’s method*]{}, is described as following : one picks a point $(X_t, X_{t+1})$ and executes backward iteration on $X_{t+1}$ resulting in two preimages $\hat{X}_t^{Left}$ and $\hat{X}_t^{Right}$, one of which closest to $X_t$ is selected as a filtered point of time $t$. This filter shrinks noise by a factor of two (i.e. Lyapunov exponent of the map) at each iteration. Andreyev et al. [@Andreyev] investigated information aspects and applications of Rosa’s method. They, however, only treated basically 1-dimensional maps since they had to operate inverse mapping directly.
Maximum likelihood method and Rosa’s method are actually identical although the former originates from the topological distortion [@Wolf] and the latter from information property. In a viewpoint of information theory [@Shannon; @Brillouin], a chaotic system itself is interpreted as an active processor of information [@Ott]. Supposing we have a measuring tool with finitely limited resolution, stretching process reveals the initial state impossible to identify with the tool at that time more precisely. If only stretching process exists, the occupied areas in state space, i.e. the energy of the system diverges to infinity as the precision increases infinitely, as Brillouin claimed in Ref. [@Brillouin]. Folding process prevents this divergence, removing some stored information inevitably, so we cannot discriminate every detail of the past merely by observing the present state. Topological distortion, therefore, induces the flow of information bits and successive recording of this flow determines more precise knowledge of the state in chaotic systems. Information flow is a general property of chaos and, for example, all hyperbolic chaotic systems are already proven to have constant positive information rates by Schittenkopf and Deco [@Schittenkopf]. This idea forms the basis of our scheme which connects two previous methods. First, we begin with fully known dynamics, just as in communication, and discuss later how to deal with given data without [*a priori*]{} knowledge.
Following Rosa et al., we start with the case of $2x \bmod 1$ map as the simplest example of stretch-fold mechanism and also of our scheme. Employing binary representation in describing states, each iteration simply shifts the decimal point one space to the right. Let us assume that we introduce noise with such a level that we can guarantee only the first effective number. If the initial state is observed to be $0.a_0 x x \ldots$ and the first and second iteration give $0.a_1 x x \ldots$ and $0.a_2 x x \ldots$, respectively, noting that digits marked by $x$ may be spurious, we can say that the initial state is in fact $0.a_0 a_1 a_2 \cdots$, effectively reducing the noise on the initial state.
The above example involves two conditions : the noise level $\delta$ is known and the dynamics is chaotic. In such cases, we ignore the spoiled parts and that converts an observed point to a set of candidate points leading to degeneracy (e.g. all the points whose first digit is $a_0$). Then we clarify what it should be by receiving information from other unspoiled parts of data. Roughly speaking, proper temporal extension can compensate spatial ambiguity [@Pethel]. If a data point $X_t$ is observed, the real value $Y_t$ should lie within a finite neighborhood $I(X_t)$, whose size comes from the noise level $\delta$. The next real value $Y_{t+1}$, evolving from $Y_t$ deterministically, also belongs to $I(X_{t+1})$ while it does not hold for every point $p_t \in I(X_t)$ and its successor, $p_{t+1}$. Noting that the inverse mapping $M^{-1}$ operates on a set of points, not on a single point where the inverse map cannot be defined, we find the $n$-th order refinement, $$\label{refinement}
I(X_t)^{new}_{(n)} = \bigcap_{i=0}^{n} M^{-i}\left\{I(X_{t+i})\right\}.$$ In terms of the previous example, $M^{-i}\left\{I(X_{i})\right\}$ with $t=0$ means the set of binary numbers whose $i$-th digit is $a_i$. As the $n$-th order refinement requires $n+1$ successive measurements, it is obvious that the diameter of a remaining set never increases so that this algorithm is convergent : $$\label{convergence}
0 < \left| I(X_t)^{new}_{(n)} \right| \leq \left| I(X_t)^{new}_{(n-1)} \right|.$$ Equation (\[refinement\]) shares similarity with (\[likelihood\]) of maximum likelihood method, while Gaussian assumption is turned out to be unnecessary in our scheme. Once $\delta$ is defined, other details of noise are irrelevant. It is also worth noting that (\[refinement\]) formalizes the basic philosophy of Rosa’s method. The difficulty in application of it is remedied by rewriting (\[refinement\]) as following : $$\label{rewriting refinement}
M^n \left\{I(X_t)^{new}_{(n)}\right\} \subset \bigcap_{i=0}^{n} M^{i}\left\{I(X_{t+n-i})\right\}$$ and this allows one to avoid calculating inverse mapping, hardly possible in high dimensional systems. We deduce that if a point does not belong to the set of the right-hand side of (\[rewriting refinement\]), it cannot lie in the set of the left-hand side. Then what has to be done is only selecting points within $I(X_t)$ which satisfy the right-hand side after $n$ times of mapping. Henceforth, we iterate all nearby grid points around the observed data which approximate $I(X_t)$ in a discrete manner, and reject false ones getting outside the next expected intervals, $I(X_{t+1})$. We repeat the same procedure only on the surviving points until the number of remaining ones are less than a certain threshold, i.e. $\left| I(X_t)^{new}_{(m)}\right| < R_{th}$. $X_t$ is then corrected to $X'_k = \left<I(X_t)^{new}_{(m)}\right>$, the average of those remaining points. The number of steps $m$, required to reach this threshold $R_{th}$, measures the performance of noise reduction and we define this quantity as [*abrasion time*]{}. Since each point has its $m$, we obtain another sequence of abrasion time $\{m_k\}_{k=1}^{N}$ after refinement. A system with short $m$ is so sensitive that wrong guesses are easily rejected, and thus it is easy to clean noise. Later in inferring dynamics without knowledge of it, we use this concept in a different context, that is, fast abrasion implies large deviation from the true dynamics.
Figure 1 demonstrates the result of this scheme for Lorenz system : $$\label{Lorenz}
\left\{\begin{array}{rcl}
\dot{x} &=& \sigma (y-x)\\
\dot{y} &=& rx - y - xz\\
\dot{z} &=& xy - bz
\end{array}\right.$$
{width="40.00000%"}
where $\sigma = 10$, $r=28$ and $b=8/3$. The [*noisy*]{} orbit $\{X_k\}$ is generated in FIG. 1(a) by introducing noise of $\delta \approx 5\%$ of whole system size, which is enough to destroy most important characteristics of the attractor [@Kantz]. Our scheme corrects each point $X_k$ into $X'_k$, as depicted in (b), where $20\times20\times20$ neighboring grid points are constructed for each data point and $R_{th}$ is set to be $10$ throughout this calculation. We define [*relative variance*]{} as $$\label{relative variance}
e = \frac{\sum_{k=1}^{N}(Y_k - X'_k)^2}{\sum_{k=1}^{N}(Y_k - X_k)^2}$$ to quantify the performance of the scheme, where $e < 1$ means that noise is reduced ($e = 0$ for total noise reduction). This demonstration yields $e \approx 0.05$, which implies a high point-to-point correspondence so that this scheme can be categorized as [*detailed noise reduction*]{} following Ref. [@Farmer1991]. Similar results are obtained for Rössler system. Though Rosa et al. [@Rosa] propose that both forward and backward iterations are necessary for high dimensional systems, we do not perform backward one since this noninvertible $M$ lacks time reversal symmetry and thus information flows with only one direction.
So far the full knowledge of dynamics has been assumed for explaining convenience. Although this assumption may be valid in some area, we need to infer dynamics from given raw data in general. Farmer and Sidorowich pointed out that how much noise one can reduce is limited by the accuracy of approximation to the true dynamics [@Farmer1991]. At first, we tried to find local linear dynamics as Kotelich and Yorke did [@Kost1988], but it was not quite satisfactory since determining the size of neighborhood was troublesome, that is, too small size often decreases statistical confidence and too large one could not capture the fine structure of the attractor. Looking for alternatives consistent with the above scheme, we noted that the true dynamics would be the most accurate approximation among other candidate models and that our getting closer to the true dynamics could be expressed by longer $m$ in average.
Let us suppose that the parameter $r$ in (\[Lorenz\]), representing Rayleigh number in convection problem [@Strogatz], is unknown to us. Even if we are given the same data as FIG. 1(a), now we should test many Lorenz systems with different $r$ values until finding $r=28$. Figure 2 shows how the choice of $r$ changes the distribution of $\{m_k\}$. We depicted only two cases of $r=28$(correct) and $r=0$(wrong) though we observed the same tendency for intermediate values of $r$. The distribution looks Maxwellian in the vicinity of true dynamics and this Maxwellian region can be reached by processing raw data. We present a qualitative description with a statistical moment of the distribution. Imposing perturbed dynamics, we see that abrasion time goes to zero as our guesses are rejected soon by observations. The average abrasion time $\bar{m} = N^{-1} \sum_{k=1}^N m_k$ rises to 14.71 for $r=28$ while becomes only 5.96 for $r=0$. Let us consider two extreme cases to elucidate basic nature of the distribution : If the underlying dynamics is so trivial (e.g. stable periodic motion) that one can easily discover it, the future orbit is highly predictable and the distribution will be drawn to infinity. As a non-chaotic system contains little information, our noise reduction scheme becomes ineffective with diverging $\bar{m}$. Conversely, if dynamics looks totally unpredictable based on our knowledge, the distribution will collapse to zero point. We again see that noise is not reduced at all, since accuracy of approximation sets an upper bound of reducing performance, as stated above. Thus higher $\bar{m}$ is preferable when dynamics is unknown, while $\bar{m}$ divergence should be avoided when dynamics is known, which may seem contradictory at first. The balance between infinity and zero indicates a status between regularity and randomness, or between perfect predictability and unpredictability. In other words, $\bar{m}$ depends both on the system we observe and the information we have on that system.
{width="40.00000%"}
From the above arguments, we suggest an algorithm for inferring dynamics : One obtains enough time signals, possibly including noise, and chooses appropriate basis functions specified by a number of parameters. After rough estimation of the parameters, by means of fitting and smoothing algorithms, the higher $\bar{m}$ discovered around those values, the better dynamics inferred. In a brief numerical experiment, we set $(\dot{x}, \dot{y}, \dot{z}) = \vec{M}(x,y,z)$, where components of $\vec{M}$ are second-order polynomials of $x$, $y$, and $z$ with unknown coefficients and we observe that even a crude search can reduce noise with approaching the true dynamics (FIG. 3). Tests of 200 random samples around our rough guess give maximum $\bar{m}=5.37$ (only about 37% of that of true dynamics), but the relative variance $e$ becomes less than 0.7. Advanced parameter searching techniques is expected to yield desirable performance. Such error-tolerance property of [*$\bar{m}$-method*]{} is supposed to be due to a sort of shadowing effect : a deviated parameter operates as dynamical noise since it is coupled to the dynamics, and an incorrect model can be shadowed by less dynamical-noisy orbits (i.e. with less deviated parameters) within some distances [@Hammel1990; @Farmer1991].
{width="40.00000%"}
In summary, we suggested a nonlinear noise reduction scheme using ideas of information theory, which requires two conditions of chaoticity and well-defined noise level. Since information flow gradually reveals more precise knowledge, it formalizes the problem into rejection of hypotheses instead of minimization. Topological consideration and information-theoretic analysis combined in our scheme provide a concise and easily applicable way for noise reduction. Noise was readily decreased to less than a twentieth for fully known Lorenz system. We introduced abrasion time and proposed its average $\bar{m}$ as a quantifier for inferring dynamics. This $\bar{m}$-method was checked by performing noise reduction, since the accuracy of this inference fundamentally sets a limit on noise reducing capability. It readily yielded the expected noise reduction.
We would like to thank P. -J. Kim, S. -O. Jeong, H. -K. Park, C. Doe, T. -W. Ko, S. Chae and H. -T. Moon for their helpful comments. This work is supported by grant No. R01-1999-000-00019-0 from the Basic Research Program of Korea Science and Engineering Foundation.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Differential distributions for heavy quark production depend on the heavy quark mass and other momentum scales, which can yield additional large logarithms and inhibit accurate predictions. Logarithms involving the heavy quark mass can be summed in heavy quark parton distribution functions in the ACOT factorization scheme. A second class of logarithms involving the heavy-quark transverse momentum can be summed using an extension of Collins-Soper-Sterman (CSS) formalism. We perform a systematic summation of logarithms of both types, thereby obtaining an accurate description of heavy-quark differential distributions at all energies. Our method essentially combines the ACOT and CSS approaches. As an example, we present angular distributions for bottom quarks produced in parity-conserving events at large momentum transfers $ Q $ at the $ ep $ collider HERA.'
author:
- 'P. M. Nadolsky$ ^{1} $, N. Kidonakis$ ^{2} $, F. I. Olness$ ^{1} $, C.-P. Yuan$ ^{3} $'
date: 14th October 2002
title: |
Resummation of transverse momentum and mass logarithms\
in DIS heavy-quark production
---
Introduction\[sec:Intro\]
=========================
In recent years, significant attention was dedicated to exploring properties of heavy-flavor hadrons produced in lepton-nucleon deep inelastic scattering (DIS). On the experimental side, the Hadron-Electron Ring Accelerator (HERA) at DESY has generated a large amount of data on the production of charmed [@Adloff:1996xq; @Breitweg:1997mj; @Adloff:1998vb; @Breitweg:1999ad; @Adloff:2001zj] and bottom mesons [@Adloff:1999nr; @Breitweg:2000nz; @BottomDISH1; @BottomDISZEUS1; @BottomZEUS2]. At present energies (of order 300 GeV in the $ ep $ center-of-mass frame), a substantial charm production cross section is observed in a wide range of Bjorken $ x $ and photon virtualities $ Q^{2} $, and charm quarks contribute up to 30% to the DIS structure functions.
On the theory side, Perturbative Quantum Chromodynamics (PQCD) provides a natural framework for the description of heavy-flavor production. Due to the large masses $ M $ of the charm and bottom quarks ($ M^{2}\gg \Lambda _{QCD}^{2} $), the renormalization scale can be always chosen in a region where the effective QCD coupling $ \alpha _{S} $ is small. Despite the smallness of $ \alpha _{S} $, perturbative calculations in the presence of heavy flavors are not without intricacies. In particular, care in the choice of a factorization scheme is essential for the efficient separation of the short- and long-distance contributions to the heavy-quark cross section. This choice depends on the value of $ Q $ as compared to the heavy quark mass $ M $. The key issue here is whether, for a given renormalization and factorization scale $ \mu _{F}\sim Q $, the heavy quarks of the $ N $-th flavor are treated as *partons* in the incoming proton, *i.e.*, whether one calculates the QCD beta-function using $ N $ active quark flavors and introduces a parton distribution function (PDF) for the $ N $-th flavor. A related, but separate, issue is whether the mass of the heavy quark can be neglected in the hard cross section without ruining the accuracy of the calculation.
Currently, several factorization schemes are available that provide different approaches to the treatment of these issues. Among the mass-retaining factorization schemes, we would like to single out the fixed flavor number factorization scheme (FFN scheme), which includes the heavy-quark contributions exclusively in the hard cross section [@Gluck:1982cp; @Gluck:1988uk; @Nason:1989zy; @Laenen:1992cc; @Laenen:1993zk; @Laenen:1993xs]; and massive variable flavor number schemes (VFN schemes), which introduce the PDFs for the heavy quarks and change the number of active flavors by one unit when a heavy quark threshold is crossed [@Collins:1998rz; @Aivazis:1994pi; @Kniehl:1995em; @Buza:1998wv; @Thorne:1998ga; @Thorne:1998uu; @Cacciari:1998it; @Chuvakin:1999nx; @Kramer:2000hn]. Further details on these schemes can be found later in the paper. Here we would like to point out that, were the calculation done to all orders of $ \alpha _{S} $, the FFN and massive VFN schemes would be exactly equivalent. However, in a finite-order calculation the perturbative series in one scheme may converge faster than that in the other scheme. In particular, the FFN scheme presents the most economic way to organize the perturbative calculation near the heavy quark threshold, *i.e.*, when $ Q^{2}\approx M^{2} $. At the same time, it becomes inappropriate at $ Q^{2}\gg M^{2} $ due to powers of large logarithms $ \ln \left( Q^{2}/M^{2}\right) $ in the hard cross section. In the VFN schemes, these logarithms are summed through all orders in the heavy-quark PDF with the help of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [@Dokshitzer:1977sg; @Gribov:1972ri; @Altarelli:1977zs]; hence the perturbative convergence in the high-energy limit is preserved. In their turn, the VFN schemes may converge slower at $ Q^{2}\approx M^{2} $, mostly because of the violation of energy conservation in the heavy-quark PDF’s in that region. Recently an optimal VFN scheme was proposed that compensates for this effect [@Tung:2001mv].
In this paper, we would like to concentrate on the analysis of semi-inclusive differential distributions (*i.e.*, distributions depending on additional kinematical variables besides $ x $ and $ Q $). We will argue that finite-order calculations in any factorization scheme do not satisfactorily describe such distributions due to additional large logarithms besides the logarithms $ \ln (Q^{2}/M^{2}) $. To obtain stable predictions, all-order summation of these extra logarithmic terms is necessary.
The extra logarithms are of the form $ (\alpha _{S}^{n}/q_{T}^{2})\ln ^{m}(q_{T}^{2}/Q^{2}) $, $ 0\leq m\leq 2n-1 $, where $ q_{T}=p_{T}/z $ denotes the transverse momentum $ p_{T} $ of the heavy hadron in the $ \gamma ^{*}p $ center-of-mass (c.m.) reference frame rescaled by the variable $ z\equiv (p_{A}\cdot p_{H})/(p_{A}\cdot q) $. Here $ p_{A}^{\mu },\, q^{\mu }, $ and $ p_{H}^{\mu } $ are the momenta of the initial-state proton, virtual photon, and heavy hadron, respectively. Our definitions for the $ \gamma ^{*}p $ c.m. frame and hadron momenta are illustrated by Fig. \[fig:GammaP\]. The resummation of these logarithms is needed when the final-state hadron escapes in the current fragmentation region (*i.e.*, close the direction of the virtual photon in the $ \gamma ^{*}p $ c.m. frame, where the rate is the largest). In the current fragmentation region, the ratio $ q_{T}^{2}/Q^{2} $ is small; therefore, the terms $ \ln ^{m}(q_{T}^{2}/Q^{2}) $ compensate for the smallness of $ \alpha _{S} $ at each order of the perturbative expansion. If hadronic masses are neglected, such logarithms can be summed through all orders in the impact parameter space resummation formalism [@Collins:1993kk; @Meng:1996yn; @Nadolsky:1999kb; @Nadolsky:2000ky], which was originally introduced to describe angular correlations in $ e^{+}e^{-} $ hadroproduction [@Collins:1981uk; @Collins:1982va] and transverse momentum distributions in the Drell-Yan process [@Collins:1985kg].[^1] Here the impact parameter $ b $ is conjugate to $ q_{T} $. The results of Refs. [@Collins:1993kk; @Meng:1996yn; @Nadolsky:1999kb; @Nadolsky:2000ky] are immediately valid for semi-inclusive DIS (SIDIS) production of light hadrons ($ \pi ,K,... $) at $ Q $ of a few GeV or higher, and for semi-inclusive heavy quark production at $ Q^{2}\gg M^{2} $. To describe heavy-flavor production at $ Q^{2}\sim M^{2} $, the massless $ q_{T} $-resummation formalism must be extended to include the dependence on the heavy-quark mass $ M $.
In this paper, we perform such extension in the Aivazis-Collins-Olness-Tung (ACOT) massive VFN scheme threshold region [@Tung:2001mv]. We adopt a “bottom-up” approach to the development of such mass-dependent resummation.[^2] We start by separately reviewing the massive VFN scheme in the inclusive DIS and $ q_{T} $ resummation in the massless SIDIS. We then discuss a combination of these two frameworks in a joint resummation of the logarithms $ \ln (Q^{2}/M^{2}) $ and $ \ln (q_{T}^{2}/Q^{2} $). As a result, we obtain a unified description of fully differential heavy-hadron distributions at all $ Q^{2} $ above the heavy quark threshold. It is well known that the finite-order calculation does not satisfactorily treat the current fragmentation region for any choice of the factorization scheme. In contrast, the proposed massive extension of the $ q_{T} $-resummation accurately describes the current fragmentation region in the whole range $ Q^{2}>M^{2} $.
The present study is interesting for two phenomenological reasons. Firstly, the quality of the differential data will improve greatly within the next few years. By 2006, the upgraded collider HERA will accumulate an integrated luminosity of $ 1\mbox {\, fb}^{-1} $ [@Greenshaw:2002wu], *i.e.*, more than eight times the final integrated luminosity from its previous runs. Studies of the heavy quarks in DIS are also envisioned at the proposed high-luminosity Electron Ion Collider [@EICWhiteBook] and THERA [@Abramowicz:2001qt]. Eventually these experiments will present detailed distributions both at small ($ Q^{2}\approx M^{2} $) and large ($ Q^{2}\gg M^{2} $) momentum transfers.
Secondly, the knowledge of the differential distributions is essential for the accurate reconstruction of inclusive observables, such as the charm component of the structure function $ F_{2}(x,Q^{2}) $. At HERA,
As an example, we apply the developed method to the leading-order flavor-creation and flavor-excitation processes in the production of bottom mesons at HERA. We find that the resummed cross section for this process can be described purely by means of perturbation theory due to the large mass of the bottom quark. Our predictions can be tested in the next few years once the integrated luminosity at HERA approaches $ 1\mbox {\, fb}^{-1}. $ Essentially the same method can be applied to charm production. In that case, however, the resummed cross section is sensitive to the nonperturbative large-$ b $ contributions due to the smaller mass of the charm quarks, and the analysis is more involved. Since the goal of this paper is to discuss the basic principles of the massive $ q_{T} $-resummation, we leave the study of the charm production and other phenomenological aspects for future publications.
The paper has the following structure. Section \[sec:ACOT\] reviews the application of the ACOT factorization scheme [@Aivazis:1994pi] and its simplified version [@Collins:1998rz; @Kramer:2000hn] to the description of the inclusive DIS structure functions. Section \[sec:CSS\] recaps basic features of the $ b $-space resummation formalism **in *massless* SIDIS. Section \[sec:MassiveCSS\] discusses modifications in the resummed cross section to incorporate the dependence on the heavy-quark mass $ M $. In Section \[sec:PhotonGluon\], we present a detailed calculation of the mass-dependent resummed cross sections in the leading-order flavor-creation and flavor-excitation processes. Section \[sec:NumericalResults\] presents numerical results for polar angle distributions in the production of bottom quarks at HERA. Appendix \[Appendix:Chg\] contains details on the calculation of the $ {\cal O}(\alpha _{S}) $ mass-dependent part of the resummed cross section. In Appendix \[Appendix:FO\], we present explicit expressions for the $ {\cal O}(\alpha _{S}) $ finite-order contributions from the photon-gluon channel. Finally, Appendix \[Appendix:KinematicalCorrection\] discusses in detail the optimization of the ACOT scheme when it is applied to the differential distributions in the vicinity of the threshold region.
Overview of the factorization scheme \[sec:ACOT\]
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Factorization in the presence of heavy quarks
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In this Section, we discuss the application of the Aivazis-Collins-Olness-Tung (ACOT) factorization scheme [@Aivazis:1994pi] to inclusive DIS observables, for which this scheme yields accurate predictions both at asymptotically high energies and near the heavy-quark threshold.
In the inclusive DIS, the factorization in the presence of heavy flavors is established by a factorization theorem [@Collins:1998rz], which we review under a simplifying assumption that only one heavy flavor $ h $ with the mass $ M $ is present. Let $ A $ denote the incident hadron. According to the theorem, the contribution $ F_{h/A}(x,Q^{2}) $ of $ h $ to a DIS structure function $ F(x,Q^{2}) $ (where $ F(x,Q^{2}) $ is one of the functions $ F_{1}(x,Q^{2}) $ or $ F_{2}(x,Q^{2}) $ in parity-conserving DIS) can be written as $$\begin{aligned}
& & F_{h/A}(x,Q^{2})=\sum _{a}\int _{\chi _{a}}^{1}\frac{d\xi }{\xi }C_{h/a}\left( \frac{\chi _{a}}{\xi },\frac{\mu _{F}}{Q},\frac{M}{Q}\right) \nonumber \\
& & \times f_{a/A}\left( \xi ,\, \left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) +{\mathcal{O}}\left( \frac{\Lambda _{QCD}}{Q}\right) .\label{F} \end{aligned}$$ Here the summation over the internal index $ a $ includes both light partons (gluons $ G $ and light quarks), as well as the heavy quark $ h $. This representation is accurate up to the non-factorizable terms that do not depend on $ M $ and can be ignored when $ Q\gg \Lambda _{QCD} $. The non-vanishing term on the r.h.s. is written as a convolution integral of parton distribution functions $ f_{a/A}\left( \xi ,\, \{\mu _{F}/m_{q}\}\right) $ and coefficient functions $ C_{h/a}(\chi _{a}/\xi ,\, \mu _{F}/Q,\, M/Q) $. The convolution is realized over the hadron light-cone momentum fraction $ \xi $ carried by the parton $ a $. The coefficient function depends on the flavor-dependent “scaling variable” $ \chi _{a} $ discussed below. The parton distributions and coefficient functions are separated by an *arbitrary* factorization scale $ \mu _{F} $ such that $ f_{a/A} $ depends only on $ \mu _{F} $ and quark masses $ \{m_{q}\}\equiv m_{u},m_{d},...,M $; and $ C_{h/a} $ depends only on $ \mu _{F},\, M, $ and $ Q $. As a result of this separation, all logarithmic terms $ \alpha _{S}^{n}\ln ^{k}\left( \mu _{F}/m_{q}\right) $ *with light-quark masses* are included in the PDF’s, where they are summed through all orders using the DGLAP equation. Note that in the massless approximation such logarithms appear in the guise of collinear poles $ 1/\epsilon ^{k} $ in the procedure of dimensional regularization. The logarithms $ \ln ^{k}(\mu _{F}/M) $ with *the heavy-quark mass* $ M $ are included either in $ C_{h/a} $ or $ f_{a/A} $ depending on the factorization scheme in use.
In the reference frame where the momentum of the incident hadron $ A $ in the light-cone coordinates is $$p_{A}^{\mu }=
\left\{ p_{A}^{+},\frac{m_{A}^{2}}{2p_{A}^{+}},{{\bf 0}}_{T}\right\} ,$$ (where $ p^{\pm }\equiv \left( p^{0}\pm p^{3}\right) /\sqrt{2} $), the quark PDF can be defined in terms of the quark field operators $ \psi_q (x) $ as [@Collins:1982uw] $$\begin{aligned}
& & f_{q/A}\left( \xi ,\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) =\overline{\sum _{spin}}\, \overline{\sum _{color}}\int \frac{dy^{-}}{2\pi }e^{-i\xi p_{A}^{+}y^{-}}\nonumber \\
& & \times \langle p_{A}|\bar{\psi_q }(0,y^{-},{{\bf 0}}_{T})\nonumber \\
& & \times {\cal P}\exp \left\{ -ig\int _{0}^{y^{-}}dz^{-}{\mathscr A}^{+}(0,z^{-},{{\bf 0}}_{T})\right\} \nonumber \\
& & \times \frac{\gamma ^{+}}{2}\psi_q (0)|p_{A}\rangle .\label{f} \end{aligned}$$ Here $ {\cal P}\exp {\{...\}} $ is the path-ordered exponential of the gluon field $ {\mathscr A}_{\nu }(x) $ in the gauge $ \eta \cdot {\mathscr A}=0 $. The r.h.s. is averaged over the spin and color of $ A $ and summed over the spin and color of $ q $. A similar definition exists for the gluon PDF. The dependence of $ f_{a/A}(\xi ,\{\mu _{F}/m_{q}\}) $ on $ \mu _{F} $ is induced in the process of renormalization of ultraviolet (UV) singularities that appear in the bilocal operator on the r.h.s. of Eq. (\[f\]). In general, the PDF is a nonperturbative object; however, it can be calculated in PQCD when $ \mu _{F}\gg \Lambda _{QCD} $, and the incident hadron $ A $ is replaced by a parton. This feature opens the door for the calculation of $ F_{h/A}(x,Q^{2}) $ for any hadron $ A $ through the conventional sequence of calculating $ C_{h/a}(\chi _{a}/\xi ,\mu _{F}/Q,M/Q) $ in parton-level DIS and convolving it with the phenomenological parameterization of the nonperturbative PDF $ f_{a/A}(\xi ,\{\mu _{F}/m_{q}\}) $. In the inclusive DIS, it is convenient to choose $ \mu _{F}\sim Q $ to avoid the appearance of the large logarithm $ \ln (\mu _{F}/Q) $ in $ C_{h/a}(\chi _{a}/\xi ,\mu _{F}/Q,M/Q) $.
The factorized representation (\[F\]) is valid in all factorization schemes. The specific factorization scheme is determined by (a) the procedure for the renormalization of the UV singularities; and (b) the prescription for keeping or discarding terms with positive powers of $ M/Q $ in the coefficient function $ C_{h/a} $. The choice (a) determines if the logarithms $ \ln ^{k}(\mu _{F}/M) $ are resummed in the heavy-flavor PDF or not. With respect to each of two issues, the choice can be done independently. For instance, the $ \overline{MS} $ factorization scheme uses the dimensional regularization to handle the UV singularities, but does not uniquely determine the choice (b). Hence, it is not necessary in this scheme to always neglect $ M $ in the coefficient function and expose the heavy-quark mass singularities as poles in the dimensional regularization.
The ACOT scheme belongs to the class of the variable flavor number (VFN) factorization schemes [@Collins:1978wz] that change the renormalization prescription when $ \mu _{F} $ crosses a threshold value $ \mu _{thr} $. It is convenient to choose $ \mu _{thr} $ for the flavor $ h $ to be equal to $ M $, since the logarithms $ \ln ^{k}\left( \mu _{F}/M\right) $ vanish at that point. If $ \mu _{F}<M $, all graphs with internal heavy-quark lines are renormalized by zero-momentum subtraction. If $ \mu _{F}>M $, these graphs are renormalized in the $ \overline{MS} $ scheme. The masses of the light quarks are neglected everywhere, and graphs with only light parton lines are always renormalized in the $ \overline{MS} $ scheme.
The physical picture behind the ACOT prescription is simple: the heavy quark is excluded as a constituent of the hadron for sufficiently low energy (an $ N-1 $ flavor subscheme), but the heavy quark is included as a constituent for sufficiently high energies (an $ N $ flavor subscheme). The renormalization by zero-momentum subtraction below the threshold leads to the explicit decoupling of the heavy-quark contributions from light parton lines. As one consequence of the decoupling, all perturbative components of the heavy-quark PDF vanish at $ \mu _{F}<M $, so that a nonzero heavy-quark PDF may appear only through nonperturbative channels, such as the “intrinsic heavy quark mechanism” [@Brodsky:1980pb]. Since the size of such nonperturbative contributions remains uncertain, they are not considered in this study. At $ \mu _{F}>M $, a non-zero heavy-quark PDF $ f_{h/A} $ is introduced, which is evolved together with the rest of the PDFs with the help of the mass-independent $ \overline{MS} $ splitting kernels. The initial condition for $ f_{h/A}(\xi ,\mu _{F}) $ is obtained by matching the factorization subschemes at $ \mu _{F}=M $. At order $ \alpha _{S} $, this condition is trivial: $$f_{h/A}(\xi ,\mu _{F}=M)=0.$$ At higher orders, the initial value of $ f_{h/A}(\xi ,\mu _{F}) $ is given by a superposition of light-flavor PDF’s [@Buza:1998wv]. A simple illustration of these issues is given in Appendix \[Appendix:Chg\].
The ACOT scheme possesses another important property: the coefficient function $ C_{h/a} $ in this scheme has a finite limit as $ Q\rightarrow \infty $, which coincides with the expression for the coefficient function obtained in the massless $ \overline{MS} $ scheme with $ N $ active flavors. This happens because the mass-dependent terms in $ C_{h/a} $ contain only positive powers of $ M/Q $, while the quasi-collinear logarithms $ \ln (\mu _{F}/M) $ are resummed in $ f_{h/A}(\xi ,\mu _{F}) $. As a consequence of the introduction of $ f_{h/A}(\xi ,\mu _{F}) $, the coefficient function $ C_{h/a} $ includes subprocesses of three classes:
- *flavor excitation*, where the parton $ a $ is a heavy quark;
- gluon *flavor creation*, where $ a $ is a gluon;
- and light-quark *flavor creation*, where $ a $ is a light quark.
In contrast, in the FFN scheme [@Gluck:1982cp; @Gluck:1988uk; @Nason:1989zy; @Laenen:1992cc; @Laenen:1993xs; @Laenen:1993zk] only the flavor-creation processes are present. The lowest-order diagrams for each class are shown in Fig. \[fig:diag2\].
The subsequent parts of the paper consider the processes shown in Figs. \[fig:diag2\]a and \[fig:diag2\]b. Note that we count the order of diagrams according to the explicit power of $ \alpha _{S} $ in the coefficient function, *i.e.*, $ {\cal O}(\alpha _{S}^{0}) $ in Fig. \[fig:diag2\]a, $ {\cal O}(\alpha _{S}^{1}) $ in Fig. \[fig:diag2\]b, and $ {\cal O}(\alpha _{S}^{2}) $ in Fig. \[fig:diag2\]c. This counting does not apply to the whole structure function $ F_{h/A}(x,Q^{2}) $ in Eq. (\[F\]) when the heavy-quark PDF is itself suppressed by $ \alpha _{S}/\pi $ near the mass threshold [@Olness:1988ep; @Barnett:1988jw; @Sullivan:2001ry]. In that region, an $ {\cal O}(\alpha _{S}^{n}) $ *flavor-excitation* contribution has roughly the same order of magnitude as the $ {\cal O}(\alpha _{S}^{n+1}) $ *flavor-creation* contribution. We return to this issue in the discussion of numerical results in Section \[sec:NumericalResults\], where we interpret the combination of the $ {\cal O}(\alpha _{S}^{0}) $ *flavor-excitation* contribution (Fig. \[fig:diag2\]a) and $ {\cal O}(\alpha _{S}^{1}) $ *flavor-creation* contribution (Fig. \[fig:diag2\]b) as a first approximation at $ Q\approx M $.
Simplified ACOT Formalism
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Of several available versions of the ACOT scheme, our calculation utilizes its modification advocated by Collins [@Collins:1998rz], which we identify as the Simplified ACOT (S-ACOT) formalism [@Kramer:2000hn]. It has the advantage of being easy to state and of allowing relatively simple calculations. This simplicity could be crucial for implementing the massive VFN prescription at the next-to-leading order in the global analysis of parton distributions.
In brief, this prescription is stated as follows.
> *Simplified ACOT (S-ACOT) prescription*: Set $ M $ to zero in the calculation of the coefficient functions $ C_{h/a} $ for the incoming heavy quarks: that is, $$C_{h/h}\left( \frac{\chi _{h}}{\xi },\frac{\mu _{F}}{Q},\frac{M}{Q}\right) \rightarrow C_{h/h}\left( \frac{\chi _{h}}{\xi },\frac{\mu _{F}}{Q},0\right) .$$
It is important to note that this prescription is not an approximation; it correctly accounts for the full mass dependence [@Collins:1998rz]. It also tremendously reduces the complexity of flavor-excitation structure functions, as they are given by the light-quark result. In the specific case considered here, the heavy quark mass in the S-ACOT scheme should be retained only in the $ \gamma ^{*}+G\rightarrow h+\bar{h} $ subprocess (Fig. \[fig:diag2\]b). Another important consequence will be discussed in Section \[sec:MassiveCSS\], where we show that the S-ACOT scheme leads to a simpler generalization of the $ q_{T} $-resummation to the mass-dependent case.
\[sec:DISrescalingVariable\]The scaling variable
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Finally, we address the issue of the most appropriate variables $ \chi _{a} $ ($ a=G,u,d,s,... $) in the convolution integral (\[F\]). In a massless calculation, $ \chi _{a} $ are just equal to Bjorken $ x $, since all momentum fractions $ \xi $ between $ x $ and unity are allowed by energy conservation. This simple relation does not hold in the massive case. For instance, in the charged-current heavy quark production $ W^{\pm }+q\rightarrow h, $ where $ h $ is present in the final, but not the initial, state, a simple kinematical argument leads to the conclusion that the longitudinal variable in the flavor-excitation processes should be rescaled by a mass-dependent factor, as $ \chi _{h}=x\left( 1+M^{2}/Q^{2}\right) $ [@Barnett:1988jw].
In the flavor-excitation subprocesses of the neutral-current heavy quark production (*e.g.*, $ \gamma ^{*}+h\rightarrow h $), typically no rescaling correction was made. The presence of a heavy quark in *both* the initial and final states of the hard scattering suggested that no kinematical shift was necessary, *i.e.*, $ \chi _{h}=x $. This assumption has been recently questioned by a new analysis [@Tung:2001mv]. Specifically, Tung *et al.* note that the heavy quarks in the hadron come predominantly from gluons splitting into quark-antiquark pairs. Hence the heavy quark $ h $ initiating the hard process must be accompanied by the unobserved $ \bar{h} $ in the beam remnant. When both $ h $ and $ \bar{h} $ are present, the hadron’s light-cone momentum fraction carried by the incoming parton cannot be smaller than $ x\left( 1+4M^{2}/Q^{2}\right) $, which is larger than the minimal momentum fraction $ \xi _{min}=x $ allowed by the single-particle inclusive kinematics. The factor of $ 4M^{2} $ arises from the threshold condition for $ h $ and $ \bar{h} $. This effect can be accounted for by evaluating the flavor-excitation cross sections at the scaling variable $ \chi _{h}=x\left( 1+4M^{2}/Q^{2}\right) $.
In brief, the rule proposed in Ref. [@Tung:2001mv] is to use $ \chi _{a}=x\left( 1+4M^{2}/Q^{2}\right) $ in flavor-excitation processes (Fig. \[fig:diag2\]a) and $ \chi _{a}=x $ in flavor-creation processes (Figs. \[fig:diag2\]b and \[fig:diag2\]c) when calculating inclusive cross sections.
Massless transverse momentum resummation\[sec:CSS\]
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We now turn to the differential distributions of the heavy-flavor cross sections. Specifically, we consider the production of a heavy-quark hadron $ H $ via the process $ e(\ell )+A(p_{A})\rightarrow H(p_{H})+e(\ell' )+X. $ This reaction is illustrated in Fig. \[fig:GammaP\] for the specific case when $ A $ is a proton. In much of the discussion, we will find it convenient to amputate the external lepton legs and work with the photon-hadron process $ \gamma ^{*}(q)+A(p_{A})\rightarrow H(p_{H})+X $ in the photon-hadron c.m. frame. Given the conventional DIS variables $ Q^{2}=-q^{2} $ and $ x=Q^{2}/(2p_{A}\cdot q) $, as well as the Lorentz invariant $ S_{eA}\equiv (\ell +p_{A})^{2} $, we decompose the electron-level cross section into a sum over the functions $ A_{\rho }(\psi ,\varphi ) $ of the lepton azimuthal angle $ \varphi $ and boost parameter $ \psi \equiv \cosh ^{-1}\left( 2xS_{eA}Q^{-2}-1\right) $ [@Meng:1992da; @Nadolsky:1999kb]: $$\begin{aligned}
\frac{d\sigma (e+A\rightarrow e+H+X)}{dxdQ^{2}d{{\bf p}}_{H}} & \propto & \sum _{\rho }V_{\rho }(q,p_{A},{{\bf p}}_{H})\nonumber \\
& \times & A_{\rho }(\psi ,\varphi ).\end{aligned}$$ This procedure is nothing else but the decomposition over the virtual photon’s helicities [@Cheng:1971mx; @Lam:1978pu; @Olness:1987mv]; hence it is completely analogous to the tensor decomposition familiar from the inclusive DIS. As a result of this procedure, the dependence on the kinematics of the final-state lepton is factorized into the functions $ A_{\rho }(\psi ,\varphi ), $ while the hadronic dynamics affects only the functions $ V_{\rho } $. In parity-conserving SIDIS, the only contributing angular functions are $$\begin{aligned}
A_{1}(\psi ,\varphi ) & = & 1+\cosh ^{2}\psi ,\nonumber \\
A_{2}(\psi ,\varphi ) & = & -2,\nonumber \\
A_{3}(\psi ,\varphi ) & = & -\cos \varphi \sinh 2\psi ,\nonumber \\
A_{4}(\psi ,\varphi ) & = & \cos 2\varphi \sinh ^{2}\psi .\label{As} \end{aligned}$$
In Section \[sec:ACOT\] we found that the ACOT prescription resums logarithms of the form $ \ln (M^{2}/Q^{2}) $. For the inclusive observables, this procedure provides accurate predictions throughout the full range of $ x $ and $ Q^{2} $. More differential observables may contain additional large logarithms in the high-energy limit. In particular, we already mentioned the logarithms of the type $ (q_{T}^{-2})\alpha _{S}^{n}\ln ^{m}(q_{T}^{2}/Q^{2}),\, 0\leq m\leq 2n-1 $, which appear when the polar angle $ \theta _{H} $ of the heavy hadron $ H $ in the $ \gamma ^{*}A $ c.m. frame becomes small (cf. Fig. \[fig:GammaP\]). Here we chose the $ z $-axis to be directed along the momentum $ {{\bf q}} $ of the virtual photon $ \gamma ^{*} $. When $ M^{2}\ll Q^{2} $, the scale $ q_{T} $ is related to $ \theta _{H} $ as $$q_{T}^{2}=Q^{2}\left( \frac{1}{x}-1\right) \frac{1-\cos \theta _{H}}{1+\cos \theta _{H}};$$ hence $$\lim _{\theta _{H}\rightarrow 0}q_{T}^{2}=Q^{2}\left( \frac{1}{x}-1\right) \left( \frac{\theta _{H}^{2}}{4}+...\right) \rightarrow 0.$$
The resummation of these logarithms of soft and collinear origin can be realized in the formalism by Collins, Soper, and Sterman (CSS) [@Collins:1981uk; @Collins:1982va; @Collins:1985kg; @Collins:1989PQCD]. The result can be expressed as a factorization theorem, which states that in the limit $ Q^{2}\gg q_{T}^{2},\{m^{2}_{q}\},\Lambda ^{2}_{QCD} $ the cross section is $$\begin{aligned}
& & \left. \frac{d\sigma (e+A\rightarrow e+H+X)}{d\Phi }\right| _{q_{T}^{2}\ll Q^{2}}=\frac{\sigma _{0}F_{l}}{2S_{eA}}A_{1}(\psi ,\varphi )\nonumber \\
& & \times \int \frac{d^{2}{{\bf b}}}{(2\pi )^{2}}e^{i{{\bf q}}_{T}\cdot {{\bf b}}}\widetilde{W}_{HA}(b,Q,x,z)\nonumber \\
& & +{\mathcal{O}}\left( \frac{q_{T}}{Q},\left\{ \frac{m_{q}}{Q}\right\} ,\frac{\Lambda _{QCD}}{Q}\right) .\label{Wmassless} \end{aligned}$$ In this equation, $ b $ is the impact parameter (conjugate to $ {q_{T}}$), $ d\Phi \equiv dxdQ^{2}dzdq_{T}^{2}d\varphi $, $ z\equiv (p_{A}\cdot p_{H})/(p_{A}\cdot q) $, and $ \sigma _{0} $ and $ F_{l} $ are constant factors **given in** Eq. (\[sigma0Fl\]). As before, $ \{m_{q}\} $ collectively denotes all quark masses, $ \{m_{q}\}\equiv m_{u},m_{d},...,M. $ At large $ Q^{2} $, the $ b $-space integral in Eq. (\[Wmassless\]) is dominated by contributions from the region $ b^{2}\lesssim 1/Q^{2} $. In this region, the hadronic form factor $ \widetilde{W}_{HA}(b,Q,x,z) $ can be factorized in a combination of parton distribution functions $ f_{a/A}(\xi _{a},\mu _{F}) $, fragmentation functions $ D_{H/b}(\xi _{b},\mu _{F}) $, and the partonic form factor $ \widehat{\widetilde{W}}_{ba} $: $$\begin{aligned}
& & \widetilde{W}_{HA}\left( b,Q,x,z\right) =\sum _{a,b}\, \int _{x}^{1}\frac{d\xi _{a}}{\xi _{a}}\int _{z}^{1}\frac{d\xi _{b}}{\xi _{b}}\nonumber \\
& & \times D_{H/b}(\xi _{b},\mu _{F})\widehat{\widetilde{W}}_{ba}\left( b,Q,{\widehat{x}},{\widehat{z}},\mu _{F}\right) \nonumber \\
& & \times f_{a/A}(\xi _{a},\mu _{F}),\label{WHA} \end{aligned}$$ where $$\begin{aligned}
& & {\widehat{{\widetilde{W }} }}_{ba}(b,Q,{\widehat{x}},{\widehat{z}},\mu _{F})=\sum _{j}e_{j}^{2}\, e^{-{\cal S}(b,Q,C_{1},C_{2})}\nonumber \\
& & \times {\mathcal{C}}^{out}_{b/j}\left( {\widehat{z}},\mu _{F}b;\frac{C_{1}}{C_{2}}\right) {\mathcal{C}}_{j/a}^{in}\left( {\widehat{x}},\mu _{F}b;\frac{C_{1}}{C_{2}}\right) .\label{W0} \end{aligned}$$ Here $ {\widehat{x}}\equiv x/\xi _{a},\, {\widehat{z}}\equiv z/\xi _{b} $. The indices $ a,b $ in Eq. (\[WHA\]) are summed over all quark flavors and gluons; the summation over $ j $ in Eq. (\[W0\]) is over the quarks only. The fractional charge of a quark $ j $ is denoted as $ e^{2}e_{j}^{2} $. The parton distributions and fragmentation functions are separated from the partonic form factor $ {\widehat{{\widetilde{W }} }}_{ba} $ at the factorization scale $ \mu _{F} $. The Sudakov factor $ {\cal S}(b,Q,C_{1},C_{2}) $ is an all-order sum of logarithms $ \ln ^{m}{(q_{T}^{2}/Q^{2})} $. It is given by an integral between scales $ C^{2}_{1}/b^{2} $ and $ C^{2}_{2}Q^{2} $ (where $ C_{1} $ and $ C_{2} $ are constants of order 1) of two functions $ {\cal A}\left( \alpha _{S}(\bar{\mu });C_{1}\right) $ and $ {\cal B}(\alpha _{S}(\bar{\mu });C_{1},C_{2}) $ appearing in the solution of equations for renormalization- and gauge-group invariance: $$\begin{aligned}
& & {\cal S}=\int _{C_{1}^{2}/b^{2}}^{C_{2}^{2}Q^{2}}\frac{d\bar{\mu }^{2}}{\bar{\mu }^{2}}\Biggl [\ln \left( \frac{C_{2}^{2}Q^{2}}{\bar{\mu }^{2}}\right) {\cal A}\left( \alpha _{S}(\bar{\mu });C_{1}\right) \nonumber \\
& & +{\cal B}(\alpha _{S}(\bar{\mu });C_{1},C_{2})\Biggr ].\label{Smassless} \end{aligned}$$ The functions $ {\cal C}^{in} $, $ {\cal C}^{out} $ contain perturbative corrections to contributions from the incoming and outgoing hadronic jets, respectively. To evaluate the Fourier-Bessel transform integral, $ \widetilde{W}_{HA}\left( b,Q,x,z\right) $ should be also defined at $ b\gtrsim 1\mbox {\, GeV}^{-1} $, where the perturbative methods are not trustworthy. The continuation of $ \widetilde{W}_{HA}\left( b,Q,x,z\right) $ to the large-$ b $ region is realized with the help of some phenomenological model, as discussed, *e.g.*, in Refs. [@Collins:1985kg; @Qiu:2000hf; @Kulesza:2002rh].
As noted above, the resummed cross section in Eq. (\[Wmassless\]), which we shall label as $ {\sigma _{{\widetilde{W }}}}$, is derived in the limit $ {q_{T}}^{2}\ll Q^{2} $. In the region $ {q_{T}}^{2}\gtrsim Q^{2} $, the standard finite-order (FO) perturbative result, $ {\sigma _{FO}}$, is appropriate. While $ {\sigma _{{\widetilde{W }}}}$ and $ {\sigma _{FO}}$ represent the correct limiting behavior, we cannot simply add these two terms to obtain the total cross section, $ {\sigma _{TOT}}$, as we would be “double-counting” the contributions common to both terms.
The solution is to subtract the overlapping contributions between $ {\sigma _{{\widetilde{W }}}}$ and $ {\sigma _{FO}}$. This overlapping contribution (the *asymptotic piece* $ \sigma _{ASY} $) is obtained by expanding the $ b $-space integral in $ {\sigma _{{\widetilde{W }}}}$ out to the finite order of $ {\sigma _{FO}}$. Thus, the complete result is given by $$\label{sigmaTOT}
\frac{d{\sigma _{TOT}}}{d\Phi }=\frac{d{\sigma _{{\widetilde{W }}}}}{d\Phi }+\frac{d{\sigma _{FO}}}{d\Phi }-\frac{d{\sigma _{ASY}}}{d\Phi }.$$ At small $ q_{T} $, where terms $ \ln ^{m}(q_{T}^{2}/Q^{2}) $ are large, $ {\sigma _{FO}}$ cancels well with $ {\sigma _{ASY}}$, so that the total cross section is approximated well by the $ b $-space integral: $ {\sigma _{TOT}}\approx {\sigma _{{\widetilde{W }}}}$. At $ q^{2}_{T}\gtrsim Q^{2} $, where the logarithms are no longer dominant, the $ b $-space integral $ {\sigma _{{\widetilde{W }}}}$ cancels with $ {\sigma _{ASY}}$, so that the total cross section is equal to $ {\sigma _{FO}}$ up to higher order corrections: $ {\sigma _{TOT}}\approx {\sigma _{FO}}$. This interplay of $ \sigma _{\widetilde{W}},\, \sigma _{FO}, $ and $ \sigma _{ASY} $ in $ \sigma _{TOT} $ is illustrated in Fig. \[fig:TotWPertAsy\]a.
As we will be referring to these different terms frequently throughout the rest of the paper, let us present a recap of their roles.
- $ {\sigma _{{\widetilde{W }}}}$ is the small-$ q_{T} $ resummed term as given by the CSS formalism in Eq. (\[Wmassless\]); sometimes called “the CSS term” [@Balazs:1997xd]. This expression contains the all-order sum of large logarithms of the form $ \ln ^{m}{(q_{T}^{2}/Q^{2})} $, which is presented as a Fourier-Bessel transform of the $ b $-space form factor $ \widetilde{W}(b,Q,x,z). $ It is a good approximation in the region $ q_{T}^{2}\ll Q^{2} $.
- $ {\sigma _{FO}}$ is the *finite-order* (FO) term; sometimes called “the perturbative term”. It contains the complete perturbative expression computed to the relevant order of the calculation $ n $. As such, this term contains logarithms of the form $ \ln ^{m}{(q_{T}^{2}/Q^{2})} $ only out to $ m=2n-1 $. It also contains terms that are not important in the limit $ q_{T}^{2}/Q^{2}\rightarrow 0 $, but dominate when $ q^{2}_{T}\sim Q^{2} $. Hence, it provides a good approximation in the region $ {q_{T}}^{2}\gtrsim Q^{2} $.
- $ {\sigma _{ASY}}$ is the *asymptotic (ASY)* term. It contains the expansion of $ {\sigma _{{\widetilde{W }}}}$ out to the same order $ n $ as in $ \sigma _{FO} $. As such, this term contains logarithms of the form $ \ln ^{m}{(q_{T}^{2}/Q^{2})} $ only out to $ m=2n-1 $. It is precisely what is needed to eliminate the “double-counting” between the $ {\sigma _{{\widetilde{W }}}}$ and $ {\sigma _{FO}}$ terms in Eq. (\[sigmaTOT\]).
- $ {\sigma _{TOT}}$ is the *total* (TOT) resummed cross section; sometimes called “the resummed term”. It is constructed as $ {\sigma _{TOT}}={\sigma _{{\widetilde{W }}}}+{\sigma _{FO}}-{\sigma _{ASY}}$. In the region $ q_{T}^{2}\ll Q^{2} $, $ {\sigma _{ASY}}$ precisely cancels the large terms present in the $ {\sigma _{FO}}$ contribution, so that $ {\sigma _{TOT}}\approx {\sigma _{{\widetilde{W }}}}$. In the region $ q_{T}^{2}\gtrsim Q^{2} $, $ {\sigma _{ASY}}$ approximately cancels the $ {\sigma _{{\widetilde{W }}}}$ term leaving $ {\sigma _{FO}}$ as the dominant representation of the total cross section: $ {\sigma _{TOT}}\approx {\sigma _{FO}}$. Hence, when calculated to a sufficiently high order of $ \alpha _{S} $, $ \sigma _{TOT} $ serves as a good approximation at all $ q_{T} $.
In a practical calculation in low orders of PQCD, one may want to further improve the cancellation between $ \sigma _{\widetilde{W}} $ and $ \sigma _{ASY} $ at $ q_{T}^{2}\gtrsim Q^{2} $. This improvement can be achieved by introducing a kinematical correction in these terms that accounts for the reduction of the allowed phase space for the longitudinal variables $ x $ and $ z $ at non-zero $ q_{T} $. The purpose of this kinematical correction is quite similar to the purpose of the inclusive scaling variable discussed in Subsection \[sec:DISrescalingVariable\]: it removes contributions from the unphysically small $ x $ and $ z $ that make the difference $ \sigma _{\widetilde{W}}-\sigma _{ASY} $ non-negligible as compared to $ \sigma _{FO} $. Note that the resummed cross sections with and without the kinematical correction are formally equivalent to one another up to higher-order corrections. Further discussion of this issue can be found in Appendix \[Appendix:KinematicalCorrection\], which introduces the kinematical correction for the resummed heavy-quark $ q_{T} $ distributions.
Extension of the CSS Formalism to heavy-quark production\[sec:MassiveCSS\]
==========================================================================
In the previous Section, we presented a procedure for the resummation of distributions $ d\sigma /dq_{T}^{2} $ in the limit when $ Q^{2} $ is much larger than all other momentum scales, $ Q^{2}\gg q_{T}^{2},\, \{m_{q}^{2}\}. $ We now are ready to discuss its extension to the case when the heavy-quark mass is not negligible. For simplicity, we again assume that only one heavy flavor $ h $ has the mass $ M $ comparable with $ Q $: $ Q^{2}\sim M^{2}\gg \Lambda _{QCD}^{2} $. The generalization for several heavy flavors can be realized through the conventional sequence of factorization subschemes, in which the heavy quarks become active partons at energy scales above their mass, and are treated as non-partonic particles at energy scales below their mass.
We start by rewriting Eq. (\[WHA\]) in a form analogous to Eq. (4.3) of Ref. [@Collins:1985kg], where the form factor $ \widetilde{W} $ was given for the Drell-Yan process:
$$\begin{aligned}
& & \widetilde{W}_{HA}\left( b,Q,x,z\right) =\sum _{j}e_{j}^{2}\, \overline{{\mathscr {P}}}^{out}_{H/j}\left( z,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) \overline{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) \nonumber \\
& & \times \exp \Biggl \{-\int _{C_{1}^{2}/b^{2}}^{C_{2}^{2}Q^{2}}\frac{d\bar{\mu }^{2}}{\bar{\mu }^{2}}\Biggl [\ln \left( \frac{C_{2}^{2}Q^{2}}{\bar{\mu }^{2}}\right) {\cal A}\left( \alpha _{S}(\bar{\mu });\left\{ \frac{\bar{\mu }}{m_{q}}\right\} ;C_{1}\right) +{\cal B}\left( \alpha _{S}(\bar{\mu });\left\{ \frac{\bar{\mu }}{m_{q}}\right\} ;C_{1},C_{2}\right) \Biggr ]\Biggr \}.\label{W43} \end{aligned}$$
Here the function $ \overline{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},C_{1}/C_{2}\right) $ describes contributions associated with the incoming hadronic jet. As illustrated in Appendix \[Appendix:Chg\], $ \overline{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},C_{1}/C_{2}\right) $ is related to the $ k_{T} $-dependent parton distribution $ {\mathscr {P}}_{j/A}^{in}\left( x,k_{T},\{m_{q}\}\right) $. Similarly, the function $ \overline{{\mathscr {P}}}^{out}_{H/j}\left( z,b,\{m_{q}\},C_{1}/C_{2}\right) $ describes contributions associated with the outgoing hadronic jet [@Collins:1982va]. It is related to the $ k_{T} $-dependent fragmentation function $ {\mathscr {P}}^{out}_{H/j}\left( z,k_{T},\{m_{q}\}\right) $. The functions $ {\cal A} $ and $ {\cal B} $ are the same as in Eq. (\[Smassless\]), except that now they retain the dependence on the quark masses $ \{m_{q}\}=m_{u},m_{d},m_{s},...,M $.
Eq. (\[WHA\]) presents a special case of Eq. (\[W43\]). It is valid at short distances, *i.e.*, when $ 1/b $ is much larger than any of the quark masses $ m_{q} $. In contrast, Eq. (\[W43\]) is valid at all $ b $ [@Collins:1985kg]. As shown in Ref. [@Collins:1982uw], the transition from Eq. (\[W43\]) to Eq. (\[WHA\]) is possible because the functions $ \overline{{\mathscr {P}}}_{j/A}^{in} $ and $ \overline{{\mathscr {P}}}^{out}_{H/j} $ factorize when $ b_{0}^{2}/b^{2}\gg \{m_{q}^{2}\}: $ $$\begin{aligned}
& & \overline{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) \rightarrow \sum _{a}\int _{x}^{1}\frac{d\xi _{a}}{\xi _{a}}\nonumber \\
& \times & {\mathcal{C}}_{j/a}^{in}\left( {\widehat{x}},\mu _{F}b;\frac{C_{1}}{C_{2}}\right) f_{a/A}\left( \xi _{a},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) ;\nonumber \\
& & \overline{{\mathscr {P}}}^{out}_{H/j}\left( z,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) \rightarrow \sum _{b}\int _{z}^{1}\frac{d\xi _{b}}{\xi _{b}}\nonumber \\
& \times & D_{H/b}\left( \xi _{b},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) {\mathcal{C}}_{b/j}^{out}\left( {\widehat{z }},\mu _{F}b;\frac{C_{1}}{C_{2}}\right) .\label{PbFactorization} \end{aligned}$$ Here we introduced a frequently encountered constant $ b_{0}\equiv 2e^{-\gamma _{E}}\approx 1.123 $. We see that the form-factor $ {\widetilde{W }}_{HA} $ is well-defined both for non-zero quark masses and in the massless limit. Hence, it does not contain negative powers of the quark masses or logarithms $ \ln \left( m_{q}/Q\right) $, with the exception of the collinear logarithms resummed in the parton distributions and fragmentation functions.
We will now argue that the factorization rule similar to Eq. (\[PbFactorization\]) should also apply in heavy-flavor production when $ M^{2} $ is not negligible compared to $ b_{0}^{2}/b^{2} $. Indeed, the factorization of the functions $ \overline{{\mathscr {P}}}_{j/A}^{in} $ and $ \overline{{\mathscr {P}}}^{out}_{H/j} $ in the limit $ b_{0}^{2}/b^{2}\gg \{m_{q}^{2}\} $ [@Collins:1982uw] closely resembles the factorization of the inclusive DIS structure functions in the limit $ Q^{2}\gg \{m_{q}^{2}\} $ [@Amati:1978wx; @Amati:1978by; @Libby:1978bx; @Ellis:1978sf; @Ellis:1979ty]. In both cases the factorization occurs because the dominant contributions to the cross section come from “ladder” cut diagrams with subgraphs containing lines of drastically different virtualities. More precisely, the leading regions in such diagrams can be decomposed into hard subgraphs, which contain highly off-shell parton lines; and quasi-collinear subgraphs, which contain lines with much lower virtualities and momenta approximately collinear to $ p_{A}^{\mu } $ (in the case of $ F_{h/A}(x,Q^{2}) $ or $ \overline{{\mathscr {P}}}_{j/A}^{in} $) or $ p_{H}^{\mu } $ (in the case of $ \overline{{\mathscr {P}}}^{out}_{H/j} $). In the functions $ \overline{{\mathscr {P}}}_{j/A}^{in} $ and $ \overline{{\mathscr {P}}}^{out}_{H/j} $, additional soft gluon subgraphs are present, but they eventually do not affect the proof of the factorization [@Collins:1982uw]. The hard subgraphs contribute to the inclusive coefficient function $ C_{h/a} $ in Eq. (\[F\]), as well as functions $ {\cal C}_{j/a}^{in} $ or $ {\cal C}_{b/j}^{out} $ in Eq. (\[PbFactorization\]). The quasi-collinear subgraphs, which are connected to the hard subgraphs through one on-shell parton on each side of the momentum cut, contribute to the PDF’s (in the inclusive DIS and SIDIS) or FF’s (in SIDIS).
The hard subgraphs are characterized by typical transverse momenta $ k_{T}^{2}\gtrsim \mu _{F}^{2}\gg \Lambda _{QCD}^{2}, $ while the PDF’s and FF’s are characterized by transverse momenta $ k_{T}^{2}\lesssim \mu _{F}^{2} $. The factorization scale $ \mu _{F} $ is of order $ Q $ in the inclusive DIS structure functions and $ b_{0}/b $ in the functions $ \overline{{\mathscr {P}}}_{j/A}^{in} $ and $ \overline{{\mathscr {P}}}^{out}_{H/j} $. As discussed in Section \[sec:ACOT\], the factorization in the inclusive DIS can be extended to the case when $ Q $ is comparable to the heavy-flavor mass $ M $, $ Q^{2}\sim M^{2}\gg \Lambda _{QCD}^{2} $. Given the close analogy between the inclusive DIS structure functions and the functions $ \overline{{\mathscr {P}}}_{j/A}^{in}, $ $ \overline{{\mathscr {P}}}^{out}_{H/j} $ , it is natural to assume that the latter factorize when $ b_{0}^{2}/b^{2}\sim M^{2}\gg \Lambda _{QCD}^{2} $ as well: $$\begin{aligned}
& & \overline{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) =\sum _{a}\int _{x}^{1}\frac{d\xi _{a}}{\xi _{a}}\nonumber \\
& \times & {\mathcal{C}}_{j/a}^{in}\left( {\widehat{x}},\mu _{F}b,bM,\frac{C_{1}}{C_{2}}\right) f_{a/A}\left( \xi _{a},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) ;\nonumber \\
& & \overline{{\mathscr {P}}}^{out}_{H/j}\left( z,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) =\sum _{b}\int _{z}^{1}\frac{d\xi _{b}}{\xi _{b}}\nonumber \\
& \times & D_{H/b}\left( \xi _{b},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) {\mathcal{C}}_{b/j}^{out}\left( {\widehat{z }},\mu _{F}b,bM,\frac{C_{1}}{C_{2}}\right) .\nonumber \\
& & \label{PbFactorizationHeavy} \end{aligned}$$
The main difference between Eqs. (\[PbFactorization\]) and (\[PbFactorizationHeavy\]) is contained in the functions $ {\cal C}^{in}_{j/a} $ and $ {\mathcal{C}}_{b/j}^{out} $, which now explicitly depend on $ M. $ These functions can be calculated according to their definitions given in Ref. [@Collins:1981uk]. The unrenormalized expressions for the $ {\cal C} $-functions contain ultraviolet singularities. To cancel these singularities, we introduce counterterms according to the procedure described in Section \[sec:ACOT\]: that is, graphs with internal heavy-quark lines are renormalized in the $ \overline{MS} $ scheme if $ \mu _{F}\sim b_{0}/b>M$ and by zero-momentum subtraction if $ b_{0}/b<M $. This choice leads to the explicit decoupling of diagrams with heavy quark lines at $ b\gtrsim b_{0}/M $. In particular, the decoupling implies that contributions to Eq. (\[PbFactorizationHeavy\]) with $ j,a, $ or $ b $ equal to $ h $ are power-suppressed at $ b>b_{0}/M. $
We now consider other sources of the dependence on $ M $ in $ d\sigma /d\Phi . $ Firstly, according to Eq. (\[W43\]), there is a dependence on $ M $ in the Sudakov functions $ {\cal A}(\alpha _{S}(\bar{\mu });\bar{\mu }/M;C_{1}) $ and $ {\cal B}(\alpha _{S}(\bar{\mu });\bar{\mu }/M;C_{1},C_{2}) $. Due to the decoupling, the mass-dependent terms in the Sudakov factor vanish at $ b\gtrsim b_{0}/M $, except for perhaps terms of truly nonperturbative nature, like the intrinsic heavy quark component [@Brodsky:1980pb]. As mentioned above, in this paper such nonperturbative component is ignored. Secondly, there may also be mass-dependent terms *in the finite-order cross section,* which are not associated with the leading contributions resummed in the $ \widetilde{W} $-term: those are the terms that contribute to the remainder in Eq. (\[Wmassless\]). The terms of both types are correctly included in $ d\sigma _{TOT}/d\Phi $. Indeed, the terms of the first type appear in all three terms $ d\sigma _{{\widetilde{W }}}/d\Phi $, $ d\sigma _{FO}/d\Phi $, and $ d\sigma _{ASY}/d\Phi $. Two out of three contributions (in $ d\sigma _{{\widetilde{W }}}/d\Phi $ and $ d\sigma _{ASY}/d\Phi $, or $ d\sigma _{FO}/d\Phi $ and $ d\sigma _{ASY}/d\Phi $) cancel with one another, leaving the third contribution uncancelled in $ d\sigma _{TOT}/d\Phi $. The terms of the second type are contained only in $ d\sigma _{FO}/d\Phi $, so that they are automatically included in $ d\sigma _{TOT}/d\Phi $.
The treatment of the massive terms simplifies more if we adapt the S-ACOT factorization scheme, in which the heavy quark mass is set to zero in the hard parts of the flavor-excitation subprocesses. As a result, $ M $ is neglected in the flavor-excitation contributions to the hard cross section $ \sigma _{FO}, $ asymptotic term $ \sigma _{ASY} $, and $ {\cal C} $-functions in the $ \widetilde{W} $-term. The mass-dependent terms are further omitted in the perturbative Sudakov factor $ {\cal S} $. At the same time, all mass-dependent terms are kept in $ \sigma _{FO} $, $ \sigma _{ASY} $, and $ {\cal C} $-functions for gluon-initiated subprocesses.
As we will demonstrate in the next section, in this prescription the cross section $ \sigma _{TOT} $ resums the soft and collinear logarithms, when these logarithms are large, and reduces to the finite-order cross section, when these logarithms are small. In particular, at $ Q\sim M $ the finite-order flavor-creation terms approximate well the heavy-quark cross section. Hence we expect that $ \sigma _{TOT} $ reproduces the finite-order flavor-creation part at $ Q\sim M $ (Fig. \[fig:TotWPertAsy\]b). For this to happen, the flavor-excitation cross section should cancel well with the subtraction $ \propto \ln (\mu _{F}/M) $ from the flavor-creation cross section; and $ \sigma _{{\widetilde{W }}} $ should cancel well with $ \sigma _{ASY} $.We find that these cancellations indeed occur in the numerical calculation, so that at $ Q\approx M $ $ \sigma _{TOT} $ agrees well with the flavor-creation contribution to $ \sigma _{FO} $. Similarly, $ \sigma _{TOT} $ reproduces the massless resummed cross section when $ Q\gg M $ (Fig. \[fig:TotWPertAsy\]a). It also smoothly interpolates between the two regions of $ Q $.
To summarize our method, the total resummed cross section in the presence of heavy quarks is calculated as $$\label{sigmaTot2}
\frac{d\sigma _{TOT}}{d\Phi }=\frac{d\sigma _{\widetilde{W}}}{d\Phi }+\frac{d\sigma _{FO}}{d\Phi }-\frac{d\sigma _{ASY}}{d\Phi },$$ *i.e.*, using the same combination of the $ \widetilde{W} $-term, finite-order cross section, and asymptotic cross section as in the massless case. All three terms on the r.h.s. of Eq. (\[sigmaTot2\]) are calculated in the S-ACOT scheme. The $ \widetilde{W} $-term is calculated as $$\begin{aligned}
& & \left( \frac{d\sigma (e+A\rightarrow e+H+X)}{d\Phi }\right) _{{\widetilde{W }}}=\frac{\sigma _{0}F_{l}}{S_{eA}}\frac{A_{1}(\psi ,\varphi )}{2}\nonumber \\
& & \times \int \frac{d^{2}{{\bf b}}}{(2\pi )^{2}}e^{i{{\bf q}}_{T}\cdot {{\bf b}}}\widetilde{W}_{HA}(b,Q,M,x,z),\label{Wmassive} \end{aligned}$$ where the form-factor $ \widetilde{W}_{HA}(b,Q,M,x,z) $ is
$$\begin{aligned}
& & \widetilde{W}_{HA}\left( b,Q,M,x,z\right) =\sum _{a,b}\, \int _{\chi _{a}}^{1}\frac{d\xi _{a}}{\xi _{a}}\int _{z}^{1}\frac{d\xi _{b}}{\xi _{b}}D_{H/b}\left( \xi _{b},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) f_{a/A}\left( \xi _{a},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) \nonumber \\
& & \times \sum _{j=u,\bar{u},d,\bar{d}...}e_{j}^{2}{\mathcal{C}}^{out}_{b/j}\left( {\widehat{z}},\mu _{F}b,bM;\frac{C_{1}}{C_{2}}\right) {\mathcal{C}}_{j/a}^{in}\left( \frac{\chi _{a}}{\xi _{a}},\mu _{F}b,bM;\frac{C_{1}}{C_{2}}\right) e^{-S_{ba}(b,Q,M)},\label{WHA2} \end{aligned}$$
and $$\begin{aligned}
& & S_{ba}(b,Q,M)\equiv \int _{C_{1}^{2}/b^{2}}^{C_{2}^{2}Q^{2}}\frac{d{\overline{\mu }}^{2}}{{\overline{\mu }}^{2}}\nonumber \\
& & \times \Biggl [{{\cal A}}(\alpha _{S}({\overline{\mu }});C_{1})\ln \left( \frac{C_{2}^{2}Q^{2}}{{\overline{\mu }}^{2}}\right) \nonumber \\
& & +{{\cal B}}(\alpha _{S}({\overline{\mu }});C_{1},C_{2})\Biggr ]+S_{ba}^{NP}(b,Q,M).\end{aligned}$$ As in the factorization of inclusive DIS structure functions (cf. Section \[sec:ACOT\]), we find it useful to replace Bjorken $ x $ by scaling variables $$\label{chih1}
\chi _{h}=x\left( 1+\frac{M^{2}}{z(1-z)Q^{2}}\right)$$ in $ \sigma _{FO} $ for the flavor-excitation subprocesses, and $$\label{chih2}
\chi ^{'}_{h}=x\left( 1+\frac{M^{2}+z^{2}q_{T}^{2}}{z(1-z)Q^{2}}\right)$$ in $ \sigma _{\widetilde{W}} $ and $ \sigma _{ASY} $. The purpose of these scaling variables is to enforce the correct threshold behavior of terms with incoming heavy quarks. Eqs. (\[chih1\]) and (\[chih2\]) are derived in detail in Appendix \[Appendix:KinematicalCorrection\].
\[sec:PhotonGluon\]Massive resummation for photon-gluon fusion
==============================================================
We now analyze contributions to the total resummed cross section $ d\sigma _{TOT}/d\Phi $ from the $ {\cal O}(\alpha _{S}^{0}) $ heavy-flavor excitation subprocess $ \gamma ^{*}(q)+h(p_{a})\rightarrow h(p_{b}) $ (Fig. \[fig:diag2\]a) and $ {\cal O}(\alpha _{S}) $ photon-gluon fusion subprocess $ \gamma ^{*}(q)+G(p_{a})\rightarrow h(p_{b})+\bar{h}(p_{s}) $ (Fig. \[fig:diag2\]b). Since we work in the S-ACOT scheme, only the $ {\cal O}(\alpha _{S}) $ fusion subprocess retains the heavy quark mass, so that we concentrate on that process first. The expression for the $ \gamma ^{*}h $ contribution, which is the same as in the massless case, is given in Eq. (\[FiniteOrderLO\]). In the following we outline the main results, while details are relegated to Appendices.
Mass-Generalized Kinematical Variables
--------------------------------------
Our approach will be to first generalize the kinematical variables from the massless resummation formalism to “recycle” as much of the results from Refs. [@Meng:1992da; @Meng:1996yn; @Nadolsky:1999kb] as possible.
Throughout the derivation, the mass of the incident hadron will be neglected: $ p_{A}^{2}=0 $. We will use the standard DIS variables $ x,\, Q^{2}, $ and $ z, $ defined by $$\begin{aligned}
x & \equiv & \frac{Q^{2}}{2p_{A}\cdot q};\, \, \, Q^{2}\equiv -q^{2};\, \, \, z\equiv \frac{p_{A}\cdot p_{H}}{p_{A}\cdot q}.\label{xHadron} \end{aligned}$$ Since we will be interested in the transverse momentum distributions (or equivalently, the angular distributions), we next define the transverse momentum in a frame-invariant manner. The four-vector $ q_{t}^{\mu } $ of the transverse momentum must be orthogonal to both of the hadrons, so that we have the conditions $ q_{t}\cdot p_{A}=0 $ and $ q_{t}\cdot p_{H}=0 $. In the massless case, $ q^{\mu }_{t} $ is simply defined by subtracting off the projections of the photon’s momentum $ q^{\mu } $ onto $ p_{A} $ and $ p_{H} $. This is slightly modified in the massive case to become $$\begin{aligned}
q_{t}^{\mu } & = & q^{\mu }-\left( \frac{p_{H}\cdot q}{p_{A}\cdot p_{H}}-M_{H}^{2}\frac{p_{A}\cdot q}{(p_{A}\cdot p_{H})^{2}}\right) p_{A}^{\mu }\nonumber \\
& - & \frac{p_{A}\cdot q}{p_{A}\cdot p_{H}}p_{H}^{\mu }.\end{aligned}$$ Here $ M_{H} $ denotes the mass of the heavy hadron. We find for $ q_{T}^{2}\equiv -q_{t}^{\mu }q_{t\mu }: $ $$\label{qTHadron}
q_{T}^{2}=Q^{2}+2\frac{p_{H}\cdot q}{z}-\frac{M_{H}^{2}}{z^{2}}.$$
The kinematical variables at the parton level can be introduced in an analogous manner. Let $ \xi _{a} $ denote the fraction of the large ’$ - $’ component of the incoming hadron’s momentum $ p_{A} $ carried by the initial-state parton $ a $ (*i.e.*, $ \xi _{a}\equiv p_{a}^{-}/p_{A}^{-} $);[^3] and $ \xi _{b} $ denote the fraction of the large ’$ + $’ component of the final-state parton’s momentum $ p_{b} $ carried by the outgoing hadron $ H $ (*i.e.*, $ \xi _{b}\equiv p^{+}_{H}/p^{+}_{b} $). We also assume that $ \xi _{b} $ relates the transverse momenta of $ b $ and $ H $, as $ (p_{T})_{H}=\xi _{b}(p_{T})_{b}. $ Since all incoming partons are massless in the S-ACOT factorization scheme, we find the following relations between the hadron-level variables $ x,\, z,\, q_{T} $ and their parton-level analogs $ \widehat{x},{\widehat{z}},\, \widehat{q}_{T} $: $$\begin{aligned}
{\widehat{x }} & \equiv & \frac{Q^{2}}{2\left( p_{a}\cdot q\right) }=\frac{x}{\xi _{a}};\\
{\widehat{z}}& \equiv & \frac{\left( p_{a}\cdot p_{b}\right) }{(p_{a}\cdot q)}=\frac{z}{\xi _{b}};\\
\widehat{q}_{T} & = & q_{T},\label{qthqt} \end{aligned}$$ where in the derivation of Eq. (\[qthqt\]) we used the first equality in Eq. (\[qTvscosthetab\]).
If we introduce a massive extension of $ \widehat{q}_{T}^{2} $ called $ \widetilde{q}_{T}^{2} $ and defined by $$\label{qTtilde}
{\widetilde{q }}_{T}^{2}\equiv \widehat{q}^{2}_{T}+\frac{M^{2}}{{\widehat{z}}^{2}},$$ then the form of $ \widetilde{q}_{T}^{2} $ in terms of the Lorentz invariants is identical to the massless case: $$\label{wtqT2}
{\widetilde{q }}_{T}^{2}=Q^{2}+2\frac{p_{h}\cdot q}{{\widehat{z}}}.$$ We also generalize the usual Mandelstam variables $ \{{\widehat{s}},{\widehat{t}},{\widehat{u}}\} $ to what we label the “mass-dependent” Mandelstam variables $ \{{\widehat{s}},{\widehat{t}}_{1},{\widehat{u}}_{1}\} $: $$\begin{aligned}
{\widehat{s}}& = & (q+p_{a})^{2},\\
{\widehat{t}}_{1} & \equiv & {\widehat{t}}-M^{2}=(q-p_{h})^{2}-M^{2},\\
{\widehat{u}}_{1} & \equiv & {\widehat{u}}-M^{2}=(p_{a}-p_{h})^{2}-M^{2}.\end{aligned}$$
By using the variables $ \widetilde{q}_{T}^{2} $, $ {\widehat{s}}, $ $ {\widehat{t}}_{1} $, and $ {\widehat{u}}_{1} $ instead of their counterparts $ q_{T}^{2}, $ $ {\widehat{s}}$, $ {\widehat{t}}, $ and $ {\widehat{u}}$, we shall be able to cast many of the massive relations in the form of the massless ones. For example, the expressions for the “mass-dependent” Mandelstam variables $ \{{\widehat{s}},{\widehat{t}}_{1},{\widehat{u}}_{1}\} $ in terms of the DIS variables can be written as $$\begin{aligned}
{\widehat{s}}& = & Q^{2}\frac{(1-{\widehat{x}})}{{\widehat{x}}};\\
{\widehat{t}}_{1} & = & -Q^{2}\frac{{\widehat{z}}}{{\widehat{x}}};\\
{\widehat{u}}_{1} & = & Q^{2}({\widehat{z}}-1)-{\widetilde{q }}_{T}^{2}{\widehat{z}}=-Q^{2}\frac{(1-{\widehat{z}})}{{\widehat{x}}}.\end{aligned}$$ Note how we made use of the generalized transverse momentum variable $ \widetilde{q}_{T}^{2} $. These relationships have the same form as their massless counterparts. As a result, the denominators of the mass-dependent propagators, which are formed from the invariants $ {\widehat{s}},{\widehat{t}}_{1} $, and $ {\widehat{u}}_{1} $, retain the same form as the denominators of the massless propagators, which are formed from the invariants $ {\widehat{s}},{\widehat{t}}, $ and $ {\widehat{u}}$.
\[subsec:GammaPframe\]Relations between $ \{E_{H},\cos \theta _{H}\}\protect $in the $ \gamma ^{*}A\protect $ c.m. frame and $ \{z,{q_{T}}^{2}\}\protect $
----------------------------------------------------------------------------------------------------------------------------------------------------------
It is useful to convert between the final-state energy $ E_{H} $, polar angle $ \theta _{H} $ and the Lorentz invariants $ \{z,{q_{T}}^{2}\} $. Given the $ \gamma ^{*}A $ c.m. energy $ W^{2}\equiv (q+p_{A})^{2}=Q^{2}(1-\nolinebreak x)/x $ and $ p\equiv |{{\bf p}}_{H}|=\sqrt{E_{H}^{2}-M_{H}^{2}} $, one easily finds the following constraints on $ E_{H}, $ $ p $, and $ \cos \theta _{H} $: $$\begin{aligned}
& M_{H}\leq E_{H}\leq \frac{W}{2}\left( 1+\frac{M_H^{2}}{W^{2}}\right) , & \label{EHrange} \\
& 0\leq p\leq \frac{W}{2}\left( 1-\frac{M_H^{2}}{W^{2}}\right) , & \label{prange} \end{aligned}$$ and $$\label{cosThetaHrange}
-1\leq \cos \theta _{H}\leq 1.$$
Given $ E_{H} $ and $ \cos \theta _{H} $, we can determine $ z $ and $ q_{T}^{2} $ as $$\begin{aligned}
z & = & \frac{1}{W}\left( E_{H}+p\cos \theta _{H}\right) ;\label{z} \\
q^{2}_{T} & = & \frac{\left( p^{2}_{T}\right) _{H}}{z^{2}}=W^{2}\frac{p^{2}\left( 1-\cos ^{2}\theta _{H}\right) }{\left( E_{H}+p\cos \theta _{H}\right) ^{2}}.\label{qTvscosthetab} \end{aligned}$$ From Eqs. (\[EHrange\]-\[cosThetaHrange\]) the bounds on $ z $ can be found as $$\frac{M_{H}^{2}}{W^{2}}\leq z\leq 1.$$ Note that the first equality in Eq.** **(**\[qTvscosthetab\]**) identifies **$ q_{T} $** as the the transverse momentum of **$ H $** rescaled by the final-state fragmentation variable **$ z $**. Hence **$ q_{T} $** can be also interpreted as the leading-order transverse momentum of the fragmenting parton. Similarly, ${\widetilde{q }}_T = M_T/{\widehat{z}}$ can be interpreted as the rescaled transverse mass $M_T$ of the heavy quark. It also follows from Eqs. (\[z\],\[qTvscosthetab\]) that the two-variable distribution with respect to the variables $ z $ and $ q_{T} $ coincides with the two-variable distribution with respect to $ E_{H} $ and $ \theta _{H} $: $$\label{zqt2EcosThetaH}
\frac{d\sigma }{dxdQ^{2}dzdq_{T}}=\frac{d\sigma }{dxdQ^{2}dE_{H}d\theta _{H}}.$$ As a result, the distributions in the theoretical variables $ z $ and $ q_{T} $ are directly related to the distributions in $ E_{H} $ and $ \theta _{H} $ measured in the experiment.
Despite the simplicity of the relation (\[zqt2EcosThetaH\]), $ z $ and $ q_{T} $ are quite complicated functions of $ E_{H} $ and $ \cos \theta _{H} $ individually. This feature is different from the massless case, where there exists a one-to-one correspondence between $ q_{T} $ and $ \cos \theta _{H} $ for the fixed $ \gamma ^{*}A $ c.m. energy $ W $: $$\label{cosThetaHM0}
\left. \cos \theta _{H}\right| _{M_{H}=0}=\frac{W^{2}-q_{T}^{2}}{W^{2}+q_{T}^{2}}.$$ This relationship does not hold in the massive case, in which *one* value of $ q_{T} $ corresponds to *two* values of $ \cos \theta _{H} $. Indeed, Eq. (\[qTvscosthetab\]) can be expressed as $$\label{qT2W}
\frac{q_{T}^{2}}{W^{2}}=\frac{(1-\lambda ^{2})\left( 1-\cos ^{2}\theta _{H}\right) }{\left( 1+\sqrt{1-\lambda ^{2}}\cos \theta _{H}\right) ^{2}},$$ where, according to Eq. (\[EHrange\]), the variable $ \lambda \equiv M_{H}/E_{H} $ varies in the following range: $$\frac{2M_{H}}{W\left( 1+M_H^{2}/W^{2}\right) }\leq \lambda \leq 1.$$ Eq. (\[qT2W\]) can be solved for $ \cos \theta _{H} $ as $$\begin{aligned}
& & \cos \theta _{H}=\frac{1}{\left( q_{T}^{2}+W^{2}\right) \sqrt{1-\lambda ^{2}}}\nonumber \\
& & \times \Biggl (-q_{T}^{2}\pm W\sqrt{\left( 1-\lambda ^{2}\right) \left( q_{T}^{2}+W^{2}\right) -q^{2}_{T}}\Biggr ).\label{SolutionscosThetaH} \end{aligned}$$ When the energy $ E_{H} $ is much larger than $ M_{H} $ ($ \lambda \rightarrow 0 $) the solution with the “$ + $” sign in Eq. (\[SolutionscosThetaH\]) turns into the massless solution (\[cosThetaHM0\]). The solution with the “$ - $” sign reduces to $ \cos \theta _{H}=-1. $
The physical meaning of the relationship between $ q_{T} $ and $ \cos \theta _{H} $ can be understood by considering plots of $ q_{T}/W $ vs. $ \theta _{H} $ for various values of $ \lambda $ (Fig. \[fig:eta\_vs\_thetaB\]). Let us identify *the current fragmentation region* as that where $ \cos \theta _{H} $ is close to $ +1 $ ($ \theta _{H}=0 $), and *the target fragmentation region* as that where $ \cos \theta _{H} $ is close to $ -1 $ ($ \theta _{H}=\pi $). Firstly, $ q_{T}=0 $ if $ \cos \theta _{H}=1 $ or $ \cos \theta _{H}=-1 $. Secondly, near the threshold ($ \lambda \rightarrow 1 $) the ratio $ q_{T}/W $ is vanishingly small and symmetric with respect to the replacement of $ \theta _{H} $ by $ (\pi -\theta _{H}) $. Thirdly, as $ \lambda $ decreases, the distribution $ q_{T}/W $ vs. $ \theta _{H} $ develops a peak near $ \theta _{H}=180^{\circ } $. This peak is positioned at $ \cos \theta _{H}=-\sqrt{1-\lambda ^{2}} $, and its height is $ q_{T}/W=\left( 1-\lambda ^{2}\right) ^{1/2}/\lambda ^{2} $. For $ \theta _{H}\ll 180^{\circ } $, the distribution rapidly becomes insensitive to $ \lambda $; more so for smaller $ \theta _{H} $.
In the limit $ \lambda \rightarrow 0 $, the peak at $ \theta _{H}=180^{\circ } $ turns into a singularity. This singularity resides at the point $ z=0 $ and corresponds to hard diffractive hadroproduction. The analysis of this region requires diffractive parton distribution functions [@Trentadue:1994ka; @Berera:1994xh; @Graudenz:1994dq; @Berera:1996fj; @deFlorian:1996fd] and will not be considered here. For $ \theta _{H}\neq 180^{\circ } $, one recovers a one-to-one correspondence between $ q_{T}/W $ and $ \cos \theta _{H} $ of the massless case. We see that there is a natural relationship between $ q_{T} $ and $ \cos \theta _{H} $, which becomes especially simple in the massless limit. In the following, we concentrate on the limit $ q_{T}\rightarrow 0 $ *and* $ z\neq 0 $, which corresponds to the current fragmentation region $ \theta _{H}\rightarrow 0 $.
Factorized cross sections\[subsec:FactorizedXSections\]
-------------------------------------------------------
Next, we consider the factorization of the hadronic cross section. Given the hadron-level phase space element $ d\Phi \equiv dxdQ^{2}dzdq_{T}^{2}d\varphi $ and its parton-level analog $ d{\widehat{\Phi }}\equiv d{\widehat{x}}dQ^{2}d{\widehat{z}}d{\widehat{q }}_{T}^{2}d{\widehat{\varphi }}, $ all three terms on the r.h.s. of Eq. (\[sigmaTot2\]) can be written as $$\begin{aligned}
& & \frac{d\sigma }{d\Phi }=\sum _{a,b}\int _{z}^{1}\frac{d\xi _{b}}{\xi _{b}}\int _{\chi _{a}}^{1}\frac{d\xi _{a}}{\xi _{a}}\nonumber \\
& \times & D_{H/b}\left( \xi _{b},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) f_{a/A}\left( \xi _{a},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) \nonumber \\
& & \times \frac{d{\widehat{\sigma }}}{d{\widehat{\Phi }}}\left( \frac{\chi _{a}}{\xi _{a}},\frac{z}{\xi _{b}},\frac{q_{T}}{Q},\frac{\mu _{F}}{Q},\frac{M}{Q}\right) .\label{Factorization} \end{aligned}$$
Let us first consider the finite-order cross section $ d{\widehat{\sigma }}_{FO}/d{\widehat{\Phi }} $ . The explicit expression for this cross section at the lepton level can be found in Appendix \[Appendix:FO\]. We are interested in extracting the leading contribution in this cross section in the limit $ Q\rightarrow \infty $ with other scales fixed. Specifically, we concentrate on the behavior of the phase-space $ \delta - $function that multiplies the matrix element $ \left| {\cal M}\right| ^{2} $: $$\begin{aligned}
& & \frac{d\widehat{\sigma }_{FO}}{d\widehat{\Phi }}\propto \delta \left( \widehat{s}+\widehat{t}+\widehat{u}+Q^{2}-2M^{2}\right) \left| {\mathcal{M}}\right| ^{2}\nonumber \\
& & =\delta \left( \widehat{s}+\widehat{t}_{1}+\widehat{u}_{1}+Q^{2}\right) \left| {\mathcal{M}}\right| ^{2}\nonumber \\
& & =\delta \left( \left( \frac{1}{\widehat{x}}-1\right) \left( \frac{1}{\widehat{z}}-1\right) -\frac{{\widetilde{q }}_{T}^{2}}{Q^{2}}\right) \left| {\mathcal{M}}\right| ^{2}.\end{aligned}$$ Here we used the mass-generalized variable $ {\widetilde{q }}_{T}^{2} $ introduced in Eq. (\[wtqT2\]). Note that in terms of the variables $ {\widehat{x }},{\widehat{z}}$, and $ \widetilde{q}_{T}^{2} $ this expression takes the same form as its massless version. In the limit $ Q\rightarrow \infty $, and $ {\widehat{x}},{\widehat{z}}, $ and $ {\widetilde{q }}_{T} $ fixed, the $ \delta $-function can be transformed using the relationship $$\begin{aligned}
\lim _{\varepsilon \rightarrow 0}\delta \left( y_{1}y_{2}-\varepsilon \right) & \approx & \frac{\delta (y_{1})}{[y_{2}]_{+}}+\frac{\delta (y_{2})}{[y_{1}]_{+}}\nonumber \\
& - & \log (\varepsilon )\delta (y_{1})\delta (y_{2}).\end{aligned}$$ This transformation yields $$\begin{aligned}
& & \lim _{Q\rightarrow \infty }\delta \left( \widehat{s}+\widehat{t}+\widehat{u}+Q^{2}-2M^{2}\right) \propto \nonumber \\
& & \frac{\delta (1-{\widehat{x}})}{[1-{\widehat{z}}]_{+}}+\frac{\delta (1-{\widehat{z}})}{[1-{\widehat{x}}]_{+}}\\
& & -\log \left( \frac{\widetilde{q}_{T}^{2}}{Q^{2}}\right) \delta \left( 1-{\widehat{x}}\right) \delta (1-{\widehat{z}}).\end{aligned}$$ This asymptotic expression for the $ \delta $-function is exactly of the same form as in the massless case up to the replacement $ {\widetilde{q }}_{T}^{2}\rightarrow q_{T}^{2} $.
Furthermore, in the above limit the matrix element $ |{\mathcal{M}}|^{2} $ itself contains singularities when $ Q^{2}\gg {\widetilde{q }}_{T}^{2} $. In particular, the largest structure function $ {\widehat{V }}_{1} $ in the $ \gamma ^{*}G $-fusion subprocess (cf. Eq. (\[V1jg\])) contains contributions proportional to $$\frac{1}{(M^{2}-{\widehat{t }})(M^{2}-{\widehat{u}})}\propto \frac{1}{t_{1}u_{1}}\propto \frac{1}{\widetilde{q}_{T}^{2}},$$ and $$\frac{M^{2}}{{\widehat{t}}^{2}_{1}{\widehat{u}}^{2}_{1}}\propto \frac{M^{2}}{{\widetilde{q }}_{T}^{4}}=\frac{{\widehat{z}}^{4}M^{2}}{\left( {\widehat{z}}^{2}q_{T}^{2}+M^{2}\right) ^{2}}.$$ When $ M $ is not negligible, these contributions are finite and comparable with other terms. However, in the limit when *both* $ M $ and $ q_{T} $ are much less than $ Q $, the terms of the first type diverge as $ 1/q_{T}^{2} $. The terms of the second type vanish at $ q_{T}\neq 0 $ and yield a finite contribution at $ q_{T}=0 $. These non-vanishing contributions are precisely the ones that are resummed in the $ \widetilde{W} $-term; in the total resummed cross section $ \sigma _{TOT} $, they have to be subtracted in the form of the asymptotic cross section $ \sigma _{ASY} $ to avoid the double-counting between $ \sigma _{FO} $ and $ \sigma _{{\widetilde{W }}} $.
To precisely identify these terms, we calculate them from their definitions, as described in Appendix \[Appendix:Chg\]. Since the $ {\cal O}(\alpha _{S} $) $ \gamma ^{*}G $ subprocess is finite in the soft limit, it contributes only to the function $ {\cal C}^{in}_{h/G}(x,\mu _{F}b,bM) $ and not to the Sudakov factor. The $ {\cal O}(\alpha _{S}/\pi ) $ coefficient in this function is $$\begin{aligned}
& & {\mathcal{C}}^{in(1)}_{h/G}({\widehat{x}},\mu _{F}b,bM)=T_{R}x(1-x)\left( 1+c_{1}(bM)\right) \nonumber \\
& & +P^{(1)}_{h/G}(x)\left( c_{0}(bM)-\ln \Bigl (\frac{\mu _{F}b}{b_{0}}\Bigr )\right) \end{aligned}$$ if $ \mu _{F}\geq M $, and $$\begin{aligned}
{\mathcal{C}}^{in(1)}_{h/G}({\widehat{x}},\mu _{F}b,bM) & = & \left. {\mathcal{C}}^{in(1)}_{h/G}({\widehat{x}},\mu _{F}b,bM)\right| _{\mu _{F}\geq M}\nonumber \\
& + & P^{(1)}_{h/G}(\widehat{x})\ln \frac{\mu _{F}}{M}\end{aligned}$$ if $ \mu _{F}<M $. Here $ P^{(1)}_{h/G}(\xi ) $ is the $ \overline{MS} $ splitting function: $ P_{h/G}^{(1)}(\xi )=T_{R}\left( 1-2\xi +2\xi ^{2}\right) , $ with $ T_{R}=1/2. $ The functions $ c_{0}(bM), $ and $ c_{1}(bM) $ denote the parts of the modified Bessel functions $ K_{0}(bM) $ and $ bM\, K_{1}(bM) $ that vanish when $ b\ll 1/M $. They are defined in Eqs. (\[c0\]) and (\[c1\]), respectively.
We now have all terms necessary to calculate the combination $ ({\cal C}^{in(0)}_{h/h}\otimes f_{h/A})(x)+({\cal C}^{in(1)}_{h/G}\otimes f_{G/A})(x) $, which serves as the first approximation to the function $ \overline{{\mathscr {P}}}_{h/A}^{in}\left( x,b,M,C_{1}/C_{2}\right) $. We find that this combination possesses two remarkable properties: it smoothly vanishes at $ \mu ^{2}_{F}=b^{2}_{0}/b^{2}\ll M^{2} $ and is differentiable with respect to $ \ln (\mu _{F}/M) $ at the point $ \mu _{F}=M $. As a result, the form factor $ \widetilde{W}(b,Q,x,z) $ for the combined $ {\cal O}(\alpha _{S}^{0}) $ flavor-excitation and $ {\cal O}(\alpha _{S}^{1}) $ flavor-creation channels is a smooth function at all $ b $, which is strongly suppressed at $ b^{2}\gg b_{0}^{2}/M^{2} $. The physical consequence is that, for a sufficiently heavy quark, the $ b $-space integral can be performed over the large-$ b $ region without introducing an additional suppression of the integrand by nonperturbative contributions. We use this feature in Section \[sec:NumericalResults\], where we calculate the resummed cross section for bottom quark production, which does not depend on the nonperturbative Sudakov factor.
Finally, by expanding the form-factor $ \widetilde{W}_{HA} $ in a series of $ \alpha _{S}/\pi $ and calculating the Fourier-Bessel transform integral in Eq. (\[Wmassive\]), we find the following asymptotic piece for the $ \gamma ^{*}G $ fusion channel: $$\begin{aligned}
& & \left( \frac{d{\widehat{\sigma }}(e+G\rightarrow e+h+\bar{h})}{d{\widehat{\Phi }}}\right) _{ASY}=\frac{\sigma _{0}F_{l}}{4\pi S_{eA}}\frac{\alpha _{S}}{\pi }\nonumber \\
& & \times A_{1}(\psi ,\varphi )\delta (1-{\widehat{z}})\nonumber \\
& & \times \left[ \frac{P^{(1)}_{h/G}({\widehat{x}})}{{\widehat{{\widetilde{q }} }}_{T}^{2}}+\frac{M^{2}{\widehat{x}}(1-{\widehat{x}})}{{\widehat{{\widetilde{q }} }}_{T}^{4}}\right] .\label{sigma_ASY} \end{aligned}$$ When $ Q\sim M, $ $ d{\widehat{\sigma }}_{ASY}/d{\widehat{\Phi }} $, which is a regular function at all $ q_{T} $, cancels well with $ d{\widehat{\sigma }}_{{\widetilde{W }}}/d{\widehat{\Phi }}. $ In the limit $ Q\rightarrow \infty $, $ d{\widehat{\sigma }}_{ASY}/d{\widehat{\Phi }} $ precisely cancels the asymptotic terms that appear in the finite-order cross section $ d{\widehat{\sigma }}_{FO}/d{\widehat{\Phi }}. $
Numerical Results \[sec:NumericalResults\]
==========================================
(a)(b)
(c)(d)
In this Section, we apply the resummation formalism to the production of bottom quarks at HERA. The calculation is done for the electron-proton c.m. energy of $ 300 $ GeV and bottom quark mass $ M=4.5 $ GeV. For simplicity we assume that the masses of the $ B $-hadrons coincide with the mass of the bottom quark $ M $. We also neglect the mixing of photons with $ Z^{0} $-bosons at large $ Q $.
In the following, we discuss polar angle distributions in the $ \gamma ^{*}p $ frame for $ x=0.05 $ and various values of $ Q $. The cross section is calculated in the lowest-order approximation as discussed in Section \[sec:PhotonGluon\].[^4] The calculation was realized using the CTEQ5HQ PDF’s [@Lai:1999wy] and Peterson fragmentation functions [@Peterson:1983ak] with $ \varepsilon =0.0033 $ [@BottomDISH1]. The finite-order cross section $ d\sigma _{FO}/d\Phi $ and asymptotic cross section $ d\sigma _{ASY}/d\Phi $ were calculated at the factorization scale $ \mu _{F}=Q $. The scale-related constants in the $ \widetilde{W} $-term were chosen to be $ C_{1}=2e^{-\gamma _{E}}=b_{0} $ and $ C_{2}=1 $, and the factorization scale was $ \mu _{F}=b_{0}/b $. The $ \widetilde{W} $-term included the $ {\cal O}(\alpha _{S}^{0}) $ $ {\cal C} $-functions $ {\mathcal{C}}^{in(0)}_{h/h}({\widehat{x}},\mu _{F}b,C_{1}/C_{2}) $, $ {\mathcal{C}}^{out(0)}_{h/h}({\widehat{z}},\mu _{F}b,C_{1}/C_{2}) $ and $ {\cal O}(\alpha _{S}^{1}) $ initial-state function $ {\mathcal{C}}^{in(1)}_{h/G}({\widehat{x}},\mu _{F}b,bM) $. In addition, it included the perturbative Sudakov factor (\[Smassless\]), unless stated otherwise. The Sudakov factor was evaluated at order $ {\cal O}(\alpha _{S}) $, which was sufficient for this calculation given the order of other terms. The functions in the Sudakov factor were evaluated as $${\cal A}(\mu ;C_{1})=C_{F}\frac{\alpha _{S}(\mu )}{\pi },$$ and $${{\cal B}}(\mu ;C_{1},C_{2})=-\frac{3C_{F}}{2}\frac{\alpha _{S}(\mu )}{\pi }.$$
According to the discussion in Section \[sec:PhotonGluon\], our calculation ignores unknown nonperturbative contributions in the $ {\widetilde{W }} $-term. In the numerical calculation, we also need to define the behavior of the light-quark PDF’s at scales $ \mu _{F}=b_{0}/b<1\mbox {\, GeV} $. Due to the strong suppression of the large-$ b $ region by the $ M $-dependent terms in the $ {\cal C} $-functions (cf. the discussion after Eq. (\[PinhG1\])), the exact procedure for the continuation of the PDF’s to small $ \mu _{F} $ has a small numerical effect. We found it convenient to “freeze” the scale $ \mu _{F} $ at a value of about $ 1 $ GeV by introducing the variable $ b_{*}=b/\sqrt{1+\left( b/b_{max}\right) ^{2}} $ [@Collins:1985kg] with $ b_{max}=b_{0}\mbox {\, GeV}^{-1}\approx 1.123\mbox {\, GeV}^{-1} $. Other procedures [@Qiu:2000hf; @Kulesza:2002rh] for continuation of $ {\widetilde{W }}_{HA}(b,Q,x,z) $ to large values of $ b $ may be used as well. Due to the small sensitivity of the resummed cross section to the region $ b^{2}\gg b_{0}^{2}/M^{2} $, all these continuation procedures should yield essentially identical predictions.
\(a) (b)
Fig. \[fig:NumericalResults\] demonstrates how various terms in Eq. (\[sigmaTot2\]) are balanced in an actual numerical calculation. Near the threshold ($ Q=5 $ GeV, Fig. \[fig:NumericalResults\]a) the cross section $ d\sigma _{TOT}/(dxdQ^{2}d\theta _{H}) $ should be well approximated by the $ {\mathcal{O}}(\alpha _{S}) $ flavor-creation diagram $ \gamma ^{*}+G\rightarrow h+\bar{h} $. We find that this is indeed the case, since the $ {\widetilde{W }} $-term, which does not contain large logarithms, cancels well with its perturbative expansion $ d\sigma _{ASY}/d\Phi $. As a result, the full cross section is practically indistinguishable from the finite-order term.
At higher values of $ Q, $ we start seeing deviations from the finite-order result. Fig. \[fig:NumericalResults\]b shows the differential distribution at $ Q=15 $ GeV, *i.e.*, approximately at $ Q^{2}/M^{2}=10 $. At this energy, $ d\sigma _{TOT}/(dxdQ^{2}d\theta _{H}) $ still agrees with $ d\sigma _{FO}/(dxdQ^{2}d\theta _{H}) $ at $ \theta _{H}\gtrsim 10^{\circ }, $ but is above $ d\sigma _{FO}/(dxdQ^{2}d\theta _{H}) $ at $ \theta _{H}\lesssim 10^{\circ }. $ The excess is due to the difference $ d\sigma _{{\widetilde{W }}}/(dxdQ^{2}d\theta _{H})-d\sigma _{ASY}/(dxdQ^{2}d\theta _{H}), $ *i.e.*, due to the higher-order logarithms.
Away from the threshold ($ Q=50 $ GeV), $ d\sigma _{TOT}/(dxdQ^{2}d\theta _{H}) $ is substantially larger than the finite-order term at $ \theta _{H}\lesssim 10^{\circ } $, where it is dominated by $ d\sigma _{{\widetilde{W }}}/(dxdQ^{2}d\theta _{H}) $. In this region, $ d\sigma _{FO}/(dxdQ^{2}d\theta _{H}) $ is canceled well by $ d\sigma _{ASY}/(dxdQ^{2}d\theta _{H}) $. Note, however, that contrary to the experience from the massless case, $ d\sigma _{FO}/(dxdQ^{2}d\theta _{H}) $ and $ d\sigma _{ASY}/(dxdQ^{2}d\theta _{H}) $ are not singular at $ \theta _{H}\rightarrow 0 $ due to the regularizing effect of the heavy quark mass in the heavy-quark propagator at $ \theta _{H}\lesssim 3^{\circ } $. Figs. \[fig:NumericalResults\]c and \[fig:NumericalResults\]d also compare the distributions with and without the $ {\cal O}(\alpha _{S}^{1}) $ perturbative Sudakov factor, respectively. Note that at the threshold the flavor-excitation terms responsible for $ {\cal S} $ are of a higher order as compared to the $ {\cal O}(\alpha _{S}^{0}) $ flavor-excitation and $ {\cal O}(\alpha _{S}^{1}) $ flavor-creation terms. Correspondingly, near the threshold the impact of $ {\cal S} $ is expected to be minimal. This expectation is supported by the numerical calculation, in which the difference between the curves with and without the $ {\cal O}(\alpha _{S}) $ perturbative Sudakov factor is negligible at $ Q=5 $ GeV, and is less than a few percent and $ Q=15 $ GeV. In contrast, at $ Q=50 $ GeV the distribution with the $ {\cal O}(\alpha _{S}) $ Sudakov factor is noticeably lower and broader than the distribution without it: at some values of $ \theta _{H} $, the difference in cross sections reaches $ 40\% $. The influence of the Sudakov factor on the integrated rate is mild: the inclusive cross section $ d\sigma /(dxdQ^{2}) $ calculated without and with the $ {\cal O}(\alpha _{S}) $ Sudakov factor is equal to $ 330 $ and $ 320\mbox {\, fb/GeV}^{2} $, respectively. Due to the enhancement at small $ \theta _{H} $, these resummed inclusive cross sections are larger than the finite-order rate $ d\sigma _{FO}/(dxdQ^{2})\approx 260\mbox {\, fb/GeV}^{2} $ by about $ 25\% $.
It is interesting to compare our calculation with the massless approximation for the $ \gamma ^{*}G $ contribution. Fig. \[fig:MassiveVsMassless\] shows the finite-order and resummed cross sections calculated in the massive and massless approaches. In contrast to the massive $ \sigma _{TOT} $, the massless $ \sigma _{TOT} $ must include the nonperturbative Sudakov factor $ S^{NP} $, which is not known *a priori* and is usually found by fitting to the data. To have some reference point, we plot the massless $ \sigma _{TOT} $ with $ S^{NP}(b)=b^{2}M^{2}/b_{0}^{2}\approx 16b^{2} $, so that, in analogy to the massive case, the region of $ b\gtrsim b_{0}/M $ in the massless $ {\widetilde{W }}(b,Q,x,z) $ is suppressed. Since the heavy-quark mass has other effects on the shape of $ {\widetilde{W }}(b,Q,x,z) $ besides the cutoff in the $ b $-space, we expect the shape of the massless and massive resummed curves be somewhat different. This feature is indeed supported by Fig. \[fig:MassiveVsMassless\]b, where at small $ \theta _{H} $ both resummed curves are of the same order of magnitude, but differ in detail. Furthermore, the shape of the massless $ \sigma _{TOT} $ can be varied by adjusting $ S^{NP} $. At the same time, the massive resummed cross section is uniquely determined by our calculation.
At sufficiently large $ \theta _{H} $, both the massless and massive resummed cross sections reduce to their corresponding finite-order counterparts. The massless cross section significantly overestimates the massive result near the threshold and at intermediate values of $ Q $. For instance, at $ Q=15 $ GeV (Fig. \[fig:MassiveVsMassless\]a) the massless cross section is several times larger than the massive cross section in the whole range of $ \theta _{H}. $ In contrast, at $ Q=50 $ GeV (Fig. \[fig:MassiveVsMassless\]b) the massless $ \sigma _{FO} $ agrees well with the massive $ \sigma _{FO} $ at $ \theta _{H}\gtrsim 20^{\circ } $ and overestimates the massive $ \sigma _{FO} $ at $ \theta _{H}\lesssim 20^{\circ }. $ The massive $ \sigma _{TOT} $ is above the massless $ \sigma _{FO} $ at $ 3^{\circ }\lesssim \theta _{H}\lesssim 10^{\circ } $ and below it at $ \theta _{H}\lesssim 3^{\circ }. $
The presence of two critical angles ($ \theta _{H}\sim 3^{\circ } $ and $ \sim 10^{\circ } $) in $ \sigma _{TOT} $ can be qualitatively understood from the following argument. The rapid rise of $ \sigma _{TOT} $ over the massive $ \sigma _{FO} $ begins when the small-$ q_{T} $ logarithms $ \ln ^{m}\left( {\widetilde{q }}_{T}^{2}/Q^{2}\right) $ become large — say, when $ {\widetilde{q }}_{T}^{2} $ is less than one tenth of $ Q^{2} $. Given that the Peterson fragmentation function peaks at about $ z\sim 0.95 $, and that $ Q=50 $ GeV, $ M=4.5 $ GeV, the condition $ {\widetilde{q }}_{T}^{2}\sim \, 0.1\, Q^{2} $ corresponds to $ \theta _{H}\sim 8^{\circ }, $ which is close to the observed critical angle of $ 10^{\circ }. $ Note that in that region $ q_{T}^{2}\gg M^{2}/z^{2}. $ On the other hand, when $ q_{T}^{2} $ is of order $ M^{2}/z^{2} $, the growth of the logarithms $ \ln \left( {\widetilde{q }}_{T}^{2}/Q^{2}\right) $ is inhibited by the non-zero mass term $ M^{2}/z^{2} $ in $ {\widetilde{q }}_{T}^{2} $. The condition $ q_{T}^{2}\sim M^{2}/z^{2} $ corresponds to $ \theta _{H}\sim 2.5^{\circ } $, which is approximately where the mass-dependent cross section turns down.
Conclusion and outlook
======================
In this paper, we presented a method to describe polar angle distributions in heavy quark production in deep inelastic scattering. This method is realized in the simplified ACOT factorization scheme [@Collins:1998rz; @Kramer:2000hn] and uses the impact parameter space ($ b $-space) formalism [@Collins:1981uk; @Collins:1982va; @Collins:1985kg] to resum transverse momentum logarithms in the current fragmentation region. We discussed general features of this formalism and performed an explicit calculation of the resummed cross section for the $ {\cal O}(\alpha _{S}^{0}) $ flavor-excitation and $ {\cal O}(\alpha _{S}^{1}) $ flavor-creation subprocesses in bottom quark production. According to the numerical results in Section \[sec:NumericalResults\], the multiple parton radiation effects in this process become important at $ Q\gtrsim 15 $ GeV (or approximately at $ Q^{2}/M^{2}\gtrsim 10 $). At $ Q=50 $ GeV, the multiple parton radiation increases the inclusive cross section by about $ 25\% $ as compared to the finite-order flavor-creation cross section.
Many aspects of the resummation in the presence of the heavy quarks are similar to those in the massless resummation. In particular, it is possible to organize the calculation in the massive case in a close analogy to the massless case by properly redefining the Lorentz invariants (in particular, by replacing the Lorentz-invariant transverse momentum $ q_{T} $ in the logarithms by the rescaled transverse mass $ {\widetilde{q }}_{T}=\sqrt{q_{T}^{2}+M^{2}/{\widehat{z}}^{2}} $). The total resummed cross section is presented as a sum of the $ b $-space integral $ \sigma _{{\widetilde{W }}} $ and the finite-order cross section $ \sigma _{FO} $, from which we subtract the asymptotic piece $ \sigma _{ASY} $. Constructed in this way, the resummed cross section reduces to the finite-order cross section at $ Q\approx M $ and reproduces the massless resummed cross section at $ Q\gg M $.
At the same time, there are important differences between the light- and heavy-hadron cases. For instance, the light hadron production is sensitive to the coherent QCD radiation with a wavelength of order $ 1/\Lambda _{QCD} $, which is poorly known and has to be modeled by the phenomenological “nonperturbative Sudakov factor”. In contrast, in the heavy-hadron case such long-distance radiation is suppressed by the large value of $ M $. Hence, for a sufficiently heavy $ M $, as in bottom quark production, the resummed cross section can be calculated without introducing the nonperturbative large-$ b $ contributions. It will be interesting to test the hypothesis about the absence of such long-distance contributions experimentally. Given the size of the differential cross sections obtained in Section \[sec:NumericalResults\], accurate tests of this approach will be feasible once the integrated luminosity of the HERA II run approaches $ 1\mbox {\, fb}^{-1} $. The same calculation can be done for charm production. However, in that case the region $ b\gtrsim 1\, \mbox {GeV}^{-1} $ is not as suppressed, and the nonperturbative Sudakov factor has to be included.
The extension of our calculation to higher orders is feasible in the near future, since many of its ingredients are already available in the literature [@Harris:1995tu; @Nadolsky:1999kb; @Amundson:2000vg]. Furthermore, in a forthcoming paper we will study the additional effects of threshold resummation [@Kidonakis:1996aq; @Kidonakis:1997gm; @Kidonakis:1998bk; @Kidonakis:1998nf; @Kidonakis:1999ze; @Kidonakis:2000ui] in DIS heavy-quark production, so that both transverse momentum and threshold logarithms are taken into account. We conclude that the combined resummation of the mass-dependent logarithms $ \ln (M^{2}/Q^{2}) $ and transverse momentum logarithms $ \ln (q_{T}^{2}/Q^{2}) $ is an important ingredient of the theoretical framework that aims at matching the growing precision of the world heavy-flavor data.
Acknowledgements {#acknowledgements .unnumbered}
================
Authors have benefited from discussions with J. C. Collins, J. Smith, D. Soper, G. Sterman, W.-K. Tung, and other members of the CTEQ Collaboration. We also appreciate discussions of related topics with A. Belyaev, B. Harris, R. Vega, W. Vogelsang, and S. Willenbrock. We thank S. Kretzer for the correspondence on the scaling variable in the ACOT factorization scheme and R. Scalise for the participation in early stages of the project. The work of P. M. N. and F. I. O. was supported by the U.S. Department of Energy, National Science Foundation, and Lightner-Sams Foundation. The research of N. K. has been supported by a Marie Curie Fellowship of the European Community programme “Improving Human Research Potential” under contract number HPMF-CT-2001-01221. The research of C.-P. Y. has been supported by the National Science Foundation under grant PHY-0100677.
\[Appendix:Chg\]\[Appendix:Wterm\]Calculation of the mass-dependent $ {\cal C}\protect $-function
=================================================================================================
In this Appendix, we derive the $ {\cal O}(\alpha _{S}) $ part of the function $ {\cal C}^{in}_{h/G}(x,\mu _{F}b,bM) $. This is the only $ {\cal O}(\alpha _{S}) $ term in the heavy-quark $ \widetilde{W} $-term that explicitly depends on the heavy-quark mass $ M $. This function appears in the factorized small-$ b $ expression for the “$ b $-dependent PDF” $ \overline{{\mathscr {P}}}_{h/A}^{in}\left( x,b,\{m_{q}\},C_{1}/C_{2}\right) $: $$\begin{aligned}
& & \overline{{\mathscr {P}}}_{h/A}^{in}\left( x,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) =\int _{x}^{1}\frac{d\xi _{a}}{\xi _{a}}\nonumber \\
& & \times {\mathcal{C}}_{h/a}^{in}\left( {\widehat{x}},\mu _{F}b,bM;\frac{C_{1}}{C_{2}}\right) f_{a/A}\left( \xi _{a},\left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) .\nonumber \\
& & \label{PbFactorizationHeavy2} \end{aligned}$$
To perform this calculation, we consider a more elementary form of Eq. (\[PbFactorizationHeavy2\]), which represents the leading regions in Feynman graphs in the limit $ Q\rightarrow \infty $. This elementary form can be found in Ref. [@Collins:1982va], where it was derived in the case of $ e^{+}e^{-} $ hadroproduction. The function $ \overline{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},C_{1}/C_{2}\right) $ is decomposed as $$\begin{aligned}
& & \overline{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) =\left| {{\mathscr H}}_{j}\left( \frac{C_{1}}{C_{2}b}\right) \right| \nonumber \\
& & \times \widetilde{U}\left( b\right) ^{1/2}\, \widehat{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},\mu ,\frac{C_{1}}{C_{2}}\right) .\label{barPin} \end{aligned}$$ Here $ {{\mathscr H}}_{j} $ denotes the “hard vertex”, which contains highly off-shell subgraphs. $ \widetilde{U} $ denotes soft subgraphs attached to $ {{\mathscr H}}_{j} $ through gluon lines. $ {\widehat{{\mathscr {P}}}}_{j/A}^{in}(x,b,\{m_{q}\},C_{1}/C_{2}) $ consists of subgraphs corresponding to the propagation of the incoming hadronic jet. The jet part $ {\widehat{{\mathscr {P}}}}_{j/A}^{in}(x,b,\{m_{q}\},C_{1}/C_{2}) $ is related to the $ k_{T} $-dependent PDF $ {\mathscr {P}}_{j/A}^{in}(x,k_{T},\{m_{q}\},\zeta _{A}) $, defined as $$\begin{aligned}
& & {\mathscr {P}}_{j/A}^{in}(x,k_{T},\{m_{q}\},\zeta _{A})=\overline{\sum _{spin}}\, \overline{\sum _{color}}\int \frac{dy^{-}d^{2}{{\bf y}}_{T}}{(2\pi )^{3}}\nonumber \\
& & \times e^{-ixp_{A}^{+}y^{-}+i{{\bf k}}_{T}\cdot {{\bf y}}_{T}}\nonumber \\
& & \times \langle p_{A}|\bar{\psi }_{j}(0,y^{-},{{\bf y}}_{T})\frac{\gamma ^{+}}{2}\psi _{j}(0)|p_{A}\rangle \label{Pin} \end{aligned}$$ in the frame where $ p_{A}^{\mu }=\left\{ p_{A}^{+},0,{{\bf 0}}_{T}\right\} , $ $ p_{a}^{\mu }=\left\{ xp_{A}^{+},M^{2}/(2xp_{A}^{+}),{{\bf k}}_{T}\right\} , $ and $ p_{A}^{+}\rightarrow \infty $. This definition is given in a gauge $ \eta \cdot {\mathscr A}=0 $ with $ \eta ^{2}<0 $. The $ k_{T} $-dependent PDF depends on the gauge through the parameter $ \zeta _{A}\equiv (p_{A}\cdot \eta )/|\eta ^{2}| $.
Let $ {\widetilde{{\mathscr {P}}}}_{j/A}^{in}(x,b,\{m_{q}\},\zeta _{A}) $ be the $ b $-space transform of $ {\mathscr {P}}_{j/A}^{in}(x,k_{T},\{m_{q}\},\zeta _{A}) $ taken in $ d $ dimensions: $$\begin{aligned}
& & {\widetilde{{\mathscr {P}}}}_{j/A}^{in}(x,b,\zeta _{A},\{m_{q}\})\equiv \int d^{d-2}{{\bf k}}_{T}e^{i{{\bf k}}_{T}\cdot {{\bf b}}}\nonumber \\
& & \times {\mathscr {P}}_{j/A}^{in}(x,k_{T},\zeta _{A},\{m_{q}\}).\label{wtPin} \end{aligned}$$ Note that our definition $ {\widetilde{{\mathscr {P}}}}_{j/A}^{in}(x,b,\zeta _{A},\{m_{q}\}) $ differs from the definition in Ref. [@Collins:1982uw] by a factor $ (2\pi )^{2-d}. $ The jet part $ {\widehat{{\mathscr {P}}}}_{a/A}^{in}(x,b,\{m_{q}\},C_{1}/C_{2}) $ is related to $ {\widetilde{{\mathscr {P}}}}_{j/A}^{in}(x,b,\{m_{q}\},\zeta _{A}) $ in the limit $ \zeta _{A}\rightarrow \infty $: $$\begin{aligned}
& & \widehat{{\mathscr {P}}}_{j/A}^{in}\left( x,b,\{m_{q}\},\frac{C_{1}}{C_{2}}\right) =\lim _{\zeta _{A}\rightarrow \infty }\Biggl \{e^{{\cal S}'(b,\zeta _{A};C_{1}/C_{2})}\nonumber \\
& & \times {\widetilde{{\mathscr {P}}}}_{j/A}^{in}(x,b,\{m_{q}\},\zeta _{A})\Biggr \},\label{limzeta} \end{aligned}$$ where $ {\cal S}'(b,\zeta _{A};C_{1}/C_{2}) $ is a partial Sudakov factor, $$\begin{aligned}
& & {\cal S}'(b,\zeta _{A};C_{1}/C_{2})\equiv \int _{C_{1}/b}^{C_{2}\zeta _{A}^{1/2}}\frac{d\bar{\mu }}{\bar{\mu }}\nonumber \\
& & \times \Biggl [\ln \left( \frac{C_{2}\zeta ^{1/2}}{\bar{\mu }}\right) \gamma _{\mathscr K}(\alpha _{S}(\bar{\mu }))\nonumber \\
& & -{\mathscr K}\left( b;\alpha _{S}\left( \frac{C_{1}}{b}\right) ,\frac{C_{1}}{b}\right) -{\mathscr G}\left( \frac{\bar{\mu }}{C_{2}};\alpha _{S}(\bar{\mu }),\bar{\mu }\right) \Biggr ].\nonumber \\
& & \label{Sprime} \end{aligned}$$ The definitions of the functions $ \gamma _{\mathscr K}, $ $ {\mathscr K} $, and $ \mathscr G $ can be found in Ref. [@Collins:1981uk].
We now have all necessary ingredients for the calculation of the $ {\cal O}(\alpha _{S}/\pi ) $ function $ {\cal C}^{in(1)}_{h/G}(x,\mu _{F}b,bM) $. Setting $ j=h $ and $ A=G, $ and expanding Eqs. (\[PbFactorizationHeavy2\],\[barPin\],\[limzeta\]), and (\[Sprime\]) in powers of $ \alpha _{S}/\pi $, we find $$\begin{aligned}
{\cal C}^{in(1)}_{h/G}(x,\mu _{F}b,bM) & = & \lim _{\zeta _{A}\rightarrow \infty }\left\{ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M,\zeta _{A})\right\} \nonumber \\
& - & f_{h/G}^{(1)}\left( x,\mu _{F}/M\right) ,\label{Cin1hG} \end{aligned}$$ where the superscript in parentheses denotes the order of $ \alpha _{S}/\pi $. In the derivation of this equation, we used the following easily deducible equalities: $$\begin{aligned}
& & \mathscr {H}_{h}^{(0)}={\widetilde{U }}^{(0)}=1,\\
& & \left( {\cal S}^{'}\right) ^{(0)}={\widetilde{{\mathscr {P}}}}_{h/G}^{in(0)}={\cal C}_{h/G}^{in(0)}=f_{h/G}^{(0)}=0,\\
& & {\cal C}_{h/h}^{in(0)}(x)=f_{G/G}^{(0)}(x)=\delta (x-1).\end{aligned}$$
The r.h.s. of Eq. (\[Cin1hG\]) can be calculated with the help of the definitions for $ f_{h/G}^{(1)}\left( x,\mu _{F}/M\right) $ and $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M,\zeta _{A}) $ in Eqs. (\[f\]) and (\[Pin\],\[wtPin\]), respectively. A further simplification can be achieved by observing that at $ {\cal O}(\alpha _{S}/\pi ) $ the limit $ \eta ^{2}\rightarrow 0 $ in $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M,\zeta _{A}) $ can be safely taken before the limit $ \zeta _{A}\rightarrow \infty $, and, furthermore, for $ \eta ^{2}=0 $ the function $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M,\zeta _{A}) $ does not depend on $ \zeta _{A} $. Correspondingly, both objects can be derived in the lightlike gauge from a single cut diagram shown in Fig. \[fig:pdf\_diag\], where the double line corresponds to the factor $ \gamma ^{+}\delta (p_{A}^{_{+}}-p^{+}_{a}-k^{'+})/2 $ in the case of $ f_{h/G}^{(1)}\left( x,\mu _{F}/M\right) $ and $ \gamma ^{+}\delta (p_{A}^{_{+}}-p^{+}_{a}-k^{'+})e^{i{{\bf k}}_{T}^{'}\cdot {{\bf b}}}/2 $ in the case of $ \lim _{\zeta _{A}\rightarrow \infty }{\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M,\zeta _{A}) $.
The difference between $ \lim _{\zeta _{A}\rightarrow \infty }{\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M,\zeta _{A})\equiv {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) $ and $ f_{h/G}^{(1)}\left( x,\mu _{F}/M\right) $ resides in the extra exponential factor $ e^{i{{\bf k}}_{T}^{'}\cdot {{\bf b}}} $ in $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) $. Remarkably, this factor strongly affects the nature of $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) $. The loop integral over $ {{\bf k}}_{T}^{'} $ in $ f_{h/G}^{(1)}\left( x,\mu _{F}/M\right) $ contains a UV singularity, which is regularized by an appropriate counterterm. In the ACOT scheme, the UV singularity is regularized in the $ \overline{MS} $ scheme if $ \mu _{F}\geq M $, and by zero-momentum subtraction if $ \mu _{F}<M $. The result for the heavy-quark PDF $ f_{h/G}^{(1)}\left( x,\mu _{F}/M\right) $ is $$\begin{aligned}
f^{(1)}_{h/G}\left( x,\frac{\mu _{F}}{M}\right) & = & \left\{ \begin{array}{c}
P_{h/G}^{(1)}(x)\ln \left( \mu _{F}/M\right) ,\, \mu _{F}\geq M;\\
0,\, \mu _{F}<M.
\end{array}\right. \nonumber \\
& & \label{f1} \end{aligned}$$ As expected, $ f_{h/G}^{(1)}\left( x,\mu _{F}/M\right) $ exhibits the threshold behavior at $ \mu _{F}=M $.
In contrast, the UV limit in the loop integral of $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) $ is regularized by the oscillating exponent $ e^{i{{\bf k}}_{T}^{'}\cdot {{\bf b}}} $. Since no UV singularity is present in $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) $, it does not depend on $ \mu _{F} $ and, therefore, does not change at the threshold. It is given by $$\begin{aligned}
{\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) & = & P^{(1)}_{h/G}(x)K_{0}(bM)\nonumber \\
& + & T_{R}x(1-x)bMK_{1}(bM).\label{PinhG1} \end{aligned}$$ Here $ K_{0}(bM) $ and $ K_{1}(bM) $ are the modified Bessel functions [@AbramowitzStegun], which satisfy the following useful properties: $$\begin{aligned}
& & \lim _{bM\rightarrow \infty }K_{0}(bM)=\lim _{bM\rightarrow \infty }bMK_{1}(bM)=0;\label{K0infty} \\
& & K_{0}(bM)\rightarrow -\ln \left( bM/b_{0}\right) \mbox {\, as\, }bM\rightarrow 0;\\
& & bM\, K_{1}(bM)\rightarrow 1\mbox {\, as\, }bM\rightarrow 0.\label{K10} \end{aligned}$$ The “infrared-safe” part $ {\mathcal{C}}^{in(1)}_{h/G}(x,\mu _{F}b,bM) $ of $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) $ is obtained by subtracting $ f^{(1)}_{h/G}\left( x,\mu _{F}/M\right) $ as in Eq. (\[Cin1hG\]): $$\begin{aligned}
& & \left. {\mathcal{C}}^{in(1)}_{h/G}(x,\mu _{F}b,bM)\right| _{\mu _{F}\geq M}=T_{R}x(1-x)\nonumber \\
& & \times \left( 1+c_{1}(bM)\right) \nonumber \\
& & +P^{(1)}_{h/G}(x)\left( c_{0}(bM)-\ln \Bigl (\frac{\mu _{F}b}{b_{0}}\Bigr )\right) ;\\
& & \left. {\mathcal{C}}^{in(1)}_{h/G}(x,\mu _{F}b,bM)\right| _{\mu _{F}<M}={\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M)\nonumber \\
& & =\left. {\mathcal{C}}^{in(1)}_{h/G}(x,\mu _{F}b,bM)\right| _{\mu _{F}\geq M}\nonumber \\
& & +P^{(1)}_{h/G}(x)\ln \frac{\mu _{F}}{M}.\label{C1injG} \end{aligned}$$ In these equations, $ c_{0}(bM) $ and $ c_{1}(bM) $ are the parts of $ K_{0}(bM) $ and $ bM\, K_{1}(bM) $ that vanish at $ bM\rightarrow 0 $ (cf. Eqs. ( \[K0infty\]-\[K10\])): $$\begin{aligned}
c_{0}(bM) & \equiv & K_{0}(bM)+\ln \frac{bM}{b_{0}};\label{c0} \\
c_{1}(bM) & \equiv & bMK_{1}(bM)-1.\label{c1} \end{aligned}$$ If $ \mu _{F} $ is chosen to be of order $ b_{0}/b $, no large logarithms appear in $ {\mathcal{C}}^{in(1)}_{h/G}(x,\mu _{F}b,bM) $ at $ b\rightarrow 0 $. At large $ Q $, the small-$ b $ region dominates the integration in Eq. (\[Wmassive\]), so that $ {\mathcal{C}}^{in}_{h/G}({\widehat{x}},\mu _{F}b,bM) $ effectively reduces to its massless expression [@Meng:1996yn; @Nadolsky:1999kb]: $$\begin{aligned}
& & \left. {\mathcal{C}}^{in(1)}_{h/G}(x,\mu _{F}b,bM)\right| _{b\rightarrow 0}=T_{R}x(1-x)\nonumber \\
& & -P^{(1)}_{h/G}(x)\ln \Bigl (\frac{\mu _{F}b}{b_{0}}\Bigr ).\end{aligned}$$
The above manipulations can be interpreted in the following way. At small $ b $ ($ b=b_{0}/\mu _{F}\leq b_{0}/M $), we subtract from $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) $ its infrared-divergent part $ P^{(1)}_{h/G}(x)\ln (\mu _{F}/M), $ which is then included and resummed in the heavy-quark PDF $ f_{h/G}(x,\mu _{F}/M) $. The convolution of the resulting $ {\cal C} $-function with the PDF remains equal to $ {\widetilde{{\mathscr {P}}}}_{h/G}^{in(1)}(x,b,M) $ up to higher-order corrections: $$\begin{aligned}
& & \sum _{a=h,G}{\cal C}^{in}_{h/a}\otimes f_{a/G}=\overline{{\mathscr {P}}}^{in(1)}_{h/G}(x,\mu _{F}b,bM)+{\mathcal{O}}(\alpha _{S}^{2}).\nonumber \\
& & \label{cfs} \end{aligned}$$ At large $ b $ ($ b>b_{0}/M $), the heavy-quark PDF $ f_{h/G} $ is identically equal to zero. To preserve the relationship (\[cfs\]) below the threshold, one should include the above logarithmic term in the function $ {\mathcal{C}}^{in(1)}_{h/G}(x,\mu _{F}b,bM) $, as shown in Eq. (\[C1injG\]). The addition of an extra term $ P^{(1)}_{h/G}(x)\ln \left( \mu _{F}/M\right) $ to $ {\mathcal{C}}^{in(1)}_{h/G}(x,\mu _{F}b,bM) $ at $ \mu _{F}<M $ enforces the smoothness of the form-factor $ {\widetilde{W }}(b,Q,x,z) $ in the threshold region, which, in its turn, is needed to avoid unphysical oscillations of the cross section $ d\sigma /dq_{T}^{2} $.
\[Appendix:FO\]The finite-order cross section
==============================================
This Appendix discusses the finite-order cross section $ d{\widehat{\sigma }}_{FO}/d{\widehat{\Phi }} $ that appears in the factorized hadronic cross section (\[Factorization\]). For the $ {\cal O}(\alpha _{S}^{0}) $ subprocess $ e+h\rightarrow e+h $, this cross section is the same as in the massless case: $$\begin{aligned}
& & \left( \frac{d{\widehat{\sigma }}(e+h\rightarrow e+h)}{d{\widehat{\Phi }}}\right) _{FO}=\frac{\sigma _{0}F_{l}}{S_{eA}}\frac{A_{1}(\psi ,\varphi )}{2}\nonumber \\
& & \times e_{j}^{2}\delta ({{\bf q}}_{T})\delta (1-{\widehat{x}})\delta (1-{\widehat{z}}),\label{FiniteOrderLO} \end{aligned}$$ where, in accordance with the notations of Ref. [@Nadolsky:1999kb], $$\begin{aligned}
\sigma _{0} & \equiv & \frac{Q^{2}}{4\pi S_{eA}x^{2}}\Bigl (\frac{e^{2}}{2}\Bigr ),\nonumber \label{sigma0Fl} \\
F_{l} & \equiv & \frac{e^{2}}{2}\frac{1}{Q^{2}}.\end{aligned}$$
The contribution of the gluon-photon fusion channel is
$$\begin{aligned}
& & \left( \frac{d{\widehat{\sigma }}(e+G\rightarrow e+h+\bar{h})}{d{\widehat{\Phi }}}\right) _{FO}=\frac{\sigma _{0}F_{l}}{4\pi S_{eA}}\frac{\alpha _{S}}{\pi }e_{Q}^{2}\delta \left( \left( \frac{1}{{\widehat{x}}}-1\right) \left( \frac{1}{{\widehat{z}}}-1\right) -\frac{{\widetilde{q }}_{T}^{2}}{Q^{2}}\right) \frac{{\widehat{x}}(1-{\widehat{x}})}{{\widehat{z}}^{2}}\nonumber \\
& & \qquad \qquad \times T_{R}\sum _{\rho =1}^{4}{\widehat{V }}_{\rho }({\widehat{x}},Q^{2},{\widehat{z}},q_{T}^{2},M^{2})A_{\rho }(\psi ,\varphi ),\label{FiniteOrder} \end{aligned}$$
where $ A_{\rho }(\psi ,\varphi ) $ denote orthonormal functions of the leptonic azimuthal angle $ \varphi $ and boost parameter $ \psi $ given in Eq. (\[As\]). The structure functions $ {\widehat{V }}_{\rho }({\widehat{x}},Q^{2},{\widehat{z}},q_{T}^{2},M^{2}) $ are calculated to be $$\begin{aligned}
{\widehat{V }}_{1} & = & \frac{1}{{\widehat{x}}^{2}{\widetilde{q }}_{T}^{2}}\left( 1-2{\widehat{x}}{\widehat{z}}+2{\widehat{x}}^{2}{\widehat{z}}^{2}-4\frac{M^{2}{\widehat{x}}^{2}}{Q^{2}}\right) \nonumber \\
& + & \frac{2{\widehat{z}}}{{\widehat{x}}}\frac{1}{Q^{2}}\left( 5{\widehat{x}}{\widehat{z}}-{\widehat{x}}-{\widehat{z}}\right) +\nonumber \\
& + & \kappa _{1}\left( 4\frac{{\widetilde{q }}_{T}^{2}}{Q^{2}}{\widehat{z}}^{2}+2-8{\widehat{z}}+8{\widehat{z}}^{2}-4\frac{M^{2}}{Q^{2}}\right) ,\label{V1jg} \\
{\widehat{V }}_{2} & = & 8\frac{1}{Q^{2}}{\widehat{z}}^{2}-4\frac{M^{2}}{Q^{2}}\frac{1}{{\widetilde{q }}_{T}^{2}}+4\kappa _{1}\left( -1+{\widehat{z}}\right) {\widehat{z}},\\
{\widehat{V }}_{3} & = & \frac{2{\widehat{z}}}{{\widehat{x}}}\frac{q_{T}}{Q{\widetilde{q }}_{T}^{2}}\left( -1+2\left( 1+\frac{{\widetilde{q }}_{T}^{2}}{Q^{2}}\right) {\widehat{x}}{\widehat{z}}\right) \nonumber \\
& + & 4\kappa _{1}{\widehat{z}}\left( -1+2{\widehat{z}}\right) \frac{q_{T}}{Q},\\
{\widehat{V }}_{4} & = & 4\frac{q_{T}^{2}}{Q^{2}{\widetilde{q }}_{T}^{2}}{\widehat{z}}^{2}+4\frac{q_{T}^{2}}{Q^{2}}{\widehat{z}}^{2}\kappa _{1}.\label{V4jg} \end{aligned}$$ In Eqs. (\[V1jg\]-\[V4jg\]), $$\kappa _{1}\equiv \frac{M^{2}(1-{\widehat{x}})}{{\widehat{z}}^{2}{\widehat{x}}{\widetilde{q }}_{T}^{4}}.$$
\[Appendix:KinematicalCorrection\]Kinematical correction
========================================================
In this Appendix, we derive the kinematical corrections (\[chih1\]) and (\[chih2\]) that are introduced in the flavor-excitation contributions to $ \sigma _{FO} $, as well as in $ \sigma _{\widetilde{W}} $ and $ \sigma _{ASY} $. Let us first consider the $ {\cal O}(\alpha _{S}) $ cross section for the photon-gluon fusion, which we write as $$\begin{aligned}
& & \left( \frac{d\sigma (e+A\rightarrow e+H+X)}{d\Phi }\right) _{\gamma ^{*}G,FO}=\nonumber \\
& & \int \frac{d\xi _{b}}{\xi _{b}}\int \frac{d\xi _{a}}{\xi _{a}}D_{H/h}\left( \xi _{b}\right) f_{G/A}\left( \xi _{a}\right) \nonumber \\
& & \times \delta \left( \left( \frac{1}{{\widehat{x}}}-1\right) \left( \frac{1}{{\widehat{z}}}-1\right) -\frac{{\widetilde{q }}_{T}^{2}}{Q^{2}}\right) \beta (\Phi ).\label{FiniteOrder2} \end{aligned}$$ Here $ \beta ({\widehat{\Phi }}) $ includes all terms in the parton-level cross section $ (d{\widehat{\sigma }}/d{\widehat{\Phi }})_{FO} $ except for the $ \delta - $function (cf. Eq. (\[FiniteOrder\])): $$\begin{aligned}
& & \beta ({\widehat{\Phi }})=\frac{\sigma _{0}F_{l}}{4\pi S_{eA}}\frac{\alpha _{S}}{\pi }e_{h}^{2}\frac{{\widehat{x}}(1-{\widehat{x}})}{{\widehat{z}}^{2}}\nonumber \\
& \times & T_{R}\sum _{\rho =1}^{4}{\widehat{V }}_{\rho }({\widehat{x}},Q^{2},{\widehat{z}},q_{T}^{2},M^{2})A_{\rho }(\psi ,\varphi ).\end{aligned}$$
The $ \delta $-function in Eq. (\[FiniteOrder2\]) can be reorganized as $$\begin{aligned}
& & \delta \left( \left( \frac{1}{{\widehat{x}}}-1\right) \left( \frac{1}{{\widehat{z}}}-1\right) -\frac{{\widetilde{q }}_{T}^{2}}{Q^{2}}\right) =\nonumber \\
& & \frac{z\, Q^{2}}{\sqrt{\widehat{W}^{4}-4M^{2}\left( q_{T}^{2}+\widehat{W}^{2}\right) }}\Biggl [\delta \left( \xi _{b}-\xi _{b}^{+}\right) \nonumber \\
& & +\delta \left( \xi _{b}-\xi _{b}^{-}\right) \Biggr ],\label{deltaxib} \end{aligned}$$ where $$\xi _{b}^{\pm }\equiv z\frac{\widehat{W}^{2}\pm \sqrt{\widehat{W}^{4}-4M^{2}(q_{T}^{2}+\widehat{W}^{2})}}{2M^{2}},$$ and $ {\widehat{W }}^{2}\equiv Q^{2}\left( 1-{\widehat{x}}\right) /{\widehat{x}}$. We see that the mass-dependent phase space element contains two $ \delta $-functions $ \delta (\xi _{b}-\xi _{b}^{+}) $ and $ \delta (\xi _{b}-\xi _{b}^{-}) $, which can be used to integrate out the dependence on $ \xi _{b} $ in Eq. (\[FiniteOrder2\]).
It can be further shown that in the massless limit the solutions $ \xi _{b}=\xi _{b}^{-} $ and $ \xi _{b}=\xi _{b}^{+} $ correspond to the heavy quarks produced in the current and target fragmentation regions, respectively. When $ M\rightarrow 0 $, the relationship (\[deltaxib\]) simplifies to $$\begin{aligned}
& & \delta \left( \left( \frac{1}{{\widehat{x}}}-1\right) \left( \frac{1}{{\widehat{z}}}-1\right) -\frac{q_{T}^{2}}{Q^{2}}\right) =\nonumber \\
& & \frac{z\, Q^{2}}{\widehat{W}^{2}}\left[ \delta \left( \xi _{b}-\xi _{b}^{0+}\right) +\delta \left( \xi _{b}-\xi _{b}^{0-}\right) \right] ,\label{deltaxibM0} \end{aligned}$$ where $$\begin{aligned}
\xi _{b}^{0+} & = & z\left( \frac{\widehat{W}^{2}}{M^{2}}-\frac{q_{T}^{2}+\widehat{W}^{2}}{{\widehat{W }}^{2}}+{\mathcal{O}}(M^{2})\right) ,\\
\xi _{b}^{0-} & = & z\left( \frac{q_{T}^{2}+\widehat{W}^{2}}{\widehat{W}^{2}}+{\mathcal{O}}(M^{2})\right) .\end{aligned}$$ In this limit, the solution $ \xi _{b}^{0+} $ diverges (and, therefore, will not contribute) unless $ z $ is identically zero. However, according to Eq. (\[z\]) and the last paragraph in Subsection B of Section \[sec:PhotonGluon\], at $ z=0 $ the observed final-state hadron appears among remnants of the target ($ \theta _{H}\approx 180^{\circ } $ in the $ \gamma ^{*}A $ c.m. frame), *i.e.*, *away* from the region of our primary interest (small and intermediate $ \theta _{H} $). Hence, in the limit $ \theta _{H}\rightarrow 0 $ all dominant logarithmic contributions as well as their all-order sums (the flavor-excitation cross section and $ \widetilde{W} $-term) arise only from terms proportional to $ \delta (\xi _{b}-\xi _{b}^{-}) $. The contributions proportional to $ \delta (\xi _{b}-\xi _{b}^{+}) $ in the current fragmentation region are suppressed.
The integration over $ \xi _{b} $ with the help of Eq. (\[deltaxib\]) leads to the following expression for the cross section (\[FiniteOrder2\]):
$$\begin{aligned}
& & \left( \frac{d\sigma (e+A\rightarrow e+H+X)}{dxdQ^{2}dzdq_{T}^{2}d\varphi }\right) _{\gamma ^{*}G}=\int _{\xi ^{min}_{a}}^{\xi _{a}^{max}}\frac{d\xi _{a}}{\xi _{a}}f_{G/A}(\xi _{a},\mu _{F})\frac{Q^{2}}{\sqrt{\widehat{W}^{4}-4M^{2}\left( q_{T}^{2}+\widehat{W}^{2}\right) }}\nonumber \\
& & \times \left[ \left. {\widehat{z}}D_{H/h}(\xi _{b},\mu _{F})\beta ({\widehat{\Phi }})\right| _{\xi _{b}=\xi _{b}^{+}}+\left. {\widehat{z}}D_{H/h}(\xi _{b},\mu _{F})\beta ({\widehat{\Phi }})\right| _{\xi _{b}=\xi _{b}^{-}}\right] .\label{Fobeta2} \end{aligned}$$
Here the lower and upper integration limits $ \xi _{a}^{min} $ and $ \xi _{a}^{max} $ are determined by demanding the argument of the square root in Eq. (\[Fobeta2\]) be non-negative and $ \xi _{b}\leq 1 $; that is, $$\begin{aligned}
\xi _{a}^{min} & = & x\left( 1+\frac{2M\left( M+\sqrt{M^{2}+q_{T}^{2}}\right) }{Q^{2}}\right) ,\nonumber \\
\xi ^{max}_{a} & = & \min \left[ x\left( 1+\frac{M^{2}+z^{2}q_{T}^{2}}{z(1-z)Q^{2}}\right) ,1\right] \label{ximinximaxPlus} \end{aligned}$$ for $ \xi _{b}=\xi _{b}^{+} $, and $$\begin{aligned}
\xi _{a}^{min} & = & x\left( 1+\frac{1}{z(1-z)}\frac{M^{2}+z^{2}q_{T}^{2}}{Q^{2}}\right) ,\nonumber \\
\xi ^{max}_{a} & = & 1\label{ximinximaxMinus} \end{aligned}$$ for $ \xi _{b}=\xi _{b}^{-} $. We see that, according to the exact kinematics of heavy flavor production, the heavy quark pairs are produced only when the light-cone momentum fraction $ \xi _{a} $ is not less than $ \xi _{a}^{min} $ (where $ \xi _{a}^{min}\geq x $) and not more than $ \xi _{a}^{max} $ (where $ \xi _{a}^{max}\leq 1 $). The exact values of $ \xi _{a}^{min} $ and $ \xi _{a}^{max} $ are different for the branches with $ \xi _{b}=\xi _{b}^{+} $ and $ \xi _{b}=\xi _{b}^{-}. $
Turning now to the flavor-excitation contributions $ \gamma ^{*}+h\rightarrow h+X $, we find that in those the integration over $ \xi _{a} $ *a priori* covers the whole range $ x\leq \xi _{a}\leq 1. $ Indeed, in those contributions the heavy antiquark in the remnants of the incident hadron is ignored, so that the reaction can go at a lower c.m. energy $ \widehat{W} $ than it is allowed by the exact kinematics. Since the PDF’s grow rapidly at small $ x $, the naively calculated total cross section $ \sigma _{TOT} $ tends to contain large contributions from the unphysical region of small $ x $ and disagree with the data. To fix this problem, we use Eq. (\[ximinximaxMinus\]) to derive the following scaling variable in the *finite-order* flavor-excitation contributions: $$\label{chi21}
\chi _{h}=x\left( 1+\frac{1}{z(1-z)}\frac{M^{2}}{Q^{2}}\right) .$$ This variable takes into account the fact that the incoming heavy quark in the flavor-excitation process appears from the contributions with $ \xi _{b}=\xi _{b}^{-} $ in the flavor-creation process, and that the transverse momentum $ zq_{T} $ of this quark in the *finite-order* cross section is identically zero.
Similarly, we notice that the $ \widetilde{W} $-term $ \sigma _{\widetilde{W}} $ and its finite-order expansion $ \sigma _{ASY} $ contain the “$ b $-dependent PDF’s” $ \overline{{\mathscr {P}}}_{h/A}^{in}\left( x,b,M,C_{1}/C_{2}\right) $, which correspond to the incoming heavy quarks with a *non-zero* transverse momentum. According to Eq. (\[ximinximaxMinus\]), the available phase space in the longitudinal direction is a decreasing function of the transverse momentum $ zq_{T} $, and it is desirable to implement this phase-space reduction to improve the cancellation between $ \sigma _{\widetilde{W}} $ and $ \sigma _{ASY} $ at large $ q_{T} $. In our calculation, this feature is implemented by evaluating $ \sigma _{\widetilde{W}} $ and $ \sigma _{ASY} $ at the scaling variable $$\label{chi22}
\chi '_{h}=x\left( 1+\frac{1}{z(1-z)}\frac{M^{2}+z^{2}q_{T}^{2}}{Q^{2}}\right) ,$$ which immediately follows from Eq. (\[ximinximaxMinus\]).
Despite the apparent complexity of the scaling variables (\[chi21\]) and (\[chi22\]), they satisfy the following important properties:
1. They are straightforwardly derived from the exact kinematical constraints on the variable $ \xi _{a} $ in Eqs. (\[ximinximaxPlus\]) and (\[ximinximaxMinus\]).
2. They remove contributions from unphysical values of $ x $ *at all values of $ Q $ and $ q_{T} $*, thus leading to better agreement with the data.
3. In the limit $ Q^{2}\gg M^{2}, $ the variable $ \chi _{h} $ in $ \sigma _{FO} $ reduces to $ x $ (cf. Eq. (\[chi21\])), so that the standard factorization for the massless finite-order cross sections is reproduced.
4. In the limit $ Q^{2}\gg M^{2} $ and $ Q^{2}\gg q_{T}^{2} $, the variable $ \chi _{h}^{'} $ in $ \sigma _{\widetilde{W}} $ and $ \sigma _{ASY} $ reduces to $ x $ (cf. Eq. (\[chi22\])), so that the exact resummed cross section is reproduced.
Finally, consider the integration of the cross section (\[Fobeta2\]) over $ z, $ $ q^{2}_{T} $, and $ \varphi $ to obtain the $ {\cal O}(\alpha _{S}) $ $ \gamma ^{*}G $ contribution to an inclusive DIS function $ F(x,Q^{2}) $. We find that $$\begin{aligned}
\left. F(x,Q^{2})\right| _{\gamma ^{*}G,{\cal O}(\alpha _{S})} & = & \int _{\xi _{a}^{'}}^{1}\frac{d\xi _{a}}{\xi _{a}}C^{(1)}_{H/G}\left( \frac{x}{\xi _{a}},\frac{\mu _{F}}{Q},\frac{M}{Q}\right) \nonumber \\
& \times & f_{G/A}\left( \xi _{a},\, \left\{ \frac{\mu _{F}}{m_{q}}\right\} \right) ,\end{aligned}$$ where the lower limit of the integral over $ \xi _{a} $ is given by $$\xi _{a}^{'}=x\left( 1+\frac{4M^{2}}{Q^{2}}\right)$$ for both solutions $ \xi _{b}=\xi _{b}^{+} $ and $ \xi _{b}=\xi _{b}^{-} $. This value of $ \xi _{a}^{'} $ can be easily found from Eqs. (\[ximinximaxPlus\]) and (\[ximinximaxMinus\]), given that $ q_{T}\geq 0 $, $ 0\leq z\leq 1 $, and $ z(1-z)\leq 1/4 $ in the interval $ 0\leq z\leq 1 $. Since in the $ \gamma ^{*}G $ contribution the integration over $ \xi _{a} $ is constrained from below by $ \xi _{a}^{'}>x $, it makes sense to implement a similar constraint in the flavor-excitation contributions by introducing the scaling variable $ \chi _{h}=x(1+4M^{2}/Q^{2}). $ This variable is precisely the one that appears in the recent version of the ACOT scheme with the optimized treatment of the inclusive structure functions in the threshold region [@Tung:2001mv]. Our scaling variables extend the idea of Ref. [@Tung:2001mv] to the semi-inclusive and resummed cross sections.
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[^1]: The similarity between the multiple parton radiation in semi-inclusive DIS and the other two processes was known for a long time; see, for instance, an early paper [@Dokshitzer:1978dr].
[^2]: An alternative “top-down” approach will require the analysis of leading regions in the high-energy limit and derivation of the evolution equations that retain terms with positive powers of $ M/Q $. Such analysis could involve methods similar to those discussed in Ref. [@Collins:1999dz].
[^3]: We remind the reader that the analysis is performed in the $ \gamma ^{*}A $ c.m. frame, where the incident hadron moves in the $ -z $ direction.
[^4]: The generalization of our approach to higher orders is straightforward. The next-order calculation should include the $ {\cal O}(\alpha _{S}) $ flavor-excitation and $ {\cal O}(\alpha ^{2}_{S}) $ flavor-creation channels, which should appear together to ensure the smoothness of the form factor $ {\widetilde{W }}(b,Q,M,x,z) $ and its suppression at $ b\gtrsim 1/M $.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'A recent high-resolution $\alpha$, $X$-ray, and $\gamma$-ray coincidence-spectroscopy experiment offered a first glimpse of excitation schemes of isotopes along $\alpha$-decay chains of $Z=115$. To understand these observations and to make predictions about shell structure of superheavy nuclei below $^{288}115$, we employ two complementary mean-field models: self-consistent Skyrme Energy Density Functional approach and the macroscopic-microscopic Nilsson model. We discuss the spectroscopic information carried by the new data. In particular, candidates for the experimentally observed $E1$ transitions in $^{276}$Mt are proposed. We find that the presence and nature of low-energy $E1$ transitions in well-deformed nuclei around $Z=110, N=168$ strongly depends on the strength of the spin-orbit coupling; hence, it provides an excellent constraint on theoretical models of superheavy nuclei. To clarify competing theoretical scenarios, an experimental search for $E1$ transitions in odd-$A$ systems $^{275,277}$Mt, $^{275}$Hs, and $^{277}$Ds is strongly recommended.'
author:
- 'Yue Shi (石跃)'
- 'D.E. Ward'
- 'B.G. Carlsson'
- 'J. Dobaczewski'
- 'W. Nazarewicz'
- 'I. Ragnarsson'
- 'D. Rudolph'
title: Structure of Superheavy Nuclei Along Element 115 Decay Chains
---
[UTF8]{}[gbsn]{}
Introduction
============
Superheavy nuclei at the limit of nuclear mass and atomic number pose a formidable challenge to both experiment and theory. The low cross sections for production of these nuclei, in the picobarn range or less, offer limited structural information. Moreover, the $\alpha$-decay chains of nuclei synthesized in experiments using a $^{48}$Ca beam with actinide targets [@(Oga07); @(Oga11); @(Oga12); @(Oga12a); @(Oga13); @(Dul10); @(Gat11); @(Hof12); @(Rud13a)] terminate by spontaneous fission before reaching the known region of the nuclear chart. This poses a problem with the unambiguous identification of the new isotopes, and more direct techniques to determine $Z$ and $A$ must be employed [@(Rud13a)]. Theoretical predictions of the shell structure of superheavy nuclei are also difficult, as the interplay between the electrostatic repulsion and nuclear attraction, combined with a very high density of single-particle (s.p.) states, make the results of calculations extremely sensitive to model details [@(Cwi96); @(Kru00); @(Ben01); @(Cwi05); @(Ben13); @(Shi13a)].
In a recent experimental study [@(Rud13a); @(Rud13b)], unique structural information on low-lying states in superheavy nuclei below $^{288}115$ has been obtained. Of particular interest is the finding that some of the measured transitions in the nucleus assigned to be $^{276}$Mt have $E1$ character, thus suggesting opposite parities of the connected states. The new data offer an exciting opportunity to constraint theoretical models in this region for the first time. Indeed, previous macroscopic-microscopic [@(Cwi94); @(Par04); @(Par05)] and self-consistent studies [@(Cwi99); @(Ben00b)] have shown that the number of opposite-parity s.p. orbitals around the Fermi level is fairly limited, and this is consistent with the Nilsson model analysis of Ref. [@(Rud13a)].
Because of the above-mentioned sensitivity to model details, robust predictions in this region are difficult to make as one is dealing with large extrapolations. To this end, when aiming at reliable predictions, it is advisable to use a model that performs well in the neighboring region where experimental information is more abundant. Furthermore, since the quadrupole deformations of $\alpha$-decay daughters of $^{288}115$ are expected to increase gradually with decreasing $Z$ and $A$ along the $\alpha$-decay chain [@(Cwi94); @(Par04); @(Par05); @(Cwi99); @(Ben00b); @(Cwi05); @(Ben13)], shape polarization is going to play a role when determining the energies of low-lying states.
In this work, we study the low-lying states in the superheavy nuclei below $^{288}115$, using the locally-optimized self-consistent Skyrme Energy Density Functional (SEDF) and Nilsson-Strutinsky (NS) frameworks. To assess the robustness of these results, we also carry out calculations using a globally-optimized SEDF model.
Models
======
The SEDF approach is a variant of nuclear density functional theory, which offers a global, self-consistent description of nuclear properties across the nuclear landscape [@(Ben03); @(Erl12)]. The recent self-consistent study of Ref. [@(Shi13a)] offers a locally optimized SEDF parameterization [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} that meets our local-extrapolability requirements: it reproduces one-quasiparticle (1-q.p.) states in $^{251}$Cf and $^{249}$Bk (the two heaviest systems where 1-q.p. energies are experimentally well known), predicts crucial deformed shell gaps at $N = 152$ and $Z = 100$, and describes rotational bands in Fm, No, and Rf isotopes. The parameter set [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} has been obtained by adjusting the spin-orbit coupling constants of a global SEDF parametrization [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} [@(Kor12)] that performs well for heavy nuclei and large deformations. We shall also use [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} in this study. The calculations follow closely Ref. [@(Shi13a)]. The Skyrme Hartree-Fock-Bogolyubov (SHFB) equations were solved using the symmetry-unrestricted solver [[<span style="font-variant:small-caps;">hfodd</span>]{}]{} (v2.52j) [@(Sch12)] by expanding 1-q.p. wave functions in 680 deformed harmonic-oscillator (HO) basis states. To compute 1-q.p. excitations in odd-$A$ nuclei, we blocked relevant orbits around the Fermi level as described in Ref. [@(Sch10)]. The strengths of the pairing force for neutrons and protons were adjusted to the odd-even mass staggering in $^{251}$Cf and $^{249}$Bk and the kinematic moment of inertia of $^{252}$No.
The SEDF results are compared with those of the Nilsson-Strutinsky (NS) approach of Ref. [@(Car06)] with the modified harmonic oscillator (MO) potential and pairing as in Ref. [@(Nil69)]. The shell-independent MO parameters ($\kappa_p = 0.058$, $\mu_p = 0.63$, $\kappa_n = 0.0526$, and $\mu_n
= 0.457$) have been locally optimized to the actinide nuclei [@(Roz86)] and applied to, e.g., $^{228,230}$Pa [@(Her89)] and $^{242}$Am [@(Hay10)].
The 2-q.p.-plus-rotor calculations for odd-odd nuclei were carried out using the MO model of Ref. [@(Rag88)]. The moments of inertia were chosen according to a phenomenological relation of Ref. [@(Gro62)]. The BCS pairing was treated as in Ref. [@(Nil69)], with the monopole pairing strengths taken as 95% of the values for even-even nuclei. No residual proton-neutron interaction was considered.
Results
=======
We first discuss properties of the even-even nuclei belonging to the $\alpha$-decay chain of $^{296}$120. Their ground states form q.p. vacua for neighboring odd-$A$ and odd-odd systems. The calculated quadrupole moments are shown in Fig. \[figQ2\]. Both SEDF models predict a similar smooth increase of quadrupole deformation along the $\alpha$-chain. In the NS calculations, $^{296}$120 is nearly spherical, $^{292}$118 and $^{288}$Lv are very weakly deformed, $^{284}$Fl and $^{280}$Cn are spherical, and the shapes of the lightest daughters have deformations close to those predicted by SEDF. These results suggest that a direct comparison between SEDF and NS models is most meaningful for $Z < 112$.
![\[figQ2\] (Color online) Quadrupole moments $Q_2$ of even-even nuclei forming the $\alpha$-decay chain $^{296}120\rightarrow \cdots
\rightarrow ^{264}$Rf, calculated with [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} and [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} SEDF models and the NS approach. ](Figure1.pdf){width="1.0\columnwidth"}
![ \[figEspNS\] (Color online) Nilsson diagram for neutrons (top) and protons (bottom) for nuclei along the $\alpha$-decay chain of $^{296}$120 using the MO potential of Ref. [@(Roz86)]. The orbits are labelled by the standard asymptotic Nilsson numbers [@(Nil69)]. The positive/negative parity levels are marked by solid/dashed lines. The Fermi levels of nuclei in Fig. \[figQ2\] are indicated by dots. The quadrupole moment was determined from shape deformations $\epsilon_2$ and $\epsilon_4$ [@defors]: $Q_{20}=0.8 AR_0^2
(\epsilon_2 + 0.5 \epsilon^2_2+0.758\epsilon^2_4-\epsilon_2 \epsilon_4)$ with $A=280$, $R_0 = r_0 A^{1/3}$, and $r_0= 1.217$fm. ](Figure2.pdf){width="0.8\columnwidth"}
It is instructive to begin the discussion from the Nilsson s.p. diagram of the MO potential shown in Fig. \[figEspNS\]. The main features of this diagram, such as the appearance of spherical shell gaps at $Z=114$ and $N=184$, have remained unchanged since the late 1960s [@(Gus67); @(Mos69)]. It is worth noting that the s.p. spectrum of the MO model, with its pronounced spherical shell gaps at $Z=114$ and $N=184$ and resulting Nilsson orbits, is fairly close to that of more realistic Woods-Saxon [@(Cwi94); @(Par04); @(Par05)] and Folded-Yukawa [@(Bol72); @(Nix72); @(Mol94)] potentials, see Refs. [@(Cwi96); @(Ben99a)] for more discussion.
The deformed shell structure of nuclei at the end of the $\alpha$-decay chain of $^{296}$120 (or $^{288}115$) is relatively simple: both in neutrons and protons there appears one unique-parity, high-$\Omega$ Nilsson state ($\nu$\[716\]13/2 and $\pi$\[615\]11/2) surrounded by levels of opposite parity, such as neutron (\[613\]5/2, \[611\]3/2), (\[606\]11/2, \[604\]9/2) and proton (\[503\]7/2, \[505\]9/2), (\[510\]1/2, \[512\]3/2) pseudo-spin doublets, respectively.
The spherical shell structure in superheavy nuclei strongly depends on the spin-orbit splitting, which governs the size of the $Z=114$ gap (cf. Table 4 of Ref. [@(Cwi96)] and discussion therein). Also, the coupling between Coulomb interaction and nuclear interaction is expected to impact the predictions. To consider both effects, we studied s.p. canonical states obtained with [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} and [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} SEDF models, which differ in the spin-orbit sector and treat the electrostatic energy self-consistently.
The s.p. energies of [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} along the $\alpha$-decay chain of $^{296}120$ are depicted in Fig. \[figEspEDF\].
![\[figEspEDF\] (Color online) Single-neutron (top) and single-proton (bottom) canonical energies of [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} for nuclei along the $\alpha$-decay chain of $^{296}$120 as in Fig. \[figQ2\]. The orbits are labelled by the standard asymptotic Nilsson numbers corresponding to the dominant components of the SHFB canonical wave functions. The positive/negative parity levels are marked by solid/dashed lines. The Fermi levels are indicated by thick dotted lines.](Figure3.pdf){width="\columnwidth"}
The s.p. neutron spectrum is dominated by deformed gaps at $N=152$ and 162, and a large spherical shell gap at $N=184$. In the deformed region $160 \le N \le
168$, the Nilsson states close to the Fermi level are primarily ${N_{\rm osc}}=6$ levels and one unique-parity, high-$\Omega$ intruder level $\nu$\[716\]13/2 originating from the spherical 1$j_{15/2}$ shell. The structure of the proton Nilsson diagram in Fig. \[figEspEDF\] is dominated by deformed gaps at $Z=100$, 102, and 108, and a spherical subshell closure at $Z=114$. The unique-parity, high-$\Omega$ intruder level $\pi$\[615\]11/2 originating from the spherical 1$i_{13/2}$ shell is surrounded by several ${N_{\rm osc}}=5$ Nilsson orbitals.
The general pattern of s.p. states predicted by [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} is not that far from that in Fig. \[figEspNS\] of the MO potential. However, there are differences in the spherical shell structure, which will impact detailed predictions for deformed superheavy nuclei belonging to $Z=115$ $\alpha$-decay chains. In particular, MO predicts larger spherical shell gaps at $Z=114$, $N=148$, and $N=178$. In [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{}, the splitting between the 1$j_{15/2}$ and 1$i_{11/2}$ spherical neutron shells is very small. This results in an upward shift of the (\[606\]11/2, \[604\]9/2) doublet.
![\[figEspEDForg\] (Color online) Similar as in Fig. \[figEspEDF\] but for [[<span style="font-variant:small-caps;">unedf1</span>]{}]{}](Figure4.pdf){width="\columnwidth"}
As seen in Fig. \[figEspEDForg\], in the case of [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} the unique-parity $\nu$1$j_{15/2}$ and $\pi$1$i_{13/2}$ shells are shifted up by a few hundred keV, which results in a significant reduction of spherical $N=164$ and $Z=114$ shell closures [@(Shi13a)]. The change in the spin-orbit potential also impacts positions of deformed levels. In particular, the deformed neutron gap at $N=152$ is reduced, and that at $N=162$ opens up. In the proton sector, the deformed Nilsson state \[615\]11/2 appears just below the significantly increased $Z=116$ gap, close to the \[505\]9/2 and \[510\]1/2 levels. The second proton intruder state \[624\]9/2 shows up just below the deformed proton gap at $Z=108$.
One-quasi-particle energies
---------------------------
To get more insights, we computed the energies of 1-q.p. excitations for odd-$Z$, even-$N$ superheavy nuclei that form the $\alpha$-decay chains of $^{287}_{116}$Lv$_{171}$ and $^{289}_{116}$Lv$_{173}$ (Tables \[tabN\] and \[tabN-orig\]), and $^{287}115_{172}$ and $^{293}$117$_{176}$ (Tables \[tabZ\] and \[tabZ-ori\]); the theoretical error on 1-q.p. excitations due to the adopted size of the HO basis is less than 60keV when going from 680 stretched HO states to 969 states. The results for $^{287}_{116}$Lv$_{171}$ and $^{293}$117$_{176}$ are shown in Figs. \[figN\] and \[figZ\], respectively.
[clccc]{} Nucleus & Config. & $E_x$ (MeV) & $Q_{20}$(b) & $\beta_2$\
\
$^{275}_{110}$Ds$_{165}$ & \[611\]3/2 & 0 & 29.8 & 0.23\
& \[613\]5/2 & 0.085 & 29.7 & 0.23\
& **\[716\]13/2** & 0.151 & 29.9 & 0.23\
& \[611\]1/2 & 0.305 & 29.8 & 0.23\
& \[604\]9/2 & 0.619 & 29.4 & 0.22\
\
$^{279}_{112}$Cn$_{167}$ & \[611\]3/2 & 0 & 28.3 & 0.21\
& \[613\]5/2 & 0.013 & 28.3 & 0.21\
& \[611\]1/2 & 0.121 & 28.3 & 0.21\
& \[604\]9/2 & 0.306 & 28.0 & 0.21\
& **\[716\]13/2** & 0.627 & 28.5 & 0.22\
\
$^{281}_{112}$Cn$_{169}$ & \[611\]1/2 & 0 & 27.2 & 0.20\
& \[604\]9/2 & 0.145 & 27.4 & 0.21\
& \[613\]5/2 & 0.159 & 27.1 & 0.20\
& \[611\]3/2 & 0.237 & 26.2 & 0.20\
& \[606\]11/2 & 0.606 & 24.8 & 0.19\
\
$^{283}_{114}$Fl$_{169}$ & \[611\]1/2 & 0 & 25.4 & 0.19\
& \[604\]9/2 & 0.165 & 25.0 & 0.19\
& \[613\]5/2 & 0.184 & 24.9 & 0.19\
& \[611\]3/2 & 0.208 & 24.5 & 0.18\
& \[606\]11/2 & 0.399 & 23.6 & 0.18\
\
$^{285}_{114}$Fl$_{171}$ & \[611\]1/2 & 0 & 21.0 & 0.15\
& \[613\]5/2 & 0.051 & 19.0 & 0.14\
& \[611\]3/2 & 0.056 & 19.0 & 0.14\
& \[606\]11/2 & 0.058 & 21.0 & 0.15\
& **\[707\]15/2** & 0.232 & 19.6 & 0.14\
\
$^{287}_{116}$Lv$_{171}$ & \[613\]5/2 & 0 & 14.2 & 0.10\
& \[611\]3/2 & 0.085 & 14.2 & 0.10\
& **\[707\]15/2** & 0.219 & 14.8 & 0.10\
& \[611\]1/2 & 0.314 & 14.3 & 0.10\
& \[622\]3/2 & 0.378 & 13.1 & 0.09\
& \[604\]9/2 & 0.495 & 16.2 & 0.12\
\
$^{289}_{116}$Lv$_{173}$ & \[613\]5/2 & 0 & 11.8 & 0.08\
& \[611\]3/2 & 0.029 & 11.8 & 0.08\
& \[611\]1/2 & 0.055 & 11.9 & 0.08\
& \[604\]7/2 & 0.372 & 11.2 & 0.08\
& \[602\]5/2 & 0.397 & 11.3 & 0.08
Although s.p. energies are not experimental observables, those around the Fermi level carry information about the low-lying q.p. configurations in neighboring odd-$A$ and odd-odd nuclei.
[clccc]{} Nucleus & Config. & $E_x$ (MeV) & $Q_{20}$(b) & $\beta_2$\
\
$^{275}_{110}$Ds$_{165}$ & \[613\]5/2 & 0 & 30.2 & 0.23\
& \[611\]3/2 & 0.050 & 30.2 & 0.23\
& **\[716\]13/2** & 0.121 & 30.2 & 0.23\
& \[611\]1/2 & 0.364 & 30.2 & 0.23\
& \[604\]9/2 & 0.561 & 29.8 & 0.23\
\
$^{279}_{112}$Cn$_{167}$ & \[611\]3/2 & 0 & 28.1 & 0.21\
& \[613\]5/2 & 0.044 & 28.1 & 0.21\
& \[611\]1/2 & 0.074 & 28.2 & 0.21\
& **\[716\]13/2** & 0.197 & 28.3 & 0.21\
& \[604\]9/2 & 0.201 & 27.8 & 0.21\
\
$^{281}_{112}$Cn$_{169}$ & \[611\]1/2 & 0 & 26.3 & 0.19\
& \[604\]9/2 & 0.109 & 26.2 & 0.19\
& \[611\]3/2 & 0.152 & 25.7 & 0.19\
& \[613\]5/2 & 0.157 & 26.0 & 0.19\
& \[606\]11/2 & 0.267 & 25.0 & 0.19\
\
$^{283}_{114}$Fl$_{169}$ & \[611\]1/2 & 0 & 25.5 & 0.19\
& \[604\]9/2 & 0.114 & 25.1 & 0.19\
& \[611\]3/2 & 0.147 & 24.7 & 0.18\
& \[613\]5/2 & 0.160 & 24.6 & 0.18\
& \[606\]11/2 & 0.195 & 24.3 & 0.18\
\
$^{285}_{114}$Fl$_{171}$ & \[604\]9/2 & 0 & 22.8 & 0.17\
& \[611\]1/2 & 0.009 & 22.2 & 0.17\
& \[611\]3/2 & 0.139 & 21.5 & 0.16\
& \[606\]11/2 & 0.169 & 22.2 & 0.17\
& \[613\]5/2 & 0.184 & 21.4 & 0.16\
& **\[707\]15/2** & 0.723 & 20.4 & 0.15\
\
$^{287}_{116}$Lv$_{171}$ & \[604\]9/2 & 0 & 21.8 & 0.16\
& \[611\]1/2 & 0.003 & 21.2 & 0.15\
& \[611\]3/2 & 0.113 & 20.5 & 0.15\
& \[613\]5/2 & 0.175 & 20.4 & 0.15\
& \[606\]11/2 & 0.204 & 21.2 & 0.15\
& **\[707\]15/2** & 0.661 & 19.2 & 0.14\
\
$^{289}_{116}$Lv$_{173}$ & \[611\]1/2 & 0 & 18.1 & 0.13\
& \[611\]3/2 & 0.256 & 17.7 & 0.13\
& \[613\]5/2 & 0.394 & 17.8 & 0.13\
& \[604\]9/2 & 0.649 & 18.4 & 0.13\
& **\[707\]15/2** & 0.796 & 18.0 & 0.13\
Odd-odd nuclei
--------------
By combining the low-lying 1-q.p. excitations, one can deduce possible 2-q.p. states in the odd-odd nuclei that form $\alpha$-decay chains of $^{288}115$. It is worth noting that there exist detailed calculations of 1-q.p. excitations in the heaviest elements using the macroscopic-microscopic Woods-Saxon model [@(Cwi94); @(Par04); @(Par05)]; unfortunately, they cannot be used to assess the neutron s.p. structure in the region of interest as the range of neutron numbers covered ($N\le 161$) in these papers is too limited.
[clccc]{} Nucleus & Config. & $E_x$ (MeV) & $Q_{20}$(b) & $\beta_2$\
\
$^{279}_{111}$Rg$_{168}$ & \[512\]3/2 & 0 & 28.3 & 0.21\
& \[510\]1/2 & 0.122 & 28.3 & 0.21\
& **\[615\]11/2** & 0.284 & 28.4 & 0.21\
& \[505\]9/2 & 0.392 & 27.7 & 0.20\
& \[521\]1/2 & 0.633 & 26.6 & 0.19\
\
$^{281}_{111}$Rg$_{170}$ & \[512\]3/2 & 0 & 26.0 & 0.20\
& \[510\]1/2 & 0.165 & 26.1 & 0.20\
& \[505\]9/2 & 0.263 & 25.8 & 0.20\
& **\[615\]11/2** & 0.277 & 26.1 & 0.20\
& \[521\]1/2 & 0.412 & 25.3 & 0.19\
\
$^{283}$113$_{170}$ & \[512\]3/2 & 0 & 24.5 & 0.18\
& \[510\]1/2 & 0.057 & 24.5 & 0.18\
& \[505\]9/2 & 0.090 & 24.4 & 0.18\
& \[503\]7/2 & 0.453 & 22.9 & 0.16\
& \[521\]1/2 & 0.629 & 23.2 & 0.16\
\
$^{285}$113$_{172}$ & \[512\]3/2 & 0 & 18.7 & 0.14\
& \[503\]7/2 & 0.096 & 17.6 & 0.13\
& \[550\]1/2 & 0.146 & 17.2 & 0.13\
& \[510\]1/2 & 0.177 & 19.4 & 0.15\
& \[505\]9/2 & 0.380 & 19.7 & 0.15\
& **\[615\]11/2** & 0.645 & 18.2 & 0.14\
\
$^{287}115_{172}$ & \[512\]3/2 & 0 & 14.4 & 0.10\
& \[550\]1/2 & 0.042 & 12.6 & 0.09\
& \[503\]7/2 & 0.127 & 14.2 & 0.10\
& **\[606\]13/2** & 0.206 & 14.2 & 0.10\
& \[510\]1/2 & 0.388 & 14.1 & 0.10\
\
$^{289}$115$_{174}$ & \[550\]1/2 & 0 & 11.3 & 0.08\
& \[512\]3/2 & 0.002 & 11.5 & 0.08\
& **\[606\]13/2** & 0.137 & 12.2 & 0.09\
& \[503\]7/2 & 0.230 & 11.6 & 0.08\
& \[510\]1/2 & 0.338 & 11.6 & 0.08\
\
$^{293}$117$_{176}$ & \[512\]3/2 & 0 & 7.8 & 0.05\
& \[550\]1/2 & 0.06 & 7.5 & 0.05\
& \[510\]1/2 & 0.218 & 7.8 & 0.05\
& \[503\]5/2 & 0.310 & 7.0 & 0.04\
& \[503\]7/2 & 0.778 & 8.1 & 0.06
[clccc]{} Nucleus & Config. & $E_x$ (MeV) & $Q_{20}$(b) & $\beta_2$\
\
$^{279}_{111}$Rg$_{168}$ & \[512\]3/2 & 0 & 27.8 & 0.21\
& **\[615\]11/2** & 0.075 & 27.8 & 0.21\
& \[505\]9/2 & 0.119 & 27.6 & 0.21\
& \[510\]1/2 & 0.252 & 27.8 & 0.21\
& **\[624\]9/2** & 0.793 & 27.6 & 0.21\
\
$^{281}_{111}$Rg$_{170}$ & **\[615\]11/2** & 0 & 25.3 & 0.19\
& \[512\]3/2 & 0.021 & 25.3 & 0.19\
& \[505\]9/2 & 0.087 & 25.4 & 0.19\
& \[510\]1/2 & 0.218 & 25.3 & 0.19\
& \[521\]1/2 & 0.601 & 23.9 & 0.18\
\
$^{283}$113$_{170}$ & \[510\]1/2 & 0 & 24.3 & 0.18\
& \[512\]3/2 & 0.024 & 24.1 & 0.18\
& **\[615\]11/2** & 0.055 & 24.3 & 0.18\
& \[503\]7/2 & 0.480 & 22.7 & 0.17\
& \[550\]1/2 & 0.702 & 22.7 & 0.17\
\
$^{285}$113$_{172}$ & \[510\]1/2 & 0 & 21.2 & 0.16\
& \[512\]3/2 & 0.044 & 21.0 & 0.15\
& **\[615\]11/2** & 0.089 & 21.1 & 0.15\
& \[505\]9/2 & 0.287 & 22.8 & 0.16\
& \[550\]1/2 & 0.411 & 19.8 & 0.15\
\
$^{287}$115$_{172}$ & \[510\]1/2 & 0 & 20.2 & 0.15\
& \[503\]7/2 & 0.055 & 20.1 & 0.15\
& \[512\]3/2 & 0.193 & 20.1 & 0.15\
& **\[615\]11/2** & 0.253 & 20.4 & 0.15\
& \[550\]1/2 & 0.682 & 19.1 & 0.14\
\
$^{289}$115$_{174}$ & \[503\]7/2 & 0 & 17.1 & 0.12\
& \[512\]3/2 & 0.024 & 16.9 & 0.12\
& \[510\]1/2 & 0.085 & 17.1 & 0.12\
& **\[615\]11/2** & 0.272 & 17.3 & 0.12\
& \[550\]1/2 & 0.438 & 16.5 & 0.12\
\
$^{293}$117$_{176}$ & \[510\]1/2 & 0 & 11.4 & 0.08\
& **\[606\]13/2** & 0.003 & 11.5 & 0.08\
& \[512\]3/2 & 0.03 & 11.2 & 0.08\
& \[550\]1/2 & 0.197 & 9.7 & 0.06\
& \[503\]5/2 & 0.26 & 11.1 & 0.08\
& \[503\]7/2 & 0.266 & 11.7 & 0.08\
& **\[615\]11/2** & 0.588 & 11.4 & 0.08\
![\[figN\] (Color online) 1-q.p. spectra for nuclei forming the $\alpha$-decay chain $^{287}$Lv $\rightarrow \cdots \rightarrow$ $^{275}$Ds predicted with [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} (upper sequence) and [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} (lower sequence). $Q_{\alpha}$ values for g.s.$\rightarrow$g.s. transitions are marked. The binding energy differences between different nuclei are shifted arbitrarily, whereas the excitation energies within a given nucleus are shown to scale. ](Figure5.pdf){width="0.7\columnwidth"}
![\[figZ\] (Color online) Similar as in Fig. \[figN\] but for the $\alpha$-decay chain $^{293}117$ $\rightarrow \cdots \rightarrow$ $^{277}$Mt. ](Figure6.pdf){width="0.7\columnwidth"}
[clccc]{} Nucleus & Config. & $E_x$ (MeV) & $Q_{20}$(b) & $\beta_2$\
\
&&[[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{}&&\
$^{275}_{109}$Mt$_{166}$ & **\[615\]11/2** & 0 & 29.7 & 0.23\
& \[512\]3/2 & 0.243 & 29.5 & 0.23\
& \[521\]1/2 & 0.402 & 29.2 & 0.22\
& \[512\]5/2 & 0.500 & 29.7 & 0.23\
& \[510\]1/2 & 0.512 & 29.7 & 0.23\
$^{277}_{109}$Mt$_{168}$ & **\[615\]11/2** & 0 & 28.3 & 0.23\
& \[521\]1/2 & 0.174 & 27.4 & 0.21\
& \[512\]3/2 & 0.269 & 28.3 & 0.23\
& \[510\]1/2 & 0.480 & 28.3 & 0.23\
& \[512\]5/2 & 0.532 & 28.3 & 0.23\
$^{275}_{108}$Hs$_{167}$ & \[613\]5/2 & 0 & 28.9 & 0.22\
& \[611\]3/2 & 0.016 & 28.9 & 0.22\
& \[611\]1/2 & 0.117 & 28.9 & 0.22\
& \[604\]9/2 & 0.235 & 28.2 & 0.22\
& **\[716\]13/2** & 0.479 & 29.4 & 0.23\
$^{277}_{110}$Ds$_{167}$ & \[611\]3/2 & 0 & 28.6 & 0.22\
& \[613\]5/2 & 0.046 & 28.9 & 0.22\
& \[611\]1/2 & 0.107 & 28.7 & 0.22\
& \[604\]9/2 & 0.335 & 28.6 & 0.22\
& **\[716\]13/2** & 0.564 & 29.2 & 0.22\
\
&&[[<span style="font-variant:small-caps;">unedf1</span>]{}]{}&&\
$^{275}_{109}$Mt$_{166}$ & \[512\]3/2 & 0 & 30.0 & 0.23\
& **\[615\]11/2** & 0.159 & 29.6 & 0.23\
& \[505\]9/2 & 0.167 & 29.0 & 0.22\
& \[510\]1/2 & 0.173 & 30.1 & 0.23\
& **\[624\]9/2** & 0.318 & 29.9 & 0.23\
$^{277}_{109}$Mt$_{168}$ & \[512\]3/2 & 0 & 28.6 & 0.22\
& **\[615\]11/2** & 0.131 & 28.2 & 0.21\
& \[505\]9/2 & 0.136 & 27.5 & 0.21\
& \[512\]5/2 & 0.182 & 28.6 & 0.22\
& **\[624\]9/2** & 0.300 & 28.5 & 0.21\
$^{275}_{108}$Hs$_{167}$ & \[613\]5/2 & 0 & 29.4 & 0.22\
& \[611\]3/2 & 0.067 & 29.4 & 0.22\
& \[611\]1/2 & 0.104 & 29.5 & 0.22\
& **\[716\]13/2** & 0.173 & 29.7 & 0.23\
& \[604\]9/2 & 0.242 & 29.1 & 0.22\
$^{277}_{110}$Ds$_{167}$ & \[611\]3/2 & 0 & 28.9 & 0.22\
& \[613\]5/2 & 0.035 & 29.0 & 0.22\
& \[611\]1/2 & 0.090 & 29.0 & 0.22\
& **\[716\]13/2** & 0.157 & 29.2 & 0.22\
& \[604\]9/2 & 0.227 & 28.7 & 0.22\
Let us look into the structure of $^{276}$Mt in some detail. The structural information relevant to this nucleus is contained in the 1-q.p. spectra of its odd-$A$ neighbors $^{275,277}$Mt, $^{275}$Hs, and $^{277}$Ds, provided in SEDF Table \[tabNZMT\]. All low-lying 1-q.p. states in these nuclei correspond to very similar quadrupole mass deformation of $\beta_2\approx
0.22$, which facilitates comparison with the Nilsson diagram of Fig. \[figEspEDF\]. The lowest 1-q.p. proton states are the unique-parity \[615\]11/2 and ${N_{\rm osc}}=5$ excitations \[512\]3/2, \[521\]1/2, \[510\]1/2, and \[512\]5/2. The 1-q.p. neutron structure corresponds to the \[716\]13/2 intruder and ${N_{\rm osc}}=6$ \[611\]3/2, \[613\]5/2, \[611\]1/2, and \[604\]9/2 Nilsson orbits. The most significant difference between the two SEDF models is the appearance of the \[505\]9/2 1-q.p. proton excitation low in energy in [[<span style="font-variant:small-caps;">unedf1</span>]{}]{}. According to the MO model of Fig. \[figEspNS\], the lowest 1-q.p proton excitations are the \[615\]11/2, \[521\]1/2, and \[505\]9/2 Nilsson orbits, while the lowest neutron states are: \[606\]11/2, \[604\]9/2, \[611\]3/2, \[613\]5/2, and \[716\]13/2. It is interesting to note that the structure of 1-q.p. proton states predicted for $^{275}$Mt in Ref. [@(Par04)] falls between predictions of [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} and [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{}. In addition, within the Woods-Saxon model of Ref. [@(Cha77)], the proton states \[615\]11/2 and \[505\]9/2 are the two lowest orbitals for Mt over a large range of deformations.
![\[p+rotMt\] (Color online) Results of 2-q.p.-plus-rotor NS calculations for $^{276}$Mt. States connected with lines have the same dominating single-particle configurations; they can be interpreted as rotational bands built on 2-q.p. bandheads indicated. Solid/dashed lines mark bands with positive/negative parity. Positive/negative parity neutron configurations are indicated by asterisks/triangles. The candidates for stretched $\Delta \Omega=0,\pm 1$ $E1$ transitions are marked by thick arrows. ](Figure7.pdf){width="\columnwidth"}
As noted in [@(Rud13a)], there is a very limited choice of q.p. configurations that could generate the observed $E1$ transitions in $^{276}$Mt. If one insists on a strict conservation of the $\Omega$ quantum number for protons and neutrons, no low-energy $E1$ transitions are predicted by [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{}. Formally, one can construct states that can be connected by an $\Delta \Omega=0,\pm 1$, parity changing operator, e.g., $\{\pi[615]11/2\otimes\nu[613]5/2\}_{3^+}$ and $\{\pi[521]1/2\otimes\nu[613]5/2\}_{2^-,3^-}$, but a significant Coriolis coupling would be required to produce a measurable $E1$ rate. The situation is fairly straightforward with [[<span style="font-variant:small-caps;">unedf1</span>]{}]{}. Here, the stretched $E1$ transition $\pi[505]9/2 \rightarrow \pi[615]11/2$ can explain the data, with the neutron spectator orbital being \[611\]3/2 or \[613\]5/2 or \[611\]1/2. The NS approach predicts two scenarios: the proton $\pi[615]11/2 \rightarrow \pi[505]9/2$ transition as in [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} and the neutron $\nu[716]13/2 \rightarrow
\nu[606]11/2$ transition. According to 2-q.p.-plus-rotor calculations shown in Fig. \[p+rotMt\] both scenarios are equally likely. It is interesting to note that the splitting between the $I=1$ and $I=2$ members of the lowest $K^{\pi}=1^-$ ($\pi[505]9/2\otimes\nu[606]11/2$) band is only 43keV, see Fig. \[p+rotMt\]. This is particularly close to the energy difference $\approx47$keV between the states suggested experimentally [@(Rud13a)].
![\[p+rotBh\] (Color online) Similar as in Fig. \[p+rotMt\] but for $^{272}$Bh. ](Figure8.pdf){width="\columnwidth"}
To analyse the case of $^{272}$Bh, we have calculated the 1-q.p. spectra of $^{271,273}$Bh, $^{273}$Hs, and $^{271}$Sg, see Table \[tabNZBH\]. The resulting level scheme is very complex. Indeed, as seen in in Fig. \[p+rotBh\], 2-q.p.-plus-rotor calculations predict quite a few candidates for low-energy $E2$ and $M1$ transitions, and this is consistent with experiment [@(Rud13a)]. The results displayed in Figs. \[p+rotMt\] and \[p+rotBh\] clearly demonstrate that a small change in ordering of Nilsson orbits can influence the decay scenarios.
[clccc]{} Nucleus & Config. & $E_x$ (MeV) & $Q_{20}$(b) & $\beta_2$\
\
&&[[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{}&&\
$^{271}_{107}$Bh$_{164}$ & \[512\]5/2 & 0 & 30.8 & 0.24\
& \[521\]1/2 & 0.076 & 30.3 & 0.23\
& **\[615\]11/2** & 0.244 & 30.4 & 0.23\
& **\[624\]9/2** & 0.415 & 30.8 & 0.24\
& \[514\]7/2 & 0.682 & 31.1 & 0.24\
$^{273}_{107}$Bh$_{166}$ & **\[615\]11/2** & 0 & 29.3 & 0.23\
& \[512\]5/2 & 0.002 & 29.6 & 0.23\
& \[521\]1/2 & 0.072 & 29.5 & 0.23\
& **\[624\]9/2** & 0.243 & 29.6 & 0.23\
& \[512\]3/2 & 0.574 & 29.4 & 0.23\
& \[514\]7/2 & 0.692 & 29.8 & 0.23\
$^{271}_{106}$Sg$_{165}$ & \[611\]3/2 & 0 & 30.1 & 0.23\
& **\[716\]13/2** & 0.156 & 30.2 & 0.23\
& \[613\]5/2 & 0.223 & 30.1 & 0.23\
& \[611\]1/2 & 0.311 & 30.1 & 0.23\
& \[604\]9/2 & 0.429 & 29.2 & 0.23\
$^{273}_{108}$Hs$_{165}$ & \[611\]3/2 & 0 & 30.1 & 0.23\
& \[613\]5/2 & 0.118 & 30.0 & 0.23\
& **\[716\]13/2** & 0.142 & 30.3 & 0.23\
& \[611\]1/2 & 0.314 & 30.1 & 0.23\
& \[604\]9/2 & 0.518 & 29.2 & 0.22\
\
&&[[<span style="font-variant:small-caps;">unedf1</span>]{}]{}&&\
$^{271}_{107}$Bh$_{164}$ & \[512\]5/2 & 0 & 31.6 & 0.24\
& **\[624\]9/2** & 0.028 & 31.5 & 0.24\
& \[512\]3/2 & 0.338 & 31.3 & 0.24\
& \[521\]1/2 & 0.554 & 30.9 & 0.24\
& \[505\]9/2 & 0.714 & 30.0 & 0.23\
$^{273}_{107}$Bh$_{166}$ & **\[624\]9/2** & 0 & 30.7 & 0.23\
& \[512\]5/2 & 0.076 & 30.7 & 0.23\
& \[512\]3/2 & 0.306 & 30.6 & 0.23\
& \[521\]1/2 & 0.377 & 29.6 & 0.23\
& \[505\]9/2 & 0.599 & 28.7 & 0.22\
$^{271}_{106}$Sg$_{165}$ & \[611\]3/2 & 0 & 31.1 & 0.24\
& \[613\]5/2 & 0.007 & 30.7 & 0.24\
& **\[716\]13/2** & 0.11 & 31.1 & 0.24\
& \[611\]1/2 & 0.252 & 31.0 & 0.24\
& \[604\]9/2 & 0.635 & 33.2 & 0.25\
$^{273}_{108}$Hs$_{165}$ & \[613\]5/2 & 0 & 30.7 & 0.24\
& \[611\]3/2 & 0.048 & 30.8 & 0.24\
& **\[716\]13/2** & 0.119 & 30.8 & 0.24\
& \[611\]1/2 & 0.300 & 30.8 & 0.24\
& \[604\]9/2 & 0.508 & 30.4 & 0.23\
The calculated $Q_{\alpha}$ values depend, of course, on the structure of parent and daughter states [@(Cwi99)] (see Figs. \[figN\] and \[figZ\]). The agreement with the measured values for the heaviest elements is reasonable, usually better than 1MeV. This is comparable with other calculations [@(Ben13); @(Pra12); @(War12); @(Sta13)].
Conclusions
===========
In summary, we studied shell structure of superheavy nuclei within the self-consistent SHFB approach and macroscopic-microscopic NS model. Detailed predictions have been made for the quasi-proton and quasi-neutron structures of nuclei belonging to the $\alpha$-decay chains of $^{287}115$, $^{287}$Lv, $^{289}$Lv, and $^{293}$117. The [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} and [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} SEDF models differ in the strength of the spin-orbit term, and this impacts detailed predictions for the deformed nuclei around $Z=110$ and $N=168$. The recent observation of low-energy $E1$ transitions in $^{276}$Mt [@(Rud13a)] provides a stringent constraint on theoretical models. Indeed, the recently proposed [[<span style="font-variant:small-caps;">unedf1</span>$^{\rm SO}$]{}]{} parametrization that performs well in the transfermium region does not offer a simple explanation of the $E1$ data, whereas the global [[<span style="font-variant:small-caps;">unedf1</span>]{}]{} parametrization explains the data in terms of the proton $\pi[505]9/2 \rightarrow \pi[615]11/2$ transition. The MO models suggests two competing scenarios: a proton transition similar to that of [[<span style="font-variant:small-caps;">unedf1</span>]{}]{}, and an alternative neutron $\nu [716]13/2 \rightarrow \nu [606]11/2$ $E1$ transition. To confirm or disprove these scenarios, theory strongly recommends a search for $E1$ transitions in neighboring odd-$A$ systems $^{275,277}$Mt, $^{275}$Hs, and $^{277}$Ds. Experimentally, this calls for high-resolution $\alpha$-photon coincidence spectroscopy of decay chains starting from $^{293}$117, $^{287,289}$115, or $^{285,287}$Fl, respectively. However, the observation of these systems is hampered either by relatively low production cross-sections or large spontenous fission branches on the way to the nuclei of structural interest [@(Oga07); @(Oga11); @(Oga12); @(Oga12a); @(Oga13); @(Dul10); @(Gat11); @(Hof12); @(Rud13a)]. A solution to this spectroscopic puzzle may significantly contribute to our understanding of shell structure in superheavy nuclei, and the strength of the spin-orbit splitting in particular.
Discussions with S. [Å]{}berg are gratefully acknowledged. This work was supported by the U.S. Department of Energy (DOE) under Contracts No. DE-FG02-96ER40963 (University of Tennessee), No. DE-SC0008499 (NUCLEI SciDAC Collaboration), No. DE-NA0001820 (the Stewardship Science Academic Alliances program); by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme; by the Polish National Science Center under Contract No. 2012/07/B/ST2/03907; and by the Swedish Research Council. An award of computer time was provided by the National Institute for Computational Sciences (NICS) and the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program using resources of the OLCF facility.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We use linear response analysis and the fluctuation-dissipation theorem to derive the energy loss of a heavy quark in the SU(2) classical Coulomb plasma in terms of the $l=1$ monopole and non-static structure factor. The result is valid for all Coulomb couplings $\Gamma=V/K$, the ratio of the mean potential to kinetic energy. We use the Liouville equation in the collisionless limit to assess the SU(2) non-static structure factor. We find the energy loss to be strongly dependent on $\Gamma$. In the liquid phase with $\Gamma\approx 4$, the energy loss is mostly metallic and soundless with neither a Cerenkov nor a Mach cone. Our analytical results compare favorably with the SU(2) molecular dynamics simulations at large momentum and for heavy quark masses.'
address: |
Department of Physics and Astronomy\
State University of New York, Stony Brook, NY, 11794
author:
- Sungtae Cho
- Ismail Zahed
title: |
Classical Strongly Coupled QGP:\
VII. Energy Loss
---
Introduction
============
Parton energy loss at RHIC is widely viewed as a way to probe the properties of the medium created during the first few fm/c of the collision. The medium is suspected to be a strongly coupled liquid [@shuryak2] with near perfect fluidity and strong energy loss.
There have been a number of calculations involving parton collisional and radiative energy loss at RHIC with the chief consequence of jet quenching [@dumitruetal]. The measured jet quenching at RHIC exceeds most current theoretical predictions, most of which are based on a weakly coupled quark-gluon plasma (wQGP).
The QCD matter probed numerically using lattice simulations and at RHIC using heavy ion collisions, is likely to be dominated by temperatures in the few $T_c$ range making it de facto non-perturbative. Non-perturbative methods are therefore welcome for analyzing the QCD matter conditions in this temperature range. An example being the holographic method as a tool for jet quenching analysis .
In this letter, we follow the approach suggested in to model the strongly coupled quark and gluon plasma, by classical colored constituents interacting via strong Coulomb interactions. This model has been initially analyzed using Molecular Dynamics (MD) simulations mostly for the SU(2) version with species of constituents (gluons). The MD results reveal a strongly coupled liquid at $\Gamma\approx 4$ the ratio of the mean kinetic to Coulomb energy (modulo statistical fluctuations). The fractional energy loss is also found to be considerably larger than most leading order QCD estimates.
Here, we will provide the analytical framework to analyze the MD simulation results for partonic energy loss in the cQGP. In section 2, we outline a formal derivation of the energy loss in the cQGP for arbitrary values of $\Gamma$. In section 3, we use linear response theory and the fluctuation-dissipation theorem to tie the energy loss to the non-static colored structure factor. In section 4, we derive explicitly the non-static structure factor using the Liouville equation. Some useful aspects of the plasmon excitations in the cQGP are discussed in section 5. In section 6, we analyze the energy loss for both charm and bottom for $\Gamma=$2,3 and 4 in the liquid phase and compare them to the recent SU(2) MD simulations . In section 7, we discuss the relevance of our results to RHIC and holographic QCD.
Energy Loss
===========
Consider an SU(2) colored particle of charge $q^a$ travelling with velocity $v$ in the strongly coupled colored plasma [@gelmanetal]. The equation of motion of this [*extra*]{} particle in phase space follows from the Poisson bracket
$$\frac{d{\boldsymbol{p}}_i}{dt}=-\{H,{\boldsymbol{p}}_i\}=q^a\cdot {\boldsymbol{E}}_{in}^a
\label{eq001p}$$
with the longitudinal colored electric field
$${\boldsymbol{E}}_{in}^a=-{\boldsymbol{\nabla}}\sum_{i}\frac{Q^a_i(t)}{|{\boldsymbol{r}}-{\boldsymbol{r}}_i(t)|}=-{\boldsymbol{\nabla}}_i\Phi^a_{in}(t,{\boldsymbol{r}}) \label{eq002p}$$
We note that in [@gelmanetal] the SU(2) plasma is considered mostly electric with massive constituents $m\beta\approx 3$. As a result the transverse electric contribution is absent in (\[eq002p\]). Also, (\[eq001p\]) does not involve the magnetic part of the Lorentz force for the same reasons. The latter is irrelevant for the energy loss per travel length ${\boldsymbol{r}}={\boldsymbol{v}}t$
$$\frac{dK}{dr}={\boldsymbol{v}} q^a{\boldsymbol{E}}_{in}^a(t,{\boldsymbol{r}}={\boldsymbol{v}} t)
\label{eq003p}$$
even in the ultrarelativistic case since the magnetic force does not perform work.
The induced colored Coulomb potential $\Phi_{ind}$ follows from the total colored potential $\Phi_{\rm tot}$ through
$$\Phi^a_{tot}(\omega,{\boldsymbol{k}})=\Phi^a_{ind}(\omega,{\boldsymbol{k}})+\Phi^a_{ex}(\omega,{\boldsymbol{k}})=\frac{\Phi^a_{ex}(\omega,{\boldsymbol{k}})}{\epsilon_L(\omega,{\boldsymbol{k}})} \label{eq004p}$$
The last relation defines the longitudinal dielectric constant with ${\boldsymbol{\Phi}}_{ex}(\omega,{\boldsymbol{k}})=\frac{4\pi}{k^2}2\pi{\boldsymbol{q}}
\delta(\omega-{\boldsymbol{k}}\cdot{\boldsymbol{v}})$, the colored potential caused by the [*extra*]{} particle in the probe approximation (ignoring back reaction). Thus
$$\Phi^a_{ind}(t,{\boldsymbol{r}})=q^a\int \frac{d{\boldsymbol{k}}}{(2\pi)^3}
\left(\frac{1}{\epsilon_L(k\cdot v, {\boldsymbol{k}})}-1\right)\,
\frac{4\pi}{k^2}e^{i{\boldsymbol{k}}\cdot{\boldsymbol{r}}-i{\boldsymbol{k}}\cdot{\boldsymbol{v}} t}
\label{eq005p}$$
Using (\[eq002p\]) and (\[eq003p\]) we have for the energy loss of a fast moving probe SU(2) charge
$$-\frac{dK}{dr}=-\frac{{\boldsymbol{q}}^2}{\pi v^2}\int
\frac{dk}{k}\int_{-kv}^{kv}\omega d\omega \Im\left(\frac{1}{
\epsilon_{L}(\omega+i0,{\boldsymbol{k}})}\right)
\label{eq006p}$$
after using the analytical property of $\epsilon_L(z,k)=\epsilon_L(-z^*,-k)$ which follows from the causal character of the longitudinal dielectric function as detailed below. (\[eq006p\]) is identical in form to the one derived for the Abelian one component colored Coulomb plasma in [@ichimaru], to the exception of the SU(2) classical Casimir ${\boldsymbol{q}}^2$ in (\[eq006p\]). It is different in content through the longitudinal dielectric constant $\epsilon_L$ which now should be derived for a colored SU(2) Coulomb plasma. Our derivation is [*fully*]{} non-Abelian in the probe approximation.
![Static structure factors for $\Gamma=2,3,4$[]{data-label="structure_G020304"}](structure_G020304.ps){width="55.00000%"}
\[S011\]
![${\bf S}_{01}(q)$: molecular dynamics simulation (a) and analytic (b). See text.[]{data-label="structure"}](S01_allaL.ps "fig:"){width="49.00000%"} ![${\bf S}_{01}(q)$: molecular dynamics simulation (a) and analytic (b). See text.[]{data-label="structure"}](structure_l2.ps "fig:"){width="49.00000%"}
Below we show that for the SU(2) colored Coulomb plasma at strong Coulomb coupling, (\[eq006p\]) reads
$$-\frac{dK}{dr}=-\frac{{\boldsymbol{q}}^2}{\pi v^2}\int \,
\frac{dk}{k^3}\,\frac{k_D^2\,{\bf S}_{01}(k)}{1-{\bf S}_{01}(k)}
\int_{-kv}^{+kv}\, d\omega\,\omega\,\Im\left(\frac{1}{
\epsilon_{1}(\omega+i0,{\boldsymbol{k}})}\right)
\label{LIN12}$$
with $k_D^2$ the SU(2) Debye wave number squared. Here
$$\begin{aligned}
\epsilon_1(z,{\boldsymbol{k}})=1-n\,{\bf c}_{D1}({\boldsymbol{k}})\,{\bf W}(z/\omega_T)
\label{LIN12X}\end{aligned}$$
with the thermal frequency $\omega_T=v_Tk$ and velocity $v_T=\sqrt{T/m}$ and
$${\bf W}(z/\omega_T)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}dt\,\frac{t}{t-z/\omega_T}\,{e^{-t^2/2}}
\label{LIN13}$$
The l=1 static structure factor ${\bf S}_{01}$
$${\bf S}_{01}(k)=\left<\left|\sum^N_{j=1}e^{i{\boldsymbol{k}}\cdot {\boldsymbol{r}}_j}\,Q_j^a\right|^2\right> \label{S01}$$
satisfies the generalized Ornstein-Zernicke equation
$${\bf S}_{01}(k)=\frac 1{1-n{\bf c}_{D1}(k)}$$
in the colored Coulomb plasma with 1 species density $n=N/V$. In Fig. \[S011\] we show analytical results for (\[S01\]) around the liquid point . In Fig. \[structure\]a we show the behavior of (\[S01\]) using SU(2) molecular dynamics simulations with the dimensionless wavenumber $q=k\,a_{WS}$ where $a_{WS}$ is the Wigner-Seitz radius through $1/n=4\pi\,a_{WS}^3/3$. In Fig. \[structure\]b we show the analytical results for the same range of $\Gamma$ in .
The $l=1$ contribution $\epsilon_1$ plays the role of a generalized longitudinal dielectric constant in the SU(2) Coulomb plasma. Indeed, for weak Coulomb coupling $\Gamma\ll 1$, $-n{\bf
c}_{D1}\approx k_D^2/k^2$ so that ${\bf S}_{01}\approx
k^2/(k^2+k_D^2)$. The energy loss (\[LIN12\]) reduces to (\[eq006p\]) with $\epsilon_L\rightarrow \epsilon_1$. At weak coupling $\epsilon_1$ in (\[LIN12X\]) is the standard Vlasov dielectric function in [@ichimaru]. The only difference is in the SU(2) Debye wave number.
Linear Response
===============
To construc the longitudinal dielectric constant for the SU(2) Coulomb plasma we will make use of the Liouville kinetic equations for the time dependent structure factors derived in . For that we recall that in linear response, the induced color charge density $\rho_{ind}^a=\nabla\cdot
E_{ind}^a/4\pi$ ties with the external potential $\Phi_{ext}^b$ through the retarded correlator
$$\rho_{ind}^a(t,{\boldsymbol{r}})=i\int\,dt'\,d{\boldsymbol{r}}'\, \left<{\bf
R}\left({\bf J}_0^a(t,{\boldsymbol{r}}){\bf J}_0^b(t',{\boldsymbol{r}}')\right)\right>\,{\Phi}_{ext}^b (t',{\boldsymbol{r}}') \label{LIN1}$$
where ${\bf J}_0^a$ are the pertinent color charge densities. In Fourier space we have
$$\Phi_{ind}^a=-\frac{4\pi}{k^2}\,\Delta^{ab}_R(\omega,{\boldsymbol{k}})\,\Phi_{ext}^b \label{LIN2}$$
with
$$\Delta_R^{ab}(\omega, {\boldsymbol{k}}) =-i\int \,e^{-i\omega t+i{\boldsymbol{k}}\cdot
{\boldsymbol{r}}}\, \left<{\bf R}\left({\bf J}_0^a(t,{\boldsymbol{r}}){\bf
J}_0^b(t',{\boldsymbol{r}}')\right)\right> \label{LIN3}$$
A comparison of (\[LIN2\]) with (\[eq004p\]) yields
$$\left(\frac 1{\epsilon_L(\omega, {\boldsymbol{k}})}-1\right)\,\delta^{ab}=-\frac{4\pi}{k^2}\,\Delta_R^{ab}(\omega,
{\boldsymbol{k}}) \label{LIN5X}$$
which defines the longitudinal dielectric constant.
The retarded correlator (\[LIN3\]) is in general a quantum object, we now show how to extract it from the correlations in the classical and strongly coupled SU(2) colored Coulomb plasma. For that, we note that the colored charge density in the SU(2) phase space is
$${\bf J}_0^a(t,{\boldsymbol{r}})=\int\,dQ\,d{\boldsymbol{p}}\, Q^a\delta f(t,{\boldsymbol{r}},
{\boldsymbol{p}}, {\boldsymbol{Q}} ) \label{CHARGE1}$$
and that the SU(2) charge-charge correlator is
$$\left<{\bf J}_0^a(t,{\boldsymbol{r}}){\bf J}_0^b(t',{\boldsymbol{r}}')\right> =\frac
13 \delta^{ab} \int\,dQ\,dQ'\,d{\boldsymbol{p}}\,d{\boldsymbol{p}}'\,{\boldsymbol{Q}}\cdot {\boldsymbol{Q}}'\, {\bf S}(t-t', {\boldsymbol{r}}-{\boldsymbol{r}}', {\boldsymbol{p}}{\boldsymbol{p}}' ,{\boldsymbol{Q}}\cdot {\boldsymbol{Q}}') \label{CHARGE2}$$
where global time, space and color invariances were used thanks to the statistical averaging. The time dependent structure factor ${\bf S}=\langle\delta f\delta f\rangle$ was defined in . Using the color Legendre transform of ${\bf
S}$ yields
$$\left<{\bf J}_0^a(t,{\boldsymbol{r}}){\bf J}_0^b(t',{\boldsymbol{r}}')\right>
=\delta^{ab} \int\,d{\boldsymbol{p}}\,d{\boldsymbol{p}}'\,
{\bf S}_1(t-t', {\boldsymbol{r}}-{\boldsymbol{r}}', {\boldsymbol{p}}{\boldsymbol{p}}')
\label{CHARGE3}$$
Only the $l=1$ partial wave in the Legendre transform of the color part of ${\bf S}$ contributes to the SU(2) charge-charge correlation function.
The fluctuation-dissipation theorem in the [*classical limit*]{} ties the retarded correlator $\Delta_R$ in (\[LIN3\]) to the Fourier transform of the classical phase space fluctuations (\[CHARGE3\]) as
$$\Im\Delta^{ab}_R(\omega, {\boldsymbol{k}})=\delta^{ab}\frac{n\omega}{2T}\,{\bf S}_1(\omega, {\boldsymbol{k}})\equiv
-\delta^{ab}\frac{n\omega}{T}\,\Im{\bf S}_1(z, {\boldsymbol{k}})
\label{LIN5}$$
The last relation follows from ${\bf S}_1(\omega, {\boldsymbol{k}})=-2\,{\rm
Im}\,{\bf S}_1(\omega, {\boldsymbol{k}})$ between the Laplace transform and Fourier transform of ${\bf S}_1$ with $z=\omega+i0$.
Non-Static Structure Factor
===========================
We have shown in that the l-color partial wave of the Laplace transform of ${\bf S}_l$ obeys the Liouville equation
$$z{\bf S}_l(z{\boldsymbol{k}} ;{\boldsymbol{p}} {\boldsymbol{p}}')-\int d{\boldsymbol{p}}_1 \Sigma_l(z{\boldsymbol{k}}
;{\boldsymbol{p}} {\boldsymbol{p}}_1)\,{\bf S}_l(z{\boldsymbol{k}} ;{\boldsymbol{p}}_1 {\boldsymbol{p}}')={\bf
S}_{0l}({\boldsymbol{k}} ;{\boldsymbol{p}} {\boldsymbol{p}}') \label{eq007e}$$
${\bf S}_{0l}$ is the $l$ static structure factor introduced in
$${\bf S}_{0l}({\boldsymbol{k}} ;{\boldsymbol{p}} {\boldsymbol{p}}')=n\,f_0({\boldsymbol{p}})\,\delta({\boldsymbol{p}}-{\boldsymbol{p}}')+n^2 f_0({\boldsymbol{p}})\,f_0({\boldsymbol{p}}')\,{\bf h}_l({\boldsymbol{k}})
\label{LIN7}$$
with the Maxwell-Boltzmann distribution $f_0({\boldsymbol{p}})$. The structure factor ${\bf h}_l({\boldsymbol{k}})$ relates to the standard structure factor ${\bf S}_{0l}(k)$ by the generalized Ornstein-Zernicke equations
$$\frac 1n \,\int \,d{\boldsymbol{p}}\,d{\boldsymbol{p}}' \,{\bf S}_{0l}(k ;{\boldsymbol{p}} {\boldsymbol{p}}') ={\bf S}_{0l}(k)=1+n\,{\bf h}_l(k)=\left(1-n\,{\bf
c}_{Dl}(k)\right)^{-1} \label{LIN8}$$
The self-energy kernel $\Sigma_l$ in (\[eq007e\]) splits into a static and collisional contribution in each color partial wave $l$ .
We note that
$${\bf S}_l(zk)=\frac 1n\,\int d{\boldsymbol{p}}\,d{\boldsymbol{p}}' \,{\bf S}_l(z{\boldsymbol{k}} ;{\boldsymbol{p}} {\boldsymbol{p}}') \label{LIN6}$$
with $l=1$ is what is needed in (\[LIN5\]). For that, we solve (\[eq007e\]) in the collisionless limit with
$$\Sigma_l(z {\boldsymbol{k}} ;{\boldsymbol{p}} {\boldsymbol{p}}')\approx \frac{1}{m}{\boldsymbol{k}}\cdot{\boldsymbol{p}}\,\delta({\boldsymbol{p}}-{\boldsymbol{p}}')-\frac{1}{m}{\boldsymbol{k}}\cdot{\boldsymbol{p}}\,n\,f_0({\boldsymbol{p}})\,{\bf c}_{Dl}({\boldsymbol{k}}) \label{LIN7}$$
We recall that the SU(2) color part of the Liouville operator is a genuine 3-body force that only enters the collisional contribution. . Inserting (\[LIN7\]) into (\[eq007e\]) and using (\[LIN7\]) and (\[LIN8\]) yield in the collisionless limit
$${\bf S}_l(z,{\boldsymbol{k}})=\frac{{\bf S}_{0l}(k)}{\epsilon_l(z,{\boldsymbol{k}})}\,\int\, d{\boldsymbol{p}}\,\frac{f_0(p)}{z-{\boldsymbol{k}}\cdot {\boldsymbol{p}}/m}
\label{LIN9}$$
with
$$\epsilon_l (z,{\boldsymbol{k}})=1+n\,{\bf c}_{Dl} (k)\int\,d{\boldsymbol{p}}\,
\frac{{\boldsymbol{k}}\cdot {\boldsymbol{p}}/m}{z-{\boldsymbol{k}}\cdot{\boldsymbol{p}}/m}\,f_0(p)
\label{LIN10}$$
and
$$\int\, d{\boldsymbol{p}}\,\frac{f_0(p)}{z-{\boldsymbol{k}}\cdot{\boldsymbol{p}}/m}=\frac{1}{\omega}\Big(1-{\bf W}(z/\omega_T)\Big)$$
If we insert (\[LIN9\]) into (\[LIN5\]) and then use (\[LIN5X\]), we find for $l=1$
$$\Im\,\frac 1{\epsilon_L(z, {\boldsymbol{k}})}=-\frac{k_D^2}{k^2}\,\frac{1}{n{\bf c}_{D1}(k)} \,\,\Im\,\frac
1{\epsilon_1(z, {\boldsymbol{k}})} \label{LIN11}$$
Inserting (\[LIN11\]) into (\[eq006p\]) yields the announced relation (\[LIN12\]).
SU(2) Plasmon
=============
Before analyzing the energy loss in (\[LIN12X\]) for heavy charged probes, it is instructive to discuss the zeros of the longitudinal dielectric constant $\epsilon_1(\omega, k)=0$ in (\[LIN12X\]) as they reflect on the longitudinal excitations in the $l=1$ channel. For that, we need the behavior of ${\bf W}(x)$ as defined in (\[LIN13\]) with $x=\omega/v_Tk$ for small and large ratio $k/k_D$. $v_T=\sqrt{T/m}$ is the velocity of the the particles in the SU(2) heat bath. In weak coupling QCD $m\approx gT$, while in strong coupling $m\approx \pi T$.
In general,
$${\bf W}(x)={\bf W}_R(x)+i{\bf
W}_I(x)=1-xe^{-x^2/2}\,\psi(x)+i\sqrt{\pi\over 2}\,x\,e^{-x^2/2}
\label{W1}$$
with $\psi(x)=\int_0^x\,dy\,e^{y^2/2}$ the incomplete exponential function. For $k\ll k_D$ or $x\gg 1$,
$${\bf W}(x)\approx -\frac{1}{x^2}+i{\sqrt{\pi}\over 2} \,xe\,^{-x^2}
\label{LIN14}$$
while for $k\gg k_D$ or $x\ll 1$
$${\bf W}(x)\approx 1-x^2+i{\sqrt{\pi}\over 2} \,xe\,^{-x^2}
\label{LIN14X}$$
So in the long wavelength limit with $k\ll k_D$, (\[LIN12X\]) expands to
$$\epsilon_1(\omega,{\boldsymbol{k}})\approx 1+\frac{n{\bf
c}_{D1}(k)}{x^2}\left(1-i{\sqrt{\pi}\over 2}x^3e^{-x^2/2}\right)
\label{LIN15}$$
For small $k$, $n{\bf c}_{D1}(k)\approx {\bf S}_{01}(k)\approx
k_D^2/k^2$ whatever the coupling in the SU(2) colored plasma. Thus
$$\epsilon_1(\omega,{\boldsymbol{k}})\approx
1-\frac{\omega_p^2}{\omega^2}\left(1-i{\sqrt{\pi}\over
2}x^3e^{-x^2/2}\right) \label{LIN16}$$
with the plasmon frequency $\omega_p=v_T\,k_D$. So for $k\ll
k_D$, the zero of (\[LIN16\]) is
$$\omega^2_1(k)\approx\omega_p^2\,\left(1-i{\sqrt{\pi}\over
2}\,\frac{k_D^3}{k^3}e^{-k_D^2/2k^2}\right) \label{LIN17}$$
The SU(2) colored Coulomb plasma supports a plasmon with frequency $\omega_p$ with an exponentially small width $e^{-\omega^2/2v_T^2k^2}$ both at weak and strong SU(2) Coulomb coupling $\Gamma$. This result agrees with our analytic and leading kinetic analysis in the hydrodynamical limit . The current analysis provides the non-analytic imaginary part as well.
The high $k\gg k_D$ limit is metallic whatever $\Gamma$ with
$$\epsilon_1(x)\approx 1-in{\bf c}_{D1}(k)\,x{\sqrt{\pi\over
2}}\,e^{-x^2/2} \label{LIN17X}$$
with a metallic conductivity $x=\omega/\omega_T=\omega/v_Tk$
$$\sigma_1(\omega, {\boldsymbol{k}})=\frac{n\,{\bf
c}_{D1}(k)}{\sqrt{32\pi}}\,{{\omega^2}\over\,v_Tk}\,e^{-\omega^2/2v_T^2k^2}
\label{META}$$
We note that the plasmon branch disappears at high $k$ in (\[LIN17X\]) as the plasma turns metallic i.e. a collection of free colored SU(2) particles with a classical thermal spectrum. Also the plasmon in (\[LIN16\]) broadens substantially at $k\approx k_D$ with its real part comparable to its imaginary part. This point causes the plasmon contribution to drop from the energy loss in the colored SU(2) Coulomb plasma as we show below.
![$-v^2 dK/drdxdq$ versus $v/v_T$ for charm and bottom quark for fixed $q$. See text.[]{data-label="2dplot"}](3d_C10.ps "fig:"){width="50.00000%"} ![$-v^2 dK/drdxdq$ versus $v/v_T$ for charm and bottom quark for fixed $q$. See text.[]{data-label="2dplot"}](3d_B10.ps "fig:"){width="49.00000%"}
![$-v^2 dK/drdxdq$ versus $v/v_T$ for charm and bottom quark for fixed $q$. See text.[]{data-label="2dplot"}](2d_C10.ps "fig:"){width="50.00000%"} ![$-v^2 dK/drdxdq$ versus $v/v_T$ for charm and bottom quark for fixed $q$. See text.[]{data-label="2dplot"}](2d_B10.ps "fig:"){width="49.00000%"}
Charm and Bottom Loss
=====================
Inserting (\[LIN12X\]) into (\[LIN12\]) and using the explicit form (\[W1\]) yields the energy loss in the SU(2) Coulomb plasma
$$\begin{aligned}
-\frac{dK}{dr}=&&\frac{g^2\,C_F}{4\pi}\frac{\omega_p^2}{v^2}\int_0^{k_{max}}\,{dk}\,\frac{1}{k}\,\nonumber\\
&&\times\frac{1}{\sqrt{2\pi}}\int_{-v/v_T}^{v/v_T}\,dx\,e^{x^2/2}\,
\Bigg(\bigg((1-n{\bf c}_{D1}(k))\,e^{x^2/2}/x+n{\bf c}_{D1}(k)\,\psi(x)\bigg)^2+\pi\,n^2{\bf c}^2_{D1}(k)/2\Bigg)^{-1}\nonumber\\
\label{CB1}\end{aligned}$$
For an SU(2) probe charge after the substitution ${\boldsymbol{q}}^2\rightarrow g^2C_F/4\pi$ with $C_F$ the SU(2) Casimir. We note that (\[CB1\]) is cutoff in the infrared by the Debye wave number since ${\bf S}_{01}(k)\approx k^2/k_D^2$. So the main contribution to the energy loss in (\[CB1\]) stems from the region $k>k_D$ for which the SU(2) plasmon is too broad to contribute as we noted earlier. Most of the loss stems from the metallic part of the SU(2) plasma which is the analogue as rescattering against the [*free*]{} thermal spectrum explicit in (\[META\]).
In Fig. \[3dplot\] we display the integrand in (\[CB1\]) versus the jet velocity $v/v_T$ and the dimensionless momentum $q=ka_{WS}$. This is a weighted plot of the longitudinal spectral function along the jet velocity. The two wings at small $q$ are the two plasmons peaks, which progressively turns into the thermal distribution at larger $q$. In Fig. \[2dplot\] we show the same integrand for fixed $q$ versus the jet velocity $v/v_T$ normalized to the thermal velocity $V_T$. We note again the 2 plasmon poles around $v\approx v_T$ at small $q$. The vanishing of the termal distribution at $q=0$ follows from the extra $x^2$ weight arising from the denominator of (\[CB1\]) for ${\bf c}_{D1}(k)\approx 0$ at large $k$.
Since the loss is colored with only $l=1$ contributing and is metallic with only $k>k_D$ contributing, we do not see colored Cherenkov radiation stemming from plasmon emission , nor the ubiquitous Mach cone stemming from coupling to the sound mode [@casalderryetal]. While the sound mode contributes to ${\bf S}$ in (\[CHARGE2\]) it drops in the statistical averaging as only $l=1$ or plasmon channel contributes. The energy loss in the classical colored SU(2) Coulomb plasma is mostly metallic with $k>k_D$ and soundless due to the color quantum numbers of the fast moving probe charge.
A qualitative estimate for the energy loss follows by using ${\bf S}_{01}(k)\approx k^2/(k^2+k_D^2)$ and saturating the integrand by $k>k_D$,
$$-\frac{dK}{dr}\approx \frac{g^2\,C_F}{4\pi}\frac{\omega_p^2}{v^2}
\,\left(\sqrt{2\over \pi}\int_{0}^{v/v_T}\,x^2\,e^{-x^2/2}
\right)\,\,\ln{\left(\frac{k_{max}}{k_D}\right)} \label{CB2}$$
The upper divergence is manifest in (\[CB1\]) at $k\gg k_D$ since ${\bf S}_{01}(k)\approx 1$ and ${\bf c}_{D1}(k)\approx
k_D^2/k^2$ through the generalized Ornstein-Zernicke equation for all Coulomb couplings. The upper cutoff $k_{max}\approx 2\gamma \,
mv$ which is set by the maximum momentum transfer to the thermal particle of mass $m$ in the rest frame of the probe particle $M\gg
m$. Typically $M$ is charm and bottom, while $m\approx gT$ in weak coupling and $m\approx \pi T$ in strong coupling for a QCD plasma near the critical point. For the former $v/v_T\approx v\sqrt{g}$ (weak coupling) while for the latter $v/v_T\approx v\sqrt{\pi}$ (strong coupling). For $v/v_T\gg 1$ (\[CB2\]) reduces further to
$$-\frac{dK}{dx}\approx \frac{g^2\,C_F}{4\pi}\frac{\omega_p^2}{v^2}\,\,\ln{\left(\frac{2\gamma
\,m\,v}{k_D}\right)}
\label{CB3}$$
For the SU(2) colored Coulomb plasma. Aside from the Casimirs, this result is analogous to the energy loss in the classical and Abelian Coulomb plasma [@ichimaru; @thoma].
To assess the energy loss for varying Coulomb coupling $\Gamma=({g^2C_2}/{4\pi})({\beta}/{a_{WS}})$, we will rewrite the energy loss (\[CB1\]) as
![Energy loss for charm (left) and bottom (right) in the cQGP: $\Gamma=2,3,4$.[]{data-label="charm_bottom1"}](Eloss_C020304.ps "fig:"){width="49.00000%"} ![Energy loss for charm (left) and bottom (right) in the cQGP: $\Gamma=2,3,4$.[]{data-label="charm_bottom1"}](Eloss_B020304.ps "fig:"){width="49.00000%"}
![Logarithmic energy loss for charm (left) and bottom (right) in absolute units. See text.[]{data-label="charm_bottom_log"}](Eloss_Clog.ps "fig:"){width="49.00000%"} ![Logarithmic energy loss for charm (left) and bottom (right) in absolute units. See text.[]{data-label="charm_bottom_log"}](Eloss_Blog.ps "fig:"){width="49.00000%"}
![Logarithmic energy loss for charm (left) and bottom (right) in absolute units. See text.[]{data-label="charm_bottom_log"}](Eloss_CBlog.ps){width="55.00000%"}
$$\begin{aligned}
-\frac{dK}{dr}&=&3\Gamma^2\left(\frac{C_F}{C_2}\right)\frac{v_T^2}{v^2}\,\frac{T}{a_{WS}}
\int_0^{q_{max}}\,{dq}\,\frac{1}{q}\,\nonumber\\
&\times&\frac{1}{\sqrt{2\pi}}\int_{-v/v_T}^{v/v_T}\,dx\,e^{x^2/2}\,
\Bigg(\bigg((1-n{\bf c}_{D1}(q))\,e^{x^2/2}/x+n{\bf c}_{D1}(q)\,\psi(x)\bigg)^2+\pi\,n^2{\bf c}^2_{D1}(q)/2\Bigg)^{-1}\nonumber\\
\label{CB4}\end{aligned}$$
where $q=k a_{WS}$ and $a_{WS}$ is the Wigner-Seitz radius. The units for the energy loss per length in (\[CB4\]) follows from $T/a_{WS}$. For SU(2), $C_F=3/4$ for a heavy quark, and $C_2=2$ for thermal constituent gluons of mass $m\approx \pi\,T$. $a_{WS}=({3}/{4\pi
n})^{1/3}=(\frac{3}{4\pi}\frac{\beta^3}{0.244\times
3})^{1/3}=0.6883\beta$ for a density dominated by black-body (gluon) radiation $n=0.244(N_c^2-1)/\beta^3=0.244\times
3/\beta^3$.
In Fig. \[charm\_bottom1\] we show the dimensionless energy loss following from (\[CB4\]) for charm and bottom as a function of the probe momentum $\gamma Mv$, for different $\Gamma=2,3,4$ around the SU(2) liquid point. The numerics have been carried using the analytic structure factor of Fig. \[structure\_G020304\]. The energy loss is normalized to the total kinetic energy in length $L$, $E/L=(\gamma-1)M/L$. Since the quark velocity is maintained constant, the energy loss is seen to exceed 1 for $\Gamma=4$. The loss is very sensitive to the Coulomb coupling $\Gamma$ in the liquid phase.
In Fig. \[charm\_bottom\_log\] we show the energy loss on a logarithmic momentum scale for both charm and bottom. The upper curve (black) is the total loss from (\[CB4\]), while the lower curve (red) is just the metallic loss following from (\[CB3\]). The difference is a measure of the energy loss due to collisions with the low momentum part of the excitational spectrum of the SU(2) plasma which is plasmon dominated. These are the wings shown in Fig. \[2dplot\]. Charm and bottom jets with low momenta say $p\approx $ 3 GeV experience energy loss through broad plasmons. The energy loss for jets with $p$ larger than 10 GeV is mostly linear and therefore metallic.
In Fig. \[charm\_bottom2\] we compare our analytical results for the energy loss (red curve) to recent SU(2) numerical simulations (black curve) using the same model . We note that the numerical simulations in are quoted for the mean-potential to kinetic energy ratio $V/K\approx 3$ which happens to fluctuate by about $\pm 1$ inside the simulation box. To make the comparison meaningful, it is better to use in the notation of $\Gamma=1/Ta_{WS}\approx 4.8$ for $n=1/\lambda^3$ and $\lambda=1/3T$ with $\lambda$ the minimum of the potential in the same notations. With this in mind, our analytical results at $\Gamma=4$ compare favorably with the molecular dynamics simulations at large momenta and for heavier quark masses (say bottom). Most of the discrepancy with the simulations is at low momentum where the effects of the hard core in are the largest.
![Energy loss: (red) analytical versus (black) SU(2) molecular dynamics .[]{data-label="charm_bottom2"}](Eloss_Cmol04.ps "fig:"){width="49.00000%"} ![Energy loss: (red) analytical versus (black) SU(2) molecular dynamics .[]{data-label="charm_bottom2"}](Eloss_Bmol04.ps "fig:"){width="49.00000%"}
![Energy loss: (red) analytical versus (black) SU(2) molecular dynamics .[]{data-label="charm_bottom2"}](Eloss_Hmol04.ps){width="49.00000%"}
Conclusions
===========
We have analyzed the energy loss of fast moving charm and bottom quarks in an SU(2) color Coulomb plasma for a broad range of the Coulomb coupling. The Coulomb character of the underlying interaction retained classically make the energy loss entirely described by the longitudinal part of the dielectric function. We have used linear response theory to derive an explicit expression for the imaginary part of the dielectric function in terms of the Laplace transform of the time-dependent structure factor in the SU(2) Coulomb plasma.
We have shown that the probe initial color and statistical averaging causes the longitudinal dielectric function to select the $l=1$ color channel of the time-dependent structure factor which is the plasmon channel. The sound channel dominates the low momentum of the $l=0$ color channel, and decouples from the longitudinal part of the dielectric function. While the SU(2) plasmon survives at strong coupling, its width for $k>k_D$ is substantial and therefore causes it to thermally decay.
The energy loss of fast moving charm and bottom quarks is mostly due to the metallic aspect of the SU(2) colored Coulomb plasma which is dominated by thermal particles. There is no colored Cerenkov cone as the plasmon is dwarfed in the metallic limit, nor a colorless Mach cone as the sound decouples due to the probe initial colors. The energy loss is soundless. Our results are of course only classical. They apply for a broad range of $\Gamma$ near the liquid point. The comparison to the MD simulations show that our energy loss is about comparable at higher momenta and for heavier quarks where the effects of the numerical hard core is small. As initially reported in , the energy loss is sizable.
Strong coupling assessment of jet energy loss in gauge theories have been carried out in the context of holographic QCD [@herzogetal]. The fact that a Mach cone was reported in these calculations , maybe due to the fact that the probe jet is actually colorless. Indeed, most of the holographic jets are inserted with an external hand that maintains a constant velocity and perhaps even balance the color charge. Clearly colorless (mesonic) jets of the type $\overline{Q}Q$ do couple to the sound channel in our case through the ${\bf S}_{00}(k)$ structure factor , and are expected to be trailed by a Mach cone.
Finally, to carry our analysis of charm and bottom at RHIC and perhaps even LHC, require an assessment of the heavy quark composition in the prompt phase of the heavy ion collision which we have not carried out. Also, we need to address more carefully the correspondence between our classical SU(2) QGP and the quantum SU(3) QGP. These issues will be addressed next.
We thank Kevin Dusling for discussions. This work was supported in part by US DOE grants DE-FG02-88ER40388 and DE-FG03-97ER4014.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We theoretically analyze antibunching of the phonon field in an optomechanical oscillator employing the membrane-in-the-middle geometry. More specifically, a single-mode mechanical oscillator is quadratically coupled to a single-mode cavity field in the regime in which the cavity dissipation is a dominant source of damping, and adiabatic elimination of the cavity field leads to an effective cubic nonlinearity for the mechanics. We show analytically in the weak coupling regime that the mechanics displays a chaotic phonon field for small optomechanical cooperativity, whereas an antibunched single-phonon field appears for large optomechanical cooperativity. This opens the door to control of the second-order correlation function of a mechanical oscillator in the weak coupling regime.'
author:
- 'H. Seok'
- 'E. M. Wright'
title: Antibunching in an optomechanical oscillator
---
Introduction
============
Cavity optomechanics is a forefront research field in which the motional degrees of freedom of a mechanical oscillator are coupled to optical fields inside an optical cavity, stemming from the interplay through cavity resonance and radiation pressure forces [@Review1; @Review2; @Review3; @Milburn_book2]. Recent progress in nano- and micro- fabrication techniques have led to impressive milestones including the cooling of a mechanical oscillator to the motional ground state [@Cooling1; @Cooling2], optomechanically induced transparency [@OMIT1], coherent coupling of optical and mechanical modes [@Coherent1; @Coherent2], entanglement between optical and mechanical resonators [@Entanglement1], and optically induced interaction between mechanical oscillators [@Muti_mechancis1]. Cavity optomechanics has numerous applications such as precision measurement of the position of a mirror allowing for a gravitational wave detection [@Sensitivity; @LIGO1], a realization of macroscopic quantum objects [@Macroscopic1], and as a fundamental platform for exploring coupling to other quantum systems [@Hybrid1; @Hybrid2; @Hybrid3].
To date almost all experiments and treatments of cavity optomechanics are based on linearized optomechanical interactions in the sense that the interaction is linear in both the field and mechanical variables, and are therefore based on single photon-phonon interactions [@Review1; @Review2; @Review3; @Milburn_book2]. The intrinsic optomechanical interaction is, however, nonlinear, which comes to the fore in the single-photon strong coupling regime. The nonlinear nature of the optomechanical interaction gives rise to a variety of features previously explored in nonlinear quantum optics [@Nonlinear_optics], including photon blockade effects [@Antibunched_photon1], the generation of non-Gaussian states [@nonGaussian], and nonclassical antibunched mechanical resonators [@Antibunched_phonon1; @Antibunched_phonon2; @Antibunched_phonon3; @Antibunched_phonon4]. Since the single-photon radiation pressure is too small to realize the nonlinear strong coupling regime in nanofabricated optomechanical systems, several proposals have studied the possibility of an enhanced optomechanical nonlinearity [@Enhancement1; @Enhancement2], and thus sub-Poissonian phonon field, based on an optomechanical system employing two optical modes in the weak coupling regime [@Antibunched_phonon5].
In this paper, we theoretically analyze an approach for producing an antibunched phonon field based on the membrane-in-the-middle geometry, and in the weak coupling regime [@Middle1; @Middle2]. In particular, a single-mode mechanical oscillator is quadratically coupled to a single-mode cavity field in the regime where the cavity damping is a dominant source of dissipation, resulting in an effective cubic nonlinearity after adiabatic elimination of the cavity field. We show that the mechanical oscillator is coupled to an effective optical reservoir at zero temperature in addition to its own mechanical heat bath at finite temperature. To avoid the difficulties that arise from the multiplicative noise that appears from the use of the Heisenberg-Langevin equations, we here employ the Schrödinger picture. Then we demonstrate analytically that the mechanics displays a chaotic phonon field with small multiphoton optomechanical cooperativity, whereas an antibunched single-phonon appears for large multiphoton cooperativity.
This remainder of this paper is organized as follows: Sec. \[sec:Model System\] describes the model system, and Sec. \[sec:Density operator formalism\] derives the relevant master equation for the mechanical system. In Sec. \[sec:Results\], we employ the complex $P$ representation to investigate the steady-state behaviors of the mechanical oscillator for both the high and low temperature regimes, and the appearance of antibunching. Finally Sec.\[Conclusions\] gives our summary and conclusions.
Model System {#sec:Model System}
============
We consider a membrane-in-the-middle optomechanical system in which the single-mode of an optical resonator is quadratically-coupled to a single mechanical mode of effective mass $m$ and frequency $\omega_m$. The net Hamiltonian governing the optomechanical system is $$\hat H = \hat H_{\rm opt} + \hat H_{\rm mech} + \hat H_{\rm om}+ \hat H_{\rm loss},$$ where $$\hat H_{\rm opt} = \hbar\omega_c \hat a^\dag \hat a + i\hbar(\eta e^{-i\omega_{L}t}\hat a^\dag- \eta^* e^{i\omega_L t}\hat a) ,$$ is the Hamiltonian for the single-mode optical field driven by a monochromatic field of frequency $\omega_{L}$ at pumping rate $\eta$, and $$\hat H_{\rm mech} = \hbar\omega_m \hat b^\dag\hat b ,$$ is the Hamiltonian for the free mechanical mode. The optomechanical interaction is given by $$\hat H_{\rm om} = \hbar g_0\hat a^\dag \hat a (\hat b+\hat b^\dag)^2,$$ where $g_0>0$ is the quadratic single-photon optomechanical coupling coefficient, that we choose as positive to avoid any issues of mechanical instability [@Previous_paper]. Finally, $\hat H_{\rm loss}$ describes the interaction of the cavity field and mechanical modes with their associated reservoirs and accounts for dissipation.
Density operator formalism {#sec:Density operator formalism}
==========================
The Heisenberg-Langevin equations of motion for our problem can involve multiplicative quantum noise in the presence of nonlinear interactions [@Previous_paper]. To circumvent these problems we here explore the dynamics of the optomechanical system in the Schr[ö]{}dinger picture since the equation of motion describing the optomechanical system is then strictly linear in the density operator. The dynamics of the optomechanical system under the influence of thermal fluctuations in the quantum regime can then be described by the master equation [@Meystre_book] $$\begin{aligned}
\dot{\tilde \rho} &=&-\frac{i}{\hbar}[\hat H_{\rm opt}+\hat H_{\rm mech}+\hat H_{\rm om}, \tilde \rho]+\frac{\kappa}{2}{\cal D}[\hat a]\tilde\rho \nonumber \\
&&+\frac{\gamma}{2}{\bar n}_{\rm th}{\cal D}[\hat b^\dag]\tilde\rho+\frac{\gamma}{2}({\bar n}_{\rm th}+1){\cal D}[\hat b]\tilde\rho, \end{aligned}$$ where $\tilde \rho$ is the density operator for the combined optomechanical system, and the dissipation terms ${\cal D}[\hat o]\tilde\rho$ are of the standard Lindblad form $${\cal D}[\hat o]\tilde\rho = (2\hat o \tilde\rho \hat o^{\dag}-\hat o^\dag \hat o \tilde \rho- \tilde\rho\hat o^\dag \hat o).$$ These account for damping of the cavity field with decay rate $\kappa$ due to the coupling to a zero-temperature optical reservoir, and damping of the mechanical oscillator with decay rate $\gamma$ due to interaction with a mechanical reservoir at temperature $T$. The thermal occupation number of the mechanical bath is denoted by ${\bar n}_{\rm th}=[{\rm exp}(\hbar\omega_m/k_BT)-1]^{-1}$.
Master equation in the interaction picture
------------------------------------------
To proceed it is convenient to introduce the unitary operator $\hat U_1$ that transforms to a frame rotating at the driving frequency $\omega_L$ for the cavity field $$\hat U_1 =e^{-i\omega_L\hat a^\dag\hat a t} ,$$ and the unitary displacement operator $\hat U_2$ capturing the steady-state mean amplitude of the cavity field resulting from the external pump $$\hat U_2 =e^{(\alpha\hat a^\dag -\alpha^* \hat a)},$$ with steady-state intracavity amplitude $\alpha$ given by $$\alpha=\frac{\eta}{-i\Delta_c+\kappa/2} \equiv \sqrt{n_c}.$$ Without loss of generality $\alpha$ is here chosen as real by judicious choice of the phase of the pumping rate $\eta$, and $\Delta_c=\omega_L-\omega_c$ is the detuning of the pump laser from the resonance. The master equation for the transformed density operator $\bar\rho = \hat U_2^\dag \hat U_1^\dag \tilde \rho \hat U_1 \hat U_2$ then becomes $$\begin{aligned}
\dot{\bar \rho} &=&i\Delta_c[\hat a^\dag\hat a, \bar \rho]-i\omega_m^\prime[\hat b^\dag\hat b, \bar \rho]-ig_0n_c[\hat b^{\dag 2}+\hat b^2, \bar \rho] \nonumber \\
&&-ig[(\hat a+\hat a^\dag)(\hat b^{\dag}+\hat b)^2, \bar \rho] -ig_0[\hat a^\dag\hat a(\hat b^{\dag}+\hat b)^2, \bar \rho] \nonumber \\
&&+\frac{\kappa}{2}{\cal D}[\hat a]\bar\rho +\frac{\gamma}{2}{\bar n}_{\rm th}{\cal D}[\hat b^\dag]\bar\rho+\frac{\gamma}{2}({\bar n}_{\rm th}+1){\cal D}[\hat b]\bar\rho, \label{master_temp}\end{aligned}$$ where $\omega_m^\prime = \omega_m + 2g_0n_c$ is the shifted frequency of the mechanical oscillator, and $g=g_0\sqrt{n_c}$. This frequency shift proportional to the intracavity photon number comes from the quadratic optomechanical interaction, as opposed to the displacement of the mechanical equilibrium position that rises for the case of linear optomechanical coupling.
In the regime in which the mean cavity photon number $n_c$ is much larger than the photon fluctuations, the fifth term on the right-hand-side of Eq. (\[master\_temp\]) may be neglected: This term is a factor $1/n_c$ smaller than the third term and a factor $1/\sqrt{n_c}$ smaller than the fourth term, these also arising from the quadratic interaction. Following this approximation leads to an optomechanical interaction that is linear in the cavity field operators.
In order to investigate the mechanics in the deep quantum regime, we proceed by assuming that the external pump is red-detuned by twice the effective mechanical frequency, $\Delta_c=-2\omega_m^\prime$. Then a further simplification follows by invoking the rotating-wave approximation in the interaction picture implemented by the unitary transformation $\hat U_3 = e^{i(\Delta_c\hat a^\dag\hat a-\omega_m^\prime\hat b^\dag\hat b)t}$, and the resulting master equation becomes $$\begin{aligned}
\dot{\rho} &=&
-ig[\hat a^\dag\hat b^2+\hat b^{\dag 2}\hat a, \rho] \nonumber \\
&&+\frac{\kappa}{2}{\cal D}[\hat a]\rho +\frac{\gamma}{2}{\bar n}_{\rm th}{\cal D}[\hat b^\dag]\rho+\frac{\gamma}{2}({\bar n}_{\rm th}+1){\cal D}[\hat b]\rho, \label{master}\end{aligned}$$ where $\rho = \hat U_3^{\dag}\bar\rho\hat U_3$. We note that the third term on the right-hand-side of Eq. (\[master\_temp\]) has been neglected on the basis that it is off-resonant and counter-rotating if $g_0n_c \ll \omega_m^\prime$, and we have checked numerically that this term is indeed negligible in the weak coupling regime. Physically the Hamiltonian representing the Schr[ö]{}dinger evolution in Eq. (\[master\]) reads $$\hat H = \hbar g(\hat a^\dag\hat b^2+\hat b^{\dag 2}\hat a),$$ and is identical to the interaction picture Hamiltonian describing a parametric amplifier in quantum optics and is well-known to generate two photons in the subharmonic mode ($\hat b$) destroying a photon in the pump mode ($\hat a$) [@Milburn_book1]. It is thus expected that two phonons of the mechanics can be destroyed by creating a single photon which is eventually leaked out the optical resonator by the cavity field dissipation at rate $\kappa$.
Reduced density operator for the mechanics
------------------------------------------
In the regime where cavity dissipation is the dominant source of damping, the state of the cavity field tends to approach to a coherent state in a timescale of $1/\kappa$ and thus the density operator describing the optomechanical system can be approximated as a product state $$\rho(t) \approx \rho_o(t) \otimes \rho_m(t),$$ where $\hat \rho_o$ is the reduced density operator for the cavity field and $\hat \rho_m$ is the reduced density operator for the mechanics. One should keep in mind that on a timescale slower than $1/\kappa$, the dynamics of the optomechanical system is dependent of that of the mechanical oscillator whereas the dynamics of the cavity field is instantaneously followed by that of the mechanics due to the fast dissipation of the cavity field. Specifically, the reduced density operator for the cavity field describes the vacuum state, $\hat \rho_o= (|0\rangle\langle 0|)_o$ in that we are already in the displaced field picture.
In order to properly eliminate the reduced density operator for the cavity field and to derive the effective master equation for the mechanical oscillator, we follow the approach used for eliminating the density operator for the pump mode of a parametric amplifier in quantum optics or for the cavity field in cavity QED, see e.g. [@Carmichael_book]. The dynamics of the reduced density operator for the mechanics is then described by the effective master equation $$\begin{aligned}
\frac{d\rho_m}{dt} &=&\frac{\Gamma_{\rm opt}}{2}{\cal D}[\hat b^2]\rho_m \nonumber\\
&&+\frac{\gamma}{2}{\bar n}_{\rm th}{\cal D}[\hat b^\dag]\rho_m +\frac{\gamma}{2}({\bar n}_{\rm th}+1){\cal D}[\hat b]\rho_m,\end{aligned}$$ where $\Gamma_{\rm opt}$ is the nonlinear optomechanical damping rate given by $$\Gamma_{\rm opt} = \frac{8g^2}{\kappa}.$$ Note that this rate is identical to the maximum value of the optomechanical damping rate for $\kappa \ll \omega_m$ [@Review1]. The first term on the right-hand-side of the effective master equation accounts for two-phonon damping of the mechanical oscillator and the damping rate is proportional to the cavity photon number, indicating that the mechanical oscillator experiences the optical reservoir at zero temperature through the cavity field. In other words, the intracavity photon number can be used as a control parameter for the nonlinear optomechanical coupling strength of the mechanics to optical reservoir. That is, the dynamics and steady-state properties of mechanical oscillator are affected by two independent heat baths: The optical bath at zero temperature via two phonon processes and mechanical bath at finite temperature via one phonon processes.
It is convenient to scale time to the inverse of the mechanical decay rate, $\tau=\gamma t$, in terms of which the effective master equation for the mechanics then becomes $$\begin{aligned}
\frac{d \rho_m}{d\tau} &=&\frac{C}{2}{\cal D}[\hat b^2]\rho_m \nonumber \\
&&+\frac{1}{2}{\bar n}_{\rm th}{\cal D}[\hat b^\dag]\rho_m +\frac{1}{2}({\bar n}_{\rm th}+1){\cal D}[\hat b]\rho_m,\end{aligned}$$ where the multiphoton cooperativity $C$ is given by $$C=\frac{\Gamma_{\rm opt}}{\gamma}=\frac{8g^2}{\gamma\kappa} .$$ The multiphoton cooperativity is dimensionless and is a measure of the relative coupling strengths of the mechanical oscillator to the cavity-filtered optical bath and mechanical heat bath. Large cooperativity compared to the thermal occupation number $\bar n_{\rm th}$ indicates that mechanical oscillator is more influenced by the optical bath than the mechanical bath and the dynamics of the mechanics is highly nonlinear.
Results {#sec:Results}
=======
We next turn to the analytic solution of the master equation for the mechanics in the high and low temperature regimes. For this purpose we employ well known phase-space methods that we now discuss briefly as applied to our case.
Phase-space methods
-------------------
As is well-known, a nonlinear quantum mechanical problem can be mapped into a classical stochastic process by an appropriate phase space representation. We proceed to derive the equation of motion for the mechanical system in the complex $P$ representation. Expanding the density operator for the mechanics as $$\rho_m = \int \frac{|\mu\rangle\langle \nu^*|}{\langle \nu^*|\mu\rangle}P(\mu, \nu) {\rm d}\mu{\rm d}\nu,$$ and making use of the quantum correspondence appropriate for the complex $P$ representation [@Generalized_P_represenation] $$\begin{aligned}
\hat b \rho_m &\leftrightarrow& \mu P(\mu, \nu), \\
\hat b ^\dag \rho_m &\leftrightarrow& \left(\nu-\frac{\partial}{\partial \mu}\right) P(\mu, \nu), \\
\rho_m \hat b ^\dag &\leftrightarrow& \nu P(\mu, \nu), \\
\rho_m \hat b &\leftrightarrow& \left(\mu-\frac{\partial}{\partial \nu}\right) P(\mu, \nu),\end{aligned}$$ the master equation for the mechanics takes the form of the Fokker-Planck equation $$\begin{aligned}
\frac{dP({\bf \chi})}{d\tau}&=& -\sum_i \frac{\partial}{\partial \chi_i}[A({\bf\chi})]_i P({\bf \chi}) \nonumber \\
&&+\frac{1}{2}\sum_{i,j}\frac{\partial}{\partial \chi_i}\frac{\partial}{\partial \chi_j}[D({\bf\chi})]_{i,j}P({\bf \chi}), \label{FP_equation}\end{aligned}$$ where ${\bf \chi} =(\mu, \nu)^T$, the drift vector $A({\bf\chi})$ is given by $$A({\bf\chi}) =
\begin{pmatrix}
-\frac{1}{2}\mu -C\nu\mu^2 \\
-\frac{1}{2}\nu -C\mu\nu^2
\end{pmatrix},$$ and the diffusion matrix $D({\bf\chi})$ is $$D({\bf\chi}) =
\begin{pmatrix}
-C\mu^2 & {\bar n}_{\rm th} \\
{\bar n}_{\rm th} & -C\nu^2
\end{pmatrix}.$$ We remark that Eq. (\[FP\_equation\]) is identical to that of the complex $P$ distriubution for single-mode optical field in a cavity that involves cubic-nonlinear dispersive medium [@Drummond_bistability]. We further note that there are two diffusion sources for the complex $P$ distribution function, thermal fluctuations due to mechanical heat bath represented by the off-diagonal elements of the diffusion matrix, and additional quantum fluctuations due to the optomechanical interaction represented by the diagonal elements. Given the steady-state complex distribution function $P_s$ all normally-ordered steady-state moments can be calculated as $$\langle (\hat b^{\dag})^n(\hat b)^{n'} \rangle_{\rm ss} = \int {\rm d}\mu~(\mu^*)^n(\mu)^{n'}P_s(\mu, \mu^*). \label{moment}$$
High temperature regime
-----------------------
In the regime where the thermal fluctuations are the dominant source of diffusion, $\bar{n}_{\rm th}\gg C$, we are able to neglect quantum fluctuations resulting from the optomechanical interaction so that the diffusion matrix can be approximated as $$D({\bf\chi}) \approx
\begin{pmatrix}
0 & {\bar n}_{\rm th} \\
{\bar n}_{\rm th} & 0
\end{pmatrix}.$$ Then setting the left-hand-side of Eq. (\[FP\_equation\]) to zero for steady-sate, and employing the usual potential condition [@Milburn_book1], the distribution function is readily found as $$P_s(\mu, \nu) = {\cal N} \exp\left({-\frac{1}{ \bar{n}_{\rm th}}\mu\nu}\right) \exp\left({-\frac{C}{\bar{n}_{\rm th}}\mu^2\nu^2}\right),$$ where ${\cal N}$ is a normalization constant. Note that this complex $P$ distribution is bounded and well behaved in the domain in which $\nu=\mu^*$, namely, the Glauber-Sudarshan $P$ representation can be used [@Drummond_para_amp]. The corresponding Glauber-Sudarshan $P$ distribution becomes $$P_s(\mu, \mu^*) = {\cal N} \exp\left({-\frac{1}{ \bar{n}_{\rm th}}|\mu|^2}\right) \exp\left({-\frac{C}{\bar{n}_{\rm th}}|\mu|^4}\right), \label{GS_distribution}$$ From this result we see that for $C\ll1$ the Glauber-Sudarshan $P$ distribution approaches that for a thermal mixture with occupation number $\bar n_{th}$ $$P_s(\mu, \mu^*) \approx \frac{1}{{\bar n}_{\rm th}\pi} \exp\left({-\frac{1}{ \bar{n}_{\rm th}}|\mu|^2}\right),$$ as expected in the limit of small multiphoton cooperativity [@Quantum_noise]. On the other hand the Glauber-Sudarshan $P$ distribution can be approximated as $$P_s(\mu, \mu^*) \approx \frac{2}{\pi^{3/2}}\sqrt{\frac{C}{\bar{n}_{\rm th}}}\exp\left({-\frac{C}{\bar{n}_{\rm th}}|\mu|^4}\right),$$ in the limit of large multiphoton cooperativity $C\gg1$.
In Fig. \[fig:cooling\] we plot the steady-state mean phonon number obtained from Eq. (\[moment\]) $$\langle\hat b^\dag \hat b\rangle_{\rm ss}\equiv n_{\rm ss}= -\frac{1}{2C}+\sqrt{\frac{\bar{n}_{\rm th}}{\pi C}}\frac{\exp\left({-\frac{1}{4C\bar{n}_{\rm th}}}\right)
}{{\rm erfc}\left(\sqrt{\frac{1}{4C\bar{n}_{\rm th}}}\right)},$$ versus the multiphoton cooperativity $C$ for different thermal phonon numbers $\bar n_{th}$. Here ${\rm erfc}(x)=1-{\rm erf}(x)$ is the complementary error function. The results show that the mechanics, in a thermal state of mean occupation number ${\bar n}_{\rm th}$ at low multiphoton cooperativity, is cooled down as the multiphoton cooperativity $C$ is increased. Indeed the steady-state mean phonon number approaches $$n_{\rm ss} \approx \sqrt{\frac{\bar n_{\rm th}}{{\pi C}}}.$$ in the limit of large multiphoton cooperativity $C\gg 1$.
To probe further we calculate the second-order correlation function defined as $$g^{(2)}(0) \equiv \frac{\langle\hat b^{\dag 2}\hat b^2\rangle_{\rm ss}}{\langle\hat b^{\dag}\hat b\rangle_{\rm ss}^2} ,$$ this being plotted in Fig. \[fig:correlation\] as a function of the multiphoton cooperativity for different thermal occupation numbers. This figure makes clear that in the regime where $C\ll1$ the second-order correlation function $g^{(2)}(0)$ becomes $2$, a feature of a thermal state. On the other hand, $g^{(2)}(0)$ approaches $\pi/2$ for large multiphoton cooperativity, indicating that the steady-state of the mechanics is chaotic. This tendency stems from the fact that the linear thermal fluctuations overwhelm the nonlinear two-phonon optomechanical cooling. As a result, the phonon distribution is always bunched in the high temperature regime, and the variance of the phonon number distribution for the mechanical oscillator is in-between those of the mechanics in a thermal equilibrium and a coherent state with the same mean phonon number. This is illustrated in Fig. \[fig:number\_distribution\] which shows the steady-state phonon number distribution $P(n)$ of the mechanical oscillator (green circles) for ${\bar n}_{\rm th}= 10^4,~C= 10^2$, along with the cases of a thermal state (red triangles) and a coherent state (blue squares) for comparison.
Low temperature regime
----------------------
In order to explore the possibility of an antibunched phonon field, a key signature that the mechanical system is in a truly quantum state, we proceed to examine the low temperature regime. We have obtained the steady-state complex $P$ distribution following the procedures outlined in Ref. [@Nonlinear_damping], but for the sake of clarity in presentation we relegate the details to the Appendix and concentrate on the results here. Specifically, we find that the complex $P$ distribution is given by $$\begin{aligned}
P_s(\mu, \nu) &=& \frac{2Ae^{2\mu\nu}}{(1+2\bar{n}_{\rm th}-C)\mu\nu}
{}_2F_1\left(1,1;\tfrac{1+2\bar{n}_{\rm th}}{C};\tfrac{\bar{n}_{\rm th}}{C\mu\nu}\right) \nonumber \\
&&+\frac{2Ae^{2\mu\nu}}{\bar{n}_{\rm th}}\sum_{r=1}^{\infty}\frac{(-2\mu\nu)^r}{rr!}\times \nonumber \\
&&{}_2F_1\left(1,2+r-\tfrac{1+2\bar{n}_{\rm th}}{C};1+r;\tfrac{C\mu\nu}{\bar{n}_{\rm th}}\right), \label{distribution_quantum}\end{aligned}$$ where ${}_2F_1(a,b;c;z)$ is the hypergeometric function. The corresponding expression for the steady-state mean phonon number of the mechanics is given by Eq. (\[mean\_phonon\_quantum\]), and is plotted in Fig. \[fig:phonon\_quantum\] as a function of the multiphoton cooperativity $C$, and for a variety of thermal occupation numbers. These results show that the mechanics is cooled down near the motional ground state in the regime where $C\gg1$. In this regime the optomechanical two-phonon damping is dominant so that only the ground and first-excited states are significantly populated (see steady-state phonon distribution indicated by blue rhombi in Fig. \[fig:population\_quantum\]). Furthermore, the population of the mechanics in the first-excited state tends to increase with increasing temperature. These results are in accordance with the numerical calculations based on the Fock-state representation [@Quadratic1].
The expression for the second-order correlation function $g^{(2)}(0)$ of the mechanics is given by Eq. (\[g2\_quantum\]). Fig. \[fig:g2\_quantum\] shows a color coded plot of the second-order correlation function of the mechanical oscillator as a function of both the mutiphoton cooperativity $(C)$ and the thermal occupation number $(\bar n_{th})$. The plot reveals that the phonon distribution of the mechanics is antibunched $(g^{(2)}(0)<1)$ when $C> 2{\bar n}_{\rm th}+1$, whereas it is bunched $(g^{(2)}(0)>1)$ when $C<2{\bar n}_{\rm th}+1$. Physically, the mechanics tends to experience one phonon absorption and emission processes, and its phonon distribution is superpoissonian, if the mechanical thermal and quantum noise sources are dominant, $C<2{\bar n}_{\rm th}+1$. However, in the regime where the optomechanical coupling is stronger than thermal decoherence, $C> 2{\bar n}_{\rm th}+1$, the mechanics has a tendency to experience two-phonon absorption and emission processes and its phonon distribution becomes antibunched. As expected, when $C= 2{\bar n}_{\rm th}+1$ the steady-state of the mechanical oscillator becomes a coherent state with a mean phonon number $$n_{\rm ss}= \frac{{\bar n}_{\rm th}}{2{\bar n}_{\rm th}+1},$$ and the second-correlation function becomes unity. This situation is indicated by the thick solid line in Fig. \[fig:g2\_quantum\]: Regions of parameter space above this line yield steady-state bunching whereas below this line antibunching is realized. Fig. \[fig:population\_quantum\] shows three representative plots of the phonon number distributions of the mechanics indicating that only the ground and first-excited states are significantly populated if $C\gg{\bar n}_{\rm th}+1$ (blue rhombi), the distribution being Poissonian if $C=2{\bar n}_{\rm th}+1$ (green circles), and the distribution becoming nearly exponential if $C\ll2{\bar n}_{\rm th}+1$ (red circles).
We finish by noting that in the regime where the mechanical heat bath is at zero temperature thermal effects are completely negligible compared to the quantum fluctuations, $\bar n_{\rm th}=0$, and the diffusion matrix $D({\bf\chi})$ reads $$D({\bf\chi})=
\begin{pmatrix}
-C\mu^2 & 0 \\
0 & -C\nu^2
\end{pmatrix}.$$ This situation was previously studied extensively in the context of quantum optics [@Nonlinear_damping] and the steady-state complex $P$ distribution is given by $$\begin{aligned}
P_s(\mu, \nu) =\frac{1-\frac{1}{C}}{2\pi^2}e^{2\mu\nu}\sum_{r=0}^{\infty} \frac{(-2\mu\nu)^{r-1}}{(r+1-\frac{1}{C})r!}.\end{aligned}$$ In this case the mechanical oscillator is coupled to an optical reservoir at zero temperature by the nonlinear optomechanical coupling, and is also coupled to the mechanical heat bath at zero temperature by the intrinsic linear interaction. Then the steady-state of the mechanical oscillator is the motional ground state, as expected, and thus the mean phonon number $n_{\rm ss}=0$ and the second-order correlation function $g^{(2)}(0)=0$ [@Nonlinear_damping].
Summary and conclusions {#Conclusions}
=======================
We have analytically investigated the steady-state of a vibrating membrane coupled to a single-mode optical field via a quadratic optomechanical interaction, and in the weak coupling limit. The mechanics was shown to experience an effective cubic nonlinearity in the limit that the cavity dissipation rate is much larger than both the optomechanical coupling and mechanical damping rates, allowing for adiabatic elimination of the cavity field. Our key result is that the steady-state phonon field is chaotic if the multiphoton cooperativity obeys $C<2{\bar n_{\rm th}}+1$ whereas it antibunched if $C>2{\bar n_{\rm th}}+1$.
There are of course barriers to realizing antibunching of a phonon field, but recent developments make this more feasible. The requirement of large optomechanical cooperativity has been realized in high-frequency optomechanical oscillators [@Cooperativity], with a quoted maximum value of $146,000$. In addition, the demonstration of a Hanbury-Brown-Twiss type experiment [@Phonon_counting] for a phonon field in a nanomechanical resonator paves the way to measuring the second-order correlation. Thus our calculation opens the door to control of the second-order correlation of the mechanical oscillator in the weak coupling regime, and the observation of phonon antibunching.
This work is supported by the Korea National Reserach Foundation (NRF) NRF-2015R1C1A1A01052349.
Steady-state complex $P$ distribution in the low temperature regime {#appendix A}
===================================================================
In order to find the steady-state complex $P$ distribution of the mechanics, we follow the procedures outlined in Ref. [@Nonlinear_damping]. Equation (\[FP\_equation\]) can be written as $$\begin{aligned}
\frac{dP}{d\tau}&=& \frac{\partial}{\partial\mu}\left[\frac{\mu}{2}+C\mu^2\nu-\frac{C}{2}\frac{\partial}{\partial\mu}\mu^2+\frac{\bar{n}_{\rm th}}{2}\frac{\partial}{\partial\nu}\right]P\nonumber \\
&&+ \frac{\partial}{\partial\nu}\left[\frac{\nu}{2}+C\nu^2\mu-\frac{C}{2}\frac{\partial}{\partial\nu}\nu^2+\frac{\bar{n}_{\rm th}}{2}\frac{\partial}{\partial\mu}\right]P.\end{aligned}$$ The steady-state complex $P$ distribution can in general be obtained from $$\begin{aligned}
\left[\frac{\mu}{2}+C\mu^2\nu-\frac{C}{2}\frac{\partial}{\partial\mu}\mu^2+\frac{\bar{n}_{\rm th}}{2}\frac{\partial}{\partial\nu}\right]P_s&=&f(\nu), \label{b1}\\
\left[\frac{\nu}{2}+C\nu^2\mu-\frac{C}{2}\frac{\partial}{\partial\nu}\nu^2+\frac{\bar{n}_{\rm th}}{2}\frac{\partial}{\partial\mu}\right]P_s&=&g(\mu), \label{b2} \end{aligned}$$ where $f(\nu)$ and $g(\mu)$ must satisfy generalized potential conditions [@Nonlinear_damping]. To find the form of these functions we write $P_s(\mu, \nu)$ as $$P_s(\mu, \nu) = \frac{Q(\mu, \nu)}{(C\mu\nu-\bar{n}_{\rm th})^2},$$ then Eqs. (\[b1\]) and (\[b2\]) can be written as $$\begin{aligned}
\frac{\partial R(\mu, \nu)}{\partial \mu} &=& e^{-2\mu\nu}(C\mu\nu-\bar{n}_{\rm th})^{1-\frac{1+2\bar{n}_{\rm th}}{C}}F(\mu, \nu), \\
\frac{\partial R(\mu, \nu)}{\partial \nu} &=& e^{-2\mu\nu}(C\mu\nu-\bar{n}_{\rm th})^{1-\frac{1+2\bar{n}_{\rm th}}{C}}G(\mu, \nu),\end{aligned}$$ where we define for typographical simplicity, $$\begin{aligned}
R(\mu, \nu) &=& e^{-2\mu\nu}(C\mu\nu-\bar{n}_{\rm th})^{-\frac{1+2\bar{n}_{\rm th}}{C}}Q(\mu, \nu), \\
F(\mu, \nu) &=& -2\frac{C\nu^2 f(\nu)+\bar{n}_{\rm th} g(\mu)}{C\mu\nu+\bar{n}_{\rm th}}, \\
G(\mu, \nu) &=& -2\frac{C\mu^2 g(\mu)+\bar{n}_{\rm th} f(\nu)}{C\mu\nu+\bar{n}_{\rm th}}.\end{aligned}$$ The generalized potential condition $$\frac{\partial^2 R(\mu, \nu)}{\partial \nu\partial \mu}= \frac{\partial^2 R(\mu, \nu)}{\partial \mu\partial \nu}$$ can be written as $$\begin{aligned}
\frac{\partial}{\partial \nu} \left[e^{-2\mu\nu}(C\mu\nu-\bar{n}_{\rm th})^{1-\frac{1+2\bar{n}_{\rm th}}{C}}F(\mu, \nu)\right] \nonumber \\
= \frac{\partial}{\partial \mu}\left[e^{-2\mu\nu}(C\mu\nu-\bar{n}_{\rm th})^{1-\frac{1+2\bar{n}_{\rm th}}{C}}G(\mu, \nu)\right] \label{potential_condition}.\end{aligned}$$ Equation (\[potential\_condition\]) is satisfied for $$\begin{aligned}
f(\nu) &=& \frac{A}{\nu}, \\
g(\mu) &=& \frac{A}{\mu},\end{aligned}$$ where $A$ is a constant. Thus, the steady-state complex $P$ distribution is given by after some algebra $$P_s(\mu, \nu) = e^{2\mu\nu}(C\mu\nu-\bar{n}_{\rm th})^{\frac{1+2\bar{n}_{\rm th}}{C}-2}[B+AI(\mu, \nu)],$$ where $B$ is a constant of integration and $I(\mu, \nu)$ is the indefinite integral $$I(\mu, \nu) = -2\int {\rm d}\mu~\frac{e^{-2\mu\nu}}{\mu}(C\mu\nu-\bar{n}_{\rm th})^{1-\frac{1+2\bar{n}_{\rm th}}{C}}. \label{I}$$ This integral may be calculated using a power-series expansion of the exponential function and the resulting steady-state complex $P$ distribution reads $$\begin{aligned}
P_s(\mu, \nu) &=& Be^{2\mu\nu}(C\mu\nu-\bar{n}_{\rm th})^{\frac{1+2\bar{n}_{\rm th}}{C}-2} \nonumber\\
&&+\frac{2Ae^{2\mu\nu}}{(1+2\bar{n}_{\rm th}-C)\mu\nu}
{}_2F_1\left(1,1;\tfrac{1+2\bar{n}_{\rm th}}{C};\tfrac{\bar{n}_{\rm th}}{C\mu\nu}\right) \nonumber \\
&&+\frac{2Ae^{2\mu\nu}}{\bar{n}_{\rm th}}\sum_{r=1}^{\infty}\frac{(-2\mu\nu)^r}{rr!}\times \nonumber \\
&&{}_2F_1\left(1,2+r-\tfrac{1+2\bar{n}_{\rm th}}{C};1+r;\tfrac{C\mu\nu}{\bar{n}_{\rm th}}\right).\end{aligned}$$ It should be noted that the two constants $A$ and $B$ are chosen from the normalization condition and the requirement that the phonon number distribution be nonnegative. Using the complex $P$ distribution function, all normal-ordered moments in the steady state can be obtained from $$\langle (\hat b^{\dag})^n(\hat b)^{n'} \rangle_{\rm ss} = \int {\rm d}\mu{\rm d}\nu~(\nu)^n(\mu)^{n'}P_s(\mu, \nu). \label{moment_complex_P}$$ Making the change of variables $$\begin{aligned}
N&=& \mu\nu, \\
z&=& \mu,\end{aligned}$$ and choosing a circular contour around the origin for the $z$ line integral, and a Hankel contour for the $N$ line integral [@Math_book2], one can find the normalization condition, the mean phonon number, the second-order correlation, and so on from Eq. (\[moment\_complex\_P\]). The normalization condition reads $$\begin{aligned}
\frac{1}{4\pi^2}&=& -B\frac{e^{\frac{2{\bar n}_{\rm th}}{C}}}{2\Gamma\left(2-\frac{1+2{\bar n}_{\rm th}}{C}\right)} \left(\frac{C}{2}\right)^{\frac{1+2{\bar n}_{\rm th}}{C}-2} \nonumber \\
&&-A\frac{2\Gamma\left(\frac{1+2{\bar n}_{\rm th}}{C}\right)}{1+2{\bar n}_{\rm th}-C}\sum_{k=0}^{\infty}\frac{(2{\bar n}_{\rm th}/C)^k}{\Gamma\left(\frac{1+2{\bar n}_{\rm th}}{C}+k\right)}.\end{aligned}$$ The populations of the $m$-th number state are given by $$\begin{aligned}
P_{m} &=& -B\sum_{k=0}^{m}\frac{4\pi^2e^{\frac{{\bar n}_{\rm th}}{C}}C^{\frac{1+2{\bar n}_{\rm th}}{C}-2} \left(\frac{{\bar n}_{\rm th}}{C}\right)^{m-k}}{\Gamma(m-k+1)\Gamma(k+1)\Gamma\left(2-k-\frac{1+2{\bar n}_{\rm th}}{C}\right)} \nonumber \\
&&-A\frac{8\pi^2\Gamma\left(\frac{1+2{\bar n}_{\rm th}}{C}\right)}{(1+2{\bar n}_{\rm th}-C)m!}\times \nonumber \\
&&\sum_{k=m}^{\infty}\frac{\Gamma(k+1)({\bar n}_{\rm th}/C)^k}{\Gamma\left(\frac{1+2{\bar n}_{\rm th}}{C}+k\right)\Gamma(k+1-m)}.\end{aligned}$$ In order for the phonon number distribution to be nonnegative, $B=0$ for $C\neq1+2\bar{n}_{\rm th}$ and $A=0$ for $C=1+2\bar{n}_{\rm th}$ due to the oscillatory behavior of the $\Gamma$ function.
If $C\neq1+2\bar{n}_{\rm th}$, normalization constant $A$ is given by $$A = -\frac{1+2\bar{n}_{\rm th}-C}{8\pi^2\Gamma\left(\frac{1+2{\bar n}_{\rm th}}{C}\right) \displaystyle\sum_{k=0}^{\infty}\frac{(2{\bar n}_{\rm th}/C)^k}{\Gamma\left(\frac{1+2{\bar n}_{\rm th}}{C}+k\right)}}.$$ The mean phonon number is given by $$n_{\rm ss}= \frac{1}{2}\frac{\displaystyle\sum_{k=0}^\infty \frac{k}{\Gamma\left(\frac{1+2{\bar n_{\rm th}}}{C}+k\right)}\left(\frac{2{\bar n_{\rm th}}}{C}\right)^k}{\displaystyle\sum_{k=0}^\infty \frac{1}{\Gamma\left(\frac{1+2{\bar n_{\rm th}}}{C}+k\right)}\left(\frac{2{\bar n_{\rm th}}}{C}\right)^k}, \label{mean_phonon_quantum}$$ and the second-order correlation function $g^{(2)}(0)$ is $$g^{(2)}(0)= \frac{\displaystyle\sum_{k, k\prime}^\infty \frac{k(k-1)}{\Gamma\left(\frac{1+2{\bar n_{\rm th}}}{C}+k\right)\Gamma\left(\frac{1+2{\bar n_{\rm th}}}{C}+k^\prime\right)}\left(\frac{2{\bar n_{\rm th}}}{C}\right)^{k+k^\prime}}{\displaystyle\sum_{k, k\prime}^\infty \frac{kk^\prime}{\Gamma\left(\frac{1+2{\bar n_{\rm th}}}{C}+k\right)\Gamma\left(\frac{1+2{\bar n_{\rm th}}}{C}+k^\prime\right)}\left(\frac{2{\bar n_{\rm th}}}{C}\right)^{k+k^\prime}}. \label{g2_quantum}$$
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Planets residing in circumstellar habitable zones (CHZs) offer our best opportunities to test hypotheses of life’s potential pervasiveness and complexity. Constraining the precise boundaries of habitability and its observational discriminants is critical to maximizing our chances at remote life detection with future instruments. Conventionally, calculations of the inner edge of the habitable zone (IHZ) have been performed using both 1D radiative-convective and 3D general circulation models. However, these models lack interactive 3D chemistry and do not resolve the mesosphere and lower thermosphere (MLT) region of the upper atmosphere. Here we employ a 3D high-top chemistry-climate model (CCM) to simulate the atmospheres of synchronously-rotating planets orbiting at the inner edge of habitable zones of K- and M-dwarf stars (between $T_{\rm eff} =$ 2600 K and 4000 K). While our IHZ climate predictions are in good agreement with GCM studies, we find noteworthy departures in simulated ozone and HO$_{\rm x}$ photochemistry. For instance, climates around inactive stars do not typically enter the classical moist greenhouse regime even with high ($\ga 10^{-3}$ mol mol$^{-1}$) stratospheric water vapor mixing ratios, which suggests that planets around inactive M-stars may only experience minor water-loss over geologically significant timescales. In addition, we find much thinner ozone layers on potentially habitable moist greenhouse atmospheres, as ozone experiences rapid destruction via reaction with hydrogen oxide radicals. Using our CCM results as inputs, our simulated transmission spectra show that both water vapor and ozone features could be detectable by instruments NIRSpec and MIRI LRS onboard the James Webb Space Telescope.'
author:
- Howard Chen
- 'Eric T. Wolf'
- Zhuchang Zhan
- 'Daniel E. Horton'
title: |
Habitability and Spectroscopic Observability of Warm M-dwarf Exoplanets\
Evaluated with a 3D Chemistry-Climate Model
---
Introduction {#sec:intro}
============
For the first time in human history, it is possible to find and characterize nearby rocky and potentially habitable worlds. Recent discoveries of Proxima Centauri b, the TRAPPIST-1 system, and LHS 1140b [@AngladaEt2016NATURE; @GillonEt2017NATURE; @DittMannEt2017NATURE] show that remote examination of small rocky planets is within reach. Terrestrial planets, such as these, are expected to be common (${\sim}15\%$) in circumstellar habitable zones (CHZs) of low-mass stars [@TarterEt2007; @Dressing+Charbonneau2015ApJ] $-$ systems that are especially amenable to spectroscopic observation due to their high transit frequencies, low star-to-planet brightness contrasts, prolonged main sequence lifetimes, and abundance in both the solar neighborhood and the projected Transiting Exoplanet Survey Satellite (TESS) sample [@HenryEt2006AJ; @BarclayEt2018ApJS]. In fact, TESS has already found small and Earth-sized planets transiting cool stars (e.g., @VanderspekEt2019ApJL [@DragomirEt2019ApJL]). Upcoming characterization efforts by the James Webb Space Telescope, ground-based 30-meter extremely large telescopes, and direct imaging missions will likely attempt to detect habitability indicators (e.g., N$_2$, H$_2$O$_v$) and/or biosignatures (O$_2$, O$_3$, CH$_4$, N$_2$O, CO$_2$; @SaganEt1993NATURE) on these K- and M-dwarf systems. Indeed, atmospheric characterization of increasingly smaller planets ($R_p \la 4~R_\oplus$) is already underway (e.g., @WakefordEt2017SCI [@BennekeEt2019NAtAstro; @BennekeEt2019arXiv]).
Future target selection and characterization efforts will benefit from improved understanding of and constraints on CHZ boundaries. Earliest estimates of the CHZ made use of energy balance models (EBMs) [@Hart1979Icarus], which established the dependence of HZ widths on stellar spectral type. Follow on studies, using 1D radiative-convective models, identified two boundaries of the inner habitable zone: one defined by the onset of a water-enriched stratosphere and another defined by a radiative equilibrium threshold [@KastingEt1984Icarus; @Kasting1988Icarus]. These early simulations assumed a fully saturated troposphere with a fixed moist adiabatic lapse rate and static clouds. Subsequently, @KastingEt2015ApJL found that as the absorbed stellar flux increases, the stratosphere moistens and warms significantly which could allow water vapor to efficiently escape to space. Other studies, also using 1D models, provided additional insights, finding for example, that CHZ widths could change according to atmospheric composition and/or atmospheric pressure (e.g., @VladiloEt2013ApJ [@ZsomEt2013ApJ; @Ramirez+Kaltenegger2017ApJL]).
In more recent years, idealized and state-of-the-art estimates of the CHZ have utilized 3D general circulation models (GCMs) to place physics-based constraints on CHZ boundaries (e.g., @AbeEt2011AsBio [@YangEt2014ApJL; @WayEt2016GRL]). GCM predictions improved upon 1D model projections by way of explicit simulation of large-scale circulation and key climate system feedbacks. For instance, incorporation of atmospheric dynamics into models of slowly-rotating planets resulted in climatic behaviors that can only be resolved in 3D, e.g., substellar cloud formation and convergence caused by changes in the Coriolis force [@YangEt2013ApJL; @KopparapuEt2016ApJ; @WayEt2018ApJS]. Follow-up studies, using similar GCMs, found that habitable planets around M-dwarf stars have moist stratospheres despite mild global mean surface temperatures (e.g., @FujiiEt2017ApJ [@KopparapuEt2017ApJ]). These results stood in contrast to previous inverse modeling approaches with 1D radiative-convective models (e.g., @Kasting1988Icarus), where a surface temperature of 340 K was deemed the threshold for the classical “moist greenhouse" regime.
Despite these advances, exoplanet GCM studies have traditionally not accounted for photochemical and atmospheric chemistry-climate interactions $-$ components recently found by both 1D [@LincowskiEt2018ApJ; @KozakisEt2018ApJ] and 3D models [@ChenEt2018ApJL] to be critical for habitability and biosignature prediction. The addition of photochemistry to prognostic atmospheric models allows for interactions between high energy photons and gaseous molecules. This often leads to the breaking of molecular bonds and creation of free radicals and ions, which have significant impacts on atmospheric composition and associated habitability. Determination of water loss, in particular, requires knowledge of where water vapor photodissociation occurs in the mesosphere and lower thermosphere (MLT), and is dependent on dynamical, photochemical, and radiative processes. To simulate the speciation, reaction, and transport of various gaseous constituents (e.g., H$_2$O$_v$) and their photochemical byproducts (e.g., H, H$_2$), coupled 3D chemistry-climate models (CCMs) are needed. CCMs are also able to simulate photochemically important species such as ozone, allowing for prognostic assessments of chemistry-climate system feedbacks. As ozone is primarily derived from molecular oxygen, prognostic ozone calculations enable consideration of O$_2$-rich atmospheres with active oxygenating photosynthesis on the surface. Lastly, the large number of chemical species calculated by CCMs provide a rich tapestry for calculating transmission spectra, compared to the simplified atmospheric compositions generally considered in GCMs.
Earlier climate models of tidally-locked Earth-like planets adopted vertical resolutions of 15-25 layers with equally spaced 30-50 mbar levels [@JoshiEt1997Icarus; @Merlis+Schneider2010; @EdsonEt2011Icar]. While this setup fully resolved the general structure of the troposphere where the majority of weather takes place [@Held+Suarez1994], it neglected the upper stratosphere and thermosphere. Critically, interactive simulation of photochemistry and atmospheric chemistry requires a high model-top (i.e., a model whose atmosphere reaches into the mid-thermosphere (${\sim} 150$ km)), as highly energetic photons initially and primarily interact with a planet’s upper atmosphere. While most radiatively active species are stable against dissociation in the troposphere, species become vulnerable to photolysis above the tropopause and to photoionization above the stratopause (${\sim}1$ mbar) and mesopause (${\sim}10^{-2}$ mbar). Apart from key dissociative processes in the MLT region, high-top atmospheric dynamics are also important as they influence the transport of gaseous molecules. Vertical velocity in the vicinity of the tropopause is not an isolated process, but influenced by momentum sources in the stratosphere and lower thermosphere [@HoltonEt1995]. Further, mean meridional circulation of the lower stratosphere, which can affect the distribution of chemical tracers, is driven primarily by the drag provided by planetary and gravity wave momentum deposition in the stratosphere and mesosphere [@HaynesEt1991; @HoltonEt1995; @Romps+Kuang2009GRL]. A high model-top is therefore essential to simulate chemical interactions and their associated dynamical processes in the MLT region.
Based on conventional theory, endmembers of habitability are represented by (i) CO$_2$-rich icehouse climates at the outer edge of the habitable zone (OHZ; e.g., @Paradise+Menou2017ApJ) and (ii) moist greenhouse climates at the inner edge of the habitable zone (IHZ; e.g., @KopparapuEt2016ApJ). In this study, we use a 3D high-top CCM to investigate the latter regime, namely the moist greenhouse limits of IHZ planets orbiting M-dwarf stars. The paper is organized as follows: In Section 2, we describe our model and experimental setup. In Section 3, we present and analyze our results. Section 4 discusses implications of our results, caveats, and relevance to observations. Finally, Section 5 summarizes key findings, provides concluding remarks, and suggests next step.
[ccccccccccc]{}
09F26T & 2600 & 0.000501 & 0.0886 & 0.9 & 4.43 & 268.36 & 2.95e-06 & 4.01e-06 & 341.90 & 0.32\
[**10F26T**]{} & 2600 & 0.000501 & 0.0886 & 1.0 & 4.11 & 282.39 & 4.22e-06 & 3.68e-06 & 314.14 & 0.23\
11F26T & 2600 & 0.000501 & 0.0886 & 1.1 & 3.82 & runaway & - & - & - &-\
10F30T & 3000 & 0.00183 & 0.143 & 1.0 & 8.53 & 270.39 & 4.99e-06 & 3.70e-06 & 337.27 & 0.38\
[**11F30T**]{} & 3000 & 0.00183 & 0.143 & 1.1 & 7.91 & 278.76 & 7.52e-06 & 3.87e-06 & 329.97 & 0.33\
12F30T & 3000 & 0.00183 & 0.143 & 1.2 & 7.41 & runaway & - & - & -&-\
10F33T & 3300 & 0.00972 & 0.249 & 1.0 & 23.13 & 266.73 & 4.63e-06 & 3.70e-06 & 333.57 & 0.45\
15F33T & 3300 & 0.00972 & 0.249 & 1.5 & 16.23 & 284.33 & 2.61e-04 & 5.47e-06 & 342.75 & 0.46\
[**16F33T**]{} & 3300 & 0.00972 & 0.249 & 1.6 & 15.46 & 297.75 & 2.05e-03 & 1.22e-04 & 343.39 & 0.47\
17F33T & 3300 & 0.00972 & 0.249 & 1.7 & 14.77 & runaway & - & - & - & -\
10F40T & 4000 & 0.0878 & 0.628 & 1.0 & 69.39 & 265.76 & 2.66e-06 & 3.69e-06 & 319.54 & 0.48\
17F40T & 4000 & 0.0878 & 0.628 & 1.7 & 47.64 & 282.88 & 3.40e-04 & 6.23e-06 & 330.04 & 0.52\
[**19F40T**]{} & 4000 & 0.0878 & 0.628 & 1.9 & 43.87 & 301.41 & 1.18e-02 & 7.27e-05 & 342.75 & 0.55\
20F40T & 4000 & 0.0878 & 0.628 & 2.0 & 42.22 & runaway & - & - & - &-\
19FSolarUV$^a$ & 4000 & 0.0878 & 0.628 & 1.9 & 43.87 & 303.38 & 1.30e-02 & 1.05e-03 & 409.62 & 0.51\
19FADLeoUV$^b$ & 4000 & 0.0878 & 0.628 & 1.9 & 43.87 & 315.29 & 9.65e-02 & 7.45e-02 & 382.15 & 0.45\
Model Description & Numerical Setup {#sec:method}
===================================
We employ the National Center for Atmospheric Research (NCAR) Whole Atmosphere Community Climate Model (WACCM) to investigate the putative atmospheres of rocky exoplanets. WACCM is a 3D global CCM that simulates interactions of atmospheric chemistry, radiation, thermodynamics, and dynamics. We set the Community Atmosphere Model v4 (CAM4) as the atmosphere component of WACCM. CAM4 uses native Community Atmospheric Model Radiative Transfer (CAMRT) radiation scheme [@kiehl1983co2], the Hack scheme for shallow convection [@Hack1994JGR], the Zhang-McFarlane scheme for deep convection [@zhang1995sensitivity], and the Rasch-Kristjansson (RK) scheme for condensation, evaporation and precipitation [@zhang2003modified]. For a complete model description see @neale2010description and [@MarshEt2013JGR].
WACCM includes an active hydrological cycle and prognostic photochemistry and atmospheric chemistry reaction networks. The chemical model is version 3 of the Modules for Ozone and Related Chemical Tracers (MOZART) chemical transport model [@KinnisonEt2007JGR]. The module resolves 58 gas phase species including neutral and ion constituents linked by 217 chemical and photolytic reactions. The land model is a diagnostic version of the Community Land Model v4 with the 1850 control setup including prescribed surface albedo, surface CO$_2$, vegetation, and forced cold start. The oceanic component is a 30-meter deep thermodynamic slab model with zero dynamical heat transport and no sea-ice. Even though a fully dynamic ocean is ideal, @YangEt2019ApJa recently demonstrated that the inclusion of such does not significantly impact the climate of moist greenhouse IHZ planets. Furthermore, the presence of significant North-South oriented continents minimizes the effects of ocean heat transport on tidally locked worlds [@YangEt2019ApJa; @DelGenioEt2019AsBio].
WACCM includes a number of improved and expanded high-top atmospheric physics and chemical components. Processes in the MLT region are based on the thermosphere-ionosphere-mesosphere electrodynamics (TIME) GCM [@RobleEt1994GRL]. Key processes included are: neutral and high-top ion chemistry (ion drag, auroral processes, and solar proton events) and their associated heating reactions. In this study, we do not use prognostic ion chemistry (e.g., WACCM-D; Verronen et al. 2016) for the sake of computational efficiency. In terms of atmospheric dynamics, WACCM allows the emergence of gravity waves (important for governing large-scale flow patterns and chemical transport) by orographic sources, convective overturning, or strong velocity shears [@neale2010description]. As we assume Earth-like topography in all our simulations, orography may also provide a means of direct forcing on planetary scale Rossby waves, which act to increase the asymmetry of atmospheric circulation and turbulent flow (the absence of topography such as on an idealized aquaplanet would minimize this effect and thus induce greater circulation symmetry). Topography also drives gravity waves which deposit energy into the mesosphere, affecting its temperature structure and circulation [@BardeenEt2010JGR]. Molecular diffusion via gravitational separation of different molecular constituents [@BanksET1973] is an extension to the nominal diffusion parameterization in CAM4. Below 65 km (local minimum in shortwave heating and longwave cooling), WACCM retains CAM4’s radiation scheme. Above 65 km, WACCM expands upon both longwave (LW) and shortwave (SW) radiative parameterizations from those of CAM3 and CAM4 [@CollinsEt2006]. WACCM uses thermodynamic equilibrium (LTE) and non-LTE heating and cooling rates in the extreme ultraviolet (EUV) and infrared (IR)[@FomichevE1998JGR]. In the SW (0.05 nm to 100 $\mu$m; @Lean2000GRL [@Solomon+Qian2005JGR]), radiative heating and cooling are sourced from photon absorption, as well as photolytic and photochemical reactions. To our knowledge, we are the first to apply these models and modules in the context of exoplanets.
Earth’s atmospheric structure is typically defined using the vertical temperature gradient. In WACCM simulations the atmospheric structure is dependent on the atmospheric gases, planetary rotation period, and bolometric stellar flux $-$ thus the simulated vertical temperature gradient is different from that of Earth. However, to facilitate comparison, we refer to simulated atmospheric layers using the typical pressure levels of Earth’s atmosphere: That is, troposphere refers to regions extending from the surface to 200 mbar, the stratosphere 200 mbar to 1 mbar, the mesosphere 1 mbar to 0.001 mbar, and the thermosphere 0.001 mbar to $5{\ensuremath{\times 10^{-6}}}$ mbar. The latter two layers comprise the so-called MLT region (see Section 1). Specifically, the mesosphere extends from ${\sim}1$ mbar to the mesopause (roughly the homopause, at ${\sim}0.001$ mbar), where temperature minima caused by CO$_2$ radiative cooling are typically found. Above the mesosphere, the thermosphere encompasses the heterosphere zone, in which diffusion plays an increasingly greater role and the chemical composition of the atmosphere varies in accordance with the atomic and molecular mass of each species. This region (i.e., the thermosphere) extends from the mesopause to model-top at pressures of $5.1 {\ensuremath{\times 10^{-6}}}$ hPa (${\sim}145$ km). Shifts in atmospheric structure occur and depend on the planetary rotation period and the bolometric stellar flux.
We configure the described model components to simulate the atmospheres of tidally-locked (trapped in 1:1 spin-orbit resonance) planets across a range of stellar spectral energy distributions, bolometric stellar fluxes, and planetary rotation periods (Table 1). We construct stellar spectral energy distributions (SEDs) using the PHOENIX synthetic spectra code assuming stellar metallicities of \[Fe/H\] = 0.0, alpha- enhancements of \[$\alpha$/M\] = 0.0, surface gravities log g = 4.5, and stellar effective temperatures ($T_{\rm eff}$) of 2600 K (TRAPPIST- 1-like), 3000 K (Proxima Centauri-like), 3300 K (AD Leo-like), and 4000 K (late K-dwarf). All synthetic stellar SEDs are assumed to be in states of quiescence. The corresponding rotation periods obeys Kepler’s 3rd law, as in @KopparapuEt2016ApJ, which is given by:
$$P_{\rm years} = \left[ \left( \frac{L_*/L_\odot}{F_p/F_\oplus}\right)^{3/4}\right] \left(M_*/M_\odot \right)^{1/2}$$
where $L/L_\odot$ is the stellar luminosity in solar units, $F_p/F_\oplus$ is the incident stellar flux in units of present-Earth flux (1360 W m$^{-2}$), and $M/M_\odot$ is the stellar mass in solar units.
We set the mass and radius of all our simulations to those of present-day Earth. We set the orbital parameters (obliquity, eccentricity, and precession) to zero. We use present Earth’s continental configuration and topography, which is a reasonable starting point as fully ocean-covered planets are unlikely to support an active climate-stabilizing carbon-silicate cycle and allow build-up of O$_2$ [@AbbotEt2012ApJ; @Lingam+Loeb2019AJ]. We place the substellar point stationary over the Pacific at 180$^{\rm o}$ longitude and turn off the quasi-biennial oscillation forcing in all simulations, as this prescription is based on observations of Earth.
We assume initially Earth-like preindustrial surface concentrations of gases N$_2$ (0.78 by volume), O$_2$ (0.21), CH$_4$ ($7.23 \times 10^{-7}$), N$_2$O ($2.73 \times 10^{-9}$), and CO$_2$ ($2.85 \times 10^{-4}$). The existence of an O$_2$-rich atmosphere implies active oxygenating photosynthesis on the surface[^1] H$_2$O$_v$ and O$_3$ are spatially and temporally variable gases but are initialized at preindustrial Earth values. The surface atmospheric pressure is 101325 Pa (1013.25 mbar). We use the native broadband radiation model of CAM4 and do not include new absorption coefficients as done in @KopparapuEt2017ApJ due to the extensive effort required to derive new coefficient values for CH$_4$, N$_2$O, and other IR absorbers included in the chemical transport model, which is beyond the scope of this work.
WACCM simulations are run at horizontal resolutions of $1.9\degree \times 2.5\degree$ (latitude by longitude) with 66 vertical levels, model top of $6 \times 10^{-6}$ hPa (150 km), and a model timestep of 900 seconds. We increase the total stellar flux by intervals of $0.1~F_p/F_\oplus$ and modify the rotation period according to Equation 1. We follow previous work [@YangEt2014ApJL; @KopparapuEt2016ApJ] and assume that the maximum flux for which a planet can maintain thermal equilibrium, i.e., top of atmosphere (TOA) radiation balance, defines the incipient stage of a runaway greenhouse. We refer to climatically-stable (i.e., in thermal equilibrium) simulations as converged simulations, while those that are climatically unstable (i.e., out of thermal equilibrium) are deemed to be in incipient runaway states. With the exception of Figure 1, all converged simulations have been run for 30 Earth years of model time and the results presented here are averaged over the last 10.
Note that our model simulation naming convention (Table 1) follows from simulated stellar flux and effective temperature: [*XXFYYT*]{}, where [*XX*]{} represents the total stellar flux (relative to the Earth’s) and [*YY*]{} represents the effective temperature of the host star. For instance, 13F26T represents an experiment in which the stellar flux is set to 1.3 $F_\oplus$ and the host star has an effective temperature of 2600 K.
Results {#sec:results}
=======
We present the first simultaneous 3D investigation of climate and atmospheric chemistry in temperate, moist greenhouse, and incipient runaway greenhouse atmospheres on synchronously-rotating planets. The results section is structured as follows: First, we examine the onset of runaway greenhouse conditions in our simulations. Next, we discuss moist greenhouse conditions and their associated climatic and chemical properties. We then investigate isolated changes in stellar spectral type and bolometric stellar flux. Specifically, we analyze the effects of different stellar spectral types, increased incident stellar flux, and UV radiation on our results. Next, we show prognostic water vapor and hydrogen mixing ratios and discuss new escape rates calculated with interactive chemistry. Lastly, we present simulated atmospheric transmission spectra and secondary eclipse thermal emission spectra using our CCM results as inputs and discuss their observational implications.
{width="1.8\columnwidth"}
Climate Behaviors in Runaway Greenhouse States
----------------------------------------------
Runaway greenhouse states are energetically unstable climate conditions in which the net absorption of stellar radiation exceeds the ability of water vapor rich and thus thermally opaque atmospheres to emit radiation to space, as described by the Simpson-Nakajima limit [@NakajimaEt1992JRG]. A subset of our simulations reaches this runaway threshold, which we define as the innermost boundary of the HZ. We illustrate incipient runaway behavior and contrast it with a climatically-stable case by providing timeseries of TOA radiation balance and surface temperature from three representative simulations, initialized from the same climatic state, i.e., global-mean surface temperature (T$_s$) of ${\sim} 291$ K (Figure \[fig:eb\]). From this initial state, simulations diverge, largely according to the intensity of stellar flux and cloud development. For example, in the climatically-stable 15F33T simulation (Figure \[fig:eb\], red curves), the TOA radiation balance begins negative, but stabilizes about zero as the day-side cloud fraction initially decreases, but then oscillates near 95% coverage. This radiation balance allows the global-mean T$_s$ to stabilize at 283 K. In contrast to this climatically stable pathway, “incipient runaway greenhouse" [@WolfEt2019ApJ] conditions are simulated for planets at both lower (12F26T) and higher (20F40T) stellar fluxes around late M-dwarf (12F26T) and late K-dwarf (20F40T) stars. In the 20F40T simulation, the planet rapidly transitions into an incipient runaway greenhouse state as the stellar flux is sufficiently high that it causes the collapse of the substellar cloud-albedo shield (Figure \[fig:eb\], gold curves). No equilibrium T$_s$ is achieved in this simulation. Similarly, in the lower flux 12F26T case (Figure \[fig:eb\], blue curves), the global-mean T$_s$ does not achieve an equilibrium. Initially global-mean T$_s$ decreases similar to 15F33T, but the higher rotation rate reduces the dayside cloud shield, allowing the TOA radiation imbalance to turn positive, which leads to the incipient stage of a runaway thermal state
{width="1.9\columnwidth"}
Climate and Chemistry near the IHZ: Temperate & Moist Greenhouse States
-----------------------------------------------------------------------
While runaway greenhouse states delineate the optimistic IHZ, both temperate and moist greenhouse climates may be situated at or near the IHZ limit. In this study, we are primarily interested in the chemistry and climatic conditions habitable planets located at the IHZ. To identify this boundary, four host star type simulations were run with incremental ($0.1~F_p/F_\oplus$) increases in stellar flux (Table 1). Simulations not pushed into the incipient runaway state described in Section 3.1, i.e., simulations with flux $0.1~F_p/F_\oplus$ less than runaway conditions, define our “IHZ limit" cohort: 10F26T (temperate), 11F30T (temperate), 16F33T (moist greenhouse), and 19F40T (moist greenhouse). Temperate atmospheres have low, Earth-like stratospheric water vapor content (typically $\la 1{\ensuremath{\times 10^{-5}}}$ mol mol$^{-1}$) and global-mean T$_s$ below 285 K (Table 1). Moist greenhouse atmospheres emerge when the stratospheric H$_2$O$_v$ mixing ratio are sufficiently high, i.e., $\ga 3{\ensuremath{\times 10^{-3}}}$ mol mol$^{-1}$ such that water-loss via diffusion-limited escape could occur at a geologically significant rate in the thermosphere [@Kasting1988Icarus]. If the water-loss is sufficiently slow (i.e., $\ga 5$ Gyrs), then rapid desiccation of a planet’s oceans is prevented and its surfaces can remain habitable.
Amongst our simulations, only IHZ limit climates around early-to-mid M-dwarfs ($T_{\rm eff} = 3300$ and 4000 K) meet the moist greenhouse criterion (Figure \[fig:summary\]b, 16F33T and 19F40T). Simulations around these stars but with lesser stellar flux, i.e., 15F33T and 17F40T, do not achieve sufficient stratospheric H$_2$O$_v$ to place them in the moist greenhouse regime (Figure \[fig:summary\]b). Simulations 16F33T and 19F40T have stratospheric H$_2$O$_v$ mixing ratios of $2.05{\ensuremath{\times 10^{-3}}}$ and $1.179{\ensuremath{\times 10^{-2}}}$ mol mol$^{-1}$ respectively (Table 1), yet their global mean T$_s$ does not exceed 310 K (Figure 2a), which indicates that the surface may be habitable despite the high stratospheric water vapor content. Temperate IHZ limit climates (experiments 10F26T and 11F30T) simulated around late M-dwarfs ($T_{\rm eff} = 2600$ and 3000 K) do not enter the moist greenhouse regime with incremental ($0.1~F_p/F_\oplus$) increases in stellar flux (Figure \[fig:summary\], red and gold). Instead, they abruptly transition into incipient runaway greenhouse states (e.g., 12F26T; Figure \[fig:eb\], blue curve). Similar conclusions were reached by @KopparapuEt2017ApJ using CAM4 with updated H$_2$O$_v$ absorption coefficients but excluding interactive chemistry.
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We demonstrate differences in climate and chemistry amongst the IHZ limit simulations across the four host star types by showing contour plots of surface temperature, high cloud fraction, upper atmospheric wind fields, TOA outgoing longwave radiation (OLR), and ozone mixing ratios averaged between $10^{-4}$ and 100 mbar (Figure \[fig:map\]). These results exhibit the convolved effects of stellar $T_{\rm eff}$, incident flux, and planetary rotation, as all three parameters are correlated. Following @ChenEt2018ApJL, we define a metric to assess the day-to-nightside gas mixing ratio contrasts:
$$r_{\rm diff} = \frac{r_{\rm day} - r_{\rm night}}{r_{\rm globe} }$$
where $r_{\rm day}$ is the dayside hemispheric mixing ratio mean, $r_{\rm night}$ the nightside mean, and $r_{\rm globe}$ the global mean. The degree of anisotropy is loosely encapsulated in this parameter, which is shown in Figures \[fig:map\] and \[fig:map\_uv\] and will be discussed throughout the paper.
Substantial differences in surface temperature distributions can be found amongst the four IHZ limit simulations. With increasing stellar $T_{\rm eff}$, day-to-nightside T$_s$ gradients decrease (Figures \[fig:map\]a-d). This is caused by increased day-to-nightside heat redistribution at higher incident fluxes. Consistent with previous GCM studies that exclude interactive chemistry (e.g., @KopparapuEt2017ApJ), we find that on slowly rotating planets, the weaker Coriolis force allows formation of optically thick substellar cloud decks by way of buoyant updrafts (Figures \[fig:map\]e-h). In addition, meridional overturning cells expand to higher latitudes when the Rossby radius of deformation approaches the diameter of the planet [@DelGenioEt1993Icar], which decrease the pole-to-equator temperature gradient. These two consequences (i.e., formation of dayside clouds and reduction of temperature gradient) of slow rotation allow planets around early M-dwarfs to maintain habitable climates at higher fluxes.
Atmospheric dynamics regulate cloud patterns, circulation symmetry, and transport of airmasses on a planet. For slowly-rotating cases, substellar OLR is reduced by the high opacity of deep convective cloud decks, which induce a strong warming effect (experiments 16F33T and 19F40T; Figure \[fig:map\]k-l). As we move from 16F13T, to 11F30T, then to 10F26T, the dynamical state gradually transitions from divergent circulation to one dominated by tropical Rossby waves and zonal jets. The resultant elevated high-to-low latitude momentum transport can be seen in the streamlines and OLR patterns (Figure \[fig:map\]i-j; see also @Matsuno1966 [@Gill1980]), and is likely caused by shear between Kelvin and Rossby waves [@Showman+Polvani2011ApJ]. Comparing our results to the circulation regimes studied by @Haqq-MisraEt2018ApJ, we find that simulations 16F33T and 19F40T are situated in their slow rotating regime (Figure \[fig:map\]i-j), in which thermally driven radial flows dominate. Simulation 10F26T (Figure \[fig:map\]i) is consistent with the rapid rotator characterized by strong zonal jet streams and a weaker substellar rising motion, while 11F30 belongs in the so-called Rhines rotator regime, in which the OLR and radial flows are shifted eastward by the emergence of turbulence (Figure \[fig:map\]j). For the simulation in latter Rhines rotator regime, the meridional extent of Rossby waves is just under the planetary radius value, thus horizontal flow is a combination of superrotation and thermal-driven circulation (similar in terms of dynamical behavior to the “transition regime" found by @CaroneEt2015MNRAS).
An additional consideration, allowed by the coupling of chemistry and dynamics, is the role of stratospheric circulation in the transportation of airmasses (and thus photochemically produced species and aerosols). Atmospheres around late M-dwarfs (simulations 10F26T and 11F30T) display superrotation that induces standing tropical Rossby waves, thereby confining the majority of the produced ozone near the equator (Figure \[fig:oz\_trans\]a and b). @CaroneEt2018MNRAS explained this by the weakening of the extratropical Rossby wave and reduced efficiency of stratospheric wave breaking; here we confirm their hypothesis by directly accounting for ozone photochemistry and transport. In atmospheres with high circulation symmetry (simulations 16F33T and 19F40T), tropical jets are effectively damped. This leads to increased strength of stratospheric meridional overturning circulations (i.e., a thermally driven version of the Brewer-Dobson circulation) and that of the Walker circulation, which allows equator-to-pole and day-to-nightside dispersal of ozone (Figure \[fig:oz\_trans\]c and d). Lastly, ozone day-to-night mixing ratio contrasts ($r_{\rm diff}$) depend on both chemical (e.g., reaction with OH) and dynamical (e.g., strength of Rossby and Kelvin waves) factors and highlight the interplay between transport, photochemical, and photolytic processes (Figure \[fig:map\]m-p; see also @ChenEt2018ApJL).
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Temperate Atmospheres: Effects of Changes in Stellar SED
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Host star spectral-type can influence attendant planet atmospheres through changes in stellar $T_{\rm eff}$ and planetary rotation period, as the latter two variables are correlated through Kepler’s third law. Our coupled CCM simulations demonstrate that the primary climatic effects of different input stellar SEDs are modulations in greenhouse gas radiative forcing, while photochemistry (e.g., driver of water photolysis) is not substantially impacted.
The red-shifted spectra of low-mass stars have consequences on climate and chemistry by way of an increased water vapor greenhouse effect and reduction in dayside cloud cover (Figure \[fig:sed\]a-c). Specifically, greater IR absorption by atmospheres around stars with $T_{\rm eff} = 2600$ K and 3000 K increases both atmospheric temperature and the amount of precipitable water, and decrease radiative cooling efficiency aloft (Figure \[fig:sed\]a and b, red curve). Reduction in the efficiency of radiative cooling and dayside cloud fractions lead to the water vapor greenhouse effect offsetting that of cloud albedo, and results in higher T$_s$ for 10F26T compared to simulations around early M-dwarfs (Figure \[fig:sed\]b). Moreover, potential increased concentrations of other greenhouses gases such as CH$_4$ and N$_2$O on late M-dwarf planets would also contribute to increased T$_s$ [@SeguraEt2005AsBio; @RugheimerEt2015ApJ].
{width="2.1\columnwidth"} {width="2.1\columnwidth"}
Further insight into the effects of different stellar $T_{\rm eff}$ can be observed in the global-mean vertical profiles (Figure \[fig:sed\]e-h). Below 80 km ($10^{-2}$ mbar), atmospheric temperature increases monotonically with decreasing stellar $T_{\rm eff}$ (Figure \[fig:sed\]e). This relationship exists because longwave absorption increases and Rayleigh scattering decreases with the redness of the host star. Above 80 km ($10^{-2}$ mbar), however, the dependence of temperature on stellar $T_{\rm eff}$ is reversed due to the increasingly important role of O$_2$ photodissociation by shortwave photons at higher altitudes (Figure \[fig:sed\]e). Both H$_2$O$_v$ shortwave heating and strength of vertical advection increase with lower $T_{\rm eff}$ due to the higher NIR fluxes. At pressures less than $10^{-4} - 10^{-5}$ mbar, temperatures rise rapidly (Figure \[fig:sed\]e) by way of thermospheric O$_2$ and O absorption of soft X-ray and EUV, while water vapor mixing ratios decline due to photodissociation to H, H$_2$, and OH (Figure \[fig:sed\]f).
Ozone photochemistry is modulated by incident UV flux, chemical reaction pathways, and ambient meteorological conditions ($P$, $T$). Our predicted ozone mixing ratios above 70 km ($10^{-1}$ mbar) reduce with decreasing stellar $T_{\rm eff}$ (Figure \[fig:sed\]g). Elevated OH production through water vapor photosys leads to greater photochemical removal of O$_3$ by OH. OH destruction of ozone is maximized near the boundary layer for the 10F26T experiment as increased surface temperatures (due to lower substellar albedos) leads to larger H$_2$O$_v$ inventories. At pressures less than $10^{-2}$ mbar, ozone mixing ratios rise as mean free paths between molecules increase dramatically with altitude.
Atmospheric hydrogen is primarily produced from the photolysis of high-altitude water vapor. At temperate conditions, H mixing ratios in both meridional and vertical profiles (Figure \[fig:sed\]d and Figure \[fig:sed\]h) are not substantially impacted by shifts in stellar SED, assuming quiescent stars. This is seen by the fact that all four simulations have close to zero H mixing ratios until ${\sim}10^{-3}$ mbar, at which point they rise to ${\sim} 10^{-7}$ mol mol$^{-1}$ (Figure \[fig:sed\]h). Increased efficiency in water vapor photolysis above the mesosphere (${\sim}10^{-2}$ mbar) is evidenced by the exponential dependence of H mixing ratios on altitude. A transition in H mixing ratios above 80 km ($10^{-2}$ mbar) is caused by the rapid increase in water vapor photolysis rates and hence stronger dependence on pressure altitude. We should point out however, that the seemingly minor effects of stellar $T_{\rm eff}$ stem from our choice of input SED-types. The PHOENIX stellar model data we employed are inactive in the UV and EUV bands, regardless of the spectral type. As many M-dwarfs are active in the Ly-$\alpha$ line fluxes ($115 < \lambda < 310$ nm; @FranceEt2013ApJ), we next explore how changes in these assumptions affect our findings (Section 3.5).
{width="2.1\columnwidth"} {width="2.1\columnwidth"}
Moist Greenhouse Atmospheres: Effects of Increasing Stellar Flux
----------------------------------------------------------------
Increasing stellar flux can affect both climatic and photochemical variables. For instance, we find that both atmospheric temperature and water vapor mixing ratios increase monotonically with increasing incident flux as reported by previous studies (e.g., @KastingEt1984Icarus [@KastingEt2015ApJL]), whereas photochemically important species and their derivatives such as ozone and hydrogen display nonlinear behavior.
Water vapor concentrations and surface climate are both strong functions of stellar flux. With increasing stellar flux at fixed stellar $T_{\rm eff}$ (= 4000 K) we find that water vapor quickly becomes a major constituent from the stratosphere (${\sim}100$ mbar) to the thermosphere ($5{\ensuremath{\times 10^{-5}}}$ mbar). For example, total precipitable water increases by a factor of 5 for every interval change of incident flux (Figure \[fig:flux\]c). However, the corresponding surface temperature rises much more gradually, wherein a change in stellar flux (from experiment 10F40T to 17F40T, or from $F_p = 1.0~F_\oplus$ to $F_p = 1.6~F_\oplus$) only causes an average T$_s$ increase of ${\sim}20$ K in the substellar hemisphere due to stabilizing cloud feedbacks (Figure \[fig:flux\]a).
The rapid rise in water vapor mixing ratios in the upper atmosphere ($\ga 1$ mbar) can be attributed to positive feedbacks between flux, H$_2$O$_v$ IR heating, and vertical motion. H$_2$O$_v$ NIR absorption and cloud feedbacks are amplified by increased stellar flux and incident IR. With increased water vapor, the tropospheric lapse rate decreases and the moist convection zone expands. These two shifts lead to displacement of the cold trap to higher altitudes (Figure \[fig:flux\]e) and thus more efficient H$_2$O$_v$ vertical advection. Greater H$_2$O$_v$ vertical transport increases its concentration in the upper atmosphere, increasing the strength of greenhouse effect and atmospheric temperature.
While we find considerable increases in H$_2$O$_v$ mixing ratios in the stratosphere, the increase in surface temperatures is less dramatic. For instance, in experiment 19F40T, the stratospheric water vapor mixing ratio has reached $10^{-2}$ mol mol$^{-1}$ (Figure \[fig:flux\]f) yet the global-mean surface temperature is still just 300 K (Figure \[fig:flux\]e). This result agrees with previous findings of the so-called habitable moist greenhouse in which the stratosphere becomes highly saturated while the troposphere is stabilized by optically thick clouds [@KopparapuEt2017ApJ]. In habitable moist greenhouse states (e.g., 19F40T; Figure \[fig:flux\]e-h), surface habitability, to the first order, depends on the rate of water escape from the upper atmosphere. If water escape is sufficiently slow in these conditions, then insofar as the surface climate remains stable, the planet could host life on a timescale that raises the possibility of remote detection.
Stellar flux can indirectly affect ozone photochemistry via an increase in atmospheric water vapor dissociation and changes in ambient conditions such as atmospheric temperature. Vertical profiles of ozone show a minimum in the ozone mixing ratio in the highest total incident flux simulation (Figure \[fig:flux\]g, 19F40T, red curve), and a maximum for the simulations receiving the least (Figure \[fig:flux\]g, 10F40T, blue curve). The resultant thinner ozone layer at high fluxes is caused by the increased removal rate via photochemical reactions with dayside HO$_{\rm x}$ and NO$_{\rm x}$ (primarily OH and NO species) for simulation 19F40T. Reduction in ozone between the boundary layer and altitude at 5.0 mbar is due to photochemical removal by OH, while the ozone maximum is shifted to 1.0 mbar (Figure \[fig:flux\]g, gold curve), indicating a change in the location of highest gross ozone production rate.
Elevated water vapor mixing ratios lead to more atomic hydrogen via photodissociation. At temperate conditions, the most efficient altitude of water vapor photolysis is at pressure levels less than $10^{-3}$ mbar, as implied by the H mixing ratio shift (Figure \[fig:flux\]h). With higher incident fluxes however, e.g., 19F40T, the inflection of H mixing ratios change with altitude indicating higher photodissociation efficiencies with height. Notably, we find that our prognostic H mixing ratios are almost never twice the amount of H$_2$O$_v$, as assumed in previous studies (e.g., @KastingEt1993Icarus [@KopparapuEt2013ApJ; @KopparapuEt2017ApJ]). This suggests that previous climate modeling works on the moist greenhouse state have overestimated water-loss rates. Lesser simulated H mixing ratios are the result of a variety of processes, including the oxidation of H by O$_2$ and photochemical shielding by O$_3$, CH$_4$, and N$_2$O. In the next section we explore the dependence of photochemistry on stellar UV radiation inputs. With simulated hydrogen, we then provide revised calculations of water loss and estimate the longevity of our exoplanetary oceans.
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Dependence on Stellar UV Activity
---------------------------------
Stellar UV radiation can affect atmospheric chemistry, photochemistry, surface habitability, and based on our findings, observability. Here, we investigate the 3D effects of different stellar UV activity assumptions on tidally-locked planets with Earth-like atmospheres. To test the effects of UV radiation we run simulations in which the UV bands ($\lambda < 300$ nm) of the fiducial $T_{\rm eff} = 4000$ K (experiment 19F40T) star are swapped with those of (a) active VPL AD Leonis data (19FADLeoUV; @SeguraEt2005AsBio) and (b) UV data obtained by doubling the Solar UV spectrum (19FSolarUV; @lean1995reconstruction). Active stellar SEDs are joined with the VIS/NIR portion of the spectra beyond 300 nm by linearly merging the last UV datapoint with the first optical ($\lambda > 300$ nm) datapoint in the PHOENIX stellar model. The aim in this section is to test how including UV activity could alter the conclusions in Section 3.1 Only changes in the UV wavelengths of the SEDs are tested as we are primarily interested in the isolated effects of UV photons, rather than those in other wavelengths. Follow-up work will make use of HST + XMM/Chandra-based M dwarf spectra with observed UV bands from @FranceEt2013ApJ, @YoungbloodEt2016ApJ, and @LoydEt2016ApJ.
UV radiation may drive changes in atmospheric dynamics, in addition to atmospheric chemistry. With changes in our fiducial late K-dwarf ($T_{\rm eff} = 4000$ K) SED, we find increases in the vertical velocities at the substellar point from 0.10 m s$^{-1}$ (inactive star;), to 0.14 m s$^{-1}$ ($2\times$ Solar UV), then to 0.18 m s$^{-1}$ (AD Leo UV; Figure \[fig:wind\_uv\]a-c), indicating stronger ascent of substellar updrafts. In addition, horizontal and thus day-to-nightside transport are enhanced as evidenced by the higher wind velocities. While mesospheric (1 mbar) zonally-averaged wind speeds forced by the quiescent M-dwarf are modest ${\sim}$25 m s$^{-1}$ (Figure \[fig:wind\_uv\]a), equatorial winds driven by elevated day-to-nightside temperature gradients can reach as high as 60 m s$^{-1}$ for the simulations forced by the AD Leo UV SED (Figure \[fig:wind\_uv\]c). Near-surface winds converge toward the substellar point due to large-scale updrafts in all three cases, but are not significantly altered by changes in UV radiation (Figure \[fig:wind\_uv\]d-f).
Global distributions of photochemically important species and their byproducts are also affected by stellar UV activity (Figure \[fig:map\_uv\]). Substellar updrafts (due to radiative heating) of chemical constituents and antistellar downdrafts (due to radiative cooling) should result in higher ozone mixing ratios on the dayside. For the simulations forced by the $2\times$ Solar UV and AD Leo UV SED (Figure \[fig:map\_uv\]b-c), this effect is heightened by increased UV in the wavelengths responsible for ozone production (shortward of 220 nm). In contrast, lower ozone production rates on the quiescent simulation lead to reduced dayside ozone (Figure \[fig:map\_uv\]a).
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The amount of dayside OH and H is directly related to the input UV, with chemical transport playing a small role. With a higher UV radiation than that received by the baseline, the OH distributions become increasingly concentric (19FSolarUV: 192% and 19FADLeoUV: 201%; Figure \[fig:map\_uv\]e-f) due to its short lifetime and the greater contribution from water vapor photodissociation. In contrast at higher UV levels, H mixing ratio distributions begin to lose their concentric shapes and reduced $r_{\rm diff}$ (19FSolarUV: 32.9%, and 19FADLeoUV: 28.8%; Figure \[fig:map\_uv\]g-i). These different responses are explained by the enhanced dispersal of H by atmospheric transport as seen by the slight eastward shift of H mixing ratio distribution in the AD Leo UV case (Figure \[fig:map\_uv\]i). Increased horizontal advection migrates the effects of enhanced dayside photolytic removal, reflected in the decreasing $r_{\rm diff}$ of hydrogen with greater UV input. Note that our inactive stellar SEDs result in more reduced dayside ozone than those reported by @ChenEt2018ApJL, which likely stems from the lack of a fully resolved stratosphere-MLT region (e.g., Brewer-Dobson circulation) in the low-top out-of-the-box version of CAM4. These discrepancies in day-to-nightside chemical gradients illustrate the need for model inter-comparisons of exoplanetary climate predictions (e.g., @YangEt2019ApJb).
Unsurprisingly, the three different UV radiation schemes produce atmospheric temperature profiles that are substantially different (Figure \[fig:vert\_uv\]a). Elevated incident UV fluxes translate to higher shortwave heating and thus atmospheric temperatures due to FUV and EUV absorption by atomic and molecular oxygen. A “harder" UV spectrum is also able to penetrate more deeply into the atmosphere.
Apart from different thermal structures, we find orders of magnitude differences in H$_2$O$_v$, O$_3$, and H mixing ratio profiles (Figure \[fig:vert\_uv\]b, c, and d)$-$ indicating that stellar activity can have strong ramifications for water loss and atmospheric chemistry for moist greenhouse atmospheres. Enhanced AD Leo EUV and UV induced shortwave heating increases stratospheric and mesospheric (between 100 and 1 mbar) temperatures and water vapor mixing ratios (red curve; Figure \[fig:vert\_uv\]a and b). However, the altitude at which photolysis is maximized moves lower due to the more energetic shortwave photons (Figure \[fig:vert\_uv\]b). For ozone mixing ratios, production outpaces destruction resulting in a thicker ozone layer for simulations around more active stars (gold and red curves, Figure \[fig:vert\_uv\]c), while the upper ozone layers remain desiccated.
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Finally, different input UV assumptions also alter the altitude at which water vapor photolysis is most efficient, which can determine the thermospheric H mixing ratio and hence water escape rate. Without stellar activity, the H mixing ratios remain low $7.27{\ensuremath{\times 10^{-5}}}$ mol mol$^{-1}$ (red curve, Figure \[fig:vert\_uv\]d). With the inclusion of stellar activity, both altered UV simulations are pushed into the true moist greenhouse regime with H mixing ratios of $1.04{\ensuremath{\times 10^{-3}}}$ and $7.45{\ensuremath{\times 10^{-2}}}$ mol mol$^{-1}$ respectively (gold and red curves, Figure \[fig:vert\_uv\]d). This implies that although planets around inactive stars may only experience minor water loss, both active Solar and AD Leo SEDs could cause attendant planets to suffer rapid water loss.
Inclusion of stellar UV activity may modify conclusions regarding host star spectral type dependent IHZ boundaries, as moist greenhouse atmospheres around early M-dwarfs ($T_{\rm eff} \sim 4000 $K) are more vulnerable to photodissociation than temperate climates around late M-dwarfs ($T_{\rm eff} \sim 2600$ K). With the present simulations, it is challenging to further this possibility as our grid of stellar effective temperature values are rather coarse (i.e., only four $T_{\rm eff}$s between 2600 and 4000 K).
![\[fig:survival\] Ocean survival timescale as a function of stellar $T_{\rm eff}$ (2600, 3000, 3300, 4000 K), UV activity ($2 \times$ Solar and AD Leo), and their corresponding global-mean T$_s$. Our results suggest that only simulations around active M-dwarfs enter the classical moist greenhouse regime as defined by @KastingEt1993Icarus. Blue shading indicates timescales less than the age of the Earth.](survival.pdf){width="1.1\columnwidth"}
Prognostic Hydrogen & Ocean Survival Timescales
-----------------------------------------------
The ability of a given planet to host a viable habitat is linked to the survivability of its ocean, i.e., the so-called “ocean loss timescale". Conventional estimates of the ocean loss timescale have used 1D climate models and GCMs that rely on prescribed H mixing ratios calculated by doubling their model-top H$_2$O$_v$. Here we reassess previous estimates by using directly simulated H mixing ratios and thermospheric temperature profiles drawn from our CCM simulations. We find that our ocean survival timescales are substantially higher than previously published estimates for quiescent stars, and are critically dependent on the stellar activity level. We demonstrate this by estimating water loss rates with Jeans’ diffusion-limited escape scheme. While an over-simplification due neglect of hydrodynamics, this first order estimate is typically used to interpret climate model results of moist greenhouse atmospheres (e.g., @KastingEt1993Icarus [@KopparapuEt2013ApJ; @KopparapuEt2017ApJ; @Wolf+Toon2015JGR]). With our prognostic hydrogen mixing ratios at each stellar $T_{\rm eff}$ and incident flux combination (Figure 2), we can calculate new escape rates of hydrogen [@Hunten1973JAS]:
$$\Phi(H) \approx \frac{b Q_{\rm H}}{H}$$
where $Q_{\rm H}$ is thermospheric hydrogen mixing ratio (model top), $H$ is the atmospheric scale height $kt/mg$, and $b$ is the binary Brownian diffusion coefficient given by:
$$b = 6.5 \times 10^7 T_{\rm thermo}^{0.7}$$
where $T_{\rm thermo}$ is the thermospheric temperature of the atmosphere (taken at 100 km altitude).
We find that IHZ planets with Earth-like atmospheric compositions experiencing water loss should be more resilient to desiccation than previously reported. For example, all simulations around inactive stars have ocean survival timescales well above 10 Gyrs (Figure \[fig:survival\]), even for those with H$_2$O$_v$ mixing ratios above $3{\ensuremath{\times 10^{-3}}}$ mol mol$^{-1}$ (i.e., classical moist greenhouse). With realistic UV SED however, the oceans are predicted to be lost quickly ($< 1$ Gyr) via the molecular diffusion of H to space. This result stands in contrast to previous estimates using diagnostic H mixing ratios to calculate the escape rates, finding much shorter ocean loss timescales across all host star spectral types (see e.g., Figure 5 in @KopparapuEt2017ApJ). Clearly, a careful assessment of a star’s activity level is critical for determining whether planets around M-dwarfs will lose their oceans to space.
[cccccc]{}
K16 & 2600 & N/A & N/A & ${\sim} 1.2$\
K16 & 3000 & N/A & N/A & ${\sim}1.4$\
K16 & 3300 & 276 & $4 \times 10^{-5}$ & 1.65\
K16 & 4000 & 295 & $6 \times 10^{-3}$ & 1.9\
K17 & 2600 & 301 & $6 \times 10^{-5}$ & 1.0\
K17 & 3000 & 280 & $5 \times 10^{-4}$ & 1.15\
K17 & 3300 & 294 & $7 \times 10^{-3}$ & 1.25\
K17 & 4000 & 303 & $1 \times 10^{-2}$ & 1.5\
Bin18 & 2550 & 285 & $2.1 \times 10^{-3}$ & 0.9\
Bin18 & 3050 & 279 & $1.1 \times 10^{-4}$ & 1.0\
Bin18 & 3290 & 288 & $9 \times 10^{-4}$ & 1.15\
Bin18 & 3960 & 309 & $3.9 \times 10^{-3}$ & 1.35\
This Study & 2600 & 282 & $5 \times 10^{-6}$ & 1.0\
This Study & 3000 & 278 & $7 \times 10^{-6}$ & 1.1\
This Study & 3300 & 298 & $2 \times 10^{-5}$ & 1.6\
This Study & 4000 & 301 & $1 \times 10^{-2}$ & 1.9\
Discussion
==========
This study builds upon previous efforts to study planets near the IHZ, but with the added complexity of interactive 3D photochemistry and atmospheric chemistry, and by self-consistently simulating the atmosphere into the lower thermosphere ($5{\ensuremath{\times 10^{-6}}}$ mbar). In comparison with studies that employed self-consistent stellar flux-orbital period relationships, our runaway greenhouse limits are further out (from the respective host stars) than those of @KopparapuEt2016ApJ, but closer in than those of @KopparapuEt2017ApJ and @BinEt2018EPSL. For example, @KopparapuEt2016ApJ found that the critical flux threshold for thermally stable simulation orbiting a 3000 K star occurs at ($F_{\rm crit}$) ${\sim} 1.3 F_\oplus$, which is approximately 0.2 $F_\oplus$ higher than predicted in this study (Table 2). This discrepancy may be attributable to (i) the inclusion of ozone and its radiative effects in WACCM and (ii) the presence of non-condensable greenhouse gas species. While @KopparapuEt2016ApJ include 1 bar of N$_2$ plus 1 ppm of CO$_2$, this study includes additional modern Earth-like CH$_4$ and N$_2$O concentrations, thus yielding IHZ limits that are further away from the host star in comparison to @KopparapuEt2016ApJ. Note that both studies use the same radiative transfer scheme, cloud physics, and convection scheme. Simulated climates around stars with higher $T_{\rm eff}$ show much smaller differences stemming from lack of ozone heating and reduced degree of inversion, leading to comparable Bond albedos at the inner edge. However, greater disparities are found between our study and @KopparapuEt2017ApJ. For example, runaway greenhouse occurs at fluxes ($F_{\rm crit}$) ${\sim}0.35 F_\oplus$ higher for simulations across nearly all M-class spectral types (Table 2). This is explained by the finer spectral resolution in the IR and updated H$_2$O absorption by @Wolf+Toon2013Asbio and @KopparapuEt2017ApJ, which cause the stratosphere to warm and moisten substantially at a much lower stellar flux. Further, the native radiative transfer of CAM4 is shown to be too weak, both in the longwave and shortwave, with respect to water vapor absorption [@YangEt2016ApJ]. Differences between previous GCM calculations of the IHZ around Sun-like stars (e.g., CAM4; @Wolf+Toon2015JGR and LMD: @LeconteEt2013NATURE) can also be attributed to treatment of moist physics and clouds [@YangEt2019ApJb].
Our predictions of water loss and habitability implications show greater divergence from previous GCM studies$-$an outcome that is not unexpected given different initial atmospheric compositions and the addition of model chemistry. For quiescent stars, model top H mixing ratio predictions (hence water loss rates) presented here are orders of magnitude lower than previous work with simplified atmospheric compositions and without interactive chemistry. Implications of our results are favorable to the survival of surface liquid water for planets around quiescent M-dwarfs. For example, a recent study of the temporal radiation environment of the LHS 1140 system suggests that the planet receives relatively constant NUV ($177 - 283$ nm) flux $<2\%$ compared to that of the Earth [@SpenelliEt2019arXiv]. Our results suggest that LHS 1140b is likely stable against complete ocean desiccation due to the low UV activity of the host star, which bodes well for its habitability. Note however, that since our WACCM simulations assume a hydrostatic atmosphere, escape of H$_2$O$_v$ is only roughly approximated. Furthermore, during the super-luminous pre-main sequence stages of M-dwarfs (<100 Myr), high amounts of X-ray/EUV irradiation may cause an early desiccation and/or runway greenhouse of planetary atmospheres in the IHZ [@Luger+Barnes2015AsBio]. Even so, rocky planets around M-dwarfs may still possess active hydrological cycles through acquisition of cometary materials [@Tian+Ida2015NatGeo] as well as extended deep mantle cycling and the emergence of secondary atmospheres [@Komacek+Abbot2016ApJ]. Despite the super-luminous stages of M-dwarfs, existence of abundant water inventories is shown to be plausible using numerical TTV analysis, for example, in the TRAPPIST-1 system . For this pilot CCM study of the IHZ, we focus on main-sequence stars to be consistent with previous work modeling moist greenhouse states (e.g., @KastingEt1993Icarus [@KopparapuEt2017ApJ]). Further study is warranted examining stellar activity levels, including enhanced UV flux, time-dependent stellar flares, and sun-like proton events, and their roles in driving water-loss in habitable planet atmospheres.
Coupled CCMs, such as the one employed here, are advantageous for helping to improve/inform 1D model simulations. Previous work (e.g., ) have shown that the constant vertical diffusion coefficients assumed in 1D models (e.g., @HuEt2012ApJ [@Kaltenegger+2010ApJ]) may be invalid for different chemical compounds. Here, we find that the efficiency of global-mean vertical transport is not only species-dependent, but also host star dependent, as the magnitude of meridional overturning circulation and degree of vertical wave mixing are inherently tied to the planetary rotation rate and stellar $T_{\rm eff}$ (Figure \[fig:oz\_trans\]), both of which are constrained for synchronously rotating planets. Although a detailed comparison of the full set of our chemical constituent profiles with those of 1D is beyond the scope of this study, our results show that 3D CCMs could offer a basis to improve prediction of the 1D vertical distribution of photochemically important species (e.g., ozone). This task is especially important in the transition regime (i.e., for planets around stars with $2900 \la T_{\rm eff} \la 3400$ K), where stratospheric circulation patterns can shift substantially (i.e., emergence of anti-Brewer-Dobson cells; @CaroneEt2018MNRAS), leading to the further breakdown of a fixed vertical diffusivity assumption by 1D models.
In this study we focus solely on simulations with Earth-like ocean coverage and landmass distributions. However, water inventories vary with accretion and escape history. If a planet is barren (i.e., without a substantial surface liquid water inventory), then moist convection is inhibited and the water vapor greenhouse effect is suppressed. This can result in the delay of a runaway greenhouse [@AbeEt2011AsBio] and the onset of moist bistability wherein surface water could condense in colder reservoirs . A similar effect could occur if the substellar point is located above a large landmass instead of an ocean basin [@LewisEt2018ApJ]. This has relevance for atmospheric chemistry and habitability as it could suppress substellar moisture, leading to lower production rates of H in the thermosphere (above $10^{-2}$ mbar) and OH in the stratosphere (between 100 and 1 mbar).
Similarly, we fix our substellar point over an ocean basin and assume circular orbits locked in 1:1 spin-orbit resonance. In reality, the substellar point and stellar zenith angle could be nonstationary [@LeconteEt2015SCI] and planetary orbits could be eccentric without the stabilizing influence of a gas giant [@TsiganisEt2005NAT]. Nonstationary solar zenith angles could affect atmospheric circulation by modulating efficiency of moist convection, while eccentricity could drive planetary climate by as evidenced by Earth’s geological record [@HortonEt2012PPP]. However, these considerations are arguably secondary as existence of thermal tides are theoretical in the context of exoplanets and planets in the RV samples that have eccentricity greater than 0.1 are not common [@ShenEt2008ApJ]. Thus, we believe our simplification of fixed substellar point and perfect circular orbit should be valid for the majority of actual planetary systems.
Apart from surface climate, continued habitability is contingent upon the formation and retention of an ozone layer to shield excessive stellar UV-C ($200 < \lambda < 280$ nm) radiation and energetic particle bombardment. A thin ozone layer is hazardous to DNA due to surface exposure to high doses of UV radiation (e.g, @OMalley+Kaltenegger2017MNRAS). Alternatively, UV radiation may be critical in instigating complex prebiotic chemistry (e.g., @RanjanEt2017ApJ). As our simulations enter the moist greenhouse regime, we find that their atmospheres have orders of magnitude lower ozone mixing ratios than those in temperate climates (Figure \[fig:flux\]g), implying that the UV fluxes reaching the planetary surface may be high and therefore potentially threatening to surface life. Further, we find that both stellar UV activity and efficiency of day-to-nightside ozone transport could control the degree of UV flux penetration on the dayside surface. Thus, future constraints on the width of this “complex life habitable zone” (HZCL; @SchwietermanEt2019ApJ) will need to evaluate its dependencies on stellar flux, spectral type, and stellar activity, and will benefit from the use 3D CCMs.
Ultimately, CCM predictions of planetary habitability near the IHZ depend on the water accretion history [@RaymondEt2006Icarus], stellar XUV evolution [@Luger+Barnes2015AsBio], orbital parameters [@KilicEt2017ApJ], and the spatial distribution of surface water [@WayEt2016GRL; @KodamaEt2018JGR]. These considerations should be investigated in a more extensive CCM-based parameter space study (similar in spirit to e.g., @Komacek+Abbot2019ApJ) to better understand the climate, chemistries, and habitability potentials of IHZ planets around M-dwarfs.
{width="1.65\columnwidth"} {width="1.65\columnwidth"} {width="1.65\columnwidth"} {width="1.65\columnwidth"}
Observational Implications & Detectability
------------------------------------------
Follow-up characterization efforts by future instruments will likely target planets around M-dwarfs. To contextualize our CCM results within an observational framework, we calculate transmission spectra, secondary eclipse thermal emission spectra, and their simulated observations using the Simulated Exoplanet Atmosphere Spectra (SEAS) model (Zhan et al. in revision). SEAS is a radiative transfer code that calculates the attenuation of photons by molecular absorption and Rayleigh/Mie scattering as the photons travel through a hypothetical exoplanet atmosphere. The simulation approach is similar to previous work by @KemptonEt2017ApJL and @KemptonEt2009ApJ. The molecular absorption cross-section for O$_2$, H$_2$O, CO$_2$, CH$_4$, O$_3$, and H are calculated using the HITRAN2016 molecular line-list database [@GordonEt2017]. The SEAS transmission spectra are validated through comparison of its simulated Earth transmission spectrum with that of real Earth counterparts measured by the Atmospheric Chemistry Experiment (ACE) data set [@BernathEt2015GRL]. For more details on SEAS, please see Section 3.4 of Zhan et al., (in revision).
To compute atmospheric spectra, we use a subset of our CCM results: (i) Earth around the Sun, (ii) tidally-locked planet around an M8V star (10F26T), (iii) tidally-locked planet around a quiescent M2V star (19F40T), and (iv) tidally-locked planet around an active M2V star (19FADLeoUV). These simulations are chosen to illustrate the spectral feature differences between rapidly-rotating planets (10F26T and Earth-Sun) and slowly-rotating planets (19F40T and 19FADLeoUV). These simulations also demonstrate the consequences of different stellar UV activity levels on the spectral shapes: from low (10F26T and 19F40T) to mid (Earth-Sun), to high (19FADLeoUV) UV inputs. Lastly, planets orbiting late K-dwarfs, such as 19F40T, are argued to be at an advantage over those around mid-to-late M-dwarfs for biosignature potential [@Arney2019ApJL; @Lingam+Loeb2017ApJL; @Lingam+Loeb2018JCAP], and thus our primary focus on the host stars with $T_{\rm eff}$ of 4000 K.
CCM inputs for the SEAS model include: simulated temperatures and mixing ratios of gaseous constituents (i.e., N$_2$, CO$_2$, H$_2$O$_v$, O$_2$, O$_3$, CH$_4$, and N$_2$O), converted to 1D vertical time-averaged profiles. Transmission spectra were generated using the terminator mean values, while the emission spectra used the dayside-mean. SEAS assumes the premise of clouds, rather than using CCM results, in order to facilitate comparison with previous work [@MorleyEt2013ApJ; @HengEt2016ApJL]. For transmission, we explore three scenarios: uniform grey cloud at 10 mbar with 1.0 opacity (or optical depth), uniform grey cloud at 100 mbar with 0.5 opacity, and no clouds. For thermal emission, we assumed a 50% patchy grey cloud at 10 mbar with 0.5 opacity. These selections are made based on the fact that the atmosphere molecular absorption path length for stellar radiation passing through the rim of the planet atmosphere is ${\sim} 10\times$ the molecular absorption pathlength for blackbody radiation traveling from the surface of the planet. Parameterized clouds are used rather than CCM simulated clouds, as the former can set an upper/lower bound to our detection threshold and thus facilitate comparison with previous work. For instance, the 10 mbar with 1.0 opacity case mimics that of an “upper atmosphere uniform haze", e.g., @Kawashima+Ikoma2018ApJ. Moreover, GCM simulation of clouds remains an active area of research and thus simulated clouds come with inherent uncertainties. In future efforts, we will endeavor to include simulated clouds and their inherent uncertainties from a suite of GCMs into CCM-SEAS.
We validated our simulated Earth atmosphere transmission spectra with measurements of Earth’s atmosphere through the ACE program [@BernathEt2015GRL] and emission spectra with MODTRAN [@Berk2014SPIE]. Minor differences are due to exclusion of trace gases in the Earth atmosphere with a column average mixing ratio less than 1 ppmv.
Our simulated observations assume that the system is 2 pc from the observer. The planet of consideration is an Earth-sized and Earth mass planet. We also assume the use of a JWST-like, 6.5 m telescope, and with 25% throughput and an approximated noise multiplier of 50% which accounts for potentially unknown stellar variability and/or instrumental effects. While the spectral resolution of JWST is R = 100 at 1 - 5 $\mu$m (NIRSpec) and R = 160 at 5 - 12 $\mu$m (MIRI), in practice this “high" resolution (as compared to Hubble WFC3) is not prioritized for detection/distinction of H$_2$O$_v$ and O$_3$ molecules in exoplanet atmospheres due to the unique and broad spectra features these two molecules have (Zhan, et al. submitted). Therefore, detection can be optimized by using a larger bin width to increase the signal-to-noise ratio (SNR) of the H$_2$O$_v$ and O$_3$ molecules at the expense of reducing the resolution just enough to distinguish the molecules in consideration. We use the empirical formula of:
$$l_n = l_0\left(\frac{\lambda_n}{\lambda_0}\right)^m$$
where $l_0$ is 0.1 $\mu$m and $\lambda_0$ is 1 $\mu$m. i.e, the bin width $l_n$, at $\lambda_n$ = 10 $\mu$m, is = 1 $\mu$m. Note that we neglect the systematic noise of JWST, which is projected to be on the order of ${\sim}10$ ppm [@GreeneEt2016ApJ]. Implementation of the noise floor into the JWST simulator will slightly weaken our predicted features. Given the planet is at 2 pc and the high stratospheric H$_2$O concentration however, the H$_2$O features will likely still be detectable.
We find that the detection of H$_2$O$_v$ in moist greenhouse atmospheres can be achieved with high SNR confidence using NIRSpec. We also find that O$_3$ detection at the 9.6 $\mu$m window can potentially reach an SNR of 3 using MIRI LRS for planets around active stars (e.g., a Sun-like star or active M-dwarf). For the transmission spectra (Figure \[fig:obs1\][^2]), we compare our simulated results with those of @FujiiEt2017ApJ and @KopparapuEt2017ApJ, and find that our CCM-SEAS results are in agreement with the potential of characterizing water vapor features between 2.5 and 8 $\mu$m.
Slowly-rotating planets orbiting the IHZ of M2V stars (e.g., 19F40T and 19FADLeoUV) have moist greenhouse atmospheres and higher stratospheric water vapor content (red curves in Figure 5), which raises their signals in comparison to rapidly-rotating planets and stratospherically dry planets (e.g., 10F26T). In a similar vein, detection of water vapor can be achieved at high SNR confidence only for the moist greenhouse case, where H$_2$O$_v$ is four orders of magnitude more abundant than in the Earth’s stratosphere. Detection of water vapor is difficult for an Earth-like atmosphere in transmission as the majority of water vapor is concentrated below the tropopause. Transmission spectra of IHZ planets generated with 1D models (e.g., @LincowskiEt2018ApJ) do not display these prominent water vapor features as large-scale hydrological circulation processes are not resolved in 1D. We also find that the results of parameterized clouds match those of previous work [@StevensonEt2016ApJL; @WakefordEt2019arXiv] such that they inhibit detection of molecular features (Figure \[fig:obs1\]).
Oxygenated-atmospheres around active stars (i.e., Earth and 19FADLeoUV) should have more pronounced ozone features due to increased ozone production rates compared to their quiescent counterparts. We find that although the H$_2$O$_v$ features are not significantly altered by stellar UV activity, the O$_3$ features are. In addition, we test the effects of total integration time (10 hr vs 100 hr) on the predicted observations to explore the “most optimistic" scenario (Figure 11). These integration times translate to 4 to 7 and 40 to 70 transits respectively for a typical M-dwarf system. We find that detectability of the ozone feature is substantially improved with a higher integration time (Figure \[fig:obs1\]).
For secondary eclipse thermal emission spectra, we find that the 9.6 $\mu$m O$_3$ feature is located near the emission peak of the planet (${\sim}300$ K blackbody). Despite this finding, detection of this feature via secondary eclipse could be challenging for Earth-sized planets near 4000 K stars due to low SNR confidence. Potentially low SNRs are a result of the constraints of JWST’s cryogenic lifetime (necessary for mid-IR observations) of 5 years. Further, as the total number of transit hours that can accumulate is less than 100, the maximum achievable SNR confidence given our simulation parameters would be less than 3. For a super-Earth (e.g. $1.75~R_\oplus$) or a hotter (but non-habitable) planet around a late M-dwarf star however, secondary eclipse measurements of these features could be achievable [@KollEt2019arXiv; @MalikEt2019arXiv].
Conclusion {#sec:conclusion}
==========
In this study, we carried out numerical simulations of climate and chemistries of tidally-locked planets with a 3D CCM. Our results show that the maintenance of ozone layers, water photodissociation efficiency, and the onset of moist and incipient runaway greenhouse states depend on the incident stellar flux, stellar spectral-type, and importantly, UV radiation. By directly simulating photochemically important species such as ozone, we find that their abundance and distribution depend on the host star spectral type. The strength of the stratospheric overturning circulation, for example, increases with stellar $T_{\rm eff}$, leading to higher efficiency in the divergent transport of airmasses and thus photochemically produced species and aerosols.
Critically, we find that only climates around active M-dwarfs enter the classical moist greenhouse regime, wherein hydrogen mixing ratios are sufficiently high such that water loss could evaporate the surface ocean within 5 Gyrs. For those around quiescent M-dwarfs, hydrogen mixing ratios do not exceed that of water vapor. As a consequence, we find that planets orbiting quiescent stars have much longer ocean survival timescales than those around active M-dwarfs. Thus, our results suggest that improved constraints on the UV activity of low-mass stars will be critical in understanding the long-term habitability of future discovered exoplanets (e.g., in the TESS sample; @GuntherEt2019arXiv).
Stellar UV radiation has pronounced effects on atmospheric circulation and chemistry. Our 3D CCM simulations show that vertical and horizontal winds in the upper atmosphere (${\sim}1$ mbar) are strengthened with higher UV fluxes. Global distributions of O$_3$, OH, and H are the result of long-term averaged tradeoffs between dynamical, photolytic, and photochemical processes$-$resulting in substantially different day-to-nightside contrasts with incident UV radiation. Thus, coupling dynamics and photochemistry will be necessary to better understand the spatial distributions and temporal variability (e.g., @OlsonEt2018ApJL) of biogenic compounds and their byproducts.
Using a radiative transfer model with our CCM results as inputs, we show that detecting prominent water vapor and ozone features on M-dwarf planets during primary transits is possible by future instruments such as the JWST [@BeichmanEt2014PASP]. However, secondary eclipse observations are more challenging due to the predicted low SNR confidence.
H.C. thanks the R. K. Kopparapu, S. D. Domagal-Goldman, and the Exoplanet Journal Club at the University of Chicago for stimulating discussions as well as J. Yang and M. Lingam for helpful comments. H.C. and D.E.H. acknowledge support from the Future Investigators in NASA Earth and Space Science and Technology (FINESST) Graduate Research Award 80NSSC19K1523. E.T.W. thanks NASA Habitable Worlds Grant 80NSSC17K0257 for support. Z.Z. thanks the MIT BOSE Fellow program, the Change Happens Foundation, and the Heising-Simons Foundation for partial funding of this work. We acknowledge and thank the computational, storage, data analysis, and staff resources provided by the QUEST high performance computing facility at Northwestern University, which is jointly supported by the Office of the Provost, Office for Research, and Northwestern University Information Technology.
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[^1]: We note that some doubt has been raised as to whether biotic O$_2$ can build-up on planets around M-dwarfs due to the potential paucity of photosynthetically active radiation [@LehmerEt2018ApJ; @LingamEt2019MNRAS].
[^2]: The planet radius is independent of planetary system, and is easier for comparison with other theory work, as ppm depend on planet/star radius ratio.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We explore application of multi-armed bandit algorithms to statistical model checking (SMC) of Markov chains initialized to a set of states. We observe that model checking problems requiring maximization of probabilities of sets of execution over all choices of the initial states, can be formulated as a multi-armed bandit problem, for appropriate costs and rewards. Therefore, the problem can be solved using multi-fidelity hierarchical optimistic optimization (MFHOO). Bandit algorithms, and MFHOO in particular, give (regret) bounds on the sample efficiency which rely on the smoothness and the near-optimality dimension of the objective function, and are a new addition to the existing types of bounds in the SMC literature. We present a new SMC tool—[[[HooVer]{}]{}]{}—built on these principles and our experiments suggest that: Compared with exact probabilistic model checking tools like Storm, [[[HooVer]{}]{}]{} scales better; compared with the [*statistical*]{} model checking tool PlasmaLab, [[[HooVer]{}]{}]{} can require much less data to achieve comparable results.'
author:
- |
Negin Musavi$^1$, Dawei Sun$^1$, Sayan Mitra$^1$, Geir Dullerud$^1$, and Sanjay Shakkottai$^2$\
[{nmusavi2,daweis2,mitras,dullerud}@illinois.edu]{}\
${}^1$University of Illinois at Urbana Champaign\
[email protected]\
${}^2$University of Texas at Austin
bibliography:
- 'references.bib'
- 'sayan1.bib'
title: |
Optimistic Optimization for Statistical Model Checking with Regret Bounds\
Extended Abstract
---
Introduction {#sec:intro}
============
The multi-armed bandit problem is an idealized mathematical model for sequential decision making in unknown random environments and it has been used to study exploration-exploitation trade-offs. In the problem setup, each arm $x \in {\mathcal{X}}$ of the bandit is associated with a cost $\lambda_x$ of playing and an unknown reward distribution $M_x$. In order to maximize the final reward with a given cost budget, the algorithm plays some arm, collects the stochastically generated reward, and decides on the next arm, until the cost budget is exhausted. Starting from the motivation of designing clinical trials in the 1930s [@thompson1935theory; @thompson1933likelihood; @robbins1952some], there has been major developments in the Bandit theory over the last few decades (see, for example the books [@munos:hal-2014; @Bubeck:2011; @Bubeck12]). Several different strategies have addressed this problem and strong connections have been drawn with other fields such as online learning. In this paper, we explore how Bandit algorithms can be used for model checking of stochastic systems. A requirement $R$ for a stochastic system ${\mathcal{M}}$ usually checks whether the measure of executions of ${\mathcal{M}}$ satisfying certain temporal formulas cross certain thresholds [@younes2002probabilistic; @GburekB18]. Model checking for such requirements can be solved by calculating the exact measure of the executions that satisfy the relevant subformulas of $R$ [@BustanRV04; @HermannsWZ08; @JansenKOSZ07; @Kwiatkowska:book2004]. In this paper, we focus on the alternative [*statistical model checking (SMC)*]{} approach which samples some executions of ${\mathcal{M}}$ and uses hypothesis testing to infer whether the samples provide statistical evidence for the satisfaction (or violation) of $R$ [@younes2002probabilistic; @Sen:2005:SMC; @Younes05]. Execution data is a costly resource[^1], therefore, a number of SMC approaches minimize the [*expected number of samples*]{} needed for verification, for example, using sequential probability ratio tests, Chernoff bound, and Student’s t-distribution. Several SMC tools have been developed (for example, Ymer [@Ymer], VESTA [@VESTA:SenVA05], MultiVesta [@MVesta], PlasmaLab [@Boyer:2013:PFD], MODES [@MODES], UPPAAL [@UPPAAL], and MRMC [@MRMC] ), and they have been used to verify many systems [@DDavidLLMPVW11; @BaierHHK02; @JhaCLLPZ09; @KyleHC15; @PalaniappanG0HT13; @ZulianiPC13; @Meseguer:2006:SAD; @Martins:2011:SMC]. Most SMC algorithms crucially rely on fully stochastic models that never make nondeterministic choices. Although recent progress has been made towards verifying Markov Decision Processes with restricted types of schedulers [@lassaigne2015approximate; @DBLP:conf/tacas/HartmannsH14; @henriques2012statistical], SMC for MDPs remain a challenge problem (see [@Agha:2018:SSM] and [@Legay:2015] for recent surveys).
We will focus on stochastic models that are essentially Discrete Time Markov Chains, except that they are initialized from a (possibly very large) set of states. In other words, these are Markov Decision Processes (MDPs) where the adversary gets to initialize[^2]. Further, we restrict our attention to safety requirements[^3]. That is, we study problems that require maximizing (or minimizing) the probability of hitting certain unsafe states, starting from any initial state. Further, this class of models and requirements capture many practical problems like online monitoring where the initialization has to consider worst case error bounds in state estimation, for example, from sensing and perception. We observe that this optimization of a probability measure over a set of initial choices, can coincide with the multi-armed bandit problem for appropriately defined costs and rewards. By building the connection with the Bandit literature, we not only gain algorithmic ideas, but also new types of theoretical (regret) bounds on the sample efficiency of the algorithms. These bounds rely on the smoothness and the near-optimality dimension of the objective function, and are fundamentally different from the existing performance bounds in the SMC literature.
[*Hierarchical optimistic optimization (HOO) [@Bubeck:2011]*]{} is a bandit algorithm that builds a tree on a search space ${\mathcal{X}}$ by using the so called principle of optimism in the face of uncertainty. It is a black-box optimization method that applies an upper confidence bound (UCB) on a tree search method for finding the optimal points over the uncertain domain. The UCB in the tree search approach takes care of the trade-off between exploiting the most promising parts of the domain and exploring the most uncertain parts of the domain. [*Multi-fidelity hierarchical optimistic optimization(MFHOO)*]{} [@sen2019noisy] is a multi-fidelity HOO based method that allows noisy and biased observations from the uncertain domain. The performance of MFHOO is measured by how the [*regret*]{}—the gap between the actual maximum and the computed—scales with the number of samples. A key feature of these algorithms is that they can work with black-box functions and the regret guarantees only rely on certain smoothness parameters and the near-optimality dimension of the problem (see Definition \[def:near\_opt\_MFHOO\]). The standard theoretical assumptions required by off-the-shelf bandit algorithms in order to get performance guarantees do not precisely fit our verification problem, and that in-depth analysis and modification is required to get these guarantees in our setting; In addition, to apply these algorithms several functions need to be judiciously determined, a priori, and are at the heart of how the algorithms will perform. These choices are non-trivial and multi-faceted, and we develop and provide such functions explicitly in the context of our SMC problem in order to demonstrate successful application.
The key contributions of the paper are as follows.
First, we show how the MFHOO algorithm, can be used for statistical model checking with provable regret bounds. In the process, we define an appropriate notions of fidelity, bias-functions, and also modify the existing near-optimality dimension required for regret bounds of MFHOO to accommodate the non-smoothness of the typical functions we have to optimize for SMC. Second, we have built a new SMC tool called [[[HooVer]{}]{}]{} using MFHOO [@sen2019noisy]. We have created a practically inspired [@simonepresent; @FanQM18] suite of benchmark [NiMC]{}models that can be useful for safety analysis of driver assistance features in vehicles for standards such as ISO26262 [@iso26262]. Using the benchmarks we have carried out a detailed performance analysis of [[[HooVer]{}]{}]{} and our results suggest that the proposed approach can indeed make use of simulations more effectively than existing SMC approaches. A fair comparison of [[[HooVer]{}]{}]{} with other discrete-state model checkers like Prism [@HKNP06], Storm [@dehnert2017storm], and PlasmaLab [@legay2016plasma] is complicated as it relies on a continuous state models. We created discretized models for comparison, and observed that: Compared with exact probabilistic model checking tools like Storm, [[[HooVer]{}]{}]{} is faster, more memory efficient and scales better, and thus it can be used to check models with very large initial state space; Compared with [*statistical*]{} model checking tools like PlasmaLab, [[[HooVer]{}]{}]{} requires much less data to achieve comparable results. Finally, to our knowledge, this is the first work connecting statistical model checking with the Bandits theory; specifically, the hierarchical tree search using the principle of optimism in the face of uncertainty. Thus we believe that the exposition of these algorithms (Section \[sec:multi-fid-background\]) engender new applications in verification and synthesis algorithms.
Model and problem statement {#sec:prelims}
===========================
Consider a Euclidean space ${\mathcal{X}}= {{{\relax\ifmmode \mathbb R\else $\mathbb R$\fi}}}^m$ and let ${{{\relax\ifmmode
{\mathbb R}_{\geq 0} \else ${\mathbb R}_{\geq 0}$
\fi}}}$ denote the non-negative real numbers. For any real-valued function $p$ of ${\mathcal{X}}$ its support is the set ${{\rm{supp}\mathit{(p)}}} \coloneqq \{ x\in{\mathcal{X}}\ | \ p(x) \neq 0\}.$ A [*discrete probability distribution*]{} over ${\mathcal{X}}$ is a function $p:{\mathcal{X}}\rightarrow[0,1]$ such that ${{\rm{supp}\mathit{(p)}}}$ is countable, and $\sum_{x \in {{\rm{supp}\mathit{(p)}}}}p(x) =1.$ We use ${{{\relax\ifmmode \mathbb P\else $\mathbb P$\fi}}}({\mathcal{X}})$ to denote the set of discrete probability distributions over ${\mathcal{X}}$. For a finite set $\mathcal{S}$, $|\mathcal{S}|$ denotes the cardinality of $\mathcal{S}$.
Nondeterministically initialized Markov chains
----------------------------------------------
\[def:mpis\] A [*Nondeterministically initialized Markov chains ([NiMC]{})*]{} ${\mathcal{M}}$ is defined by a triple $(\mathcal{X}, \Theta, P)$, where:
${\mathcal{X}}= {{{\relax\ifmmode \mathbb R\else $\mathbb R$\fi}}}^m$ is the state space;
$\Theta \subseteq {\mathcal{X}}$ is the set of possible initial states; and
$P:{\mathcal{X}}\rightarrow {{{\relax\ifmmode \mathbb P\else $\mathbb P$\fi}}}({\mathcal{X}})$ is the [*probability transition function*]{}.
That is, from state $x \in {\mathcal{X}}$, the next state is chosen according to the discrete distribution $P(x)$. The probability of transitioning from state $x$ to state $x' \in {\mathcal{X}}$ is $P(x)(x')$, which we write more compactly as ${{P}_{{x,x'}}}$. An [*execution*]{} $\alpha$ of length $k$ for the [NiMC]{}${\mathcal{M}}$ is a sequence of states $\alpha = \{x_0, x_1, \ldots , x_k\}$, where $x_0 \in \Theta$, and for each $i>0$, ${{P}_{{x_{i-1},x_{i}}}} > 0.$ We denote the set of all length $k$ executions of ${\mathcal{M}}$ starting from $x_0$ as ${{{\rm{Execs}}_{\mathit{x_0}}}}(k)$. The probability of an execution $\alpha$, given $x_0$, is $ \prod_{i=1}^k {{P}_{{x_{i-1},x_i}}}=:p(\alpha) $. Given a set ${\mathcal{U}}\subseteq {\mathcal{X}}$, we say an execution $\alpha$ *hits* ${\mathcal{U}}$ if there exists $x\in \alpha$ such that $x\in{\mathcal{U}}$. We denote the subset of executions starting from $x_0$, of length $k$, that hit ${\mathcal{U}}$ by ${{{\rm{Execs}}_{x_0}}}(k, {\mathcal{U}})$. From a given initial state $x_0 \in \Theta$ the probability of ${\mathcal{M}}$ hitting an unsafe state within $k$ steps is given by: $$\begin{aligned}
{{p_{k,{\mathcal{U}}}{(x_0)}}} = \sum_{\alpha \in {{{\rm{Execs}}_{x_0}}}( k, {\mathcal{U}})} p(\alpha).
\label{eq:sumexecs}\end{aligned}$$ Note that if $x_0\in{\mathcal{U}}$ then ${{{\rm{Execs}}_{x_0}}}( k, {\mathcal{U}})={{{\rm{Execs}}_{\mathit{x_0}}}}(k)$ and ${{p_{k,{\mathcal{U}}}{(x_0)}}}=1$. We are interested in finding the [*worst case*]{} probability of hitting unsafe states from any initial state of ${\mathcal{M}}$. This can be regarded as determining, for each $k$, the value $$\begin{aligned}
\label{eq:SMCproblem}
\max_{x_0 \in \Theta} {{p_{k,{\mathcal{U}}}{(x_0)}}} .\end{aligned}$$ Importantly, we would like to solve this optimization problem without relying on detailed information about the probability transition function $P$. Further, our solution should not rely on precisely computing ${{p_{k,{\mathcal{U}}}{(x_0)}}}$ for a given $x_0 \in \Theta$, but instead only the use of [*noisy observations*]{}.
Example: Single-lane platoon with two speeds ([[$\mathsf{SLplatoon2}$]{}]{}) {#example1}
----------------------------------------------------------------------------
We present an [NiMC]{}of a platoon of $m$ cars on a single lane ([[$\mathsf{SLplatoon2}$]{}]{}). Variations of this model are used in all our experiments later in Section \[sec:experiments\]. Each car probabilistically decides to “cruise” or “brake” based on its current gap with the predecessor. These types of models are used for risk analysis of Automatic Emergency Braking (AEB) systems [@simonepresent; @FanQM18]. The probabilistic parameters of the model are derived from data collected from overhead traffic enforcement cameras on roads. The uncertainty in the initial positions (and gaps) arise from perception inaccuracies, which are modeled as worst-case ranges.
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![*Left*: Model of [[$\mathsf{SLplatoon2}$]{}]{}. *Right*: Probability of hitting ${\mathcal{U}}$ vs. the initial $\mathit{gap}$ with different execution lengths $k$. Here, initial $s_1 \in (20,23), s_2 \in (0,2)$, $\mathsf{vb} = 1$, $\mathsf{vc} = 2$, $\mathsf{pfar} = 0.85$, $\mathsf{pnear} = 0.15$, $\mathsf{near} = 3$ and $\delta = 1$ []{data-label="fig:singlelane"}](probs_time_horizon_Example2_New.jpg){width="\textwidth"}
Let $s_{i}$ be the position of $i$^th^ car in the sequence. Initially, $s_i$ takes a value in an interval on the $x$-axis such that $s_{1} > s_{2} >\ldots > s_{m}$. The pseudocode in Figure \[fig:singlelane\] specifies the probabilistic transition rule that updates the position of all the cars synchronously. Car $1$ always moves at a constant breaking speed of $\mathsf{vc}$. The variable $\mathit{gap}_i$ is the distance of $i$ to the predecessor $i-1$, for each $i=2,...,m$. If $\mathit{gap}_i$ is less than the constant threshold $\mathsf{near}$, then $i$ continues to cruise with probability $\mathsf{pnear}$ and it brakes with probability $1- \mathsf{pnear}.$ Similarly, if $\mathit{gap}_i$ greater than $\mathsf{near}$, then $i$ continues to cruise with probability $\mathsf{pfar}$ and brakes with probability $1- \mathsf{pfar}.$ It is straightforward to connect the above description to a formal definition of a [NiMC]{}. The state space ${\mathcal{X}}=\mathbb{R}^m$. The set of initial states $\Theta$ is a hyperrectangle in ${\mathcal{X}}$ (such that $s_1 > s_2 \ldots > s_m$). For any state, $x\in {\mathcal{X}}$ the probability transition function is given by the equations in lines \[ln:lead\_p\], \[ln:other\_p1\], \[ln:other\_p2\], \[ln:other\_p3\] and \[ln:other\_p4\]. We define the set of unsafe states ${\mathcal{U}}=\{(s_{1},s_{2},...,s_{m})\in {\mathcal{X}}\ |\ \exists i\in\{2,...,m\},\ \mbox{ such that }\mathit{gap}_i \leq \delta\}\subseteq {\mathcal{X}}$ for some constant collision threshold $\delta$. Given that cars start their motion at any initial state from $\Theta$, the goal is to find the maximum probability of hitting the unsafe set ${\mathcal{U}}$. For $m=2$, $\mathsf{vb} = 1$, $\mathsf{vc} = 2$, $\mathsf{pfar} = 0.85$ $\mathsf{pnear} = 0.15$, $\mathsf{near} = 3$, $\delta = 1$, and initial $s_1 \in (20,23), s_2 \in (0,2)$, Figure \[fig:singlelane\] shows estimates of probabilities of hitting the unsafe set from different initial separations between cars. As our intuition suggests, for large enough time horizons the probability of hitting the unsafe set approaches $1$ from all initial states, but, for smaller time horizons the maximum probability of unsafety arises when the initial gap is smaller.
Background: Hierarchical Optimistic Optimization {#sec:multi-fid-background}
================================================
[*Multi-Fidelity Hierarchical Optimistic Optimization (MFHOO)*]{} [@sen2019noisy] is an black-box optimization algorithm from the [*multi-armed bandits*]{} literature [@munos:hal-2014; @Bubeck:2011; @Bubeck12]. The setup is the following: suppose we want to maximize the function $f:{\mathcal{X}}\rightarrow \mathbb{R}$, which is assumed to have a unique global maximum. Let $f^* = \underset{x\in{\mathcal{X}}}{\sup}$ $f(x)$. MFHOO allows the choice of evaluating $f$ at different fidelities with different costs. This flexibility matters for SMC because it will be beneficial to evaluate the probability of unsafety ${{p_{k,x_0}{({\mathcal{U}})}}}$ for certain initial states more precisely, for example, with longer number of simulations, while for other initial states a less precise evaluation may be adequate.
Thus, MFHOO has access to a biased function $f_z(x)$ that depends on fidelity parameter $z \in [0,1]$. Setting $z = 0$ gives the lowest fidelity (and lowest cost) and $z=1$ corresponds to full fidelity (and highest cost). At full fidelity, $f_1(x) = f(x)$, and the evaluation is unbiased. More generally, $|f_z(x)-f(x)|\leq\zeta(z)$ and evaluating $f_z(x)$ costs $\lambda(z)$, where the functions $\zeta, \lambda:[0,1]\rightarrow\mathbb{R}_{>0}$ are respectively, non-increasing and non-decreasing, and called the [*bias*]{} and the [*cost*]{} functions [@sen2019noisy].
A bandit algorithm chooses a sequence of sample points (arms) $x_1, x_2, \ldots \in {\mathcal{X}}$, evaluates them at fidelities $z_1, z_2, \ldots$, and receives the corresponding sequence of noisy observations (rewards) ${y}_1, {y}_2, \ldots$. We assume that each ${y}_j$ is drawn from a unknown distribution $M_{z_j,x_j}$ satisfying $f_{z_j}(x_j)= \int u dM_{z_j,x_j}(u)$. Further the distribution has a sub-Gaussian component, with variance $\sigma^2$, which captures uncertainty in the observations. The algorithm [*actively chooses*]{} $x_{j+1}$ based on past choices $x_1, \ldots, x_j$ and observations ${y}_1, \ldots, {y}_j$. When the budget $\Lambda$ is exhausted, the algorithm decides the optimal point $\bar{x}_{n(\Lambda)} \in {\mathcal{X}}$ and the optimal value $f(\bar{x}_{n(\Lambda)})$ with the aim of minimizing [*regret*]{}, which is defined as $$R(\Lambda)=f^*-f(\bar{x}_{n(\Lambda)}).$$ The MFHOO algorithm (Algorithm \[Al:MFHOO\]) for selecting $x_{j+1}$ estimates $f^*$ by building a binary tree in which each height in the tree represents a partition of the state space ${\mathcal{X}}$. The algorithm maintains estimates of an upper-bound on $f$ for each partition subset, and uses the principle of optimism for choosing the next sample $x_{j+1}$. That is, it chooses the samples in those partitions where the estimated upper-bounds are the highest.
Each node in the constructed tree is labeled by a pair of integers $(h, i)$, where $h$ is the height of the node in the tree, and $i$ satisfying $0\leq i\leq 2^h$ is its position within height level $h$. The root is labeled $(0,1)$, and each node $(h,i)$ can have two children $(h+1, 2i-1)$ and $(h+1, 2i)$. Node $(h, i)$ is associated with subset a of ${\mathcal{X}}$, denoted by ${{\mathcal{P}_{h,i}}}$, where ${{\mathcal{P}_{h,i}}}={{\mathcal{P}_{h+1,2i-1}}}\cup {{\mathcal{P}_{h+1,2i}}}$, and for each $h$ these disjoint subsets satisfy $\cup_{i=1}^{2^h} {{\mathcal{P}_{h,i}}} = {\mathcal{X}}$. Therefore, larger values of $h$ represent finer partitions of ${\mathcal{X}}$. For each node $(h,i)$ in the tree, the algorithm maintains the following information:
${{\mathit{count}_{h,i}}}$: the number of times the node is visited;
$\hat{f}_{h,i}$: the empirical mean of observations over points visited in ${{\mathcal{P}_{h,i}}}$;
$U_{h,i}$: an initial estimate of the maximum of $f$ over ${{\mathcal{P}_{h,i}}}$; and
$B_{h,i}$: a tighter and optimistic upper bound on the maximum of $f$ over ${{\mathcal{P}_{h,i}}}.$
The algorithm proceeds as follows. The $\mathit{tree}$ starts out with a single node, the root $(0,1)$, initializes the $B$-values of its two children $B_ {1,1}$ and $B_{1,2}$ to $+\infty$, and initializes the cost $C$ to $0$. At a high-level, in each iteration of the **while** loop (line \[ln:mf\_outerwhile\]), the algorithm adds a new node $(\mathit{hnew},\mathit{inew})$ in the $\mathit{tree}$ and updates all of the above quantities for several nodes in $\mathit{tree}$. In more detail, first a $\mathit{path}$ from the root to a leaf is found by traversing the child with the higher $B$-value (with ties broken arbitrarily). Let the child with the higher $B$-value of the traversed leaf be $(\mathit{hnew},\mathit{inew})$ (line \[ln:mf\_path\]). An arbitrary point $x$ in the partition of this node ${{\mathcal{P}_{\mathit{hnew},\mathit{inew}}}}$ is chosen (line \[ln:mf\_choose\]). Then, this point is evaluated at fidelity $z_{hnew}=\zeta^{-1}(\nu\rho^{hnew})$ and a reward ${y}$ is received (line \[ln:mf\_recieve\]). Next, $\mathit{tree}$ is extended by inserting $(\mathit{hnew},\mathit{inew})$ in the $\mathit{tree}$ (line \[ln:mf\_insert\]) and for all the nodes $(h,i)$ in $\mathit{path}$ including $(\mathit{hnew},\mathit{inew})$, the ${{\mathit{count}_{h,i}}}$ and the empirical mean $\hat{f}_{h,i}$ are updated (line \[ln:mf\_update-path\]). Finally, in line \[ln:mf\_update-tree\], for all nodes $(h,i)$ in $\mathit{tree}$, $U_{h,i}$ and $B_{h,i}$ are updated using the smoothness parameters $\nu_{1}>0$ and $\rho\in(0,1)$ which will discussed later in Section \[sec:regret-mfhoo\] and the parameter $\sigma$. Once the sampling budget $\Lambda$ is exhausted, a leaf with maximum $B$-value is returned by the Algorithm \[Al:MFHOO\] [@sen2019noisy].
**input**: Budget: $\Lambda$, parameter: $\sigma$, bias $\zeta(.)$, cost $\lambda(.)$, smoothness params $\nu>0$ and $\rho\in(0,1)$ $\mathit{tree} =\{(0,1)\}$, $B_{1,1}=B_{1,2}=\infty$, $C=0$ $\label{ln:mf_outerwhile}$ $(\mathit{path}, (\mathit{hnew},\mathit{inew})) \gets \mathit{Traverse}(\mathit{tree})$ $\label{ln:mf_path}$ $\textbf{choose}$ $x\in{{\mathcal{P}_{\mathit{hnew},\mathit{inew}}}}$ $\label{ln:mf_choose}$ $\textbf{query}$ $x$ at fidelity $z_{hnew} = \zeta^{-1}(\nu\rho^{hnew})$ and get observation ${y}$ $\label{ln:mf_recieve}$ $C \leftarrow C + \lambda(z_{hnew})$ $\label{ln:mf_cost}$ $\mathit{tree.Insert}((\mathit{hnew},\mathit{inew}))$ $\label{ln:mf_insert}$ $\label{ln:mf_update-path}$ ${{\mathit{count}_{h,i}}} \leftarrow {{\mathit{count}_{h,i}}} + 1$ $\hat{f}_{h,i} \leftarrow (1- 1/{{\mathit{count}_{h,i}}})\hat{f}_{h,i} + {y}/{{\mathit{count}_{h,i}}}$ $B_{\mathit{hnew}+1,2\mathit{inew}-1} \leftarrow +\infty$, $B_{\mathit{hnew}+1,2\mathit{inew}} \leftarrow +\infty$ from leaves up to root: $\label{ln:mf_update-tree}$ $U_{h,i} \leftarrow \hat{f}_{h,i} +\mathit{sqrt}(2\sigma^2\ln n/{{\mathit{count}_{h,i}}})+\nu\rho^h + \zeta(z_{h})$ $B_{h,i} \leftarrow \min\{U_{h,i}, \max\{B_{h+1,2i+1}, B_{h+1,2i}\}\}$\
$\underset{(h,i)\in \mathit{tree}}{\rm argmax}\ B_{h,i}$
Regret bounds for MFHOO {#sec:regret-mfhoo}
-----------------------
In this section, we summarize the assumptions and results from [@sen2019noisy]. In order to analyze the regret bounds for the MFHOO algorithm, the following assumption on the smoothness of $f(x)$ [@sen2019noisy] is made.
\[Ass:MFHOO\] There exist $\nu>0$ and $\rho \in (0, 1)$ such that for all $\mathcal{P}_{h,i}$ satisfying $f^* - \underset{x\in\mathcal{P}_{h,i}}{\sup}\ f(x) \leq c\nu\rho^h$ (for a constant $c\geq0$), we have that $f(x)\geq f^* - \max\{2c, c + 1\}\nu\rho^h$ , for all $x\in\mathcal{P}_{h,i}$.
This assumption connects the function $f$ to the partitioning rule of the binary tree. We now define the concept of [*near-optimality dimension*]{} which plays an important role in the analysis of black-box optimization algorithms [@Bubeck:2011; @sen2019noisy; @grill2015black; @valko2013stochastic]. It measures the dimension of sets that are close to optimal. The regret bound for MFHOO uses this notion. First, given a partitioning scheme ${{\mathcal{P}_{h,i}}}$ over ${\mathcal{X}}$, we define $\mathcal{N}_h(\epsilon)$ as the number of $\epsilon$-near optimal partitions, that is, the number of partitions $\mathcal{P}_{h,i}$ such that satisfies $\sup_{{x\in\mathcal{P}_{h,i}}} f(x)\geq f(x^*)-\epsilon$.
\[def:near\_opt\_MFHOO\] The near-optimality dimension of $f$ with respect to parameters $(\nu, \rho)$ is: $d(\nu, \rho)=\inf\{d'\in\mathbb{R}_{>0}:\exists B>0,\ s.t.\ \forall h \geq 0,\ \mathcal{N}_h(2\nu\rho^h) \leq B\rho^{-d'h}\}.$
With Assumption \[Ass:MFHOO\], the regret bound for MFHOO is proved in [@sen2019noisy].
\[Th:regret\_MFHOO\] If Algorithm \[Al:MFHOO\] runs with parameters $\nu$ and $\rho$ that satisfy Assumption \[Ass:MFHOO\] and a cost budget of $\Lambda$, then the simple regret is bounded $$\begin{aligned}
R(\Lambda) = O\bigg( (\frac{B\log n(\Lambda)}{n(\Lambda)})^{\frac{1}{d(\nu,\rho)+2}}\bigg),\end{aligned}$$ where $n(\Lambda) = \max\{n:\sum_{h=1}^{n} \lambda(z_h) \leq \Lambda\}$. Here, $z_h = \zeta^{-1}(\nu\rho^h)$.
According to the Theorem \[Th:regret\_MFHOO\], regret is minimized if the near-optimality dimension $d(\nu, \rho)$ is minimized. If the smoothness parameters that minimize the near-optimlaity dimension $d(\cdot,\cdot)$ are known, then MFHOO achieves the minimum regret of Theorem \[Th:regret\_MFHOO\].
Statistical Model Checking with Optimistic Optimization {#sec:smc-mfhoo}
=======================================================
We aim to solve the statistical model checking problem of maximizing ${{p_{k,{\mathcal{U}}}{(x)}}}$ of Equation (\[eq:SMCproblem\]) for a given [NiMC]{}${\mathcal{M}}$ and a time horizon $k$, using MFHOO. In order to apply the MFHOO algorithm, one has to make several critical choices regarding the objective function, the budget, the cost, the parameters for fidelity and smoothness, and the multi-fidelity bias function. In this section we discuss the rationale behind our choices.
Objective function, budget, cost, and fidelity {#sec:objAndfid}
----------------------------------------------
#### Fidelity parameter $z$.
Consider a [NiMC]{}$\mathcal{M} = ({\mathcal{X}}, \Theta, P)$ with the unsafe set ${\mathcal{U}}\subseteq {\mathcal{X}}$. We have to maximize ${{p_{k_{\mathit{max}},{\mathcal{U}}}{(x)}}}$ over all initial states $x \in \Theta$, and for a long time horizon $k_{\mathit{max}}$. Given $x\in \Theta$, the fidelity of evaluating ${{p_{k_{\mathit{max}},{\mathcal{U}}}{(x)}}}$ will depend on the actual length of the simulations drawn for creating the observation $y$ for the state $x$. Suppose we fix $k_{\mathit{min}}$ as the shortest simulation to be used. We define the fidelity of an observation (or evaluation) with simulations of length $k \in [k_{\mathit{min}}, k_{\mathit{max}} ]$ as $z=(k-k_{min})/(k_{min}-k_{max})$.
#### Objective function $f$ and observations.
A natural choice for the objective function would be to define $f_{z=1}(x) := {{p_{k_{max},{\mathcal{U}}}{(x)}}}$, for any initial state $x \in \Theta$. Computing this probability, however, is infeasible when the probability transition function $P_{\mathcal{M}}$ is unknown. Even if $P_{\mathcal{M}}$ is known, calculating ${{p_{k,{\mathcal{U}}}{(x)}}}$ involves summing over many executions (as in (\[eq:sumexecs\])). Instead, we take advantage of the fact that MFHOO can work with noisy observations. For any initial $x \in \Theta$, and execution $\alpha \in {{{\rm{Execs}}_{x}}}(k)$ we define the observation: $$\begin{aligned}
{Y}= 1 \ \mbox{if} \ \alpha \in {{{\rm{Execs}}_{x}}}(k,{\mathcal{U}}), \ \mathit{and} \ = 0 \ \mathit{otherwise}.
$$ Recall that for a fixed initial states $x$, ${\mathcal{M}}$ is a Markov chain and defines a probability distribution over the set of executions ${{{\rm{Execs}}_{x}}}(k)$ as given by Equation (\[eq:sumexecs\]). Thus, given an initial state $x$, ${Y}=1$ with probability ${{p_{k,{\mathcal{U}}}{(x)}}}$, and ${Y}=0$ with probability $1-{{p_{k,{\mathcal{U}}}{(x)}}}$. That is, ${Y}$ is a Bernoulli random variable with mean ${{p_{k,{\mathcal{U}}}{(x)}}}$ at fidelity $z$. In MFHOO, once an initial state $x \in {{\mathcal{P}_{\mathit{hnew},\mathit{inew}}}}$ is chosen (line \[ln:mf\_choose\]), we simulate ${\mathcal{M}}$ upto $k$ steps several times starting from $x$ and calculate the empirical mean of $Y$, which serves as the noisy observation $y$ at fidelity $z$.
#### Cost $\lambda(z)$ and budget $\Lambda$.
In our setup the cost function $\lambda(z)$ for any $z$ is the computational time required to simulate an execution $\alpha \in {{{\rm{Execs}}_{x}}}(k)$ for any $x \in \Theta$, where $k$ is the execution length corresponding to the fidelity $z$. The budget $\Lambda$ is a computational cost budget. The next proposition states that with these choices, our goal to maximize ${{p_{k_{\mathit{max}},{\mathcal{U}}}{(x)}}}$ over $ \Theta$ can be achieved using MFHOO.
\[opt-gap\] Given smoothness parameters $\rho$ and $\nu$ satisfying Assumption \[Ass:MFHOO\] and budget $\Lambda$, suppose Algorithm \[Al:MFHOO\] returns $\bar{x}_{\Lambda} \in \Theta$. Then, $R(\Lambda) = {{p_{k_{\mathit{max}},{\mathcal{U}}}{(x^*)}}} - {{p_{k_{\mathit{max}},{\mathcal{U}}}{(\bar{x}_{\Lambda})}}}$, where $R(\Lambda)$ bound is given by Theorem \[Th:regret\_MFHOO\].
Thus, the point $\bar{x}_{\Lambda} \in \Theta$ returned by Algorithm \[Th:regret\_MFHOO\] is such that the gap between its probability ${{p_{k,{\mathcal{U}}}{(\bar{x})}}}$ of hitting ${\mathcal{U}}$ and the true maximum probability ${{p_{k,{\mathcal{U}}}{(x^*)}}}$ is bounded by Theorem \[Th:regret\_MFHOO\] in terms of the available budget $\Lambda$.
Multi-fidelity bias function {#sec:mfbf}
----------------------------
Recall that the bias function $\zeta(z)$ gives an upper bound $|f(x)-f_{z}(x)|\leq \zeta(z)$, over all $x \in \Theta$ and for any fidelity $z \in [0,1]$. We will derive a bias function satisfying $|{{p_{k_{\mathit{max}},{\mathcal{U}}}{(x_0)}}}-{{p_{k,{\mathcal{U}}}{(x_0)}}}|\leq \zeta(z)$. Of course, a guarantee like this is only possible for known models. Therefore, for this section we will assume that the [NiMC]{}${\mathcal{M}}$ is known. We also assume ${\mathcal{X}}$ is finite and ${\mathcal{U}}$ is [*the*]{} absorbing set for ${\mathcal{M}}$ (i.e., all other states are transient; $x\in{\mathcal{U}}$ if and only if $P_{xx}=1$).
Fixing an initial state $x$, ${\mathcal{M}}$ is a reducible Markov chain. Let $q$ be the number of transient states and $u = |{\mathcal{U}}|$. Then, the probability transition function can be represented by the $(q+u)\times(q+u)$ matrix $P$: $$\begin{aligned}
\label{eq:p}
P=
\left(
\begin{array}{c|c}
Q & R \\
\hline
{\bf 0} & I
\end{array}
\right)
\ \mathit{and} \
P^{t}=
\left(
\begin{array}{c|c}
Q^{t} & \sum_{k=0}^{t-1} Q^{k}R
\\\hline
{\bf 0} & I
\end{array}
\right).\end{aligned}$$ Here $Q$ is an $q\times q$ matrix, $I$ is an $u\times u$ identity matrix, ${\bf 0\/}$ is an $u\times q$ zero matrix, and $R$ is a nonzero $q\times u$ matrix. The following standard result gives the absorption probabilities in terms of $Q$.
\[thm:3\] Suppose $B = NR$ is a $q\times u$ matrix, where $N=\sum_{j=0}^{\infty} Q^{j}=(I-Q)^{-1}$ and $R$ is as defined in (\[eq:p\]). Then $B_{ij}$ is the probability, that starting from state $s_i$, the chain ${\mathcal{M}}$ is absorbed in $s_j \in {\mathcal{U}}$.
We now state and prove the theorem that defines a multi-fidelity bias function.
\[th:mf\_bias\_func\] For any $x \in \Theta$ and any time horizon $k < k_{max}$, $$|{{p_{k_{\mathit{max}},{\mathcal{U}}}{(x)}}}-{{p_{k,{\mathcal{U}}}{(x)}}}| \leq \frac{\kappa_{\infty}(E)(k_{\mathit{max}}-k)}{(k_{\mathit{max}}-(k_{\mathit{max}}-1)\lambda_{\mathit{max}})(k-(k-1)\lambda_{\mathit{max}})},$$ where $\lambda_{max}$ is the maximum eigenvalue, $E$ is the matrix of eigenvectors of the matrix $Q$, and $\kappa_{\infty}(E)$ is the ($\infty$-norm) condition number of $E$.
This theorem can be proved using the spectral decomposition of matrix $Q$ and $\infty$-norm bounds (for detailed analysis please refer to Appendix \[app:proof\]).
\[re:mf\_bias\] Using the fidelity parameter $z=(k_{max}-k)/(k_{max}-k_{min})$ in the upper bound given by Theorem \[th:mf\_bias\_func\], the bias function can be rewritten as: $$\begin{aligned}
\zeta(z) &= \frac{g_{1}(1-z)}{g_{2}z+g_{3}}, \mathit{where} \ g_{1} = \kappa_{\infty}(E)(k_{max}-k_{min}) \\
g_{2} &= ((1-\lambda_{max})k_{max} - \lambda_{max})(k_{max}-k_{min})(1-\lambda_{max}) \nonumber \\
g_{3} &= ((1-\lambda_{max})k_{max} - \lambda_{max})((1-\lambda_{max})k_{min} - \lambda_{max}). \nonumber\end{aligned}$$
This bias function upper bounds the gap between the probability of hitting the unsafe set for a point in the initial set for different time horizons. Thus, it can be used in the analysis of the theoretical regret bound of the Algorithm \[Al:MFHOO\] given in Theorem \[Th:regret\_MFHOO\]. If the transition matrix $P$ is known, then the bias function can be used in updating the $U$-values of nodes in the Algorithm \[Al:MFHOO\]. Note, this bias function depends on parameters $\kappa_{\infty}(E)$ and $\lambda_{max}$. In the case where the full transition model of the [NiMC]{} is unknown, but there is access to $\kappa_{\infty}(E)$ and $\lambda_{max}$, this function can be utilized in the algorithm. More generally, in problems with unknown $P$ and no access to the parameters $\kappa_{\infty}(E)$ and $\lambda_{max}$, we consider a linear parameterized bias function $\zeta(z)=b(1-z)$ with an unknown parameter $b$, and adaptively estimate the parameter $b$ [@sen2019noisy].
Non-smoothness and non-uniqueness of hitting probabilities {#sec:Assumptions}
----------------------------------------------------------
In this section, we start with the observation that in general the function ${{p_{k,{\mathcal{U}}}{(x)}}}$ over $x \in \Theta$ is not a continuous function and has infinitely many maxima. Thus this function does not satisfy the Assumption \[Ass:MFHOO\] and the assumption of finite number of maxima, that are required for the regret bounds of Theorem \[Th:regret\_MFHOO\]. However, a modified definition of near-optimality dimension we show that the bounds given by Theorem \[Th:regret\_MFHOO\] will hold. Finally, we will compare the theoretical regret with the actual regret achieved by MFHOO.
Consider a discrete time linear system with state space ${\mathcal{X}}=\mathbb{R}^m$, set of initial states $\Theta \subseteq{{\mathcal{X}}}$ and set of unsafe states given by ${\mathcal{U}}:= \{x\in {\mathcal{X}}\ | \ c^{T} x\leq a\}$ for some vector $c \in {{{\relax\ifmmode \mathbb R\else $\mathbb R$\fi}}}^m$ and $a \in {{{\relax\ifmmode \mathbb R\else $\mathbb R$\fi}}}$. The system evolves according to: $$\label{eq:LTI}
x_{t+1} = Ax_{t} +B W_{t},$$ where $x_{t}\in {\mathcal{X}}$ is the state vector at time $t$. $W_{t} \in {{{\relax\ifmmode \mathbb R\else $\mathbb R$\fi}}}^r$ is a discrete random vector with probability mass distribution $p_{W}$ at sequence $t$, and $A$ and $B$ are $m \times m$ and $m \times r$ matrices, respectively.
Consider two initial states $x_0, x_0' \in \Theta,$ such that $x_0' = x_0 + \epsilon$, for some small $\epsilon$. Writing out the probabilities of hitting unsafe states of this system explicitly for the two initial states and finding the gap between these probabilities we can observe that, for small enough $\epsilon$, the gap between the probability is zero[^4]. By increasing the $\epsilon$, the gap can take a nonzero value. This means that the probability of hitting the unsafe set of the system (\[eq:LTI\]) is not a continuous function in terms of the initial states and has infinitely many maxima. This is because random variable $W_t$ is a discrete-type random variable. [[$\mathsf{SLplatoon2}$]{}]{} is a concrete example of this kind. Assumption \[Ass:MFHOO\] asserts that there exist smoothness parameters such that for all $h \geq 0$ in the tree, and all the partitions that are near optimal, the gap between $f^*$ and the value of the function over those partitions should be bounded. From the above discussion we see that this may not hold for all depths $h \geq 0$. In addition, the regret bounds in Theorem \[Th:regret\_MFHOO\] are for function with finite number of maxima. If there are infinitely many maxima, then for given smoothness parameters the number of $2\nu\rho^h$-near-optimal partitions $\mathcal{N}_{h}(2\nu\rho^h)$ would not be bounded for all $h\geq 0$. However, we observe that any instance of MFHOO runs on a finite budget and the final constructed [*tree*]{} has a maximum height $h_{max}$. For this $h_{\mathit{max}}$, there exist smoothness parameters $\nu$ and $\rho$ that satisfy the Assumption \[Ass:MFHOO\], and we can modify the near-optimality dimension in the Definition \[def:near\_opt\_MFHOO\] as:
\[def:mod\_near\_opt\] $h_{\mathit{max}}$-bounded near-optimality dimension of $f$ with respect to $(\nu, \rho)$ is: $d_{m}(\nu, \rho)=\inf\{d'\in\mathbb{R}_{>0}:\exists B>0,\ \forall h \in [0,h_{max}],
\mathcal{N}_h(2\nu\rho^h) \leq B\rho^{-d'h}\}$.
With this modified definition, there exists a $B$ satisfying the above condition and the corresponding $d_m$ can be used to recover the regret bound of the Theorem \[Th:regret\_MFHOO\].
#### Regret in theory and in practice.
Given a budget $\Lambda$, let $\bar{x}_{\Lambda} \in \Theta$ be the point that the Algorithm \[Al:MFHOO\] returns after the budget is exhausted. Then the regret $R(\Lambda)$ would be ${{p_{k_{max},{\mathcal{U}}}{(x^*)}}} - {{p_{k_{max},{\mathcal{U}}}{(\bar{x}_{\Lambda})}}}$. We compare the theoretical regret bound and the actual regret for the [[$\mathsf{SLplatoon2}$]{}]{} model. Consider an instance of [[$\mathsf{SLplatoon2}$]{}]{} with $m =2$, initial states $s_1 \in (50,70)$ and $s_2 \in (0,20)$. Intuitively, the partition corresponding to $s_1 \in (50,51) $ and $s_2 \in (19,20)$ would maximize the probability of hitting ${\mathcal{U}}$. In Figure \[fig:analysis\_slane\] ([*top-left*]{}), the mean of actual regret obtained by running MFHOO for various smoothness parameters is presented. As the budget increases, as expected, the actual regret decreases monotonically and approximately approaches to $0$. For budget $\Lambda = 30$, the partitioning stops at $h_{max} = 15$. We can derive the number of $2\nu\rho^h$-near-optimal partitions for any $\nu$ and $\rho$. Setting $\nu = 0.08$ $\rho = 0.77$ (actual values used to generate regret results), we see that for $h\in[9,15]$, the maximum number of $2\nu\rho^h$-near-optimal partitions $\mathcal{N}_{h}(2\nu\rho^h) = \lceil 2^h/400 \rceil$. This is because the total number of partitions at depth $h$ is equal to $2^h$, and for $h\in[9,15]$, the $2\nu\rho^h$-near-optimal partitions belong to the area corresponding to $s_1 \in (50,51) $ and $s_2 \in (19,20)$, whose area is $1/400$. According to Definition \[def:mod\_near\_opt\], for different values of $B$, different $h_{\mathit{max}}$-bounded near-optimality dimensions $d_{m}$ can be obtained. We are looking for the pair of $(d_{m},B)$ values that minimize the theoretical regret bound of Theorem \[Th:regret\_MFHOO\]. Figure \[fig:analysis\_slane\] ([*top-right*]{}) represents the number of $2\nu\rho^h$-near-optimal partitions for $h\in[9,15]$ that are upper bounded by $B\rho^{-d_{m}h}$ for different values of $d_{m}$ and their corresponding $B$. Figure \[fig:analysis\_slane\] ([*bottom*]{}) also, represent the theoretical upper bound vs. $d_{m}$ and their corresponding $B$ for different values of smoothness parameters used to generate the actual regret. As it is seen, the best theoretical regret bounds that can be achieved is approximately $0.08$.
![[[$\mathsf{SLplatoon2}$]{}]{} with the same parameters as in Section \[example1\] and initial states $s_1 \in (50,70)$ and $s_2 \in (0,20)$. *Top-left*: Actual regret of running MFHOO averaged for various smoothness parameters. *Top-right*: Number of $2\nu\rho^h$ partitions for various $d_{m}$ values and their corresponding $B$ with $\nu = 0.08$ and $\rho = 0.77$. *Bottom-left*: Plot of $g(d_{m})$ vs. $d_{m}$ for various smoothness parameters, where $R(\Lambda) = O(g(d_{m}))$ as in the Theorem \[Th:regret\_MFHOO\]. *Bottom-right*: Plot of $g(B)$ vs. $B$ for various smoothness parameters, where $R(\Lambda) = O(g(B))$ as in the Theorem \[Th:regret\_MFHOO\].[]{data-label="fig:analysis_slane"}](actual_regret_slane.jpg "fig:"){width="50.00000%"} ![[[$\mathsf{SLplatoon2}$]{}]{} with the same parameters as in Section \[example1\] and initial states $s_1 \in (50,70)$ and $s_2 \in (0,20)$. *Top-left*: Actual regret of running MFHOO averaged for various smoothness parameters. *Top-right*: Number of $2\nu\rho^h$ partitions for various $d_{m}$ values and their corresponding $B$ with $\nu = 0.08$ and $\rho = 0.77$. *Bottom-left*: Plot of $g(d_{m})$ vs. $d_{m}$ for various smoothness parameters, where $R(\Lambda) = O(g(d_{m}))$ as in the Theorem \[Th:regret\_MFHOO\]. *Bottom-right*: Plot of $g(B)$ vs. $B$ for various smoothness parameters, where $R(\Lambda) = O(g(B))$ as in the Theorem \[Th:regret\_MFHOO\].[]{data-label="fig:analysis_slane"}](cells_slane.jpg "fig:"){width="50.00000%"} ![[[$\mathsf{SLplatoon2}$]{}]{} with the same parameters as in Section \[example1\] and initial states $s_1 \in (50,70)$ and $s_2 \in (0,20)$. *Top-left*: Actual regret of running MFHOO averaged for various smoothness parameters. *Top-right*: Number of $2\nu\rho^h$ partitions for various $d_{m}$ values and their corresponding $B$ with $\nu = 0.08$ and $\rho = 0.77$. *Bottom-left*: Plot of $g(d_{m})$ vs. $d_{m}$ for various smoothness parameters, where $R(\Lambda) = O(g(d_{m}))$ as in the Theorem \[Th:regret\_MFHOO\]. *Bottom-right*: Plot of $g(B)$ vs. $B$ for various smoothness parameters, where $R(\Lambda) = O(g(B))$ as in the Theorem \[Th:regret\_MFHOO\].[]{data-label="fig:analysis_slane"}](Regret_d_slane.jpg "fig:"){width="50.00000%"} ![[[$\mathsf{SLplatoon2}$]{}]{} with the same parameters as in Section \[example1\] and initial states $s_1 \in (50,70)$ and $s_2 \in (0,20)$. *Top-left*: Actual regret of running MFHOO averaged for various smoothness parameters. *Top-right*: Number of $2\nu\rho^h$ partitions for various $d_{m}$ values and their corresponding $B$ with $\nu = 0.08$ and $\rho = 0.77$. *Bottom-left*: Plot of $g(d_{m})$ vs. $d_{m}$ for various smoothness parameters, where $R(\Lambda) = O(g(d_{m}))$ as in the Theorem \[Th:regret\_MFHOO\]. *Bottom-right*: Plot of $g(B)$ vs. $B$ for various smoothness parameters, where $R(\Lambda) = O(g(B))$ as in the Theorem \[Th:regret\_MFHOO\].[]{data-label="fig:analysis_slane"}](Regret_B_slane.jpg "fig:"){width="50.00000%"}
[[[HooVer]{}]{}]{} tool and experimental evaluation {#sec:experiments}
===================================================
We have implemented a prototype tool called [[[HooVer]{}]{}]{} which uses MFHOO for solving the SMC problem of (\[eq:SMCproblem\]). We compare the performance of [[[HooVer]{}]{}]{} with that of Prism [@HKNP06], Storm [@dehnert2017storm], and PlasmaLab [@legay2016plasma] on several benchmarks we have created.
Benchmark models {#sec:allbenchmarks}
----------------
We have created several [NiMC]{}models for evaluation of probabilistic and statistical model checking tools. The benchmarks are variants of [[$\mathsf{SLplatoon2}$]{}]{} (Section \[example1\]). The complete models are described in Appendix \[sec:app:benchmarks\]. The executable models are available from [@SMCOO-online:2020]. [[$\mathsf{SLplatoon3}$]{}]{} models a sequence of $m$ cars on a single lane. At each step, each car can choose to move with one of [*three*]{} speeds: $\mathsf{vbrake}$, $\mathsf{vcruise}$, and $\mathsf{vspeedup}$. The first vehicle moves at a constant speed. For all the others, the speed is chosen probabilistically, according to distributions that depend on their distance to the preceding vehicle. For example, if the longitudinal distance $s_{i-1} - s_i$ is less than a threshold $\mathsf{th\_close}$ then the speed for vehicle $i$ is chosen according to a probability distribution $\mathsf{pclose}$, and so on. The state variables, the initial states, and the unsafe set are defined in the same way as in [[$\mathsf{SLplatoon2}$]{}]{}.
[[$\mathsf{MLplatoon}$]{}]{} models $m$ cars on $\ell$ lanes. At every step, each car probabilistically chooses to either move forward like a vehicle in [[$\mathsf{SLplatoon3}$]{}]{}, or it changes to its left or right lane. These actions are chosen probabilistically, according to probability distributions that depend on their distances to the vehicles on its current lane, as well as left and right lanes. In addition to the longitudinal position $s_i$, each vehicle $i$ has a second state variable $y_i \in \{1, \ldots, \ell\}$ which keeps track of its current lane. The initial value of each $s_i$ is chosen as in the case of [[$\mathsf{SLplatoon2}$]{}]{}, and $y_i$ is set to 1. The unsafe set is defined based on the distance to the preceding car [*on the same lane*]{}.
[[[HooVer]{}]{}]{} implementation and metrics {#sec:eval-mfhoo}
---------------------------------------------
Our implementation of the [[[HooVer]{}]{}]{} tool uses the MFHOO implementation presented in [@sen2019noisy] to solve the model checking problem of Equation (\[eq:SMCproblem\]). It works in two stages: First, it uses MFHOO to find the best partition ${{\mathcal{P}_{h,i}}}$ and a putative “best" (most unsafe) initial point $x \in {{\mathcal{P}_{h,i}}}$ with the maximum estimate for the probability of hitting the unsafe set ${\mathcal{U}}$. In the second stage, [[[HooVer]{}]{}]{} uses additional simulations to do a Monte Carlo estimation of the probability ${{p_{k,{\mathcal{U}}}{(x)}}}$. In the experiments reported below, a constant number of $26$K simulations are used in all experiments in the second stage.
To achieve the theoretically optimal performance, MFHOO requires the smoothness parameters $\rho$ and $\nu$ which are unknown for our benchmarks. To circumvent this [[[HooVer]{}]{}]{} chooses several parameter configurations ($3$ sets in our experiments), runs an instance of MFHOO for each, and returns the result with the highest hitting probability. For each instance of MFHOO, we set a time budget which is the maximum time allowed to be consumed by the simulator.
#### Metrics.
We report the regret of [[[HooVer]{}]{}]{}. In order to calculate the regret, first we have to calculate the actual maximum hitting probability for each benchmark. This is computed using Prism [@HKNP06] which uses numerical and symbolic analysis. The regret is the difference between the exact probability and the estimated probability. Then, we report the running time and memory usage. The memory usage is measured by calculating the total size of the Python object which contains the constructed tree and all other data of MFHOO. We also report the number of [*queries*]{} for each method, which is total number of simulations used in both stage 1 and 2. All the experiments are conducted on a Linux workstation with two Xeon Silver 4110 CPUs and 32 GB RAM.
Table \[tab:results\] shows the running time, the memory usage, the number of queries, and the resulting actual regret for [[[HooVer]{}]{}]{} using MFHOO as well as [[[HooVer]{}]{}]{}(1) using HOO. On every benchmark, [[[HooVer]{}]{}]{} gives low regrets. With the same simulation budget, [[[HooVer]{}]{}]{} devotes longer simulations in the interesting parts $\Theta$, as a consequence, it is usually faster than [[[HooVer]{}]{}]{}(1) as shown in Figure \[fig:rescaling\_comparison\].
Model Method Time(s) Memory(MB) \#queries Regret
------- ----------------------- --------- ------------ ----------- -------------------------
Storm 68.88 1019 NA 0
PlasmaLab 37.73 NA 749711 0.0017 $\pm$ 0.0020
[[[HooVer]{}]{}]{}(1) 59.37 9.13 31381 0.0011 $\pm$ 0.0025
[[[HooVer]{}]{}]{} 24.03 6.23 **29415** **0.0002 $\pm$ 0.0020**
Storm 89.10 1974 NA 0
PlasmaLab 22.61 NA 749711 0.0022 $\pm$ 0.0021
[[[HooVer]{}]{}]{}(1) 44.52 7.30 30315 0.0038 $\pm$ 0.0113
[[[HooVer]{}]{}]{} 26.46 4.53 **28520** **0.0010 $\pm$ 0.0018**
Storm NA OOM NA NA
PlasmaLab 49.14 NA 749711 0.0032 $\pm$ 0.0026
[[[HooVer]{}]{}]{}(1) 110.64 9.37 31555 0.0016 $\pm$ 0.0025
[[[HooVer]{}]{}]{} 76.35 5.27 **28955** **0.0007 $\pm$ 0.0043**
: Comparison of [[[HooVer]{}]{}]{}, Storm and PlasmaLab. [[[HooVer]{}]{}]{}(1) corresponds to using HOO (i.e. the full fidelity algorithm) in stage 1 of [[[HooVer]{}]{}]{}. For regret of [[[HooVer]{}]{}]{} and PlasmaLab, we run the experiments for 10 times and report the “mean $\pm$ std". For all the experiments, $|\overline{\Uptheta}| = 4096$.[]{data-label="tab:results"}
Comparison with other model checkers {#sec:eval-mfhoo}
------------------------------------
We compare the performance of [[[HooVer]{}]{}]{} with other model checkers. Prism [@HKNP06] and Storm [@dehnert2017storm] are leading probabilistic model checkers for Markov chains and MDPs and compute exact probability of reaching the unsafe states. As Storm has the same functionality as Prism and we found it to be much more efficient than Prism in all our experiments, we only compare [[[HooVer]{}]{}]{} with Storm. PlasmaLab [@legay2016plasma] is one of the few [*statistical*]{} model checkers that can handle MDPs. For probability estimation problems, PlasmaLab uses smart sampling algorithm [@d2015smart] to efficiently assign the simulation budgets to each scheduler and then estimates the probability for the putative “best" scheduler.
#### Discretizing and scaling the benchmarks.
The theory for MFHOO is based on a continuous state spaces, however, most model checking tools, including the ones mentioned above, are designed for discrete state space models and they do not support floats as state variables. Therefore, a direct comparison of the approaches on the same model is infeasible, and we created equivalent, discretized (quantized) versions of the benchmarks.
In [[[HooVer]{}]{}]{}, the algorithm keeps partitioning $\Theta$ hierarchically and stops at a depth $h$, which can be considered as searching over all the $2^h$ partitions at depth $h$. In order to make a fair comparison, we make sure that the discrete version of the benchmark has $2^h$ initial states, i.e. $|\overline{\Theta}| = 2^h$ where $\overline{\Theta}$ is the discretized initial state space.
Before stating the discretizing process, we give two key observations of the benchmarks. First, considering the nature of our benchmarks, it is obvious that if we scale the velocities, distance thresholds and initial distances by a constant factor, the probability of hitting unsafe set doesn’t change. Second, taking [[$\mathsf{SLplatoon2}$]{}]{} as an example, we set all the constants (velocities and distance thresholds) in the model as integers, which leads to the function ${{p_{k_{\mathit{max}},{\mathcal{U}}}{(x)}}}$ shown in Figure \[fig:singlelane\]. It’s clear that for any state $x \in {\mathcal{X}}$, there exists an integer state $\overline{x}$ such that ${{p_{k_{\mathit{max}},{\mathcal{U}}}{(x)}}} = {{p_{k_{\mathit{max}},{\mathcal{U}}}{(\overline{x})}}}$. If we make sure that for each integer interval in the original continuous space, the discretized space has an value in that interval, then the maximum probability doesn’t change. With these observations, we discretize the benchmark as follows. First, we sample $2^h$ points uniformly in the continuous initial state space. Due to the second observation mentioned above, if $2^h$ is larger than the number of integer points in the original space, the maximum probability doesn’t change. Then, we find an integer scaling factor $c$ such that by multiplying $c$ all the $2^h$ points become integer points. Other constants in the model are also multiplied by $c$. According to the first observation, scaling with a constant doesn’t change the probability. Thus, we now have a model where all state variables are integers and it has the same maximum probability as the original model. Then we evaluate Storm and PlasmaLab on this new model.
All of the model checkers mentioned above support the Prism [@HKNP06] language. Thus, we implement each benchmark in Prism language, and then we check the equivalence between the Prism implementation and the Python implementation by calculating and comparing the probability ${{p_{k_{\mathit{max}},{\mathcal{U}}}{(x)}}}$ at all integer points $x$.
For Storm, we report the running time and the memory usage. These metrics are directly measured by the software itself. For PlasmaLab, we report the running time. We do not report the memory usage of PlasmaLab because it doesn’t provide an interface for that and it is hard to track the actual memory used by the algorithm inside the JAVA virtual machine. We also report the regret for PlasmaLab. We use the term “regret" here just for simplicity, which also refers to the difference between the estimated probability and the exact probability.
The results of [[[HooVer]{}]{}]{}, PlasmaLab and Storm are summarized in Table \[tab:results\]. We show in Figure \[fig:rescaling\_comparison\] how the performance of different methods changes as the $|\overline{\Uptheta}|$ increases. As the size of the initial state space increases, the memory usage of Storm grows quickly, which limits its application on large models. In contrast, [[[HooVer]{}]{}]{} and PlasmaLab scale well. The running time of PlasmaLab is determined by the parameters of the smart sampling [@d2015smart] algorithm. We use the same parameters regardless of $|\overline{\Uptheta}|$, and thus the running time of PlasmaLab is almost constant.
As shown in Table \[tab:results\], Storm fails to check the [[$\mathsf{MLplatoon}$]{}]{} model due to the memory limitation. Compared with PlasmaLab, [[[HooVer]{}]{}]{} requires more running time. However, we note that PlasmaLab is a considerably more mature tool than [[[HooVer]{}]{}]{}. As shown in Table \[tab:results\], compared to PlasmaLab, [[[HooVer]{}]{}]{} requires much fewer queries to reach comparable regrets, which attests to the sample efficiency of our proposed method.
![*Left*: Comparison of [[[HooVer]{}]{}]{} and [[[HooVer]{}]{}]{}(1) on [[$\mathsf{SLplatoon3}$]{}]{}. (Here, we only consider the number of queries in stage 1, because that for stage 2 is just a constant.) *Middle*: Performance of different methods as the $|\overline{\Uptheta}|$ increases for [[$\mathsf{SLplatoon3}$]{}]{}. *Right*: Performance of different methods as the $|\overline{\Uptheta}|$ increases for [[$\mathsf{MLplatoon}$]{}]{}.[]{data-label="fig:rescaling_comparison"}](comparison_HOO_MFHOO.pdf "fig:"){width="32.00000%"} ![*Left*: Comparison of [[[HooVer]{}]{}]{} and [[[HooVer]{}]{}]{}(1) on [[$\mathsf{SLplatoon3}$]{}]{}. (Here, we only consider the number of queries in stage 1, because that for stage 2 is just a constant.) *Middle*: Performance of different methods as the $|\overline{\Uptheta}|$ increases for [[$\mathsf{SLplatoon3}$]{}]{}. *Right*: Performance of different methods as the $|\overline{\Uptheta}|$ increases for [[$\mathsf{MLplatoon}$]{}]{}.[]{data-label="fig:rescaling_comparison"}](rescaling_SLplatoon3_time_mem.pdf "fig:"){width="32.00000%"} ![*Left*: Comparison of [[[HooVer]{}]{}]{} and [[[HooVer]{}]{}]{}(1) on [[$\mathsf{SLplatoon3}$]{}]{}. (Here, we only consider the number of queries in stage 1, because that for stage 2 is just a constant.) *Middle*: Performance of different methods as the $|\overline{\Uptheta}|$ increases for [[$\mathsf{SLplatoon3}$]{}]{}. *Right*: Performance of different methods as the $|\overline{\Uptheta}|$ increases for [[$\mathsf{MLplatoon}$]{}]{}.[]{data-label="fig:rescaling_comparison"}](rescaling_MLplatoon_time_mem.pdf "fig:"){width="32.00000%"}
Conclusions {#sec:conc}
===========
In this paper, we formulated the statistical model checking problem for a special type of MDP models as a multi-armed bandit problem and showed how a hierarchical optimistic optimization algorithm from [@sen2019noisy] can be used to solve it. In the process, we modified the existing definition of near-optimality dimension to accommodate the non-smoothness of the typical functions we have to optimize and recovered the regret bounds of the algorithm. We created several benchmarks, developed a SMC tool [[[HooVer]{}]{}]{}, and experimentally established the sample efficiency and scalability of the method.
In order to enlarge the area of application of this method, it is necessary to study more general temporal properties for [NiMC]{} beyond bounded time safety. In this paper, we focused on a very special type of MDP models with nondeterminism only in the initial set. The results suggest that it will be interesting to study black-box optimization algorithms for model checking more general MDP models.
Multi-Fidelity Parallel Optimistic Optimization (MFPOO) {#app:MFPOO}
=======================================================
The optimal smoothness parameters are generally not known. Algorithm \[Al:MFPOO\] [@sen2019noisy] adaptively searches for the optimal smoothness parameters by releasing several MFHOO instances with various $\nu$ and $\rho$ values.
: $\nu_{max}$ and $\rho_{max})$, $\Lambda(.)$, $\lambda(.)$ Let $N=(1/2)D_{max}\log(\Lambda/\log{\Lambda})$, where $D_{max}=\log{2}/\log({1/\rho_{max}})$ Spawn MFHOO with parameters $(\nu_{max}, \rho_i = \rho_{max}^{N/(N-i-1)})$ with budget $(\Lambda-N\lambda(1))/N$ Let $\hat{x}_{\Lambda,i}$ be the point returned by the $i^{th}$ MFHOO instance for $i \in \{0, .., N-1\}$. Evaluate all $\{\hat{x}_{\Lambda,i}\}_i$ at $z = 1$.\
The point $\hat{x}_{\Lambda} = \hat{x}_{\Lambda, i^*}$ where $i^*=\underset{i}{\mathrm{argmax}}\ f(x_{\Lambda,i}) + \epsilon_i$.
The following theorem gives the upper bound for the algorithm \[Al:MFPOO\].
\[Th:regret\_MFPOO\] If Algorithm \[Al:MFPOO\] is run with parameters $(\nu_{max}\geq \nu^*)$, $(\rho_{max}\geq \rho^*)$ and given a total cost budget $\Lambda$, then the simple regret of at least one of the MFHOO instances spawned by Algorithm \[Al:MFPOO\] is bounded a follows $$\begin{aligned}
R(\Lambda) = O\left( (\dfrac{\nu_{max}}{\nu^*})^{D_{max}}(\dfrac{B\log n(\Lambda/\log{\Lambda})}{n(\Lambda/\log{\Lambda})})^{\frac{1}{d(\nu^*,\rho^*)+2}}\right).\end{aligned}$$
It is noted that if the Algorithm \[Al:MFHOO\] is run at full fidelity case $z=1$, then it will be equivalent to a budgeted HOO algorithm. At this case the simple regret bound will be: $$R(\Lambda) = O\bigg((\dfrac{\log{(\Lambda/\lambda(1))}}{{\Lambda/\lambda(1)}})^{\frac{1}{d(\nu,\rho)+2}}\bigg).$$
Proofs {#app:proof}
======
*(Theorem \[th:mf\_bias\_func\])* We define a new Markov chain $\mathcal{M}(k)$ that starts from $x$ (like ${\mathcal{M}}$) and stops by going entering a new absorbing state called [*Stop*]{}, in $k$ steps, in expectation. That is, from any transient state $s_i$, the one step transition probability of hitting $\mathit{Stop}$ is $\frac{1}{k}$. For any other transient state $s_j$, the one step probability of transitioning from $s_i$ to $s_j$ is $(1-\frac{1}{k})P_{ij}$. Therefore, the probability transition matrix $P_k$ for the new Markov chain $\mathcal{M}(k)$ is as follows. $$\begin{aligned}
P_k= \left(
\begin{array}{c|cc}
Q_k & R_k & \frac{1}{k}\textbf{1}\ \\
\hline
{\bf 0} & I & {\bf 0} \\
{\bf 0} & {\bf 0} & 1 \\
\end{array}
\right) =
\left(
\begin{array}{c|cc}
(1-\frac{1}{k})Q & (1-\frac{1}{k})R & \frac{1}{k}\textbf{1}\ \\
\hline
{\bf 0} & I & {\bf 0} \\
{\bf 0} & {\bf 0} & 1 \\
\end{array}
\right),\end{aligned}$$ where $\textbf{1}$ is a $q\times 1$ vector whose elements are $1$, and the $\mathit{Stop}$ state is the last state. $P_k$ is a $(q+u+1)\times (q+u+1)$ row stochastic matrix. Then after $t$ steps, $P_{k}^{t}$ would be $$\begin{aligned}
P_k^t= \left(
\begin{array}{c|cc}
Q_k^t & \sum_{j=0}^{t-1} Q_k^{j}R_k & \frac{1}{k}\sum_{j=0}^{t-1} Q_k^{j}\textbf{1}\
\\\hline
{\bf 0} & I & {\bf 0} \\
{\bf 0} & {\bf 0} & 1 \\
\end{array}
\right).\end{aligned}$$
For the Markov chain $\mathcal{M}(k)$, ${{p_{k,{\mathcal{U}}}{(s_i)}}}$ is the probability that the chain is absorbed to any of the states in the set ${\mathcal{U}}$, given that chain starts at a transient state $s_i$. Using Theorem \[thm:3\], ${{p_{k,{\mathcal{U}}}{(s_i)}}}$ and ${{p_{k_{\mathit{max}},{\mathcal{U}}}{(s_i)}}}$ can be written for ${\mathcal{M}}$ and $\mathcal{M}(k_{\mathit{max}})$, respectively: $$\begin{aligned}
{{p_{k,{\mathcal{U}}}{(s_i)}}} = c^TN_{k}R_{k}\textbf{1}, \ \
{{p_{k_{\mathit{max}},{\mathcal{U}}}{(s_i)}}} = c^TN_{k_{\mathit{max}}}R_{k_{\mathit{max}}}\textbf{1},\end{aligned}$$ where $c$ is a selection $m\times1$ vector whose $i$^th^ entry is $1$ and all other entries are $0$. $N_k = (I-Q_k)^{-1}$ and $\textbf{1}$ is a $u\times 1$ vector of $1$’s. Subtracting, we have $$\begin{aligned}
\label{eq:prob_bound}
|{{p_{k_{max},{\mathcal{U}}}{(s_i)}}}-{{p_{k,{\mathcal{U}}}{(s_i)}}}| = |c^T(N_{k_{max}}R_{k_{max}}-N_{k}R_{k})\textbf{1}|.\end{aligned}$$ Without loss of generality we can assume that $Q$ is diagonalizable[^5]. Let $D$ the diagonal matrix of eigenvalues of $Q$, i.e. $D = \mbox{diag}(\lambda_i)$, where $\lambda_i$ are the eigenvalues of $Q$ and let $E$ be the matrix such that $Q = EDE^{-1}$. Using this and definition of $Q_k$, we can rewrite $N_k=(I-Q_{k})^{-1}$ as $$\begin{aligned}
\label{eq:Nn}
N_{k} = E(I-(1-\frac{1}{k})D)^{-1}E^{-1} = E\ \mbox{diag}\bigg(\frac{k}{k-(k-1)\lambda_i}\bigg)\ E^{-1}.\end{aligned}$$ Rewriting $N_{k_{\mathit{max}}}$ in the same way the right hand side of (\[eq:prob\_bound\]) can be expressed as $$\begin{aligned}
\label{eq:prob_bound_final}
& = |\ c^T\ E\ \mbox{diag}\bigg(\frac{k_{max}-k}{(k_{max}-(k_{max}-1)\lambda_i)(k-(k-1)\lambda_i)}\bigg)\ E^{-1}\ R\ \textbf{1}\ |\\
& \leq \frac{\|c\|_{\infty}\|E\|_{\infty}\|E^{-1}\|_{\infty}\|R\|_{\infty}\|\textbf{1}\|_{\infty}(k_{max}-k)}{(k_{max}-(k_{max}-1)\lambda_{max})(k-(k-1)\lambda_{max})}
\\
&\leq \frac{\kappa_{\infty}(E)(k_{max}-k)}{(k_{max}-(k_{max}-1)\lambda_{max})(k-(k-1)\lambda_{max})}. \label{eq-bias-fn}\end{aligned}$$ In deriving the last two inequalities, we used $\|c\|_{\infty} = 1$, $\kappa_{\infty}(E) = \|E\|_{\infty}\|E^{-1}\|_{\infty}$ and $\|\textbf{1}\|_{\infty} = 1$. In addition, we have used the fact that $R$ is a row-stochastic matrix, that is, its rows sum up to less than unity, and thus, $\|R\|_{\infty} \leq 1$.
Benchmarks {#sec:app:benchmarks}
==========
All our benchmarks are variants of [[$\mathsf{SLplatoon2}$]{}]{} discussed in Section \[example1\]. The executable models are available from [@SMCOO-online:2020]. [[$\mathsf{SLplatoon3}$]{}]{} models a platoon of $m$ cars where in every time step, each car can choose to move with one of [*three*]{} speeds: $\mathsf{vbrake}$, $\mathsf{vcruise}$, and $\mathsf{vspeedup}$. The first vehicle always moves at some constant speed. For all the others, these three actions are chosen probabilistically, according to probability distributions that depend on their distance to the preceding vehicle. For example, if the distance $s_{i-1} - s_i$ is less than a threshold constant $\mathsf{th\_close}$ then the speed is chosen according to a probability distribution $\mathsf{pclose}$.
The state variables of the model are defined as follows: for the $i^{th}$ car, we denote $s_i \in \mathbb{R}$ as the position along the lane. With out loss of generality, we assume $s_1> s_2 > \ldots > s_m$. We also define an auxiliary variable $\mathit{gap}_i$ for all $i>1$ as the distance to the preceding car, i.e. $\mathit{gap}_i = s_{i-1} - s_i$. The set of initial states and the unsafe set have the same definition as in [[$\mathsf{SLplatoon2}$]{}]{}.
The constants in the model are defined as follows: $\mathsf{th\_far}$ and $\mathsf{th\_close}$ are some distance thresholds; $\mathsf{vbrake}$, $\mathsf{vcruise}$ and $\mathsf{vspeedup}$ are some velocities; $\mathsf{pclose}$, $\mathsf{pfine}$ and $\mathsf{pfar}$ are probability distributions for different modes. For example, $\mathsf{pclose}$ is the probability distribution over three actions, “brake", “cruise" and “speed up", and we denote $\Pr(v=\mathsf{vbrake}) = \mathsf{pclose}[brake]$ as the probability of choosing action “brake" when this probability distribution is used.
With these variables, the behavior of each car at every time step is described in Fig. \[fig:Slplatoon3\].
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[ ]{}
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[[$\mathsf{MLplatoon}$]{}]{} models a platoon of $m$ cars on $\ell$ lanes where in every time step, each car can choose to move with one of [*five*]{} actions: moving forward with speed $\mathsf{vbrake}$, $\mathsf{vcruise}$ or $\mathsf{vspeedup}$, or chaging to the left or right lane. These actions are chosen probabilistically, according to probability distributions that depend on their distances to the vehicles on its current lane, left lane or the right lane.
The state variables of the model are defined as follows: for the $i^{th}$ car, we denote $s_i \in \mathbb{R}$ as the position along the lane and $y_i \in \{1, \ldots, \ell\}$ as the ID of the current lane. Then, we define some auxiliary variables that can be derived from the state variables. We denote $\mathit{d\_ahead}[i]$ as the distance to the preceding car on the same lane. If the $i^{th}$ car is the leading car on its current lane, then $\mathit{d\_ahead}[i] = \infty$. Then, we denote $\mathit{d\_left}[i]$ as the minimal $s$-distance (i.e. only considering the difference of the $s$ variables) to the cars on the left lane. If there is no car on the left lane, then $\mathit{d\_left}[i] = \infty$. If the $i^{th}$ car is on the left-most lane, i.e. $y_i = 1$, then $\mathit{d\_left}[i] = -\infty$. Similarly, we define $\mathit{d\_right}[i]$.
The definition of the initial state space of this model is a little bit different due to the $y$ variables. When choosing the initial state, all the $y$ variables are set to 1, i.e. all cars start from the left-most lane. Then, $(s_1, \ldots, s_m)$ is picked from a rectangle in $\mathbb{R}^n$ just as what we have done in [[$\mathsf{SLplatoon2}$]{}]{} and [[$\mathsf{SLplatoon3}$]{}]{}. Finally, we define the unsafe set ${\mathcal{U}}= \{(s_1, \ldots, s_m)\ |\ \exists i \in \{1, \ldots, m\}, \mathit{d\_ahead}[i] < \mathsf{unsafe\_rule}\}$.
The constants in the model are defined as follows: $\mathsf{th\_far}$, $\mathsf{th\_close}$ and $\mathsf{th\_clear}$ are some thresholds; $\mathsf{pturn}$ is the probability of chaning to the target lane if allowed to do that; $\mathsf{vbrake}$, $\mathsf{vcruise}$, $\mathsf{vspeedup}$, $\mathsf{pclose}$, $\mathsf{pfine}$ and $\mathsf{pfar}$ are defined with the same meaning as in [[$\mathsf{SLplatoon3}$]{}]{}.
With these variables, the behavior of each car at every time step is described in Fig. \[fig:Mlplatoon\].
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Example illustrating non-smoothness of hitting probabilities {#sec:app:nonsmoothness}
============================================================
In this section, we present an illustrative example to show that in general the function ${{p_{k,{\mathcal{U}}}{(x)}}}$ over $x \in \Theta$ is not a continuous function and has infinitely many maxima.
Consider a discrete time linear system with state space ${\mathcal{X}}=\mathbb{R}^m$, set of initial states $\Theta \subseteq{{\mathcal{X}}}$ and set of unsafe states given by ${\mathcal{U}}:= \{x\in {\mathcal{X}}\ | \ c^{T} x\leq a\}$ for some vector $c \in {{{\relax\ifmmode \mathbb R\else $\mathbb R$\fi}}}^m$ and $a \in {{{\relax\ifmmode \mathbb R\else $\mathbb R$\fi}}}$. The system evolves according to: $$\label{eq:LTI}
x_{t+1} = Ax_{t} +B W_{t},$$ where $x_{t}\in {\mathcal{X}}$ is the state vector at time $t$. $W_{t} \in {{{\relax\ifmmode \mathbb R\else $\mathbb R$\fi}}}^r$ is a discrete random vector with probability mass distribution $p_{W}$ at sequence $t$, and $A$ and $B$ are $m \times m$ and $m \times r$ matrices, respectively. Starting from a initial state $x_{0}\in\Theta$, the state of the system at any time $t$ can be written as $$x_{t} = A^{t}x_{0} +\sum_{i=0}^{t-1} A^{i}B W_{i}.$$ Suppose the system in (\[eq:LTI\]) hits the unsafe set first at sequence $t_{h}$, i.e. $t_{h}=min\{t\geq 0: x_t\in {\mathcal{U}}\}$. Other trajectories may reach the unsafe set at a time after $t_{h}$ or may not reach the unsafe set at all. Assume that the trajectories reaching the unsafe set will stay there after. Starting from $x_{0}$, suppose we are interested in the probability of hitting the unsafe set for the system in (\[eq:LTI\]) at sequence $n$, where $t_{h} < n$. Then the probability of reaching the unsafe set at sequence $n$ starting from $x_{0}$ can be expressed as $$\begin{aligned}
\mbox{Pr}\{x_{n}\in {\mathcal{U}}\} & = \mbox{Pr}\{c^{T} x_{n} \leq a\} = \mbox{Pr}\{c^{T} A^{n}x_{0} +\sum_{i=0}^{n-1} c^{T}A^{i}B W_{i} \leq a\} \\
& = \mbox{Pr}\{\sum_{i=0}^{n-1} c^{T}A^{i}B W_{i} \leq a - c^{T} A^{n}x_{0}\}.
\end{aligned}$$ Since $W_i$s are discrete-type random variable, $\sum_{i=0}^{n-1} c^{T}A^{i}B W_{i}$ is so, i.e. there is a finite set of values $\{v_{i} : i \in J\subset{\mathbb{N}}\}$ such that $\mbox{Pr}\{\sum_{i=0}^{n-1} c^{T}A^{i}B W_{i} \in\{v_{i} : i \in J\}\} = 1$. Therefore, $$\begin{aligned}
\mbox{Pr}\{x_{n}\in {\mathcal{U}}\} & = \sum_{k : v_{k} \leq a - c^{T} A^{n}x_{0}} \mbox{Pr}\{\sum_{i=0}^{n-1} c^{T}A^{i}B W_{i} = v_{k}\}.
\end{aligned}$$ Similarly the probability of reaching the unsafe set at sequence $n$ starting from initial state $x_{0}^{'} = x_{0} + \epsilon \in \Theta$ for some $\epsilon \in \mathbb{R}^m$ can be expressed as $$\begin{aligned}
\mbox{Pr}\{x'_{n}\in {\mathcal{U}}\} & = \sum_{k : v_{k} \leq a - c^{T} A^{n}x_{0} - c^{T} A^{n}\epsilon} \mbox{Pr}\{\sum_{i=0}^{n-1} c^{T}A^{i}B W_{i} = v_{k}\}.
\end{aligned}$$ Let $v_{min} = min(a - c^{T} A^{n}x_{0} - c^{T} A^{n}\epsilon,\ a - c^{T} A^{n}x_{0})$ and $v_{max} = max(a - c^{T} A^{n}x_{0} - c^{T} A^{n}\epsilon,\ a - c^{T} A^{n}x_{0})$, then we can write $$\label{eq:prob_sens}
\begin{aligned}
|\mbox{Pr}\{x_{n}\in {\mathcal{U}}\} - \mbox{Pr}\{x'_{n}\in {\mathcal{U}}\}| & = \sum_{k : v_{\mathit{min}} < v_{k} \leq v_{\mathit{max}}} \mbox{Pr}\{\sum_{i=0}^{n-1} c^{T}A^{i}B W_{i} = v_{k}\}.
\end{aligned}$$ Based on (\[eq:prob\_sens\]), for small enough $\epsilon$, in a norm sense, $v_{max} - v_{min}$ could be small such that, there exists no $v_{k}$ in the interval $(v_{min},\ v_{max})$ to make $|\mbox{Pr}(x_{n}\in {\mathcal{U}}) - \mbox{Pr}(x'_{n}\in {\mathcal{U}})| \neq 0$. In other words, for small enough $\epsilon$, $|\mbox{Pr}(x_{n}\in {\mathcal{U}}) -\mbox{Pr}(x'_{n}\in {\mathcal{U}})| = 0$. By increasing the $\epsilon$, once the gap $v_{max} - v_{min}$ is large enough, there exist a $v_{k}$ in the interval $(v_{min},\ v_{max})$ with nonzero $\mbox{Pr}(\sum_{i=0}^{n-1} c^{T}A^{i}B W_{i} = v_{k})$. This means that there will be discontinuities in the function of probability of hitting the unsafe set of the system (\[eq:LTI\]) in terms of the initial states. Here, discontinuities origin in the type of the random variable $W_t$, which is a discrete-type random variable. In addition, the hitting probability function would have infinitely many maxima. An example of this kind of function can be seen in Figure \[fig:singlelane\]. For the systems of this kind, neither the local smoothness Assumption \[Ass:MFHOO\], nor the assumption of finite number of maxima (or unique maximum) does not hold.
Regret: Theory and practice for single car example ( [[$\mathsf{Singlecar}$]{}]{})
==================================================================================
We define the [NiMC]{}of a single car. Consider a single car that moves in $\mathbb{R}^2$. Let $(s_1, s_2)$ be the position vector of the car in $\mathbb{R}^2$. The transition model of the car is as: $$(s_1, s_2) \leftarrow \left\{
\begin{array}{ll}
(s_1 + 1, s_2 + 1) & \ \ \mbox{w.p.} \ \ p\\
(s_1 + 1, s_2 - 1) & \ \ \mbox{w.p.} \ \ 1-p,\\
\end{array}\right.$$ where $p\in[0,1]$ and is unknown. While the state space ${\mathcal{X}}=\mathbb{R}^2$, the set of initial states $\Theta=\{(s_1,s_2)\in{\mathcal{X}}\ |\ s_1=0\ and\ s_2\in(-b, b)\}$, where $b\in\mathbb{R}_{>0}$. Let set of unsafe states be ${\mathcal{U}}=\{(s_1, s_2)\in{\mathcal{X}}\ |\ |s_2|\geq b\}$. Given that car starts its motion at a initial state in $\Theta$, the goal is to find the maximum probability of hitting the unsafe set ${\mathcal{U}}$.
![[[$\mathsf{Singlecar}$]{}]{} model with $b = 10$ and $p = 0.5$. Probability of hitting the unsafe set as a function of initial $s_2$ through different time horizons.[]{data-label="fig:scar_prob_func"}](probs_time_horizon_new.jpg){width="50.00000%"}
Figure \[fig:scar\_prob\_func\] represents the probability of hitting the unsafe set for states in the initial set through different time horizons for $b = 10$ and $p=0.5$. For large enough time horizons the probability of hitting the unsafe set for the initial states of set $\Theta$ approaches $1$, but, for smaller time horizons the maximum probability of hitting the unsafe set arises for initial states that are close to the unsafe set, meaning all initial states such that $s_1 = 0,\ s_2 \in (-10, -9]$ or $s_1 = 0,\ s_2 \in [9, 10)$. The probability of hitting the unsafe set, as it is seen in the Figure \[fig:scar\_prob\_func\], is not continuous and it has infinitely many maxima.
Now, Consider [[$\mathsf{Singlecar}$]{}]{} model with parameters $k=100$, $b = 10$ and $p = 0.5$. Clearly any $s2 \in [9,10) \cup (-10,-9]$ maximize the probability of hitting the unsafe set. In Figure \[fig:analysis\_slane\] ([*top-left*]{}) the mean of actual regret obtained by running the Algorithm \[Al:MFHOO\] with various smoothness parameters is presented. As budget increases the actual regret obtained by the algorithm approaches to $0$. For $\Lambda = 37.5$, the partitioning stops at $h_{max} = 13$. We can derive the number of $2\nu\rho^h$-near-optimal partitions for any $\nu$ and $\rho$. Let’s continue the analysis by setting $\nu = 0.02$ $\rho = 0.94$ which are used in generating the results of actual regret. Then for $h\in[6,13]$, the number of $2\nu\rho^h$-near-optimal partitions $\mathcal{N}_{h}(2\nu\rho^h) = \lceil 2^h/10 \rceil$. This is because the total number of partitions at depth $h$ is equal to $2^h$ and for $h\in[6,13]$, the $2\nu\rho^h$-near-optimal partitions belong to the interval $[9,10) \cup (-10,9]$ whose length is $1/10$. According to the Definition \[def:mod\_near\_opt\], for different values of $B$, different near-optimality dimensions can be obtained. We are looking for the pair of values $d_{m}$ and $B$ that minimizes the theoretical regret bound in Theorem \[Th:regret\_MFHOO\]. In Figure \[fig:analysis\_slane\] ([*top-right*]{}) the number of $2\nu\rho^h$-near-optimal partitions are presented for $h\in[6,13]$ that are upper bounded by $B\rho^{-d_{m}h}$ for different values of $d_{m}$ and their corresponding $B$. In Figure \[fig:analysis\_slane\] ([*bottom*]{}) the theoretical upper bound vs. $d_{m}$ and their corresponding $B$ for different values of smoothness parameters used in generating the actual regret are represented. As it is seen, the Best theoretical regret bounds that can be achieved is approximately $0.12$.
![[NiMC]{} [[$\mathsf{Singlecar}$]{}]{} with $k=100$, $b = 10$ and $p = 0.5$ *Top-left*: Actual regret fro running MFHOO averaged over various smoothness parameters. *Top-right*: Number of $2\nu\rho^h$ partitions for various $d_{m}$ values and their corresponding $B$ with $\nu = 0.02$ and $\rho = 0.94$. *Bottom-left*: Plot of $g(d_{m})$ vs. $d_{m}$ for various smoothness parameters, where $R(\Lambda) = O(g(d_{m}))$ as in the Theorem \[Th:regret\_MFHOO\]. *Bottom-right*: Plot of $g(B)$ vs. $B$ for various smoothness parameters, where $R(\Lambda) = O(g(B))$ as in the Theorem \[Th:regret\_MFHOO\].[]{data-label="fig:analysis_scar"}](actual_regret.jpg "fig:"){width="50.00000%"} ![[NiMC]{} [[$\mathsf{Singlecar}$]{}]{} with $k=100$, $b = 10$ and $p = 0.5$ *Top-left*: Actual regret fro running MFHOO averaged over various smoothness parameters. *Top-right*: Number of $2\nu\rho^h$ partitions for various $d_{m}$ values and their corresponding $B$ with $\nu = 0.02$ and $\rho = 0.94$. *Bottom-left*: Plot of $g(d_{m})$ vs. $d_{m}$ for various smoothness parameters, where $R(\Lambda) = O(g(d_{m}))$ as in the Theorem \[Th:regret\_MFHOO\]. *Bottom-right*: Plot of $g(B)$ vs. $B$ for various smoothness parameters, where $R(\Lambda) = O(g(B))$ as in the Theorem \[Th:regret\_MFHOO\].[]{data-label="fig:analysis_scar"}](cells_new.jpg "fig:"){width="50.00000%"} ![[NiMC]{} [[$\mathsf{Singlecar}$]{}]{} with $k=100$, $b = 10$ and $p = 0.5$ *Top-left*: Actual regret fro running MFHOO averaged over various smoothness parameters. *Top-right*: Number of $2\nu\rho^h$ partitions for various $d_{m}$ values and their corresponding $B$ with $\nu = 0.02$ and $\rho = 0.94$. *Bottom-left*: Plot of $g(d_{m})$ vs. $d_{m}$ for various smoothness parameters, where $R(\Lambda) = O(g(d_{m}))$ as in the Theorem \[Th:regret\_MFHOO\]. *Bottom-right*: Plot of $g(B)$ vs. $B$ for various smoothness parameters, where $R(\Lambda) = O(g(B))$ as in the Theorem \[Th:regret\_MFHOO\].[]{data-label="fig:analysis_scar"}](Regret_d_scar.jpg "fig:"){width="50.00000%"} ![[NiMC]{} [[$\mathsf{Singlecar}$]{}]{} with $k=100$, $b = 10$ and $p = 0.5$ *Top-left*: Actual regret fro running MFHOO averaged over various smoothness parameters. *Top-right*: Number of $2\nu\rho^h$ partitions for various $d_{m}$ values and their corresponding $B$ with $\nu = 0.02$ and $\rho = 0.94$. *Bottom-left*: Plot of $g(d_{m})$ vs. $d_{m}$ for various smoothness parameters, where $R(\Lambda) = O(g(d_{m}))$ as in the Theorem \[Th:regret\_MFHOO\]. *Bottom-right*: Plot of $g(B)$ vs. $B$ for various smoothness parameters, where $R(\Lambda) = O(g(B))$ as in the Theorem \[Th:regret\_MFHOO\].[]{data-label="fig:analysis_scar"}](Regret_B_scar.jpg "fig:"){width="50.00000%"}
[^1]: Generating execution data involves running simulations or performing tests.
[^2]: Finite number nondeterministic action choices can be encoded in the choice of the initial state.
[^3]: All the results in the paper generalize to bounded time properties of the form $P_{\geq \theta}(\psi)$ where $\theta$ is a threshold constant and $\psi$ is a path formula. Generalizing to nested probabilistic operators and unbounded time properties will require further research.
[^4]: The full analysis is given in Appendix \[sec:app:nonsmoothness\].
[^5]: Even if $Q$ is not diagonalizable, we can use Jordan decomposition and that will not affect the analysis beyond this point.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report on the dramatic effect of random point defects, produced by proton irradiation, on the superfluid density $\rho_{s}$ in superconducting Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ single crystals. The magnitude of the suppression is inferred from measurements of the temperature-dependent magnetic penetration depth $\lambda(T)$ using magnetic force microscopy. Our findings indicate that a radiation dose of 2$\times$10$^{16}$cm$^{-2}$ produced by 3 MeV protons results in a reduction of the superconducting critical temperature $T_{c}$ by approximately 10$\%$. In contrast, $\rho_{s}(0)$ is suppressed by approximately 60$\%$. This break-down of the Abrikosov-Gorkov theory may be explained by the so-called “Swiss cheese model”, which accounts for the spatial suppression of the order parameter near point defects similar to holes in Swiss cheese. Both the slope of the upper critical field and the penetration depth $\lambda(T/T_{c})/\lambda(0)$ exhibit similar temperature dependences before and after irradiation. This may be due to a combination of the highly disordered nature of Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ with large intraband and simultaneous interband scattering as well as the $s^\pm$-wave nature of short coherence length superconductivity.'
author:
- Jeehoon Kim
- 'N. Haberkorn'
- 'M. J. Graf'
- 'I. Usov'
- 'F. Ronning'
- 'L. Civale'
- 'E. Nazaretski'
- 'G. F. Chen'
- 'W. Yu'
- 'J. D. Thompson'
- 'R. Movshovich'
title: 'Magnetic penetration-depth measurements of a suppressed superfluid density of superconducting Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ single crystals by proton irradiation'
---
Introduction
============
Proximity of the superconducting and magnetic states in iron-based superconductors has stimulated extensive studies of the gap nature,[@Hashimoto; @Ishida; @Martin] order-parameter symmetry,[@Tanatar; @Kuroki; @Nakayama] and the pairing mechanisms in these materials.[@Mazin] The response of the superconducting condensate to impurities is sensitive to the symmetry of the superconducting state, and their influence has been widely investigated to gain better understanding of the nature of the order parameter in both low- and high-temperature unconventional superconductors. [@Anderson; @Pethick; @Annett; @Nakajima; @Kim]
The Abrikosov-Gor’kov (AG) theory[@Abrikosov] explains the effects of impurities in the low-$T_{c}$ superconductors, where a large superconducting coherence length $\xi$ effectively averages the suppression of order parameter at the impurity sites over many impurities, leading to a uniformly suppressed order parameter. However, the AG theory breaks down when applied to the effect of disorder on superconducting properties in the cuprates superconductors,[@Annett] where $\xi$ is short and comparable to the average spacing between disorder centers. The order parameter is therefore suppressed locally at the impurity site and has a chance to recover between impurities. The influence of disorder on the superfluid density $\rho_{s}$ in cuprates is well described by the so called “Swiss cheese” model, which considers spatial dependence of the order parameter and its strong suppression near defects.[@Byers; @Balatsky1; @Flatte; @Salkola; @Hettler; @Zhu; @Balatsky2] In iron-based systems, where superconductivity exhibits both $s$-wave characteristics and a small coherence length, the situation is between the low-temperature and high-temperature superconductors. Consequently these systems pose an intriguing question of how the effect of disorder on $T_{c}$ and the superfluid density in these compounds compares to that in conventional BCS superconductors and cuprates.[@Nakajima]
Recently, two irradiation experiments on Co-doped BaFe$_{2}$As$_{2}$ (Co-122) were performed to study the influence of disorder.[@Kim; @Nakajima] The temperature-dependent penetration depth measurements suggested an $s^{\pm}$ state, with strong nonmagnetic scattering in the unitary limit,[@Kim] whereas transport measurements showed an $s^{++}$ state with weak scattering in the Born limit.[@Nakajima] Both experiments showed a relatively small suppression of $T_{c}$ caused by nonmagnetic impurities induced by irradiation; these findings are consistent with an $s^{++}$ state, since superconductivity with a sign changing order parameter is quite sensitive to nonmagnetic impurities.[@Bang; @Vorontsov; @Gordon] Reports in several iron-arsenide systems by different experimental techniques are consistent with theoretical predictions of $s$ wave, potentially nodal s-wave or sign reversing s-wave.[@Kuroki; @Mazin]
In this work we investigate the influence of random point defects introduced by proton irradiation on $\lambda(T)$ in Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ (CNFA) single crystals. We use the magnetic force microscopy (MFM) technique to determine absolute values of $\lambda(T)$.[@Jeehoon; @PRB; @Jeehoon; @Jeehoon; @MgB2; @Xu1995; @Coffey1995] The CNFA single crystals, showing homogeneity, have been grown with a self-flux technique. Details of the sample preparation and characterization can be found elsewhere.[@Haberkorn]
Experiment
==========
The 3 MeV protons are known to produce between one and a few tens of atomic displacements,[@Civale] creating random point defects as well as nanoclusters with typical dimensions of few nanometers. The CNFA sample was irradiated with the total proton dose of 2$\times$10$^{16}$ cm$^{-2}$, which corresponds to an average distance ($d$) between defects of 2.8 nm.[@Haberkorn; @2012] The sample was cleaved, and its thickness measured to be around 28 $\mu$m, which is smaller than the penetration range of 40 $\mu$m for the 3 MeV proton beam. Electrical resistivity in both unirradiated and irradiated samples were measured using a standard four-probe technique. The sample was mounted in a rotatable probe and measurements were performed in magnetic fields varying between 0 and 9 T. MFM measurements described here were performed in a home-built low-temperature MFM apparatus.[@Nazaretski; @RSI; @2009] Three samples, CNFA, irradiated CNFA (ICNFA), and a Nb reference film were loaded and investigated in a comparative experiment within a single cool-down. The magnetic stray field calibration was performed by imaging vortices in a Nb reference as a function of applied magnetic field.[@Jeehoon; @PRB] Measurements of $\lambda$ were performed using the Meissner response technique.[@Jeehoon; @PRB; @Jeehoon; @Jeehoon; @MgB2] The Meissner response curves were first measured as a function of the tip-sample separation in the Nb reference with known $\lambda(T)$. Subsequently, the cantilever was moved to a sample of interest and the Meissner response curves were acquired. Direct comparison of measured curves yields the absolute value of $\lambda$ in a sample under investigation. Details of experimental technique are described elsewhere.[@Jeehoon; @PRB; @Jeehoon; @Jeehoon; @MgB2]
The reference Nb thin film ($T_{c}\approx$ 8.8 K) has a thickness of 300 nm and was grown by electron beam deposition. The $T_{c}$ of CNFA from transport measurements is 19.4 K and that of ICNFA is 17.8 K. The width of the superconducting transition did not change after irradiation. No upturn in resistivity was observed at low temperatures, indicating that irradiation by protons results in the formation of nonmagnetic point-like scattering centers.[@Martin; @2009] The MFM measurements were performed using a high-resolution Nanosensors cantilever[@Nanosensors] that was polarized along the tip axis in a 3 T magnetic field. Both Nb and CNFA samples were zero-field cooled for Meissner experiments; a magnetic field of a few Oe was applied above $T_{c}$, followed by cooling for vortex imaging experiments.
![\[f:Hc2\] (color online) Temperature dependent H$_{c2}$ along the $c$ axis and within the $ab$ plane in unirradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ (black triangles), taken from Ref. 24, and proton irradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ (red circles).](Figure1.eps){width="8.5cm"}
Results
=======
H$_{c2}(T)$ measurements
------------------------
Figure \[f:Hc2\] shows the upper critical field $H_{c2}(T)$ with ${\bf H}\parallel c$ and ${\bf H}\perp c$ (within the [*ab*]{} plane) for CNFA and ICNFA. $H_{c2}(T)$ is linear in both samples and for both directions, with average slopes of $\beta^{ab} = -\frac{\partial H^{ab}_{c2}}{\partial T}\big
|_{T_{c}}=$ 4 T/K and $\beta^c = -\frac{\partial
H^{c}_{c2}}{\partial T}\big |_{T_{c}}=$ 2.2 T/K for CNFA, and $\beta^{ab} = -\frac{\partial H^{ab}_{c2}}{\partial T}\big
|_{T_{c}}=$ 3.8 T/K and $\beta^c = -\frac{\partial
H^{c}_{c2}}{\partial T}\big |_{T_{c}}=$ 2.3 T/K for ICNFA. A modest superconducting anisotropy parameter $\gamma =
{{\beta^{ab}}\over{\beta^c}} = 1.85-1.65$ for both CNFA and ICNFA samples points toward a three-dimensional behavior. The superconducting coherence length $\xi$ can be expressed in the Ginzburg-Landau region as $\xi_{GL}(T) \approx
\xi_0/\sqrt{1-T/T_c}$. In the case of a one-band model or two weakly coupled bands with similar Fermi surface properties and pairing interactions the zero-temperature in-plane coherence length $\xi^{ab}_{0}$ and out-of-plane coherence length $\xi^{c}_{0}$ are given by the slope of the upper critical field:[@Haberkorn; @Bauer2009] $(\xi^{ab}_{0})^2\approx{\Phi_{0}}/{2\pi T_{c} \beta^{c}}$ and $(\xi^{c}_{0})^2\approx{\Phi_{0}}/{2\pi T_{c} \beta^{ab}}$. We obtain the zero-temperature Ginzburg-Landau values of $\xi_{CNFA}^{ab}(0)$ = 2.8 nm and $\xi_{ICNFA}^{ab}(0)$ = 2.8 nm, which are similar in magnitude to the short-coherence length cuprate and PuCoGa$_5$ superconductors. Within our measurement uncertainty no appreciable change of the coherence length took place after irradiation, although $T_{c}$ is suppressed by 10%.
![\[f:vortex\] (color online) Single vortex images in (a) the Nb reference, (b) the unirradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$, and (c) the irradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$. (d) Comparison of single vortex profiles obtained from (a), (b), and (c). All images were obtained under the same experimental conditions in a single-cool down with the tip lift height of 300 nm at 4 K. The color scale bar refers to (a)-(c).](Figure2.eps){width="9.0cm"}
![\[f:absolute\] (color online) Meissner response curves obtained from (a) the Nb reference (blue diamonds), (b) the unirradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ (green diamonds), and (c) the irradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ (red diamonds) at 4 K. The different slopes of the Meissner curves obtained from each sample indicate a systematic change of $\lambda$. The inset: The Meissner curves for the unirradiated and irradiated samples are shifted along the horizontal axis to overlay the Meissner curve for the reference Nb sample. The difference of the penetration depths $\Delta\lambda$ can be obtained from the values of the shift.](Figure3.eps){width="8.5cm"}
$\lambda(T)$ measurements
-------------------------
Prior to measurements of the absolute values of $\lambda(T)$, vortex images were obtained under the same experimental conditions for all samples. These measurements yield information about homogeneity of CNFA and ICNFA samples on a submicron scale ($\sim 100$ nm). The well-formed single vortices in Nb and CNFA suggest the homogeneity of the sample; however, the irregular shape of single vortex in ICNFA (elongated vortex in the diagonal direction of the image) suggests the presence of inhomogeneity in the superfluid density on a sub-micron scale, which may be related to impurities introduced from irradiation. We employed the following imaging procedure: First, a single vortex in the Nb sample was obtained at 4 K after the stray field calibration of the MFM system.[@Jeehoon; @PRB] Second, the MFM tip was moved on to CNFA and a single vortex image obtained, and third, a single vortex image was obtained after the tip was moved on to ICNFA as shown in Figs. \[f:vortex\](a), (b), and (c). The line profile for each of the single vortices is shown in Fig. \[f:vortex\](d). The intensity of the vortex center in different samples correlates with the magnitude of $\lambda$, since all images were taken under the same conditions and with the same tip. Lower intensity corresponds to a larger $\lambda$; therefore, $\lambda$ in ICNFA is much larger than that in the Nb reference. In addition, the magnitude of $\lambda$ among the superconducting samples can be inferred from the relative size of a single vortex: The larger the size, the larger is $\lambda$. Therefore, $\lambda$ in ICNFA, showing the largest vortex size, is the biggest among them.
To extract absolute values of $\lambda$ in ICNFA we performed the Meissner response measurements as described above. The Meissner curves as a function of the tip-sample separation were obtained in all three samples, Nb, CNFA, and ICNFA (see Fig. \[f:absolute\]). The decay rate of the frequency shift $\delta f$ as a function of the tip-sample separation $z$ provides the relative magnitude of $\lambda$, [*i.e.*]{}, the higher the rate $d(\delta f)/dz$ the larger the $\lambda$. In bulk and thick films, the Meissner response force obeys a universal power-law dependence with tip-to-sample distance.[@Xu1995; @Coffey1995] The force is given by $F_{Meissner}=A\times f(\lambda+z)$, where $z$ is the tip-to-sample distance, $A$ is a pre-factor containing information about the geometry of the magnetic tip, and $f(z)\sim
1/z^3$. By shifting the $f^{ICNFA}(z)$ data with respect to distance in order to overlay it with the $f^{Nb}(z)$ curve, one can obtain the absolute values of $\lambda_{ICNFA}(T)=\lambda_{Nb}(T)+\Delta\lambda(T)$, where $\Delta\lambda(T)$ is the magnitude of the shift. The shift $\Delta\lambda$ between the Nb and ICNFA data equals 320 nm, resulting in $\lambda_{ICNFA}(0)=\lambda_{Nb}(0)+\Delta\lambda(0)
= 110\;\text{nm} + 320\;\text{nm}=430$ nm. Using the same procedure we also obtained $\lambda_{CNFA}(0)=\lambda_{Nb}(0)+\Delta\lambda(0) =
110\;\text{nm}+150\;\text{nm} =260$ nm. Our experimental error is around $10\%$ and depends on the magnitude of $\lambda$ and the system noise level. A key result of this work is that the $\lambda(0)$ values before and after irradiation differ significantly. This is in stark contrast to both the coherence length $\xi$, which shows little change after irradiation, as well as the small suppression in $T_c$ of 10%. The Meissner force MFM measurements of the ICNFA sample were performed after cleaving followed by irradiation. The ICNFA sample was remeasured after polishing. Both measurements showed the same $\lambda$ within experimental uncertainty. This indicates that irradiation does not noticeably affect sample quality. Therefore we can neglect the degradation of the sample surface for Meissner screening currents. It should be noted that our parameter free method of using the Nb reference sample is based on the assumption of a universal scaling function $F(z)$ for the Meissner force. This approach is valid for type- superconductors, where the electromagnetic response is local, [*i.e.*]{}, $\kappa = \lambda/\xi \gg 1$. Here we neglected higher order corrections in $1/\kappa$. Our Nb film has $\kappa = \lambda/\xi=110 nm/10 nm\approx 10$. The large $\kappa$ value in Nb allows direct comparison of Meissner responses between the Nb reference and CNFA, which results in good agreement by overlaying the Meissner curves, shown as insets in Fig. \[f:absolute\] and Figs. \[f:lambda-T\](a)-(b). Our novel method of using a reference sample is justified a posteriori because the Meissner curves would not overlay with one another just by shifting them.
![\[f:lambda-T\] (color online) Temperature dependent Meissner curves for (a) the unirradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ and (b) the irradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ samples. Insets in (a) and (b) show overlaid temperature dependent Meissner curves at 4 K, validating our procedure for extracting $\lambda(T)$. (c) Temperature dependent $\lambda(T)$ in both unirradiated and irradiated Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ samples determined from (a) and (b). (d) $\lambda(T)$ from (c) normalized by the $T=0$ value as a function of the normalized temperature.](Figure4.eps){width="9.0cm"}
The temperature-dependent Meissner response curves measured in both CNFA and ICNFA samples are shown in Figs. \[f:lambda-T\](a) and (b). The gradual variation of the Meissner curves as a function of temperature indicates a systematic change of $\lambda(T)$. The insets in (a) and (b) show the Meissner curves obtained at different temperatures but shifted to lie on top of the Meissner curve taken at $T=4$ K; the curves overlay each other very well. The shift value for a given $T$ to $T=4$ K along the horizontal axis allows one to calculate $\lambda(T)$ at $T$. The resulting $\lambda(T)$ and normalized $\lambda(T)/\lambda(0)$ in both samples are shown in Figs. \[f:lambda-T\](c) and (d), respectively. Results indicate that $\lambda(T)$ increases after proton irradiation; however, the dependence of $\lambda(T)/\lambda(0)$ on the normalized temperature $T/T_c$ is the same for both samples within our experimental uncertainty. The penetration depth exhibits the typical power-law behavior $\Delta\lambda(T)/\lambda(0) \sim T^n$ with $n \approx 2$ reported previously for doped iron-arsenide superconductors.[@Martin]
Discussion
==========
The radiation dose of 2$\times$10$^{16}$ cm$^{-2}$ produced by 3 MeV protons in a Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ sample causes the suppression of the superfluid density $\rho_{s}(0)\approx
1/\lambda^{2}(0)$ by about 60$\%$ whereas $T_{c}$ is only suppressed by 10$\%$. We plot the value of the normalized $\rho_s(0)$ for ICNFA as a solid circle in the Uemura plot[@Uemura; @1989] of disordered superconductors in Fig. \[f:swiss\], as well as theoretical results of one-band AG for d-wave pairing (solid line) and two-band AG calculations for $s^{\pm}$ pairing (red open circles).[@Vorontsov] Also shown are the BdG (Bogoliubov-de Gennes) calculations for $d$-wave pairing (red hatched circles), a Swiss cheese model far from the AG theory. [@Das] Our result bears similarity to the data for self-irradiated PuCoGa$_5$ [@Ohishi2007] and He-irradiated YBCO high-temperature superconductor, showing that $T_{c}$ is strongly immune to disorder relative to $\rho_s(0)$,[@Basov1994; @Moffat1997] contrary to the conventional AG theory for $d$-wave paring. By analogy we argue that the break-down of the AG theory is accounted for by the Swiss cheese model within the BdG lattice theory of short-coherence length superconductors,[@Das] which shows an abrupt suppression of the order parameter near point defects. This model describes the spatial dependence of the local density of states and the order parameter in the vicinity (within a few lattice constants) of a point-like nonmagnetic impurity in the strong scattering limit, similar to holes in Swiss cheese. Franz and coworkers[@Franz] also reported the break down of the AG theory and strong suppression of $\rho_{s}(0)$ for $d$-wave paring. The effect is stronger in samples with small $\xi/a_{0}$ ratio ($a_{0}$ is the lattice constant). In the opposite limit, the AG theory is valid and the order parameter is then suppressed uniformly in the entire sample because $\xi \gg a_0, d $, with $d$ the average distance between impurities. In our sample the ratio of $\xi_{0}/a_{0}$ is approximately 7, $\xi\approx 2.8$ nm and $d$ is about 2.8 nm, justifying the Swiss cheese scenario.
It is worth noting that the $T$ dependence of $\lambda(T)$ remains the same after irradiation as shown in Fig. \[f:lambda-T\] (d), while it changes in cuprates.[@Szotek; @Prohammer1991; @Kim1994] This discrepancy may result from the nature of the multiband $s$-wave paring as well as the highly disordered nature of CNFA on the Ca/Na sites which lie above and below the iron layer. The fact that the temperature behavior of $\lambda(T)$ is robust after irradiation may be ascribed to large intraband scattering with $s^\pm$ pairing and that the system itself is already in the “dirty” limit prior to irradiation, consistent with its short coherence length and power-law dependence of $\lambda(T)$. Additional disorder (mostly in the iron layer) by proton irradiation therefore has little impact on the temperature behavior of $\lambda(T)$, while added interband scattering is detrimental to (increases) the absolute magnitude of $\lambda(0)$.
![\[f:swiss\] (online color) Uemura plot of the superfluid density in disordered short coherence length superconductors. $T_{c0}$ and $\rho_{s0}$ are values obtained from a pristine crystal; $T_{c}$ and $\rho_{s}$ are those measured after irradiation. The solid circle represents the proton irradiated CNFA obtained in this work. For comparison we plot results of the one-band AG and BdG (Swiss cheese) calculations[@Das] for $d$-wave pairing, two-band AG $s^\pm$-wave calculations[@Vorontsov], and experimental results for self-irradiated PuCoGa$_5$ [@Ohishi2007] and helium irradiated YBCO samples. [@Basov1994; @Moffat1997]](Figure5.eps){width="7.5cm"}
The pair-breaking effect due to nonmagnetic scattering in the AG theory can be quantitatively analyzed using the normalized scattering rate in conjunction with $\lambda$ given by: $g^{\lambda}={\hbar\Delta\rho_{0}}/({2\pi
k_{B}T_{c0}\mu_{0}\lambda^{2}_{0}})$, where $\Delta\rho_{0}$ is residual resistivity change induced by irradiation, $\Delta\rho_{0}=\rho^{irr}_{0}-\rho^{unirr}_{0}$, $T_{c0}$ is the critical temperature before irradiation, and $\lambda_{0}$ is the penetration depth of the unirradiated sample.[@Nakajima] The parameter $g^{\lambda}$ and $T_{c0}$ are expressed as $\text
{ln}(T_{c0}/T_{c})=\psi(1/2+g^\lambda
T_{c0}/(2T_{c}))-\psi(1/2)$, where $\psi(x)$ is the digamma function, based on the $s^{\pm}$ scenario.[@Chubukov] This pair-breaking result for $T_c$ is similar to that for conventional $s$-wave with magnetic impurities or $d$-wave with nonmagnetic impurities. Here the critical scattering rate parameter, where superconductivity vanishes, is $g=g^{\pm}\approx$ 0.28 in the $s^{\pm}$ pairing state. The extrapolated critical scattering parameter, obtained using $\Delta\rho_{0}=$30 $\mu\Omega$ cm and $\lambda=$260 nm, is $g^{\lambda}_{exp}\approx 3.7$. This value is much larger than that expected in the $s^{\pm}$ scenario, quantifying the break-down of the AG theory in irradiated iron-arsenide superconductors, where the approximation of the uniformly impurity-averaged Green’s function is not valid. Similar results were reported in Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ irradiated by protons[@Kim] and illustrate the generality of the Swiss cheese model for pair-breaking in this large class of high-temperature superconductors.
Conclusion
==========
We reported the influence of random point disorder produced by proton irradiation on the superfluid density in Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$. It leads to a dramatic change of $\lambda(0)$ after irradiation, in contrast to the small variation of $T_{c}$ and predictions by the AG theory. Both $\xi(T)$ and $\lambda(T)$ show similar temperature behavior before and after irradiation. This behavior may be understood within the Swiss cheese model, the pair-breaking nature of $s^\pm$ interband superconductivity, and a short coherence length, which considers the spatial dependence of the order parameter and its strong suppression near defects at the atomic scale. Finally, the extracted normalized scattering rate, in conjunction with the absolute value of $\lambda(T)$, is much larger than the critical scattering rate for the $s^{\pm}$ pairing, confirming the break-down of the AG theory in these disordered superconductors. Further detailed multiband BdG model calculations combined with systematic doping and irradiation studies may shed light on the suppression of superconductivity in this large class of iron-based superconductors.
Work at LANL was supported by the US Department of Energy, Basic Energy Sciences, Division of Materials Sciences and Engineering. Work at Brookhaven was supported by the US Department of Energy under Contract No. DE-AC02-98CH10886. Work by G.F.C. and W.Y. (fabrication of samples) was supported by the NSFC under Grants No. 10974254 and No. 11074304, and by the National Basic Research Program of China under Grants No. 2010CB923000 and No. 2011CBA00100. N.H. is member of CONICET (Argentina).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Bloch oscillations are a fundamental property of wavepackets subject to an external field in a lattice. The period of oscillation is set by the magnitude of this field, and it is independent of the shape and the nature of the band in which the wavepacket is created. Here we show that Bloch oscillations can be directly related to a topological invariant. This invariant characterizes the evolution operators describing the wavepacket evolution under a gauge transformation. Using a general Floquet framework to describe quantum walks, we unveil a new class of sub-oscillations within a Bloch period, whose number is given by the topological invariant. Our findings allow a direct implementation in photonic setups, which provide a new protocol to measure certain topological invariants.'
author:
- 'Lavi K. Upreti'
- 'C. Evain'
- 'S. Randoux'
- 'P. Suret'
- 'A. Amo'
- 'P. Delplace'
bibliography:
- 'Windingbib.bib'
title: Topological swing in Bloch oscillations
---
Bloch oscillations, the oscillatory motion of an electron subject to a constant electric field in a periodic potential, are one of the most fascinating effects of adiabatic quantum transport. Initially introduced by Zener in the context of quantum electrons in crystals [@Bloch1929; @Zener1934], Bloch oscillations are found in a wide variety of physical systems such as semiconductor superlattices [@Voisin1988; @Feldmann1992; @Leo1992], trapped cold atoms [@Dahan1996] and photonics systems [@Pertsch1999; @Wimmer2015]. From fractional Bloch oscillations [@Claro2003; @Corrielli2013] to super Bloch oscillations [@Alberti2009; @Haller2010], their declension reveals the richness of wavepackets dynamics in periodic structures, captured within a simple semiclassical picture. A recent case of interest has been the study of Bloch oscillations in bands with nontrivial Berry curvature [@Longhi2007; @Szameit2010; @Atala2013; @Cominotti2013; @Flaschner2016; @Li2016] or Berry-Zak phases [@Holler2018]. In this case, the geometrical or topological properties of the bands alter the oscillation dynamics and can give rise to intricate wavepacket evolutions.
The appearance of anomalous velocities and other topological effects in a lattice subject to periodic driving is another striking property of wavepacket dynamics. From quantized Thouless pumping to Floquet topological insulators, periodically driving a parameter in lattice Hamiltonians has proved to be a very powerful way of generating nontrivial topological features. A fundamental question is whether Bloch oscillations can appear in Hamiltonians subject to periodic driving and whether such oscillations can possess topological traits. Bloch oscillations have indeed been considered in the context of cyclic driving in photonic [@Wimmer2015; @Zhang2018] but their eventual relation to topological invariants remains to be established.
In a lattice Hamiltonian subject to a periodic drive, different regimes can be encountered depending on the frequency-scale of the driving in comparison to the hopping energy between sites. At high frequencies, a clever driving of the Hamiltonian gives rise to an artificial gauge field and the appearance of bands characterized by a non-zero Chern number [@GomezDelplace; @Goldman2014]. At frequencies comparable to the hopping energies, the cyclic driving can give rise to anomalous topological phases, characterized by topological gap invariants with unidirectional [@Rudner2013] or helical [@CarpentierPRL2015] edge states. At further lower frequencies, in the adiabatic regime, phenomena such as Thouless pumping results in the quantized drift of particles [@Thouless1983; @Kitagawa2010]. The drift originates from the anomalous group velocity intimately related to the Berry curvature of the Bloch bands, and can be used to measure it [@Dudarev2004; @Pettini2011; @Ozawa2014; @Lohse2016; @Nakajima2016; @Wimmer2017]. Remarkably, the manifestation of the Berry curvature through the motion of wavepackets is one of the few existing tools to probe the geometrical and topological properties of the bulk bands [@Aidelsburger2014], even in non-periodic systems [@Kraus2012; @Baboux2017]. These examples show the intimate relationship between periodic driving and the emergence of geometrical and topological properties of wavepackets in a lattice.
Here we establish a connection between Bloch oscillations and the topological features associated with periodic driving. We report a new topological property of the motion of a wavepacket in a lattice subject to a cyclic driving that manifests in a new class of Bloch *sub-*oscillations. We use a general Floquet framework to describe 1D quantum walks that is accessible in current photonic setups. Remarkably, both the usual Bloch oscillations and the sub-oscillations are found to be governed by a winding number of the evolution operator and are, therefore, of topological origin. Our framework provides a clear picture of the interplay between this topological *swing* and standard Bloch oscillations. It establishes a strong parallel between Bloch oscillations and Thouless pumping as they both reflect two complementary topological aspects of wavepackets dynamics.
The model we consider is sketched in Fig. \[fig:sc4\]. It consists of an oriented scattering network that describes a periodic discrete-time evolution of a quantum state or a wavepacket. The links between the nodes of the network have a preferential orientation (from top to bottom) that accounts for the direction of the flow with the time of an input signal. It is thus formally equivalent to a 1D quantum walk and describes, for instance, the evolution of a light pulse injected in a 1D lattice of birefringent beamsplitters [@Kitagawa2012], periodically coupled waveguides [@Bellec2017] or coupled fiber rings [@Wimmer2017; @Weidemann2020]. At each time step $j$ and position $l$, the scattering of the wave amplitudes along the rightward $\alpha_{l}^{j}$ and leftward $\beta_{l}^{j}$ links occurs at a node of coordinates $(l,j)$, and is parametrized by a dimensionless parameter $\theta_j$ entering the unitary matrix $$\begin{aligned}
\label{eq:sj}
S_{j} = \begin{pmatrix}
\cos\theta_{j} & i\sin\theta_{j}\\
i\sin\theta_{j}& \cos\theta_{j}
\end{pmatrix} \ .
\end{aligned}$$ In addition to these scattering processes, we introduce a phase shift $\phi_j$ carried along by the states in each link, as in [@Wimmer2017]. Without any loss of generality, we consider a non-zero phase shift for the leftward states only (in blue in Fig. \[fig:sc4\]). The key point is that the value of this phase is allowed to vary at each time step $j$ within a cyclic time period of $N$ steps. It can, therefore, be regarded as a periodic driving parameter of the scattering matrix. The outgoing amplitudes at time $j+1$ are related to the incoming amplitudes at time $j$ as $$\begin{aligned}\label{eq:timestep}
\alpha_{l}^{j+1}&=(\cos \theta_{j} \alpha_{l+1}^{j} + i \sin \theta_{j} \beta_{l+1}^{j}) {{\rm e}}^{i \phi_{j}}\\
\beta_{l}^{j+1}&=(i \sin \theta_{j} \alpha_{l-1}^{j} + \cos \theta_{j} \beta_{l-1}^{j}) \, .
\end{aligned}$$
![2D oriented scattering lattice where the vertical axis plays the role of time. A dashed rectangle emphasizes the unit cell of this lattice.[]{data-label="fig:sc4"}](figure1){width="0.7\columnwidth"}
Assuming discrete translation invariance along the $x$-direction, the system can be treated in the Bloch-Floquet formalism. The corresponding (Floquet) unitary evolution operator after a periodic sequence of $N$ steps reads: $$\begin{aligned}
\label{eq:floquet_def}
&U_F(k,\{\phi_j \} )= (B_{\text{mod}(N,2)}S_N D_{N}) ....(B_0S_2D_{2})(B_1S_1D_{1}), \\
&\ B_1=\begin{pmatrix}
1 & 0\\
0 & {{\rm e}}^{-i k}
\end{pmatrix},\quad
B_0=\begin{pmatrix}
{{\rm e}}^{i k} & 0\\
0 & 1
\end{pmatrix}, \quad
D_{j}=\begin{pmatrix}
{{\rm e}}^{i \phi_{j}} & 0\\
0 & 1
\end{pmatrix}\nonumber
\end{aligned}$$ where $k$ is the (dimensionless) quasimomentum in the $x$-direction. The eigenvalues of Eq. \[eq:floquet\_def\] decompose as $\lambda={{\rm e}}^{i \varepsilon}$, where $\varepsilon$ will be hereafter referred to as the (dimensionless) quasienergy. The phases $\phi_j$ are chosen to be proportional to a phase of reference $\phi$ by a rational number, i.e. $\phi_j=(m_j/n_j)\phi$. The Floquet operator $U_F(k,\phi)$ then depends on two periodic variables, the quasimomentum $k$ and the phase $\phi$, which can be considered as a synthetic dimension. Thus, the quasienergies $\varepsilon(k,\phi)$ span a synthetic 2D Brillouin zone (BZ). Note that for two-steps period ($S_1$ and $S_2$ in Fig. \[fig:sc4\], $ N=2 $), the system reduces to former models studied experimentally in topological photonics, such as arrays of coupled waveguides with $\phi=0$, where topological anomalous edge states were observed [@Bellec2017], and coupled fiber loops with $\phi_2=-\phi_1$ (zero net phase over a period), where the Berry curvature of the synthetic bands was measured [@Wimmer2017].
An interesting situation arises when imposing a pattern of phase shifts $\phi_j$ along a time period of $N$ steps such that the net phase $\phi_{net}\equiv\ \sum_{j=1}^N \phi_j $ does not vanish. This is the phase gained after a period, and it can be interpreted as a periodic kick when considering the dynamics of a wavepacket. The condition $\phi_{net}\neq 0$ breaks inversion symmetry in the synthetic dimension $\phi$. Hence, in the full synthetic BZ, it also breaks the generalized inversion symmetry $U_F(-k,-\phi) = \sigma_x U_F(k,\phi) \sigma_x$, with $ \sigma_{x} $ being the standard Pauli matrix (see Ref. \[\]). Remarkably, this symmetry breaking leads to a winding of all the quasienergy bands with respect to $\phi$, as illustrated in Fig. \[fig:winding\](a). A similar quasienergy winding was reported when considering periodically driven trapped cold atoms with a different protocol [@Gong2016].
![(a)Quasienergy spectrum with a winding $ \nu_{\phi} = -2$ obtained for a scattering model with two-steps per period for $ \theta_{1} =\pi/4 $, $ \theta_{2} =\pi/4-0.6 $, $\phi_1=\phi$ and $\phi_2=-2\phi$. (b) Values of $\nu_\phi$ for integer values of $m_i/n_i$.[]{data-label="fig:winding"}](figure2.pdf){width="\columnwidth"}
{width="100.00000%"}
For simplicity, let us keep our focus on two-steps period ($N=2$), so that the two distinct phase shifts at each step read $ \phi_1 =(m_1/n_1)\phi $ and $ \phi_2 = (m_2/n_2)\phi $. The size of the BZ in the $\phi$ dimension thus depends on the choice of $m_i$ and $n_i$. Let us introduce the period $\Phi$ of the quasienergy with respect to the phase variable i.e. $\varepsilon(k,\phi+\Phi)=\varepsilon(k,\phi)$. This period is related to the least common multiple (LCM) of a combination of $m_i$ and $n_i$ in the following way $\Phi=4\pi\, \text{LCM}\left[(m_1/n_1 -m_2/n_2)^{-1},(m_1/n_1 + m_2/n_2)^{-1}\right]$ (see Ref. ). This allows us to define the winding of the quasienergies along $\phi$ as $$\begin{aligned}
\label{eq:nuphi}
\nu_{\phi} \equiv \sum_{p=1}^N \frac{1}{2\pi }\int_{0}^{\Phi} \dd\phi \frac{\partial \varepsilon_p(k,\phi)}{\partial \phi} \ .
\end{aligned}$$ This winding number is a topological property of the Floquet evolution operator, as it reads as an element of the homotopy group $\pi_1[U(N)]=\mathbb{Z}$ $$\begin{aligned}
\label{eq:nuphi2}
\nu_{\phi}
= \frac{1}{2\pi i}\int_{0}^{\Phi} \dd\phi \tr \Big[U_{F}^{-1}\partial_{\phi} U_{F}\Big] \in 2\mathbb{Z}\ .
\end{aligned}$$ Note that this is an even integer in our specific case due to the even number of bands (two) in our model. A direct calculation leads to the simple result $$\begin{aligned}
\nu_{\phi} = \dfrac{\Phi}{2\pi} \left(\dfrac{m_1}{n_1} + \dfrac{m_2}{n_2}\right),
\label{eq:winding}
\end{aligned}$$ which remarkably does not depend either on $k$ (since the winding of a quasienergy band $\varepsilon_p(k,\phi)$ along $\phi$ must be the same for any $k$) or on the scattering amplitudes $\theta_j$. Instead, it is proportional to the net phase $(\phi_1+\phi_2)/\phi$ that breaks inversion symmetry. A phase diagram representing the different possible values of $\nu_\phi$ as a function of $m_i/n_i$ is shown in Fig. \[fig:winding\](b).
A striking consequence of the winding of the quasienergy bands is the unconventional dynamics of the wavepackets in position space when adiabatically increasing the coordinate $\phi$. In the following, we show how these dynamics reveal a new kind of Bloch oscillations described by the winding number $\nu_\phi$. Figure \[fig:osc\](b) shows the $j$-time evolution of a Gaussian wavepacket injected at $j=0$ in the blue band of Fig. \[fig:winding\](a) at $k=0$, when $\phi$ is adiabatically increased from 0 to $\Phi=4 \pi$, with $\phi(j)=\gamma_0 j$ where the rate $\gamma_0=2\pi/2000$ \[see Fig. \[fig:osc\](a)\]. To compute the spatio-temporal dynamics, we apply Eq. to the initial wavepacket. The wavepacket periodically oscillates in space coordinate while keeping $k$ constant. This can be readily seen in Fig. \[fig:osc\](c)-(f), where we show the 2D Fourier transform of the wavepacket after having evolved to the time step indicated by the horizontal lines in Fig. \[fig:osc\](b). These panels provide a phenomenological understanding of the mechanism behind the oscillations: as $\phi$ is adiabatically increased, the band dispersions are displaced in a diagonal direction in $(k,\varepsilon)$ space \[green arrows in Fig. \[fig:osc\](c)-(f)\], a direct consequence of the winding of the bands (see also figure \[fig:winding\] (a)). Therefore, the group velocity $v_g=\frac{\partial\varepsilon}{\partial k}$ of a wavepacket with a given $k$ changes sign when $\phi(j)$ increases, resulting in oscillations in the spatial coordinate.
It is worth stressing that two distinct drivings are present in our model: (i) a fast cyclic driving of the phases $\phi_1$, $\phi_2$ within a Floquet period, which confers a nontrivial winding to the bands; (ii) a slow adiabatic increase of the phase $\phi$ which results in the oscillations. An analytical calculation of the centre of mass trajectory $X_c(t,k)$ of the wavepacket initially injected at a given $k$ can be inferred from the group velocity of the quasienergy bands in parameter space (see Ref. ): $$\begin{aligned}
X_c(t,k) = \gamma_0\int_0^t \dd \tau\, v_g(\phi(\tau),k),
\label{eq:classique}
\end{aligned}$$ where the continuous-time variable $t$ extrapolates the discrete one $j$. This semiclassical trajectory is shown in black dashed lines in Fig. \[fig:osc\](b), which fits the simulation plot perfectly.
More importantly, the observed oscillatory phenomenon establishes a direct connection between the winding of the bands in our Floquet-Bloch model and the usual Bloch oscillations in a periodic crystal subject to a constant electric field. Indeed, the adiabatic increase of the phase shift $\phi$ at a rate $\gamma_0$ when the time steps $j$ increases is analogous to a time-dependent vector potential that induces a (fictitious) electric field [@Krieger1986] $E$ and, therefore, should result in Bloch oscillations. This was already noticed in the case of a single-step time evolution ($N=1$) by Wimmer and co-workers [@Wimmer2015], who reported a gauge transformation relating the dynamics of a wavepacket in a lattice subject to a static potential gradient (i.e., a constant electric field), and the dynamics in a lattice subject to an adiabatic increase of the parameter $\phi$ (see also [@Supplementary]). To establish the connection between the winding of the quasienergy bands and Bloch oscillations, we note that, according to Eq. , the time periodicity $T_B$ of the center of mass motion $X_c$ is inherited from the periodicity of the quasienergy with respect to $\phi$. Accounting for the rate $\gamma_0$ between time and phase variables, one infers that $T_B = \Phi /\gamma_0$. This directly relates the time period of the oscillations to the winding number associated to the quasienergy bands via Eq. as $$\begin{aligned}
T_B = \frac{2\pi}{\gamma_0} \frac{\nu_\phi}{\frac{m_1}{n_1}+\frac{m_2}{n_2}},
\label{eq:period}
\end{aligned}$$ where negative values of $\nu_\phi$ correspond to mirror symmetric trajectories to those with $|\nu_\phi|$.
In Eq. , we recognize the usual period $T_B$ for Bloch oscillations induced by an average constant electric field $E=(E_1+E_2)/2$ where $E_j=\frac{m_j}{n_j}\frac{\gamma_0}{2}$ is the fictitious electric field applied during the time step $j$ (see Ref. [@Supplementary] for more details), except that in Eq. , this standard relation is modified by the winding number $\nu_\phi$. In particular, the period $T_B=2\pi/ E$ of the usual Bloch oscillations is recovered for $\left|\nu_\phi\right|=2$, a situation in which each band winds once, as reported in Figs. \[fig:winding\](a) and \[fig:osc\](b).
Beyond this standard case, our model predicts a novel kind of topological oscillations: higher winding numbers may not only change the period $T_B$, but also yield more complex oscillations with additional turning points within $T_B$. Two examples are shown in Fig. \[fig:osc\](g-h) for values of $m_i$, $n_i$ resulting in bands of windings $\nu_\phi=6$ and $8$, respectively, and same oscillating period $T_B$ as in Fig. \[fig:osc\](b). Remarkably, in a period $T_B$, the number of turning points is found to be precisely $\mathcal{N}_{t}=|\nu_\phi |$ (see Ref. [@Supplementary]). This result confers a topological nature to Bloch oscillations. Note that the standard ones simply have two turning points per period (see Fig. \[fig:osc\](b)), in agreement with $\mathcal{N}_t=2=|\nu_\phi|$.
![(a) Two-steps scattering network with the next nearest coupling in the second step. A dashed black rectangle emphasizes the unit cell of this lattice. (b) Quasienergy bulk spectrum for the model depicted in (a) with $\theta_1=\pi/4$, $\theta_2=\pi/4-0.6$ and $\phi_1=-\phi_2$. (c) Quantized displacement of the mean particle position with associated winding numbers $\nu_k$.[]{data-label="fig:sck"}](figure4){width="48.00000%"}
So far, we have considered windings of the bands induced by periodic pumping in the synthetic dimension. We now show that a winding of the quasienergy bands along the $k$-direction can similarly be induced when inversion symmetry is broken in the spatial dimension, and it results in a different topological phenomenon: quantized displacement of the mean particle position. This effect can be straightforwardly implemented in scattering network models by connecting next-nearest neighbor nodes, as sketched in Fig. \[fig:sck\](a) for a two-step time evolution (see Ref. [@Supplementary] for the step evolution equations). For the sake of generality, we have kept the phase $\phi_{j}$, which we take as $\phi_1=\phi=-\phi_2$, such that $\nu_\phi=0$, i.e., there are no Bloch oscillations. The corresponding quasienergy bands, displayed in Fig. \[fig:sck\](b), show a winding along $k$ for each $\phi$. This feature is captured by a winding number of the Floquet operator along $k$, analogous to that defined in Eqs. - for $\phi$. More generally, when considering even further long range couplings, this winding number is found to read [@Supplementary]: $$\begin{aligned}
\nu_{k} = \dfrac{\kappa}{2\pi} \left(\dfrac{r_1}{s_1} + \dfrac{r_2}{s_2}\right)
\label{eq:windingk}
\end{aligned}$$ where $\kappa$ is the periodicity of the bands in $k$ and ${r_j}/{s_j}$ is related to the range of the couplings between nodes to the left or to the right at each time step $j$. For the case illustrated in Fig. \[fig:sck\](a), ${r_1}/{s_1}=1$, ${r_2}/{s_2}=-2$.
In the spirit of the seminal work of Thouless [@Thouless1983], and as revisited by Kitagawa *et al.* [@Kitagawa2010] within the Floquet formalism, this winding number $\nu_{k}$ can be related to the mean displacement of particles after $P$ Floquet periods $T$, in a state where all the bands are uniformly excited, that is [@Supplementary]: $$\begin{aligned}
\Delta x =-P \dfrac{2\pi}{\kappa}\nu_k
\end{aligned}$$ Despite this apparent similarity, this quantized transport property differs from the usual Thouless pumping that results from an *adiabatic* driving of the system. In that case, the quantization can be expressed as a Chern number of the slowly driven instantaneous filled states parametrized over the effective 2D BZ $(k,t)$. This Chern number was later reinterpreted as a sum of the winding numbers in $k$ over the filled bands [@Kitagawa2010]. In the adiabatic regime, if this sum runs over all the bands, as in our case, then the Chern numbers of each band sum up to zero, and there is no drift. Quantized drifts obtained for our *non-adiabatic* scattering model are shown in Fig. \[fig:sck\](c). More generally, quasienergy windings along both $\phi$ and $k$ coordinates can coexist, leading to quite complex drifted Bloch oscillations for wavepackets shown in the Supplementary Material [@Supplementary].
Our study unveils the topological aspects of Bloch oscillations and extends them to a family of oscillatory phenomena accessible in artificial systems such as arrays of photonic waveguides and coupled fibers. It generalizes straightforwardly to periodically driven lattices of ultracold atoms where a protocol to generate quasienergy windings and oscillations was proposed [@Gong2016], although neither the winding number $\nu_\phi$ nor its relation to the number of Bloch sub-oscillations was identified. This direct relation between the number of turning points within an oscillation period and the winding number of the bands provides a new protocol to measure topological invariants in systems described by a quantum walk.
*Acknowledgements*– The authors are thankful to Benoit Douçot for his very constructive comments. This work was supported by the French Agence Nationale de la Recherche (ANR) under grant Topo-Dyn (ANR-14-ACHN-0031), the Labex CEMPI (ANR-11-LABX-0007), the CPER Photonics for Society P4S, the I-Site ULNE project NONTOP and the *Métropole Européenne de Lille* via the project TFlight.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In recent papers, the theory of representations of finite groups has been proposed to analyzing the violation of Bell inequalities. In this paper, we apply this method to more complicated cases. For two partite system, Alice and Bob each make one of $d$ possible measurements, each measurement has $n$ outcomes. The Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements $d$, but the quantum bound is dependent both on $n$ and $d$. Even so, when $d$ is large enough, the quantum bound is only dependent on $d$. The subset of probabilities for four parties based on the choice of six orbits under group action is derived and its violation is described. Restricting the six orbits to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived. Moreover, all the corresponding nonlocal games are analyzed.'
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[**Keywords:**]{} Bell inequality; Group Theory; Multipartite Systems; Nonlocal Game
Introduction
============
The Bell inequalities are compelling examples of essential differences between quantum and classical physics [@bell]. They are characterized by three parameters, the number of parties($N$), the number of measurement settings($M$) and the number of outcomes for each measurement($K$) [@clauser; @clauser1; @kaz; @WW; @CGLMP; @SLK; @cabello]. The famous Bell inequality, the Clauser-Horne-Shimony-Holt(CHSH) inequality [@clauser], is for the case $N=M=K=2$. The usual form of the CHSH inequality is: $$S=E(QS)+E(RS)+E(RT)-E(QT)\leq 2,$$ where $Q,R$ are measurements sent to Alice by a referee, $S,T$ are are measurements sent to Bob by same referee, Alice and Bob perform their measurements simultaneously and then return their results $+1$ or $-1$ to the referee, $E(\cdot)$ is the expectation value of the product of the outcomes of the experiment. But, in quantum mechanics, the upper bound $2\sqrt{2}$ of $S$ can be attained which is larger than $2$, and CHSH violation is therefore predicted by the theory of quantum mechanics. For general $N$ and $M=K=2$, the Bell inequalities were structured by Werner and Wolf [@WW]. For $N=2$, $M=2$, and general $K$, the Bell inequalities were found by Collins, Gisin, Linden, et al. [@CGLMP], then Son, Lee and Kim generalized this situation to multipartite arbitrary dimensional systems with $M=2$ [@SLK].
Recently, there appeared interesting papers [@ugur], [@ugur1], [@Bolonek] and [@BS]. In these papers the method of group representations theories has been proposed as a tool to analyzing the quantum mechanical violation of Bell inequalities.
In this paper, we apply the group theory to analyzing the violation of Bell inequalities for more complicated cases. In Sec. \[sec2\], the scenario for two parties is considered. Alice and Bob share some state ${\left|\phi\right>}$, and Alice performs one of $d$ measurements sent by a referee on her part of the state, Bob does similar operation. Then Alice and Bob return their measurement results $v_A(s)$ and $v_B(t)$ to referee, $v_A(s)$ and $v_B(t)$ take values in set $\{0,1,\cdots, n-1 \}$, arbitrarily $n$ is a natural number. Two chosen orbits under an group action approach a Bell inequality. In this section, we will see that the classical bound is independent of $n$, but the quantum bound is dependent both on $n$ and $d$. More interesting conclusion is that when the number of measurements is large enough, the quantum bound is only dependent on $d$. In Sec. \[sec3\], the cases of $N=4$, $M=2$ and $K=4$ was analyzed. The Bell inequality is constructed based on the choice of group orbits. In Sec. \[sec4\], restricting the six orbits in Sec. \[sec3\] to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived for three partite system. For all scenarios, the corresponding nonlocal games are all analyzed.
Two partite systems {#sec2}
===================
Suppose we have two parties, Alice and Bob, and their joint states are elements of a tensor product Hilbert space $\mathbb{C}^n\otimes {\mathbb{C}}^n$, with each ${\mathbb{C}}^n$ is spanned by the orthonormal basis $\{{\left|0\right>}, {\left|1\right>}, \cdots, {\left|n-1\right>} \}$. Each of them can measure one of $d$ observables, and for each observable the possible values for the result of the measurement are $0$, $1$, $\cdots$ or $n-1$. Alice’s observables are $a_{j}$, Bob’s are $b_{j}$, $j=0,1,\cdots,d-1$.
For the $n\times n$ translation operator $T$, there has a spectral decomposition $$\begin{aligned}
\label{Tn}
T &=& | w_0 \rangle\langle w_0|+e^{-i2\pi /n} | w_1 \rangle\langle w_1|+e^{-i4\pi /n} | w_2 \rangle\langle w_2| \nonumber\\
&& + \cdots + e^{i4\pi /n} | w_{n-2} \rangle\langle w_{n-2}|+ e^{i2\pi /n} | w_{n-1} \rangle\langle w_{n-1}|,\end{aligned}$$ where $$\label{w_j2}
|w_j \rangle=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1} e^{i2\pi jk /n}{\left|j\right>}, \hspace{5mm} \langle w_j|w_k\rangle=\delta_{jk}$$ for any $j,k=0,1,\cdots,n-1$. Chose an operator $U$ such that $U^d=T$. Thus $\{j\rightarrow U^j \mid j=0,1,\cdots, nd-1 \}$ is a representation of the cyclic group $\mathbb{Z}_{nd}$, the group of integers ${\mathbb{Z}}$ modulo $nd$. We have $d$ basis $\{{\left|v_j^{k}\right>}=U^k{\left|j\right>} | j=0,1,\cdots,n-1 \}$, for any $k=0,1,\cdots,d-1$, corresponding to Alice’s observables $a_k$ and Bob’s observables $b_k$. $a_k=\sum_{j=0}^{n-1}j|v_j^{k}\rangle\langle v_j^{k}|$, similarly for $b_k$, $k=0,1,\cdots,d-1$.
Under the group action $\alpha:\{U^j\otimes U^j|j=0,1,\cdots,nd-1\}\times {\mathbb{C}}^n\otimes {\mathbb{C}}^n\longrightarrow {\mathbb{C}}^n\otimes {\mathbb{C}}^n$, $\alpha(U^j\otimes U^j,{\left|\psi\right>})=U^j\otimes U^j{\left|\psi\right>}$, we choose two states: $$\label{2orbits}
{\left|0v_1^0\right>}\hspace{5mm} {\text and} \hspace{5mm} {\left|0v_1^1\right>},$$ each orbit has $nd$ elements, and the two orbits are distinct with each other. The sum of probabilities corresponding to these states give some Bell inequalities. From local realistic theory, they read $$\small{
\begin{split}
&\sum_{k=0}^{n-1}\sum_{j=0}^{d-1} P(a_j=k,b_j=k\oplus1)+\sum_{k=0}^{n-1}\sum_{j=0}^{d-2} P(a_j=k,b_{j+1}=k\oplus1)\\
& +\sum_{k=0}^{n-1} P(a_{d-1}=k,b_{0}=k\oplus2) \leq 2d-1,
\end{split}}$$ where $\oplus$ means the addition modulo $n$, $P(a_j=k,b_j=k\oplus1)$ means the probability of the event when Alice take measure $a_j$ and obtain value $k$, Bob take measure $b_j$ and obtain value $k\oplus1$. But in quantum mechanics, the bound of the sum of these probabilities can attain a larger value.
In order to get the quantum mechanics bound, we need to find a special state ${\left|\phi\right>}$, such that the expectation value $\langle \phi |O | \phi\rangle$ is maximum, where $$\label{Belleq O2}
O= \sum_{j=0}^{nd-1}(U\otimes U)^j({\left|0v_1^0\right>}{\left<0v_1^0\right|}+{\left|0v_1^1\right>}{\left<0v_1^1\right|})(U^{\dag}\otimes U^{\dag})^j.$$ When ${\left|\phi\right>}$ is the eigenstate of $O$ corresponding to the maximum eigenvalue, the expectation value $\langle \phi |O | \phi\rangle$ attain the maximum one. So the question is reduced to how to calculate the maximum eigenvalue of $O$.
Note that the eigenstates of $U\otimes U$ are states of the form ${\left|w_kw_l\right>}$ for $k,l=0,1,2,\cdots, n-1$, and all eigenvalues are degenerate. There has a spectral decomposition for $ U\otimes U$, $$U\otimes U= \sum_\lambda \lambda P_\lambda,$$ where $P_\lambda$ is the projector onto the eigenspace of $U\otimes U$ with eigenvalue $\lambda$, and $P_\lambda$ satisfy properties $\sum_\lambda P_\lambda=Id$ and $P_\lambda P_{\lambda'} =\delta_{\lambda\lambda'}P_\lambda$. Thus the operator $O$ can be simplified as follows, $$\begin{aligned}
O &=& \sum_{j=0}^{nd-1}(U\otimes U)^j({\left|0v_1^0\right>}{\left<0v_1^0\right|}+{\left|0v_1^1\right>}{\left<0v_1^1\right|})(U^{\dag}\otimes U^{\dag})^j \nonumber\\
&=& \sum_{j=0}^{nd-1}(\sum_\lambda \lambda P_\lambda)^j({\left|0v_1^0\right>}{\left<0v_1^0\right|}+{\left|0v_1^1\right>}{\left<0v_1^1\right|})(\sum_{\lambda'} \lambda'^* P_{\lambda'} ^{\dag})^j \nonumber\\
&=& \sum_{j=0}^{nd-1} \sum_\lambda \lambda^j P_\lambda ({\left|0v_1^0\right>}{\left<0v_1^0\right|}+{\left|0v_1^1\right>}{\left<0v_1^1\right|}) \sum_{\lambda'} ( \lambda'^*)^j P_{\lambda'}\nonumber\\
&=& nd\sum_\lambda P_\lambda ({\left|0v_1^0\right>}{\left<0v_1^0\right|}+{\left|0v_1^1\right>}{\left<0v_1^1\right|}) P_{\lambda}.\end{aligned}$$ Therefore, in order to calculate the eigenvalues of $O$, we only need to diagonalize it within the subspaces corresponding to the eigenvalues of $U\otimes U$. Denote by $L(\lambda)$ the subspace spanned by all eigenvectors $\{u_\lambda^{\lambda_j} \}$ of $U\otimes U$ corresponding to the eigenvalue $\lambda$. The eigenvector corresponding to the maximum eigenvalue of $O$ lies in the subspace $P_\lambda ({\left|0v_1^0\right>}{\left<0v_1^0\right|}+{\left|0v_1^1\right>}{\left<0v_1^1\right|}) P_{\lambda}$ when $L(\lambda)$ has maximum dimension.
[**The case I**]{}
To be connivent, we suppose that $n>1$ is odd. And choose $$\begin{aligned}
U &=& | w_0 \rangle\langle w_0|+e^{-i2\pi /dn} | w_1 \rangle\langle w_1|+e^{-i4\pi /dn} | w_2 \rangle\langle w_2| \nonumber\\
&& + \cdots + e^{i4\pi /dn} | w_{n-2} \rangle\langle w_{n-2}|+ e^{i2\pi /dn} | w_{n-1} \rangle\langle w_{n-1}|.\end{aligned}$$
In this case, the eigenvector corresponding to the maximum eigenvalue of $O$ lies in the subspace when $\lambda=1$.
That is to say, we shall calculate the maximum eigenvalue of $P_{1}({\left|0v_1^0\right>}{\left<0v_1^0\right|}+{\left|0v_1^1\right>}{\left<0v_1^1\right|})P_{1}$, denoting this operator by $R$. Set $$\begin{aligned}
{\left|\psi_1\right>}=P_{1}{\left|0v_1^0\right>}, \hspace{5mm} {\left|\psi_2\right>}=P_{1}{\left|0v_1^1\right>}.\end{aligned}$$ Suppose that the eigenvectors of $R$ corresponding to eigenvalue $\mu$ is $\sum_{j=1}^{2}x_j{\left|\psi_j\right>}$, then for any $k=1,2$, we have $\sum_{j=1}^{2}x_j\langle \psi_k | \psi_j\rangle=\mu x_k$. Denote the matrix $M=(\langle \psi_k | \psi_j\rangle)_{k,j=1}^2$. The eigenvalues of $M$ are exactly the ones of $R$. With the help of Wolfram Mathematica 8.0, we quickly obtain the matrix $M$ as the following form $$\left(
\begin{array}{cc}
\frac{1}{n} & \frac{1}{n^2} (1+X) \\
\frac{1}{n^2} (1+X) & \frac{1}{n} \\
\end{array}
\right),$$ where $X=\cos\frac{(d-2) \pi }{2 d} \csc\frac{\pi }{d n}-\cos\frac{(d n-2) \pi }{2 d n} \csc\frac{\pi }{d n}$. The maximum eigenvalue of $M$ is $\lambda_{max}^M=(1+n+X)/n^2$. Thus the maximum eigenvalue $\lambda_{max}^O$ of $O$ is $$\begin{aligned}
\lambda_{max}^O&=&dn\lambda_{max}^M \nonumber\\
&=& d(1+n+\cos\frac{(d-2)\pi}{2d}\csc\frac{\pi}{dn}-\cos\frac{(dn-2)\pi}{2dn}\csc\frac{\pi}{dn})/n.\end{aligned}$$ Set $y_1=\frac{1}{n}$, $x_1=\frac{1}{d}$, then $0<y_1\leq \frac{1}{3}$, $0<x_1\leq \frac{1}{2}$. Define the functions $$\begin{aligned}
f(x_1,y_1) &=& 1+y_1+y_1(\sin (x_1\pi)\csc( x_1y_1\pi)-\sin( x_1y_1\pi)\csc( x_1y_1\pi)), \\
g(x_1) &=& 2-x_1 .\end{aligned}$$ The imagine of $f(x_1,y_1)$ and $g(x_1)$ are shown in Figure \[fig1\] which are drawn by Wolfram Mathematica 8.0.
![The imagine of $f(x_1,y_1)$ and $g(x_1)$ []{data-label="fig1"}](CnCnodd.pdf "fig:"){width="40.00000%"}\
$f(x_1,y_1)$ is the curved surface and $g(x_1)$ is plane surface. Clearly, $f(x_1,y_1)>g(x_1)$ in definition domain $0<y_1\leq \frac{1}{3}$, $0<x_1\leq \frac{1}{2}$. Equivalently, $\lambda_{max}^O>2d-1$ for any odd $n$ and any number of outcomes $d\geq 2$. Therefore, the quantum bound violates the classical bound.
[**The case II**]{}
For the case $n$ is even, $n\geq 2$. We choose $$\begin{aligned}
U &=& | w_0 \rangle\langle w_0|+e^{-i2\pi /dn} | w_1 \rangle\langle w_1|+e^{-i4\pi /dn} | w_2 \rangle\langle w_2| + \cdots\nonumber\\
&& + e^{-i(n-2)\pi /dn} | w_{\frac{n-2}{2}} \rangle\langle w_{\frac{n-2}{2}}|+ e^{i\pi /d} | w_{\frac{n}{2}} \rangle\langle w_{\frac{n}{2}}| + e^{i(n-2)\pi /dn} | w_{\frac{n+2}{2}} \rangle\langle w_{\frac{n+2}{2}}| \nonumber\\
&& + \cdots + e^{i4\pi /dn} | w_{n-2} \rangle\langle w_{n-2}|+ e^{i2\pi /dn} | w_{n-1} \rangle\langle w_{n-1}|.\end{aligned}$$
In this case, the eigenvector corresponding to the maximum eigenvalue of $O$ lies in the subspace when $\lambda=e^{i2\pi /nd}$.
We shall calculate the maximum eigenvalue of the operator $$\label{R2even}
R = P_{e^{i2\pi /nd}}{\left|0v_1^0\right>}{\left<0v_1^0\right|}P_{e^{i2\pi /nd}}+P_{e^{i2\pi /nd}}{\left|0v_1^1\right>}{\left<0v_1^1\right|})P_{e^{i2\pi /nd}},$$ Denote $$\begin{aligned}
{\left|\psi_1\right>}=P_{e^{i2\pi /nd}}{\left|0v_1^0\right>}, \hspace{5mm}
{\left|\psi_2\right>}=P_{e^{i2\pi /nd}}{\left|0v_1^1\right>}, \hspace{5mm}
M=(\langle \psi_k | \psi_j\rangle)_{k,j=1}^2,\end{aligned}$$ then the eigenvalues of $M$ are exactly the ones of $R$. With the help of Wolfram Mathematica 8.0, we get the matrix $M$ has the following form $$\left(
\begin{array}{cc}
\frac{1}{n} & \frac{1}{n^2} (1+X+e^{i\pi /d}) \\
\frac{1}{n^2} (1+X+e^{-i\pi /d}) & \frac{1}{n} \\
\end{array}
\right),$$ where $X=\cos\frac{(dn+2-2n) \pi }{2dn} \csc\frac{\pi }{d n}-\cos\frac{(d n-2) \pi }{2 d n} \csc\frac{\pi }{d n}$. The maximum eigenvalue of $M$ is $\lambda_{max}^M=(n+((1+X+\cos\frac{\pi}{d})^2+(\sin\frac{\pi}{d})^2)^{1/2})/n^2$. Thus the maximum eigenvalue $\lambda_{max}^O$ of $O$ is $$\begin{aligned}
\label{}
\lambda_{max}^O&=&dn\lambda_{max}^M \nonumber\\
&=& d(n+((1+X+\cos\frac{\pi}{d})^2+(\sin\frac{\pi}{d})^2)^{1/2})/n.\end{aligned}$$
Set $y_2=\frac{1}{n}$, $x_2=\frac{1}{d}$, then $0<y_2\leq \frac{1}{2}$, $0<x_2\leq \frac{1}{2}$. Define the functions $$\begin{aligned}
f(x_2,y_2) & = &1+y_2((1+\widetilde{X}+\cos(x_2 \pi))^2+(\sin x_2 \pi)^2)^{\frac{1}{2}}, \\
g(x_2) &=& 2-x_2 ,\end{aligned}$$ where $\widetilde{X}=\sin (x_2\pi-x_2 y_2 \pi)\csc( x_2y_2\pi)-\sin( x_2y_2\pi)\csc( x_2y_2\pi)$. From Wolfram Mathematica 8.0, one obtain the imagine of $f(x_2,y_2)$ and $g(x_2)$ in Figure \[fig2\].
![The imagine of $f(x_2,y_2)$ and $g(x_2)$ []{data-label="fig2"}](CnCneven.pdf "fig:"){width="40.00000%"}\
$f(x_2,y_2)$ is the curved surface and $g(x_2)$ is plane surface. Clearly, in definition domain $0<y_2\leq \frac{1}{2}$, $0<x_2\leq \frac{1}{2}$, we have $f(x_2,y_2)>g(x_2)$. Equivalently, $\lambda_{max}^O>2d-1$ for any odd $n$ and any outcomes $d\geq 2$. Therefore, the quantum bound violates the classical bound.
No matter $n$ is odd or even, the above results can be explained as a nonlocal game.
![The structure of nonlocal game[]{data-label="nonlocal"}](nonlocal.pdf "fig:"){width="25.00000%"}\
We have that Alice and Bob each receive a bit $s$ and $t$ respectively from a referee, with each bit equally likely to be $0,\ 1,\ \cdots$, or $d-1$. After Alice and Bob perform measurements on their own part respectively, they send measurement results $v_A(s)$ and $v_B(t)$ back to the referee, $v_A(s)$ and $v_B(t)$ take values in the set $\{0,1,\cdots,n-1 \}$. The structure of the nonlocal game are shown in Figure \[nonlocal\]. The winning conditions are listed in Table \[Tab1\].
Specifically, the values of $(s,t)$ are listed on the first row, and the corresponding bit values $v_A(s)$ and $v_B(t)$ respectively sent by Alice and Bob are listed on the second row. This game is won if the bit values $v_B(t)-v_A(s)=2\ {\text mod} \ d$ when $(s,t)=(d-1,0)$ and if the bit values $v_B(t)-v_A(s)=1\ {\text mod} \ d$ for all other allowed choice of $(s,t)$.
The maximum classical probability of winning this game can be achieved if Alice always returning the bit value $0$ and Bob always returning the bit value $1$. So the maximum classical probability is $\frac{2d-1}{2d}$.
In the quantum strategy, Alice and Bob share the state ${\left|\phi\right>}$ which is the eigenstate of $O$ corresponding to its maximum eigenvalue $\lambda_{max}^O$. If they receive values $s$ and $t$ from the referee respectively, Alice measure $a_s$, Bob measure $a_t$, and then they send the measurement results to referee. The probability of winning this game is then $\frac{\lambda_{max}^O}{2d}$. From Figure \[fig1\] and Figure \[fig2\], we know that the value of quantum bound is larger than the value of classical bound.
Furthermore, we note that, no matter $n$ is odd or even, the classical bound of Bell inequality Eq.(\[Belleq O2\]) is $2d-1$ which is independent of the choice of $n$. And the quantum bound is decided by both the values of $n$ and $d$.
For $n$ is odd, fix a $x_1$, we compute the partial derivative of function $f(x_1,y_1)$ with respect to $y_1$: $$\frac{\partial f(x_1,y_1) }{\partial y_1}=\csc(\pi x_1 y_1) \sin(\pi x_1) - \pi x_1 y_1 \cot(\pi x_1 y_1) \csc(\pi x_1 y_1)\sin(\pi x_1).$$ In the domain $0<y_1\leq \frac{1}{3}$, $0<x_1\leq \frac{1}{2}$, the trigonometric functions $\csc(\pi x_1 y_1)$, $\sin (\pi x_1)$ and $\cot(\pi x_1 y_1)$ are all exceed $0$, so the derivative function $\frac{\partial f(x_1,y_1) }{\partial y_1}$ always exceed $0$ in the definition domain. Thus the continuous function $f(x_1,y_1)$ is a monotonic increasing function for any fixed $x_1$. That is to say, the quantum bound $\lambda_{max}^O$ is a monotonic decreasing function with respect to $n$ for a fixed $d$, the number of measurements.
Even so, when $d$ is large enough, the quantum bound is independent of the choice of $n$. If each measurement has $3$ outcomes, i.e. $y_1=\frac{1}{3}$, $$\begin{aligned}
f(x_1,\frac{1}{3}) &=& 1+\frac{1}{3}+\frac{1}{3}(\sin (x_1\pi)\csc \frac{x_1\pi}{3}-\sin \frac{x_1\pi}{3}\csc \frac{x_1\pi}{3}) \nonumber\\
&=& \frac{2}{3} (2+\cos\frac{2 \pi x}{3}).\end{aligned}$$ If $n$ is large enough, i.e. $y_1\rightarrow 0$, we evaluate the limit value $$\begin{aligned}
f(x_1,0) &:=& \lim_{y_1\rightarrow 0}f(x_1,y_1) \nonumber\\
&=& 1+\frac{1}{\pi x}\sin(\pi x).\end{aligned}$$ We draw the graphics of functions $f(x_1,\frac{1}{3})$ and $f(x_1,0)$ in Figure \[fig3\],
![The imagine of $f(x_1,0)$ and $f(x_1,\frac{1}{3})$ []{data-label="fig3"}](indep.pdf "fig:"){width="40.00000%"}\
function $f(x_1,\frac{1}{3})$ is solid, and function $f(x_1,0)$ is dotted. From this figure, we see that when $x_1$ is near to $0$, $f(x_1,0)$ approach $f(x_1,\frac{1}{3})$.
When $n$ is even, we have similar analysis. That is to say, when the number of measurements is large enough, the violation of Bell inequality is determined by the number of measurements $d$ and independent of $n$, the number of outcomes.
Four partite system {#sec3}
===================
For a four partite system, Alice, Bob, Charlie and Danniel share joint state ${\left|\psi\right>}$ which are elements of the Hilbert space $\mathbb{C}^4\otimes {\mathbb{C}}^4 \otimes {\mathbb{C}}^4 \otimes {\mathbb{C}}^4$, with each ${\mathbb{C}}^4$ is spanned by the orthonormal basis $\{{\left|0\right>}, {\left|1\right>}, {\left|2\right>}, {\left|3\right>} \}$. Each of them can take one of two measurements, and for each measurement the possible values for the outcomes are $0$, $1$, $2$ or $3$. Alice’s observable operators are $a_{0}$ and $a_1$, Bob’s are $b_{0}$ and $b_{1}$, Charlie’s are $c_{0}$ and $c_{1}$, and Danniel’s are $d_{0}$ and $d_{1}$. The orthonormal basis $\{{\left|0\right>}, {\left|1\right>}, {\left|2\right>}, {\left|3\right>} \}$ correspond to the observable operators $a_0$, $b_0$, $c_0$ and $d_0$, $a_0=|1\rangle\langle 1| + 2|2\rangle\langle 2|+3|3\rangle\langle 3|$, similarly for $b_0$, $c_0$ and $d_0$. Next we define the second basis.
Since the $4\times 4$ translation operator $T$ is an orthogonal matrix under any orthonormal basis, we have $$\label{T4}
T= | w_0 \rangle\langle w_0| +e^{-i\pi /2} | w_1 \rangle\langle w_1 |+e^{i\pi} | w_2 \rangle\langle w_2 |+ e^{i\pi /2}| w_3 \rangle\langle w_3 |,$$ where $$\label{w_j4}
|w_j \rangle=\frac{1}{2}\sum_{k=0}^3 e^{i\pi jk /2}{\left|j\right>}, \hspace{5mm} \langle w_j|w_k\rangle=\delta_{jk}$$ for any $j,k=0,1,2,3$. Choose operator $$\label{U}
U= | w_0 \rangle\langle w_0| +e^{-i\pi /4} | w_1 \rangle\langle w_1 |+e^{i\pi /2} | w_2 \rangle\langle w_2 |+ e^{i\pi /4}| w_3 \rangle\langle w_3 |,$$ then the space $\{j\rightarrow U^j \mid j=0,1,\cdots, 7 \}$ is a representation of the cyclic group ${\mathbb{Z}}_8$, the group of integers ${\mathbb{Z}}$ modulo $8$. We can define the second basis $\{ v_j=U^j {\left|j\right>} \mid j=0,1,2,3\}$, and this basis corresponds to the observable operators $a_1$, $b_1$, $c_1$ and $d_1$, $a_1=|v_1\rangle\langle v_1| + 2|v_2\rangle\langle v_2|+3|v_3\rangle\langle v_3|$, similarly for $b_1$, $c_1$ and $d_1$. Denote $V=U\otimes U\otimes U\otimes U$, then $\{j\rightarrow V^j \mid j=0,1,\cdots, 7 \}$ is also a representation of the cyclic group ${\mathbb{Z}}_8$. We choose the following six states: $$\label{6orbits}
{\left|000v_1\right>}, \hspace{5mm} {\left|0v_0v_00\right>}, \hspace{5mm} {\left|0v_00v_0\right>}, \hspace{5mm} {\left|v_0v_303\right>}, \hspace{5mm} {\left|0v_1v_11\right>}, \hspace{5mm} {\left|0v_220\right>},$$ where state ${\left|000v_1\right>}$ means ${\left|0\right>} \otimes {\left|0\right>} \otimes{\left|0\right>} \otimes{\left|v_1\right>}$. Each of these six states in (\[6orbits\]) has an orbit with $8$ elements under the action of $\{ V^j | j=0,1,\cdots, 7 \}$, and the six orbits are distinct. The set of all states in the six orbits leads to a violation of Bell inequality. The specific analysis is given as follows.
For each state in the six orbits, it corresponds to a particular choice of measurements and the corresponding measurement results. For example, for the state ${\left|000v_1\right>}$, it corresponds to the fact that Alice measuring $a_0$ and obtaining $0$, Bob measuring $b_0$ and obtaining $0$, Charlie measuring $c_0$ and obtaining $0$, Danniel measuring $d_1$ and obtaining $1$.
From local realistic theory, the sum of these $48$ events give a Bell inequality, it reads $$\footnotesize{
\begin{split}
&\sum_{j=0}^3 P(a_0=b_0=c_0=j,d_1=j\oplus1)+\sum_{j=0}^3 P(a_1=b_1=c_1=j,d_0=j\oplus2)\\
& +\sum_{j=0}^3 P(a_0=b_1=c_1=d_0=j) +\sum_{j=0}^3 P(a_1=d_1=j,b_0=c_0=j\oplus1)\\
& +\sum_{j=0}^3 P(a_0=b_1=c_0=d_1=j) +\sum_{j=0}^3 P(a_1=c_1=j,b_0=d_0=j\oplus1) \\
& +\sum_{j=0}^3 P(a_1=c_0=j,b_1=d_0=j\oplus3) +\sum_{j=0}^3 P(a_0=j,b_0=c_1=j\oplus3,d_1=j\oplus2)\\
& +\sum_{j=0}^3 P(a_0=d_0=j,b_1=c_1=j\oplus1) +\sum_{j=0}^3 P(a_1=d_1=j,b_0=c_0=j\oplus1) \\
& +\sum_{j=0}^3 P(a_0=d_0=j,b_1=c_0=j\oplus2) +\sum_{j=0}^3 P(a_1=d_1=j,b_0=j\oplus3,c_1=j\oplus2)\leq 2,
\end{split}}$$ where $\oplus$ means the addition modulo $4$, $P(a_0=b_0=c_0=j,d_1=j\oplus1)$ means the probability of the event when Alice, Bob and Charlie take measure $a_0$, $b_0$ and $c_0$ respectively and obtain the same value $j$, Danniel take measure $d_1$ and obtain value $j\oplus1$. But in quantum bound can attain the value $2.021$.
In order to maximize the sum of probabilities corresponding to these $48$ states in quantum mechanics, we need to find a state ${\left|\phi\right>}$, such that the expectation value $\langle \phi |O | \phi\rangle$ is maximum, where $$\label{expect O4}
O= \sum_{j=0}^7V^jL(V^{\dag})^j,$$ and $$\label{expect L4}
\begin{split}
& L={\left|000v_1\right>}{\left<000v_1\right|}+{\left|0v_0v_00\right>}{\left<0v_0v_00\right|}+{\left|0v_00v_0\right>}{\left<0v_00v_0\right|}+\\
&\qquad {\left|v_0v_303\right>}{\left<v_0v_303\right|}+{\left|0v_1v_11\right>}{\left<0v_1v_11\right|}+{\left|0v_220\right>}{\left<0v_220\right|}.
\end{split}$$ The maximum value of $\langle \phi |O | \phi\rangle$ occurs when ${\left|\phi\right>}$ is the eigenstate of $O$ corresponding to its maximum eigenvalue. So the question is reduced to how to calculate the maximum eigenvalue of $O$.
Note that the eigenstates of $V$ are states of the form ${\left|w_kw_lw_mw_n\right>}$ for $k,\ l,\ m,\ n=0,\ 1,\ 2,\ 3$, and the eigenvalues are $\pm1$, $e^{\pm i\pi/4}$, $e^{\pm i\pi/2}$ and $e^{\pm 3i\pi/4}$. The spectral decomposition $V= \sum_\lambda \lambda P_\lambda$, leads to a simplification of $O$, $O = 8\sum_\lambda P_\lambda L P_{\lambda}.$ The eigenvector corresponding to the maximum eigenvalue of $O$ lies in the subspace when $V$ has eigenvalue $e^{i\pi/2}$. Set $$\begin{aligned}
{\left|\psi_1\right>}=P_{e^{i\pi/2}}{\left|000v_1\right>}, \hspace{5mm} {\left|\psi_2\right>}=P_{e^{i\pi/2}}{\left|0v_0v_00\right>}, \hspace{5mm} {\left|\psi_3\right>}=P_{e^{i\pi/2}}{\left|0v_00v_0\right>}, \nonumber\\ {\left|\psi_4\right>}=P_{e^{i\pi/2}}{\left|v_0v_303\right>}, \hspace{5mm} {\left|\psi_5\right>}=P_{e^{i\pi/2}}{\left|0v_1v_11\right>}, \hspace{5mm}
{\left|\psi_6\right>}=P_{e^{i\pi/2}}{\left|0v_220\right>},\end{aligned}$$ and $R=P_{e^{i\pi/2}}LP_{e^{i\pi/2}}$, then the operator can be written as, $$\label{R4}
R = \sum_{j=1}^6 {\left|\psi_j\right>}{\left<\psi_j\right|}.$$
Note that the eigenvectors of $R$ can be expressed as $\sum_{j=1}^{6}x_j{\left|\psi_j\right>}$, then there exists eigenvalue $\mu$ such that $\sum_{k=1}^6 {\left|\psi_k\right>}{\left<\psi_k\right|}\sum_{j=1}^{6}x_j{\left|\psi_j\right>}=\mu\sum_{j=1}^{6}x_j{\left|\psi_j\right>}$. Rewrite the eigenvalue equation, it becomes $$\sum_{j=1}^{6}x_j\langle \psi_k | \psi_j\rangle\sum_{k=1}^{6}{\left|\psi_k\right>}=\mu\sum_{k=1}^{6}x_k{\left|\psi_k\right>}.$$ Equivalently, for any $k=1,2,\cdots,6$, we have $\sum_{j=1}^{6}x_j\langle \psi_k | \psi_j\rangle=\mu x_k$. Denote the matrix $M=(\langle \psi_k | \psi_j\rangle)_{k,j=1}^6$, then the eigenvalues of $M$ are exactly the ones of $R$. With the help of Wolfram Mathematica 8.0, we quickly obtain the matrix $256M$ as the following form $$\tiny{
\left(
\begin{array}{cccccc}
\medskip
44 & -2-\sqrt{2}+ i\sqrt{2} & -2-\sqrt{2}+ i\sqrt{2} & -\sqrt{2}-i(2+\sqrt{2}) & 4-3\sqrt{2}+i(\sqrt{2}-2) & 4-4i \\
\medskip
-2-\sqrt{2}- i\sqrt{2} & 44 & 8+4\sqrt{2} & i 4\sqrt{2} & 8i & -\sqrt{2}+ i(2-\sqrt{2}) \\
\medskip
-2-\sqrt{2}- i \sqrt{2}& 8+4\sqrt{2} & 44 & 0 & 0 & -\sqrt{2}+ i(2-\sqrt{2}) \\
\medskip
-\sqrt{2}+i(2+\sqrt{2})& - i 4\sqrt{2} & 0 & 44 & 0 & -2+\sqrt{2}- i\sqrt{2} \\
\medskip
4-3\sqrt{2}- i(\sqrt{2}-2) & -8 i & 0 & 0 & 44 & 2+\sqrt{2}+ i(4+3\sqrt{2}) \\
4+4 i & -\sqrt{2}- i(2-\sqrt{2}) & -\sqrt{2}- i(2-\sqrt{2}) & -2+\sqrt{2}+ i\sqrt{2} & 2+\sqrt{2}- i(4+3 \sqrt{2}) & 44 \\
\end{array}
\right),}$$ and obtain the numerical approximate value of the maximum eigenvalue of $M$ is $64.667/256$. Thus the maximum eigenvalue of $O$ is $8\times\frac{64.667}{256}\approx2.021>2$. So we get a violation.
The above results can be explained as a nonlocal game. The bits $s$, $t$, $u$ and $v$ are sent to Alice, Bob, Charlie and Danniel respectively from the same referee, take value $0$ or $1$ with equally likely possibility. Alice take measure $a_s$ on her part, Bob take measure $b_t$ on his part, Charlie take measure $c_u$ on her part, Danniel take measure $d_v$ on his part. Then they each send a bit value $v_A(s)$, $v_B(t)$, $v_C(u)$ and $v_D(v)$ back to the referee, the bit values take values in the set $\{0,\ 1,\ 2,\ 3\}$. The winning conditions are listed in Table \[Tab2\].
Specifically, the values of $(s,t,u,v)$ are listed on the left, and the corresponding outcomes $v_A(s)$, $v_B(t)$, $v_C(u)$ and $v_D(v)$ sent by Alice, Bob, Charlie and Danniel are listed on the right. The maximum classical probability of winning this game can be achieved when the outcomes $v_A(s)$, $v_B(t)$, $v_C(u)$ and $v_D(v)$ take same value, $(s,t,u,v)$ take value $(0110)$ or $(0110)$. So the maximum classical probability is $\frac{2}{12}\approx 0.1667$.
In the quantum strategy, Alice, Bob, Charlie and Danniel share the state $\phi$ which is the eigenstate of $O$ corresponding to its maximum eigenvalue $2.021$. If they receive values $s$, $t$, $u$ and $v$ from the referee respectively, Alice measure $a_s$, Bob measure $a_t$, Charlie measure $a_u$ and Danniel measure $a_v$, then they send the measurement results to referee. With this strategy, the probability of winning this game is $0.1684$.
Three partite system {#sec4}
====================
Alice, Bob and Charlie make measurements, each party make one of $2$ possible measurements, and each measurement has $4$ outcomes. As above, the orthonormal basis $\{{\left|0\right>}, {\left|1\right>}, {\left|2\right>}, {\left|3\right>} \}$ correspond to the observables $a_0$, $b_0$ and $c_0$, $a_0=|1\rangle\langle 1| + 2|2\rangle\langle 2|+3|3\rangle\langle 3|$, similarly for $b_0$ and $c_0$. The orthonormal basis $\{{\left|v_0\right>}, {\left|v_1\right>}, {\left|v_2\right>}, {\left|v_3\right>} \}$ correspond to the observables $a_1$, $b_1$ and $c_1$, $a_1=|v_1\rangle\langle v_1| + 2|v_2\rangle\langle v_2|+3|v_3\rangle\langle v_3|$, similarly for $b_1$ and $c_1$.
We restrict the six orbits in Eq.(\[6orbits\]) to three partite system by forgetting the last party, and guarantee the bound of probabilities from local realistic theory invariant. We get $4$ orbits in ${\mathbb{C}}^4\otimes{\mathbb{C}}^4\otimes{\mathbb{C}}^4$, which will get a violation of Bell inequality. The representative elements of the $4$ orbits are: $$\label{4orbits}
{\left|0v_0v_0\right>}, \hspace{5mm} {\left|v_0v_30\right>}, \hspace{5mm} {\left|0v_1v_1\right>}, \hspace{5mm} {\left|0v_22\right>}.$$ Each of the four states in Eq.(\[4orbits\]) has an orbit with $8$ elements under the action of $\{U^j\otimes U^j\otimes U^j| j=0,1, \cdots, 7 \}$, and the four orbits are distinct. From local realistic theory, the sum of these $32$ states give a Bell inequality. It reads $$\footnotesize{
\begin{split}
&\sum_{j=0}^3 P(a_0=b_1=c_1=j)+\sum_{j=0}^3 P(a_1=j,b_0=c_0=j\oplus1)\\
& +\sum_{j=0}^3 P(a_1=c_0=j,b_1=j\oplus3) +\sum_{j=0}^3 P(a_0=j,b_0=c_1=j\oplus3)\\
& +\sum_{j=0}^3 P(a_0=j,b_1=c_1=j\oplus1) +\sum_{j=0}^3 P(a_1=j,b_0=c_0=j\oplus2) \\
& +\sum_{j=0}^3 P(a_0=j,b_1=c_0=j\oplus2) +\sum_{j=0}^3 P(a_1=j,b_0=j\oplus3,c_1=j\oplus2)\leq 2,
\end{split}}$$ where $\oplus$ means the addition modulo $4$. But the quantum bound of the sum of these probabilities can attain the value $2.075$.
Similarly, we need to find a special state ${\left|\phi\right>}$, such that the expectation value $\langle \phi |O | \phi\rangle$ is maximum, where $$\label{expect O3}
O= \sum_{j=0}^7(U\otimes U\otimes U)^jL(U^{\dag}\otimes U^{\dag}\otimes U^{\dag})^j,$$ and $$\label{expect L3}
L={\left|0v_0v_0\right>}{\left<0v_0v_0\right|}+ {\left|v_0v_30\right>}{\left<v_0v_3\right|}+{\left|0v_1v_1\right>}{\left<0v_1v_1\right|}+{\left|0v_22\right>}{\left<0v_22\right|}.$$
Note that the eigenstates of $U\otimes U\otimes U$ are states of the form ${\left|w_kw_mw_n\right>}$ for $k,\ m,\ n=0,\ 1,\ 2,\ 3$, and the eigenvalues are $\pm1$, $e^{\pm i\pi/4}$, $e^{\pm i\pi/2}$ and $e^{\pm 3i\pi/4}$. For the spectral decomposition of $U$, $ U\otimes U\otimes U= \sum_\lambda \lambda P_\lambda$, the operator $O$ can be simplified as $$O = 8\sum_\lambda P_\lambda L P_{\lambda}.$$ The eigenvector corresponding to the maximum eigenvalue of $O$ lies in the subspace when $U\otimes U\otimes U$ has eigenvalue $e^{i\pi/2}$. Set $$\begin{aligned}
{\left|\psi_1\right>}=P_{e^{i\pi/2}}{\left|0v_0v_0\right>}, \hspace{5mm} {\left|\psi_2\right>}=P_{e^{i\pi/2}}{\left|v_0v_30\right>}, \nonumber \\ {\left|\psi_3\right>}=P_{e^{i\pi/2}}{\left|0v_1v_1\right>}, \hspace{5mm} {\left|\psi_4\right>}=P_{e^{i\pi/2}}{\left|0v_22\right>}.\end{aligned}$$ Denote the operator $P_{e^{i\pi/2}}LP_{e^{i\pi/2}}$ by $R$, then $R$ can be write as $R = \sum_{j=1}^4 {\left|\psi_j\right>}{\left<\psi_j\right|}.$ Suppose that the eigenvectors of $R$ corresponding to eigenvalue $\mu$ is $\sum_{j=1}^{4}x_j{\left|\psi_j\right>}$, then the eigenvalue equation can be rewritten as $$\label{}
\sum_{j=1}^{4}x_j\langle \psi_k | \psi_j\rangle\sum_{k=1}^{4}{\left|\psi_k\right>}=\mu\sum_{k=1}^{4}x_k{\left|\psi_k\right>}.$$ Denote the matrix $M=(\langle \psi_k | \psi_j\rangle)_{k,j=1}^4$. The eigenvalues of $M$ are exactly the ones of $R$. With the help of Mathematics, we obtain the matrix $M$ as the following form $$\small{
\frac{1}{64}\left(
\begin{array}{cccc}
12 & i \sqrt{2} & 2 i & 1+i \\
-i \sqrt{2} & 12 & -1-i & \sqrt{2} \\
-2 i & -1+i & 12 & 2 i+i \sqrt{2} \\
1-i & \sqrt{2} & -2 i-i \sqrt{2} & 12 \\
\end{array}
\right),}$$ and obtain the numerical approximate value of the maximum eigenvalue of $M$ is $16.597/64$. Thus the maximum eigenvalue of $O$ is $8\times\frac{16.597}{64}\approx2.075>2$.
This results can also be phrased as a nonlocal game. Alice, Bob and Charlie each receive a bit $s$, $t$ and $u$ respectively from same referee. $s$, $t$ and $u$ take value $0$ or $1$ with equally likely possibility. Then Alice make measurement $a_s$, Bob make measurement $b_t$ and Charlie make measurement $c_u$. They return the measurement results $v_A(s)$, $v_B(t)$ and $v_C(u)$ back to the referee respectively, the outcomes take values in the set $\{0,\ 1,\ 2,\ 3\}$. The winning conditions are listed in Table \[Tab3\].
Specifically, the values of $(s,t,u)$ are listed on the left, and the corresponding measurement results $v_A(s)$, $v_B(t)$ and $v_C(u)$ sent by Alice, Bob and Charlie are listed on the right. Note that, the states listed in (\[4orbits\]) is obtained by the states in (\[6orbits\]) by forgetting the last bit and an additional condition. The Table \[Tab3\] is obtained by deleting the first two rows and the last two rows in the left table of Table \[Tab2\], and forgetting the last bit values. The maximum classical probability of winning this game is $\frac{2}{8}\approx 0.25$.
In the quantum strategy, Alice, Bob and Charlie share the state $\phi$ which is the eigenstate of $O$ corresponding to its maximum eigenvalue $2.075$. If they receive values $s$, $t$ and $u$ from the referee respectively, then Alice measure $a_s$, Bob measure $a_t$, Charlie measure $a_u$, and they send the measurement results to referee. With this strategy, the probability of winning this game is $0.2594$.
Acknowledgement {#acknowledgement .unnumbered}
---------------
This work is supported by NSFC 11571119 and NSFC 11475178.
[99]{} J.S. Bell, *Physica* **1**, 195 (1964) J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, *Phys. Rev. Lett.* **23**, 880 (1969) J.F. Clauser, M.A. Horne, *Phys. Rev. D* **10**, 526 (1974) D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, A. Zeilinger, *Phys. Rev. Lett.* **85**, 4418 (2000) R.F. Werner, M.M. Wolf, *Phys. Rev. A* **64**, 032112 (2001) D. Collins, N. Gisin, N. Linden, S. Massar, S. Popescu, *Phys. Rev. Lett.* **88**, 040404 (2002) W. Son, J. Lee, M.S. Kim, *Phys. Rev. Lett.* **96**, 060406 (2006) A. Cabello, S. Severini, A. Winter, *Phys. Rev. Lett.* **112**, 040401 (2014) V. Ugǔr Gűney, M. Hillery, *Phys. Rev. A* **90**, 062121 (2014) V. Ugǔr Gűney, M. Hillery, *Phys. Rev. A* **91**, 052110 (2015) K. Bolonek-Lasoń, *Phys. Rev. A* **94**, 022107 (2016) K. Bolonek-Lasoń, S. Sobieski, *Quantum Inf. Process* **16**, 38 (2017)
[^1]: [email protected]
[^2]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the problem of bounding mean first passage times for a class of continuous-time Markov chains that captures stochastic interactions between groups of identical agents. The quantitative analysis of such probabilistic population models is notoriously difficult since typically neither state-based numerical approaches nor methods based on stochastic sampling give efficient and accurate results. Here, we propose a technique that extends recently developed methods using semi-definite programming to determine bounds on mean first passage times. We further apply the technique to hybrid models and demonstrate its accuracy and efficiency for some examples from biology.'
author:
- 'Michael Backenköhler${}^{1}$, Luca Bortolussi${}^{1,2}$, Verena Wolf${}^{1}$'
bibliography:
- 'paper.bib'
title: |
Bounding Mean First Passage Times in\
Population Continuous-Time Markov Chains
---
Introduction
============
Population Continuous-Time Markov Chains (PCTMCs) provide a widely used framework to capture stochastic interactions between groups of identical agents. This subclass of Continuous-Time Markov Chains (CTMCs) is used to describe the stochastic dynamics of systems in various domains. Prominent applications are chemical reaction networks in quantitative biology [@BuchWolkenhauer], epidemic spreading [@porter2016dynamical], performance analysis of technical and information systems [@bortolussi2013; @gast2019] as well as the behavior of collective adaptive systems [@bernardo2016].
For the quantitative analysis of CTMCs, many approaches have been developed, where properties of interest are often expressed in terms of temporal logics such as CSL [@aziz1996verifying; @baier2000model; @baier2003model], MTL [@chen2011time], and timed-automata specifications [@chen2009quantitative; @mikeev2013fly]. In addition, there exist efficient software tools [@hinton2006prism; @kwiatkowska2011prism; @dehnert2017storm]. A central problem in this context is the computation of reachability probabilities.
Popular exact methods for CMTCs rely on numerical approaches that explicitly consider each system state individually. A major problem is that these methods cannot scale in the context of population models with large copy numbers of agents. A popular alternative to tackle this problem is statistical model checking, which is based on stochastic simulation [@david2015statistical]. For PCTMCs arising in the context of chemical reaction networks, trajectories of the process are usually generated using the Stochastic Simulation Algorithm (SSA) [@gillespie77]. However, since the number of possible interactions grows with the number of agents, stochastic simulations of PCTMCs are time-consuming. Moreover, they are subject to inherent statistical uncertainty and give only statistically estimated bounds.
Recent work concentrates on numerical methods for PCTMCs that approximate the statistical moments of the system without the need to consider the probability of each state. For groups of identically behaving agents, it is possible to derive systems of differential equations for the evolution of the statistical population moments [@bogomolov2015adaptive; @schnoerr2017survey; @bortolussi2013model; @engblom2006computing; @schnoerr2015; @gast2019]. However, as the system of exact moment equations is not closed, approximation schemes typically rely on certain assumptions about the underlying probability distribution. For example, one might employ a “low dispersion closure” which assumes that higher-order moments are the same as those of a normal distribution [@hespanha2008moment]. Such approximations are, by nature, ad-hoc and do typically not come with any guarantees. Here, we do not need such closure schemes and retain guaranteed results up to the numerical accuracy of the computations.
Moment-based methods often scale well in terms of population sizes. However, it is not possible to control the effects of such approximations, which in some cases can lead to large errors [@schnoerr2015]. This issue reverberates on the application of these methods to compute reachability probabilities and mean first passage times [@hayden2012fluid; @bortolussi2013model; @bortolussi2014stochastic]. Moreover, they can suffer from numerical instabilities, in particular, when the maximum order of the considered moments has to be increased to more appropriately describe the underlying distribution. Here, we build on recent work on moment bounds [@sakurai2017convex; @dowdy2018dynamic] to propose a method to compute bounds for Mean First Passage Times (MFPTs) in PCTMCs. For a set of states, the MFPT within a fixed time horizon $T$ directly characterizes the probability of reaching that set within $T$ time units. Thus, safe upper and lower bounds on MFPTs can constitute a core component for the verification of properties in PCTMCs. Our approach is based on a martingale formulation of the stopped process that we derive from the exact moment equations. From this formalization, we deduce a set of linear moment constraints from which we derive upper and lower moment bounds using semi-definite programming (SDP). Monotone sequences of both upper and lower bounds can be obtained by increasing the order of the relaxation. Crucially, no closure approximations are introduced. Therefore the bounds are strict up to the numerical accuracy of the SDP solver. For our numerical investigations, we concentrate on examples from biology and find encouraging results already for a small number of moments. For instance, in one of our case studies $100,\!000$ SSA runs are necessary to achieve a relative width of 0.9% for the MFPT confidence interval. The SDP solver, however, returns a guaranteed interval with a relative width of 0.3%.\
In summary, this paper presents the following novel contributions:
- the derivation of moment constraints for bounding first passage times and reachability probabilities using a convex programming scheme
- the extension of this scheme to stochastic hybrid systems exhibiting multimodal behavior
- a scaling strategy for improved robustness during optimization
The paper is structured as follows: Section \[sec:related\] covers work related to the analysis of first passage times in PCTMCs and recent work on moment bounds. Section \[sec:bg\] introduces the PCTMC framework and its semantics. In Section \[sec:moments\] we derive a martingale from the moment dynamics of a PCTMC. Based on this process, in Section \[sec:mfpt\_bounds\] we formulate linear and semi-definite constraints to state a semi-definite program to compute bounds on the MFPT and reachability probabilities. In Section \[sec:evaluation\], we discuss the practical considerations of the SDP implementation and provide results on a set of case studies. Finally, in Section \[sec:conclusion\] we provide concluding remarks and directions of future work.
Related Work {#sec:related}
============
Considerable effort has been directed at the analysis of first passage time distributions in PCTMCs. Most works can either focus on an explicit state-space analysis [@barzel2008calculation; @munsky2009specificity; @kuntz2019exit; @kuntz2018approximation] or employ approximation techniques for which, in general, no error bounds can be given [@schnoerr2017efficient; @hayden2012fluid; @bortolussi2014stochastic]. For some model classes such as kinetic proofreading, analytic solutions are possible [@munsky2009specificity; @bel2009simplicity; @iyer2016first].
Barzel and Biham [@barzel2008calculation] propose a recursive scheme that consists of one equation for each state, expressing the average time the system needs to transition from that state to the target state. Kuntz et al. [@kuntz2018approximation] propose to employ moment bounds in a linear programming approach to compute exit time distribution using state-space truncation schemes. In Ref. [@kuntz2019exit] the authors propose a finite state-space projection scheme to bound first passage time distributions
Hayden et al. [@hayden2012fluid] use moment closure approximations and Chebychev’s inequality to gain an understanding of first passage time dynamics. Schnoerr et al. [@schnoerr2017efficient] also employ a moment closure approximation and further approximate threshold functions to derive an approximate first passage time distribution. Bortolussi and Lanciani [@bortolussi2014stochastic] use a mean-field approximation which is required to reach the target region.
Recently, several groups independently suggested the use of semi-definite optimization for the computation of moment bounds for the limiting distribution [@ghusinga2017exact; @dowdy2018bounds; @kuntz2017rigorous; @sakurai2017convex]. In this approach, the differential equations describing the moment dynamics are set to zero and form linear constraints. Alongside, semi-definite constraints can be placed on the *moment matrices*. These give a semi-definite program that can be solved efficiently.
This approach has been extended to the transient case [@dowdy2018dynamic; @sakurai2019bounding]. The approach is similar in both works and is a cornerstone of the MFPT analysis presented here. They differ mainly by the fact that Sakurai and Hori apply a polynomial time-weighting [@sakurai2019bounding], while Dowdy and Barton use an exponential one [@dowdy2018dynamic]. We adopt the former approach because it can be naturally adapted to the description of densities over time. The resulting forms can also be adapted to statistical estimation problems [@backenkohler2019control].
Semi-definite programming has been applied to a wide range of problems, including stochastic processes in the context of financial mathematics [@lasserre2006pricing; @kashima2009polynomial]. For good introductions and overviews of application areas, we refer the reader to Parrilo [@parrilo2003semidefinite] and, more recently, Lasserre [@lasserre2010moments].
Particularly relevant for this work is the application of convex optimization to first passage times. Helmes et al. [@helmes2001computing] formulated a linear program using the Hausdorff moment conditions to bound moments of the first passage time distribution in Markovian processes. Semi-definite optimization has been successfully applied in financial mathematics by Kashima and Kawai [@kashima2009polynomial], as well as Lasserre et al. [@lasserre2006pricing] to bound prices of exotic options. Here, the approach by Lasserre is adapted to PCTMCs.
Preliminaries {#sec:bg}
=============
A Population Continuous-Time Markov Chain (PCTMC) describes the interactions among a set of species $S_1,\dots, S_{n_S}$ in a well-stirred reactor[^1]. Since we assume that all reactant molecules are equally distributed in space, we only keep track of the overall copy number of molecules of each species. Therefore the state-space is $\mathcal{S}\subseteq\mathbb{N}^{n_S}$. The interactions are expressed as *reactions* with a certain gain and loss of molecules, given by the non-negative integer vectors $\vec{v}_j^{-}$ and $\vec{v}_j^{+}$ for some reaction $j$, respectively. Such a reaction is denoted as $$\label{eq:reaction}
\sum_{i=1}^{n_S} v_{ji}^{-} S_i
\xrightarrow{a_j}
\sum_{i=1}^{n_S} v_{ji}^{+} S_i\,.$$ The reaction rate constant $a_j>0$ determines the propensity function $\alpha_j$ of the reaction. If just a constant is given, *mass-action* propensities are assumed, where for $\vec x\in\mathcal{S}$ we define $$\label{eq:stoch_mass_action}
\alpha_j(\vec{x})\coloneqq a_j\prod_{i=1}^{n_S}\binom{x_i}{v_{ji}^{-}}\,.$$ The system’s behavior is described by a stochastic process $\{{\vec{X}}_t\}_{t\geq 0}$. We denote the abundance of a given species $S_i$ in $\vec X_t$ by $X_t^{(S_i)}$. The propensity $\alpha_j(\vec x)$ gives the infinitesimal probability of a reaction occurring, given a state $\vec x$. That is, for $\vec v_j=\vec v_j^+-\vec v_j^-$ and a small time step $\delta t >0$, $$\label{eq:reaction_prob}
\Pr({\vec{X}}_{t+\delta t}=\vec{x}+\vec{v}_j\mid \vec{X}_t=x)
=\alpha_j(\vec{x})\delta t + o(\delta t)\,.$$ Therefore, given a system of $n_R$ reactions, the semantics of $\vec X_t$ is given by a continuous-time Markov chain (CTMC) on $\mathcal{S}$ with infinitesimal generator matrix $Q$ with entries $$\label{eq:cme_generator}
Q_{\vec x, \vec y} = \begin{cases}
\sum_{j:\vec x+\vec v_j = y}\alpha_j(\vec x)\,,&\text{if}\;\vec x\neq
\vec y,\\[1ex]
-\sum_{j=1}^{n_R} \alpha_j(\vec x)\,, &\text{otherwise.}
\end{cases}$$ Accordingly, given an initial distribution on $\mathcal{S}$, the time-evolution of the process’ distribution is given by the Kolmogorov forward equation. For a single state, it is commonly referred to as the *chemical master equation* (CME) $$\label{eq:cme}
\frac{d\pi}{d t} (\vec x,t) =
\sum_{j=1}^{n_R}\left(
\alpha_j(\vec x-\vec v_j)\pi(\vec x-\vec v_j,t) - \alpha_j(\vec x)\pi(\vec x,t)
\right)\,,$$ where $\pi(\vec x,t)=\Pr(\vec X_t=\vec x)$ and $\Pr(\vec X_0=\vec x)=\pi(\vec{x}, 0)$.
In this work, we are interested in *first passage times* of such processes. That is the time, the process first enters a set of target states $B\subseteq \mathcal{S}$. Naturally, the analysis of first passage times is equivalent to the analysis of times at which the process exits the complement $\mathcal{S}\setminus B$. More formally, the first passage time $\tau$ for some target set $B$ is defined as the random variable $$\label{eq:fpt_def}
\tau = \inf\{t\geq 0\mid \vec X_t \in B\}$$
Consider the following simple non-linear PCTMC as an example.
\[model:dim\] We first examine a simple dimerization model on an unbounded state-space with reactions $$\varnothing\xrightarrow{\mu}M,\quad 2M\xrightarrow{\delta}D$$ and initial condition $X_0^{(M)}=X_0^{(D)}=0$. The semantics is given by a CTMC $\vec{X}_t=(X_t^{(M)}, X_t^{(D)})^{{\top}}$, where $(S_1, S_2)=(M,D)$. The reaction propensities according to are $\alpha_1(\vec{x})=\mu$ and $\alpha_2(\vec{x})=\delta\, x^{(M)} (x^{(M)} -
1)$. The change vectors $v_1^-={(0,0)}^{{\top}}$, $v_1^+={(1,0)}^{{\top}}$, $v_2^-={(2,0)}^{{\top}}$, and $v_2^{+}={(0,1)}^{{\top}}$. Consequently, $v_1={(1,0)}^{{\top}}$ and $v_2={(-2, 1)}^{{\top}}$.
In this example, we are interested in the time at which $M$ exceeds some threshold $H$. With the framework presented in the sequel, one can bound the expected value of this time. Further, it is possible to impose a time-horizon $T$, and find bounds on the probability of $ X_t^{(M)}\geq H$ for some $0\leq t\leq T$. The employed framework is centered around semi-definite relaxations of the generalized moment problem [@lasserre2010moments]. These require linear constraints on the moments of measures. In the following section, we derive such constraints.
Martingale Formulation {#sec:moments}
======================
Next, we will discuss equations for the evolution of the statistical moments of the process and a related martingale formulation. This is later used to derive linear constraints on the moments of appropriate measures that can be used to bound MFPTs.
In particular, we consider the dynamics of *raw moments* ${\ensuremath{{\mathbb{E}}\left(\vec X_t^{\vec m}\right)}}$ for $\vec m\in \mathbb N^{n_S}$ and a fixed probability measure. The order of a moment ${\ensuremath{{\mathbb{E}}\left({\vec X}^{\vec m}\right)}}$ is given by its exponent sum, i.e. $\sum_i m_i$. We can derive the time evolution of the raw moments ${\ensuremath{{\mathbb{E}}\left(\vec X_t^{\vec m}\right)}}$ directly from the CME in . Note, that the notion of the expected value can be generalized to any measure $\mu$ on a Borel-measurable space $(M, \mathcal{B}(M))$. There the $m$-th raw moment is $\int_M x^{m}\,d\mu(x)$.
Let $f$ be a polynomial function, $t\ge0$. We can easily derive ordinary differential equations (ODEs) to describe the dynamics of ${\ensuremath{{\mathbb{E}}\left(f(\vec{X}_t)\right)}}$. Specifically, $$\label{eq:mom_ode}
\frac{d}{dt}{\ensuremath{{\mathbb{E}}\left(f(\vec X_t)\right)}} = \sum_{j=1}^{n_R}{\ensuremath{{\mathbb{E}}\left(\left(f({\vec X_t +
\vec{v_j}}) - f(\vec X_t)\right)\alpha_j(\vec X_t)\right)}}\,.$$
When choosing $f(X_t)=X_t^m$ and $\vec m=(1,0)$ and $\vec m=(2,0)$, for Model \[model:dim\], for example, we get the following system of ODEs for the change of the first and second statistical moment of species $M$ $$\begin{aligned}
\frac{d}{dt}{\ensuremath{{\mathbb{E}}\left({X}_t\right)}} &= \mu{\ensuremath{{\mathbb{E}}\left({X}_t^0\right)}} -
2{\delta}\left({\ensuremath{{\mathbb{E}}\left({X}_t^2\right)}}-{\ensuremath{{\mathbb{E}}\left({X}_t\right)}}\right)\label{eq:dim_exp_ode}\\[1ex]
\frac{d}{dt}{\ensuremath{{\mathbb{E}}\left({X}_t^2\right)}} &= \mu(2{\ensuremath{{\mathbb{E}}\left({X}_t\right)}} + 1) - 4\delta\left({\ensuremath{{\mathbb{E}}\left({X}_t^3\right)}} -
2{\ensuremath{{\mathbb{E}}\left({X}_t^2\right)}} + {\ensuremath{{\mathbb{E}}\left({X}_t\right)}}\right)\,,\end{aligned}$$ where we let $X_t=X_t^{(M)}$ for ease of notation. These ODEs cannot be integrated because the system is not closed. The right-hand side for moment ${\ensuremath{{\mathbb{E}}\left(X_t^m\right)}}$ always contains ${\ensuremath{{\mathbb{E}}\left(X_t^{m+1}\right)}}$. To solve an initial value problem, one typically resorts to ad-hoc approximations of the highest order moments to close the system. Here we do *not* need such approximations because we do not numerically integrate such equations.
Multiplying with some polynomial function $w(t)$ and integrating on $[0, T]$ yields [@dowdy2018dynamic; @sakurai2019bounding] $$\label{eq:exp_constraint}
\begin{split}
& w(T)\,{\ensuremath{{\mathbb{E}}\left(f(\vec X_T)\right)}}
- w(0)\,{\ensuremath{{\mathbb{E}}\left(f(\vec X_{0})\right)}}
- \int_{0}^{T}\frac{dw(t)}{dt}{\ensuremath{{\mathbb{E}}\left(f(\vec X_t)\right)}}\,dt\\
=&\sum_{j=1}^{n_R}\int_{0}^{T}w(t)\,
{\ensuremath{{\mathbb{E}}\left(\left(f{(\vec X_t + \vec v_j)} - f(\vec X_t)\right)\alpha_j(\vec X_t)\right)}}\,dt.
\end{split}$$ If we now assume that ${\ensuremath{{\mathbb{E}}\left(|\vec X^{\vec m}_t|\right)}}$ and $\left|w(t)\right|$ remain finite for all $t\in[0,T]$, $\vec{m}\in\mathbb{N}^{n_S}$ we can interchange summation and integral of a monomial ${\vec{x}}^{\vec{m}}$ and pull all expectation operators outside, i.e. for a polynomial $g$ $$\begin{split}
\int_{0}^Tg(t){\ensuremath{{\mathbb{E}}\left(\vec X_t^{\vec m}\right)}}\,dt
=&\;{\ensuremath{{\mathbb{E}}\left(\int_{0}^Tg(t){\vec X}_t^{\vec m}\,dt\right)}}.
\end{split}$$ Hence, for pulling the expectation operator outside yields a martingale $\{Z_T\}_{T\geq 0}$, where $$\label{eq:martingale}
\begin{split}
Z_T\coloneqq&\,w(T)f(\vec X_T) - w(0)f(\vec X_{0}) -
\int_{0}^T\frac{dw(t)}{dt}f(\vec X_t)\,dt\\
&-\sum_{j=1}^{n_R}\int_{0}^Tw(t)
(f(\vec X_t+\vec v_j) - f(\vec X_t))\alpha_j(\vec X_t)\,dt\,,
\end{split}$$ with respect to $\vec X_t$. When choosing $w(t)=t^k$ with $k\in\mathbb N$ and $f(\vec x)={\vec x}^{\vec m}$ it takes the form $$\label{eq:basic_poly_martingale}
Z_T^{(\vec m, k)}=
T^k \vec X_T^{\vec m}
- 0^k \vec X_{0}^{\vec m}
+ \sum_{i}c_i\int_0^T t^{k_i} \vec X_t^{\vec m_i}\,dt$$ where $(\vec m_i)_i$, $(k_i)_i$, and $(c_i)_i$ are finite sequences resulting from the substitution of $f$ and $w$ and expansion of . We will use this martingale in the following section to derive linear constraints for the semi-definite program used to bound MFPTs.
If we apply this to our previous example , letting $m=1$ and $k=1$ we obtain the following process for Model \[model:dim\]. $$\begin{aligned}
Z_T^{(1,1)} = TX_T - \int_0^T X_t\,dt - \mu \int_0^T t\,dt - 2\delta
\int_0^T t X_t\,dt +
2{\delta}\int_0^TtX_t^2\,dt,\end{aligned}$$ where the sequences above are $(m_i)_i=(1,0,1,2)$, $(k_i)_i=(0,1,1,1)$, and $(c_i)_i=(-1,-\mu, -2\delta,2\delta)$.
Bounds for Mean First Passage Times {#sec:mfpt_bounds}
===================================
We now turn to the analysis of first passage times within some time-bound $T>0$. Given some set $B\subset \mathcal{S}$ the first passage time is given by the random variable $$\tau=\inf\{t\geq 0\mid \vec X_t \in B\}\land T\,.$$ For this work, we only look at threshold hitting times, i.e. we set a threshold $H$ for species $S$ and thus $B=\{\vec{x}\mid x^{(S)}\geq
H\}$.[^2] In the sequel, we will use $\tau$ as a stopping time in our martingale formulation and consider $Z_\tau^{(\vec m, k)}$ instead of $Z_T^{(\vec m, k)}$. Since defines a martingale, $Z_{\tau}^{(\vec m, k)}$ remains a martingale by Doob’s optional sampling theorem [@gihmantheory]. In particular, this implies that ${\mathbb{E}}(Z_{\tau}^{(\vec m, k)})=0$.
Linear Moment Constraints
-------------------------
To simplify our presentation, we fix an initial state $\vec x_0$, i.e. $P(\vec X_0=\vec x_0)=1$. Using ${\mathbb{E}}(Z_{\tau}^{(\vec m, k)})=0$ and the form for $Z_{\tau}^{(\vec m, k)}$ yields the following linear constraint on expected values. $$\label{eq:constraint}
0 = \,{\ensuremath{{\mathbb{E}}\left({\tau}^k\vec X_{\tau}^{\vec m}\right)}}
- 0^k\vec x_0^{\vec m}
+ \sum_{i}c_i{\ensuremath{{\mathbb{E}}\left(\int_{0}^{\tau} t^{k_i} \vec X_t^{\vec m_i}\,dt\right)}}\,,$$ where $0^0=1$. For the ease of exposition, we now turn to first passage times of one-dimensional processes w.r.t. an upper threshold $H$. In particular, we will consider moments $X^m$ of a one-dimensional process for $m=0,1,2\ldots$. The approach proposed in the sequel, however, can be straightforwardly extended to multi-dimensional processes and more complex target sets $B$.
Consider again Model \[model:dim\] and assume that we are interested in the time at which species $M$ exceeds threshold $M$. Since the abundance of $D$ does not influence $M$, we can ignore species $D$ and treat the process as one-dimensional. Figure \[fig:decomposition\] shows three example trajectories: Two reach an upper threshold $H=10$, while one reaches the final time horizon $T=4$.
![The relationship between the occupation measure $\xi$ and the exit location probability measures $\nu_1$ and $\nu_2$. The shaded area indicates the structure of the occupation measure. Three example trajectories are additionally plotted with their exit location highlighted. The plots are based on $10,\!000$ sample trajectories.[]{data-label="fig:decomposition"}](decomp1.pdf)
We notice, that expresses a relationship between the process dynamics up to the hitting time via expected values of the time-integrals and the final process state at the hitting time via ${\ensuremath{{\mathbb{E}}\left(\tau^k {X}_{\tau}^{{m}}\right)}}$. In particular, we can distinguish between the following two positive measures [@lasserre2010moments Chapter 9.2]:
- *Expected Occupation Measure* $\xi$ supported on $[0,H]\times [0,T]$: $$\label{eq:ex_occ_measure}
\xi(A\times C) \coloneqq {\ensuremath{{\mathbb{E}}\left(\int_{[0,\tau]\cap {C}}{1}_{\in A}(X_t)\,dt\right)}},$$
- *Exit Location Probability* supported on $(\{H\}\times[0,T]) \cup
([0,H]\times\{T\})$: $$\label{eq:exit_loc_measure}
\nu(A\times C)\coloneqq \Pr((X_{\tau},\tau)\in A\times C),$$
where $A\times C$ is a measurable set, i.e. $A$ and $C$ are elements of the Borel $\sigma$-algebras on $[0,H]$ and $[0,T]$, respectively.
Using Figure \[fig:decomposition\], one can gain an intuition for these two measures. The expected occupation measure is shaded in blue. As the name implies $\xi(A\times C)$ tells us how much time the process spends in $A$ up to $\tau$ restricting to the time instants belonging to $C$. In particular, $\xi([0,H]\times [0,T])={\ensuremath{{\mathbb{E}}\left(\tau\right)}}$. The exit location probability $\nu$, while being a two-dimensional distribution, can be viewed as a composition of a density describing the time at which the process reaches $H$ (if it does) and a probability mass function on the states of the process if the time-horizon is reached without exceeding $H$. We split the measure $\nu$ into $\nu_1$ and $\nu_2$ by conditioning on $\tau=T$. Thus, $\nu_1(C)\coloneqq\Pr(\tau\in C, \tau<T)$ and $\nu_2(A)\coloneqq\Pr(X_T\in
A, \tau=T)$. To refer to the moments of these measures, we define *partial moments* $${\ensuremath{{\mathbb{E}}\left(g({X}); f({Y}) = y\right)}}\coloneqq
{\ensuremath{{\mathbb{E}}\left(g({X})\mid f({Y})=y\right)}}\Pr(f({Y})=y)\,,$$ for some polynomial $g$ and some indicator function $f$. Then $${\ensuremath{{\mathbb{E}}\left(\tau^k X_{\tau}^m\right)}}=T^k{\ensuremath{{\mathbb{E}}\left(X_{\tau}^m;\tau=T\right)}} + H^m{\ensuremath{{\mathbb{E}}\left({\tau}^k;\tau < T, X_{\tau}=H\right)}}\,.$$ Therefore the linear moment constraints have the form $$\begin{split}
0 = \,&T^k{\ensuremath{{\mathbb{E}}\left(X_{\tau}^m;\tau=T\right)}} + H^m{\ensuremath{{\mathbb{E}}\left({\tau}^k;\tau < T, X_{\tau}=H\right)}}\\
&- 0^kx_0^{m}
+ \sum_{i}c_i{\ensuremath{{\mathbb{E}}\left(\int_{0}^{\tau} t^{k_i}X_t^{m_i}\,dt\right)}}\,.
\end{split}$$ Next, we consider infinite sequences of partial moments by letting $k$ and $m$ range over the natural numbers. Let $\vec{y}_1=(y_{1k})_k$, $\vec{y}_2=(y_{2m})_m$, and $\vec{z}=(z_{km})_m$ denote the moment sequences of $\nu_1$, $\nu_2$, and $\xi$, respectively. $$y_{1k} \coloneqq {\ensuremath{{\mathbb{E}}\left({\tau}^k;\tau < T\right)}},\quad y_{2m}\coloneqq{\ensuremath{{\mathbb{E}}\left(X_{\tau}^m;\tau=T\right)}},\quad
z_{km}\coloneqq{\ensuremath{{\mathbb{E}}\left(\int_0^{\tau}t^k X_t^m\,dt\right)}}\,$$ Thus, the variable corresponding to $z_{00}={\ensuremath{{\mathbb{E}}\left(\tau\right)}}$ becomes the objective of the optimization problem that we describe in the sequel.
Semi-Definite Constraints
-------------------------
It is a necessary condition for a positive measure that the *moment matrices* are positive semi-definite. A matrix $M\in\mathbb{R}^{n\times n}$ is positive semi-definite, denoted by $M\succeq 0$ if and only if $${\vec v}^T M{\vec v} \geq 0\quad \forall \vec v\in\mathbb{R}^n\,.$$ As an example, let us consider a one-dimensional random variable $X$ with moment sequence $\vec x\in\{\vec{y}_1,\vec{y}_2,\vec{z}\}$. For moment order $r$, the entries of the $(r+1)\times (r+1)$ moment matrix $M_r(\vec x)$ are given by the raw moments. In particular, $(M_r)_{ij}=x_{i+j-2}$ for $i,j\in\mathbb{N}_r$ where $\mathbb{N}_r=\{0,1,\dots,r\}$ and the maximum order in the matrix is $2r$. For instance, $$\label{eq:m1_dim}
M_1(\vec x) =
\begin{bmatrix}
x_0 & x_1 \\
x_1 & x_2
\end{bmatrix}$$ needs to be positive semi-definite. By Sylvester’s criterion this means $\det
M_1\geq 0$ and $x_0\geq 0$. We can easily see, that this entails $$\det M_1={\ensuremath{{\mathbb{E}}\left(X^2\right)}}-{\ensuremath{{\mathbb{E}}\left(X\right)}}^2
\geq 0\,.$$ This restriction is natural since the variance is always non-negative.
It is crucial to restrict the measures $\xi$, $\nu_1$, and $\nu_2$ to their supports. This can be done, by defining polynomials that are non-negative on the intended support of the measure. For example, $\mu_2$ has support $[0,H]$. We can now define $$u_H(t,x) = Hx - x^2, \quad x\in \mathbb R$$ as a polynomial that is non-negative on $[0,T]$. Using such polynomials, we can construct *localizing matrices*, which have to be positive semi-definite [@lasserre2010moments]. Applying $u_H$ to the moment matrix in we obtain $$M_1(u_H, \vec{y_2})=
\begin{bmatrix}
Hy_{20} - y_{22} & Hy_{21} - y_{23} \\
Hy_{21} - y_{23} & Hy_{22} - y_{24}
\end{bmatrix}$$ with the constraint $M_1(u_H, \vec{y_2})\succeq 0$, where the application of a polynomial such as $u_H$ to a moment matrix is formally defined for the multidimensional case in Section \[subsec:multidim\]. Similarly, let $u_T(t, x) = Tt-t^2$ to restrict $\nu_1$ to $[0,T)$.
A semi-definite program to bound MFPTs
--------------------------------------
With the linear constraints on the measures and and the semi-definite constraints discussed in the previous sections, we can now formulate a semi-definite program (SDP). An SDP is an optimization over the cone of positive semi-definite $n \times n$-matrices $\mathcal{X}$ under linear constraints: $$\label{eq:sdp_canonical}
\begin{split}
\min_{X\in\mathcal{X}} \hspace{1em} & \sum_{i,j} A_{ij}^{(0)}X_{ij} \\
\text{such that} \hspace{1em}
& X\succeq 0\\
& \sum_{i,j} A_{ij}^{(k)}X_{ij} \leq b_k, \quad k=1,\dots,m
\end{split}$$ with constant matrices $A^{(i)}\in \mathbb{R}^{n\times n}$, $i=0,\dots,m$ and constants $b_k\in\mathbb{R}$, $k=1,\dots,m$. Such a problem is convex and can be solved efficiently [@vandenberghe2010cvxopt].
The derived linear equations and linear matrix inequalities can now be used to formulate an SDP. The full optimization problem has infinitely many constraints because there are infinitely many moments. We relax this problem by constructing the SDP using by choosing a finite order $r$ for the moment matrices $M_r$. With each moment sequence $\vec x$ we associate a sequence proxy variables $\vec{x'}$ used in the optimization problem. Now we can state the SDP relaxation to the MFPT problem for any order $0<r<\infty$ $$\label{eq:sdp_for_fpt}
\begin{split}
\min / \max \hspace{1em}& z_{00}^{\prime} \\
\text{such that}\hspace{1em} & M_r(\vec{z'})\succeq 0,
M_r({u}_T, \vec{z'})\succeq 0, M_r({u}_H, \vec{z'})\succeq 0\\
& M_r(\vec{y_1'}) \succeq 0, M_r({u}_T,\vec{y_1'}) \succeq 0\\
& M_r(\vec{y_2'}) \succeq 0, M_r({u}_H, \vec{y_2'}) \succeq 0\\
& 0= y_{1k}' H^m - y_{2m}'T^k - 0^k x_0^m +\sum_i c_i z_{k_i m_i}', \quad\forall m, k
\end{split}$$ This problem can be solved using off-the-shelf SDP solvers such as MOSEK [@mosek], CVXOPT [@vandenberghe2010cvxopt], or SCS [@scs].
Multi-Dimensional Generalization {#subsec:multidim}
--------------------------------
For a general multi-dimensional moment sequence $\vec{y}={({\ensuremath{{\mathbb{E}}\left(\vec X^{\vec{m}}\right)}})}_{\vec{m}\in\mathbb{N}^{n_s}}$, the moment matrix is [@lasserre2010moments] $$M_r(\vec y)(\vec\alpha,\vec\beta)
=y_{\vec\alpha + \vec\beta},\quad\forall\vec{\alpha},
\vec{\beta}\in\mathbb{N}_r^n$$ where row and column indices, $\vec{\alpha}$ and $\vec\beta$, are ordered according to the canonical basis $$\label{eq:canoncial_basis}
\vec{v}_r(\vec{x}) =
{(1,x_1,x_2,\dots,x_n,x_1^2,x_1x_2,\dots ,x_1x_n,\dots ,x_1^r,\dots
,x_n^r)}^T\,.$$ Equivalently, $M_r(\vec{y})={\ensuremath{{\mathbb{E}}\left(\vec{v}_r (\vec x)\vec{v}_r(\vec x)^T\right)}}$. For a moment sequence the semi-definite restriction $M_r(\vec{y})\succeq 0$ must hold.
Measures can be restricted to semi-algebraic sets $\{\vec x\in\mathbb{R}^n \mid u_j(\vec x)\geq 0, j=1,\dots,m\}$, where $u_j$, $j=1,\dots,m$ are polynomials [@lasserre2010moments]. This is done by placing restrictions on the [localizing matrices]{}. For each polynomial $u_i\in\mathbb{R}[x]$ with coefficient vector $\vec{u}=\{u_{\vec\gamma}\}$, i.e. $u(\vec x) = \sum_{\vec{\gamma}\in\mathbb{N}^n} u_{\vec{\gamma}}
\vec{x}^{\vec{\gamma}}$, the localizing matrix is $$M_r(u, \vec{y})(\vec{\alpha}, \vec{\beta})=
\sum_{\vec\gamma\in\mathbb{N}^n}u_{\vec\gamma}
y_{\vec\gamma+\vec\alpha+\vec\beta},\quad
\forall\vec{\alpha},\vec{\beta}\in\mathbb{N}^n_r.$$ Requiring that this matrix is positive semi-definite restricts the measure to $\{\vec{x}\mid u_i(\vec{x})\geq 0\}$. This way we can, for example, restrict the moment sequence $\vec{y}$ to measures that are positive w.r.t. dimension $j$. Simply letting $u(\vec{x}) = x_j$ and requiring $M_1(\vec{u},\vec{y})\succeq 0$ for $i=1,\dots,n_S$ gives us this restriction.
Implementation and Evaluation {#sec:evaluation}
=============================
The main challenge of finding a solution to the SDP problem in is numerical stability. Usually, the moment sequences vary by many orders of magnitude. For an SDP solver to work, the moment matrices need to be re-scaled [@dowdy2018bounds] such that moments only vary by few orders of magnitude. In other scenarios such as the bounding of general transient or steady-state moments, the scaling can be particularly difficult, because the magnitude of moments is generally not known a priori. However, for the MFPT problem, we propose the following moment scaling.
Moment Scaling
--------------
Using the fact that $\mathcal{S}\setminus {B}$ is often finite, it is possible to derive trivial bounds, which can be used to scale moments. If, for example, we have a one-dimensional process $X_t$ with $X_0 = 0$ a.s. and are interested in the hitting time of an upper threshold $H>0$ until time $T>0$ for $i,k\in \mathbb N$ $$z_{ik} = {\ensuremath{{\mathbb{E}}\left(\int_0^{\tau}t^i X_t^k\,dt\right)}}\leq{\ensuremath{{\mathbb{E}}\left(\int_0^T t^i X_t^k\,dt\right)}}\leq H^k\int_0^T t^i\,dt
=\frac{T^{i+1}H^k}{i+1}.$$ Thus, we fix a scaling vector $\vec d$ with entries $d_{ik}={T^{i+1}H^k}$ in the same order as the canonical base vector . Using this scaling vector, we can define a scaling matrix $D={\vec d}{\vec{d}}^{{\top}}$. Clearly, $D \succeq 0$. Now we can formulate the optimization over a scaled version $D^{-1}M(\vec{z'})$ instead of $M(\vec{z'})$. The moment matrices of the exit location probabilities are scaled in the same way. Alternatively, one can use approximations such as moment closures or bounds obtained by lower-order relaxations.
Case Studies
------------
We implemented and solved the SDP programs described above using MOSEK [@mosek] (version 9.1.2) via the CVXPY interface [@cvxpy] (version 1.0.24).
As a first case study, we use Model \[model:dim\] with parameters $\mu=100$ and $\delta=0.1$. In this model, we are interested in the time at which the number of agents of type $M$ surpasses a threshold of 25 before some time-horizon $T$, i.e. $\tau=\inf\{t\geq 0\mid X_t \geq 25\}\land T$. First, we set no finite time horizon $T$, i.e. $T=\infty$. This is achieved by dropping the moments $\vec y_2$ of measure $\nu_2$ in the linear constraints . The empirical FPT distribution based on 100,000 SSA simulations is given in Figure \[fig:dim\_fpt\]a and the bounds, given different moment orders, are given in Figure \[fig:dim\_fpt\]b. As we can see in Figure \[fig:dim\_fpt\]b, the bounds capture the MFPT precisely for orders 5, 6. The difference between upper and lower bound decreases roughly exponentially with increasing relaxation order $r$. We found that this trend was consistent among the case studies presented here (cf. Figure \[fig:convergence\]).
![First passage times for Model 1 with $\tau=\inf\{t\geq 0\mid X_t \geq 10\}\land \infty$. The dashed red line denotes the sampled MFPT. (a) The distribution of $\tau$ estimated based on 100,000 SSA samples. (b) The bounds based on the SDP in with different moment orders.\[fig:dim\_fpt\]](fpt_dist_dim.pdf "fig:")\
(a)
![First passage times for Model 1 with $\tau=\inf\{t\geq 0\mid X_t \geq 10\}\land \infty$. The dashed red line denotes the sampled MFPT. (a) The distribution of $\tau$ estimated based on 100,000 SSA samples. (b) The bounds based on the SDP in with different moment orders.\[fig:dim\_fpt\]](fpt_bounds_dim.pdf "fig:")\
(b)
Next, we look at first passage times within a finite time-horizon $T$. In Figure \[fig:dim\_fpt\_fin\]a we summarize the bounds obtained for the MFPT over $T$. While low-order relaxations (light) give rather loose bounds, the bounds are already fairly tight when using $r=4$. In many cases, hitting probabilities, that is, the probability of reaching the threshold before time $T$, are of particular interest. This is done by switching the optimization objective in from the mass of the expected occupation measure $\xi$ to the mass of $\nu_1$. In terms of moments, the objective changes from $z_{00}$ to $y_{10}$. The need for such a scenario often arises in the context of model checking, where one might be interested in the probability of a population exceeding a critical threshold. By varying the time horizon, we are able to recover bounds on the cumulative density $F(t) = \Pr(X_s=H\mid s<t)$ of the first passage time (Fig. \[fig:dim\_fpt\_fin\]b).
![First passage times for the dimerization model with $\tau=\inf\{t\geq 0\mid X_t \geq 25\}\land T$. The results for SDP relaxations of orders 1 (light) to 6 (dark) are shown. (a) The bounds on the MFPT for differing time horizons $T$. (b) Bounds on the probability to reach the threshold before time $T$.\[fig:dim\_fpt\_fin\]](mfpt_bounds.pdf "fig:")\
(a)
![First passage times for the dimerization model with $\tau=\inf\{t\geq 0\mid X_t \geq 25\}\land T$. The results for SDP relaxations of orders 1 (light) to 6 (dark) are shown. (a) The bounds on the MFPT for differing time horizons $T$. (b) Bounds on the probability to reach the threshold before time $T$.\[fig:dim\_fpt\_fin\]](reachability_probs.pdf "fig:")\
(b)
As a second study, we consider a 2-dimensional model by combining two independent dimerizations.
\[model:double\_dim\] $$\varnothing\xrightarrow{10^4}M_1,\quad 2M_1\xrightarrow{0.1}D_1,\quad \varnothing\xrightarrow{10^4}M_2,\quad 2M_2\xrightarrow{0.1}D_2$$
As a FPT we consider the time at which either $M_1$ or $M_2$ surpasses a threshold of 200 or a time horizon of $T=10$ is reached, i.e. $$\tau=\inf\{t\geq 0\mid X_t^{(M_1)} \geq 200\}\land \inf\{t\geq 0\mid X_t^{(M_2)} \geq 200\}\land 10\,.$$ As before, we ignore the products $D_1$ and $D_2$ since they do not influence $\tau$. Still, the possible state-space reaches a size of $200^2=40,\!000$. The SSA (using $n=10,\!000$ runs) gives the estimate ${\ensuremath{{\mathbb{E}}\left(\tau\right)}}\approx 2.8378e-02$ which is captured tightly by the SDP bounds (cf. Table \[tab:bounds\]). For higher relaxation orders $r \geq 5$ numerical issues prevented the solution of the corresponding SDPs.
Hybrid Models and Multi-Modal Behaviour
---------------------------------------
The analysis of switching times is a particularly interesting case of FPTs that arises in many contexts. Often mode switching in such systems can be described a modulating Markov process whose switching rates may depend on the system state (e.g. the population sizes). In biological applications, mode switching often describes a change of the DNA state [@hasenauer2014method; @stekel2008strong] and the analysis of switching time distribution is of particular interest [@spieler2011model; @barzel2008calculation]. In the context of PCTMCs, the state-space of such models can be given as $$\mathcal{S}=
\mathbb{N}^{\tilde{n}_S}\times {\{0,1\}}^{\hat{n}_S}\,.$$ This state is modeled by a $\hat{n}_S$ population variables with binary domains. Therefore, at each time point, the state of these modulator variables is given by a set of Bernoulli random variables. When considering the moments of such a variable $X$, clearly ${\ensuremath{{\mathbb{E}}\left(X^m\right)}}={\ensuremath{{\mathbb{E}}\left(X\right)}}=\Pr(X=1)$ for all $m\geq 1$.
We apply a split of $\vec X_t$ into the high count part ${\vec{\tilde{X}}}_t$ and the binary part ${\vec{\hat{X}}}_t$ to the expectations in . Similarly, we split $\vec v_j$ and with a case distinction over the mode variable, we arrive at a similar result as in [@hasenauer2014method]: $$\label{eq:mcm}
\begin{split}
\frac{d}{dt}{\ensuremath{{\mathbb{E}}\left(\vec{\tilde{X}}^{{\vec m}}_t 1_{=\vec y}({\vec{\hat{X}}}_t)\right)}}
=&\sum_{j=1}^{n_R}
{\ensuremath{{\mathbb{E}}\left({\left(
{\vec{\tilde{{X}}}}_t+\vec{\tilde{{v}}}_j
\right)}^{\vec{m}}\alpha_j(\vec{\tilde{{X}}}_t, \vec{{y}} -
\vec{\hat{v}}_j)
1_{={\vec{y}- \vec{\hat{v}}_j}}({\vec{\hat{X}}}_t )\right)}}\\
&- \sum_{j=1}^{n_R}
{\ensuremath{{\mathbb{E}}\left({\vec{\tilde{{X}}}}_t^{\vec{m}}\alpha_j(\vec{\tilde{{X}}}_t,
{\vec{y}})
1_{=\vec y}({\vec{\hat{X}}}_t)\right)}}\,.
\end{split}$$ Similarly to the general moment case, we can derive a constraint, by multiplying with a time-weighting factor and integrating.
------------------------------------------- ------- --------- -------- -------- -------- --------
Model
1 2 3 4 5
Double Dim. (Model \[model:double\_dim\]) lower 0.0010 0.0250 0.0275 0.0280 —
upper 10.0000 0.0575 0.0323 0.0299 —
Gene Expression (Model \[model:gexpr\]) lower 4.0000 6.0028 6.2207 6.3377 6.3772
upper 10.7179 6.4619 6.4079 6.4004 6.3835
------------------------------------------- ------- --------- -------- -------- -------- --------
: MFPT bounds on Models \[model:double\_dim\] and \[model:gexpr\].\[tab:bounds\]
For simplicity, here we assume $\tilde n_S=\hat n_S=1$. Fixing appropriate sequences ${(c_i)}_i$, ${(m_i)}_i$, ${(k_i)}_i$, and ${(y_i)}_i$ the constraint has the following form. $$\label{eq:mcm_constraint}
\begin{split}
&\sum_{y\in\{0,1\}}{H}^{m}{\ensuremath{{\mathbb{E}}\left(\tau^k;{{\hat{X}}}_{\tau}= y, \tau
< T\right)}}
+
T^k{\ensuremath{{\mathbb{E}}\left({{\tilde{X}}}_T^{m};{{\hat{X}}}_T= y,\tau=T\right)}}\\
=\; & 0^k{{\tilde{x}}}_0^{m} 1_{= y}({\hat{x}}_0) +
\sum_i c_i {\ensuremath{{\mathbb{E}}\left(\int_0^{\tau}t^{k_i}{{{\tilde{X}}}_t}^{{m}_i}\,dt;
{{\hat{X}}}_t= y_i\right)}}\\
\end{split}$$ This way we can decompose the moment matrices such that for each mode $y\in\{0,1\}$, we have moment matrices composed of the respective partial moments. To this end, let $z^{(y)}_m$ be the partial moment w.r.t. ${{\hat{X}}}= y$. The moment constraint over the partial moments has a linear structure: $$0=y_{1k} {H}^{m} - y_{2m}T^k - 0^k x_0^{m}
+\sum_i c_i z^{(y_i)}_{k_i m_i}\,.$$
As an instance of a multi-modal system, we consider a simple gene expression with self-regulating negative feedback which is a common pattern in many genetic circuits [@stekel2008strong].
\[model:gexpr\] This model consists of a gene state that is either on or off, i.e. $X^{D_{\text{on}}}_t
+X^{D_{\text{off}}}_t = 1$, $\forall t\geq 0$. Therefore the system has two *modes*. $$D_{\text{on}} \xrightarrow{\tau_{0}} D_{\text{off}}, \quad
D_{\text{off}} \xrightarrow{\tau_{1}} D_{\text{on}}, \quad
D_{\text{on}} \xrightarrow{\rho} D_{\text{on}} + P, \quad$$ $$P\xrightarrow{\delta}\varnothing,\quad
P + D_{\text{on}} \xrightarrow{\gamma} D_{\text{off}}$$ The model parameters are $(\tau_0,\tau_1,\rho,\delta,\gamma)=(10,10,2,0.1,0.1)$ and $X_0^{(D_{\text{off}})}=1$, $X_0^{(P)}=0$ a.s.
As a first passage time we consider $$\tau=\inf\{t\geq 0\mid X_t^{(P)} \geq
5\}\land 20\,.$$
![The interval width, i.e. the difference between upper and lower bound, for different case studies and targeted first passage times against the order $r$ of the SDP relaxation.\[fig:convergence\]](convergence.pdf)
The results are summarized in Table \[tab:bounds\]. The estimated MFPT based on $100,\!000$ SSA samples is ${\ensuremath{{\mathbb{E}}\left(\tau\right)}}\approx 6.37795\pm0.02847$ at $99\%$ confidence level. Note that our SDP solution for $r=5$ yields tighter moment bounds than the statistical estimation. In Fig. \[fig:convergence\] we summarize our results about the decrease of the interval widths for increasing relaxation order $r$ by plotting them on a log-scale. We see an approximately exponential decrease in $r$. The semi-definite programs above were all solved within at most a few seconds.
Conclusion {#sec:conclusion}
==========
State-based methods to compute reachability probabilities and first passage times for continuous-time Markov chains are not scalable due to state-space explosion, an issue exacerbated in population models. Moment-based methods offer an alternative for PCTMCs, which scales with the number of different populations in the system, but are approximate methods with little or no control of the error. In this paper, we bridge this gap by proposing a rigorous approach to derive bounds on first passage times and reachability probabilities, leveraging a semi-definite programming formulation based on appropriate moment constraints.
Our proposed scaling mitigates numerical instabilities of the SDP solvers, which are caused by the fact that moments typically span several orders of magnitude. However, the scaling only addresses the moment matrices but not the linear constraints which still contain values with varying orders of magnitudes. We, therefore, plan as future work to introduce an appropriate scaling for the linear constraints or to redefine the moment constraints (e.g. using an exponential time weighting [@dowdy2018dynamic]). Based on this investigation, we expect to make this approach applicable to more problems including, for example, the computation of bounds of rare event probabilities. Numerical instabilities due to moment values of largely differing orders of magnitudes are a current limitation of all moment-based methods. We expect that the development of more sophisticated scaling techniques will improve approximate moment-based methods, as well.
### Acknowledgements
We would like to thank Andreas Karrenbauer for helpful comments on the usage of SDP solvers and Gerrit Großmann for the valuable comments on this manuscript. This work is supported by the DFG project “MULTIMODE”, and partially supported by the italian PRIN project “SEDUCE” n. 2017TWRCNB.
[^1]: In the sequel, we will also use other letters than $S_i$ as species names.
[^2]: Note, that this framework allows for a more general class of target sets, which are discussed in Section \[subsec:multidim\].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Matthias R. Gaberdiel\
Institut für Theoretische Physik\
ETH Zurich, 8093 Zurich, Switzerland\
\
and\
\
Roberto Volpato\
Albert-Einstein Institut für Gravitationsphysik\
Am Mühlenberg 1, 14476 Golm, Germany
title: 'Mathieu Moonshine and Orbifold K3s[^1]'
---
AEI-2012-058
[Mathieu Moonshine and Orbifold K3s]{}[^2]
Matthias R. Gaberdiel\
Institut für Theoretische Physik\
ETH Zurich, 8093 Zurich, Switzerland\
and
Roberto Volpato\
Max-Planck-Institut für Gravitationsphysik\
Am Mühlenberg 1, 14476 Golm, Germany
[**Abstract:**]{} The current status of ‘Mathieu Moonshine’, the idea that the Mathieu group $\mathbb{M}_{24}$ organises the elliptic genus of K3, is reviewed. While there is a consistent decomposition of all Fourier coefficients of the elliptic genus in terms of Mathieu $\mathbb{M}_{24}$ representations, a conceptual understanding of this phenomenon in terms of K3 sigma-models is still missing. In particular, it follows from the recent classification of the automorphism groups of arbitrary K3 sigma-models that (i) there is no single K3 sigma-model that has $\mathbb{M}_{24}$ as an automorphism group; and (ii) there exist ‘exceptional’ K3 sigma-models whose automorphism group is not even a subgroup of $\mathbb{M}_{24}$. Here we show that all cyclic torus orbifolds are exceptional in this sense, and that almost all of the exceptional cases are realised as cyclic torus orbifolds. We also provide an explicit construction of a $\mathbb{Z}_5$ torus orbifold that realises one exceptional class of K3 sigma-models.
Introduction
============
In 2010, Eguchi, Ooguri and Tachikawa observed that the elliptic genus of K3 shows signs of an underlying Mathieu $\mathbb{M}_{24}$ group action [@EOT]. In particular, they noted (see section \[sec:MatMoon\] below for more details) that the Fourier coefficients of the elliptic genus can be written as sums of dimensions of irreducible $\mathbb{M}_{24}$ representations.[^3] This intriguing observation is very reminiscent of the famous realisation of McKay and Thompson who noted that the Fourier expansion coefficients of the $J$-function can be written in terms of dimensions of representations of the Monster group [@Thompson; @CN]. This led to a development that is now usually referred to as ‘Monstrous Moonshine’, see [@Gannon2006] for a nice review. One important upshot of that analysis was that the $J$-function can be thought of as the partition function of a self-dual conformal field theory, the ‘Monster conformal field theory’ [@Borcherds; @FLM], whose automorphism group is precisely the Monster group. The existence of this conformal field theory explains many aspects of Monstrous Moonshine although not all — in particular, the genus zero property is rather mysterious from this point of view.
In the Mathieu case, the situation is somewhat different compared to the early days of Monstrous Moonshine. It is by construction clear that the underlying conformal field theory [*is*]{} a K3 sigma-model (describing string propagation on the target space K3). However, this does not characterise the corresponding conformal field theory uniquely as there are many inequivalent such sigma-models — in fact, there is an $80$-dimensional moduli space of such theories, all of which lead to the same elliptic genus. The natural analogue of the ‘Monster conformal field theory’ would therefore be a special K3 sigma-model whose automorphism group coincides with $\mathbb{M}_{24}$. Unfortunately, as we shall review here (see section \[sec:Theorem\]), such a sigma-model does not exist: we have classified the automorphism groups of all K3 sigma-models, and none of them contains $\mathbb{M}_{24}$ [@Gaberdiel:2011fg]. In fact, not even all automorphism groups are contained in $\mathbb{M}_{24}$: the exceptional cases are the possibilities (ii), (iii) and (iv) of the theorem in section \[sec:Theorem\] (see [@Gaberdiel:2011fg]), as well as case (i) for nontrivial $G'$. Case (iii) was already shown in [@Gaberdiel:2011fg] to be realised by a specific Gepner model that is believed to be equivalent to a torus orbifold by $\mathbb{Z}_3$. Here we show that also cases (ii) and (iv) are realised by actual K3s — the argument in [@Gaberdiel:2011fg] for this relied on some assumption about the regularity of K3 sigma-models — and in both cases the relevant K3s are again torus orbifolds. More specifically, case (ii) is realised by an asymmetric $\mathbb{Z}_5$ orbifold of $\mathbb{T}^4$ (see section \[sec:Z5\]),[^4] while for case (iii) the relevant orbifold is by $\mathbb{Z}_3$ (see section \[sec:Z3\]).
Cyclic torus orbifolds are rather special K3s since they always possess a quantum symmetry whose orbifold leads back to $\mathbb{T}^4$. Using this property of cyclic torus orbifolds, we show (see section \[sec:symme\]) that the group of automorphisms of K3s that are cyclic torus orbifolds is always exceptional; in particular, the quantum symmetry itself is never an element of ${\mathbb{M}}_{24}$. Although some ‘exceptional’ automorphism groups (contained in case (i) of the classification theorem) can also arise in K3 models that are not cyclic torus orbifolds, our observation may go a certain way towards explaining why only $\mathbb{M}_{24}$ seems to appear in the elliptic genus of K3.
We should mention that Mathieu Moonshine can also be formulated in terms of a mock modular form that can be naturally associated to the elliptic genus of K3 [@EOT; @Cheng:2010pq; @CD1; @CD2]; this point of view has recently led to an interesting class of generalisations [@CDH]. There are also indications that, just as for Monstrous Moonshine, Mathieu Moonshine can possibly be understood in terms of an underlying Borcherds-Kac-Moody algebra [@Govindarajan:2010fu; @Suresh; @Govindarajan:2011em; @Hohenegger:2011us].
Mathieu Moonshine {#sec:MatMoon}
=================
Let us first review the basic idea of ‘Mathieu Moonshine’. We consider a conformal field theory sigma-model with target space K3. This theory has ${\cal N}=(4,4)$ superconformal symmetry on the world-sheet. As a consequence, the space of states can be decomposed into representations of the ${\cal N}=4$ superconformal algebra, both for the left- and the right-movers. (The left- and right-moving actions commute, and thus we can find a simultaneous decomposition.) The full space of states takes then the form \[K3spec\] [H]{} = \_[i,j]{} N\_[ij]{} [H]{}\_i |[H]{}\_j , where $i$ and $j$ label the different ${\cal N}=4$ superconformal representations, and $N_{ij}\in{\mathbb N}_0$ denote the multiplicities with which these representations appear. The ${\cal N}=4$ algebra contains, apart from the Virasoro algebra $L_n$ at $c=6$, four supercharge generators, as well as an affine $\hat{\mathfrak{su}}(2)_1$ subalgebra at level one; we denote the Cartan generator of the zero mode subalgebra $\mathfrak{su}(2)$ by $J_0$.
The full partition function of the conformal field theory is quite complicated, and is only explicitly known at special points in the moduli space. However, there exists some sort of partial index that is much better behaved. This is the so-called [*elliptic genus*]{} that is defined by \_[K3]{}(,z) = [Tr]{}\_[RR]{} ( q\^[L\_0 - ]{} y\^[J\_0]{} (-1)\^F |[q]{}\^[|[L]{}\_0 - ]{} (-1)\^[|[F]{}]{} ) \_[0,1]{}(,z) . Here the trace is only taken over the Ramond-Ramond part of the spectrum (\[K3spec\]), and the right-moving ${\cal N}=4$ modes are denoted by a bar. Furthermore, $q=\exp(2\pi i \tau)$ and $y=\exp(2\pi i z)$, $F$ and $\bar{F}$ are the left- and right-moving fermion number operators, and the two central charges equal $c=\bar{c}=6$. Note that the elliptic genus does not actually depend on $\bar\tau$, although $\bar{q}=\exp(-2\pi i \bar\tau)$ does; the reason for this is that, with respect to the right-moving algebra, the elliptic genus is like a Witten index, and only the right-moving ground states contribute. To see this one notices that states that are not annihilated by a supercharge zero mode appear always as a boson-fermion pair; the contribution of such a pair to the elliptic genus however vanishes because the two states contribute with the opposite sign (as a consequence of the $(-1)^{\bar{F}}$ factor). Thus only the right-moving ground states, i.e. the states that are annihilated by all right-moving supercharge zero modes, contribute to the elliptic genus, and the commutation relations of the ${\cal N}=4$ algebra then imply that they satisfy $(\bar{L}_0 -\tfrac{\bar{c}}{24}) \phi_{\rm ground} = 0$; thus it follows that the elliptic genus is independent of $\bar{\tau}$. Note that this argument does not apply to the left-moving contributions because of the $y^{J_0}$ factor. (The supercharges are ‘charged’ with respect to the $J_0$ Cartan generator, and hence the two terms of a boson-fermion pair come with different powers of $y$. However, if we also set $y=1$, the elliptic genus does indeed become a constant, independent of $\tau$ and $\bar\tau$.)
It follows from general string considerations that the elliptic genus defines a [*weak Jacobi form of weight zero and index one*]{} [@Kawai:1993jk]. Recall that a weak Jacobi form of weight $w$ and index $m$ is a function [@EichlerZagier] \_[w,m]{} : [H]{}\_+ , (,z) \_[w,m]{}(,z) that satisfies $$\label{eq:jactmn1}
\phi_{w,m}\Bigl(\frac{a \tau + b}{c \tau + d} , \frac{z}{c \tau + d}\Bigr) =
(c \tau+d)^w \, e^{ 2 \pi i m \frac{c z^2}{c \tau + d} } \, \phi_{w,m}(\tau,z)
\qquad \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in {\rm SL}(2,{{\mathbb Z}}) \ ,
$$ $$\label{eq:jactmn2}
\phi(\tau,z+ \ell \tau + \ell') = e^{-2 \pi i m
(\ell^2 \tau+ 2 \ell z)} \phi(\tau,z) \qquad\qquad\qquad
\ell,\ell'\in {{\mathbb Z}}\ ,$$ and has a Fourier expansion $$\phi(\tau,z) = \sum_{n \geq 0,\, \ell\in {{\mathbb Z}}} c(n,\ell) q^n
y^\ell$$ with $c(n,\ell) = (-1)^w c(n,-\ell)$. Weak Jacobi forms have been classified, and there is only one weak Jacobi form with $w=0$ and $m=1$. Up to normalisation $\phi_{\rm K3}$ must therefore agree with this unique weak Jacobi form $\phi_{0,1}$, which can explicitly be written in terms of Jacobi theta functions as \_[0,1]{}(,z) = 8 \_[i=2,3,4]{} . Note that the Fourier coefficients of $\phi_{{\rm K3}}$ are integers; as a consequence they cannot change continuously as one moves around in the moduli space of K3 sigma-models, and thus $\phi_{{\rm K3}}$ must be actually independent of the specific K3 sigma-model that is being considered, i.e. independent of the point in the moduli space. Here we have used that the moduli space is connected. More concretely, it can be described as the double quotient \[moduli\] [M]{}\_[K3]{} = [O]{}(\^[4,20]{}) \\ [O]{}(4,20) / [O]{}(4) (20) . We can think of the Grassmannian on the right (4,20) / [O]{}(4) (20) as describing the choice of a positive-definite 4-dimensional subspace $\Pi \subset {\mathbb R}^{4,20}$, while the group on the left, ${\rm O}(\Gamma^{4,20})$, leads to discrete identifications among them. Here ${\rm O}(\Gamma^{4,20})$ is the group of isometries of a given fixed unimodular lattice $\Gamma^{4,20} \subset {\mathbb R}^{4,20}$. (In physics terms, the lattice $\Gamma^{4,20}$ can be thought of as the D-brane charge lattice of string theory on K3.)
Let us denote by ${\cal H}^{(0)}\subset {\cal H}_{\rm RR}$ the subspace of (\[K3spec\]) that consists of those RR states for which the right-moving states are ground states. (Thus ${\cal H}^{(0)}$ consists of the states that contribute to the elliptic genus.) ${\cal H}^{(0)}$ carries an action of the left-moving ${\cal N}=4$ superconformal algebra, and at any point in moduli space, its decomposition is of the form \[BPSdec\] [H]{}\^[(0)]{} = 20\_[h=,j=0]{} 2 \_[h=,j=]{} \_[n=1]{}\^ D\_n [H]{}\_[h=+n,j=]{} , where ${\cal H}_{h,j}$ denotes the irreducible ${\cal N}=4$ representation whose Virasoro primary states have conformal dimension $h$ and transform in the spin $j$ representation of $\mathfrak{su}(2)$. The multiplicities $D_n$ are [*not*]{} constant over the moduli space, but the above argument shows that A\_n = \_[D\_n]{} (-1)\^[|[F]{}]{} are (where $D_n$ is now understood not just as a multiplicity, but as a representation of the right-moving $(-1)^{\bar{F}}$ operator that determines the sign with which these states contribute to the elliptic genus). In this language, the elliptic genus then takes the form \[ellgenN4\] \_[K3]{}(,z) = 20 \_[h=,j=0]{}(,z) - 2 \_[h=,j=]{}(,z) + \_[n=1]{}\^ A\_n \_[h=+n,j=]{}(,z) , where $\chi_{h,j}(\tau,z)$ is the ‘elliptic’ genus of the corresponding ${\cal N}=4$ representation, \_[h,j]{} (,z) = \_[[H]{}\_[h,j]{}]{} ( q\^[L\_0-]{} y\^[J\_0]{} (-1)\^F ) , and we have used that $(-1)^{\bar{F}}$ takes the eigenvalues $+1$ and $-1$ on the $20$- and $2$-dimensional multiplicity spaces of the first two terms in (\[BPSdec\]), respectively.
The key observation of Eguchi, Ooguri & Tachikawa (EOT) [@EOT] was that the $A_n$ are sums of dimensions of ${\mathbb{M}}_{24}$ representation, in striking analogy to the original Monstrous Moonshine conjecture of [@CN]; the first few terms are $$\begin{aligned}
A_1 & = & 90 = {\bf 45} + \overline{\bf 45} \\
A_2 & = & 462 = {\bf 231} + \overline{\bf 231} \\
A_3 & = & 1540 = {\bf 770} + \overline{\bf 770} \ ,\end{aligned}$$ where ${\bf N}$ denotes a representation of ${\mathbb{M}}_{24}$ of dimension $N$. Actually, they guessed correctly the first six coefficients; from $A_7$ onwards the guesses become much more ambiguous (since the dimensions of the ${\mathbb{M}}_{24}$ representations are not that large) and they actually misidentified the seventh coefficient in their original analysis. (We will come back to the question of why and how one can be certain about the ‘correct’ decomposition shortly, see section \[sec:decompose\].) The alert reader will also notice that the first two coefficients in (\[BPSdec\]), namely $20$ and $-2$, are not directly ${\mathbb{M}}_{24}$ representations; the correct prescription is to introduce virtual representations and to write \[remain\] 20 = [**23**]{} - 3 , - 2 = - 2 . Recall that ${\mathbb{M}}_{24}$ is a sporadic finite simple group of order \[Morder\] | \_[24]{} | = 2\^[10]{} 3\^3 5 7 11 23 = 244 823 040 . It has $26$ conjugacy classes (which are denoted by 1A, 2A, 3A, $\ldots$, 23A, 23B, where the number refers to the order of the corresponding group element) — see eqs. (\[conj1\]) and (\[conj2\]) below for the full list — and therefore also $26$ irreducible representations whose dimensions range from $N=1$ to $N=10395$. The Mathieu group ${\mathbb{M}}_{24}$ can be defined as the subgroup of the permutation group $S_{24}$ that leaves the extended Golay code invariant; equivalently, it is the quotient of the automorphism group of the $\mathfrak{su}(2)^{24}$ Niemeier lattice, divided by the Weyl group. Thought of as a subgroup of ${\mathbb{M}}_{24}\subset S_{24}$, it contains the subgroup ${\mathbb{M}}_{23}$ that is characterised by the condition that it leaves a given (fixed) element of $\{1,\ldots, 24\}$ invariant.
Classical symmetries
--------------------
The appearance of a Mathieu group in the elliptic genus of K3 is not totally surprising in view of the Mukai theorem [@Mukai; @Kondo]. It states that any finite group of symplectic automorphisms of a K3 surface can be embedded into the Mathieu group ${\mathbb{M}}_{23}$. The symplectic automorphisms of a K3 surface define symmetries that act on the multiplicity spaces of the ${\cal N}=4$ representations, and therefore explain part of the above findings. However, it is also clear from Mukai’s argument that they do not even account for the full ${\mathbb{M}}_{23}$ group. Indeed, every symplectomorphism of K3 has at least five orbits on the set $\{1,\ldots,24\}$, and thus not all elements of ${\mathbb{M}}_{23}$ can be realised as a symplectomorphism. More specifically, of the $26$ conjugacy classes of ${\mathbb{M}}_{24}$, $16$ have a representative in ${\mathbb{M}}_{23}$, namely \[conj1\]
[ll]{} &\
&
where ‘geometric’ means that a representative can be (and in fact is) realised by a geometric symplectomorphism (i.e. that the representative has at least five orbits when acting on the set $\{1,\ldots,24\}$), while ‘non-geometric’ means that this is not the case. The remaining conjugacy classes do [*not*]{} have a representative in ${\mathbb{M}}_{23}$, and are therefore not accounted for geometrically via the Mukai theorem \[conj2\] . The classical symmetries can therefore only explain the symmetries in the first line of (\[conj1\]). Thus an additional argument is needed in order to understand the origin of the other symmetries; we shall come back to this in section \[sec:Theorem\].
Evidence for Moonshine {#sec:decompose}
----------------------
As was already alluded to above, in order to determine the ‘correct’ decomposition of the $A_n$ multiplicity spaces in terms of ${\mathbb{M}}_{24}$ representations, we need to study more than just the usual elliptic genus. By analogy with Monstrous Moonshine, the natural objects to consider are the analogues of the McKay Thompson series [@Thompson1]. These are obtained from the elliptic genus upon replacing \[guess\] A\_n = R\_n \_[R\_n]{} (g) , where $g\in {\mathbb{M}}_{24}$, and $R_n$ is the ${\mathbb{M}}_{24}$ representation whose dimension equals the coefficient $A_n$; the resulting functions are then (compare (\[ellgenN4\])) \[ellgentwinN4\] \_[g]{}(,z) = \_[[**23**]{} - 3 ]{} (g) \_[h=,j=0]{}(,z) - 2 \_[[**1**]{}]{}(g) \_[h=,j=]{}(,z) + \_[n=1]{}\^ \_[R\_n]{} (g) \_[h=+n,j=]{}(,z) . The motivation for this definition comes from the observation that if the underlying vector space ${\cal H}^{(0)}$, see eq. (\[BPSdec\]), of states contributing to the elliptic genus were to carry an action of ${\mathbb{M}}_{24}$, $\phi_g(\tau,z)$ would equal the ‘twining elliptic genus’, i.e. the elliptic genus twined by the action of $g$ \[twining\] \_g(,z) = [Tr]{}\_[[H]{}\^[(0)]{}]{} ( g q\^[L\_0 - ]{} y\^[J\_0]{} (-1)\^F |[q]{}\^[|[L]{}\_0 - ]{} (-1)\^[|[F]{}]{} ) . Obviously, a priori, it is not clear what the relevant $R_n$ in (\[guess\]) are. However, we have some partial information about them:
[()]{}
For any explicit realisation of a symmetry of a K3 sigma-model, we can calculate (\[twining\]) directly. (In particular, for some symmetries, the relevant twining genera had already been calculated in [@David:2006ji].)
The observation of EOT determines the first six coefficients explicitly.
The twining genera must have special modular properties.
Let us elaborate on (iii). Assuming that the functions $\phi_g(\tau,z)$ have indeed an interpretation as in (\[twining\]), they correspond in the usual orbifold notation of string theory to the contribution $$\begin{aligned}
\label{boxd}
\phi_g(\tau,z) \ \longleftrightarrow
& e\, & \boxed{\phantom{{ \tiny \begin{array}{cc}
\cdot & \cdot \hspace*{-0.2cm} \\ & \hspace*{-0.2cm} \end{array}}}} \nonumber \\
& & \hspace*{0.35cm} g\end{aligned}$$ where $e$ is the identity element of the group. Under a modular transformation it is believed that these twining and twisted genera transform (up to a possible phase) as $$\begin{array}{llllll}
& h & \boxed{\phantom{{\tiny \begin{array}{cc}
\cdot & \cdot \hspace*{-0.2cm} \\ & \hspace*{-0.2cm} \end{array}}}}
\hspace*{0.5cm}
& \stackrel{\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)} {\xrightarrow{\hspace*{2cm}}} \hspace*{0.5cm}
& h^d g^c & \boxed{\phantom{{\tiny \begin{array}{cc}
\cdot & \cdot \hspace*{-0.2cm} \\ & \hspace*{-0.2cm}\end{array}}}} \\
& & \hspace*{0.3cm} g
& & & \hspace*{0.05cm} g^a h^b
\end{array}$$ The twining genera (\[boxd\]) are therefore invariant (possibly up to a phase) under the modular transformations with $$\label{cons}
\gcd(a,o(g))=1 \qquad \hbox{and} \qquad c = 0 \ \ \hbox{mod $o(g)$,}$$ where $o(g)$ is the order of the group element $g$ and we used that for $\gcd(a,o(g))=1$, the group element $g^a$ is in the same conjugacy class as $g$ or $g^{-1}$. (Because of reality, the twining genus of $g$ and $g^{-1}$ should be the same.) Since $ad-bc=1$, the second condition implies the first, and we thus conclude that $\phi_g(\tau,z)$ should be (up to a possible multiplier system) a weak Jacobi form of weight zero and index one under the subgroup of ${\rm SL}(2,{\mathbb Z})$ $$\Gamma_0(N) = \left\{
\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in
{\rm SL}(2,{\mathbb Z}) \ : \ c = 0 \,\, \hbox{mod $(N)$} \right\} \ ,$$ where $N=o(g)$. This is a relatively strong condition, and knowing the first few terms (for a fixed multiplier system) determines the function uniquely. In order to use this constraint, however, it is important to know the multiplier system. An ansatz (that seems to work, see below) was made in [@Gaberdiel:2010ca] \[conj\] \_g(,) =e\^e\^\_g(,z) ,
a & b\
c&d
\_0(N) , where $N$ is again the order of $g$ and $h|\gcd(N,12)$. The multiplier system is trivial ($h=1$) if and only if $g$ contains a representative in ${\mathbb{M}}_{23}\subset {\mathbb{M}}_{24}$. For the other conjugacy classes, the values are tabulated in table \[t:hvalues\].
$\begin{array}{|c|ccccccccc|}\hline
\text{Class} & {\rm 2B} & {\rm 3B} & {\rm 4A} & {\rm 4C} & {\rm 6B}
& {\rm 10A} & {\rm 12A} & {\rm 12B} & {\rm 21AB} \\ \hline
h & 2 & 3 & 2 & 4 & 6 & 2 & 2 & 12 & 3\\ \hline
\end{array}$
It was noted in [@CD1] that $h$ equals the length of the shortest cycle (when interpreted as a permutation in $S_{24}$, see table 1 of [@CD1]).
Using this ansatz, explicit expressions for all twining genera were determined in [@Gaberdiel:2010ca]; independently, the same twining genera were also found (using guesses based on the cycle shapes of the corresponding $S_{24}$ representations) in [@Eguchi:2010fg]. (Earlier partial results had been obtained in [@Cheng:2010pq] and [@Gaberdiel:2010ch].)
These explicit expressions for the twining genera then allow for a very non-trivial check of the EOT proposal. As is clear from their definition in (\[ellgentwinN4\]), they determine the coefficients \_[R\_n]{} (g) This information is therefore sufficient to [*determine*]{} the representations $R_n$, i.e. to calculate their decomposition into irreducible ${\mathbb{M}}_{24}$ representations, for all $n$. We have worked out the decompositions explicitly for the first $500$ coefficients, and we have found that each $R_n$ can be written as a direct sum of ${\mathbb{M}}_{24}$ representations with non-negative integer multiplicities [@Gaberdiel:2010ca]. (Subsequently [@Eguchi:2010fg] tested this property for the first $600$ coefficients, and apparently Tachikawa has also checked it for the first $1000$ coefficients.) Terry Gannon has informed us that this information is sufficient to prove that the same will then happen for all $n$ [@Gannon]. In some sense this then proves the EOT conjecture.
Symmetries of K3 models {#sec:Theorem}
=======================
While the above considerations establish in some sense the EOT conjecture, they do not offer any insight into why the elliptic genus of K3 should exhibit an ${\mathbb{M}}_{24}$ symmetry. This is somewhat similar to the original situation in Monstrous Moonshine, after Conway and Norton had found the various Hauptmodules by somewhat similar techniques. Obviously, in the case of Monstrous Moonshine, many of these observations were subsequently explained by the construction of the Monster CFT (that possesses the Monster group as its automorphism group) [@Borcherds; @FLM]. So we should similarly ask for a microscopic explanation of these findings.
In some sense it is clear what the analogue of the Monster CFT in the current context should be: we know that the function in question [*is*]{} the elliptic genus of K3. However, there is one problem with this. As we mentioned before, there is not just one K3 sigma-model, but rather a whole moduli space (see eq. (\[moduli\])) of such CFTs. So the simplest explanation of the EOT observation would be if there is (at least) one special K3 sigma-model that has ${\mathbb{M}}_{24}$ as its automorphism group. Actually, the relevant symmetry group should commute with the action of the ${\cal N}=(4,4)$ superconformal symmetry (since it should act on the multiplicity spaces in ${\cal H}^{(0)}$, see eq. (\[BPSdec\])). Furthermore, since the two ${\cal N}=4$ representations with $h=\tfrac{1}{4}$ and $j=\tfrac{1}{2}$ are singlets — recall that the coefficient $-2$ transforms as $- 2=- 2\cdot {\bf 1}$, see (\[remain\]) — the automorphism must act trivially on the 4 RR ground states that transform in the $({\bf 2},{\bf 2})$ representation of the $\mathfrak{su}(2)_L \times \mathfrak{su}(2)_R$ subalgebra of ${\cal N}=(4,4)$. Note that these four states generate the simultaneous half-unit spectral flows in the left- and the right-moving sector; the requirement that the symmetry leaves them invariant therefore means that spacetime supersymmetry is preserved.
Recall from (\[moduli\]) that the different K3 sigma-models are parametrised by the choice of a positive-definite $4$-dimensional subspace $\Pi\subset {\mathbb R}^{4,20}$, modulo some discrete identifications. Let us denote by $G_\Pi$ the group of symmetries of the sigma-model described by $\Pi$ that commute with the action of ${\cal N}=(4,4)$ and preserve the RR ground states in the $({\bf 2},{\bf 2})$ (see above). It was argued in [@Gaberdiel:2011fg] that $G_\Pi$ is precisely the subgroup of ${\rm O}(\Gamma^{4,20})$ consisting of those elements that leave $\Pi$ pointwised fixed. The possible symmetry groups $G_\Pi$ can then be classified following essentially the paradigm of the Mukai-Kondo argument for the symplectomorphisms of K3 surfaces [@Mukai; @Kondo]. The outcome of the analysis can be summarised by the following theorem [@Gaberdiel:2011fg]:
[**Theorem:**]{}
*Let $G$ be the group of symmetries of a non-linear sigma-model on $K3$ preserving the ${\mathcal{N}}=(4,4)$ superconformal algebra as well as the spectral flow operators. One of the following possibilities holds:*
[[()]{}]{}
$G=G'. G''$, where $G'$ is a subgroup of ${\mathbb{Z}}_2^{11}$, and $G''$ is a subgroup of ${\mathbb{M}}_{24}$ with at least four orbits when acting as a permutation on $\{1,\ldots,24\}$
$G=5^{1+2}:{\mathbb{Z}}_4$
$G={\mathbb{Z}}_3^4:A_6$
$G=3^{1+4}:{\mathbb{Z}}_2.G''$, where $G''$ is either trivial, ${\mathbb{Z}}_2$ or ${\mathbb{Z}}_2^2$.
Here $G=N.Q$ means that $N$ is a normal subgroup of $G$, and $G/N\cong Q$; when $G$ is the semidirect product of $N$ and $Q$, we denote it by $N:Q$. Furthermore, $p^{1+2n}$ is an extra-special group of order $p^{1+2n}$, which is an extension of $\mathbb{Z}_p^{2n}$ by a central element of order $p$.
We will give a sketch of the proof below (see section \[sec:proof\]), but for the moment let us comment on the implications of this result. First of all, our initial expectation from above is not realised: none of these groups $G\equiv G_\Pi$ contains ${\mathbb{M}}_{24}$. In particular, the twining genera for the conjugacy classes 12B, 21A, 21B, 23A, 23B of ${\mathbb{M}}_{24}$ cannot be realised by any symmetry of a K3 sigma-model. Thus we cannot give a direct explanation of the EOT observation along these lines.
Given that the elliptic genus is constant over the moduli space, one may then hope that we can explain the origin of ${\mathbb{M}}_{24}$ by ‘combining’ symmetries from different points in the moduli space. As we have mentioned before, this is also similar to what happens for the geometric symplectomorphisms of K3: it follows from the Mukai theorem that the Mathieu group ${\mathbb{M}}_{23}$ is the smallest group that contains all symplectomorphisms, but there is no K3 surface where all of ${\mathbb{M}}_{23}$ is realised, and indeed, certain generators of ${\mathbb{M}}_{23}$ can never be symmetries, see (\[conj1\]). However, also this explanation of the EOT observation is somewhat problematic: as is clear from the above theorem, not all symmetry groups of K3 sigma-models are in fact subgroups of ${\mathbb{M}}_{24}$. In particular, none of the cases (ii), (iii) and (iv) (as well as case (i) with $G'$ non-trivial) have this property, as can be easily seen by comparing the prime factor decompositions of their orders to (\[Morder\]). The smallest group that contains all groups of the theorem is the Conway group ${\rm Co}_1$, but as far as we are aware, there is no evidence of any ‘Conway Moonshine’ in the elliptic genus of K3.
One might speculate that, generically, the group $G$ must be a subgroup of ${\mathbb{M}}_{24}$, and that the models whose symmetry group is not contained in ${\mathbb{M}}_{24}$ are, in some sense, special or ‘exceptional’ points in the moduli space. In order to make this idea precise, it is useful to analyse the exceptional models in detail. In [@Gaberdiel:2011fg], some examples have been provided of case (i) with non-trivial $G'$ (a torus orbifold $\mathbb{T}^4/{\mathbb{Z}}_2$ or the Gepner model $2^4$, believed to be equivalent to a ${\mathbb{T}}^4/{\mathbb{Z}}_4$ orbifold), and of case (iii) (the Gepner model $1^6$, which is believed to be equivalent to a ${\mathbb{T}}^4/\mathbb{Z}_3$ orbifold, see also [@Fluder]). For the cases (ii) and (iv), only an existence proof was given. In section \[sec:Z5\], we will improve the situation by constructing in detail an example of case (ii), realised as an asymmetric ${\mathbb{Z}}_5$-orbifold of a torus $\mathbb{T}^4$. Furthermore, in section \[sec:Z3\] we will briefly discuss the ${\mathbb{Z}}_3$-orbifold of a torus and the explicit realisation of its symmetry group, corresponding to cases (ii) and (iv) for any $G''$.
Notice that all the examples of exceptional models known so far are provided by torus orbifolds. In fact, we will show below (see section \[sec:symme\]) that all cyclic torus orbifolds have exceptional symmetry groups. Conversely, we will prove that the cases (ii)–(iv) of the theorem are always realised by (cyclic) torus orbifolds. On the other hand, as we shall also explain, some of the exceptional models in case (i) are not cyclic torus orbifolds.
Sketch of the proof of the Theorem {#sec:proof}
----------------------------------
In this subsection, we will describe the main steps in the proof of the above theorem; the details can be found in [@Gaberdiel:2011fg].
It was argued in [@Gaberdiel:2011fg] that the supersymmetry preserving automorphisms of the non-linear sigma-model characterised by $\Pi$ generate the group $G\equiv G_\Pi$ that consists of those elements of $O(\Gamma^{4,20})$ that leave $\Pi$ pointwise fixed. Let us denote by $L^G$ the sublattice of $G$-invariant vectors of $L\equiv \Gamma^{4,20}$, and define $L_G$ to be its orthogonal complement that carries a genuine action of $G$. Since $L^G\otimes {\mathbb{R}}$ contains the subspace $\Pi$, it follows that $L^G$ has signature $(4,d)$ for some $d\ge 0,$ so that $L_G$ is a negative-definite lattice of rank $20-d$. In [@Gaberdiel:2011fg], it is proved that, for any consistent model, $L_G$ can be embedded (up to a change of sign in its quadratic form) into the Leech lattice $\Lambda$, the unique $24$-dimensional positive-definite even unimodular lattice containing no vectors of squared norm $2$. Furthermore, the action of $G$ on $L_G$ can be extended to an action on the whole of $\Lambda$, such that the sublattice $\Lambda^G\subset\Lambda $ of vectors fixed by $G$ is the orthogonal complement of $L_G$ in $\Lambda$. This construction implies that $G$ must be a subgroup of ${\rm Co}_0\equiv {\rm Aut}(\Lambda)$ that fixes a sublattice $\Lambda^G$ of rank $4+d$. Conversely, it can be shown that any such subgroup of ${\rm Aut}(\Lambda)$ is the symmetry group of some K3 sigma-model.
This leaves us with characterising the possible subgroups of the finite group ${\rm Co}_0\equiv {\rm Aut}(\Lambda)$ that stabilise a suitable sublattice; problems of this kind have been studied in the mathematical literature before. In particular, the stabilisers of sublattices of rank at least $4$ are, generically, the subgroups of ${\mathbb{Z}}_2^{11}:{\mathbb{M}}_{24}$ described in case (i) of the theorem above. The three cases (ii), (iii), (iv) arise when the invariant sublattice $\Lambda^G$ is contained in some $\mathcal{S}$-lattice $S\subset\Lambda$. An $\mathcal{S}$-lattice $S$ is a primitive sublattice of $\Lambda$ such that each vector of $S$ is congruent modulo $2S$ to a vector of norm $0$, $4$ or $6$. Up to isomorphisms, there are only three kind of $\mathcal{S}$-lattices of rank at least four; their properties are summarised in the following table: $$\begin{array}{ccccc}
\text{Name} \qquad&\text{type}\qquad & \operatorname{rk}S \qquad & {\rm Stab}(S) \qquad & {\rm Aut}(S)\\
(A_2\oplus A_2)'(3) \qquad& 2^93^6 \qquad & 4 \qquad & {\mathbb{Z}}_3^4: A_6\qquad & {\mathbb{Z}}_2\times(S_3\times S_3).{\mathbb{Z}}_2\\
A_4^*(5) \qquad& 2^{5}3^{10} \qquad & 4 \qquad & 5^{1+2}:{\mathbb{Z}}_4 \qquad & {\mathbb{Z}}_2\times S_5\\
E_6^*(3) \qquad& 2^{27}3^{36} \qquad & 6 \qquad & 3^{1+4}:{\mathbb{Z}}_2 \qquad & {\mathbb{Z}}_2\times W(E_6) \ .
\end{array}$$ Here, $S$ is characterised by the type $2^p3^q$, which indicates that $S$ contains $p$ pairs of opposite vectors of norm $4$ (type 2) and $q$ pairs of opposite vectors of norm $6$ (type 3). The group ${\rm Stab}(S)$ is the pointwise stabiliser of $S$ in ${\rm Co}_0$ and ${\rm Aut}(S)$ is the quotient of the setwise stabiliser of $S$ modulo its pointwise stabiliser ${\rm Stab}(S)$. The group ${\rm Aut}(S)$ always contains a central ${\mathbb{Z}}_2$ subgroup, generated by the transformation that inverts the sign of all vectors of the Leech lattice. The lattice of type $2^{27}3^{36}$ is isomorphic to the weight lattice (the dual of the root lattice) of $E_6$ with quadratic form rescaled by $3$ (i.e. the roots in $E_6^*(3)$ have squared norm $6$), and ${\rm Aut}(S)/{\mathbb{Z}}_2$ is isomorphic to the Weyl group $W(E_6)$ of $E_6$. Similarly, the lattice of type $2^{5}3^{10}$ is the weight lattice of $A_4$ rescaled by $5$, and ${\rm Aut}(S)/{\mathbb{Z}}_2$ is isomorphic to the Weyl group $W(A_4)\cong S_5$ of $A_4$. Finally, the type $2^93^6$ is the $3$-rescaling of a lattice $(A_2\oplus A_2)'$ obtained by adjoining to the root lattice $A_2\oplus A_2$ an element $(e_1^*,e_2^*)\in A_2^*\oplus A_2^*$, with $e_1^*$ and $e_2^*$ of norm $2/3$. The latter ${\mathcal S}$-lattice can also be described as the sublattice of vectors of $E_6^*(3)$ that are orthogonal to an $A_2(3)$ sublattice of $E_6^*(3)$. The group ${\rm Aut}(S)/{\mathbb{Z}}_2$ is the product $(S_3\times S_3).{\mathbb{Z}}_2$ of the Weyl groups $W(A_2)=S_3$, and the ${\mathbb{Z}}_2$ symmetry that exchanges the two $A_2$ and maps $e_1^*$ to $e_2^*$.
The cases (ii)–(iv) of the above theorem correspond to $\Lambda^G$ being isomorphic to $A_4^*(5)$ (case ii), to $(A_2\oplus A_2)'(3)$ (case iii) or to a sublattice of $E_6^*(3)$ different from $(A_2\oplus A_2)'(3)$ (case iv). In the cases (ii) and (iii), $G$ is isomorphic to ${\rm Stab}(S)$. In case (iv), ${\rm Stab}(S)$ is, generically, a normal subgroup of $G$, and $G''\cong G/{\rm Stab}(S)$ is a subgroup of ${\rm Aut}(S)\cong {\mathbb{Z}}_2\times W(E_6)$ that fixes a sublattice $\Lambda^G\subseteq E_6^*(3)$, with $\Lambda^G\neq (A_2\oplus A_2)'(3)$, of rank at least $4$. The only non-trivial subgroups of ${\mathbb{Z}}_2\times W(E_6)$ with these properties are $G''={\mathbb{Z}}_2$, which corresponds to $\Lambda^G$ being orthogonal to a single vector of norm $6$ in $E_6^*(3)$ (a rescaled root), and $G''={\mathbb{Z}}_2^2$, which corresponds to $\Lambda^G$ being orthogonal to two orthogonal vectors of norm $6$.[^5] If $\Lambda^G$ is orthogonal to two vectors $v_1,v_2\in E_6^*(3)$ of norm $6$, with $v_1\cdot v_2=-3$, then $\Lambda^G\cong (A_2\oplus A_2)'(3)$ and case (iii) applies.
Symmetry groups of torus orbifolds {#sec:symme}
==================================
In this section we will prove that all K3 sigma-models that are realised as (possibly left-right asymmetric) orbifolds of ${\mathbb{T}}^4$ by a cyclic group have an ‘exceptional’ group of symmetries, i.e. their symmetries are not a subgroup of $\mathbb{M}_{24}$. Furthermore, these torus orbifolds account for most of the exceptional models (in particular, for all models in the cases (ii)–(iv) of the theorem). On the other hand, as we shall also explain, there are exceptional models in case (i) that are not cyclic torus orbifolds.
Our reasoning is somewhat reminiscent of the construction of [@Tuite:1992tt; @Tuite:1993hy] in the context of Monstrous Moonshine. Any ${\mathbb{Z}}_n$-orbifold of a conformal field theory has an automorphism $g$ of order $n$, called the *quantum symmetry*, which acts trivially on the untwisted sector and by multiplication by the phase $\exp(\frac{2\pi i k}{n})$ on the $k$-th twisted sector. Furthermore, the orbifold of the orbifold theory by the group generated by the quantum symmetry $g$, is equivalent to the original conformal field theory [@Ginsparg:1988ui]. This observation is the key for characterising K3 models that can be realised as torus orbifolds:
[*A K3 model ${\mathcal{C}}$ is a ${\mathbb{Z}}_n$-orbifold of a torus model if and only if it has a symmetry $g$ of order $n$ such that ${\mathcal{C}}/\langle g\rangle$ is a consistent orbifold equivalent to a torus model.*]{}
In order to see this, suppose that ${\mathcal{C}}_{K3}$ is a K3 sigma-model that can be realised as a torus orbifold ${\mathcal{C}}_{K3} = \tilde{\mathcal{C}}_{{\mathbb{T}}^4} /\langle \tilde{g} \rangle$, where $\tilde{g}$ is a symmetry of order $n$ of the torus model $\tilde {\mathcal{C}}_{{\mathbb{T}}^4}$. Then ${\mathcal{C}}_{K3}$ possesses a ‘quantum symmetry’ $g$ of order $n$, such that the orbifold of ${\mathcal{C}}_{K3}$ by $g$ describes again the original torus model, $\tilde{{\mathcal{C}}}_{{\mathbb{T}}^4} = {\mathcal{C}}_{K3} / \langle g \rangle$.
Conversely, suppose ${\mathcal{C}}_{K3}$ has a symmetry $g$ of order $n$, such that the orbifold of ${\mathcal{C}}_{K3}$ by $g$ is consistent, i.e. satisfies the level matching condition — this is the case if and only if the twining genus $\phi_g$ has a trivial multiplier system — and leads to a torus model ${\mathcal{C}}_{K3} / \langle g \rangle = \tilde{{\mathcal{C}}}_{\mathbb{T}^4}$. Then ${\mathcal{C}}_{K3}$ itself is a torus orbifold since we can take the orbifold of $\tilde{\mathcal{C}}_{\mathbb{T}^4}$ by the quantum symmetry associated to $g$, and this will, by construction, lead back to ${\mathcal{C}}_{K3}$.
Thus we conclude that ${\mathcal{C}}\equiv {\mathcal{C}}_{K3}$ can be realised as a torus orbifold if and only if ${\mathcal{C}}$ contains a symmetry $g$ such that (i) $\phi_{g}$ has a trivial multiplier system; and (ii) the orbifold of ${\mathcal{C}}$ by $g$ leads to a torus model $\tilde{\mathcal{C}}_{{\mathbb{T}}^4}$. It is believed that the orbifold of ${\mathcal{C}}$ by any ${\mathcal{N}}=(4,4)$-preserving symmetry group, if consistent, will describe a sigma-model with target space either a torus ${\mathbb{T}}^4$ or a K3 manifold. The two cases can be distinguished by calculating the elliptic genus; in particular, if the target space is a torus, the elliptic genus vanishes. Actually, since the space of weak Jacobi forms of weight zero and index one is 1-dimensional, this condition is equivalent to the requirement that the elliptic genus $\tilde\phi(\tau,z)$ of $\tilde {\mathcal{C}}= {\mathcal{C}}/ \langle g \rangle$ vanishes at $z=0$.
Next we recall that the elliptic genus of the orbifold by a group element $g$ of order $n=o(g)$ is given by the usual orbifold formula (,z)=\_[i,j=1]{}\^[n]{} \_[g\^i,g\^j]{}(,z) , where $\phi_{g^i,g^j}(\tau,z)$ is the twining genus for $g^j$ in the $g^i$-twisted sector; this can be obtained by a modular transformation from the untwisted twining genus $\phi_{g^d}(\tau,z)$ with $d=\gcd(i,j,n)$. As we have explained above, it is enough to evaluate the elliptic genus for $z=0$. Then \[const\] \_[g\^d]{}(,z=0)=\_[**24**]{}(g\^d) , where $\operatorname{Tr}_{\bf 24}(g^d)$ is the trace of $g^d$ over the $24$-dimensional space of RR ground states, and since (\[const\]) is constant (and hence modular invariant) we conclude that \[elliptorbif\] (,0)=\_[i,j=1]{}\^[n]{} \_[**24**]{}(g\^[(i,j,n)]{}) . According to the theorem in section \[sec:Theorem\], all symmetry groups of K3 sigma-models are subgroups of ${\rm Co}_0$ and, in fact, $\operatorname{Tr}_{\bf 24}(g^d)$ coincides with the trace of $g^d\in {\rm Co}_0$ in the standard $24$-dimensional representation of ${\rm Co}_0$. Thus, the elliptic genus of the orbifold model $\tilde {\mathcal{C}}={\mathcal{C}}/\langle g\rangle$ only depends on the conjugacy class of $g$ in ${\rm Co}_0$. The group ${\rm Co}_0$ contains $167$ conjugacy classes, but only $42$ of them contain symmetries that are realised by some K3 sigma-model, i.e. elements that fix at least a four-dimensional subspace in the standard $24$-dimensional representation of ${\rm Co}_0$. If $\operatorname{Tr}_{\bf 24}(g)\neq 0$ (this happens for $31$ of the above $42$ conjugacy classes), the twining genus $\phi_g(\tau,z)$ has necessarily a trivial multiplier system, and the orbifold ${\mathcal{C}}/\langle g\rangle$ is consistent. These classes are listed in the following table, together with the dimension of the space that is fixed by $g$, the trace over the $24$-dimensional representation, and the elliptic genus $\tilde\phi(\tau,z=0)$ of the orbifold model $\tilde {\mathcal{C}}$ (we underline the classes that restrict to ${\mathbb{M}}_{24}$ conjugacy classes):
[$$\begin{array}{c|ccccccccccccccccc}
\text{${\rm Co}_0$-class} & \text{\underline{1A}} & \text{\underline{2B}} & \text{2C} & \text{\underline{3B}} & \text{3C} & \text{4B} & \text{\underline{4E}} &
\text{4F} & \text{\underline{5B}} & \text{5C} & \text{6G} & \text{6H} & \text{6I} &
\text{\underline{6K}} &
\text{6L} & \text{6M} & \text{\underline{7B}} \\
\text{dim fix} & 24 & 16 & 8 & 12 & 6 & 8 & 10 & 6 & 8 & 4 & 6 & 6 & 6 & 8 & 4 & 4 & 6 \\
\operatorname{Tr}_{\bf 24}(g) & 24 & 8 & -8 & 6 & -3 & 8 & 4 & -4 & 4 & -1 & -4 & 4 & 5 & 2 & -2 & -1 & 3 \\
\tilde\phi(\tau,0) &24 & 24 & 0 & 24 & 0 & 24 & 24 & 0 & 24 & 0 & 0 & 24 & 24 & 24 & 0 & 0 & 24
\end{array}$$ $$\begin{array}{c|cccccccccccccc}
\text{${\rm Co}_0$-class} & \text{8D} & \text{\underline{8G}} & \text{8H} & \text{9C} & \text{10F} & \text{10G} & \text{10H} &
\text{\underline{11A}} & \text{12I} & \text{12L} & \text{12N} & \text{12O} & \text{\underline{14C}} &
\text{\underline{15D}} \\
\text{dim fix} & 4 & 6 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 \\
\operatorname{Tr}_{\bf 24}(g) & 4 & 2 & -2 & 3 & -2 & 2 & 3 & 2 & 2 & 1 & -2 & 2 & 1 & 1 \\
\tilde\phi(\tau,0) & 24 & 24 & 0 & 24 & 0 & 24 & 24 & 24 & 24 & 24 & 0 & 24 & 24 & 24
\end{array}$$]{}
Note that the elliptic genus of the orbifold theory $\tilde{\mathcal{C}}$ is always $0$ or $24$, corresponding to a torus or a K3 sigma-model, respectively. Out of curiosity, we have also computed the putative elliptic genus $\tilde\phi(\tau,0)$ for the $11$ classes of symmetries $g$ with $\operatorname{Tr}_{24}(g)=0$ for which we do not expect the orbifold to make sense — the corresponding twining genus $\phi_g$ will typically have a non-trivial multiplier system, and hence the orbifold will not satisfy level-matching. Indeed, for almost none of these cases is $\tilde\phi(\tau,0)$ equal to $0$ or $24$, thus signaling an inconsistency of the orbifold model: $$\begin{array}{c|ccccccccccc}
\text{${\rm Co}_0$-class} & \text{\underline{2D}} & \text{\underline{3D}} & \text{4D} & \text{\underline{4G}} & \text{\underline{4H}} & \text{6O} & \text{\underline{6P}} &
\text{8C} & \text{8I} & \text{\underline{10J}} & \text{\underline{12P}} \\
\text{dim fix} & 12 & 8 & 4 & 8 & 6 & 6 & 4 & 4 & 4 & 4 & 4 \\
\operatorname{Tr}_{\bf 24}(g) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\tilde\phi(\tau,0) &12 & 8 & 0 & 12 & 6 & 12 & 4 & 12 & 6 & 12 & 12
\end{array}$$ The only exception is the class ${\rm 4D}$, which might define a consistent orbifold (a torus model). It follows that a K3 model ${\mathcal{C}}$ is the ${\mathbb{Z}}_n$-orbifold of a torus model if and only if it contains a symmetry $g$ in one of the classes $$\begin{aligned}
\label{qusymm}
&{\rm 2C},\ {\rm 3C},\ {\rm 4F},\ {\rm 5C},\ {\rm 6G},\ {\rm 6L},\ {\rm 6M},\ {\rm 8H},\
{\rm 10F},\ {\rm 12N},\\
&{\rm 4B},\ {\rm 4D}, \ {\rm 6H},\ {\rm 6I},\ {\rm 8C},\ {\rm 8D},\ {\rm 9C},\
{\rm 10G},\ {\rm 10H},\ {\rm12I},\ {\rm 12L},\ {\rm 12O} \notag\end{aligned}$$ of ${\rm Co}_0$.[^6] Here we have also included (in the second line) those classes of elements $g\in {\rm Co}_0$ for which ${\mathcal{C}}/\langle g^i\rangle$ is a torus model, for some power $i>1$. Our main observation is now that none of the ${\rm Co}_0$ classes in (\[qusymm\]) restricts to a class in ${\mathbb{M}}_{24}$, i.e.
[*All K3 models that are realised as ${\mathbb{Z}}_n$-orbifolds of torus models are exceptional. In particular, the quantum symmetry is not an element of ${\mathbb{M}}_{24}$.*]{}
One might ask whether the converse is also true, i.e. whether all exceptional models are cyclic torus orbifolds. This is not quite the case: for example, the classification theorem of section \[sec:Theorem\] predicts the existence of models with a symmetry group $G\cong GL_2(3)$ (the group of $2\times 2$ invertible matrices on the field ${\mathbb{F}}_3$ with $3$ elements). The group $G$ contains no elements in the classes , so the model is not a cyclic torus orbifold; on the other hand, $G$ contains elements in the class ${\rm 8I}$ of ${\rm Co}_0$, which does not restrict to ${\mathbb{M}}_{24}$. A second counterexample is a family of models with a symmetry $g$ in the class ${\rm 6O}$ of ${\rm Co}_0$. A generic point of this family is not a cyclic torus model (although some special points are), since the full symmetry group is generated by $g$ and contains no elements in . Both these counterexamples belong to case (i) of the general classification theorem. In fact, we can prove that
[*The symmetry group $G$ of a K3 sigma-model ${\mathcal{C}}$ contains a subgroup $3^{1+4}:{\mathbb{Z}}_2$ (cases (iii) and (iv) of the theorem) if and only if ${\mathcal{C}}$ is a ${\mathbb{Z}}_3$-orbifold of a torus model. Furthermore, $G=5^{1+2}:{\mathbb{Z}}_4$ (case (ii)) if and only if ${\mathcal{C}}$ is a ${\mathbb{Z}}_5$-orbifold of a torus model.* ]{}
The proof goes as follows. All subgroups of ${\rm Co}_0$ of the form $3^{1+4}:{\mathbb{Z}}_2$ (respectively, $5^{1+2}:{\mathbb{Z}}_4$) contain an element in the class 3C (resp., 5C), and therefore the corresponding models are ${\mathbb{Z}}_3$ (resp., ${\mathbb{Z}}_5$) torus orbifolds. Conversely, consider a ${\mathbb{Z}}_3$-orbifold of a torus model. Its symmetry group $G$ contains the quantum symmetry $g$ in class 3C of ${\rm Co}_0$. (It must contain a symmetry generator of order three whose orbifold leads to a torus, and 3C is then the only possibility.) The sublattice $\Lambda^{\langle g\rangle}\subset\Lambda$ fixed by $g$ is the $\mathcal{S}$-lattice $2^{27}3^{36}$ [@Curtis]. From the classification theorem, we know that $G$ is the stabiliser of a sublattice $\Lambda^G\subset\Lambda$ of rank at least $4$. Since $\Lambda^G\subseteq\Lambda^{\langle g\rangle}$, $G$ contains as a subgroup the stabiliser of $\Lambda^{\langle g\rangle}$, namely $3^{1+4}:{\mathbb{Z}}_2$.
Analogously, a ${\mathbb{Z}}_5$ torus orbifold always has a symmetry in class 5C, whose fixed sublattice $\Lambda^{\langle g\rangle}$ is the $\mathcal{S}$-lattice $2^53^{10}$ [@Curtis]. Since $\Lambda^{\langle g\rangle}$ has rank $4$ and is primitive, $\Lambda^G=\Lambda^{\langle g\rangle}$ and the symmetry group $G$ must be the stabiliser $5^{1+2}:{\mathbb{Z}}_4$ of $\Lambda^{\langle g\rangle}$.
It was shown in [@Gaberdiel:2011fg] that the Gepner model $(1)^6$ corresponds to the case (ii) of the classification theorem. It thus follows from the above reasoning that it must indeed be equivalent to a ${\mathbb{Z}}_3$-orbifold of $\mathbb{T}^4$, see also [@Fluder]. (We shall also study these orbifolds more systematically in section \[sec:Z3\].) In the next section, we will provide an explicit construction of a ${\mathbb{Z}}_5$-orbifold of a torus model and show that its symmetry group is $5^{1+2}:{\mathbb{Z}}_4$, as predicted by the above analysis.
A K3 model with symmetry group $5^{1+2}:{\mathbb{Z}}_4$ {#sec:Z5}
=======================================================
In this section we will construct a supersymmetric sigma-model on $\mathbb{T}^4$ with a symmetry $g$ of order $5$ commuting with an ${\mathcal{N}}=(4,4)$ superconformal algebra and acting asymmetrically on the left- and on the right-moving sector. The orbifold of this model by $g$ will turn out to be a well-defined SCFT with ${\mathcal{N}}=(4,4)$ (in particular, the level matching condition is satisfied) that can be interpreted as a non-linear sigma-model on K3. We will argue that the group of symmetries of this model is $G=5^{1+2}.{\mathbb{Z}}_4$, one of the exceptional groups considered in the general classification theorem.
The torus model
---------------
Let us consider a supersymmetric sigma-model on the torus $\mathbb{T}^4$. Geometrically, we can characterise the theory in terms of a metric and a Kalb-Ramond field, but it is actually more convenient to describe it simply as a conformal field theory that is generated by the following fields: four left-moving $u(1)$ currents $\partial X^a(z)$, $a=1,\ldots,4$, four free fermions $\psi^a(z)$, $a=1,\ldots,4$, their right-moving analogues $\bar\partial X^a(\bar z),\tilde\psi^a(\bar z)$, as well as some winding-momentum fields $V_\lambda(z,\bar z)$ that are associated to vectors $\lambda$ in an even unimodular lattice $\Gamma^{4,4}$ of signature $(4,4)$. The mode expansions of the left-moving fields are X\^a(z)=\_[n]{} \_n z\^[-n-1]{} , \^a=\_[n+]{} \_n z\^[-n-]{} , where $\nu=0,1/2$ in the R- and NS-sector, respectively. Furthermore, we have the usual commutation relations \[commutations\] \[\_m\^a,\_n\^b\]=m \^[ab]{} \_[m,-n]{} {\_m\^a,\_n\^b}=\^[ab]{} \_[m,-n]{} . Analogous statements also hold for the right-moving modes $\tilde\alpha_n$ and $\tilde \psi_n$. The vectors $\lambda\equiv (\lambda_L,\lambda_R)\in\Gamma^{4,4}$ describe the eigenvalues of the corresponding state with respect to the left- and right-moving zero modes $\alpha^a_0$ and $\tilde\alpha^a_0$, respectively. In these conventions the inner product on $\Gamma^{4,4}$ is given as (,’) = \_L \_L’ - \_R \_R’ .
### Continuous and discrete symmetries
Any torus model contains an $\hat{\mathfrak{su}}(2)_1\oplus \hat{\mathfrak{su}}(2)_1\oplus
\hat{\mathfrak{u}}(1)^4$ current algebra, both on the left and on the right. Here, the $\hat{\mathfrak{u}}(1)^4$ currents are the $\partial X^a$, $a=1,\ldots,4$, while $\hat{\mathfrak{su}}(2)_1\oplus \hat{\mathfrak{su}}(2)_1=\hat{\mathfrak{so}}(4)_1$ is generated by the fermionic bilinears $$\begin{aligned}
\label{su(2)1}
a^3:=\bar\psi^{(1)}\psi^{(1)}+\bar\psi^{(2)}\psi^{(2)}\qquad
a^+:=\bar\psi^{(1)}\bar\psi^{(2)}\qquad a^-:=-\psi^{(1)}\psi^{(2)}\ ,\\
\label{su(2)2}
\hat a^3:=\bar\psi^{(1)}\psi^{(1)}-\bar\psi^{(2)}\psi^{(2)}\qquad
\hat a^+:=\bar\psi^{(1)}\psi^{(2)}\qquad \hat a^-:=-\psi^{(1)}\bar\psi^{(2)}\ ,\end{aligned}$$ where $$\begin{aligned}
\psi^{(1)}& =\tfrac{1}{\sqrt{2}}\, (\psi^1+i\psi^2)\qquad \qquad \qquad
& \psi^{(2)}=\tfrac{1}{\sqrt{2}}\, (\psi^3+i\psi^4)\\
\bar{\psi}^{(1)}& =\tfrac{1}{\sqrt{2}}\, (\psi^1-i\psi^2)\qquad \qquad \qquad
& \bar{\psi}^{(2)}=\tfrac{1}{\sqrt{2}}\, (\psi^3-i\psi^4)\ .\end{aligned}$$ At special points in the moduli space, where the $\Gamma^{4,4}$ lattice contains vectors of the form $(\lambda_L,0)$ with $\lambda_L^2=2$, the bosonic $\mathfrak{u}(1)^4$ algebra is enhanced to some non-abelian algebra ${\mathfrak{g}}$ of rank $4$. There are generically $16$ (left-moving) supercharges; they form four $(\bf 2,\bf 2)$ representations of the ${\mathfrak{su}}(2)\oplus {\mathfrak{su}}(2)$ zero mode algebra from (\[su(2)1\]) and (\[su(2)2\]). Altogether, the chiral algebra at generic points is a *large* ${\mathcal{N}}=4$ superconformal algebra.
We want to construct a model with a symmetry $g$ of order $5$, acting non-trivially on the fermionic fields, and commuting with the *small* ${\mathcal{N}}=4$ subalgebras both on the left and on the right. A small ${\mathcal{N}}=4$ algebra contains an $\hat{\mathfrak{su}}(2)_1$ current algebra and four supercharges in two doublets of $\mathfrak{su}(2)$. The symmetry $g$ acts by an ${\rm O}(4,{\mathbb{R}})$ rotation on the left-moving fermions $\psi^a$, preserving the anti-commutation relations . Without loss of generality, we may assume that $\psi^{(1)}$ and $\bar\psi^{(1)}$ are eigenvectors of $g$ with eigenvalues $\zeta$ and $\zeta^{-1}$, where $\zeta$ is a primitive fifth root of unity \^5=1 , and that the $\hat{\mathfrak{su}}(2)_1$ algebra preserved by $g$ is . This implies that $g$ acts on all the fermionic fields by \[fermtransf\] \^[(1)]{} \^[(1)]{} , |\^[(1)]{}\^[-1]{} |\^[(1)]{} , \^[(2)]{}\^[-1]{} \^[(2)]{} , |\^[(2)]{} |\^[(2)]{} . Note that the action of $g$ on the fermionic fields can be described by $e^{\frac{2\pi i k}{5}\hat a^3_0}$ for some $k=1,\ldots,4$, where $\hat a^3$ is the current in the algebra . The four $g$-invariant supercharges can then be taken to be \[bostransf\] \_[i=1]{}\^2J\^[(i)]{}|\^[(i)]{} , \_[i=1]{}\^2|J\^[(i)]{}\^[(i)]{} , (|J\^[(1)]{}|\^[(2)]{}-|J\^[(2)]{}|\^[(1)]{}) , (J\^[(1)]{}\^[(2)]{}-J\^[(2)]{}\^[(1)]{}) , where $J^{(1)},\bar J^{(1)},J^{(2)},\bar J^{(2)}$ are suitable (complex) linear combinations of the left-moving currents $\partial X^a$, $a=1,\ldots,4$. In order to preserve the four supercharges, $g$ must act with the same eigenvalues on the bosonic currents \[bostransf2\] J\^[(1)]{} J\^[(1)]{} , |J\^[(1)]{}\^[-1]{} |J\^[(1)]{} , J\^[(2)]{}\^[-1]{} J\^[(2)]{} , |J\^[(2)]{} |J\^[(2)]{} . A similar reasoning applies to the right-moving algebra with respect to an eigenvalue $\tilde\zeta$, with $\tilde\zeta^5=1$. For the symmetries with a geometric interpretation, the action on the left- and right-moving bosonic currents is induced by an ${\rm O}(4,{\mathbb{R}})$-transformation on the scalar fields $X^a$, $a,=1,\ldots,4$, representing the coordinates on the torus; then $\zeta$ and $\tilde\zeta$ are necessarily equal. In our treatment, we want to allow for the more general case where $\zeta\neq \tilde\zeta$.
The action of $g$ on $J^a$ and $\tilde J^a$ induces an ${\rm O}(4,4,{\mathbb{R}})$-transformation on the lattice $\Gamma^{4,4}$. The transformation $g$ is a symmetry of the model if and only if it induces an automorphism on $\Gamma^{4,4}$. In particular, it must act by an invertible integral matrix on any lattice basis. The requirement that the trace of this matrix (and of any power of it) must be integral, leads to the condition that \[zetaint\] 2(\^i+\^[-i]{}+\^i+\^[-i]{}) , for all $i\in{\mathbb{Z}}$. For $g$ of order $5$, this condition is satisfied by \[eigenv\] =e\^ =e\^ , and this solution is essentially unique (up to taking powers of it or exchanging the definition of $\zeta$ and $\zeta^{-1}$). Eq. shows that a supersymmetry preserving symmetry of order $5$ is necessarily left-right asymmetric, and hence does not have a geometric interpretation.
It is now clear how to construct a torus model with the symmetries and . First of all, we need an automorphism $g$ of $\Gamma^{4,4}$ of order five. Such an automorphism must have eigenvalues $\zeta,\zeta^2,\zeta^3,\zeta^4$, each corresponding to two independent eigenvectors $v_{\zeta^i}^{(1)},v_{\zeta^i}^{(2)}$, $i=1,\ldots, 4$, in $\Gamma^{4,4}\otimes{\mathbb{C}}$. Given the discussion above, see in particular , we now require that the vectors v\_[\^1]{}\^[(1)]{} ,v\_[\^1]{}\^[(2)]{} , v\_[\^4]{}\^[(1)]{} , v\_[\^4]{}\^[(2)]{} span a positive-definite subspace of $\Gamma^{4,4}\otimes{\mathbb{C}}$ (i.e. correspond to the left-movers), while the vectors v\_[\^2]{}\^[(1)]{} , v\_[\^2]{}\^[(2)]{} ,v\_[\^3]{}\^[(1)]{} , v\_[\^3]{}\^[(2)]{} span a negative-definite subspace of $\Gamma^{4,4}\otimes{\mathbb{C}}$ (i.e. correspond to the right-movers). An automorphism $g$ with the properties above can be explicitly constructed as follows. Let us consider the real vector space with basis vectors $x_1,\ldots,x_4$, and $y_1,\ldots,y_4$, and define a linear map $g$ of order $5$ by g(x\_i)=x\_[i+1]{} ,g(y\_i)=y\_[i+1]{} ,i=1,…,3 , and g(x\_4)=-(x\_1+x\_2+x\_3+x\_4) ,g(y\_4)=-(y\_1+y\_2+y\_3+y\_4) . A $g$-invariant bilinear form on the space is uniquely determined by the conditions (x\_i,x\_i)=0=(y\_i,y\_i) ,i=1,…, 4 and (x\_1, x\_2)=1 ,(x\_1,x\_3)=(x\_1,x\_4)=-1 , (y\_1, y\_2)=1 , (y\_1,y\_3)=(y\_1,y\_4)=-1 , as well as (x\_1,y\_1)=1 , (x\_i,y\_1)=0 , (i=2,3,4) . The lattice spanned by these basis vectors is an indefinite even unimodular lattice of rank $8$ and thus necessarily isomorphic to $\Gamma^{4,4}$. The $g$-eigenvectors can be easily constructed in terms of the basis vectors and one can verify that the eigenspaces have the correct signature.
This torus model has an additional ${\mathbb{Z}}_4$ symmetry group that preserves the superconformal algebra and normalises the group generated by $g$. The generator $h$ of this group acts by $$\begin{aligned}
\label{haction3a} & h(x_i):=g^{1-i}(x_1+x_4+2y_1+y_2+y_3+y_4)\ ,\\ \label{haction3b} &
h(y_i):=g^{1-i}(-2x_1-x_2-x_3-x_4-y_1-y_3-y_4)\ ,\qquad i=1,\ldots,4\ ,\end{aligned}$$ on the lattice vectors. The $g$-eigenvectors $v^{(a)}_{\zeta^i}$, $a=1,2$, $i=1,\ldots,4$ can be defined as v\^[(1)]{}\_[\^i]{}:=\_[j=0]{}\^4 \^[-ij]{}g\^j(x\_1+h(x\_1)) , v\^[(2)]{}\_[\^i]{}:=\_[j=0]{}\^4 \^[-ij]{}g\^j(x\_1-h(x\_1)) , so that h(v\^[(1)]{}\_[\^i]{})=-v\^[(2)]{}\_[\^[-i]{}]{} ,h(v\^[(2)]{}\_[\^i]{})=v\^[(1)]{}\_[\^[-i]{}]{} . Correspondingly, the action of $h$ on the left-moving fields is $$\begin{aligned}
\label{haction1}
&\psi^{(1)}\mapsto -\psi^{(2)}\ ,\quad \psi^{(2)}\mapsto \psi^{(1)}\ ,\quad
\bar\psi^{(1)}\mapsto -\bar\psi^{(2)}\ ,\quad \bar\psi^{(2)}\mapsto \bar\psi^{(1)}\ ,\\ \label{haction2}
&J^{(1)}\mapsto -J^{(2)}\ ,\quad J^{(2)}\mapsto J^{(1)}\ ,\quad \bar J^{(1)}\mapsto -\bar J^{(2)}\ ,
\quad \bar J^{(2)}\mapsto \bar J^{(1)}\ ,\end{aligned}$$ and the action on the right-moving fields is analogous. It is immediate to verify that the generators of the superconformal algebra are invariant under this transformation.
The orbifold theory
-------------------
Next we want to consider the orbifold of this torus theory by the group $\mathbb{Z}_5$ that is generated by $g$.
### The elliptic genus
The elliptic genus of the orbifold theory can be computed by summing over the ${\rm SL}(2,{\mathbb{Z}})$ images of the untwisted sector contribution, which in turn is given by \^U(,z)= \_[k=0]{}\^4\_[1,g\^k]{}(,z) , where \_[1,g\^k]{}(,z) =\_[RR]{}(g\^k q\^[L\_0-]{}|q\^[L\_0-]{} y\^[2J\_0]{}(-1)\^[F+F]{}) . The $k=0$ contribution, i.e. the elliptic genus of the original torus theory, is zero. Each $g^k$-contribution, for $k=1,\ldots,4$, is the product of a factor coming from the ground states, one from the oscillators and one from the momenta \_[1,g\^k]{}(,z)=\_[1,g\^k]{}\^[gd]{}(,z) \_[1,g\^k]{}\^[osc]{}(,z) \_[1,g\^k]{}\^[mom]{}(,z) . These contributions are, respectively, \_[1,g\^k]{}\^[gd]{}(,z)= y\^[-1]{}(1-\^k y)(1-\^[-k]{} y)(1-\^[2k]{})(1-\^[-2k]{}) =2y\^[-1]{}+2y+1 , \_[1,g\^k]{}\^[osc]{}(,z)= \_[n=1]{}\^ , and \_[1,g\^k]{}\^[mom]{}(,z)=1 , where the last result follows because the only $g$-invariant state of the form $(k_L,k_R)$ is the vacuum $(0,0)$. Thus we have \_[1,g\^k]{}(,z)=5 , where \_1(,z)=-iq\^[1/8]{}y\^[-12]{}(y-1)\_[n=1]{}\^(1-q\^n)(1-yq\^n)(1-y\^[-1]{}q\^n) , is the first Jacobi theta function. Modular transformations of $\phi_{1,g^k}(\tau,z)$ reproduce the twining genera in the twisted sector \[Z5twis\] \_[g\^l,g\^k]{}(,z)=\_[\^[(l)]{}]{} (g\^k q\^[L\_0-]{} |q\^[L\_0-]{}y\^[2J\_0]{}(-1)\^[F+F]{}) , and using the modular properties of the theta function we obtain \_[g\^l,g\^k]{}(,z)=5 , for $k,l\in{\mathbb{Z}}/5{\mathbb{Z}}$, $(k,l)\neq (0,0)$. The elliptic genus of the full orbifold theory is then \_[orb]{}(,z)=\_[k,l/5]{} \_[g\^l,g\^k]{}(,z) =\_ . Since $\phi_{g^k,g^l}(\tau,0)=5$ for all $(k,l)\neq (0,0)$, we have \[indexorb\] \_[orb]{}(,0)=\_ 5 =24 , which shows that the orbifold theory is a non-linear sigma-model on K3. In particular, the untwisted sector has $4$ RR ground states, while each of the four twisted sectors contains $5$ RR ground states. For the following it will be important to understand the structure of the various twisted sectors in detail.
### The twisted sectors {#sec:twistsect}
In the $g^k$-twisted sector, let us consider a basis of $g$-eigenvectors for the currents and fermionic fields. For a given eigenvalue $\zeta^{i}$, $i\in{\mathbb{Z}}/5{\mathbb{Z}}$, of $g^k$, the corresponding currents $J^{i,a}$ and fermionic fields $\psi^{i,b}$ (where $a,b$ labels distinct eigenvectors with the same eigenvalue) have a mode expansion J\^[i,a]{}(z)=\_[n+]{} \^[i,a]{}\_nz\^[-n-1]{} , \^[i,a]{}(z)=\_[r++]{}\^[i,a]{}\_rz\^[-r-1/2]{} , where $\nu=1/2$ in the NS- and $\nu=0$ in the R-sector. The ground states of the $g^k$-twisted sector are characterised by the conditions $$\begin{aligned}
&\alpha^{i,a}_n|m,k\rangle=\tilde\alpha^{i,a}_n|m,k\rangle = 0 \ ,\qquad
&\forall\ n>0,\ i,\ a\ ,\\
&\psi^{i,b}_r|m,k\rangle=\tilde\psi^{i,b}_r|m,k\rangle = 0 \ ,\qquad &\forall\ r>0,\ i,\ b\ ,\end{aligned}$$ where $|m,k\rangle$ denotes the $m^{\rm th}$ ground state in the $g^k$-twisted sector. Note that since none of the currents are $g$-invariant, there are no current zero modes in the $g^k$-twisted sector, and similarly for the fermions. For a given $k$, the states $|m,k\rangle$ have then all the same conformal dimension, which can be calculated using the commutation relation $[L_{-1},L_1]=2L_0$ or read off from the leading term of the modular transform of the twisted character (\[Z5twis\]). In the $g^k$-twisted NS-NS-sector the ground states have conformal dimension h= h= , while in the RR-sector we have instead h= = . In particular, level matching is satisfied, and thus the asymmetric orbifold is consistent [@Narain:1986qm]. The full $g^k$-twisted sector is then obtained by acting with the negative modes of the currents and the fermionic fields on the ground states $|m,k\rangle$.
Let us have a closer look at the ground states of the $g^k$ twisted sector; for concreteness we shall restrict ourselves to the case $k=1$, but the modifications for general $k$ are minor (see below). The vertex operators $V_\lambda(z,\bar z)$ associated to $\lambda\in\Gamma^{4,4}$, act on the ground states $|m,1\rangle$ by \_[z0]{} V\_(z,|z) |m,1=e\_ |m,1 , where $e_\lambda$ are operators commuting with all current and fermionic oscillators and satisfying \[ecomm\] e\_ e\_= (,) e\_[+]{} , for some fifth root of unity $\epsilon(\lambda,\mu)$. The vertex operators $V_\lambda$ and $V_\mu$ must be local relative to one another, and this is the case provided that (see the appendix) = C(,) , where \[Pgdef\] C(,)=\_[i=1]{}\^4(\^i)\^[(g\^i(),)]{}=\^[(P\_g(),)]{} P\_g() = \_[i=1]{}\^4 ig\^i() . The factor $C(\lambda,\mu)$ has the properties $$\begin{aligned}
\label{Clinear}
&C(\lambda,\mu_1+\mu_2)=C(\lambda,\mu_1)\, C(\lambda,\mu_2)\ ,\qquad
C(\lambda_1+\lambda_2,\mu)=C(\lambda_1,\mu)\, C(\lambda_2,\mu)\ ,\\
&C(\lambda,\mu)=C(\mu,\lambda)^{-1}\ ,\label{Cinverse}\\
&C(\lambda,\mu)=C(g(\lambda),g(\mu))\ .\label{Cinvar}\end{aligned}$$ Because of , $C(\lambda,0)=C(0,\lambda)=1$ for all $\lambda\in\Gamma^{4,4}$, and we can set e\_0=1 , so that $\epsilon(0,\lambda)=1=\epsilon(\lambda,0)$. More generally, for the vectors $\lambda$ in the sublattice R:={\^[4,4]{}C(,)=1 , \^[4,4]{}}\^[4,4]{} , we have $C(\lambda+\mu_1,\mu_2)=C(\mu_1,\mu_2)$, for all $\mu_1,\mu_2\in\Gamma^{4,4}$, so that we can set \[Rinvariance\] e\_[+]{}=e\_ ,R , \^[4,4]{} . Thus, we only need to describe the operators corresponding to representatives of the group $\Gamma^{4,4}/R$. The vectors $\lambda\in R$ are characterised by (P\_g(), ) 05 ,\^[4,4]{} , and since $\Gamma^{4,4}$ is self-dual this condition is equivalent to P\_g()5 \^[4,4]{} . Since $g$ has no invariant subspace, we have the identity 1+g+g\^2+g\^3+g\^4=0 that implies (see ) \[Pgoneminusg\] P\_g(1-g) = (1-g)P\_g =-5 . Thus, $\lambda\in R$ if and only if P\_g()=P\_g(1-g)( ) , for some $\tilde\lambda\in\Gamma^{4,4}$, and since $P_g$ has trivial kernel (see ), we finally obtain R=(1-g) \^[4,4]{} . Since also $(1-g)$ has trivial kernel, $R$ has rank $8$ and $\Gamma^{4,4}/R$ is a finite group. Furthermore, |\^[4,4]{}/R|=(1-g)=25 , and, since $5\, \Gamma^{4,4}\subset R$, the group $\Gamma^{4,4}/R$ has exponent $5$. The only possibility is \^[4,4]{}/R\_5\_5 . Let $x,y\in\Gamma^{4,4}$ be representatives for the generators of $\Gamma^{4,4}/R$. By , we know that $C(x,x)=C(y,y)=1$, so that $C(x,y)\neq 1$ (otherwise $C$ would be trivial over the whole $\Gamma^{4,4}$), and we can choose $x,y$ such that C(x,y)= . Thus, the ground states form a representation of the algebra of operators generated by $e_x$, $e_y$, satisfying e\_x\^5=1=e\_y\^5 ,e\_x e\_y=e\_y e\_x . The group generated by $e_x$ and $e_y$ is the extra-special group $5^{1+2}$, and all its non-abelian irreducible representations[^7] are five dimensional. In particular, for the representation on the $g$-twisted ground states, we can choose a basis of $e_x$-eigenvectors |m;1 e\_x|m;1=\^m|m;1 , m/5 , and define the action of the operators $e_y$ by e\_y|m;1=|m+1;1 . For any vector $\lambda\in\Gamma^{4,4}$, there are unique $a,b\in{\mathbb{Z}}/5{\mathbb{Z}}$ such that $\lambda=ax+by+(1-g)(\mu)$ for some $\mu\in \Gamma^{4,4}$ and we define[^8] e\_:=e\_x\^ae\_y\^b . Since $g(\lambda)=ax+by+(1-g)(g(\mu)-ax-by)$, by we have e\_[g()]{}=e\_ , so that, with respect to the natural action $g(e_\lambda):=e_{g(\lambda)}$, the algebra is $g$-invariant. This is compatible with the fact that all ground states have the same left and right conformal weights $h$ and $\tilde h$, so that the action of $g=e^{2\pi i(h-\tilde h)}$ is proportional to the identity.
The construction of the $g^k$-twisted sector, for $k=2,3,4$, is completely analogous to the $g^1$-twisted case, the only difference being that the root $\zeta$ in the definition of $C(\lambda,\mu)$ should be replaced by $\zeta^k$. Thus, one can define operators $e_x^{(k)},e_y^{(k)}$ on the $g^k$-twisted sector, for each $k=1,\ldots,4$, satisfying \[genrel\] (e\_x\^[(k)]{})\^5=1=(e\_y\^[(k)]{})\^5 ,e\_x\^[(k)]{} e\_y\^[(k)]{}=\^k e\_y\^[(k)]{} e\_x\^[(k)]{} . The action of these operators on the analogous basis $|m;k\rangle$ with $m\in{\mathbb{Z}}/5{\mathbb{Z}}$ is then \[gkground\] e\_x\^[(k)]{}|m;k=\^[m]{}|m;k ,e\_y\^[(k)]{}|m;k=|m+k;k .
### Spectrum and symmetries {#sec:twistsec2}
The spectrum of the actual orbifold theory is finally obtained from the above twisted sectors by projecting onto the $g$-invariant states; technically, this is equivalent to restricting to the states for which the difference of the left- and right- conformal dimensions is integer, $h-\tilde h\in{\mathbb{Z}}$. In particular, the RR ground states in each (twisted) sector have $h=\tilde h=1/4$, so that they all survive the projection. Thus, the orbifold theory has $4$ RR ground states in the untwisted sector (the spectral flow generators), forming a $({\bf 2},{\bf 2})$ representation of $\mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R$, and $5$ RR ground states in each twisted sector, which are singlets of $\mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R$. In total there are therefore $24$ RR ground states, as expected for a non-linear sigma-model on K3. (Obviously, we are here just reproducing what we already saw in (\[indexorb\]).)
Next we want to define symmetry operators acting on the orbifold theory. First we can construct operators $e_\lambda$ associated to $\lambda\in\Gamma^{4,4}$, that will form the extra special group $5^{1+2}$. They are defined to act by $e_\lambda^{(k)}$ on the $g^k$-twisted sector. The action of the untwisted sector preserves the subspaces ${\mathcal{H}}_m^{U}$, $m\in {\mathbb{Z}}/5{\mathbb{Z}}$, of states with momentum of the form $\lambda=nx+my+(1-g)(\mu)$, for some $n\in{\mathbb{Z}}$ and $\mu\in \Gamma^{4,4}$. Let us denote by $T_{m;k}$ a generic vertex operator associated with a $g^k$-twisted state, $k=1,\ldots,4$, with $e_x$-eigenvalue $\zeta^{m}$, $m\in{\mathbb{Z}}/5{\mathbb{Z}}$, and by $T_{m;0}$ a vertex operator associated with a state in ${\mathcal{H}}_m^{U}$. Consistency of the OPE implies the fusion rules \[orbifusion\] T\_[m;k]{}T\_[m’;k’]{}T\_[m+m’;k+k’]{} . These rules are preserved by the maps T\_[m;k]{}e\_ T\_[m;k]{} e\_\^[-1]{} ,\^[4,4]{} , which therefore define symmetries of the orbifold theory. As we have explained above, these symmetries form the extra-special group $5^{1+2}$.
Finally, the symmetries , , and of the original torus theory induce a ${\mathbb{Z}}_4$-group of symmetries of the orbifold. Since $h^{-1}gh=g^{-1}$, the space of $g$-invariant states of the original torus theory is stabilised by $h$, so that $h$ restricts to a well-defined transformation on the untwisted sector of the orbifold. Furthermore, $h$ maps the $g^k$- to the $g^{5-k}$-twisted sector. Eqs. and can be written as $$\begin{aligned}
&h(x_1)=2x_1+(1-g)(-x_1-x_2-x_3+y_1+y_2+y_3+y_4)\ ,\\
&h(y_1)=2y_1+(1-g)(-x_1-x_2-x_3-x_4-2y_1-y_2-y_3-y_4)\ .\end{aligned}$$ It follows that the action of $h$ on the operators $e_\lambda^{(k)}$, $k=1,\ldots,4$ must be he\_x\^[(k)]{}h\^[-1]{}=e\_[2x]{}\^[(5-k)]{} ,he\_y\^[(k)]{}h\^[-1]{}=e\_[2y]{}\^[(5-k)]{} , and it is easy to verify that this transformation is compatible with . Correspondingly, the action on the twisted sector ground states is h|m;k= |3m;5-k , and it is consistent with .
Thus the full symmetry group is the semi-direct product G=5\^[1+2]{}:\_4 , where the generator $h\in{\mathbb{Z}}_4$ maps the central element $\zeta\in 5^{1+2}$ to $\zeta^{-1}$.
All of these symmetries act trivially on the superconformal algebra and on the spectral flow generators, and therefore define symmetries in the sense of the general classification theorem. Indeed, $G$ agrees precisely with the group in case (ii) of the theorem. Thus our orbifold theory realises this possibility.
Models with symmetry group containing $3^{1+4}:{\mathbb{Z}}_2$ {#sec:Z3}
==============================================================
Most of the torus orbifold construction described in the previous section generalises to symmetries $g$ of order different than $5$. In particular, one can show explicitly that orbifolds of ${\mathbb{T}}^4$ models by a symmetry $g$ of order $3$ contain a group of symmetries $3^{1+4}:{\mathbb{Z}}_2$, so that they belong to one of the cases (iii) and (iv) of the theorem, as expected from the discussion in section \[sec:symme\].
We take the action of the symmetry $g$ on the left-moving currents and fermionic fields to be of the form and , where $\zeta$ is a now a third root of unity; analogous transformations hold for the right-moving fields with respect to a third root of unity $\tilde\zeta$. In this case, eq. can be satisifed by ==e\^ , so that the action is left-right symmetric and $g$ admits an interpretation as a geometric ${\rm O}(4,{\mathbb{R}})$-rotation of order $3$ of the torus ${\mathbb{T}}^4$. For example, the torus ${\mathbb{R}}^4/(A_2\oplus A_2)$, where $A_2$ is the root lattice of the $su(3)$ Lie algebra, with vanishing Kalb-Ramond field, admits such an automorphism.
The orbifold by $g$ contains $6$ RR ground states in the untwisted sector. In the $k^{\rm th}$ twisted sector, $k=1,2$, the ground states form a representation of an algebra of operators $e^{(k)}_{\lambda}$, $\lambda\in\Gamma^{4,4}$, satisfying the commutation relations e\^[(k)]{}\_e\^[(k)]{}\_=C(,)\^k e\^[(k)]{}\_e\^[(k)]{}\_ , where C(,) =\^[(P\_g(),)]{} ,P\_g=g+2g\^2 . As discussed in section \[sec:twistsect\], we can set e\^[(k)]{}\_[+]{}=e\_\^[(k)]{} , R,\^[4,4]{} , where R=(1-g)\^[4,4]{} . (Note that $\Gamma^{4,4}$ contains no $g$-invariant vectors). The main difference with the analysis of section \[sec:twistsect\] is that, in this case, \^[4,4]{}/R\_3\^4 . In particular, we can find vectors $x_1,x_2,y_1,y_2\in\Gamma^{4,4}$ such that C(x\_i,y\_j)=\^[\_[ij]{}]{} ,C(x\_i,x\_j)=C(y\_i,y\_j)=1 . The corresponding operators obey the relations e\^[(k)]{}\_[x\_i]{}e\^[(k)]{}\_[y\_j]{}=\^[k\_[ij]{}]{}e\^[(k)]{}\_[y\_j]{}e\^[(k)]{}\_[x\_i]{} , e\^[(k)]{}\_[x\_i]{}e\^[(k)]{}\_[x\_j]{}=e\^[(k)]{}\_[x\_j]{}e\^[(k)]{}\_[x\_i]{} ,e\^[(k)]{}\_[y\_i]{}e\^[(k)]{}\_[y\_j]{}=e\^[(k)]{}\_[y\_j]{}e\^[(k)]{}\_[y\_i]{} , as well as (e\^[(k)]{}\_[x\_i]{})\^3=1=(e\^[(k)]{}\_[y\_i]{})\^3 . These operators generate the extra-special group $3^{1+4}$ of exponent $3$, and the $k^{\rm th}$-twisted ground states form a representation of this group. We can choose a basis $|m_1,m_2;k\rangle$, with $m_1,m_2\in {\mathbb{Z}}/3{\mathbb{Z}}$, of simultaneous eigenvectors of $e^{(k)}_{x_1}$ and $e^{(k)}_{x_2}$, so that e\_[x\_i]{}\^[(k)]{}|m\_1,m\_2;k=\^[m\_i]{}|m\_1,m\_2;k , e\_[y\_i]{}\^[(k)]{}|m\_1,m\_2;k=|m\_1+k\_[1i]{},m\_2+k\_[2i]{};k . The resulting orbifold model has $9$ RR ground states in each twisted sector, for a total of $6+9+9=24$ RR ground states, as expected for a K3 model. As in section \[sec:twistsec2\], the group $3^{1+4}$ generated by $e^{(k)}_\lambda$ extends to a group of symmetries of the whole orbifold model. Furthermore, the ${\mathbb{Z}}_2$-symmetry that flips the signs of the coordinates in the original torus theory induces a symmetry $h$ of the orbifold theory, which acts on the twisted sectors by h |m\_1,m\_2;k= |-m\_1,-m\_2;k . We conclude that the group $G$ of symmetries of any torus orbifold $\mathbb{T}^4/{\mathbb{Z}}_3$ contains a subgroup $3^{1+4}: {\mathbb{Z}}_2$, and is therefore included in the cases (iii) or (iv) of the classification theorem. This obviously ties in nicely with the general discussion of section \[sec:symme\].
Conclusions
===========
In this paper we have reviewed the current status of the EOT conjecture concerning a possible $\mathbb{M}_{24}$ symmetry appearing in the elliptic genus of K3. We have explained that, in some sense, the EOT conjecture has already been proven since twining genera, satisfying the appropriate modular and integrality properties, have been constructed for all conjugacy classes of $\mathbb{M}_{24}$. However, the analogue of the Monster conformal field theory that would ‘explain’ the underlying symmetry has not yet been found. In fact, no single K3 sigma-model will be able to achieve this since none of them possesses an automorphism group that contains $\mathbb{M}_{24}$.
Actually, the situation is yet further complicated by the fact that there are K3 sigma-models whose automorphism group is [*not even a subgroup of*]{} $\ \mathbb{M}_{24}$; on the other hand, the elliptic genus of K3 does not show any signs of exhibiting ‘Moonshine’ with respect to any larger group. As we have explained in this paper, most of the exceptional automorphism groups (i.e. automorphism groups that are not subgroups of $\mathbb{M}_{24}$) appear for K3s that are torus orbifolds. In fact, all cyclic torus orbifolds are necessarily exceptional in this sense, and (cyclic) torus orbifolds account for all incarnations of the cases (ii) – (iv) of the classification theorem of [@Gaberdiel:2011fg] (see section \[sec:symme\]). We have checked these predictions by explicitly constructing an asymmetric $\mathbb{Z}_5$ orbifold that realises case (ii) of the theorem (see section \[sec:Z5\]), and a family of $\mathbb{Z}_3$ orbifolds realising cases (iii) and (iv) of the theorem (see section \[sec:Z3\]). Incidentally, these constructions also demonstrate that the exceptional cases (ii)-(iv) actually appear in the K3 moduli space — in the analysis of [@Gaberdiel:2011fg] this conclusion relied on some assumption about the regularity of K3 sigma-models.
The main open problem that remains to be understood is why precisely $\mathbb{M}_{24}$ is ‘visible’ in the elliptic genus of K3, rather than any smaller (or indeed bigger) group. Recently, we have constructed (some of) the twisted twining elliptic genera of K3 [@GPRV], i.e. the analogues of Simon Norton’s generalised Moonshine functions [@N]. We hope that they will help to shed further light on this question.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Terry Gannon, Gerald Hoehn, and Michael Tuite for useful conversations. The research of MRG is supported in parts by the Swiss National Science Foundation.
Commutation relations in the twisted sector
===========================================
The vertex operators $V_\lambda(z,\bar z)$ in the $g$-twisted sector can be defined in terms of formal exponentials of current oscillatorsE\^\_(z,|z):= (\_(\_L)\^[(r)]{}\_[r]{})(\_(\_R)\^[(r)]{}\_[r]{}) , where $(\lambda_L\cdot \alpha)^{(r)}_{r}$ and $(\lambda_R\cdot \tilde\alpha)^{(r)}_{r}$ are the $r$-modes of the currents $$\begin{aligned}
(\lambda_L\cdot \partial X)^{(r)}&:=\frac15\sum_{i=0}^4\zeta^{5ir}\lambda_L\cdot g^i(\partial X)=\frac15\sum_{i=0}^4\zeta^{5ir}g^{-i}(\lambda_L)\cdot \partial X\ ,\\
(\lambda_R\cdot \bar \partial X)^{(r)}&:=\frac15\sum_{i=0}^4\bar\zeta^{5ir}\lambda_R\cdot g^i(\bar\partial X)=\frac15\sum_{i=0}^4\bar \zeta^{5ir}g^{-i}(\lambda_R)\cdot \bar\partial X\ .\end{aligned}$$ Then we can define V\_(z,|z):=E\^-\_(z,|z) E\^+\_(z,|z) e\_ , where the operators $e_\lambda$ commute with all current oscillators and satisfy e\_ e\_=(,) e\_[+]{} , for some fifth root of unity $\epsilon(\lambda,\mu)$. The commutator factor C(,):= , can be determined by imposing the locality condition V\_(z\_1,|z\_1) V\_(z\_2,|z\_2) =V\_(z\_2,|z\_2) V\_(z\_1,|z\_1) . To do so, we note that the commutation relations between the operators $E^\pm_\lambda$ can be computed, as in [@Lepowsky1985], using the Campbell-Baker-Hausdorff formula $$\begin{gathered}
\label{commuta}
E^{+}_\lambda(z_1,\bar z_1)E^{-}_\mu(z_2,\bar z_2)
= E^{-}_\mu(z_2,\bar z_2)E^{+}_\lambda(z_1,\bar z_1)\\
\prod_{i=0}^4 [(1-\zeta^{-i}(\tfrac{z_1}{z_2})^{\frac15})^{g^i(\lambda)_L\cdot\mu_L}
(1-\bar\zeta^{-i}(\tfrac{\bar z_1}{\bar z_2})^{\frac15})^{g^i(\lambda)_R\cdot\mu_R}]\ .\end{gathered}$$ Using and $e_\lambda e_\mu=C(\lambda,\mu)e_\mu e_\lambda$, the locality condition is then equivalent to C(,)\_[i=0]{}\^4=1 , that is C(,)(- )\^[\_i g\^i()\_L\_L]{}(-)\^[\_i g\^i()\_R\_R]{}\_[i=0]{}\^4\[(\^[-i]{})\^[g\^i()\_L\_L]{}(|\^[-i]{})\^[g\^i()\_R\_R]{}\]=1 . Since $\Gamma^{4,4}$ has no $g$-invariant vector, we have the identities \_[i=0]{}\^4g\^i()\_L=0=\_[i=0]{}\^4g\^i()\_R , and hence finally obtain C(,)=\_[i=0]{}\^4(\^i)\^[g\^i(\_L)\_L-g\^i(\_R)\_R]{} .
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[^1]: Partially based on talk given by M.R.G. at the conference ‘Conformal Field Theory, Automorphic Forms and Related Topics’, Heidelberg, 19-23 September 2011.
[^2]: Partially based on talk given by M.R.G. at the conference ‘Conformal Field Theory, Automorphic Forms and Related Topics’, Heidelberg, 19-23 September 2011.
[^3]: Actually, they did not just look at the Fourier coefficients themselves, but at the decomposition of the elliptic genus with respect to the elliptic genera of irreducible ${\cal N}=4$ superconformal representations. They then noted that these expansion coefficients (and hence in particular the usual Fourier coefficients) are sums of dimensions of irreducible $\mathbb{M}_{24}$ representations.
[^4]: Since the orbifold action is asymmetric, this evades various no-go-theorems (see e.g. [@Walton]) that state that the possible orbifold groups are either $\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_4$, or $\mathbb{Z}_6$.
[^5]: The possibility $G''={\mathbb{Z}}_4$ that has been considered in [@Gaberdiel:2011fg] has to be excluded, since there are no elements of order $4$ in $W(E_6)$ that preserve a four-dimensional sublattice of $E_6^*(3)$.
[^6]: We should emphasise that for us the term ‘orbifold’ always refers to a conformal field theory (rather than a geometrical) construction. Although a non-linear sigma-model on a geometric orbifold $M/{\mathbb{Z}}_N$ always admits an interpretation as a CFT orbifold, the converse is not always true. In particular, there exist asymmetric orbifold constructions that do not have a direct geometric interpretation, see for example section \[sec:Z5\].
[^7]: We call a representation non-abelian if the central element does not act trivially.
[^8]: The ordering of $e_x$ and $e_y$ in this definition is arbitrary; however, any other choice corresponds to a redefinition $\tilde e_\lambda=c(\lambda)e_\lambda$, for some fifth root of unity $c(\lambda)$, that does not affect the commutation relations $\tilde e_\lambda\tilde e_\mu=C(\lambda,\mu)\tilde e_\mu\tilde e_\lambda$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study here the r-modes in the Cowling approximation of a slowly rotating and magnetized neutron star with a poloidal magnetic field, where we neglect any deformations of the spherical symmetry of the star. We were able to quantify the influence of the magnetic field in both the oscillation frequency $\sigma_r$ of the r-modes and the growth time $t_{gw}$ of the gravitational radiation emission. We conclude that magnetic fields of the order $10^{15}$ G at the center of the star are necessary to produce any changes. Our results for $\sigma_r$ show a decrease of up to $\sim$ 5% in the frequency with increasing magnetic field, with a $B^2$ dependence for rotation rates $\Omega/\Omega_K \gtrsim 0.07$ and $B^4$ for $\Omega/\Omega_K \lesssim 0.07$. (These results should be trusted only within slow rotation approximation and we kept $\Omega/\Omega_{K}< 0.3$.) For $t_{gw}$, we find that it is approximately 30% smaller than previous Newtonian results for non-magnetized stars, which would mean a faster growth of the emission of gravitational radiation. The effect of the magnetic field in $t_{gw}$ causes a non-monotonic effect, that first slightly increases $t_{gw}$ and then decreases it further by another $\sim$ 5%. (The value of magnetic field for which $t_{gw}$ starts to decrease depends on the rotational frequency, but it is generally around $10^{15}$G.) Future work should be dedicated to the study of the effect of viscosity in the presence of magnetic fields, in order to establish the magnetic correction to the instability window.'
author:
- Cecilia Chirenti
- Jozef Skákala
title: |
The effect of magnetic fields on the r-modes of slowly rotating\
relativistic neutron stars
---
Introduction
============
The r-mode instability was first discovered in [@Andersson-instability; @Friedman-instability] and it was predicted that the instability could become a significant source of gravitational radiation. This happens because the r-modes are generically unstable to the CFS gravitational-radiation-driven instability [@Chandrasekhar; @Friedman-Schutz1; @Friedman-Schutz2]. The r-mode instability follows immeadiately from the fact that r-modes that are prograde with respect to a distant observer are retrograde in the comoving frame for all values of the angular velocity (the canonical energy of the modes is negative). For some nice reviews see [@Kokkotas3; @Stergioulas]. However, different mechanisms to damp the instability have to be considered: one of them is viscosity [@Lindblom], another one could be sufficiently strong magnetic fields [@Rezzolla1; @Rezzolla2; @Rezzolla3]. The instability windows for non-magnetized Newtoninan stars were initially calculated in [@Lindblom2; @Andersson]. More recently, the effect of magnetic fields on r-modes was discussed in [@Rezania; @Abbassi] for a spherical shell and in [@Lander] for a neutron star with purely toroidal field, in all cases in the Newtonian context.
This paper is a first part of a project that is supposed to contribute to the understanding of the instability window for slowly rotating relativistic neutron stars with magnetic fields. We are here interested in and focused on the modification of the r-modes in the presence of magnetic fields and its effects on the gravitational wave emission. The effect of magnetic fields on the r-mode frequencies could be interesting for astrophysical objects such as magnetars, that have magnetic fields of the order of magnitude $10^{15}$ G and are very slow rotators with rotational periods of a few seconds. We consider here stars of comparable magnetic fields with rotational periods of a few milliseconds (due to a numerical difficulty: longer periods would need longer time evolutions). The issue of viscous damping of the mode, which determines the instability window of the r-mode, is further complicated by the presence of the magnetic field. Therefore we leave it for future work.
We treat the problem within the realm of perturbation theory, first by deriving general perturbation equations and then by solving them numerically with a 2D Lax-Wendroff method. The same numerical methods were used and tested in our previous paper [@Chirenti]. The advantages of the 2D dynamical evolution in this case is that it avoids both the r-mode continuous spectrum problem [@Beyer] and the need for truncating the solution at some $\ell$ (as done for instance in [@Kokkotas1; @Kokkotas2] for torsional modes of a relativistic star with a dipole magnetic field). We first calculate the r-mode frequencies for different values of the rotation parameter and magnetic fields. Then we calculate the instability growth rate due to the emission of gravitational waves as a function of the magnetic field. (This gives both general relativistic and magnetic field corrections to the results of [@Lindblom2; @Andersson].)
The paper is organized as follows: in the second section we describe our background model and in the third section we present the full perturbation equations of our model. This section is followed by the fourth section, where we compute the r-mode frequencies for different rotation rates of the star, as a function of the magnetic field. In the fifth section we compute the growth time due to the r-mode gravitational wave emission as a function of magnetic field. In the sixth section we present the concluding remarks. (Everywhere in the paper, unless explicitly mentioned, we use the units $c=G=M_{\bigodot}=1$.)
The background model
====================
We work with a slowly and uniformly rotating magnetized star with a polytropic equation of state. Our model neutron star has $M = 1.4 M_{\bigodot}$, $R = 14.08$ km, and the pressure $p$ is given by the polytropic equation of state $p=K\rho^{\Gamma}$, taken with the parameters $K=100$, $\Gamma=2$ and $\rho$ is the rest-mass density of the star. The Keplerian frequency $\Omega_{K}$ (mass shedding limit) that we use in our paper to normalize the rotation of the star is $\Omega_{K}=\sqrt{M/R^{3}}=1.3$ .
The effect of the rotation is taken up to the linear order in the rotation parameter $\Omega$, which means one considers the effect of the rotation on the spacetime metric (frame dragging function), but neglects the effect of the rotation on the stellar structure. (The deformations of the stellar fluid due to rotation are of the order $\Omega^{2}$.)
We consider a dipole magnetic field and for the realistic neutron stars with magnetic fields (up to the order of $10^{15}G$ for magnetars), one can neglect the effect of the magnetic field on both the stellar structure and the background metric (for a more detailed argumentation, see [@Kokkotas1]).
This means the model follows from a line element: $$ds^{2}=-e^{\nu}dt^{2}+e^{\lambda}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}(\theta)\left[d\phi-\omega dt\right]^{2},$$ where $\nu$, $\lambda$ and $\omega$ are functions of $r$, and a stress energy tensor given by: $$T^{\mu\nu}=(p+\epsilon+H^{2})u^{\mu}u^{\nu}+\left(p+\frac{H^{2}}{2}\right)g^{\mu\nu}-H^{\mu}H^{\nu}.$$ Here $\epsilon$ is the total energy density, the 4-velocity $u^{\mu}$ is given as $$u^{(t,r,\theta,\phi)}=\left(e^{-\nu/2},~0,~0,~\Omega e^{-\nu/2}\right),$$ and $H^{\mu}$ is related to the magnetic field $B^{\mu}$, defined as usual in terms of the electromagnetic tensor $F^{\mu\nu}$ as $$H_{\mu}=\frac{B_{\mu}}{4\pi}=-\frac{1}{8\pi}\epsilon_{\mu\nu\alpha\beta}u^{\nu}F^{\alpha\beta}.$$ Our background model is thus obtained by the TOV equations supplemented by a polytropic equation of state, Hartle’s equation for the frame dragging function $\omega$ [@Hartle]: $$\frac{e^{\frac{\nu+\lambda}{2}}}{r^4}\left( r^4 e^{-\frac{\nu+\lambda}{2}}\omega_{,r}\right)_{,r} + \frac{2}{r}(\nu_{,r} + \lambda_{,r})(\Omega - \omega) = 0\,,
\label{eq:Hartle}$$ and the magnetic field is given by the Maxwell equations. Therefore our dipole equilibrium magnetic field (for our choice of equilibrium magnetic field, the induction equation is trivially fulfilled) obeys the Maxwell equations $$F^{\mu\nu}_{~;\nu}=4\pi J^{\mu},$$ solved with a 4-current $J_{\mu}$ with the only non-zero component $$\label{eq:J_mu}
4\pi J_{\phi}(r,\theta) = -\alpha(r)\sin^2(\theta)\,,$$ (this choice is equivalent, for instance, to choosing the terms with parity $(-1)^{\ell+1}$ in the parity decomposition given in [@Ruffini] and keeping only $\ell = 1$, $m = 0$) with the radial profile $$\alpha (r) = \alpha_{0} r^{2}\epsilon^{2}(r)~,$$ that describes a ring current inside the star. Moreover, this current profile allows us to consider the magnetic field as force-free at the surface (the Lorentz force goes to zero at the surface of the star as $\epsilon^2$).
Choosing the vector potential as $A_{\phi} = -a(r)\sin^2(\theta)$ (same parity and symmetry choices as we did for $J^{\mu}$ above), the Maxwell equations give:
$$\label{eq:a_gen}
e^{-\lambda}a_{,rr} + \left[ 4\pi(p-\epsilon) r +
\frac{1-e^{-\lambda}}{r}\right]a_{,r} -
\frac{2}{r^2}a + \alpha = 0.$$
Here the components of our dipole magnetic field are obtained from $a(r)$ as: $$H^{r}(r,\theta)=\frac{a(r)}{2\pi r^{2}}\cos \theta~,$$ $$H^{\theta}(r,\theta)=\frac{a_{,r}e^{-\lambda}}{4\pi r^{2}}\sin\theta~.$$ It is known that the equation has outside the star an exact analytic solution [@Wasserman]: $$a(r)= C r^{2}\left[\ln\left(\frac{r}{r-2M}\right)-\frac{2M (r+M)}{r^{2}}\right].
\label{eq:a_exact}$$ The full solution inside and outside the star for the dipole magnetic field was computed by numerically solving eq. (\[eq:a\_gen\]) inside the star, matching the regular series expansion of the solution near the center with the numerical solution up to the surface, where we require continuity of $a(r)$ and its first derivative with the exterior analytic solution (\[eq:a\_exact\]).
Some representative plots of the behavior of the radial function $a(r)$ can be seen in figure \[fig:teste\_a\], for increasing values of the current parameter $\alpha_0$. One typical solution for the amplitude of the magnetic field inside the star is given in figure \[fig:teste\_B2\] (we note here that the solid lines are contour lines, and not magnetic field lines). Throughout the paper we refer to the value of the magnetic field at the center of the star. But since the magnetic field at the pole of a star can be determined observationally, while the value of the magnetic field at the center of star must be calculated with some model, we present in figure \[fig:teste\_Bpole\] the relation between $B_{\rm{pole}}$ and $B_{\rm{center}}$ given by our model.
In figure \[fig:teste\_omee\] we present some representative plots of the numerical solutions obtained for the frame dragging function $\omega$, from Hartle’s equation (\[eq:Hartle\]).
![The function $a(r)$ for different currents as a function of the radial coordinate divided by the radius of the star.[]{data-label="fig:teste_a"}](fig1.eps){width="1\linewidth"}
![The absolute value of the dipole magnetic field (in Gauss) corresponding to $\alpha_{0}=10$. The magnetic field is shown in the plane given by the rotational axis and a perpendicular direction to the rotational axis.[]{data-label="fig:teste_B2"}](fig2.eps){width="1\linewidth"}
![The value of the magnetic field at the magnetic pole $B_{\textrm{pole}}$ as a funtion of the magnetic field at the center of the star $B_{\textrm{center}}$ as calculated with our model. We obtained $B_{\textrm{pole}} = 0.0974 B_{\textrm{center}}$.[]{data-label="fig:teste_Bpole"}](fig3.eps){width="1\linewidth"}
![The frame dragging function $\omega$ as a function of the radial coordinate normalized by the radius of the star. (The frame dragging function up to the linear order in $\Omega$ is for a uniformly rotating star only a function of $r$ and outside the star behaves as $2J/r^{3}$ [@Hartle].)[]{data-label="fig:teste_omee"}](fig4.eps){width="1\linewidth"}
Perturbation equations
======================
We are working in the Cowling approximation and considering only barotropic perturbations, therefore the fundamental set of perturbation variables is given by $\delta p, \delta u^{\nu}, \delta H^{\nu}$. (For an analysis of the accuracy of the Cowling approximation, but only for f and p modes, see [@Kojima]. For r-modes, the Cowling approximation gives more accurate frequencies, as these modes do not involve large density variations [@Friedman].) The perturbation equations are obtained by perturbing the Euler equations ($i=r,\theta,\phi$): $$\delta \left( \{\delta^{i}_{\nu}+u^{i}u_{\nu}\}T^{\nu\beta}_{~;\beta}\right)=0~$$ and the energy conservation equation $$\delta\left(u_{\nu}T^{\nu\beta}_{~;\beta}\right)=0~$$ together with the perturbed induction equations $$\delta\left\{(u^{\mu}H^{\nu}-H^{\mu}u^{\nu})_{;\nu}\right\}=0.$$ Furthermore one uses as constraints the perturbed ideal MHD equation: $$\delta (H^{\mu}_{;\mu})=\delta \left(H_{\mu}u^{\mu}_{;\nu}u^{\nu}\right)~,$$ and the 4-velocity normalization condition $$\delta (u^{\nu}u_{\nu})=0~,$$ together with the fact that the magnetic field remains perpendicular to the 4-velocity $$\delta (u_{\nu}H^{\nu})=0~.$$ The last two equations can be subsequently used to reduce the variables to the 7 independent variables $\delta p, \delta H^{i}, \delta u^{i}$, with $i=r,\theta,\phi$.
The original form of the perturbation equations
-----------------------------------------------
There are 3 independent components of the perturbed induction equation which turn into:
- the $r$ component: $$\begin{aligned}
e^{-\nu/2}(\delta H^{r}_{,t}+\Omega\delta H^{r}_{,\phi})+(H^{r}_{,\theta}+H^{r}\cot(\theta))\delta u^{\theta}+H^{r}(\delta u^{t}_{,t}+\delta u^{\phi}_{,\phi}+\delta u^{\theta}_{,\theta})-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\left(H^{\theta}_{,\theta}+\cot(\theta)H^{\theta}\right)\delta u^{r}-H^{\theta}\delta u^{r}_{,\theta}=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
- the $\theta$ component:
$$\begin{aligned}
e^{-\nu/2}(\delta H^{\theta}_{,t}+\Omega \delta H^{\theta}_{,\phi})+H^{\theta}(\delta u^{t}_{,t}+\delta u^{r}_{,r}+\delta u^{\phi}_{,\phi})+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\left\{H^{\theta}_{,r}+H^{\theta}\left(\frac{\nu_{,r}+\lambda_{,r}}{2}+\frac{2}{r}\right)\right\}\delta u^{r}-\left(H^{r}_{,r}+\left\{\frac{\nu_{,r}+\lambda_{,r}}{2}+\frac{2}{r}\right\}H^{r}\right)\delta u^{\theta}-H^{r}\delta u^{\theta}_{,r}=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
- the $\phi$ component:
$$\begin{aligned}
e^{-\nu/2}\delta H^{\phi}_{,t}-\frac{\nu_{,r}}{2}H^{r}\delta u^{\phi}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
H^{r}\delta u^{\phi}_{,r}-H^{\theta}\delta u^{\phi}_{,\theta}-\Omega e^{-\nu/2}(\delta H^{t}_{,t}+\delta H^{r}_{,r}+\delta H^{\theta}_{,\theta}+\left(\frac{\lambda_{,r}}{2}+\frac{2}{r}\right)\delta H^{r}+\cot(\theta)\delta H^{\theta})=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\label{induc_phi}\end{aligned}$$
The perturbed energy conservation equation is independent of the magnetic fields and is given as: $$\begin{aligned}
\label{4}
\delta p_{,t}+\Omega\cdot\delta p_{,\phi}+e^{\nu/2}\cdot\Gamma p\left[e^{-\nu}r^{2}\sin^{2}(\theta)\cdot(\Omega-\omega)\cdot\delta u^{\phi}_{,t}+\delta u^{r}_{,r}+\delta u^{\theta}_{,\theta}+\delta u^{\phi}_{,\phi}\right]=~~~~~~~~~~~~~\nonumber\\
=-e^{\nu/2}\cdot\left\{\Gamma p\left[\frac{\nu_{,r}+\lambda_{,r}}{2}+\frac{2}{r}\right]-(p+\epsilon)\frac{\nu_{,r}}{2}\right\}\cdot\delta u^{r}-e^{\nu/2}\cdot\Gamma p\cdot\cot(\theta)\cdot\delta u^{\theta}~.~~~~~~~~\end{aligned}$$
The independent components of the perturbed Euler equation:
- the $r$ component $$\begin{aligned}
(\epsilon+p+H^{r2}e^{\lambda}+H^{\theta 2}r^{2})\{e^{-\nu/2}\Omega\delta u^{r}_{,\phi}+e^{-\nu/2-\lambda}r^{2}\sin^{2}(\theta) [\omega_{,r}+ \left( \nu_{,r}-\frac{2}{r} \right) (\Omega-\omega)]\delta u^{\phi}\}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\left[ (\epsilon+p+H^{r2}e^{\lambda}+H^{\theta 2}r^{2})e^{-\nu/2}-e^{\lambda-\nu/2}H^{r 2} \right] \delta u^{r}_{,t}\frac{e^{-\lambda}\nu_{,r}}{2}(\delta\epsilon+\delta p)=~~~~~~~~~~~~~~~~~~~~~\nonumber\\
=-e^{-\lambda}\delta p_{,r}+H^{r}(\delta H^{r}_{,r}+e^{-\nu/2}r^{2}H^{\theta}\delta u^{\theta}_{,t}+e^{-\nu}r^{2}\sin^{2}(\theta)(\Omega-\omega)\delta H^{\phi}_{,t}+\delta H^{\phi}_{,\phi}+\delta H^{\theta}_{,\theta})+~~~~~~~~~~~~~~~~~~~\\
+\delta H^{\theta} \left( -e^{-\lambda}r^{2}H^{\theta} \left[ \frac{\nu_{,r}}{2}+\frac{2}{r} \right] +\cot(\theta)H^{r} \right) +\delta H^{r}_{,\theta}H^{\theta}-e^{-\lambda}r^{2}H^{\theta}\delta H^{\theta}_{,r}+~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\delta H^{r} H^{r} \left\{ \frac{\lambda_{,r}}{2}+\frac{2}{r} \right\}~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
- the $\theta$ component $$\begin{aligned}
(\epsilon+p+H^{r2}e^{\lambda}+H^{\theta 2}r^{2})\{e^{-\nu/2}\Omega\delta u^{\theta}_{,\phi}-2e^{-\nu/2}\sin(\theta)\cos(\theta)\cdot(\Omega-\omega)\cdot\delta u^{\phi}\}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\left[(\epsilon+p+H^{r2}e^{\lambda}+H^{\theta 2}r^{2})e^{-\nu/2}-e^{-\nu/2}r^{2}H^{\theta 2}\right]\delta u^{\theta}_{,t}=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
=-\frac{1}{r^{2}}\delta p_{,\theta}+\delta H^{\theta}\left[H^{\theta}\cot(\theta)+H^{r}\left(\frac{\nu_{,r}}{2}+\frac{2}{r}\right)\right]+~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
+\delta H^{r}H^{\theta}\left(\frac{\lambda_{,r}}{2}+\frac{2}{r}\right)+H^{r}\delta H^{\theta}_{,r}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+H^{\theta}(e^{\lambda-\nu/2}H^{r}\delta u^{r}_{,t}+e^{-\nu}r^{2}\sin^{2}(\theta)(\Omega-\omega)\delta H^{\phi}_{,t}+\delta H^{r}_{,r}+\delta H^{\theta}_{,\theta}+\delta H^{\phi}_{,\phi})-\frac{e^{\lambda}}{r^{2}}H^{r}\delta H^{r}_{,\theta}.~~~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
- the $\phi$ component
$$\begin{aligned}
(\epsilon+p+H^{r2}e^{\lambda}+H^{\theta 2}r^{2})\{e^{-\nu/2}\delta u^{\phi}_{,t}+e^{-\nu/2}\Omega\delta u^{\phi}_{,\phi}\}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+(\epsilon+p+H^{r2}e^{\lambda}+H^{\theta 2}r^{2})e^{-\nu/2}\left[(\Omega-\omega)_{,r}+\left(\frac{2}{r}-\nu_{,r}\right)(\Omega-\omega)\right]\delta u^{r}+~~~~~~~~~~~~~~~~\nonumber\
\nonumber\\
+2(\epsilon+p+H^{r2}e^{\lambda}+H^{\theta 2}r^{2})e^{-\nu/2}\cot(\theta)
(\Omega-\omega)\delta u^{\theta}=~~~~~~~~~~~~~~~~~\nonumber\\
=-\left[\frac{1}{r^{2}\sin^{2}(\theta)}\delta p_{,\phi}+e^{-\nu}(\Omega-\omega)\delta p_{,t}\right]+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
+H^{r}\delta H^{\phi}_{,r}+H^{\theta}\delta H^{\phi}_{,\theta}-\delta H^{t}\left\{\left(\frac{2\omega}{r}+\omega_{,r}+\frac{\Omega\nu_{,r}}{2}+\nu_{,r}(\Omega-\omega)\right)H^{r}+2\omega\cot(\theta)H^{\theta}\right\}+~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\delta H^{\phi}\left(\left[\frac{\nu_{,r}}{2}+\frac{2}{r}\right]H^{r}+2\cot(\theta)H^{\theta}\right)-\Omega(H^{r}\delta H^{t}_{,r}+H^{\theta}\delta H^{t}_{,\theta})-
~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\frac{e^{\lambda}}{r^{2}\sin^{2}(\theta)}H^{r}\delta H^{r}_{,\phi}-\frac{1}{\sin^{2}(\theta)}H^{\theta}\delta H^{\theta}_{,\phi}-e^{-\nu}(\Omega-\omega)(e^{\lambda}H^{r}\delta H^{r}_{,t}+r^{2}H^{\theta}\delta H^{\theta}_{,t}).~~~~~~~~~~~~\nonumber\end{aligned}$$
The supplementary three constraints are the following:
- the perturbed ideal MHD equation $$\begin{aligned}
e^{-\nu}r^{2}\sin^{2}(\theta)(\Omega-\omega)\delta H^{\phi}_{,t}+\delta H^{r}_{,r}+\delta H^{\theta}_{,\theta}+\delta H^{\phi}_{,\phi}+\left(\frac{\lambda_{,r}}{2}+\frac{2}{r}\right)\delta H^{r}+\cot(\theta)\delta H^{\theta}=H^{r}e^{\lambda-\nu/2}\Omega\delta u^{r}_{,\phi}+~~~~~~~~~~
\nonumber\\
+\delta u^{\phi}r^{2}\sin^{2}(\theta) e^{-\nu/2}\left(H^{r}\left[(\Omega-\omega)\left(\nu_{,r}-\frac{2}{r}\right)+\omega_{,r}\right]-2H^{\theta}\cot(\theta)(\Omega-\omega)\right)+H^{\theta}r^{2}e^{-\nu/2}\Omega\delta u^{\theta}_{,\phi}.~~~~~~~~~~~~~~~
\label{idealMHD}\end{aligned}$$
- perturbed perpendicularity condition of magnetic field and 4-velocity $$e^{\nu/2}\delta H^{t}=e^{\lambda}H^{r}\delta u^{r}+r^{2}H^{\theta}\delta u^{\theta}+e^{-\nu/2}r^{2}\sin^{2}(\theta)(\Omega-\omega)\delta H^{\phi}~,$$
- and the perturbed 4-velocity normalization condition: $$\delta u^{t}=e^{-\nu}r^{2}\sin^{2}(\theta)(\Omega-\omega)\delta u^{\phi}~.$$
Let us mention that the upper equations reduce for the special case of $\Omega=\delta p=\delta u^{r}=\delta u^{\theta}=0$ to the equations shown in [@Kokkotas1].
The form of perturbation equations suitable for the numerical integration
-------------------------------------------------------------------------
For the numerical integration with the 2D Lax-Wendroff scheme we need to obtain the dynamical equations in a form containing only one time derivative in each equation. This is most convenient to achieve by proper linear combinations of the original equations and ommiting the $\sim\Omega^{2}$ terms. The 7 independent variables $\delta H^{i}, \delta u^{i}, \delta p$ , ($i=r,\theta,\phi$), are further transformed into “momentum-like” variables $$\delta \tilde H^{i}=(\epsilon+p)\delta H^{i}, ~~~~ \delta \tilde u^{i}=(\epsilon+p)\delta u^{i}~.$$ (For the introduction of these variables in the Newtonian context see [@Jones].) This transformation is done for the purpose of obtaining a simple boundary condition at the stellar surface as: $$\delta \tilde u^{i}=\delta \tilde H^{i}=\delta p=0.$$
Furthermore we apply regularity conditions at the center of the star and at the rotational axis, together with the correct symmetry conditions at the equatorial plane. (For the details about these conditions see our previous work [@Chirenti].)
Another constraint that has to be fulfilled is the time independent MHD equation, that is checked to be satisfied in each step of the calculation (up to certain determined numerical error). The time independent MHD constraint is obtained from the perturbed ideal MHD equation by subtracting the appropriate multiple of the $\phi$ component of the perturbed induction equation . The constraint reads:
$$\begin{aligned}
e^{-\nu/2}r^{2}\sin^{2}(\theta)(\Omega-\omega)\left(H^{r}\delta u^{\phi}_{,r}+H^{\theta}\delta u^{\phi}_{,\theta}\right)+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\delta H^{r}_{,r}+\delta H^{\theta}_{,\theta}+\delta H^{\phi}_{,\phi}+\left(\frac{\lambda_{,r}}{2}+\frac{2}{r}\right)\delta H^{r}+\cot(\theta)\delta H^{\theta}=H^{r}e^{\lambda-\nu/2}\Omega\delta u^{r}_{,\phi}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\nonumber\\
+\delta u^{\phi}r^{2}\sin^{2}(\theta) e^{-\nu/2}\left(H^{r}\left[(\Omega-\omega)\left(\frac{\nu_{,r}}{2}-\frac{2}{r}\right)+\omega_{,r}\right]+2H^{\theta}\cot(\theta)(\omega-\Omega)\right)+H^{\theta}r^{2}e^{-\nu/2}\Omega\delta u^{\theta}_{,\phi}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
Our integration domain occupies only the first quadrant, since we take advantage of the symmetries at the equatorial plane. Our numerical grid typically has $50 \times 50$ points in $r \times \theta$, where $r$ varies in $[0,R]$ and $\theta$ in $[0,\pi/2]$. We take usually 10.000-50.000 time steps in the evolution of the equations, depending on the rotating rate and, consequently, the frequency of the r-mode. In each time evolution we observe at least several periods of oscillation of the perturbations. We point out here that our time evolutions were so far stable, and we did not see signs of the hydromagnetic instability observed in [@LanJones] for axial-led perturbations in Newtonian gravity. The investigation of this issue in our relativistic treatment is left for a future work.
We limited the maximum rotation rate considered here by $\Omega = 0.27\Omega_K$, motivated by the results of [@Passamonti0], where they see corrections of the order $\Omega^3$ in the r-mode frequencies for $\Omega \gtrsim 0.3 \Omega_K$. We also limited our minimum rotation rate at $\Omega = 0.7\Omega_K$ because of numerical reasons, as already stated in the introduction (lower rotation rates would demand longer simulations). In the next section we discuss the numerical limits on the magnetic field.
The final equations obtained via the linear combinations and redefinitions of variables can be found in the appendix \[A1\]. In appendix \[A2\] we present simplified equations obtained from the equations in appendix \[A1\] by neglecting the coefficients of the order $\Omega H^{i}$ and $H^{2}$. These simplified equations can be useful for sufficiently weak magnetic fields and sufficiently slow rotation rates, where neglecting the $\Omega H^{i}$ and $H^{2}$ terms could be justified. We used also these equations to compute the r-mode frequencies and the results are compared in the figure \[fig:rmode1\].
The results for the r-mode frequencies (for $\ell=m=2$)
=======================================================
The r-modes ($\ell=m=2$) were computed using the equations -. We solve the system of equations with a 2D Lax-Wendroff scheme with non-constant coefficients [@Mitchell]. We refer the reader to a previous work [@Chirenti] for further details on the numerical setup used for obtaining the r-mode frequency and eigenfunction. (In [@Chirenti] it was used for non-magnetized and differentially rotating stars.)
We calculated the r-modes first for zero magnetic field. The dependence of the r-mode frequencies on the rotation parameter for the non-magnetized field case is shown in figure \[fig:rmode0\]. We compared results with [@Ruoff] for the star with $\Omega/\Omega_K = 0.27$, and found that our results match with less than 3% error. (For more results on r-modes of non-magnetized stars see also the papers [@Yoshida-Lee; @Yoshida-Yoshida; @Gaertig; @Font; @Kastaun].)
In figures \[fig:rmode1\] and \[fig:rmode2\] we present r-modes as a function of magnetic field. For comparison we present in the figure \[fig:rmode1\] also r-modes calculated via the simplified equations from the appendix \[A2\] for the star with $\Omega/\Omega_K = 0.27$. The approximation of the simplified equations is shown to break down in this case at the value of magnetic field around $2.5\times 10^{15}$ G, while the results obtained from the full equations *seem* to breakup at a larger magnetic field around $3.5\times 10^{15}$ G. For larger magnetic fields than $4\times 10^{15}$ G (where we do not entirely trust our results), we still see that the r-mode disappears completely due to the growth of another mode (possibly an Alfvén mode). We believe that the breakdown in the behavior of the r-mode frequencies is caused by the growth of this other mode and the subsequent deformation of the r-mode. This is consistent with the expectations based on the results of [@Rezzolla1; @Rezzolla2; @Rezzolla3; @Lander].
As can be seen in the figure \[fig:rmode2\], the r-mode frequencies change very little when one turns on the magnetic field. This is consistent with the observation of [@Lee] for Newtonian stars. The change of the frequencies is more pronounced for smaller values of the rotation parameter. For $\Omega/\Omega_{K}=0.07$ and magnetic field $3.3\times 10^{15}$ G, the r-mode frequency changes by a little less than 4%. In case of larger rotation $\Omega/\Omega_{K}=0.17$ the same value of magnetic field changes the r-mode frequency by 1%.
Even though the variations are small, one can still clearly observe from Fig.\[fig:rmode2\] the behavior of the frequencies with respect to the increasing magnetic field. Such a behavior seems to have remarkable features: the r-mode frequencies behave for sufficiently large $\Omega$ ($\Omega\sim 0.17 \Omega_{K}$) as $\sim B^{2}$, whereas for a smaller value of $\Omega$, ($\Omega/\Omega_{K}\approx 0.07$), the behavior of the frequencies with respect to the magnetic field is given as $\sim B^{4}$. Let us note that the $\sim B^{4}$ dependence was observed for the r-modes of the spherical shell in [@Abbassi].
In figures and we show the plots of the r-mode eigenfunctions for all the variables. (The eigenfunctions are shown in the plane given by the rotational axis and a perpendicular axis to the rotation.) The complicated interplay between r-modes and magnetic fields is more visible in the $\delta H^{\phi}$ eigenfunction, where we can see a sort of double peak. This happens for all rotation rates, and it is more pronounced for larger magnetic fields. We believe that this shows the deformation of the r-mode eigenfunction caused by other modes excited for large enough magnetic fields. For more details on the procedure used for extracting the eigenfunctions, see again [@Chirenti].
![The plot represents the r-mode frequency as a function of the rotation parameter for zero magnetic field.[]{data-label="fig:rmode0"}](fig5.eps){width="1\linewidth"}
![The plot shows the relative change of the r-mode frequency as a function of the absolute value of the magnetic field. It compares the r-mode frequency computed using the equations -, (the equations with the linearized background coefficients), (the red line), with the r-mode obtained via the non-simplified equations - (the green line). The solid line shows a quadratic fit done with the points before the breakup.[]{data-label="fig:rmode1"}](fig6.eps){width="1\linewidth"}
![The relative change of the r-mode frequency as a function of the absolute value of the magnetic field, for different values of the rotation parameter. (Computed with the full, non-simplified equations.) The lines show the quadratic (and quartic, for $\Omega/\Omega_K = 0.07$) fits.[]{data-label="fig:rmode2"}](fig7.eps){width="1\linewidth"}
{width="0.7\linewidth"}
{width="0.315\linewidth"} {width="0.315\linewidth"} {width="0.315\linewidth"}
The r-mode instability and gravitational radiation
==================================================
The r-mode instability growth times (for $\ell=m=2$) are calculated by using the usual quadrupole formula (for the details see for example [@Andersson]). The characteristic timescale is calculated from the equation: $$\frac{dE}{dt}=-\frac{2E}{t_{gw}}=-\frac{\int \rho|\delta v|^{2}dV}{t_{gw}}$$ with $\delta v^{i}=\delta u^{i}/u^{t}$, $i=r,\theta,\phi$. The energy time derivative is calculated from the quadrupole formula as: $$\frac{dE}{dt}|_{gw}=-(\sigma+m\Omega)\sum_{\ell=2}^{\infty}N_{\ell}\sigma^{2\ell+1}
(|\delta D_{\ell m}|^{2}+|\delta J_{\ell m}|^{2}),$$ with $$N_{\ell}=4\pi\frac{(\ell+1)(\ell+2)}{\ell(\ell-1)[(2\ell+1)!!]^{2}},$$ where $\delta D_{\ell m}$ and $\delta J_{\ell m}$ are the mass and the current multipoles defined as in [@Andersson]. (We use both mass and current multipoles, but because the r-modes involve only a perturbed velocity field, to the lowest order in $\Omega$, gravitational radiation through current multipoles dominates over that produced by mass multipoles [@Andersson; @Lindblom2].) In particular the multipoles can be expressed as: $$\delta D_{\ell m}=\int \delta \rho r^{\ell}Y^{*}_{\ell m}dV$$ and $$\delta J_{\ell m}=2\frac{\ell}{\ell+1}\int r^{\ell}(\rho\delta v^{i}+\delta \rho v^{i})Y^{B*}_{i~ \ell m}dV.$$ (Here $Y_{\ell m}$ and $Y^{B i}_{\ell m}$ are the multipoles defined in [@Thorne].)
We were able to fit the function $t_{gw}$ for zero magnetic fields as a functions of the rotation period $P$ as $$t_{gw}=\tau_{gw}(P/1\rm{ms})^{p_{gw}} \rm{s},$$ with the dimensionless parameters $\tau_{gw}$ and $p_{gw}$ taking the values $\tau_{gw}=13.65$ and $p_{gw}=5.83$. We compare our values for $\tau_{gw}$ and $p_{gw}$ with the values obtained in [@Lindblom; @Andersson] in table \[tab:par\] (see also figure \[fig:logtau\]).
our result ref. [@Lindblom2] ref. [@Andersson]
------------- ------------ ------------------- -------------------
$\tau_{gw}$ 13.65 18.91 20.83
$p_{gw}$ 5.83 6 5.93
: The $\tau_{gw}$ and $p_{gw}$ parameters for the case of zero magnetic fields. We are comparing our results with [@Andersson; @Lindblom2] where the calculations were done for the Newtonian polytropes with stellar parameters close to ours. We obtained 27-34 % faster emission of gravitational waves compared to the Newtonian setting.[]{data-label="tab:par"}
![The (normalized) logarithmic timescale for the r-mode instability growth as a function of the (normalized) logarithm of the period of rotation (for zero magnetic field). We compare here our results with the results of [@Andersson; @Lindblom2].[]{data-label="fig:logtau"}](fig10.eps){width="1\linewidth"}
The instability growth time scale $t_{gw}$ relative change due to the magnetic field is shown in figure . We can see that the relative change becomes positive for lower values of magnetic fields (increasing the growth time and slowing down the emission of gravitational waves) and then becomes negative for larger values of the magnetic field (with the opposite effect), causing a relative change of up to $\sim$ 5%. Similarly to the r-mode frequencies, the relative effect of the magnetic field is more pronounced for the lower rotation rates. However, to estimate the amount of gravitational waves emitted and the window of the instability we would need to calculate the viscosity damping rates and that is left for future work.
![The relative difference in the timescale for the r-mode instability growth as a function of the magnetic field.[]{data-label="fig:tauB1"}](fig11.eps){width="1\linewidth"}
Conclusions
===========
We presented here a model for a rotating magnetized star, in which we neglect the distortion of both the geometry and the fluid by a *dipolar* magnetic field. We derived the full perturbation equations in the Cowling approximation. After solving the 2D time evolution problem with a Lax-Wendroff method, we computed the r-mode frequencies using the Fourier spectrum of the solution and we were able to extract the eigenfunctions of the perturbations. The frequencies and eigenfunctions of the r-mode were then used to calculate the growth time scale due to gravitational radiation, using the Newtonian quadrupole formalism.
We found that the effect of the magnetic field on the frequencies is very small, (up to 5% for the lowest rotation rates). For lower rotation rates the frequencies follow a $B^{4}$ dependence and, for higher rotation rates, a $B^{2}$ dependence. The effect on the r-mode growth time $t_{gw}$ indicates a faster emission of gravitational waves, compared to the Newtonian non-magnetized calculations of [@Lindblom2; @Andersson]. We found that $t_{gw}$ is more significantly affected by the presence of general relativity ($\sim 30$%) and less significantly by the presence of magnetic field (up to $\sim 5$%).
Our results indicate that the relative effects of the magnetic field are more pronounced for more slowly rotating stars. Therefore it could be possible that they achieve higher values for magnetars, that have rotation periods about 1000 times larger than the ones considered here (due to numerical limitations). However, it is not trivial to estimate how large these corrections would be, given the complicated dependence on both rotation and magnetic field of the solutions.
The effect of viscosity will play of course a key role in determining the actual instability window and is left for the future work. A more realistic description of the star would also need to include work with realistic equations of state and a stellar crust, together with considering the backreaction from the production of toroidal magnetic field [@Cuofano]. This is also left for the future.
The authors are especially thankful to Luciano Rezzolla and Shin’ichirou Yoshida for many useful discussions in different stages of this project. This work was supported by FAPESP and the Max Planck Society.
The form of equations suitable for the numerical code {#A1}
=====================================================
The final dynamical equations for the numerical evolution are given as:
$$\begin{aligned}
\label{one1}
(p+\epsilon) \left[ \Omega-(\Omega-\omega)\frac{\Gamma p}{\epsilon+p+H^{2}} \right] \delta p_{,\phi}-r^{2}\sin^{2}(\theta)(\Omega-\omega)\frac{\Gamma p}{\epsilon+p+H^{2}} \left[ -H^{r}\delta \tilde H^{\phi}_{,r}-H^{\theta}\delta\tilde H^{\phi}_{,\theta}-\right.~~~~~~~~~~~~~~~~\nonumber\\
\left. \left( H^{r} \left[ \nu_{,r}+\frac{\nu_{,r}}{2}\frac{\epsilon+p}{\Gamma p}+\frac{2}{r} \right] +2\cot(\theta)H^{\theta} \right) \delta\tilde H^{\phi}+\frac{e^{\lambda}H^{r}}{r^{2}\sin^{2}(\theta)}\delta \tilde H^{r}_{,\phi}+\frac{H^{\theta}}{\sin^{2}(\theta)}\delta\tilde H^{\theta}_{,\phi} \right] +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
+(p+\epsilon)\delta p_{,t}+e^{\nu/2}\Gamma p(\delta\tilde u^{r}_{,r}+\delta\tilde u^{\theta}_{,\theta}+\delta\tilde u^{\phi}_{,\phi})+e^{\nu/2}\Gamma p\left(\frac{\lambda_{,r}}{2}+\nu_{,r}+\frac{2}{r}\right)\delta \tilde u^{r}+~~~~~~~~~~~~\nonumber\\
+e^{\nu/2}\Gamma p\cot(\theta)\delta\tilde u^{\theta}=0~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{two1}
\left[ e^{-\nu/2}\Omega-\frac{e^{\lambda}H^{r 2}(\Omega-\omega)}{\epsilon+p+H^{2}} \right] \delta\tilde H^{r}_{,\phi}-\frac{H^{r}r^{2}\sin^{2}(\theta)(\Omega-\omega)}{\epsilon+p+H^{2}} \left[ \frac{p+\epsilon}{r^{2}\sin^{2}(\theta)}\delta p_{,\phi}-H^{r}\delta\tilde H^{\phi}_{,r}-H^{\theta}\delta\tilde H^{\phi}_{,\theta}- \right.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\left. \left( H^{r} \left[ \frac{\nu_{,r}}{2}\frac{p+\epsilon}{\Gamma p}+\nu_{,r}+\frac{2}{r} \right] +2\cot(\theta)H^{\theta} \right) \delta\tilde H^{\phi}+\frac{H^{\theta}}{\sin^{2}(\theta)}\delta\tilde H^{\theta}_{,\phi}\right] +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
+e^{-\nu/2}\delta \tilde H^{r}_{,t}+[H^{r}_{,\theta}+H^{r}\cot(\theta)]\delta\tilde u^{\theta}+H^{r}(\delta\tilde u^{\phi}_{,\phi}+\delta\tilde u^{\theta}_{,\theta})-(H^{\theta}_{,\theta}+\cot(\theta)H^{\theta})\delta\tilde u^{r}-H^{\theta}\delta\tilde u^{r}_{,\theta}=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber \end{aligned}$$
$$\begin{aligned}
\label{three1}
\left[\Omega e^{-\nu/2}-\frac{(\Omega-\omega)H^{\theta 2}r^{2}}{\epsilon+p+H^{2}}\right]\delta\tilde H^{\theta}_{,\phi}-\frac{(\Omega-\omega)r^{2}\sin^{2}(\theta)H^{\theta}}{\epsilon+p+H^{2}}\left[ \frac{p+\epsilon}{r^{2}\sin^{2}(\theta)}\delta p_{,\phi}-H^{r}\delta\tilde H^{\phi}_{,r}-H^{\theta}\delta\tilde H^{\phi}_{,\theta}-\right.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\left.\left(H^{r}\left[\frac{\nu_{,r}}{2}\frac{p+\epsilon}{\Gamma p}+\nu_{,r}+\frac{2}{r}\right]+2\cot(\theta)H^{\theta}\right)\delta\tilde H^{\phi}+\frac{e^{\lambda} H^{r}}{r^{2}\sin^{2}(\theta)}\delta\tilde H^{r}_{,\phi}\right]+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
+e^{-\nu/2}\delta\tilde H^{\theta}_{,t}+H^{\theta}(\delta\tilde u^{r}_{,r}+\delta\tilde u^{\phi}_{,\phi})+\left[H^{\theta}_{,r}+H^{\theta}\left(\frac{\nu_{,r}\frac{p+\epsilon}{\Gamma p}+\lambda_{,r}}{2}+\nu_{,r}+\frac{2}{r}\right)\right]\delta\tilde u^{r}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\left[H^{r}_{,r}+H^{r}\left(\frac{\nu_{,r}\frac{p+\epsilon}{\Gamma p}+\lambda_{,r}}{2}+\nu_{,r}+\frac{2}{r}\right)\right]\delta\tilde u^{\theta}-H^{r}\delta\tilde u^{\theta}_{,r}=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{four1}
\left[ \Omega(\epsilon+p+H^{2})-(\Omega-\omega)(H^{2}+\Gamma p) \right] \delta\tilde u^{\phi}_{,\phi}+~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+ \left[ -(\epsilon+p+r^{2}H^{\theta 2})\omega_{,r}+2\omega e^{\lambda}H^{r} \left( \frac{H^{r}}{r}+\cot(\theta)H^{\theta} \right) + \right.~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\Omega e^{\lambda} \left( H^{r 2} \left[ \lambda_{,r}+\frac{\nu_{,r}}{2}\frac{(\Gamma+1)p+\epsilon}{\Gamma p} \right] +H^{r}H^{r}_{,r}+H^{\theta}H^{r}_{,\theta} \right) +~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+(\Omega-\omega) \left\{ (\epsilon+p+H^{2}) \left( \frac{2}{r}-\nu_{,r} \right) - \left( \frac{\lambda_{,r}}{2}+\frac{2}{r} \right) (H^{2}+\Gamma p)-\nu_{,r}(r^{2}H^{\theta 2}+\Gamma p)- \right.~~~~~~~~\nonumber\\
\frac{\nu_{,r}}{2}r^{2}H^{\theta 2}\frac{p+\epsilon}{\Gamma p}- \left. \left.\left( e^{\lambda}H^{r}H^{r}_{,r}+r^{2}H^{\theta}H^{\theta}_{,\theta} \right) +\nu_{,r}H^{r 2}e^{\lambda} \right\} \right] \delta\tilde u^{r}+~~~~~~~~~~~~~~~~~\nonumber\\
+\left[ (\Omega-\omega)\left( \{2(\epsilon+p)+H^{2}-\Gamma p\}\cot(\theta)+r^{2}H^{r}H^{\theta}\frac{\nu_{,r}}{2}\frac{(3\Gamma +1)p+\epsilon}{\Gamma p} \right)\right.+~~~~~~~~\\
\left.+\Omega r^{2}H^{r}H^{\theta}\frac{\nu_{,r}}{2} \left( 1+\frac{\epsilon+p}{\Gamma p} \right) +\omega r^{2} \left[ 2H^{\theta} \left( \frac{2H^{r}}{r}+H^{\theta}\cot(\theta) \right) +H^{r}H^{\theta}_{,r}+H^{\theta}H^{\theta}_{,\theta} \right] +r^{2}H^{\theta}H^{r}\omega_{,r} \right] \delta\tilde u^{\theta}+~~~~~~~~~~~~~\nonumber\\
\left[(2\Omega-\omega)r^{2}H^{\theta 2}-(\Omega-\omega)\left(H^{2}+\Gamma p\right)\right]\delta\tilde u^{\theta}_{,\theta}+(2\Omega-\omega)e^{\lambda}H^{r}H^{\theta}\delta\tilde u^{r}_{,\theta}+~~~~~~~~~~~~~~\nonumber\\
+(2\Omega-\omega)r^{2}H^{r}H^{\theta}\delta\tilde u^{\theta}_{,r}+~~~~~~~~~~~~~~~~~~~\nonumber\\
+\left[ (2\Omega-\omega) e^{\lambda}H^{r 2}-(\Omega-\omega) \left( \Gamma p+H^{2} \right) \right] \delta\tilde u^{r}_{,r}+
(\epsilon+p+H^{2})\delta\tilde u^{\phi}_{,t}+e^{\nu/2} \left[ \frac{p+\epsilon}{r^{2}\sin^{2}(\theta)}\delta p_{,\phi}-\right.~~~~~~~~~~~~\nonumber\\
H^{r}\delta\tilde H^{\phi}_{,r}-H^{\theta}\delta\tilde H^{\phi}_{,\theta}- \left. \left[ \left( \frac{\nu_{,r}}{2}+\frac{2}{r} \right) H^{r}+2\cot(\theta)H^{\theta} \right] \delta\tilde H^{\phi}+\frac{e^{\lambda}H^{r}}{r^{2}\sin^{2}(\theta)}\delta\tilde H^{r}_{,\phi}+\frac{H^{\theta}}{\sin^{2}(\theta)}\delta\tilde H^{\theta}_{,\phi} \right] =0.~~~~~~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{five1}
\Omega (\epsilon+p+H^{2})e^{-\nu/2}\delta u^{r}_{,\phi}+\frac{(\epsilon+p+H^{2})e^{-\nu/2}\Omega r^{2}H^{r}H^{\theta}}{\epsilon+p+e^{\lambda}H^{r 2}}\delta u^{\theta}_{,\phi}+~~~~~~~~~~~~~~\nonumber\\
+e^{-\nu/2}r^{2}\sin^{2}(\theta)(\epsilon+p+H^{2}) \left[ e^{-\lambda} \left( \omega_{,r}+ \left\{ \nu_{,r}-\frac{2}{r} \right\} (\Omega-\omega) \right) \right. -~~~~~~~~~~\nonumber\\
\frac{\Omega-\omega}{\epsilon+p+e^{\lambda}H^{r 2}} \left. \left\{ \frac{\nu_{,r}H^{r 2}}{2}\frac{(2\Gamma+1)p+\epsilon}{\Gamma p}+2H^{r}H^{\theta}\cot(\theta) \right\} \right] \delta\tilde u^{\phi}-~~~~~~~~\nonumber\\
\frac{H^{r}r^{2}\sin^{2}(\theta)e^{-\nu/2}(\Omega-\omega)(\epsilon+p+H^{2})}{\epsilon+p+e^{\lambda}H^{r 2}} \left[ H^{r}\delta\tilde u^{\phi}_{,r}+H^{\theta}\delta\tilde u^{\phi}_{,\theta} \right] +~~~~~~~~~\nonumber\\
+\frac{e^{-\nu/2}(\epsilon+p)(\epsilon+p+H^{2})}{\epsilon+p+e^{\lambda}H^{r 2}}\delta\tilde u^{r}_{,t}+
\frac{e^{-\lambda}(p+\epsilon)\nu_{,r}}{2}\left[1+\frac{\epsilon+p}{\Gamma p}\right]\delta p+~~~~~~~~~~\\
+e^{-\lambda}(p+\epsilon)\delta p_{,r}-\frac{H^{r}(\epsilon+p+H^{2})}{\epsilon+p+e^{\lambda}H^{r 2}}(\delta\tilde H^{r}_{,r}+\delta\tilde H^{\phi}_{,\phi}+\delta\tilde H^{\theta}_{,\theta})-\frac{H^{\theta}(\epsilon+p)}{\epsilon+p+e^{\lambda}H^{r 2}}\delta\tilde H^{r}_{,\theta}+~~~~~~~~\nonumber\\
\left[e^{-\lambda}r^{2}H^{\theta}\left(\frac{\nu_{,r}}{2}+\frac{2}{r}\right)-\cot(\theta)H^{r}-\frac{r^{2}H^{\theta}\left\{H^{r}\left(H^{\theta}\cot(\theta)+H^{r}\left[
\frac{\nu_{r}}{2}+\frac{2}{r}\right]\right)+(\epsilon+p)e^{-\lambda}\frac{\nu_{,r}}{2}\frac{(\Gamma+1)p+\epsilon}{\Gamma p}\right\}}{\epsilon+p+e^{\lambda}H^{r 2}}\right] \delta\tilde H^{\theta}+~~~~~~~~\nonumber\\
+\frac{H^{\theta}e^{-\lambda}r^{2}(\epsilon+p)}{\epsilon+p+e^{\lambda}H^{r 2}}\delta\tilde H^{\theta}_{,r}-\frac{H^{r}(\epsilon+p+H^{2})}{\epsilon+p+e^{\lambda}H^{r 2}}\left(\frac{\nu_{,r}+\lambda_{,r}}{2}+\frac{\nu_{,r}}{2}\frac{p+\epsilon}{\Gamma p}+\frac{2}{r}\right)\delta\tilde H^{r}+\frac{H^{r}H^{\theta}(p+\epsilon)}{\epsilon+p+e^{\lambda}H^{r 2}}\delta p_{,\theta}=0~~~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{six1}
(\epsilon+p+H^{2})e^{-\nu/2}\Omega\delta\tilde u^{\theta}_{,\phi}+\frac{(\epsilon+p+H^{2})e^{-\nu/2}\Omega e^{\lambda}H^{\theta}H^{r}}{\epsilon+p+r^{2}H^{\theta 2}} \delta\tilde u^{r}_{,\phi}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
(\epsilon+p+H^{2})\sin^{2}(\theta)e^{-\nu/2} \left[ 2\cot(\theta)(\Omega-\omega)+\frac{H^{r}H^{\theta}r^{2}(\Omega-\omega)\nu_{,r}}{2(\epsilon+p+r^{2}H^{\theta 2})}\frac{(2\Gamma+1)p+\epsilon}{\Gamma p}- \right.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\frac{H^{\theta}H^{r}r^{2}}{\epsilon+p+r^{2}H^{\theta 2}} \left. \left( \omega_{,r}+ \left\{ \nu_{,r}-\frac{2}{r} \right\}(\Omega-\omega) \right) \right] \delta\tilde u^{\phi}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\frac{H^{\theta}e^{-\nu/2}r^{2}\sin^{2}(\theta)(\Omega-\omega)(\epsilon+p+H^{2})}{\epsilon+p+r^{2}H^{\theta 2}} \left[ H^{r}\delta\tilde u^{\phi}_{,r}+H^{\theta}\delta\tilde u^{\phi}_{,\theta} \right]+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\frac{e^{-\nu/2}(\epsilon+p)(\epsilon+p+H^{2})}{\epsilon+p+r^{2}H^{\theta 2}}\delta\tilde u^{\theta}_{,t}+\frac{\nu_{,r}H^{r}H^{\theta}(p+\epsilon)}{2(\epsilon+p+r^{2}H^{\theta 2})}\left[1+\frac{\epsilon+p}{\Gamma p}\right]\delta p+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
+\frac{H^{\theta}H^{r}(p+\epsilon)}{\epsilon+p+r^{2}H^{\theta 2}}\delta p_{,r}-\frac{H^{\theta}(\epsilon+p+H^{2})}{\epsilon+p+r^{2}H^{\theta 2}}(\delta\tilde H^{r}_{,r}+\delta\tilde H^{\phi}_{,\phi}+\delta\tilde H^{\theta}_{,\theta})+\frac{e^{\lambda}H^{r}(\epsilon+p)}{r^{2}(\epsilon+p+r^{2}H^{\theta 2})}\delta\tilde H^{r}_{,\theta}+~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\left[ \frac{H^{r}}{\epsilon+p+r^{2}H^{\theta 2}} \left( r^{2}H^{\theta 2} \left\{ \frac{\nu_{,r}}{2}+\frac{2}{r} \right\} -e^{\lambda}\cot(\theta)H^{r}H^{\theta}-(\epsilon+p)\frac{\nu_{,r}}{2}\frac{(\Gamma+1)p+\epsilon}{\Gamma p} \right) \right. -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
H^{\theta}\cot(\theta)-H^{r} \left. \left\{\frac{\nu_{,r}}{2}+\frac{2}{r}\right\}\right]\delta\tilde H^{\theta}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\frac{H^{r}(\epsilon+p)}{\epsilon+p+r^{2}H^{\theta 2}}\delta\tilde H^{\theta}_{,r}-\frac{H^{\theta}(\epsilon+p+H^{2})}{\epsilon+p+r^{2}H^{\theta 2}}\left[\frac{\nu_{,r}+\lambda_{,r}}{2}+\frac{\nu_{,r}}{2}\frac{p+\epsilon}{\Gamma p}+\frac{2}{r}\right]\delta\tilde H^{r}+\frac{p+\epsilon}{r^{2}}\delta p_{,\theta}=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{seven1}
\Omega\left[\epsilon+p-H^{2}\right]\delta\tilde H^{\phi}_{,\phi}+\Omega H^{r}\frac{\nu_{,r}(p+\epsilon)}{2}\left(1+\frac{\epsilon+p}{\Gamma p}\right)\delta p+\Omega H^{r}(p+\epsilon)\delta p_{,r}-~~~~~~~~~\nonumber\\
\Omega H^{2}\left(\delta\tilde H^{r}_{,r}+\delta\tilde H^{\theta}_{,\theta}\right)-\Omega \left(\frac{\nu_{,r}+\lambda_{,r}}{2}+\frac{\nu_{,r}}{2}\frac{p+\epsilon}{\Gamma p}+\frac{2}{r}\right)H^{2}\delta\tilde H^{r}-\Omega \cot(\theta)H^{2}\delta\tilde H^{\theta}-~~~~~~~~~~\\
\Omega H^{\theta}(p+\epsilon)\delta p_{,\theta}+(\epsilon+p)e^{\nu/2}\left[ e^{-\nu/2}\delta\tilde H^{\phi}_{,t}-\left[\frac{1}{2}\frac{p+\epsilon}{\Gamma p}+1\right]\nu_{,r}H^{r}\delta\tilde u^{\phi}-H^{r}\delta\tilde u^{\phi}_{,r}-H^{\theta}\delta\tilde u^{\phi}_{,\theta}\right]=0.~~~~~~~~~~~\nonumber\end{aligned}$$
The equations with the linearized coefficients {#A2}
==============================================
$$\begin{aligned}
\label{one2}
\left[ \Omega(\epsilon+p)-(\Omega-\omega)\Gamma p \right] \delta p_{,\phi} +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+ (\epsilon+p)\delta p_{,t}+e^{\nu/2}\Gamma p(\delta\tilde u^{r}_{,r}+\delta\tilde u^{\theta}_{,\theta}+\delta\tilde u^{\phi}_{,\phi})+e^{\nu/2}\Gamma p \left( \frac{\lambda_{,r}}{2}+\nu_{,r}+\frac{2}{r} \right) \delta \tilde u^{r}+ ~~~~~~~~~~~~\\
+ e^{\nu/2}\Gamma p\cot(\theta)\delta\tilde u^{\theta} =0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{two2}
e^{-\nu/2}\Omega\delta\tilde H^{r}_{,\phi}+e^{-\nu/2}\delta \tilde H^{r}_{,t}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+[H^{r}_{,\theta}+H^{r}\cot(\theta)]\delta\tilde u^{\theta}+H^{r}(\delta\tilde u^{\phi}_{,\phi}+\delta\tilde u^{\theta}_{,\theta})-(H^{\theta}_{,\theta}+\cot(\theta)H^{\theta})\delta\tilde u^{r}-H^{\theta}\delta\tilde u^{r}_{,\theta}=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
$$\begin{aligned}
\label{three2}
\Omega e^{-\nu/2}\delta\tilde H^{\theta}_{,\phi}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+e^{-\nu/2}\delta\tilde H^{\theta}_{,t}+H^{\theta}(\delta\tilde u^{r}_{,r}+\delta\tilde u^{\phi}_{,\phi})+\left[H^{\theta}_{,r}+H^{\theta}\left(\nu_{,r}\left\{\frac{p+\epsilon}{2\Gamma p}+1\right\}+\frac{\lambda_{,r}}{2}+\frac{2}{r}\right)\right]\delta\tilde u^{r}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\left[H^{r}_{,r}+H^{r}\left(\nu_{,r}\left\{\frac{p+\epsilon}{2\Gamma p}+1\right\}+\frac{\lambda_{,r}}{2}+\frac{2}{r}\right)\right]\delta\tilde u^{\theta}-H^{r}\delta\tilde u^{\theta}_{,r}=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber \end{aligned}$$
$$\begin{aligned}
\label{four2}
\left[ \Omega(\epsilon+p)-(\Omega-\omega)\Gamma p) \right] \delta\tilde u^{\phi}_{,\phi}
+ ~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+ \left[ -(\epsilon+p)\omega_{,r}+(\Omega-\omega) \left\{ (\epsilon+p) \left( \frac{2}{r}-\nu_{,r} \right) - \left( \frac{\lambda_{,r}}{2}+\nu_{,r}+\frac{2}{r} \right) \Gamma p \right\} \right] \delta\tilde u^{r}+~~~~~~~~~~~~\\
+ (\Omega-\omega) \{2(\epsilon+p)-\Gamma p\}\cot(\theta) \delta\tilde u^{\theta}-
(\Omega-\omega) \Gamma p \delta\tilde u^{\theta}_{,\theta}-
(\Omega-\omega) \Gamma p \delta\tilde u^{r}_{,r}+
(\epsilon+p)\delta\tilde u^{\phi}_{,t}+e^{\nu/2} \left[ \frac{\epsilon+p}{r^{2}\sin^{2}(\theta)}\delta p_{,\phi}- \right. ~\nonumber\\
H^{r}\delta\tilde H^{\phi}_{,r}-H^{\theta}\delta\tilde H^{\phi}_{,\theta}- \left. \left[ \left( \frac{\nu_{,r}}{2}+\frac{2}{r} \right) H^{r}+2\cot(\theta)H^{\theta} \right] \delta\tilde H^{\phi}+\frac{e^{\lambda}H^{r}}{r^{2}\sin^{2}(\theta)}\delta\tilde H^{r}_{,\phi}+\frac{H^{\theta}}{\sin^{2}(\theta)}\delta\tilde H^{\theta}_{,\phi} \right] =0.~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{five2}
\Omega (\epsilon+p)e^{-\nu/2}\delta u^{r}_{,\phi}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+e^{-\nu/2}r^{2}\sin^{2}(\theta) e^{-\lambda}(\epsilon+p) \left( \omega_{,r}+ \left\{ \nu_{,r}-\frac{2}{r} \right\} (\Omega-\omega) \right) \delta\tilde u^{\phi}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+e^{-\nu/2}(\epsilon+p)\delta\tilde u^{r}_{,t}+
(\epsilon+p)\frac{e^{-\lambda}\nu_{,r}}{2} \left[ \frac{\epsilon+p}{\Gamma p} +1 \right] \delta p+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+(\epsilon+p)e^{-\lambda}\delta p_{,r}-H^{r}(\delta\tilde H^{r}_{,r}+\delta\tilde H^{\phi}_{,\phi}+\delta\tilde H^{\theta}_{,\theta})-H^{\theta}\delta\tilde H^{r}_{,\theta}+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
+ \left[ e^{-\lambda}r^{2}H^{\theta} \left( \frac{\nu_{,r}}{2}+\frac{2}{r} \right) -\cot(\theta)H^{r}-r^{2}H^{\theta}e^{-\lambda}\frac{\nu_{,r}}{2}\left\{\frac{\epsilon+p}{\Gamma p}+1\right\} \right] \delta\tilde H^{\theta}+~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+H^{\theta}e^{-\lambda}r^{2}\delta\tilde H^{\theta}_{,r}-H^{r} \left( \frac{\lambda_{,r}}{2}+\frac{\nu_{,r}}{2}\left\{\frac{\epsilon+p}{\Gamma p}+1\right\}+\frac{2}{r} \right) \delta\tilde H^{r}=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{six2}
(\epsilon+p)e^{-\nu/2}\Omega\delta\tilde u^{\theta}_{,\phi}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\sin^{2}(\theta)e^{-\nu/2} (\epsilon+p)2\cot(\theta)(\Omega-\omega) \delta\tilde u^{\phi}
+e^{-\nu/2}(\epsilon+p)\delta\tilde u^{\theta}_{,t}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
H^{\theta}(\delta\tilde H^{r}_{,r}+\delta\tilde H^{\phi}_{,\phi}+\delta\tilde H^{\theta}_{,\theta})+\frac{e^{\lambda}H^{r}}{r^{2}}\delta\tilde H^{r}_{,\theta}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\left[ H^{r}\frac{\nu_{,r}}{2}\left\{\frac{p+\epsilon}{\Gamma p}+1\right\}+H^{\theta}\cot(\theta)+H^{r} \left\{ \frac{\nu_{,r}}{2}+\frac{2}{r} \right\} \right] \delta\tilde H^{\theta}-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
H^{r}\delta\tilde H^{\theta}_{,r}-H^{\theta} \left[ \frac{\lambda_{,r}}{2}+\frac{\nu_{,r}}{2}\left\{\frac{p+\epsilon}{\Gamma p}+1\right\}+\frac{2}{r} \right] \delta\tilde H^{r}+\frac{\epsilon+p}{r^{2}}\delta p_{,\theta}=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
$$\begin{aligned}
\label{seven2}
\Omega\delta\tilde H^{\phi}_{,\phi}+\delta\tilde H^{\phi}_{,t}-e^{\nu/2} \left( \left[ \frac{1}{2}\frac{p+\epsilon}{\Gamma p}+1 \right] \nu_{,r}H^{r}\delta\tilde u^{\phi}+H^{r}\delta\tilde u^{\phi}_{,r}+H^{\theta}\delta\tilde u^{\phi}_{,\theta}\right)=0.~~~~~~~~~~~~~~~~~~~~\end{aligned}$$
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We demonstrate [*in situ*]{} fluorescence detection of $^7$Li atoms in a 1D optical lattice with single atom precision. Even though illuminated lithium atoms tend to boil out, when the lattice is deep, molasses beams without extra cooling retain the atoms while producing sufficient fluorescent photons for detection. When the depth of the potential well at an antinode is 2.4 mK, an atom remains trapped for 30 s while scattering probe photons at the rate of $1.7 \times 10^5$ s$^{-1}$. We propose a simple model that describes the dependence of the lifetime of an atom on well depth. When the number of trapped atoms is reduced, a clear stepwise change is observed in integrated fluorescence, indicating the detection of a single atom. At a photon-collecting efficiency of only 1.3% owing to small numerical aperture, the presence or absence of an atom is determined within 300 ms with an error of less than $5 \times 10^{-4}$.'
author:
- 'Hyok Sang Han, Hyun Gyung Lee, Seokchan Yoon, and D. Cho[^1]'
title: 'Fluorescence detection of single lithium atoms in an optical lattice using Doppler-cooling beams'
---
INTRODUCTION
============
As both internal and motional states of trapped atoms are controlled more and more precisely for their quantum manipulation, it has also become very important to observe those atoms [*in situ*]{} and individually. Efforts to image fluorescence from single trapped atoms started with those in a magneto-optical trap (MOT) [@Kimble]. It was extended to single atoms in a 1D optical lattice with a site-specific resolution using a diffraction-limited imaging system and careful offline analysis [@Meschede2009]. These efforts culminated when the individual sites of a 2D optical lattice were imaged using objective lenses with high numerical aperture (NA) [@Greiner2009; @Kuhr2010]. The 2D version is known as a quantum-gas microscope, and it was developed primarily to prepare samples for and read out results from quantum simulation of interacting particles. Initially, these experiments were performed using either $^{133}$Cs [@Kimble; @Meschede2009] or $^{87}$Rb [@Greiner2009; @Kuhr2010] atoms because molasses beams can be used to simultaneously image and cool heavy alkali-metal atoms. In recent years, 2D imaging techniques have been extended to fermionic atoms such as $^{6}$Li [@Greiner2015] and $^{40}$K [@Zwierlein2015; @Kuhr2015], which are better proxies for strongly-interacting electrons. However, light atoms tend to boil out before scattering sufficient photons for imaging because of their large recoil energy and poor polarization gradient cooling. To overcome this difficulty, Raman sideband cooling [@Greiner2015; @Zwierlein2015] and electromagnetically-induced-transparency (EIT) cooling [@Kuhr2015] have been employed. This complicates the apparatus and imaging process. In addition, an exposure time of longer than 1 s is required because Raman cooling and EIT cooling rely on putting atoms in low-lying dark states. The energy-lowering stimulated processes are interlaced with brief optical-pumping stages, during which photons are harvested.
In the present work, using only Doppler-cooling beams, we demonstrate [*in situ*]{} imaging of single $^7$Li atoms in a 1D optical lattice with single atom precision. Lattice depth $U_0$ turns out to be a critical parameter; above $U_0 = 1.5$ mK, there is an abrupt increase in the number of photons scattered by an atom before it escapes the lattice. A simple model of evaporation followed by Doppler cooling explains this phenomenon. Although the nearest sites are not resolved in our detection because of small NA of 0.22, our approach can be combined with either a large-NA system or spectroscopic identification of individual sites [@MeschedeQR] to facilitate quantum gas microscopy of light atoms. In our measurement at $U_0$ = 2.4 mK, the presence or absence of an atom can be determined with 99.95% probability using a 300-ms exposure time, despite the low photon-collecting efficiency.
APPARATUS
=========
A double MOT fed by a Zeeman slower is used to load lithium atoms to an optical lattice [@magic-pol]. The 1D lattice is formed in an octagonal glass chamber by focusing and retro-reflecting a Gaussian beam. See Fig. 1. The wavelength $\lambda_L$ is 1064 nm and the $e^{-2}$ intensity radius at the focus is 14 $\mu$m. Mode matching of the reflected beam is optimized by maximizing the power coupled back to the optical fiber that delivers the lattice beam. When incident power is 1.3 W, the depth $U_0$ at an antinode is 1 mK or 830$E_R$, where $E_R = h^2/2m\lambda_L^2$. A home-built ytterbium-doped fiber laser provides the single-frequency lattice beam. MOT beams with a radius of 1.6 mm are used as imaging beams. The fluorescence from lattice atoms is collected by an objective lens with NA of 0.22 and refocused to an electron-multiplying charge-coupled device (EMCCD) with unit magnification. NA of 0.22 corresponds to a photon-collecting efficiency of 1.3% and the EMCCD has a quantum efficiency of 90% at 671 nm. With further reduction by 0.9 owing to scattering and diffraction losses, one out of 100 fluorescent photons are detected [@MOT-imaging].
EXPERIMENT
==========
Our aim is to detect the fluorescence from lattice-bound atoms with single atom precision. We collect data from a region of interest (ROI), which consists of 3 by 3 pixels of the EMCCD. Each pixel measures $16 \times 16$ $\mu\rm{m}^2$, and the ROI corresponds to 100 sites at the center of the lattice. In the first part of the experiment, we attempt to determine the conditions that allow [*[in situ]{}*]{} imaging of atoms using Doppler-cooling beams. In the second part, we reduce the number of atoms to observe stepwise change in integrated fluorescence.
Typically, we load a thousand atoms to the lattice using the MOT with low-power beams of 150 $\mu$W in each direction for both trapping and repumping. An anti-Helmholtz coil is turned off and the MOT beams optimized for imaging are illuminated. For $^7$Li, the scalar polarizabilities of the $2S_{1/2}$ and $2P_{3/2}$ states at $\lambda_L$ = 1064 nm are -270 and -167 in atomic units, respectively [@JYKim; @Safranova]. The $2P_{3/2}$ state has a negative polarizability owing to its coupling to the $3S_{1/2}$ and $3D$ states, and it is a trappable state. Nevertheless, the $|2S_{1/2}, F=2 \,\rangle$ $\rightarrow$ $|2P_{3/2}, F=3 \,\rangle$ transition suffers both frequency shift and inhomogeneous broadening; the lattice beam causes a blue shift of 8 MHz in the $D2$ transition when $U_0$ is 1 mK. Detuning of the MOT trap beam is adjusted for a given $U_0$ to maximize the number of photons $N_c$ scattered by an atom before it escapes from the lattice. The repump beam is stabilized to the $|2S_{1/2}, F=1 \,\rangle$ $\rightarrow$ $|2P_{3/2}, F=2 \,\rangle$ transition with fixed detuning. Illumination of the near-resonant beams results in sites with either one or no atoms owing to photoassociative losses. We use approximately 50 atoms trapped at the central 100 sites for the fluorescence detection.
The decay time constant $\tau_{FL}$ of the fluorescence signal versus $U_0$ is plotted in Fig. 2. For this measurement, we use 25 $\mu$W in each direction for the trap and repump beams. $\tau_{FL}$ is related to the number of scattered photons as $N_c = R \tau_{FL}$, where $R$ is the photon scattering rate. Because of the inhomogeneous broadening, $R$ depends on $U_0$ and it is on the order of $10^5$ s$^{-1}$. When $U_0$ is below 1.4 mK, Li atoms escape the lattice almost instantaneously because of large recoil energy and lack of polarization gradient cooling, as noted earlier. However, above $U_0 = 1.5$ mK, $\tau_{FL}$ increases steeply and at $U_0 = 2.4$ mK it reaches 30 s, which is the vacuum-limited trap lifetime $\tau_{VC}$. This behavior may be explained by assuming that the independent processes of boil out and Doppler cooling alternate while atoms are imaged. Suppose that the lattice atoms are Doppler cooled to temperature $T_D$ and those with energy higher than $U_0$ are ejected. The ejected fraction $p$ is $(1+\theta+\theta^2/2)e^{-\theta}$ with $\theta = U_0/k_B T_D$ when the 3D harmonic potential well is isotropic [@Ketterele1995]. In the proposed model, this ejection occurs instantaneously resulting in a truncated Maxwell-Boltzmann distribution. This is followed by another round of Doppler cooling, which lasts for $\tau_{MB}$ to reestablish the distribution at $T_D$. This model leads to the following relation: $$\frac{1}{\tau_{FL}} = \frac{1}{\tau_{VC}} -\frac{\ln (1-p)}{\tau_{MB}}.$$ The fitting curve shown in Fig. 2 is for $T_D = 120$ $\mu$K, $\tau_{MB} = 300$ $\mu$s, and $\tau_{VC} = 30$ s. We note that the Doppler cooling limit for lithium is 140 $\mu$K. In the experiment on 2D imaging of $^6$Li in Ref. [@Greiner2015], $U_0$ at the antinode is estimated to be 0.8 mK from the quoted trap frequencies, and the Raman sideband cooling was indispensable for imaging.
In order to demonstrate the single-atom precision, we load only one or two atoms to the central part of the lattice by reducing the loading time and waiting until the ROI signal decreases beyond a threshold. We maintain $U_0$ at 2.4 mK and the imaging power at 25 $\mu$W. We continue to use the ROI of 3 by 3 pixels, but we take data only when the center pixel is the brightest so that the imaged atoms are within $\pm$10 sites from the minimum spot. The Rayleigh range of the lattice beam is 580 $\mu$m, and these sites can be considered identical.
Figure 3 (a) shows the evolution of the ROI signal as the lattice loses one atom at a time, starting with two atoms. The exposure time for each data point is 300 ms. In this measurement, the single atom stays trapped for 25 s. The count rate of the fluorescence from a single atom is 1.7 kHz and that from the scattered imaging beam is 3.8 kHz. We note that the fluorescence count rate corresponds to the photon scattering rate $R= 1.7 \times 10^5$ s$^{-1}$ and it is only 6% of that from a single atom in a MOT with the same imaging beam intensities [@MOT-imaging]. We attribute this reduction to the inhomogeneous broadening of the $D2$ transition in the lattice. The histogram of the time-series data is shown in Fig. 3 (b). Using the Gaussian fits, we obtain an average and standard deviation of 1180 and 35, respectively, for the zero-atom case, and 1660 and 95, respectively, for the one-atom case. As was already apparent in Fig. 3 (a), the fluorescence signal exhibits considerable noise, which was not observed in the MOT fluorescence [@MOT-imaging]. We conclude that this noise is caused by fluctuation in lattice-beam parameters such as power and mode matching. Even with the noise, when we use a proper threshold, which is 1340 counts in this case, both the sensitivity of deciding correctly the presence of a single atom and the specificity of deciding correctly the absence of an atom are larger than 99.95% for an exposure time of 300 ms. With the large NA of the quantum gas microscopes, the exposure time can be reduced by more than an order of magnitude.
Conclusion
==========
In this work, we address the problems in imaging or detecting lithium atoms in an optical lattice. When the potential well is sufficiently deep, simple Doppler cooling enables us to obtain a fluorescence signal with single-atom precision while the atoms remain trapped. We plan to combine this capability with the site-specificity obtained by using an applied magnetic field gradient to achieve site-resolved [*in situ*]{} imaging for our quantum manipulation experiment on single lithium atoms.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was supported by a grant to the Atomic Interferometer Research Laboratory for National Defense funded by the Defense Acquisition Program Administration and Agency for Defense Development in Korea.
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[^1]: e-mail address:[[email protected]]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present an eccentric precessing gas disk model designed to study the variable circumstellar absorption features detected for WD 1145+017, a metal polluted white dwarf with an actively disintegrating asteroid around it. This model, inspired by one recently proposed by Cauley et al., calculates explicitly the gas opacity for any predetermined physical conditions in the disk, predicting the strength and shape of all absorption features, from the UV to the optical, at any given phase of the precession cycle. The successes and failures of this simple model provide valuable insight on the physical characteristics of the gas surrounding the star, notably its composition, temperature and density. This eccentric disk model also highlights the need for supplementary components, most likely circular rings, in order to explain the presence of zero velocity absorption as well as highly ionized Si IV lines. We find that a precession period of $4.6\pm0.3$ yrs can successfully reproduce the shape of the velocity profile observed at most epochs from April 2015 to January 2018, although minor discrepancies at certain times indicate that the assumed geometric configuration may not be optimal yet. Finally, we show that our model can quantitatively explain the change in morphology of the circumstellar features during transiting events.'
author:
- 'M. Fortin-Archambault'
- 'P. Dufour'
- 'S. Xu (许偲艺)'
bibliography:
- 'references.bib'
title: Modeling of the Variable Circumstellar Absorption Features of WD 1145+017
---
[UTF8]{}[gbsn]{}
Introduction
============
It has become widely accepted now that the source of the heavy elements observed in the photosphere of some white dwarfs is accretion from tidally disrupted planets or planetesimals [see @JuraYoung2014 and references therein]. The discovery by @Vanderburg2015 of an ongoing disintegration event around the white dwarf WD 1145+017 has recently solidified the confidence in this scenario, opening at the same time a new window onto our understanding of this process.
This unique system shows some remarkable characteristics. Six stable periods between 4.5 and 5 hours were detected in the original K2 light curve [@Vanderburg2015]. These periods have been interpreted as being the signature of several smaller objects that have broken off from one main body orbiting the star. These fragments are thought to drift away from the main orbit and slowly disintegrate into dust and gas before being accreted onto the surface of the star [@Rappaport2016; @Veras2017]. The system is evolving constantly, with changes in the light curve on timescales ranging from minutes to months [@Gaensicke2016; @Rappaport2016; @Rappaport2018; @Gary2017]. At some point, there was even one transit deeper than 10% observed every 3.6 hours on average [@Croll2017]. As can be expected for such a system, the photosphere is highly contaminated with heavy elements [@Xu2016], and it displays the typical infrared excess from the presence of a dust disk [@Vanderburg2015].
High resolution spectroscopic data also uncovered the presence of wide asymmetric circumstellar features with linewidths of $\sim 300$ km/s [@Xu2016]. Interestingly, the circumstellar lines have evolved from being mostly red-shifted to blue-shifted within about two years [@Redfield2017]. Recently, @Cauley2018 proposed an eccentric misaligned precessing gas disk model to explain the evolution of the circumstellar features. While this model was successful in following the evolution of the Fe II 5316 ${\text{\normalfont\oldAA}}$ region over almost a two year period, it could not provide strong insights on the physical properties of the gas (abundances, temperature and density) nor provide the expected absorption profile across the whole electromagnetic spectrum.
The goal of this paper is thus to explicitly compute the circumstellar absorption profile of WD 1145+017 at any time, from the UV to the optical, for any given geometrical/physical structure given as input. As a first step, we fully explore the configuration presented by @Cauley2018 in order to test its validity, limits and finally propose modifications and avenues of research for future studies.
The observational data we used are presented in § \[sec:observations\]. An updated analysis of the photospheric chemical composition of WD 1145+017 is presented in § \[sec:photosphere\] while § \[sec:model\] describes in detail the theoretical framework for our modeling of the precessing gas disk. Our results, including an analysis of the physical properties of the disk, its precession period, the need for additional components and the circumstellar absorption behavior during transits are presented in § \[sec:results\]. Finally, we present a summary of our findings and conclusions in § \[sec:conclusion\].
Observations {#sec:observations}
============
This study uses numerous intermediate/high resolution (R = 14,000-40,000) spectra from Keck I telescope taken with the High Resolution Echelle Spectrometer (HIRESb, HIRESr) and the Echellette Spectrograph and Imager (ESI) instruments. We also use data taken with the VLT (X-SHOOTER), HST (COS, March 28, 2016 data) as well as a light curve (taken simultaneously to Keck data) obtained from the University of Arizona’s 61 inch telescope. In total, there are 17 epochs of spectroscopic observations between April 2015 and May 2018. Details concerning these observations and the data reduction procedures have already been described at length in @Xu2019 [see also their Table 2 and 3].
Photospheric Abundances {#sec:photosphere}
=======================
The first detailed photospheric abundance analysis of WD 1145+017 was presented by [@Xu2016]. This analysis, however, assumed the atmospheric parameters determined in @Vanderburg2015, namely [$T_{\rm eff}~$]{}= 15,900K and [$\log{g}$ ]{}= 8.0. Since this effective temperature determination was based on models that included an approximate amount of heavy elements (the circumstellar contamination of many lines was not known at the time) and assuming [$\log{g}$ ]{}= 8.0 [parallax measurement only became available with $Gaia$ DR2 data release, @Gaia2018], the atmospheric parameters and abundance analysis need to be revisited. Using the same method and grids presented in detail in [@Coutu2019], we now obtain [$T_{\rm eff}~$]{}= $14,500 \pm 900$ K and [$\log{g}$ ]{}= $8.11 \pm 0.02$. Using those new parameters, we then calculate grids of synthetic spectra for each element and determine chemical abundances using, as in [@Xu2016], lines not contaminated by circumstellar absorption in HIRES data taken in March and April 2016 (2016.02.03 and 2016.03.02). While the strength and position of the circumstellar features change significantly on timescales of minutes to months [@Redfield2017], we find no variation in abundances (from non-contaminated absorption lines) when determined from spectra taken at any other epochs compared to those determined from the March and April 2016 data.
![Oxygen absorption in the HIRESr data (2016.04.01). The top panel shows the O I triplet contaminated by circumstellar absorption. The bottom panel shows the non-contaminated O I 8446 ${\text{\normalfont\oldAA}}$ line that is used for abundance determination ($\log{\mathrm{O}/\mathrm{He}} = -5.12$, blue line)[]{data-label="fig:Oxygen_newabn"}](f1.png){width="\columnwidth"}
@Xu2016 reported conflicting results regarding oxygen, as the 2 available lines they detected in low resolution spectra indicated very different abundances ($\log{\mathrm{O}/\mathrm{He}} = -3.7$ for the O I 7775 ${\text{\normalfont\oldAA}}$ triplet and $\log{\mathrm{O}/\mathrm{He}} = -4.5$ for the O I 8446 ${\text{\normalfont\oldAA}}$ line, see their Figure 1). This discrepancy was concerning because it was originally thought that the high abundance from the triplet could not be explained by circumstellar absorption since these lines originate from energy levels $\sim 9$ eV above the ground level. However, there is now evidence suggesting the presence of a hot component in the system (for example, the presence of Si IV lines in the UV, see section \[subsec:Circular ring components\]). Moreover, the new higher resolution data from HIRESr ($2016.03.02$ epoch) clearly show that the red portion of the O I triplet is contaminated by circumstellar absorption (see Figure \[fig:Oxygen\_newabn\]). Hence, we decided to use only the O I $8446$ ${\text{\normalfont\oldAA}}$ line for the photospheric abundance determination and our updated value is now $\log{\mathrm{O}/\mathrm{He}} = -5.12 \pm 0.35$. With this updated oxygen abundance measurement, the mass fraction of oxygen, which was extremely high ($\sim$ 60%) according to [@Xu2016], is now compatible with the expectation from accretion of bulk Earth-like material (see Figure \[fig:Abundance\_4\]).
![Mass fraction of elements accreted in the photosphere of WD 1145+017. []{data-label="fig:Abundance_4"}](f2.pdf){width="\linewidth"}
Using HST data in the UV (2016.03.28), we are also able to obtain limits for N and S. Carbon is also possibly detected (although the lines are severely contaminated by circumstellar absorption, the presence of core features with the correct radial velocity suggests that the carbon is indeed photospheric, see Figure \[fig:Carbon\_43vs75\]).
![Possible detection of photospheric carbon (see core of strongest predicted lines) blended with circumstellar absorption features. Unidentified tickmarks indicate the position of iron lines. []{data-label="fig:Carbon_43vs75"}](f3.png){width="1.05\columnwidth"}
Our final photospheric abundance determinations (or limits) for each element are presented in Table \[tab:new\_abundances\]. Finally, we measure that the radial velocity from photospheric lines in the optical (too much contamination from circumstellar disk for reliable measurements in the UV ) is $ \rm 43 \pm 2 \: km/s$, which corresponds, once we remove the gravitational contribution of $\rm 35.3 \pm 0.8 \: km/s$, to a proper motion of $\rm 8 \pm 3 \: km/s$.
\[tab:new\_abundances\]
-------------------- ---------------------- --------------- ---------------------- -- -- --
[$T_{\rm eff}~$]{} 14,500 $\pm$ 900 $K$
[$\log{g}$ ]{} 8.11 $\pm$ 0.02
Ion log n(Z)/n(He) M$^{a}$ $\dot{M}$$^{b}$
(10$^{20}$ g) ( 10$^8$ g s$^{-1}$)
H -5.0 $\pm$ 0.20 ... ...
C $\sim -7.5$ 1.9 0.0837
N $<-7.0$ $<$6.9 $<$0.339
O -5.12 $\pm$ 0.35 600 29.9
Mg -5.91 $\pm$ 0.20 146 7.19
Al -6.89 $\pm$ 0.20 17 0.875
Si -5.89 $\pm$ 0.20 179 8.97
S $<-7.0$ $<$16 $<$0.963
Ca -7.0 $\pm$ 0.20 20 0.151
Ti -8.57 $\pm$ 0.20 0.64 0.0528
V: -9.25:$\pm$ 0.25 0.14 0.0118
Cr -7.92 $\pm$ 0.40 3.1 0.248
Mn -8.57 $\pm$ 0.20 0.73 0.0586
Fe -5.61 $\pm$ 0.20 680 52.4
Ni -7.02 $\pm$ 0.30 28 0.207
total ... 1497 100
-------------------- ---------------------- --------------- ---------------------- -- -- --
: WD 1145+017 atmospheric parameters and accretion rates
[**Notes.**]{}\
$^{a}$ Current mass in the white dwarf’s convection zone [@Dufour2017].\
$^{b}$ Accretion rate assuming a steady state.\
Gas Disk Model {#sec:model}
==============
Theoretical Framework {#sec:theoretical}
---------------------
In order to account for the varying asymmetrical circumstellar absorption features seen in the spectra of WD 1145+017, a simple elliptical precessing gas disk model similar to that proposed by [@Cauley2018] is developed. Here, we aim to explicitly compute all circumstellar absorption, from UV to optical at various times in the cycle assuming the same geometric configuration, but based on detailed opacity calculations for the physical conditions present in the disk. The disk is presented edge-on with a non-negligible width covering about half of the star surface. Note that, as noted by [@Cauley2018], this configuration is probably not the only one able to reproduce the circumstellar absorption features as the reality may certainly be more complex. However, by exploring in detail the successes and failures of this simple configuration, we can get a much deeper physical insight that will be valuable for future studies of this system. The remainder of this section provides details on the opacity calculations, the gas disk construction and free parameters that are used to adjust it to match the various epochs of observation obtained in the last 4 years.
Disk Configuration {#sec:Geometry}
------------------
Following [@Cauley2018], the disk is constructed with 14 eccentric misaligned rings in the $xy$ plan. There are 3 parameters governing the position of the rings and they each vary linearly between their value for the innermost and outermost ring. The perihelion distances vary between $15.93 \, R_{*}$ and $23.64 \, R_{*}$, the eccentricities vary between 0.25 and 0.30 and finally, we assume a $78^{\circ}$ shift between the apsidal lines of the innermost and outermost rings. Each ring has a radial width of $0.5 \, R_*$ with $R_*=0.0118 \, R_{\odot}$, as determined from our updated stellar parameters. The 14 rings are confocal and the common focus is where the star is positioned. An example of the configuration at an arbitrary moment in the precession cycle is shown in Figure \[fig:config\_geo\_t0\]. The model disk is also precessing. The precession period is equal for the 14 rings, so the rotation of the configuration is solid. The absorption from this disk can be calculated at any time $t$ (or angle of rotation of the rings) to predict the shape of the circumstellar features (see section \[subsec:Precession period\]).
![Configuration of the 14 eccentric gas rings. The dashed blue lines represent the apsidal line of the innermost and outermost rings, the red line is the line of sight and the black circle is the position of the star. []{data-label="fig:config_geo_t0"}](f4.png){width="\columnwidth"}
Opacity Calculation {#sec:opacity}
-------------------
The disk is positioned edge-on in our line of sight. To compute the total circumstellar absorption, we divide the surface of the star that is covered by the disk in a 20$\times$20 grid in the $yz$ plan. Each grid box is composed of 14 layers representing the 14 gas rings, each with an opacity specifically computed from the physical parameters at that point in the disk. We compute the radiation transmitted through the disk at each line of sight by attenuating the specific intensity from the star at that angle with an exponential $e^{-\sum_{\rm ring}^{} \kappa_{\nu} d \rm m}$, where $\kappa_{\nu}$ is the opacity for the column mass $dm$ of the ring (Doppler shifted appropriately using the velocity profile, see below) and the sum is done over the 14 ring layers. We thus do not consider emission or multiple scatterings through the rings in this simple model. The spectrum is then obtained, as usual, by integrating the specific intensities over the surface of the star (limb darkening is thus automatically taken into account in the procedure).
To compute the opacities, we need to attribute physical parameters everywhere in the disk (i.e we need the temperature, the mass density and the chemical composition to describe the gas in each cell). We first assume that the chemical composition of the disk is the same as that of the photosphere, an assumption that appears to be excellent to the first order (see section \[subsec:Density and Temperature\]). We exclude, however, hydrogen and helium since we see no circumstellar absorption from these elements (moreover, they are not expected to be a significant part of a tidally disrupted asteroid).
For the values of the mass density, we use vertical and radial structures for the disk similar to those proposed by [@Cauley2018] assuming the disk temperature approximation described in [@Melis2010]. The density structure is computed from the input value of the density of the innermost ring in the middle plane of the disk ($z=0$), $\rho_0(r_{in})$, which is a free parameter of the model. The middle plane values for the following rings are given by
$$\rho_0(r)=\rho_0(r_{in}) \times \left (\frac{r_{in}}{r} \right ) ^2
\label{eq:density_rprofile}$$
where $r$ is perihelion distance of the rings and $r_{in}$ represents the value for this innermost ring. From this, we compute the vertical scale of each ring using
$$\rho(z)=\rho_0 \; e^{-z^2/H^2}
\label{eq:density_zprofile}$$
where $\rho_0$ is density at $z=0$ for each ring and $H$ is the scale height given by [@Melis2010],
$$H=\left ( \frac{2 \, k \, T_{gas} \, D^3}{G \, M_* \, \mu} \right ) ^{1/2} ,
\label{eq:scale_height}$$
where $T_{gas}$ is the gas temperature, $D$ the distance from the star, $M_*$ the mass of the star ($0.656 \: M_{\odot}$) and $\mu=2.4 \times 10^{-23}$ g. The density profile is symmetric around the $z=0$ plane and is also equal around a ring.
For simplicity, we first use a constant temperature throughout the disk. While this approximation allows a fair representation, at certain epochs, for most circumstellar absorption features in the optical, this simple model is clearly insufficient to reproduce all features, indicating that a more complex temperature structure is needed. We thus also experiment with some temperature gradients in the gas (see section \[subsec:gradient\]).
Using the physical structure for the disk described above (temperature, density and abundances), we compute the opacity of the rings in each grid box using the public atmospheric code package <span style="font-variant:small-caps;">tlusty/synspec</span>[^1]. The synthetic spectrum code <span style="font-variant:small-caps;">synspec</span> has a special mode called “iron curtain”, which takes the temperature, electronic density and chemical composition as input to compute the opacity of an uniform slab of gas with those parameters. The electronic density, an a priori unknown quantity, is obtained by first running the atmospheric code <span style="font-variant:small-caps;">tlusty</span> with a one layer input having the temperature, mass density and abundances fixed to the desired values.
![Radial velocity profile of each ring across the line of sight for the $t=0$ yrs configuration. The colors for each ring matches the colors used in Figure \[fig:config\_geo\_t0\]. []{data-label="fig:velocity_profile_t0"}](f5.png){width="\columnwidth"}
Once the opacity in each grid cell has been calculated, Doppler velocity shift due to the revolution of the gas in the rings is applied. The line of sight velocity is simply the keplerien orbital velocity ($x$ component)
$$v=\sqrt{G M_* \left ( \frac{2}{r} - \frac{1}{a} \right )} .
\label{eq:kepler_velocity}$$
where $r$ is the distance from the star, and $a$ the semi-major axis of the ellipse.
The line of sight velocity profile for the geometric configuration displayed earlier is presented in Figure \[fig:velocity\_profile\_t0\]. We can then see that for this particular configuration, the profile is almost entirely red-shifted. As the disk precesses, different ranges of line of sight velocities can be obtained, producing, as observed for WD 1145+017, circumstellar feature shifts from $\sim$+200 km/s to -200 km/s.
Finally, although only a small component, we also include the gravitational redshift, $v_{grav}=\frac {G M_*} {c \, r}$ (see Figure \[fig:redshift\_grav\]). This additional shift slightly changes the shape of the circumstellar features as the velocity shift varies along a ring and is different for each ring.
![Gravitational redshift as a function of distance from WD 1145+017 (black curve). The blue area represents the position of the star and the red area the position of the disk. The full red lines show the position of the perihelion for the innermost and outermost rings, and the dashed red lines show the aphelion distances. []{data-label="fig:redshift_grav"}](f6.png){width="\columnwidth"}
Figure \[fig:article\_precessioncycle\] shows an example of the resulting circumstellar absorption for the 5316 ${\text{\normalfont\oldAA}}$ FeII line during the entire precession cycle of the disk. In the next sections, we compare in detail the predictions of this simple model with absorption features observed at different epochs, from both optical and UV data.
\[fig:article\_precessioncycle\]
Results {#sec:results}
=======
Density and Temperature {#subsec:Density and Temperature}
-----------------------
\[fig:article\_raiesbien6000\]
We first determine the combination of free parameters that best represent the shape and depth of the circumstellar features. We start with the HIRESb optical spectra for the 2016.04.01 epoch which presents almost entirely red-shifted features. We first find the time (angle) in the precession cycle that best reproduces the overall shape of the absorption features. This time is henceforth used as the zero point of the precession period. Once the disk is well positioned, we try different combinations of central density and gas temperature in order to reproduce the depth of all circumstellar features simultaneously. We test wide ranges from 3000 K to 30,000 K (in step of 1000 K) and $1 \times 10^{-7}$ $\rm g/cm^3$ to $5 \times 10^{-5}$ $\rm g/cm^3$. We find that the combination that best reproduces the shape of a fair amount of the circumstellar features is a central density of $(6.0 \pm 1.0) \times 10^{-6}$ $\rm g/cm^3$ and a temperature of $6000 \pm 1000 $ $\rm K$, which we will use in what follows (the quoted uncertainties are conservative values based on clearly inferior fit using adjacent grid points in our parameter space). We note that the total integrated mass for the eccentric rings assuming this structure is $2.1\times 10^{16}$ g (somewhere between the mass of Uranus and Neptune’s rings, or about a millionth of the mass of Saturn’s rings), a value that should only be considered a rough order of magnitude estimate, given all the approximations involved. Nevertheless, this is significantly smaller than the total amount of material present in the star’s photosphere (see Table \[tab:new\_abundances\]). Also, the lifetime of the gas, given by the estimated mass of the gas disk divided by the total accretion rate (derived assuming steady state), would be much less than a year, indicating that the gas must be replenished completely on a very short timescale [see also @Xu2016 for a similar discussion]. However, since it has been recently proposed that convective overshooting may also have a significant effect on the diffusion coefficients and mass of the mixed region in helium-rich white dwarfs @Cunningham2019, it is probably best not to acquiesce literally yet to interpretations based on 1D convection zone models.
In Figure \[fig:article\_raiesbien6000\], we show examples of lines that are reproduced very well with this simple model. It is interesting to note that the depths for several elements (Mg, Ca, Cr, Ti, Fe, Ni) are all reproduced simultaneously with this model, indicating that the assumed input abundances, which are the the ones found from the photospheric analysis, are an excellent first order approximation.
There are, however, regions where the model does not match as well, or predicts lines that are not observed (see Fig \[fig:article\_raiesmauvaisesdeuxtemps\]). While these lines appear to require a different gas temperature to be reproduced, we found no constant temperature model that is satisfactory.
\[fig:article\_raiesmauvaisesdeuxtemps\]
We can also compare the prediction from our model with data taken in the UV with HST only a few days before the HIRESb data. Since the number of transitions is much greater in the UV, the circumstellar and stellar features cannot be isolated and they practically form a superposed continuum of lines. Nevertheless, Figure \[fig:article\_raiesbien13000\] shows that the $6000$ $\rm K$ model reproduces most of the absorption quite successfully with the exception of a few notable lines arising from higher energy levels. In particular, we note the presence of two strong Si IV lines ($\sim 1394$ and $1402{\text{\normalfont\oldAA}}$), indicating that a much higher temperature is needed [note that similar circumstellar Si IV lines were also detected in two hotter polluted white dwafs, PG 0843+516 and SDSS 1228+1040, see @Gaensicke2012]. Increasing the temperature to $13,000$ $\rm K$ can produce such strong lines (it also increases the depth of the circumstellar carbon component) without causing too many changes in the parts that were previously well reproduced with the $6000$ $\rm K$ model (see blue line in Figure \[fig:article\_raiesbien13000\]).
\[fig:article\_raiesbien13000\]
Although the hotter disk model accounts for these particular lines, it also predicts features where there are none, for example the Fe line we see around $1396$ ${\text{\normalfont\oldAA}}$. This indicates that our constant temperature approximation is not sufficient to simultaneously explain all the circumstellar features.
We also note the presence of unaccounted for symmetric components not red-shifted (the strong lines near $\sim 1394$ and $1402{\text{\normalfont\oldAA}}$ that are not reproduced by either temperature models). These features most probably originate from a circular, or very low eccentricity, gas ring situated further out than the eccentric disk. Our analysis of these “zero shift/circular components” is presented in section \[subsec:Circular ring components\].
Disk Structure with Temperature Gradient {#subsec:gradient}
----------------------------------------
\[fig:article\_raiesmauvaisesstructuretemp\]
\[fig:figure9\_struc\_ld\]
The computation structure of our model allows us to look at the opacity in each grid box in order to track where particular lines are formed and thus assess what the physical parameters needed to form them are. We find that the denser central parts of the disk need to be hotter to form lines from higher energy levels, and the outer parts of the disk need to be cooler to create the many less energetic lines. We thus use this information to test different simple temperature structures that have hotter central regions, and cooler outer regions.
\[fig:figure8\_struc\_20160401\]
As described in section \[sec:Geometry\], the vertical width of the disk is divided in 20 boxes that have a density structure symmetric around the central plane. We explore temperature structures constructed the same way, that is they are also symmetric around the central plane, but for the sake of simplicity we keep this vertical structure constant throughout the 14 rings (thus 10 vertical layers, from the central to the outer temperature values). We also simply assume, for this little experiment, a linear variation of the temperature. We attribute a $13,000$ $\rm K$ value for the central layer and the 9 others are scaled linearly with a step of $1100$ $\rm K$, bringing the outer temperature to $3100$ $\rm K$. These parameters were obtained by testing different central temperatures and steps and comparing the resulting models with certain features in the data that are affected by these changes. The central density is kept at $6 \times 10^{-6}$ $\rm g/cm^3$, and since the vertical density structure of each ring depends on the gas temperature (see scale height Eq. \[eq:scale\_height\]), we use the mean value of the structure, which in this case is $8050$ $\rm K$. We note that this simple temperature structure is not physical, and that a detailed analysis using a temperature structure obtained from first principle, which is beyond the scope of this paper, should eventually be performed. Nevertheless, this experiment is very useful to gain some valuable insights on the physical conditions present in the disk.
Results from this non-constant temperature model are shown in Figure \[fig:article\_raiesmauvaisesstructuretemp\] and \[fig:figure9\_struc\_ld\] for the same regions presented Figure \[fig:article\_raiesmauvaisesdeuxtemps\] and \[fig:article\_raiesbien13000\]. Although the agreement is much better than the constant temperature models, there are still a few problems. In particular, the lines from higher lower energy level displayed in Figure \[fig:article\_raiesbien13000\] are still not deep enough even with the much hotter central regions of the disk. As for the regions that were well reproduced with the $6000$ $\rm K$ model in Figure \[fig:article\_raiesbien6000\], the change in temperature structure has little impact on those lines, with the exception of a few (for example, magnesium near 3840 Å, see Figure \[fig:figure8\_struc\_20160401\]). This simple model also produces an O I absorption feature similar to what is observed in the $\sim$ 7775 Å triplet (see Figure \[fig:Oxygen\_newabn\]), although it is predicted too deep.
These shortcomings are not surprising given that the structure we implemented is very simple and probably not realistic. The vertical temperature scale is probably not linear and there must also be a radial temperature scale. We also do not take into account that the rings are eccentric, meaning there are parts of the same ring that are closer to the star, and thus probably hotter, than others. It is also possible that the chemical composition of the disk is not exactly the same as that determined in the photosphere, thus affecting the relative depths of some circumstellar features from different elements. Nevertheless, considering all those approximations, this simple model does a fairly good job at reproducing the majority of the circumstellar absorption features from the UV to the optical, while providing, at the same time, some interesting insights to the real physical structure of the disk.
Zero Velocity Absorption Component {#subsec:Circular ring components}
----------------------------------
We briefly mentioned, in the previous section, the presence of components with practically zero velocity shift that were contributing significantly to the Si IV profiles. Similar stationary components, although not as strong as the silicon ones, can also be observed in the circumstellar absorption profiles of many elements in the optical (see Figure \[fig:article\_5316tick\] for an example). Such absorption features cannot be reproduced using our eccentric disk model. These features are relatively narrow, symmetric and blue-shifted by almost exactly the value, within the uncertainties, of the gravitational redshift, indicating that the source of this absorption is located at a distance where the redshift is $\sim 1$ km/s at most. These features can thus be used to confirm (or determine independently) the mass of the white dwarf. These blue-shifted (relative to the photosphere) features are present at all the epochs and appear to be stable. A similar feature was also detected for the H and K Ca lines in the spectra of the white dwarf WD 1124-293 by [@Debes2012], which they interpreted as evidence for circumstellar gas.
\[fig:article\_5316tick\]
The value of the blueshift and the symmetry of the features suggest they originate from a circular component situated beyond the farthest point of the eccentric disk (43.75 $\rm R_*$, see Figure \[fig:redshift\_grav\]). We thus experimented with adding the absorption produced by an additional gas ring with a radial width of 0.5 $\rm R_*$ and with radii between 44 and 70 $\rm R_*$ (well inside the semi-major axis of the main transiting bodies). We assumed, for simplicity, the same vertical density structure as the one for the eccentric rings, and we computed the central densities from the radial scale in respect to the inner value of the disk, which gives values between $8 \times 10^{-7}$ and $3 \times 10^{-7}$ $\rm g/cm^3$ respectively. We also tested several temperatures (assumed constant throughout the ring) similar to those we found for the eccentric disk. Note that these ring properties are just for exploratory purposes and are certainly not unique (there is not much to be gained at this point by adding another layer of free parameters).
Examples of the resulting absorption for two 8050 K rings situated respectively at 44 and 70 $\rm R_*$ are shown in Figure \[fig:article\_montrercoches\]. Both rings produce features at the desired velocity shift from the photospheric line, but the ring farthest from the star produces, as expected, narrower lines, which seem to best reproduce the observations. The closer ring, however, seems to better represent the depth of the features (except for for Ni that is not well reproduce by either models) but given this is a simple constant temperature model, we refrain from drawing any definitive conclusion at this point.
\[fig:article\_montrercoches\]
A similar circular ring model can also be used to reproduce the zero velocity Si IV component observed in the UV (1393.76 and 1402.77 ${\text{\normalfont\oldAA}}$ lines, see bottom right panel of Figure \[fig:article\_raiesbien13000\]). However, a much higher temperature is needed to produce transitions from this highly ionized species. Figure \[fig:article\_SiIVring\] shows the result from a constant 19,000 K ring model with a central density of $1 \times 10^{-6}$ $\rm g/cm^3$ situated 44 $\rm R_*$ from the star (presented here with the constant 13,000 K eccentric disk model which was doing a better job at reproducing the shifted Si IV features, see Figure \[fig:article\_raiesbien13000\]). While these parameters seem to reproduce the width and depth quite nicely, it is difficult to explain the presence of such a hot ring further away than the cooler eccentric disk. More elaborate ring models are clearly needed to fully understand the exact nature of those absorption features.
\[fig:article\_SiIVring\]
Precession period {#subsec:Precession period}
-----------------
Now equipped with an eccentric ring model that can convincingly reproduce most of the circumstellar features present in the HIRES and HST data (2016.04.01 and 2016.03.28, respectively), we next try to apply it to reproduce circumstellar features present in spectroscopic data taken at other epochs.
@Cauley2018 first estimated the precession period of the disk to be $\approx 5.3$ yrs, a value that we wish to revisit now that we have data covering a larger part of the cycle. We first try to find the position in the precession cycle that best reproduces the circumstellar features for each epoch of observation. This is done by rotating the geometric configuration presented in Figure \[fig:config\_geo\_t0\], and by calculating the corresponding absorption from the disk, until the shape of the circumstellar features are optimally reproduced. Of the remaining epochs of data available [see Table 3 of @Xu2019], we find there are 3 other times where the quality of the fit is particularly good and similar to that presented in Figures \[fig:article\_raiesmauvaisesstructuretemp\] and \[fig:figure8\_struc\_20160401\], namely 2015.04.11, 2017.03.06 and 2018.01.01 (note that epochs very near these dates are also acceptable). These four epochs are roughly 1 year apart and cover a good fraction of a whole cycle. Figure \[fig:article\_bonnesepoques\] shows a zoom on the $5316.6$ Fe II line for these dates (Figures similar to Fig. \[fig:article\_raiesmauvaisesstructuretemp\] and \[fig:figure8\_struc\_20160401\] at these dates are presented in Appendix I).
\[fig:article\_bonnesepoques\]
From the angle of rotation (relative to the 2016.04.01, our zero point) needed to reach the configuration on the cycle that best matches the observed velocity profiles, we determine the corresponding phase at each epochs. Assuming that the precession rate is constant, we can perform a linear fit to the phases and obtain the precession period of the disk (see Figure \[fig:article\_fitperiode\]). We find a precession period of $4.6\pm0.3$ yrs, a slightly shorter value than the one estimated by @Cauley2018. It is interesting to note that the best fit with our assumed configuration is obtained, as in @Cauley2018, for a retrograde precession while GR-driven precession should be prograde. It is possible that there exists a different configuration that would better fit with prograde precession or that some disk pressure terms indeed contribute to produce a retrograde precession [see @Miranda2018], but discriminating between these posibilities is beyond the scope of this paper.
\[fig:article\_fitperiode\]
Figure \[fig:article\_period46an\] shows the prediction of our eccentric disk model in the region of the Fe II $5316.6$ line, a well isolated feature that is covered by all spectroscopic data, for all 17 epochs using this fitted period and zero point. In general, the model reproduces the overall shape and velocity spread for most of the epochs, although there are admittedly times where the the match is not perfect, or not good at all. In fact, there are some epochs for which there is no angle anywhere on the cycle that can reproduce the observed velocity profile.
\[fig:article\_period46an\]
In particular data taken in November and December 2016 seem to have narrower and deeper features and are almost centered on the photospheric line, which can never be reproduce by our model. Figure \[fig:article\_2016112620180424vitesse\] shows that the velocity width for the 2016.11.26 data is around 190 km/s and the model produces a wider feature of around 265 km/s. The April and May 2018 epochs are also problematic, as the features appear to have returned toward lower velocity shift much faster than predicted by the 4.6 yr period model. Note the model can produce narrow and deep features similar to those observed, but only at much greater velocities than observed (profiles produced at times where the velocity shift is lower are much wider, see Figure \[fig:article\_precessioncycle\]).
\[fig:article\_2016112620180424vitesse\]
While it is possible that the assumed constant precession rate is at fault here, these shortcomings are most probably an indication that the real geometric configuration of the eccentric disk is actually different than the one we are exploring in this paper. It is likely that some parameters that describe this configuration, namely the perihelion distances of the rings, eccentricities and angular shift of the apsidal line between the different rings, need to be tweaked a bit in order to achieve better results (for example, narrow features at lower velocities can probably be obtained with a smaller angular shift between rings that would be a bit farther from the star). Finding the exact configuration in this manner, however, is a very time consuming task that is outside the scope of this paper. A better approach left for future studies would probably be to invert the problem and infer the disk structure from the observed velocity profiles [for example, see @Manser2016].
Transit spectra {#subsec:Transit spectra}
---------------
Simultaneous ground based photometric and spectroscopic observations of WD 1145+017 were secured on March 28, 2016 (two transits were covered with the 61’-inch telescope in Arizona while, at the same time, multiple 10 minutes sub-exposures were obtained with the Keck/ESI spectrograph). One interesting characteristic that was uncovered by these observations was that the circumstellar features were much shallower during the transit than out of transit [see @Xu2019 for details]. Using our simple eccentric model, it is possible to explain, in a semi-quantitative way, the behavior of the absorption features by simply blocking completely certain areas before integrating over the stellar surface, mimicking the passage of the asteroid (or disintegrating chunks detached from it) on our line of sight. For simplicity, we consider circular blocking objects (consisting of the solid disintegrating rocky body + optically opaque gas cloud) situated at $\sim$ 101 $\rm R_*$ (Keplerian distance for a 4.5h orbit around a $0.656 \: M_{\odot}$ star) with diameters of 5000, 11,250 and 14,700 km. We also assume that the objects orbit in the same plane as the gas disk. For each object size, we compute the expected spectrum for 7 different positions during the transit (see Figure \[fig:article\_grilletransit\]).
\[fig:article\_grilletransit\]
The first noteworthy result of this experiment is that the 5000 km transit, which blocks only 9% of the surface of the star, barely has a detectable effect on the resulting spectrum at any point during the transit. We find that in order to have a noticeable effect, the blocking object has to be big enough to block not only a large part of the disk, but also regions of the star not covered by the disk. We estimate that the diameter thus has to be at least $\sim 6900$ km for a detectable depth variation during a transit. Figure \[fig:article\_transit100\_150km\] shows the resulting spectrum in the region of the 5169.03 Fe II line at the different moments during the transit for diameters of 11,250 and 14,700 km, blocking respectively 45% and 80% of the surface of the star.
\[fig:article\_transit100\_150km\]
\[fig:article\_transitdata\_both\]
First of all, we observe that the features are systematically deeper for the 14,700 km body transit. This can be explained by the fact that this larger body blocks, relative to the smaller one, a larger fraction of the photosphere that is not covered by the eccentric disk. Note that had the transiting object blocked only part of the star that has no part of the eccentric disk in the line of sight, then the circumstellar features would have been deeper than when out of transit. Since the absorption features become shallower during transits [@Xu2019], this confirms that the disintegrating asteroid and the disk are orbiting in the same plane, as suspected (if the orbit of the blocking body was inclined in relation to the eccentric disk, the circumstellar features would first become deeper, then shallower and then deeper again before returning to normal). It is also interesting to note that the shape of the features evolves during the transit, the most red-shifted side being more affected in the beginning of the transit while the lower velocity part is affected towards the end of the transit. This is a behavior that is expected given the line of sight velocities across the disk at this particular time (see Figure \[fig:velocity\_profile\_t0\]). Unfortunately, the time resolution and signal-to-noise ratio from our March 28, 2016 observations are insufficient to detect such minute changes during the transit. Nevertheless, our simple model is able to successfully produce a reduction of the depth of the absorption features similar to those measured during the 2016.03.28 transits. Given that the light curve shows $\sim 40\%$ dips during transits, we can also rule out a body as large as 14,700 km since it would cover 80% of the star surface (note there is the very likely possibility, however, that the transiting body is not totally optically thick, in which case it could indeed be much larger). Taking our simple 11,250 km diameter blocking area, we can compare the resulting shape of the circumstellar features at various points during the transit.
Figure \[fig:article\_transitdata\_both\] shows the predicted shape of the Fe II 5169 region approximately at the maximum of the transit (corresponding to t4 in Figure \[fig:article\_transit100\_150km\]) as well as when out of transit. While the temporal resolution and signal-to-noise of the sub-exposures are not sufficient to allow a better characterization of the transiting body, it is encouraging that this simple model is sensitive to details of the physical characteristics of the transiting object. The model also predicts, as observed, that the change in depths of the various absorption features are not the same for every transition, which is a reflection of the fact that they are not all formed at the same place in the disk. Moreover, it also naturally explains the shallower transits observed in the UV [see @Xu2019 for a semi-analytical explanation of this phenomenon]. In a forthcoming publication, we will attempt to extract a more detailed comprehension of the physical conditions in the disk based on the variation in depth of the various transitions during transiting events.
Conclusion {#sec:conclusion}
==========
This paper first presented an updated analysis, based on new atmospheric parameters, of the photospheric composition of WD 1145+017. In particular, we now find that the somewhat high oxygen abundance problem reported in a previous study [@Xu2016] disappeared thanks in part to new high resolution spectroscopic data showing that some of the lines previously used for the abundance determination were contaminated by circumstellar absorption features. Our updated analysis now shows that the chemical abundance pattern observed at the photosphere is, to the first order, very close to what is expected for the accretion of a rocky body with bulk Earth composition.
We next developed an eccentric precessing disk model similar to that proposed by @Cauley2018 in order to reproduce the numerous variable circumstellar features observed over time in the spectra of WD 1145+017. One of the main advantages of this model in comparison to that of @Cauley2018 is that it predicts the shape of all circumstellar features from the UV to the optical using opacity calculations based on the assumed physical conditions of the disk. While the considered physical structure and dynamical configuration are certainly not a perfect representation of reality (in particular the very approximate temperature structure assumed), this simple model is able to reproduce the majority of the circumstellar features, from UV to optical, at numerous epochs during the precession cycle. Based on our calculations, the abundances for the circumstellar gas also appear to be about the same as those determined from the photosphere (no adjustment is needed to reproduce the relative depths from different elements). We also estimate that the total mass of the gas present in those rings is on the order of $\sim 10^{16}$ g, close to the mass of Uranus and Neptune’s ring systems.
This eccentric disk model for circumstellar absorption also highlights the need for supplementary components, as some features cannot be accounted for by it. In particular, the numerous symmetrical zero velocity features (relative to the gravitationally red-shifted photospheric lines) observed in the optical probably indicate the presence of a supplementary low eccentricity ring outside the eccentric ring system. Furthermore, an additional low eccentricity component of hot gas is also needed to explain the presence of highly ionized species such as Si IV in the UV.
We find that a precession period of $4.6\pm0.3$ yrs is able to reproduce relatively well the observed shape of the velocity profile from April 2015 to January 2018, although some epochs, notably Nov/Dec 2016 and Apr/May 2018, are more challenging. Minor adjustments on the disk parameters (perihelion distances of the rings, eccentricities and angular shift of the apsidal line between the different rings) could probably provide an even better representation of the circumstellar features, a task left for future studies.
Finally, our simple model can also be used to confirm in a more quantitative way the semi-analytical description presented in @Xu2019, that is the anti-correlation between the circumstellar line strength and the transit depth, the shallower transit in the UV and the alignment of the orbital plane for the gas disk and the blocking fragment.
To conclude, considering the numerous approximations used in the construction of this eccentric disk model, its ability to reproduce a fair number of the observed characteristics of this system over time is quite satisfying. Future studies will aim at improving the physical structure of the rings (in particular implement a more realistic temperature structure) in order to better characterize the various components in orbit around WD 1145+017. A full analysis of all spectroscopic data available, including multiple epochs of HST data, will be presented elsewhere once the model has been improved and refined to a more satisfactory level.
This work was supported in part by NSERC (Canada) and the Fund FRQNT (Québec).
Appendix I
==========
\[fig:figure11\_20150411\]
\[fig:figure8\_struc\_20150411\]
\[fig:figure11\_20180101\]
\[fig:figure8\_struc\_20180101\]
[^1]: http://nova.astro.umd.edu/index.html
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'To describe many-particle systems suspended in incompressible low-Reynolds-number *fluids*, effective hydrodynamic interactions can be introduced. Here, we consider particles embedded in *elastic* media. The effective elastic interactions between spherical particles are calculated analytically, inspired by the approach in the fluid case. Our experiments on interacting magnetic particles confirm the theory. In view of the huge success of the method in hydrodynamics, we similarly expect many future applications in the elastic case, e.g. for elastic composite materials.'
author:
- Mate Puljiz
- Shilin Huang
- 'Günter K. Auernhammer'
- 'Andreas M. Menzel'
date: 'see https://doi.org/10.1103/PhysRevLett.117.238003'
title: ' rigid inclusions in elastic media matrix-mediated interactions '
---
Hydrodynamics determines our daily life. Examples are given by the flow of air into our lungs [@zhang2002transient], drinking of beverages and digestive processes [@meng2005computer; @*ferrua2010modeling], technical applications such as microfluidic devices [@squires2005microfluidics], or shape optimization of planes, vehicles, ships, and propellers [@wald2006aerodynamics; @*campana2006shape; @*muller2014aerodynamic]. All these processes are described by the Navier-Stokes equations [@navier1822memoire; @*stokes1845theories] or variants thereof. This set of equations typically poses significant challenges during solution due to a convective nonlinearity reflecting inertial effects. Basically, turbulence is driven by the inertial term. It often renders analytical solutions impossible.
The situation changes for small dimensions and velocities or high viscosity. Then, the relative strength of inertial effects, measured by the Reynolds number, is low. The nonlinearity can be neglected. A Green’s function in terms of the so-called Oseen matrix is then available, which formally solves the problem analytically [@karrila1991microhydrodynamics; @dhont1996introduction]. In this way, semi-dilute colloidal suspensions, i.e. the dispersion of nano- to micrometer-sized particles in a fluid [@dhont1996introduction; @felderhof1977hydrodynamic; @*ermak1978brownian; @*durlofsky1987dynamic; @*zahn1997hydrodynamic; @*meiners1999direct; @*dhont2004thermodiffusion; @*rex2008influence], or microswimmer suspensions [@pooley2007hydrodynamic; @*baskaran2009statistical; @*menzel2016dynamical; @*lauga2009hydrodynamics; @*drescher2010direct; @*drescher2011fluid; @*paxton2004catalytic] are described effectively. The explicit role of the fluid is eliminated and replaced by effective hydrodynamic interactions between the suspended particles [@karrila1991microhydrodynamics; @dhont1996introduction].
Despite the success of this theoretical approach for colloidal suspensions, hardly any investigations consider a surrounding elastic solid instead of a suspending fluid. This is surprising, since, as we show below, the formalism can be adapted straightforwardly to linearly elastic matrices and is confirmed by our experiments. Our approach will, for instance, facilitate describing the response of elastic composite materials to external stimuli. Such materials consist of more or less rigid inclusions embedded in an elastic matrix. They are of growing technological interest and may serve, e.g., as soft actuators or sound attenuation devices [@an2003actuating; @*fuhrer2009crosslinking; @*bose2012soft; @*cheng2006observation; @*still2011collective; @*baumgartl2007tailoring; @*baumgartl2007erratum].
In previous theoretical studies, the physics of one single rigid or deformable inclusion was addressed [@eshelby1957determination; @*eshelby1959elastic; @*walpole1991rotated; @*walpole1991translated; @*walpole2005green; @phanthien1993rigid], also under acoustic irradiation [@oestreicher1951field; @*norris2006impedance; @*norris2008faxen]. For more than a single inclusion, mainly the so-called load problem was analyzed theoretically for a pair of rigid inclusions: one prescribes displacements of two rigid inclusions in an elastic matrix, and then determines the forces necessary to achieve these given displacements [@phanthien1994loadtransfer; @*kim1995faxen].
Here, we take the converse point of view, based on the cause-and-effect chain in our experiments: external forces are imposed onto the inclusions, or mutual forces between the inclusions are induced, for example to actuate the material or to tune its properties. In response to the forces, the inclusions are displaced. Since they cannot penetrate through the surrounding elastic matrix, they transmit the forces to the matrix and distort it. Such distortions lead to mutual long-ranged interactions between the inclusions, in analogy to hydrodynamic interactions in colloidal suspensions [@karrila1991microhydrodynamics; @dhont1996introduction; @tanaka2000simulation].
We present a basic derivation of analytical expressions for these interactions from the underlying elasticity equations. Then, we verify the theory by experiments on rigid paramagnetic particles embedded in soft elastic matrices. Mutual particle interactions are induced by an external magnetic field. As we demonstrate, theory and experiment are in good agreement, .
For simplicity, we assume a homogeneous, isotropic, infinitely extended elastic matrix, and low-amplitude deformations. Applying a bulk force density $\mathbf{f}_b(\mathbf{r})$ to the matrix, its equilibrated state satisfies the linear elastostatic Navier-Cauchy equations [@cauchy1828exercises], $$\label{eq_navier-cauchy}
\nabla^2\mathbf{u}(\mathbf{r}) + \frac{1}{1-2\nu}\nabla\nabla\cdot\mathbf{u}(\mathbf{r}) ={} -\frac{1}{\mu}\mathbf{f}_b(\mathbf{r}).$$ This is the elastic analogue to the linearized Stokes equation in low-Reynolds-number hydrodynamics [@dhont1996introduction]. Instead of velocities, $\mathbf{u}(\mathbf{r})$ here denotes the displacement field, describing the reversible relocations of the volume elements from their initial positions during deformations. $\mu$ is the shear modulus of the matrix and $\nu$ its Poisson ratio, connected to its compressibility [@landau1986theory]. Importantly, for a point force density $\mathbf{f}_b(\mathbf{r})=\mathbf{F}\delta(\mathbf{r})$ acting on the matrix, the resulting deformation field can be calculated analytically from Eq. (\[eq\_navier-cauchy\]) via Fourier transform as $\mathbf{u}(\mathbf{r})=\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r})\cdot\mathbf{F}$. Here, $$\label{greens_function}
\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}) ={} \frac{1}{{\textcolor{black}{8\pi}}\mu}\left[\frac{{\textcolor{black}{1}}}{r}\mathbf{\underline{\hat{I}}}+\frac{\mathbf{r}\mathbf{r}}{r^3}\right]$$ is the corresponding Green’s function [@landau1986theory], $\mathbf{\underline{\hat{I}}}$ the identity matrix, $r$=$|\mathbf{r}|$, and the underscore marks second-rank tensors and matrices. Still, it is practically impossible to explicitly solve Eq. (\[eq\_navier-cauchy\]) analytically in the presence of several rigid embedded particles of finite size. An iterative procedure resolves this problem, .
![ []{data-label="fig_illustration"}](figure1.pdf){width="6.5cm"}
We consider $N$ rigid spherical particles of radius $a$, with no-slip boundary conditions on their surfaces. First we only address the $i$th particle at position $\mathbf{r}_i$, subject to an external force $\mathbf{F}_i$. The embedded particle transmits this force to the surrounding matrix and induces a displacement field $$\label{eq_u_i_0}
\mathbf{u}_i^{(0)} (\mathbf{r}) = \left(1+\frac{a^2}{6}\nabla^2\right)\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}-\mathbf{r}_i)\cdot\mathbf{F}_i.$$ This field is the elastic analogue of hydrodynamic Stokes flow [@karrila1991microhydrodynamics; @dhont1996introduction], for elastic media. Inserting Eq. (\[greens\_function\]) reproduces a corresponding expression in Ref. . Eq. (\[eq\_u\_i\_0\]) is confirmed as it satisfies Eq. (\[eq\_navier-cauchy\]), shows the correct limit $\mathbf{u}_i^{(0)}(\mathbf{r})=\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}{\textcolor{black}{-\mathbf{r}_i}})\cdot\mathbf{F}_i$ for $a\rightarrow0$, and for ${\textcolor{black}{|\mathbf{r}-\mathbf{r}_i|}}=a$ is constant on the particle surface. Thus, Eq. (\[eq\_u\_i\_0\]) for ${\textcolor{black}{|\mathbf{r}-\mathbf{r}_i|}}=a$ reveals the rigid displacement $$\mathbf{U}_i^{(0)} \,=\, \mathbf{u}_i^{(0)}({\textcolor{black}{|\mathbf{r}-\mathbf{r}_i|}}=a) \,=\, \frac{{\textcolor{black}{1}}}{{\textcolor{black}{6\pi}}\mu a}\mathbf{F}_i$$ of the $i$th particle in response to $\mathbf{F}_i$ in accord with the no-slip conditions at ${\textcolor{black}{|\mathbf{r}-\mathbf{r}_i|}}=a$.
To find the effective elastic interactions between particles $i$ and $j$ ($j\neq i$), we take the induced displacement field $\mathbf{u}_i^{(0)}(\mathbf{r})$ as given. We need to determine how particle $j$ reacts to the imposed field $\mathbf{u}_i^{(0)}(\mathbf{r})$. In general, particle $j$ can be rigidly translated by a displacement vector $\mathbf{U}_j^{(1)}$ and rigidly rotated by a rotation vector $\mathbf{\Omega}_j^{(1)}$. Taking into account the no-slip conditions on the surface $\partial V_j$ of the $j$th particle, the equality $$\label{eq_balance}
\mathbf{U}_j^{(1)}+\mathbf{\Omega}_j^{(1)}\times(\mathbf{r}-\mathbf{r}_j)=\mathbf{u}_i^{(0)}(\mathbf{r})+\int_{\partial V_j}\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}-\mathbf{r}')\cdot\mathbf{f}(\mathbf{r}')\mathrm{d}S'$$ must hold for all $\mathbf{r}\in\partial V_j$. $\mathbf{f}(\mathbf{r}')$ describes the surface force density exerted by the surface of particle $j$ onto the matrix.
Such an embedded particle will translate and rotate as dictated by the surrounding matrix. We obtain the expression for $\mathbf{U}_j^{(1)}$ by integrating Eq. (\[eq\_balance\]) over $\partial V_j$. Similarly, for $\mathbf{\Omega}_j^{(1)}$, Eq. (\[eq\_balance\]) is multiplied dyadically by $\mathbf{r}-\mathbf{r}_j$, and after integration over $\partial V_j$ the antisymmetric part is extracted. , $\mathbf{u}_i^{(0)}(\mathbf{r})$ is Taylor expanded around $\mathbf{r}_j$. , we use that Eq. (\[eq\_navier-cauchy\]) for $\mathbf{r}\notin\partial V_i$ leads to . The last term in Eq. (\[eq\_balance\]) vanishes at this stage as no total net external force or torque is applied to particle $j$ at the present step of iteration. In the end, we recover the elastic analogues of the hydrodynamic [@batchelor1972hydrodynamic; @karrila1991microhydrodynamics; @dhont1996introduction] Faxén laws $$\begin{aligned}
\mathbf{U}_j^{(1)} &={} &\left(1+\frac{a^2}{6}\nabla^2\right)\mathbf{u}_i^{(0)}(\mathbf{r})\bigg|_{\mathbf{r}=\mathbf{r}_j},\label{eq_faxen-translation} \\
\boldsymbol{\Omega}_j^{(1)} &={} &\frac{1}{2}\nabla\times\mathbf{u}_i^{(0)}(\mathbf{r})\bigg|_{\mathbf{r}=\mathbf{r}_j}\label{eq_faxen-rotation}.\end{aligned}$$ This is how particle $j$ is translated and rotated in the field $\mathbf{u}_i^{(0)}(\mathbf{r})$ induced by particle $i$. Yet, elastic retroaction occurs between the particles, as described in the following.
The force densities $\mathbf{f}(\mathbf{r}')$ in Eq. (\[eq\_balance\]) that the particles exert on their environment in general will not vanish identically. Since the particles are rigid, they resist deformation . Thus, they exert counteracting stresses onto the deformed matrix. The stresslet exerted by particle $j$ onto the matrix can be denoted as $\mathbf{\underline{S}}_j=\int_{\partial V_j}\mathrm{d}S'\{[\mathbf{f}(\mathbf{r}')\mathbf{r}'+
(\mathbf{f}(\mathbf{r}')\mathbf{r}')^T]/2 - {\textcolor{black}{\mathbf{\underline{\hat{I}}}\,[\mathbf{f}(\mathbf{r}')\cdot\mathbf{r}']/3}} \}$, where $[\bullet]^T$ marks the transpose. In our case, we can directly calculate from Eq. (\[eq\_balance\]) the stresslet $\mathbf{\underline{S}}_j^{(1)}$ that particle $j$ exerts onto the matrix when it resists to the deformation described by $\mathbf{u}_i^{(0)}(\mathbf{r})$. To find the expression for $\mathbf{\underline{{S}}}_j^{(1)}$, one proceeds in the same way as described above for $\mathbf{\Omega}_j^{(1)}$ but eventually extracts the symmetric part. The latter contains the definition of $\mathbf{\underline{S}}_j^{(1)}$. We obtain $$\mathbf{\underline{S}}_j^{(1)}\! = \frac{{\textcolor{black}{10\pi}}\mu a^3}{-{\textcolor{black}{3}}}\left(1+\frac{a^2}{10}\nabla^2\right)
\!
\left[ \nabla\mathbf{u}_i^{(0)}(\mathbf{r})+\big(\nabla\mathbf{u}_i^{(0)}(\mathbf{r})\big)^{\!T} \right]\!\Big|_{\mathbf{r}_j}\!\!.
\label{eq_faxen-stresslet}$$ This stresslet leads to additional distortions of the matrix, , described by a displacement field $\mathbf{u}^{(1)}_j(\mathbf{r})$ that overlays $\mathbf{u}^{(0)}_i(\mathbf{r})$. We find $\mathbf{u}^{(1)}_j(\mathbf{r})$ from the general expression $\mathbf{u}_j(\mathbf{r})=\int_{\partial V_j}\mathrm{d}S'\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}-\mathbf{r}')\cdot\mathbf{f}(\mathbf{r}')$ by Taylor expanding the Green’s function in $\mathbf{r}'$ around $\mathbf{r}'=\mathbf{r}_j$. The definition of $\mathbf{\underline{S}}_j$ shows up as the symmetric part of the second-order term of the series, similarly to the hydrodynamic case [@batchelor1972hydrodynamic; @karrila1991microhydrodynamics], leading to $$\label{eq_u_j_1}
\mathbf{u}_j^{(1)}(\mathbf{r}) ={} -\left(\mathbf{\underline{S}}_j^{(1)}\cdot\nabla\right)\cdot\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}-\mathbf{r}_j).$$
This expression completes our first step of iteration. In the second step, it is particle $i$ that is exposed to the field $\mathbf{u}_j^{(1)}(\mathbf{r})$. Correspondingly, we find its reaction from Eqs. (\[eq\_faxen-translation\])–(\[eq\_u\_j\_1\]) by replacing $(\mathbf{u}_i^{(0)}, \mathbf{U}_j^{(1)}, \mathbf{\Omega}_j^{(1)}, \mathbf{\underline{S}}_j^{(1)}, \mathbf{u}_j^{(1)}, \mathbf{r}_j)$ with $(\mathbf{u}_j^{(1)}, \mathbf{U}_i^{(2)}, \mathbf{\Omega}_i^{(2)}, \mathbf{\underline{S}}_i^{(2)}, \mathbf{u}_i^{(2)}, \mathbf{r}_i)$. Particle $i$ now feels the consequences of its self-generated field $\mathbf{u}_i^{(0)}(\mathbf{r})$ *reflected* by particle $j$ in the form of $\mathbf{u}_j^{(1)}(\mathbf{r})$. Therefore, the procedure was termed *method of reflections* in hydrodynamics [@karrila1991microhydrodynamics; @dhont1996introduction]. We have not found in the hydrodynamic derivation [@dhont1996introduction] the above reasoning of explicitly imposing on the matrix environment the rigidity-induced stress. In principle, this refinement of the deformation field via back-and-forth reflections between the two particles can be , leading to increasingly-higher-order corrections in $a/r_{ij}$, where $r_{ij}=|\mathbf{r}_i-\mathbf{r}_j|$. For our example systems below, these iterations converge quickly, see Fig. \[fig\_two\_particles\](c), so that it is sufficient to consider contributions up to (including) order $r_{ij}^{-4}$.
Due to the linearity of Eq. (\[eq\_navier-cauchy\]), we can sum up the particle displacements obtained from the different steps of iteration. Moreover, we can consider external forces $\mathbf{F}_i$ on *all* particles and calculate the resulting net displacements $\mathbf{U}_i$ due to the mutual elastic interactions ($i=1,...,N$). These contributions superimpose. In analogy to the hydrodynamic [@dhont1996introduction] mobility matrix we express the result by an elastic *displaceability matrix* $\mathbf{\underline{M}}$: $$\begin{aligned}
\label{eq_displaceability_matrix}
\begin{pmatrix}
\mathbf{U}_1\\
\vdots\\
\mathbf{U}_N
\end{pmatrix}
={}
\begin{pmatrix}
\mathbf{\underline{M}}_{11} & \ldots & \mathbf{\underline{M}}_{1N} \\
\vdots & \vdots & \vdots \\
\mathbf{\underline{M}}_{N1} & \ldots & \mathbf{\underline{M}}_{NN}
\end{pmatrix}
\cdot
\begin{pmatrix}
\mathbf{F}_1\\
\vdots\\
\mathbf{F}_N
\end{pmatrix}.\end{aligned}$$ Limiting ourselves to contributions up to (including) order $r_{ij}^{-4}$, we find $$\begin{aligned}
\mathbf{\underline{M}}_{i=j} &\!=\!& M_0\Bigg[\mathbf{\underline{\hat{I}}} - \sum\limits_{\substack{k=1 \\ k\not=i}}^{N} \frac{{\textcolor{black}{15}}}{{\textcolor{black}{4}}}\bigg(\frac{a}{r_{ik}}\bigg)^{\!\!\!4}
\mathbf{\hat{r}}_{ik}\mathbf{\hat{r}}_{ik}
\Bigg], \label{M_II_2}\\
\mathbf{\underline{M}}_{i\not=j} &\!=\!& M_0 \frac{3}{{\textcolor{black}{4}}}\frac{a}{r_{ij}} \Bigg[ \left( {\textcolor{black}{\mathbf{\underline{\hat{I}}}+}} \mathbf{\hat{r}}_{ij}\mathbf{\hat{r}}_{ij} \right)
+ {\textcolor{black}{2}} \bigg(\frac{a}{r_{ij}} \bigg)^{\!\!\!2} \!\left({\textcolor{black}{\frac{1}{3}}}\mathbf{\underline{\hat{I}}}-\mathbf{\hat{r}}_{ij}\mathbf{\hat{r}}_{ij}\right)\! \Bigg]
\notag\\
& & {}+ \mathbf{\underline{M}}_{i\not=j}^{(3)} ,
\label{M_IJ_2}
$$ where $M_0 ={} {\textcolor{black}{1}}/{\textcolor{black}{6\pi}}\mu a$ and $\mathbf{\hat{r}}_{ij}=\mathbf{r}_{ij}/r_{ij}$ ($i,j=1,2,...,N$).
additional three-body interactions $\sim\! r^{-4}_{ij}$ calculated in full analogy to the above procedure for the two-body interaction, $$\label{M_tt_3}
\mathbf{\underline{M}}_{i\not=j}^{(3)} \!=\! M_0\frac{{\textcolor{black}{15}}}{{\textcolor{black}{8}}}
\!\sum\limits_{\substack{k=1\\k\not=i,j}}^{N}
\!\bigg(\frac{a}{r_{ik}}\bigg)^{\!\!\!2}
\bigg(\frac{a}{r_{jk}}\bigg)^{\!\!\!2}
\Big[
{\textcolor{black}{1-3}}(\mathbf{\hat{r}}_{ik}\cdot\mathbf{\hat{r}}_{jk})^2\Big]\mathbf{\hat{r}}_{ik}\mathbf{\hat{r}}_{jk}.$$ That is, the deformation field induced by a force on a first particle spreads to a second particle , from where it is reflected towards the third particle .
![[]{data-label="fig_3body"}](figure2.pdf){width="7.cm"}
![ (a) Schematic of the samples. After fabrication of the bottom gel layer (I), the paramagnetic nickel (Ni) particles are placed into the center plane (dashed), before the top layer (II) is added. The enclosing plastic molds are open to the top for optical investigation. (b) Snapshot of a system of two Ni particles (diameters 150.6$\pm$1.9 $\mathrm{\mu m}$) embedded in a soft elastic gel, here for vanishing external magnetic field. (c,d) Change in distance $\Delta r_{12}$ between the two particles when applying an external magnetic field along different directions in the particle plane via clockwise rotation. The horizontal arrow in (b) defines the angle of $0^{\circ}$. Data points in (d) were measured experimentally. The line is calculated from the theory, where shaded areas arise from uncertainties in the experimental input values. The “zoom” in (c) highlights the rapid convergence of the theory. []{data-label="fig_two_particles"}](figure3.pdf){width="8.3cm"}
Eqs. (\[eq\_displaceability\_matrix\])–(\[M\_tt\_3\]) represent the central theoretical result. To confirm and illustrate the merit of the theory, we performed experiments on small groups of paramagnetic particles embedded in a soft elastic gel matrix. Applying an external magnetic field induced mutual magnetic forces between the particles. Rotating the magnetic field tuned these forces. The resulting relative displacements of the particles were tracked by optical microscopy.
We used paramagnetic Nickel (Ni) particles obtained from Alfa Aesar ($-100+325$ mesh, purity 99.8%). The magnetic hysteresis curves (measured by a vibrating sample magnetometer, Lake Shore 7407) showed a low remanence of $\sim$ 7.5 kA/m, a low coercive field of $\sim$ 2.4 mT, and a volume magnetization of $\pm$17 kA/m under an external magnetic field of $\sim$ 216 mT. We carefully selected Ni particles of similar sizes (deviation less than 2% within each group) and a roundness $\gtrsim 0.91$ (measured by image analysis [@imagej]). These particles were embedded in the middle plane of a soft elastic polydimethylsiloxane-based [@huang2016buckling] gel, see Fig. \[fig\_two\_particles\](a). First, a bottom gel layer with a thickness of 3.3 mm and a diameter of 24 mm was prepared in a plastic mold. Second, after sufficient stiffening ($\sim$ 0.5 h), the Ni particles were carefully deposited on its top around the center. Third, a top gel layer with the same composition and size as the bottom layer was added. To ensure good connection between the two layers, at least 7 days of cross-linking were allowed. Using a 32-magnet Halbach array to generate a homogeneous magnetic field [@huang2016buckling], we applied $\sim$ 216 mT to the embedded Ni particles, which is close to saturation. Starting from the initial direction, the magnetic field was rotated clockwise for $180^{\circ}$ in 18 steps within the plane containing the Ni particles. Their center-of-mass positions were tracked by a CCD camera (MATRIX VISION mvBlueCOUGAR-S) with the zoom macro lens (Navitar Zoom 7000) mounted above the samples and subsequent image analysis [@imagej].
We measured the changes in particle distance $\Delta r_{ij}$ ($i\neq j$) for a two- and three-particle system, see Figs. \[fig\_two\_particles\] and \[fig\_three\_particles\], respectively, when rotating the external magnetic field.
 and (d), now for a three-particle system. (a) The snapshot was taken for vanishing external magnetic field (particle diameters 208.5$\pm$2.3 $\mathrm{\mu m}$). (b–d) Changes $\Delta r_{ij}$ in all three distances ($i,j=1,2,3$, $i\neq j$). []{data-label="fig_three_particles"}](figure4.pdf){width="8.3cm"}
Forces $\mathbf{F}_i$ on the particles result from mutual magnetic interactions. Due to substantial particle separations, we approximate the induced magnetic moments as point dipoles [@klapp2005dipolar; @*biller2015mesoscopic]. Thus, we find $$\label{eq_force}
\mathbf{F}_i ={} -\frac{3\mu_0 m^2}{4\pi} \sum\limits_{\substack{j=1 \\ j\not=i}}^{N} \frac{ 5\mathbf{\hat{r}}_{ij}(\mathbf{\hat{m}}\cdot\mathbf{\hat{r}}_{ij})^2 - \mathbf{\hat{r}}_{ij} - 2\mathbf{\hat{m}}(\mathbf{\hat{m}}\cdot\mathbf{\hat{r}}_{ij}) }{{\textcolor{black}{r_{ij}^{\:4}}}} ,$$ with $\mu_0$ the vacuum permeability and $\mathbf{m}=m\mathbf{\hat{m}}$ the induced magnetic moments, considered identical for all particles in the close-to-saturating homogeneous external magnetic field. Using as input parameters the experimentally determined particle positions, sizes, and magnetization, and calculated all changes $\Delta r_{ij}$ from Eqs. (\[eq\_displaceability\_matrix\])–(\[eq\_force\]). Perfect agreement between theory and experiment in Figs. \[fig\_two\_particles\] and \[fig\_three\_particles\] supports the significance of the theoretical approach .
In summary, we considered rigid spherical particles displaced against a surrounding elastic matrix by externally induced forces. We derived analytical expressions to calculate the resulting particle displacements. Mutual interactions due to induced matrix deformations are effectively included. This renders the procedure a promising tool to describe the behavior of elastic composite materials [@ilg2013stimuli; @*odenbach2016microstructure; @*menzel2016hydrodynamic]. Our experiments on paramagnetic particles in a soft elastic gel matrix and subject to tunable magnetic interactions confirm the potential of the theory.
Upon dynamic extension, a prospective application concerns macroscopic rheology [@denn2014rheology], or nano- and microrheology [@ziemann1994local; @*bausch1999measurement; @*waigh2005microrheology; @*wilhelm2008out] where the matrix properties are tested by external agitation of embedded probe particles. Also biological and medical questions are addressable in this way, for instance cytoskeletal properties [@ziemann1994local; @*bausch1999measurement; @*waigh2005microrheology; @*wilhelm2008out; @mizuno2007nonequilibrium]. An extension of the theory to include imposed torques is straightforward and will be presented in the near future.
The authors thank J. Nowak for measuring the magnetization curves and the Deutsche Forschungsgemeinschaft for support of this work through the priority program SPP 1681 (Nos. AU 321/3-2 and ME 3571/3-2).
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Supplemental material {#supplemental-material .unnumbered}
======================
[As stressed in the main text, the derivation of the displaceability matrix can likewise be performed for *compressible* systems. Following the same steps of derivation as in the main text, we present below the corresponding expressions for completeness. Apart from that, we add further experimental results and comparison with the theory for a four-particle system, in complete analogy to our presentation for the three-particle system in the main text. ]{}
Expressions for a compressible elastic matrix
---------------------------------------------
For clarity and to facilitate the comparison with the hydrodynamic fluid case, we have presented in the main text the expressions for an *incompressible* elastic system. That is, the system tends to locally preserve the volume of all its volume elements during any type of elastic deformation. However, and in contrast to the hydrodynamic fluid case [@batchelor1972hydrodynamic; @karrila1991microhydrodynamics; @dhont1996introduction], for elastic matrices it is straightforward to allow for *compressibility* in the derivation. This extended derivation proceeds in direct analogy to the one presented in the main text. We again assume a homogeneous and isotropic elastic matrix of infinite extension. Once more, we start from the linear elastostatic Navier-Cauchy equations [@cauchy1828exercises], $$\label{suppl_eq_navier-cauchy}
\nabla^2\mathbf{u}(\mathbf{r}) + \frac{1}{1-2\nu}\nabla\nabla\cdot\mathbf{u}(\mathbf{r}) ={} -\frac{1}{\mu}\mathbf{f}_b(\mathbf{r}).$$ As in the main text, $\mathbf{u}(\mathbf{r})$ denotes the displacement field, $\mu$ the shear modulus of the matrix [@landau1986theory], $\nu$ the Poisson ratio [@landau1986theory], and $\mathbf{f}_b(\mathbf{r})$ the bulk force density. Now, we do not restrict our analysis to incompressible materials that locally adhere to $\nabla\cdot\mathbf{u}(\mathbf{r})=0$, and we do not assign a specific value to $\nu$.
The resulting Green’s function for a point force density $\mathbf{f}_b(\mathbf{r})=\mathbf{F}\delta(\mathbf{r})$ then reads [@landau1986theory] $$\label{suppl_greens_function}
\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}) ={} \frac{1}{16\pi(1-\nu)\mu}\left[\frac{3-4\nu}{r}\mathbf{\underline{\hat{I}}}+\frac{\mathbf{r}\mathbf{r}}{r^3}\right].$$ Using this expression, if an external force $\mathbf{F}_i$ is acting on a rigid spherical particle $i$ of radius $a$ embedded in the matrix with no-slip boundary conditions on its surface, a displacement field $$\label{suppl_eq_u_i_0}
\mathbf{u}_i^{(0)} (\mathbf{r}) = \left(1+\frac{a^2}{6}\nabla^2\right)\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}-\mathbf{r}_i)\cdot\mathbf{F}_i$$ is induced. Eq. (\[suppl\_eq\_u\_i\_0\]) has the same form as in the main text, but $\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}$ is different, see Eq. (\[suppl\_greens\_function\]). Again, the validity of Eq. (\[suppl\_eq\_u\_i\_0\]) is confirmed as it satisfies Eq. (\[suppl\_eq\_navier-cauchy\]), shows the correct limit $\mathbf{u}_i^{(0)}(\mathbf{r})=\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}{\textcolor{black}{-\mathbf{r}_i}})\cdot\mathbf{F}_i$ for $a\rightarrow0$, and for ${\textcolor{black}{|\mathbf{r}-\mathbf{r}_i|}}=a$ is constant on the particle surface. For ${\textcolor{black}{|\mathbf{r}-\mathbf{r}_i|}}=a$, it reveals the rigid displacement $$\mathbf{U}_i^{(0)} \,=\, \mathbf{u}_i^{(0)}({\textcolor{black}{|\mathbf{r}-\mathbf{r}_i|}}=a) \,=\, \frac{5-6\nu}{24\pi(1-\nu)\mu a}\mathbf{F}_i$$ of the $i$th particle in response to $\mathbf{F}_i$.
The no-slip condition under our assumptions applies on the surface $\partial V_j$ of a particle $j$ also for compressible matrices. Thus Eq. (5) in the main text preserves its shape, i.e. $$\label{suppl_eq_balance}
\mathbf{U}_j^{(1)}\!+\mathbf{\Omega}_j^{(1)}\!\times(\mathbf{r}-\mathbf{r}_j)=\mathbf{u}_i^{(0)}(\mathbf{r})+ \!\int_{\partial V_j}\!\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}-\mathbf{r}')\cdot\mathbf{f}(\mathbf{r}')\mathrm{d}S',$$ where $\mathbf{U}_j^{(1)}$ denotes the translation of particle $j$, $\mathbf{\Omega}_j^{(1)}$ is its rotation, the displacement field $\mathbf{u}_i^{(0)}(\mathbf{r})$ is induced by particle $i$, and $\mathbf{f}(\mathbf{r}')$ denotes the surface force density that particle $j$ exerts on the surrounding matrix. The derivation of expressions for $\mathbf{U}_j^{(1)}$ and $\mathbf{\Omega}_j^{(1)}$ in the form of the Faxén laws follows the same strategy as described in the main text and leads to $$\begin{aligned}
\mathbf{U}_j^{(1)} &={} &\left(1+\frac{a^2}{6}\nabla^2\right)\mathbf{u}_i^{(0)}(\mathbf{r})\bigg|_{\mathbf{r}=\mathbf{r}_j},\label{suppl_eq_faxen-translation} \\
\boldsymbol{\Omega}_j^{(1)} &={} &\frac{1}{2}\nabla\times\mathbf{u}_i^{(0)}(\mathbf{r})\bigg|_{\mathbf{r}=\mathbf{r}_j}\label{suppl_eq_faxen-rotation}.\end{aligned}$$
Also the stresslet $\mathbf{\underline{S}}_j$ exerted by particle $j$ onto the matrix is derived in analogy to what is described in the main text. In general, for compressible systems, this stresslet is given by the expression $\mathbf{\underline{S}}_j=\int_{\partial V_j}\mathrm{d}S'[\mathbf{f}(\mathbf{r}')\mathbf{r}'+
(\mathbf{f}(\mathbf{r}')\mathbf{r}')^T]/2$. This expression slightly differs from the one introduced below Eq. (7) in the main text for incompressible systems. There, a trace-free definition was used to exclude compressions and dilations of the matrix, which needs to be the case for volume-conserving systems. It can be seen from the main text that the difference in definitions plays no actual role for our derivation. The reason is Eq. (9), where the extra term $\sim\mathbf{\underline{\hat{I}}}$ in the incompressible case only leads to a contribution $\sim\nabla\cdot\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}$. Yet, $\nabla\cdot\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}$ vanishes in the incompressible case. Therefore, following the same strategy as described in the main text, we obtain $$\begin{aligned}
\mathbf{\underline{S}}_j^{(1)} &\!=\!&{} -\frac{4\pi(1-\nu)\mu a^3}{4-5\nu}\left(1+\frac{a^2}{10}\nabla^2\right)
\!
\Bigg[\frac{1}{1-2\nu}\mathbf{\underline{\hat{I}}}\,\nabla\!\cdot\!\mathbf{u}_i^{(0)}(\mathbf{r})
\notag\\
&&{}+\frac{5}{2}\Big(\nabla\mathbf{u}_i^{(0)}(\mathbf{r})+\big(\nabla\mathbf{u}_i^{(0)}(\mathbf{r})\big)^T\Big) \Bigg]\Bigg|_{\mathbf{r}=\mathbf{r}_j}. \label{suppl_eq_faxen-stresslet}\end{aligned}$$ Likewise, the displacement field $\mathbf{u}_j^{(1)}(\mathbf{r})$ resulting from the rigidity of particle $j$ and its resistance to deformation, expressed by the stresslet $\mathbf{\underline{S}}_j^{(1)}$, is calculated as described in the main text. Eq. (\[suppl\_eq\_faxen-stresslet\]) here contains a term $\sim 1/(1-2\nu)$, which would diverge for $\nu\rightarrow 0.5$. However, it gets canceled by a counter-factor $\sim(1-2\nu)$ in the calculation. More precisely, upon inserting Eq. (\[suppl\_eq\_u\_i\_0\]) into Eq. (\[suppl\_eq\_faxen-stresslet\]), the expression $\nabla\cdot\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}$ appears; straightforward calculation of $\nabla\cdot\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}$ via Eq. (\[suppl\_greens\_function\]) leads to a factor $\sim(1-2\nu)$. In the end, $\mathbf{u}_j^{(1)}(\mathbf{r})$ has the same form as Eq. (9) in the main text, $$\label{suppl_eq_u_j_1}
\mathbf{u}_j^{(1)}(\mathbf{r}) ={} -\left(\mathbf{\underline{S}}_j^{(1)}\cdot\nabla\right)\cdot\mathbf{\hspace{.02cm}\underline{\hspace{-.02cm}G}}(\mathbf{r}-\mathbf{r}_j).$$
In the next step, again, the reaction of particle $i$ in response to the field $\mathbf{u}_j^{(1)}(\mathbf{r})$ is obtained from Eqs. (\[suppl\_eq\_faxen-translation\])–(\[suppl\_eq\_u\_j\_1\]) by replacing $(\mathbf{u}_i^{(0)}, \mathbf{U}_j^{(1)}, \mathbf{\Omega}_j^{(1)}, \mathbf{\underline{S}}_j^{(1)}, \mathbf{u}_j^{(1)}, \mathbf{r}_j)$ with $(\mathbf{u}_j^{(1)}, \mathbf{U}_i^{(2)}, \mathbf{\Omega}_i^{(2)}, \mathbf{\underline{S}}_i^{(2)}, \mathbf{u}_i^{(2)}, \mathbf{r}_i)$.
Summing up the contributions from the different steps of iteration and considering all $N$ particles simultaneously leads to an expression in the form of an elastic displaceability matrix $\mathbf{\underline{M}}$ as given in the main text: $$\begin{aligned}
\label{suppl_eq_displaceability_matrix}
\begin{pmatrix}
\mathbf{U}_1\\
\vdots\\
\mathbf{U}_N
\end{pmatrix}
={}
\begin{pmatrix}
\mathbf{\underline{M}}_{11} & \ldots & \mathbf{\underline{M}}_{1N} \\
\vdots & \vdots & \vdots \\
\mathbf{\underline{M}}_{N1} & \ldots & \mathbf{\underline{M}}_{NN}
\end{pmatrix}
\cdot
\begin{pmatrix}
\mathbf{F}_1\\
\vdots\\
\mathbf{F}_N
\end{pmatrix}.\end{aligned}$$ Limiting ourselves to contributions up to (including) order $r_{ij}^{-4}$, we find for a *compressible* system the more general expressions $$\begin{aligned}
\mathbf{\underline{M}}_{i=j} &\!=\!& M_0\Bigg\{\mathbf{\underline{\hat{I}}} - \sum\limits_{\substack{k=1 \\ k\not=i}}^{N} \frac{3}{4(4-5\nu)(5-6\nu)}\bigg(\frac{a}{r_{ik}}\bigg)^{\!\!\!4}\notag\\
&{} &\Big[ \Big(37-44\nu+10(1-2\nu)^2\Big)\mathbf{\hat{r}}_{ik}\mathbf{\hat{r}}_{ik}\notag\\
&&{}+5(1-2\nu)^2\left(\mathbf{\underline{\hat{I}}}- \mathbf{\hat{r}}_{ik}\mathbf{\hat{r}}_{ik}\right) \Big] \Bigg\}, \label{suppl_M_II_2}\\
\mathbf{\underline{M}}_{i\not=j} &\!=\!& M_0 \frac{3}{2(5-6\nu)}\frac{a}{r_{ij}} \Bigg[ \Bigg( 4(1-\nu)-\frac{4}{3} \bigg(\frac{a}{r_{ij}}\bigg)^{\!\!\!2} \Bigg) \mathbf{\hat{r}}_{ij}\mathbf{\hat{r}}_{ij} \notag\\
& & {}+ \Bigg( 3-4\nu+\frac{2}{3} \bigg(\frac{a}{r_{ij}} \bigg)^{\!\!\!2} \Bigg)\!\left(\mathbf{\underline{\hat{I}}}-\mathbf{\hat{r}}_{ij}\mathbf{\hat{r}}_{ij}\right)\! \Bigg]
+ \mathbf{\underline{M}}_{i\not=j}^{(3)} ,
\label{suppl_M_IJ_2}
\notag\\[-.2cm]&&\end{aligned}$$ where $M_0 ={} (5-6\nu)/24\pi(1-\nu)\mu a$ and $\mathbf{\hat{r}}_{ij}=\mathbf{r}_{ij}/r_{ij}$ ($i,j=1,2,...,N$). Here, the three-body interactions contribute as given by $\mathbf{\underline{M}}_{i\not=j}^{(3)}$ in the form $$\begin{aligned}
\label{suppl_M_tt_3}
\mathbf{\underline{M}}_{i\not=j}^{(3)} &\!=\!& M_0\frac{3}{8(4-5\nu)(5-6\nu)}
\sum\limits_{\substack{k=1\\k\not=i,j}}^{N}
\bigg(\frac{a}{r_{ik}}\bigg)^{\!\!2}
\bigg(\frac{a}{r_{jk}}\bigg)^{\!\!2}
\notag\\
&&\Big[
\!-\!10(1-2\nu)\Big( (1-2\nu)\big((\mathbf{\hat{r}}_{ik}\cdot\mathbf{\hat{r}}_{jk}) \mathbf{\underline{\hat{I}}}+ \mathbf{\hat{r}}_{jk}\mathbf{\hat{r}}_{ik} \big) \notag\\
&{}&+ 3 (\mathbf{\hat{r}}_{ik}\cdot\mathbf{\hat{r}}_{jk}) (\mathbf{\hat{r}}_{ik}\mathbf{\hat{r}}_{ik} + \mathbf{\hat{r}}_{jk}\mathbf{\hat{r}}_{jk}) - \mathbf{\hat{r}}_{ik}\mathbf{\hat{r}}_{jk} \Big) \notag\\
&{}&+ 3\left(7-4\nu - 15 (\mathbf{\hat{r}}_{ik}\cdot\mathbf{\hat{r}}_{jk})^2\right)\mathbf{\hat{r}}_{ik}\mathbf{\hat{r}}_{jk}
\Big]. $$
The corresponding expressions for *incompressible* systems in the main text readily follow from Eqs. (\[suppl\_eq\_displaceability\_matrix\])–(\[suppl\_M\_tt\_3\]) by setting the Poisson ratio $\nu=0.5$. Here, we derived and listed the more general expressions for *compressible* elastic matrices.
Four-particle system
--------------------
In addition to the two- and three-particle samples described in the main text, we also generated and analyzed four-particle systems. Their preparation, experimental analysis, and the corresponding comparison with the theory are in complete analogy to the three-particle system described in the main text. We recall that our theoretical description in the main text *up to the investigated order* (including $r_{ij}^{-4}$) is exact for arbitrary particle numbers. No higher-body interactions appear to this order. Therefore, Eqs. (10)–(14) in the main text also apply to systems of particle numbers $N>3$ up to (including) order $r_{ij}^{-4}$, i.e. if the particle separations are not significantly reduced. Thus, our four-particle results predominantly provide a supplement to the results presented in the main text. Our experimental and theoretical results for the four-particle system are depicted in Fig. \[fig\_4particle\]. One could continue to further increasing particle numbers in the same way.
{width=".8\textwidth"}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the long wavelength domain, typically for wavelengths , the laser fields are usually taken as independent of the spatial coordinate. However, at the gas-solid interface the electron density of the material and the incident laser fields vary sharply on a scale of few angströms. Instead of solving Maxwell equations, we present here a theoretical model, called *Electromagnetic Fields from Electron Density* (EMFED), generating a continuous vector potential $\vec{A}\left(\vec{r},t\right)$ from phenomenological relations combining the unperturbed electron density of the material system, the material constants and the laws of optics. As an application of this model, we calculate in a time dependent approach the transition probability and the induced current density between the last bulk state below the Fermi energy and the first image state of a Cu(001) metallic surface. These observables are significantly modified by the spatial variation of the vector potential at the surface. The Coulomb gauge condition ($\vec{\nabla}\cdot\vec{A}=0$), fullfilled everywhere else, breaks down near the surface. The difference between the $\mathfrak{s}$- and $\mathfrak{p}$- polarizations of the laser field partially unravels this effect.'
author:
- 'Georges Raşeev[^1]'
- Eric Charron
title: 'Spatial variation of the laser fields and electron dynamics at a gas-solid interface'
---
Introduction
============
Because of its importance for the photoelectric effect, many studies [@makinson:49; @adawi:64; @mahan:70_b; @endriz:73] have modeled the spatial variation of a laser electric field at a gas-solid (metal) interface in the long wavelength (LWL) domain (wavelengths $\lambda \geqslant 100$ Å or, equivalently, energies ). This electric field has been used to calculate the laser-matter interaction and the associated photoelectric probability. In 1975 Feibelman [@feibelman:75] has evaluated the spatial dependence of the laser vector potential for a gas-solid Jellium interface solving the Maxwell equations using a nonlocal conductivity tensor. This model has been applied by Levinson *et al.* [@levinson:79] to photoemission and the results compared to experimental measurements.
More recently, Miller *et al.* [@miller:chiang:1997] have used the one step model of Mahan [@mahan:70_b], built on a Jellium potential displaced towards the vacuum to take into account a surface state, to study the direct, indirect and surface one photon photoemission of the Ag(111) surface. As in the work presented in this paper, this model uses a laser-matter operator in the velocity gauge written as a sum of a standard $\vec{A}\cdot\vec{\nabla}$ and a surface $(\vec{\nabla}\cdot\vec{A})$ interaction term. Instead of an explicit calculation with a vector potential dependent on the spatial coordinate, the surface term has been fitted to the experimental spectrum to explain the unsymmetrical line shapes present in these data. Other models, derived from the work of Miller *et al.*, have used the velocity gauge in the analysis of a two-photon photoemission experiment of metallic silver [@pontius:petek:2005] near the surface plasmon resonance.
Using the same velocity form of the laser-matter interaction, Tergiman *et al.* [@tergiman:girardeau:97] have calculated the contribution to a photo-current of a multiphoton excitation by solving analytically the Schrödinger equation for the electrons. These authors have modified the Fermi-Dirac distribution describing the initial state of the electrons to include the contributions from electron-electron collisions. With this model, Tergiman *et al.* were able to evaluate the linear and nonlinear multiphoton contributions to the photo-current as a function of the laser fluence.
In recent years many experimental studies in photoelectron emission have used the versatile two-photon photoemission (TPPE) technique [@berthold:hofer:2004; @kubo:petek:2005; @kirchmann:wolf:2005; @rohmer:aeschlimann:bauer:2006] to study the temporal evolution of the electron dynamics and of the associated observables on clean and covered surfaces. Today the majority of theoretical simulations of TPPE are performed using the density matrix theory in its Liouville-von Neumann form [@wolf:1999; @klamroth:saalfrank:hofer:2001; @boger:weinelt:2002; @pontius:petek:2005; @boger:fauster:weinelt:2005; @mii:ueba:2005]. The density matrix formulation has the advantage of permitting the inclusion of the elastic and inelastic electron-electron collisions which constitute efficient desexitation channels when the physical system is in interaction with the bath of the solid. Note that, except for the work of Pontius *et al.* [@pontius:petek:2005], these density matrix formulations make use of the length gauge and the laser electric field is assumed to be independent of the spatial coordinate.
In this article we discuss the laser-matter interaction at a gas-solid interface in the long wavelength domain. The sharp rise of the electron density at the gas-solid interface, taking place on a sub-nanometric scale, affects the electromagnetic fields through a steep variation of the refractive index even if the particular atoms of the solid are not directly “seen" by the electromagnetic fields. We try to answer the following question: what is the influence of the spatial dependence of the laser fields on the electron excitation at the interface and on the associated observables? This question may be of importance especially in the presence of adsorbates and nano structures. In a formulation where the vector potential $\vec{A}$ is a function of $\vec{r}$, there is an additional contribution to the laser-matter interaction coming from the operator $(\vec{\nabla}\cdot\vec{A})$ mentioned above. This contribution is known as a “surface photoelectric effect” (see e.g. Desjonquères and Spanjaard [@book:desjonqueres_spanjaard]).
The above question is answered here in the framework of the Schrödinger equation, for a clean metallic surface in the context of a single excitation between a discretized state of the band of the solid and an image state. We have developed a phenomenological model, called *Electromagnetic Fields from Electron Density* (EMFED), where the electromagnetic fields are explicitly function of the coordinate $z$ normal to the surface. A simplified version of this model, where the electron density used to generate the vector potential was calculated from a Jellium solid, has already been published [@laser:matter:raseev:03; @des:quant:bejan:03; @emed:al:raseev:2005]. Here, the laser fields are calculated using macroscopic material constants, conductivity $\sigma$, dielectric function $\varepsilon$ and refractive index $\tilde{n}$, which depend on the chemical nature of the bulk material and on the laser wavelength. These material constants, taken from experimental measurements tabulated by Palik [@book:palik_opt_cont; @book:palik_opt_cont_II], are considered to be local relative to the spatial coordinate $z$, *i.e.* $\varepsilon(z,z')\simeq \varepsilon(z)$. To explicitly obtain the $z$-dependence of the laser field, the dielectric function $\varepsilon$ of the metal is related to the unperturbed electron density $\rho_e(z)$ of the metallic surface evaluated from the corresponding wave functions of the occupied states of the system. As in the Jellium calculations performed by Lang and Kohn [@density:lang:kohn:70] using DFT (see also Jenning *et al.* [@electron:jenning:1988]), the calculated electron density is normalized with respect to its value in the bulk. The dielectric function $\varepsilon(z)$ is then connected to the vector potential $\vec{A}\left(\vec{r},t\right)$ through its wave vector $\vec{k}^{ph}$ *via* the refractive index $\tilde{n}$.
Using the EMFED vector potential $\vec{A}\left(\vec{r},t\right)$, the Hamiltonian describing the interaction of the electrons of a metallic surface with the laser field is constructed. In the actual implementation of the model, the motion parallel and perpendicular to the surface are decoupled. Consequently, the wave function is expanded in terms of products of parallel and perpendicular to the surface basis functions. In the directions parallel to the surface we use a linear combination of atomic orbitals which fulfills the Bloch periodicity condition, and the set of functions perpendicular to the surface is, for simplicity, restricted to a single resonant contribution. The time dependent Schrödinger equation is projected on this basis, yielding a system of coupled first order ordinary differential equations (ODE) for the time variable. This ODE system of coupled equations is propagated for the duration of the laser pulse and the excitation probability and electron current density are calculated.
Specifically we have studied a clean Cu(001) surface where the bulk continuum near the Fermi level is discretized and we have considered a resonant excitation between this bulk Fermi and the first image states. The laser fluence used in the present simulation is in the range of the experiments by Velic *et al.* [@velic:wolf:98] for C$_6$H$_6$ on Cu(111) or Kirchmann *et al.* [@kirchmann:wolf:2005] for C$_6$F$_6$ on Cu(111), and is well within the perturbation regime.
The laser field at the interface: the EMFED model {#sec:EMFED_model}
=================================================
The vector potential of the laser field derived from the laws of optics {#sec:A_pot}
-----------------------------------------------------------------------
The geometry considered here is depicted in Figure \[fig:Geometry\]: the plane of incidence (POI) of the laser beam on the surface is defined as $(xOz)$, $z$ being the direction normal to the surface, pointing towards the metal. The three-dimensional space is therefore divided in two parts: $z < 0$ corresponds to the gas phase and $z \geqslant 0$ corresponds to the solid. The angles $\theta_i$ and $\theta_t$ stand for the incident and transmitted angles. The angle between the POI and the crystallographic direction $\vec{u}$ of the surface is $\varphi$. The incoming electromagnetic wave, whose wave vector is denoted by $\vec{k}^{ph}$, is $\mathfrak{p}$- polarized when the vector potential $\vec{A}\left(\vec{r},t\right)$ (notation $^{\mathfrak{p}}\!\vec{A}$) is in the POI, and $\mathfrak{s}$- polarized when this vector (notation $^{\mathfrak{s}}\!\vec{A}$) is normal to the POI.
![(color online) Schematic picture of reflexion and refraction of a polarized laser beam incident on a metallic (001) surface of a face centered cubic (FCC) crystal. The FCC lattice parameter $a_0$ is shown and the shortest distance between two atoms on the surface corresponds to $a_0^u=a_0^v=a_0/\sqrt{2}$ in the $u$ and $v$ directions. The vector potentials $^{\mathfrak{s}}\!\vec{A}$ and $^{\mathfrak{p}}\!\vec{A}$, respectively for ${\mathfrak{s}}$- and ${\mathfrak{p}}$- polarizations, point in the directions perpendicular and parallel to the POI or $xOz$ (see text for details).[]{data-label="fig:Geometry"}](Fig1){width="8cm"}
In the following, we label $(i)$, $(r)$ and $(t)$ the incident, reflected and transmitted components of the vector potential, taken as classical here. Taking into account the continuity of the fields at the interface (see e.g. Jackson [@book:jackson]), one writes the normal projection (coordinate $z$) of the vector potential of a $\mathfrak{p}$- polarized monochromatic laser beam as $$\begin{aligned}
\label{eq:A_z}
\left\{\begin{array}{ccll}
^{\mathfrak{p}}\!A_z(x,z,t) & = & \sin\theta_i \left[^{\mathfrak{p}}\!A^{(i)} + \,^{\mathfrak{p}}\!A^{(r)}\right] & (z < z_P)\\
^{\mathfrak{p}}\!A_z(x,z,t) & = & \sin\theta_t\; \tilde{\varepsilon}(\omega,z)
\;^{\mathfrak{p}}\!A^{(t)} & (z \geq z_P),
\end{array}
\right.\end{aligned}$$ and the tangent projection (transverse coordinate $x$) as $$\begin{aligned}
\label{eq:A_x}
\left\{\begin{array}{ccll}
^{\mathfrak{p}}\!A_x(x,z,t) & = & \cos\theta_i \left[^{\mathfrak{p}}\!A^{(i)} - \,^{\mathfrak{p}}\!A^{(r)}\right] & (z < z_P)\\
^{\mathfrak{p}}\!A_x(x,z,t) & = & \cos\theta_t\; ^{\mathfrak{p}}\!A^{(t)} & (z\ge z_P),
\end{array}
\right.\end{aligned}$$ where $z_P$ is the position of the image plane which can be different from the position $z=0$ of the geometrical surface (see next subsection). In Eq. (\[eq:A\_z\]), one should notice the appearance of the complex relative local dielectric function $\tilde{\varepsilon}(\omega,z)$, function of the $z$ coordinate.
For a $\mathfrak{s}$- polarized monochromatic laser beam, the vector potential is directed perpendicular to the POI, and therefore is parallel to the $y$ direction in our coordinate system, and $$\begin{aligned}
\label{eq:A_y}
\left\{\begin{array}{ccll}
^{\mathfrak{s}}\!A_y(x,z,t) & = & ^{\mathfrak{s}}\!A^{(i)} + \,^{\mathfrak{s}}\!A^{(r)} & (z < z_P)
\\
^{\mathfrak{s}}\!A_y(x,z,t) & = & ^{\mathfrak{s}}\!A^{(t)} & (z\ge z_P).
\end{array}
\right.\end{aligned}$$ The incident, reflected and transmitted components of the vector potential used above are simply given by the standard classical expressions $$\label{eq:A_irt}
\begin{array}{llrrl}
^{\mathfrak{pol}}\!A^{(i)}(x,z,t) & = &\! &\! \!A_{0} \!&
\!e^{i\left(k^{ph}_x x\,+\,k^{ph}_z z\,-\,\omega t\right)}
\\
^{\mathfrak{pol}}\!A^{(r)}(x,z,t) & = &\! ^{\mathfrak{pol}}\!R(\omega,z)
&\!\! A_{0}\! & \!e^{i\left(k^{ph}_x x\,-\,k^{ph}_z z\,-\,\omega t\right)}
\\
^{\mathfrak{pol}}\!A^{(t)}(x,z,t) & = &\! ^{\mathfrak{pol}}T(\omega,z)
&\!\! A_{0}\! & \!e^{i\left(k^{ph}_x x\,+\,k^{ph}_z z\,-\,\omega t\right)}
\end{array}$$ where $\mathfrak{pol}$ corresponds to the $\mathfrak{p}$ or $\mathfrak{s}$ linear polarizations. The reflection $^{\mathfrak{pol}}\!R(\omega,z)$ and transmission $^{\mathfrak{pol}}T(\omega,z)$ coefficients are given by
\[eq:A\_irt\_coeff\_p\] $$\begin{aligned}
^{\mathfrak{p}}\!R(\omega,z) &=&
\frac{\tilde{n}(\omega,z)\;\cos\;\theta_i\;-\;\cos\;\theta_t}
{\tilde{n}(\omega,z)\;\cos\;\theta_i\;+\;\cos\;\theta_t},
\\
^{\mathfrak{p}}T(\omega,z) &=& \frac{2\cos\;\theta_i}
{\tilde{n}(\omega,z)\;\cos\;\theta_i\;+\;\cos\;\theta_t},\end{aligned}$$
and
\[eq:A\_irt\_coeff\_s\] $$\begin{aligned}
^{\mathfrak{s}}\!R(\omega,z) &=&
\frac{\cos\;\theta_i\;-\;\tilde{n}(\omega,z)\cos\;\theta_t}
{\cos\;\theta_i\;+\;\tilde{n}(\omega,z)\;\cos\;\theta_t},
\\
^{\mathfrak{s}}T(\omega,z) &=& \frac{2\cos\;\theta_i}
{\cos\;\theta_i\;+\;\tilde{n}(\omega,z)\;\cos\;\theta_t},\end{aligned}$$
where the complex refraction index $\tilde{n}(\omega,z)$, also function of $z$ coordinate, is obtained from the EMFED model as described in the next subsection. The attenuation of the field in the metal is simply taken into account through a complex wave vector $$\label{eq:k_ph}
k^{ph} = \tilde{n}\;\omega/c.$$ In Eq. (\[eq:A\_irt\]), the normalization factor $A_{0}$ defines the amplitude of the incident laser vector potential. It can be calculated from the fluence $F$ or the intensity $\mathfrak{I}_0$ of the laser pulse. If the vector potential is described by a $\sin^2$ pulse shape of full width at half maximum (FWHM) $\tau$, one gets $$\begin{aligned}
\label{eq:A_0}
A_0(\mathrm{a.u.}) & = & \frac{1.66\;10^{-3}}{i\;\omega(\mathrm{a.u.})}\;\sqrt{\frac{F(\mathrm{J/m^2})}{\tau(\mathrm{fs})}}.\end{aligned}$$ The numerical relation between the fluence $F$ and the laser intensity $\mathfrak{I}_0$ for this pulse shape is $$F(\mathrm{J/m^2})=0.9708\;\mathfrak{I}_0(\mathrm{W/m^2})\;\tau(\mathrm{s}).$$ The angle of refraction $\theta_t$ is, in this phenomenological model, a complex function of the incident laser angular frequency $\omega$ and of the coordinate $z$. It is calculated from the law of Snell-Descartes $$\label{eq:theta_t}
\sin\;\theta_t = \frac{\sin\;\theta_i}{\tilde{n}(\omega,z)}\;.$$
Material constants from the conductivity and the electron density
-----------------------------------------------------------------
The equations (\[eq:A\_z\]-\[eq:A\_y\]) are standard expressions of continuity at an abrupt interface. Here the material constants, as for example the dielectric function $\tilde{\varepsilon}$ or the refractive index $\tilde{n}$, are functions of the coordinate $z$. This dependence also induces a dependence relative to $z$ of the associated wave number $k^{ph}$ (Eq. (\[eq:k\_ph\])) and of the refraction angle $\theta_t$ (Eq. (\[eq:theta\_t\])).
Following Drude’s model [@Drude], at optical and higher frequencies, both bound and conduction electrons of a metal contribute to the dielectric function through the relation $$\label{eq:Esigma}
\tilde{\varepsilon}(\omega)=\varepsilon_b(\omega) + i\;\frac{\tilde{\sigma}(\omega)}{\omega\,\varepsilon_0}\,,$$ where $\varepsilon_b(\omega)$ is the relative permittivity due to the bound electrons, $\varepsilon_0$ denotes the vacuum permittivity (only non-magnetic materials are considered here with a relative permeability $\mu=1$) and $\tilde{\sigma}(\omega)$ is the complex conductivity (see e.g. Jackson[@book:jackson], chapter 7). The Drude model also gives the following analytical expression for the conductivity $$\label{eq:conductivity}
\tilde{\sigma}(\omega) = \frac{e^2}{m^{*}\, (\gamma_0-i\;\omega)}\;\rho^s\,,$$ where $e$ is the charge of the electron, $m^*$ its effective mass, $\rho^s$ the electron density of the bulk and $\gamma_0$ the electron friction coefficient.
The above equations relate $\tilde{\varepsilon}(\omega)$ and $\rho^s$ and can be used as a template to model in a simple way the spatial dependence of the dielectric function at the surface $$\label{eq:dielectric_analytique}
\tilde{\varepsilon}(\omega,z) = 1 + \left(\tilde{\varepsilon}_s(\omega) - 1\right)\,\rho_e(z),$$ where $\tilde{\varepsilon_s}(\omega)$ is the complex relative dielectric function in the solid and $\rho_e(z)$ is the relative electron density normalized with respect to the bulk density $\rho^s$. Equation (\[eq:dielectric\_analytique\]) has the advantage of imposing a continuous variation of $\tilde{\varepsilon}(\omega,z)$ through the interface with a linear dependence on $\rho_e(z)$. In addition this expression shows the correct asymptotic limits $$\left\{
\begin{array}{cccl}
\tilde{\varepsilon}(\omega,-\infty) & = & 1 &\\
\tilde{\varepsilon}(\omega,+\infty) & = & \tilde{\varepsilon}_s(\omega) & .
\end{array}
\right.$$
Equation (\[eq:dielectric\_analytique\]) is the heart of the EMFED model which uses both the unperturbed relative electron density of the material system $\rho_e(z)$ under study and the tabulated values [@book:palik_opt_cont; @book:palik_opt_cont_II] of the associated complex refractive index of the bulk ($\tilde{n}_s^2(\omega)=\tilde{\varepsilon}_s(\omega)$).
Using the density functional theory (DFT), the spatial variation of the electron density $\rho_e(z)$ at a gas-Jellium interface has been calculated by Lang and Kohn [@density:lang:kohn:70] for several electrons densities in the bulk $\rho^s$, or Wigner-Seitz radii $r_s=(3/(4\pi\rho^s))^{1/3}$. A calculation of this electron density from the occupied states of the material system, using a smooth model Jellium potential at the interface due to Jennings *et al.*[@electron:jenning:1988], gives similar results. In the present implementation of the EMFED model to a metallic surface, we calculate this electron density from a more accurate model potential perpendicular to the surface parametrized by Chulkov *et al.* [@des:quant:chulkov:99]. The Jellium and Chulkov *et al.* potentials are displayed as red solid and black dashed lines in the lower part of Figure \[fig:total\_density\_cu\_001\]. For a given potential, the total relative electron density $\rho_e(z)$ is expressed as a discretized sum $$\begin{aligned}
\label{eq:electron_density}\displaystyle
\rho_e(z) &\varpropto&\sum_{I=1,n_F} |\eta_I(z)|^2\end{aligned}$$ running over all the occupied states up to the Fermi level. $\eta_I(z)$ denote here the wavefunctions associated with the eigenstates of the one-dimensional time-independent Schrödinger equation in the $z$ direction. This relative electron density $\rho_e(z)$ is normalized with respect to the mean average value of the density $\rho^s$ deep in the bulk. The sum is discretized because we have adopted a discrete variable representation (DVR) approach [@dvr:bacic:light:1986; @dvr:colbert:miller:1992] for the description of the continuum band structure of the metal. The electron densities calculated with the potential of Chulkov *et al.* and with the Jellium potential are displayed as red solid and black dashed lines in the upper part of Figure \[fig:total\_density\_cu\_001\]. For the Cu(001) surface studied here, the electrons at the Fermi level belong to the occupied band, and, because of the use of the DVR, the band is discretized. The last discrete state below the Fermi energy, hereafter called bulk Fermi state, and the first image state ($n=1$) are displayed in the lower part of Figure \[fig:total\_density\_cu\_001\]. These states participate in the photoexcitation considered here. Note that in the present case these states and the total electron density extend over several ängstroms in the vacuum. Note also the difference in the amplitude of the oscillations of the densities and potentials between the Chulkov *et al.* and Jellium models. This difference is due to the explicit inclusion of the atomic structure in the first but not in the second model.
The refraction index is obtained from the standand relation to the dielectric function $$\label{eq:refraction_index}
\tilde{n}(\omega,z)=\sqrt{\varepsilon(\omega,z)}\,.$$ Using the electron density (\[eq:electron\_density\]), the analytic form of the dielectric function (\[eq:dielectric\_analytique\]), and the associated refraction index one can calculate, from the expressions of subsection \[sec:A\_pot\], the vector potential $\vec{A}\left(\vec{r},t\right)$ of the laser field at the interface. This simple procedure allows us to introduce the spatial variation of the vector potential at the gas-solid interface.
The EMFED model explained above can tentatively be understood from the point of view of the solutions of the Maxwell equations. Far in the vacuum the asymptotic condition corresponds to the incident plane wave $ ^{\mathfrak{pol}}\!A^{(i)}$ of Eq. (\[eq:A\_irt\]). The propagation of the Maxwell equations towards the interface smoothly accumulates the presence of the electron density in the vector potential. At the end of the propagation far in the bulk the vector potential corresponds to the transmitted component $^{\mathfrak{pol}}\!A^{(t)}$. In the EMFED model, the reflected component contributes near the surface, therefore allowing for the continuity of the electromagnetic fields at the interface. This result is therefore qualitatively similar to a model of Feibelman [@feibelman:75_l] who solved numerically Maxwell equations for a vacuum/Jellium system.
![(color online) Lower graph (a): The electron-matter interaction potentials for Cu(001) using the Chulkov *et al.* (red solid line) or Jellium with $r_s$=2.67 a.u. (black dashed line) models. The probability densities $\left|\eta_I(z)\right|^2$ associated with the bulk Fermi and first image states accommodated by the Chulkov potential are also shown at their energy (- 0.12 eV from $E_F$; -0.574 eV from the vacuum) in blue and green respectively. Upper graph (b): the total electron densities calculated using Eq. (\[eq:electron\_density\]) with the same colors and line types. The vertical line at $z_P=-1.31\,$Å corresponds to the position of the image plane $P$ at $\rho_e(z_P)=0.5$ where the continuity of the vector potential is enforced.[]{data-label="fig:total_density_cu_001"}](Fig2_dens_pot){width="8cm"}
For the $\mathfrak{s}$- polarization, the vector potential presents a single $y$ component displayed in Figure \[fig:cu\_001\_As\_y\]. It shows a smooth variation everywhere and particularly near the image plane of the Cu(001) system at corresponding to the electron density $\rho_e(z_P)=0.5$ where, following Eqs. (\[eq:A\_z\]), (\[eq:A\_x\]) and (\[eq:A\_y\]), the continuity of the vector potential is enforced. This image plane can be shifted from $\rho_e(z_P)=0.2$ to $\rho_e(z_P)=0.8$ with nearly no consequence on the vector potential itself. Similar smooth behavior is obtained in the calculation of Feibelman [@feibelman:75_l]. The dashed lines in Figure \[fig:cu\_001\_As\_y\] correspond to a calculation with a constant vector potential in each medium where only the incident and transmitted components are taken into account.
Various derivations and models have also used the material constants in the two media together with the continuity at the interface to obtain the corresponding properties of the laser-matter interaction. For example, Feibelman [@feibelman:1989] obtained in this way the surface plasmon frequency and Liebsch and Schaich [@liebsch:schaich:1995] explained the silver anomalous dispersion of the surface plasmon frequency.
Because we are not solving explicitly the Maxwell equations for the electromagnetic fields, the problem of the gauge used here has not been touched up to now. How behaves the external electromagnetic field in the two media (vacuum and bulk) and at the interface from the point of view of the gauge of the electromagnetic wave? The vector potential can be decomposed in perpendicular and parallel components with respect to the surface plane as $\vec{A}=\vec{A}_{\perp}+\vec{A}_{\parallel}$. A simple look at Figure \[fig:Geometry\] gives $$\label{eq:fieldp}
\left\{
\begin{array}{cccl}
^\mathfrak{p}\!A_{\perp} & = & ^\mathfrak{p}\!A_z(x,z)\\
^\mathfrak{p}\!A_{\parallel} & = & ^\mathfrak{p}\!A_x(x,z)
\end{array}
\right.$$ for the $\mathfrak{p}$- polarization, and $$\label{eq:fields}
\left\{
\begin{array}{cccl}
^\mathfrak{s}\!A_{\perp} & = & 0\\
^\mathfrak{s}\!A_{\parallel} & = & ^\mathfrak{s}\!A_y(x,z)
\end{array}
\right.$$ for the $\mathfrak{s}$- polarization. In this last case of $\mathfrak{s}$- polarization, it is already clear that everywhere, and the Coulomb gauge condition is therefore verified in our model for this specific polarization. The case of $\mathfrak{p}$- polarization is clearly different since the fast variation of $^\mathfrak{p}\!A_{\perp}$ with $z$ in the vicinity of the surface implies that when approaching the metal. As a consequence, the electromagnetic fields given by the EMFED model do not verify the Coulomb gauge for the $\mathfrak{p}$- polarization. This discussion also applies to the Schrödinger equation, and the laser-matter interaction term in the Hamiltonian includes a non-Coulomb gauge term proportional to $(\vec{\nabla}\cdot\vec{A})$.
![(color online) Real and imaginary components $y$ of the vector potential $\vec{A}$ as a function of the coordinate $z$ for a $\mathfrak{s}$- polarized laser beam. Following Eq. (\[eq:A\_irt\]), the $x$ dependence in Eq. (\[eq:fields\]) corresponds to a plane wave and therefore the dependence of the vector potential relative to this variable is not shown in the present figure. The photon energy and the laser fluence are 4.169 eV and 10 $\mu$J/cm$^2$ respectively. The real and imaginary refraction indexes of the bulk metallic Cu, $n$=1.386 and $\kappa$=1.687 at 4.2 eV, are taken from the tables of Palik [@book:palik_opt_cont; @book:palik_opt_cont_II]. The black Re$[\vec{A}]$ and red Im$[\vec{A}]$ solid lines are the results obtained with the EMFED model, while the dashed lines correspond to the incident and transmitted fields in a simple approach from optics with constant vector potentials (quotted as gas/solid in the figures below).[]{data-label="fig:cu_001_As_y"}](Fig3_A_s){width="8cm"}
A time dependent model for the electronic dynamics at the interface {#sec:time_dependent_model}
===================================================================
The Hamiltonian {#sec:hamiltonian}
---------------
The total electronic Hamiltonian including the laser-matter interaction is similar to the Hamiltonian for the unperturbed system except that the electron momentum $\vec{p}$ is replaced by ($\vec{p}+e\vec{A}$), where $e$ and $\vec{A}(\vec{r},t)$ are the charge of the electron and the vector potential of the laser field. The total Hamiltonian reads $$\begin{aligned}
\label{eq:Hamiltonian}
\hat{H}(\vec{r},t) & = & \hat{H}^0(\vec{r}) + \hat{H}^L(\vec{r},t) + \hat{H}^Q(\vec{r},t),\end{aligned}$$ where the zero order Hamiltonian is $$\hat{H}^0(\vec{r}) = \frac{\vec{p}\,^2}{2\;m} + V(\vec{r}).$$ The laser-matter interaction terms of the Hamiltonian can be decomposed as
$$\begin{aligned}
\label{eq:H_L}
\hat{H}^L(\vec{r},t) &=&\frac{\hbar e}{2 i m}\bigl[2\;
\vec{A}\cdot \vec{\nabla} + (\vec{\nabla}\cdot\vec{A})\bigr]\\
\label{H_Q}
\hat{H}^Q(\vec{r},t) &=& \frac{e^2}{2m}[\vec{A}\cdot\vec{A}],\end{aligned}$$
where the parentheses in the first equation above mean that the operator $\vec{\nabla}$ acts on the vector potential $\vec{A}(\vec{r},t)$ only. The superscripts $L$ and $Q$ stand respectively for the linear and quadratic terms in $\vec{A}$.
$\hat{H}_{\parallel}^L$ $\hat{H}_{\perp}^L$ $\hat{H}_{\parallel}^Q$ $\hat{H}_{\perp}^Q$
---------------- -------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------- ------------------------- ---------------------
$\mathfrak{p}$ $2\;A_x\;\frac{\partial}{\partial x} + \left(\frac{\partial A_x}{\partial x}\right)$ $2\;A_z\;\frac{\partial}{\partial z} + \left(\frac{\partial A_z}{\partial z}\right)$ $A_x^2$ $A_z^2$
$\mathfrak{s}$ $2\;A_y\;\frac{\partial}{\partial y}$ —– $A_y^2$ —–
: \[tab:A\_p\_operator\]Decomposition of the laser-matter interaction Hamiltonian for a $\mathfrak{p}$- and $\mathfrak{s}$- polarized vector potential, respectively $^\mathfrak{p}{\vec{A}(\vec{r},t)}$ and $^\mathfrak{s}{\vec{A}(\vec{r},t)}$.
The analysis given in the preceding section has revealled a non-zero divergence of the vector potential $(\vec{\nabla}\cdot\vec{A})$ in the case of $\mathfrak{p}$- polarization. This non-Coulomb gauge contribution develops in two terms, one perpendicular , and the other parallel $(\partial A_x/\partial x)$ to the surface (see Table \[tab:A\_p\_operator\]). These two terms, called [*surface terms*]{}, give rise to the so called surface photoelectric effect (see e.g. Desjonquères and Spanjaard [@book:desjonqueres_spanjaard]). For $\mathfrak{s}$- polarization, these two terms do not appear, but Table \[tab:A\_p\_operator\] shows that two standard contributions remain in the interaction Hamiltonian: $2\;A_y\partial /\partial y$ and $A_y^2$.
Table \[tab:A\_p\_operator\] also shows that a characterization of the laser field can be performed by calculating the difference between the observables measured for $\mathfrak{p}$ and $\mathfrak{s}$- polarizations. For equivalent directions $x$ and $y$ parallel to the surface, the contributions from $A_x^2+2\;A_x\partial /\partial x$ of $\mathfrak{p}$ and $A_y^2+2\;A_y\partial /\partial y$ of $\mathfrak{s}$- polarizations are equivalent and eliminated from the interaction. Unfortunately $A_z^2+2\;A_z\partial /\partial z$ is not eliminated and therefore one cannot measure directly the influence of the surface terms $(\vec{\nabla}\cdot \vec{A})=(\partial A_x/\partial x) + (\partial A_z/\partial z)$.
The time dependent Schrödinger equation {#subsec:time_dependent_model}
---------------------------------------
The time-dependent Schrödinger equation describing the interaction of the laser field with the electrons at the gas-solid interface reads $$\label{eq:TDSE}
i\hbar\;\frac{\partial}{\partial t}\,\Psi(\vec{r},t) = \hat{H}(\vec{r},t)\,\Psi(\vec{r},t),$$ where $\Psi(\vec{r},t)$ denotes the electronic time-dependent wavefunction. This wave function is expanded on a series of time independent basis functions $\Phi_I (\vec{r})$ $$\begin{aligned}
\label{eq:expansion}
\Psi(\vec{r},t) &=& \sum_I \Phi_I (\vec{r})C_I(t) = \mathbf{\Phi} \mathbf{C},\end{aligned}$$ where $C_I(t)$ are the time dependent expansion coefficients. The second expression corresponds to a matrix notation.
Figure \[fig:Geometry\] shows that the presence of the two media breaks the periodicity in the direction $z$ normal to the surface. On the other hand, and since there is no electron density in the vacuum, we will treat the system as a periodic system in the parallel direction even in the vaccum. In the present version of the model we write the basis wave function $\Phi_I (\vec{r})$ of Eq. (\[eq:expansion\]) as a product $$\label{eq:basis_functions}
\Phi_I(\vec{r}) = \eta_I(z) \; f_I(\vec{r}_{\parallel})\,,$$ where $\eta_I(z)$, already defined with Eq. (\[eq:electron\_density\]), is a wavefunction associated with the eigenstates of the one-dimensional time-independent Schrödinger equation along the perpendicular direction $z$. Along the surface plane direction , the basis functions $f_I(\vec{r}_{\parallel})$ are written as $$\label{eq:f_I}
f_I(\vec{r}_{\parallel}) = \sum_{j}\;\; e^{i\;\vec{k}_{\parallel}^I\cdot\vec{R}_{\parallel}^j}\;\;
\chi_{I}(\vec{r}_{\parallel}-\vec{R}_{\parallel}^j)\,,$$ where $\vec{k}_{\parallel}^I$ denotes the parallel electron wave vector. In this expression, $\chi_{I}$ denotes a function defined in the elementary cell. The vectors $\vec{R}_{\parallel}^j$ allow for the translation of this localized wavefunction in the parallel direction. This generates a two-dimensional periodic function in the surface plane which fulfills the Bloch periodicity condition (see for instance Ziman [@book:ziman]). This approach is similar to the method of linear combinations of atomic orbitals (LCAO) for a solid.
Inserting the expansion (\[eq:expansion\]) in the time-dependent Schröringer equation (\[eq:TDSE\]), and projecting it on the time independent basis functions $\Phi_I(\vec{r})$ yields a coupled system of first order ordinary differential equations (ODE) for the expansion coefficients $C_I(t)$ $$\label{eq:coupled_equations}
i\hbar\frac{\partial C_I}{\partial t} = \sum_{J}\; \langle\Phi_I|\hat{H}(\vec{r},t)|\Phi_J\rangle\; C_J(t)\,.$$ The diagonal terms can be written as $$\label{eq:diagonal}
\langle\Phi_I|\hat{H}(\vec{r},t)|\Phi_I\rangle = E_I^{\perp} + E_I^{\parallel}
+ \langle\Phi_I|\hat{H}^L+\hat{H}^Q|\Phi_I\rangle\,,$$ where $E_I^{\perp}$ and $E_I^{\parallel}$ denote the energies associated with the wavefunctions $\eta_I(z)$ and $f_I(\vec{r}_{\parallel})$ respectively.
For low laser intensities, the off-diagonal laser-matter interaction matrix element can be approximated by its linear component . Here this term is an integral over the entire space. On the other hand, the current density, Eq. (\[eq:courant\_I\_int\]) below, also contains this integral but taken over a restricted domain in the perpendicular direction, namely from $-\infty$ to $z$. One can write this integral, to be used in these two expressions, as $$\label{eq:interaction}
\langle\Phi_I|\hat{H}^L|\Phi_J\rangle_z = \frac{\hbar\,e}{2\,i\,m}\;\mathcal{J}_{IJ}(z)\;e^{-i\,\omega\,t}\,,$$ where the calculation of $\mathcal{J}_{IJ}(z)$ is detailed in Appendix \[annex:A\].
Using the model explained above, two observables are calculated: the excitation probability and the electron current density. The population of the different eigenstates and therefore the excitation probability are obtained from the overlap between the states $\Phi_I(\vec{r})$ and the final wave packet $\Psi(\vec{r},t_f)$ at the end of the pulse ($t=t_f$) $$\label{eq:section_efficace_I}
P_I(t_f) = \left|\;\langle\;\Phi_I(\vec{r})\;\left|\;\Psi(\vec{r},t_f)\;\right.\rangle\;\right|^2 = \left|C_I(t_f)\right|^2.$$ The expression for the electron current density, including the linear momentum of the electron $\vec{p}$ and of the laser field $e\vec{A}$, can be written using the kinetic energy operator $\hat{T}=(\vec{p}+e\,\vec{A})^2/2\;m$ (see e.g. Messiah [@book:messiah]) as $$\begin{aligned}
\label{eq:courant_I}
I(z,t) & = & \frac{1}{i\hbar} \int_{-\infty}^z\!\!\!\!dz'\int_{S_{cell}}\!\!\!\!\!\!\!\!dS\;
\left[\Psi^{\dagger}\hat{T}\Psi-(\hat{T}\Psi)^{\dagger}\Psi\right],\end{aligned}$$ where $S_{cell}$ denotes the surface of the elementary cell along the parallel direction. This total current can be split in two parts $$\begin{aligned}
\label{eq:courant_I_0_int}
I(z,t) & = & I_0(z,t)\;+\;I_{int}(z,t)\,,\end{aligned}$$ with
$$\label{eq:courant_I_0}
I_0 = \frac{\hbar}{2im}\mathbf{C}^{\dagger}\!\!\left[
\int_{-\infty}^z\!\!\!\!\!\!dz'\!\!\int_{S_{cell}}\!\!\!\!\!\!\!\!dS\;
\left(\mathbf{\Phi}^{\dagger}\nabla^2\mathbf{\Phi}-\nabla^2\mathbf{\Phi}^{\dagger}\mathbf{\Phi}
\right)\right]\!\!\mathbf{C}$$
and $$\label{eq:courant_I_int}
I_{int} = -\frac{e}{2m} \mathbf{C}^{\dagger}\left[
e^{-i\omega t}\,\mathbf{J}_{int}(z)-e^{i\omega t}\,\mathbf{J}_{int}^{\dagger}(z)
\right]\mathbf{C}\,.$$
As in Eq. (\[eq:expansion\]), in the previous equations $\mathbf{\Phi}$ is the line vector of the basis functions and $\mathbf{C}$ is the column vector of the coefficients. The term $\mathbf{J}_{int}(z)$ is a matrix whose elements are the integrals $\mathcal{J}_{IJ}(z)$ given in Eqs. (\[eq:J\_int\_p\]) and (\[eq:J\_int\_s\]). In the calculation of $I_{int}$, the quadratic terms in $\vec{A}$ have been neglected.
The physical model and the excitation dynamics {#sec:physmod}
==============================================
Details of the physical model {#subsec:physmod}
-----------------------------
In the present model calculation we concentrate on the influence of the spatial dependence of the laser field in the long wavelength domain (section \[sec:A\_pot\]) on the observables associated with a photoexcitation process.
The model representing the material system is restricted to its simplest meaningfull form which is summarized below. First, the clean Cu(001) surface has been selected since it corresponds to a simple surface structure. The vectors $\vec{u}$ and $\vec{v}$, used to generate the (001) surface displayed in Figure \[fig:Geometry\], are not the primitive vectors of the FCC solid and the associated potential $V(u,v,z)$ is not separable. For example, in the plane of incidence $(xOz)$ there is a shift by $a_0/2$ in the $z$ direction between two adjacent columns of atoms (see the left part of Figure \[fig:Geometry\]). Starting from a DFT calculation, Chulkov *et al.* [@des:quant:chulkov:99] have obtained a model potential in the $z$ direction where the periodicity is $a_0/2$. With this potential in the $z$ direction, the tridimensional potential $V(u,v,z)$, now separable in the first approximation, is used in the present simulation.
Secondly, solving the time-independent Schrödinger equation in this potential yields the set of wave functions defined in Eqs. (\[eq:basis\_functions\]) and (\[eq:f\_I\]). If one discretizes the bands of the solid using a DVR approach [@dvr:bacic:light:1986; @dvr:colbert:miller:1992], the Chulkov potential for the e-Cu(001) system accommodates discretized bulk and image states. In the $z$ direction our grid extends from 70Å in the vacuum to the 200Å in the bulk, with 5881 grid points. As the initial state we select the last occupied discretized bulk state below the Fermi energy located at -0.12 eV from the Fermi energy already designated under the name of bulk Fermi state. The final state is the $n=1$ image state located at -0.574 eV from the vacuum energy. These two states are separated by 4.169 eV and their densities are presented in the lower part of figure \[fig:total\_density\_cu\_001\].
In the direction parallel to the surface we use a periodic wave function given in Eq. (\[eq:f\_I\]). We do not perform an explicit calculation to obtain the optimized wave functions. Instead, in the elementary cell, the localized wave functions $\chi_I(u,v)$ are two-dimensional harmonic oscillator wavefunctions. Following the usual selection rules, we have restricted this basis set to the $s$ and $p$ symmetries only
$$\begin{aligned}
\label{eq:gaussian}
\chi_s(u,v) & = & \sqrt{\frac{2\xi}{\pi}}\;e^{-\xi(u^2+v^2)}\\
\chi_{p_u}(u,v) & = & \sqrt{\frac{8}{\pi}}\,\xi\;u\;e^{-\xi(u^2+v^2)}\\
\chi_{p_v}(u,v) & = & \sqrt{\frac{8}{\pi}}\,\xi\;v\;e^{-\xi(u^2+v^2)}\end{aligned}$$
As shown in Eq. (\[eq:f\_I\]), the periodic wave function $f_I(\vec{r}_{\parallel})$ is obtained from these localized basis functions translating them by a multiple of $a_0^u$ and $a_0^v$ (see the caption of Figure \[fig:Geometry\]). On the Cu(001) surface this distance is $a_0^u=a_0^v=2.556\,$Å and the symmetry of the surface cut imposes the use of the same exponent $\xi$ in the $u$ and $v$ directions. This exponent was chosen such that the resulting wave functions are confined in the elementary cell. For simplicity, and because no optimization is performed parallel to the surface, the energies $E_I^{\parallel}$ of the $s$, $p_u$ and $p_v$ states in Eq. (\[eq:diagonal\]) are set to zero.
The vector potential $\vec{A}(\vec{r},t)$ has been obtained using the EMFED model following the prescriptions given in section \[sec:EMFED\_model\] where the electron density is calculated using the Eq. (\[eq:electron\_density\]) and the wave functions $\eta_I(z)$ (Eq. (\[eq:basis\_functions\]) are obtained using the DVR calculations performed using the grid given above. At the photon excitation energy of 4.169 eV, the complex refraction index $\tilde{n}= 1.386+1.687\,i$ of the solid is taken from the tables of Palik [@book:palik_opt_cont]. It is used to calculate the dielectric constant $\tilde{\epsilon}_s(\omega)$ of the solid and, combined with the electron density from Eq. (\[eq:electron\_density\]), the dielectric function $\tilde{\epsilon}(\omega,z)$ in Eq. (\[eq:dielectric\_analytique\]). Finally, the refraction index $\tilde{n}(\omega,z)$ is calculated from Eq. (\[eq:refraction\_index\]) and used to construct the EFMED vector potential in subsection \[sec:A\_pot\].
In the calculations, the incident angle of the laser beam is fixed at $\theta_i$=45 degrees except for the one presented in figure \[fig:observables\_theta\] where this angle is varied. For the Cu(001) surface, the observables calculated in the present work are independent of the orientation angle $\varphi$ of the POI (see Fig. \[fig:Geometry\]). We have therefore fixed the orientation of the plane of incidence (POI) parallel to the $u$ axis of the material system i.e. $\varphi$=0. With these parameters, the vector potential $\vec{A}(\vec{r},t)$ is calculated using Eqs. (\[eq:A\_z\]-\[eq:A\_y\]). We consider a pulsed laser with a peak intensity of 10$^8$ W/cm$^2$ and a temporal FWHM of $\tau=80$ fs resulting in a fluence of 10 $\mu$J/cm$^2$. This fluence is well within the perturbation regime, where space charges near the electron analyzer do not influence TPPE measurements [@velic:wolf:98; @kirchmann:wolf:2005].
The calculations are performed at the $\Gamma$ point and the dispersion is neglected. Therefore, in equation (\[eq:integral\_7\]) of the appendix, only the first term is calculated. With the present choice of basis functions parallel to the surface, the laser-matter interaction terms between the $s$ and $p_u$ or $p_v$ states, given in Eq. (\[eq:integral\_8\]), reduce to $${^u}\mathcal{D}^{00,00}_{IJ} = \int\!\!\!\!\int \chi^*_{s}\;\frac{\partial}{\partial u}\;\chi_{p_u}\;du\,dv = \sqrt{\xi}\,,$$
where $s$ and $p_u$ belong to different states $I$ perpendicular to the surface. The ODE system of coupled equations (\[eq:coupled\_equations\]) contains six states, three $s$, $p_u$ and $p_v$ for the initial and the final states. In the unambiguous cases these initial and final states will be referred simply to as $i$ and $f$. The integrals over the basis functions for the e-surface potential and the laser-matter interaction have been obtained once for all and the system of six coupled ODE (\[eq:coupled\_equations\]) has been solved at each time step by a computer program based on a predictor-corrector algorithm [@ab:shampine:burkardt:1975]. The obtained time dependent expansion coefficients permit the calculation of the excitation probability (Eq. (\[eq:section\_efficace\_I\])) and of the current density (Eqs. (\[eq:courant\_I\])-(\[eq:courant\_I\_int\])).
![(color online) Temporal evolution of the excitation probability at resonant energy $\hbar\omega= E_{f}-E_{i}=4.169$ eV for a pulsed laser FWHM of $\tau=80$ fs and intensity of . The displayed time interval corresponds to a total duration of 220 fs. Upper graph: Excitation probability calculated with the EMFED vector potential with the laser polarizations $\mathfrak{p}$ and $\mathfrak{s}$. The direction of the incident light is $\theta_i$=45 degrees and the POI is parallel to the $(100)$ direction that corresponds to $\varphi=0$ in Figure \[fig:Geometry\]. Lower graph: Excitation probability calculated with the $\mathfrak{p}$-polarization for the EMFED vector potential (black full line), for an abrupt vector potential at the interface (gas/solid, blue dashed line) and for a constant vector potential from the gas phase (gas, green dotted dashed line).[]{data-label="fig:res_excitation_probab"}](Fig4_csec){width="8cm"}
Excitation dynamics
-------------------
![(color online) Current density $I_0$, corresponding to the kinetic energy operator $\nabla^2$ obtained from the Eq. (\[eq:courant\_I\_0\]), as a function of the coordinate $z$ at the peak of the laser pulse. This current density is normalized to the maximum of $^\mathfrak{s}\!I_0$ using the normalization factor of 6.5$\times$10$^{-5}$. The characteristics of the laser are given in the caption of Figure \[fig:res\_excitation\_probab\].[]{data-label="fig:res_current_0"}](Fig5_i0_z){width="8cm"}
![Interaction current density $I_{int}$ for the $\mathfrak{p}$ polarization of the laser corresponding to the operator $2\;\vec{A}\cdot\vec{\nabla} + (\vec{\nabla}\cdot\vec{A})$ at the peak of the laser pulse as a function of the coordinate $z$ calculated from the Eq. (\[eq:courant\_I\_int\]). This current density is normalized using the factor 3.37$\times$10$^{-5}$. The characteristics of the laser pulse are given in the caption of Figure \[fig:res\_excitation\_probab\].[]{data-label="fig:res_current_int"}](Fig6_I_int){width="8cm"}
![(color online) Temporal evolution of the extrema of the current density $I_0$ at resonance $\hbar\omega=4.169$ eV for the $\mathfrak{p}$- and $\mathfrak{s}$- polarizations of the laser. The characteristics of the laser pulse are given in the caption of Figure \[fig:res\_excitation\_probab\]. The current density is normalized to the maximum of $^\mathfrak{s}\!I_{0}$ using the normalization factor 1.31$\times$10$^{-4}$. Upper graph: the displayed temporal region corresponds to the peak of the laser pulse. Lower graph: envelope of the temporal evolution of the extrema of the current density $I_0$.[]{data-label="fig:res_current_0_temp"}](Fig7_I_0_t){width="8cm"}
The excitation probability between the bulk Fermi state and the first image state has been calculated at resonance for the two laser polarizations $\mathfrak{p}$ and $\mathfrak{s}$. The evolution of this excitation probability during the laser pulse is presented in Figure \[fig:res\_excitation\_probab\]. It is the sum of the excitation probabilities of the $s$, $p_u$ and $p_v$ image states calculated using the Eq. (\[eq:section\_efficace\_I\]). This observable does not display the oscillatory pattern related to the phase of the laser field since the square in eq. (\[eq:section\_efficace\_I\]) annihilate the phase of the expansion coefficients and the rise and the stabilization of the excitation probability is smooth.
As seen in the upper part of Figure \[fig:res\_excitation\_probab\], the excitation probabilities of the $\mathfrak{s}$ and $\mathfrak{p}$- polarizations are different. The expression of the excitation probability for the $\mathfrak{s}$ polarization contains a single contribution from the $A_y$ component of the vector potential. For $\mathfrak{p}$ polarization both $A_x$ and $A_z$ contribute, giving rise to a destructive interference which explains the present lower probability for this polarization.
In the lower part of Figure \[fig:res\_excitation\_probab\], one compares the excitation probability for the $\mathfrak{p}$ polarization calculated with the vector potential of the EMFED model, with the one calculated with a discontinuous vector potential at the interface (label gas/solid, see also Fig. \[fig:cu\_001\_As\_y\]). A comparison is also performed with a vector potential independent of the spatial coordinate (label gas). The excitation probability calculated using the gas/solid discontinuous potential gives a result 25 % lower than that calculated with EMFED. The simulation using the spatially independent constant vector potential gives an excitation probability 24 times smaller than that obtained using the EMFED model. Taking into account the spatial variation of the laser field at the interface is therefore crucial for a realistic description of the electron dynamics at the interface.
Moreover we have calculated the current density decomposed into two contributions: $I_0$ and $I_{int}$. The current density $I_0$ is three orders of magnitude larger than $I_{int}$ and therefore $I_0$ presents all the characteristics of the total current density. The spatial dependence of the current density $I_0$ and $I_{int}$ at the peak of the laser pulse, obtained from the expressions (\[eq:courant\_I\_0\]) and (\[eq:courant\_I\_int\])), is presented respectively in Fig. \[fig:res\_current\_0\] and \[fig:res\_current\_int\].
The $I_0$ current density displays spatial oscillations in the $z$ direction that mimic the oscillations in the electron-solid interaction potential shown in the lower part of Fig. \[fig:total\_density\_cu\_001\]. The $I_{int}$ current density shows similar but attenuated oscillations. The Eqs. (\[eq:courant\_I\_0\]) and (\[eq:courant\_I\_int\]) show that the $I_0$ and $I_{int}$ current densities are calculated as definite integrals from $-\infty$ to $z$. Changing the upper limit $z$ adds positive or negative contributions to the current giving rise to the observed oscillations. As for the excitation probability in Figure \[fig:res\_excitation\_probab\] and for the same reasons the $\mathfrak{s}$ current density is larger than the $\mathfrak{p}$ current density. Outside the surface region the current density is zero therefore at a nanoscopic scale there is no charge migration.
The lower part of Fig. \[fig:res\_current\_0\] displays the $I_0$ current density for the different approximations of the vector potential. One sees a similar behavior to the one obtained for the excitation probability. These two observables are of different nature and one concludes that the introduction of the variation in space of the vector potential is essential for a proper modeling at the gas-solid interface.
Figure \[fig:res\_current\_int\] presents the current density $I_{int}$ for the $\mathfrak{p}$ polarization of the laser. For the $\mathfrak{s}$ polarization the corresponding current density $I_{int}$ is zero and is not displayed. This result can be explained as follows. For the $\mathfrak{p}$- polarization, the numerical derivatives of the basis functions, relative to the $z$ coordinate, do not compensate in Eq. (\[eq:courant\_I\_int\]) resulting in a non-zero interaction current density $I_{int}$. The zero contribution for the $\mathfrak{s}$- polarization is traced back to the integral Eq. (\[eq:courant\_I\_int\]) where in this case the two terms compensate each other due to the conservation of the momentum parallel to the surface (see appendix).
![(color online) Energy behavior near the resonance energy at of the excitation probability for $t\rightarrow\infty$ (a) and of the maximum of the current density $I_0$ (b) approximately located in the vacuum at $z\simeq$-2 Å. This current density is normalized to the maximum of $^\mathfrak{p}\!I_0$ using the normalization factor 6.8$\times$10$^{-5}$. The characteristics of the laser pulse are given in the caption of Fig. \[fig:res\_excitation\_probab\].[]{data-label="fig:energy_var_observables"}](Fig8_p_i0){width="8cm"}
![Excitation probability for $t\rightarrow\infty$ (a) and maximum of the current density $I_0$ (b) located in the vacuum approximately at $z\simeq$-2 Å and normalized with the factor 1.2$\times$10$^{-4}$ as a function of the incident angle $\theta_i$ of the laser beam. The calculations are performed at the excitation energy and the characteristics of the laser pulse are given in the caption of Figure \[fig:res\_excitation\_probab\]. The calculations correspond to the laser polarizations $\mathfrak{p}$ (full black line) and $\mathfrak{s}$ (dashed red line).[]{data-label="fig:observables_theta"}](Fig9_p_i0_theta){width="8cm"}
Next we present in Figure \[fig:res\_current\_0\_temp\] the temporal evolution of the extrema of $I_0$, around $z\simeq$ -2 Å. The upper graph presents a small region of the temporal evolution of these extrema near the peak of the pulse for the $\mathfrak{p}$- and $\mathfrak{s}$- polarizations. As expected, the current density oscillates at the Bohr frequency $\omega_B= (E_f-E_i)/\hbar\simeq$ 6.3 fs$^{-1}$. The calculations for the different polarizations appear in the same order as the excitation probability. Moreover, there is a small temporal dephasing between the $\mathfrak{p}$- and $\mathfrak{s}$- polarizations. A possible origin for this time shift is the destructive interference between the $^{\mathfrak{p}}\!A_z$ and $^{\mathfrak{p}}\!A_x$ contributions to the current (see Eqs. (\[eq:interaction\]), (\[eq:J\_int\_p\]) and (\[eq:J\_int\_s\])). In opposition to the excitation probability where the laser phase is lost, the observed oscillatory behavior in $I_0$ is due to the relative phase of the products of the complex expansion coefficients $C_i^*\;C_f$ or $C_f^*\;C_i$ in Eq. (\[eq:courant\_I\_0\]). The envelope of the evolution of these extrema of the current density $I_0$, presented in the lower part of Figure \[fig:res\_current\_0\_temp\], displays a smooth variation similar to the evolution of the excitation probability (Figure \[fig:res\_excitation\_probab\]). Because the interaction current $I_{int}$ is about three order of magnitude smaller than $I_0$ (see Fig. \[fig:res\_current\_0\] and \[fig:res\_current\_int\]), we do not present the similar temporal variation of this current.
A final series of results is shown in Figures \[fig:energy\_var\_observables\] and \[fig:observables\_theta\]. The first figure displays the behavior of the extrema of the calculated observables as a function of the photon energy near the resonant excitation energy between the bulk Fermi state and the $n=1$ image state. The excitation probability at t$\rightarrow\infty$ and the maximum of the current density $I_0$ display the usual behavior where a maximum appears at the resonance energy. The FWHM of these Lorentzian-like curves for the excitation probability and for the current density are 0.026 eV and 0.035 eV, in agreement with the bandwith of the pulse.
The second figure displays the evolution of the excitation probability and of the maximum current density $I_0$ with the incident angle $\theta_i$ of the laser beam (see Fig. \[fig:Geometry\]). As one expects, when the incidence is normal to the surface, the observables have the same value for the $\mathfrak{p}$- and $\mathfrak{s}$- polarizations. At grazing incidence the laser field does not penetrate in the solid and no excitation occurs. But as soon as the incident angle $\theta_i$ diminishes, for example for $\theta_i=89\,$deg these observables rise sharply.
Discussion and conclusion
=========================
This paper presents a detailed analysis of the laser-matter interaction at the gas-solid interface in the long wavelength domain, i.e. $\lambda \geqslant 100$ Å. The main goal of the present work is the identification of the role played by the spatial variation of the laser fields, due to the sudden rise of the electron density at the surface, on the electron dynamics at the gas-solid interface.
First let us briefly recall the model. For a one photon excitation, our theoretical model calculates the excitation probability and the associated electron current density between a state of the solid near the Fermi energy (called bulk Fermi state) and the $n=1$ image state. The laser-matter interaction Hamiltonian (subsection \[sec:hamiltonian\]) is calculated using the vector potential of the laser field obtained from the EMFED model as a function of the $z$ coordinate (section \[sec:EMFED\_model\]). The electron dynamics at the interface is obtained by the solution of the time dependent Schrödinger equation (subsection \[subsec:time\_dependent\_model\]).
For the Cu(001) surface, the present model makes use of a separable potential parallel and perpendicular to the surface (see subsection \[subsec:physmod\]), the last component of the potential being taken from the work of Chulkov *et al.* [@des:quant:chulkov:99].
From the total unperturbed electron density $\rho_e(\vec{r})$ of the material system, the EMFED model obtains, using the constants of the material, a realistic vector potential $\vec{A}(\vec{r},t)$ function of the spatial coordinate for an arbitrary incident angle of the photon. The only needed ingredient is an electron-matter interaction potential.
The total wave function is expanded (Eq. (\[eq:expansion\])) in a series of six discrete basis functions (see subsections \[subsec:time\_dependent\_model\] and \[subsec:physmod\]). The first three are constructed as a simple product of the ground bulk Fermi state in the perpendicular direction and a $s$ or $p$ orbital parallel to the surface. The last three are constructed similarly but with the $n=1$ image state in the perpendicular direction. The projection on these basis functions of the time dependent Schrödinger equation gives rise to a system of coupled first order ordinary differential equations in time (Eq. (\[eq:coupled\_equations\])) which are solved for each time step by a predictor corrector algorithm.
Next, the results of the present model can be summarized as follows. The analysis of Figures \[fig:res\_excitation\_probab\], \[fig:res\_current\_0\] and \[fig:res\_current\_int\] concludes that, near the gas-solid interface, a realistic model of the electron dynamics in the presence of the laser-matter interaction requires the inclusion of the spatial variation of the laser field even in the long wavelength domain.
The excitation probability (Fig. \[fig:res\_excitation\_probab\]) and the current density (Fig. \[fig:res\_current\_0\]) are different for the $\mathfrak{p}$ and $\mathfrak{s}$- polarizations of the laser, giving an indication on the so called surface photoelectric effect appearing in the neighborhood, approximately $\pm$ 3–5 Å, of the interface. Precisely, Table \[tab:A\_p\_operator\] gives the different contributions to the interaction Hamiltonian and one sees that, even if the vector potential projections $x$ and $y$ are equivalent, the $x$ scalar product contains the derivative with respect to $x$ of the vector potential whereas the $y$ scalar product does not contain a similar surface term. Therefore, from the difference between the observables for the $\mathfrak{p}$ and $\mathfrak{s}$- polarizations, in the present excitation between discrete states or an excitation giving rise to an electron emission in the vacuum, one cannot estimate precisely the surface photoelectric effect.
The current density $I_0(z,t)$ (Fig. \[fig:res\_current\_0\]) reproduces the spatial oscillations of the electron-solid potential as well as the temporal oscillations of the laser field. Contrary to the excitation probability, this observable is unravel the phase of the time evolution of the laser. The current density $I_{int}$ (Fig. \[fig:res\_current\_int\]) is directly related to the laser-matter interaction term in the Hamiltonian and, for photoemission, to the surface photoelectric effect. But this term is weak and cannot be measured experimentally.
Finally, let us discuss the approximations and the limitations of the present model. First, the separable electron-surface potential contains a contribution in the $z$ direction explicitly calculated and a contribution parallel to the surface that is not explicitly calculated. But this separable potential could be replaced by a more realistic tridimensional potential obtained using, for example, a DFT calculation. In addition, the present simulation of the excitation between discrete states can easily be extended to the photoemission and, because our wave function is expanded on a basis set, the number of these basis functions can easily be raised.
Secondly, the use of tabulated material constants in our model implies that this model is able to account for the presence of the plasma oscillations in the bulk as one of us has shown for a Jellium Al(100) surface [@emed:al:raseev:2005]. One can also account for the change in the observables at the surface plasmon frequency because in the present model the dielectric function depends on $z$. The dependence introduced here is similar to the one used by Feibelman [@feibelman:1989] when he derives the frequency of the surface plasmon.
Let us now replace the present work in a more general context. The consequences of the spatial variation of the vector potential have been highlighted through a numerical calculation of the excitation probability and of the current density. But other physical observables *qualitatively* sensitive to such a spatial variation of the vector potential, have to be identified, calculated and measured. For example, time resolved TPPE experiments, coupled with some interference measurements [@pontius:petek:2005] or combination of TPPE and photoemission electron microscope (PEEM) [@Munzinger:Aeschlimann:Bauer:2005; @rohmer:aeschlimann:bauer:2006], could be efficient experimental techniques for observables sensitive to the spatial variation of the vector potential. Particularly the last cited technique TPPE/PEEM allows for an imaging of single nano objects of a size of about 50 nm thus diminishing the averaging made over many objects in standard experiments. Combination of a laser with a STM [@stm:gerstner:00; @riedel:dujardin:05] in a single simultaneous experiment can be also used for measuring the observables related to these nano objects.
One can easily extend the present model to single photon photoemission but such a model remains oversimplified for the previously cited experimental techniques. In fact, to model a TPPE experiment, one has to take into account, in addition to the direct excitation processes and associated continua of the electron, the electron-electron relaxation at the surface and in the bulk. Modeling such an experiment, implies the combination of the EMFED vector potential with a density matrix formulation [@wolf:1999; @klamroth:saalfrank:hofer:2001; @boger:weinelt:2002; @pontius:petek:2005; @boger:fauster:weinelt:2005; @mii:ueba:2005]. Our approach calculates some of the ingredients needed for such a density matrix model, as for example the unperturbed states and the interaction matrix elements with a spatially varying vector potential. Other needed ingredients, like the elastic and inelastic parameters of the electron-electron collisions corresponding to the relaxation to the bath, can be taken from the existing literature.
Concerning other systems to be studied, different clean metallic surfaces can be selected, but adsorbates [@berthold:hofer:2004; @kirchmann:wolf:2005] or nano objects on surfaces [@kubo:petek:2005; @Munzinger:Aeschlimann:Bauer:2005; @rohmer:aeschlimann:bauer:2006] could also be very sensitive to a spatially varying vector potential. The calculation of the observables as a function of photon frequency, particularly at the bulk and surface plasmon resonance, is the next goal in our modeling. Generalization to semiconductor surfaces is also of interest. However, for these surfaces reconstruction takes place and the wave functions parallel to the surface should be functions of the slab layer.
In summary, the present model clearly shows that the spatial variation of the vector potential, in the vicinity of the gas-solid interface, significantly modifies the electron dynamics near the surface, even in the long wavelength domain. Here, we have proposed a relatively simple approach for the calculation of this spatial variation. Future developments will allow for a better estimate of its influence on the measurements performed with recent experimental techniques like two-photon photoemission (TPPE).
We thank Herve le Rouzo for many fruitfull discussions. Discussions with Doina Bejan during the early stage of this work are also warmly acknowledged. This work has been done with the financial support of the LRC of the CEA, under contract number DSM 05–33.
Evaluation of the integral $\mathcal{J}_{IJ}(z)$ {#annex:A}
================================================
To obtain $\mathcal{J}_{IJ}(z)$, let us rewrite the vector potential $\vec{A}(x,z,t)$ of Eqs. (\[eq:A\_z\])-(\[eq:A\_y\]) in section \[sec:A\_pot\] as a product of two terms related to the variables $z$, $x$ and $t$ $$\label{A_z_x_t}
^{\mathfrak{pol}}\!\vec{A}(x,z,t) = \;^{\mathfrak{pol}}\!\!\vec{\mathcal{A}}(z)\;\;
e^{i\left(k_x^{ph}\,x\,-\,\omega\,t\right)}\,,$$ Then, the interaction Hamiltonian $\hat{H}^L$ (Eq. (\[eq:H\_L\])) becomes $$\hat{H}^L = \frac{\hbar\;e}{2\;m\;i}\;\;^{\mathfrak{pol}}\!\mathcal{O}(\mathcal{A})\;\;
e^{i\left(k_x^{ph}\,x\,-\,\omega\,t\right)}$$ where
$$\begin{aligned}
^{\mathfrak{p}}\!\mathcal{O}(\mathcal{A}) &=&
\,^{\mathfrak{p}}\!\!\mathcal{A}_x\;
\biggl[2\cos\varphi\frac{\partial}{\partial u}
+ 2\sin\varphi\frac{\partial}{\partial v} + i\,k_x^{ph}\biggr]
\nonumber \\
&& + 2\,^{\mathfrak{p}}\!\!\mathcal{A}_z\frac{\partial}{\partial z}
+ \frac{\partial\,^{\mathfrak{p}}\!\!\mathcal{A}_z}{\partial z},\end{aligned}$$
and $$^{\mathfrak{s}}\!\mathcal{O}(\mathcal{A}) = 2\;^{\mathfrak{s}}\!\mathcal{A}_y\;
\biggl[-\sin\varphi \frac{\partial}{\partial u}
+ \cos\varphi \frac{\partial}{\partial v}\biggr],$$
respectively for the $\mathfrak{p}$- and $\mathfrak{s}$- polarizations. In the equations above, the operator $^{\mathfrak{pol}}\!\mathcal{O}(\mathcal{A})$ is written in the orthogonal crystallographic coordinate system {u,v,z} of the surface (see Fig. \[fig:Geometry\]), rotated by an angle $\varphi$ relative to the POI coordinate system. The angle $\varphi$ between the two coordinate systems can be varied in an experimental set-up. For the discrete transitions presented in this paper, the observables, transition probability and current density, are in principle independent of the angle $\varphi$. In photoemission, one can unravel the local surface symmetry (see for example Smith [*et al.*]{} [@smith:76]).
The integral $\mathcal{J}_{IJ}(z)$, appearing in the laser-matter interaction matrix elements (\[eq:interaction\]), can now be written in the particular case of $\mathfrak{p}$- and $\mathfrak{s}$- polarizations as
$$\begin{aligned}
\label{eq:J_int_p}
^{\mathfrak{p}}\!\mathcal{J}_{IJ}(z) & = & \bigl(\, ^{\mathfrak{p}}S_{IJ}^{\perp}(z)\,^{\mathfrak{p}}\mathcal{D}_{IJ}^{\parallel}
+\,^{\mathfrak{p}}\mathcal{D}_{IJ}^{\perp}(z)\; ^{\mathfrak{p}}\!S_{IJ}^{\parallel}\bigr)\mathcal{T}^{\parallel}_{IJ}~~~~~~\\
\label{eq:J_int_s}
^{\mathfrak{s}}\!\mathcal{J}_{IJ}(z) & = & ^{\mathfrak{s}}S_{IJ}^{\perp}(z)\,^{\mathfrak{s}}\mathcal{D}_{IJ}^{\parallel}
\mathcal{T}^{\parallel}_{IJ}.\end{aligned}$$
Taking into account the simplifications related to the LWL approximation, one can evaluate these integrals along the following lines: The factor $\mathcal{T}^{\parallel}_{IJ}$ gives a selection rule for the angular momenta parallel to the surface $$\begin{aligned}
\label{T_factor}
\mathcal{T}^{\parallel}_{IJ} &=& \sum_{j=0}^{\infty}\exp[i(\vec{k}_{\parallel}^I\;-\;\vec{k}_{\parallel}^J\;
+\;\vec{k}_{x}^{ph})\cdot\vec{R}_{\parallel}^j]
\nonumber \\
&\simeq&\delta(\vec{k}_{\parallel}^I-\vec{k}_{\parallel}^J+\vec{k}_{x}^{ph}).\end{aligned}$$ The $\vec{k}_{x}^{ph}$ contribution appears as a consequence of the presence of exponential $\exp(i\;k_x^{ph}\;x)$ in the spatially varying vector potential Eq. (\[A\_z\_x\_t\]). In the LWL approximation this exponential is a slowly varying quantity relative to the surface reciprocal vectors $k_u$ and $k_v$ and one can use the mean value theorem to factorize it from the integrals $^{\mathfrak{pol}}\!\mathcal{D}_{IJ}^{\parallel}$ or $^{\mathfrak{pol}}\!S_{IJ}^{\parallel}$. However, because $k_x^{ph}\ll k_{\parallel}$, this condition reduces to the standard condition of the conservation of the linear momentum $\vec{k}_{\parallel}^I \simeq \vec{k}_{\parallel}^J$ parallel to the surface. For $\mathfrak{p}$- polarization, the explicit expressions of the integrals Eq. (\[eq:J\_int\_p\]) are $$\begin{aligned}
\label{eq:integral_1}
^{\mathfrak{p}}\!S_{IJ}^{\perp}(z)
&=& \int_{-\infty}^z\;\eta_I^*(z')\mathcal{A}_x(z')\eta_J(z')\;dz'
\\ \label{eq:integral_2}
^{\mathfrak{p}}\!\mathcal{D}_{IJ}^{\parallel} &=& 2\,(
\cos\varphi\; \mathcal{D}_{IJ}^u + \sin\varphi\;\mathcal{D}_{IJ}^v)
\nonumber \\
&& + i\;k_x^{ph}\;S_{IJ}
\\ \label{eq:integral_3}
^{\mathfrak{p}}\!\mathcal{D}_{IJ}^{\perp}(z)&=&
\int_{-\infty}^z\;
\eta_I^*(z')\biggl[2\;\mathcal{A}_z(z')\frac{\partial}{\partial z'}
\nonumber \\
&&+\frac{\partial\mathcal{A}_z(z')}{\partial z'}\biggr]\eta_J(z')\;dz'
\\ \label{eq:integral_4}
^{\mathfrak{p}}\!S_{IJ}^{\parallel}&=& 2\,S_{IJ}(\cos\varphi + \sin\varphi) \end{aligned}$$ where $\mathcal{D}_{IJ}^u$, $\mathcal{D}_{IJ}^v$ and $S_{IJ}$ are function of the reciprocal lattice vectors $k_u$ and $k_v$. In the equations (\[eq:integral\_1\]) and (\[eq:integral\_3\]) one identifies the usual interaction terms present in the velocity gauge where the vector potential modulates the associated integrals. There are also new terms, called surface contributions, originating from the derivative of the vector potential, like the last term in the Eq. (\[eq:integral\_3\]). For $\mathfrak{s}$- polarization, the integrals of Eq. (\[eq:J\_int\_s\]) read $$\begin{aligned}
\label{eq:integral_5}
^{\mathfrak{s}}\!\mathcal{D}_{IJ}^{\parallel}&=&
2\,(- \sin\varphi\;\mathcal{D}_{IJ}^u + \cos\varphi\;\mathcal{D}_{IJ}^v)
\\ \label{eq:integral_6}
^{\mathfrak{s}}\!S_{IJ}^{\perp}(z)&=& \;\int_{-\infty}^z\;
\eta_I^*(z')\;\mathcal{A}_y(z')\eta_J(z')\;dz'\end{aligned}$$ No surface is term present for this polarization. If one retains only the interaction with the near neighbors, the integrals $\mathcal{D}_{IJ}^u$, $\mathcal{D}_{IJ}^v$ and $S_{IJ}$, appearing in Eqs.(\[eq:integral\_2\]), (\[eq:integral\_4\]) and (\[eq:integral\_5\]), present a similar structure. For example, this structure can be unravelled by writing explicitly the integral $\mathcal{D}_{IJ}^u$ $$\begin{aligned}
\label{eq:integral_7}
\mathcal{D}_{IJ}^u &=& {^u}\mathcal{D}^{00,00}_{IJ}
\\
&&+ 2\cos(k_u\,a_0^u)({^u}\mathcal{D}^{00,10}_{IJ} + {^u}\mathcal{D}^{10,00}_{IJ})
\nonumber \\
&&+ 2\cos(k_v\,a_0^v)({^u}\mathcal{D}^{00,01}_{IJ} + {^u}\mathcal{D}^{01,00}_{IJ})
\nonumber\end{aligned}$$ where the integral ${^u}\mathcal{D}^{\ell_u\ell_v,\ell'_u\ell'_v}$ reads $$\begin{aligned}
\label{eq:integral_8}
{^u}\mathcal{D}^{\ell_u\ell_v,\ell'_u\ell'_v}_{IJ} & = & \int\!\!\!\!\int \chi^*_{I}(u-\ell_u a_0^u,v - \ell_v a_0^v)\\
& & \frac{\partial}{\partial u}\;\chi_{J}(u-\ell'_u a_0^u,v - \ell'_v a_0^v)\;du\,dv.
\nonumber\end{aligned}$$ Here the integration domain has been extended over three elementary cells in each direction to calculate the interaction with the near neighbors. The interaction operator $^{\mathfrak{pol}}\!\mathcal{O}(\mathcal{A})$ contains symmetric (derivative of the vector potential) and antisymmetric contributions relative to the inversion of the system of coordinates. This last contribution is predominant. Consequently, in Eq. (\[eq:basis\_functions\]) the set of basis functions parallel to the surface should contain symmetric and antisymmetric basis functions. In the simplest case, the functions parallel to the surface should contain “s" and “p" like basis functions.
Finally note that the above simple derivation has been inspired by the one given in the appendix F of Desjonquères and Spanjaard [@book:desjonqueres_spanjaard] where the vector potential is independent of the spatial coordinates.
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[^1]: Corresponding author, E-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Both from gravitational (G) experiments and from a new theoretical approach based on a particle model it is proved that the classical invariability of the bodies, after a change of relative rest-position with respect to other bodies, it is not true. The same holds for the traditional hypotheses based on the classical one. The new relationships are strictly linear. From them it is proved that a universe expansion must be associated with a G expansion of every particle in it, in just the same proportion. It does not change the relative distances, indefinitely. From the relative viewpoint, globally, the universe must be rather static. According to the new cosmic scenario, galaxies must be evolving, indefinitely, in rather closed cycles between luminous and black states. The new kind of linear black hole must absorb radiation until it can explode after releasing new H gas that would trigger new luminous period of star clusters and galaxies. Statistically, most of the galaxies must be in cool states. The last ones should account for all of them, the higher velocities of the galaxies in clusters, the radiation coming from intergalactic space, including the low temperature black-body background observed in the CMBR.'
author:
- |
Rafael A. Vera[^1]\
Departamento de Física.\
Facultad de Ciencias Físicas y Matemáticas.\
Universidad de Concepción. Chile
title: The new cosmological scenario and dark matter fixed by general experimental facts
---
Introduction
============
The current hypotheses in gravitation and in cosmology are tacitly based on *the classical hypothesis on the absolute invariability of the bodies after a change of the rest position with respect to other bodies*. Such hypothesis comes from the fact that, according to the Equivalence Principle (EP), all of the bodies of his local system obey the same inertial and gravitational (G) laws[@1]. Therefore, all of then must change in the same proportion, i.e., every local ratio would remain constant.
In principle such changes can be tested by observers that have not changed of position in the field. However this has not been done because, on the contrary, *current G tests are tacitly based on such classical hypothesis, as shown below.*
A crucial test for the cosmological hypotheses
----------------------------------------------
To fairly test such hypothesis the observer ($A$) must stay in a fixed distance from the earth center. The observed body ($B$) must be at another distance from it.
A well-defined experiment that meet this condition is the G time dilation (GTD) experiments done with standard clocks, [@2]$^{,}$[@3]
From the results of such experiments, corrected after special relativity, it is concluded that *the relative frequency of a non local (NL) clock* $%
B$*, located at rest at a well defined distance* $\Delta r$*from earth surface, runs with a different frequency compared with the clock* $A$* at the earth surface*. Thus *the clocks are not invariable after a change of distance with respect to the earth center*.
### Discussion
It has been argued that: ”*experiments to detect difference of frequencies between the clock* $B$* and* $A$* have given negative results. What happens (in a GTD experiment) is that an electromagnetic signal controlled by clock B generally arrives at A with a frequency different from the frequency of the same type of clock there. Photons must do some work in moving around and their frequency changes*”.
This is not true because *the readings of the clocks* *do not depend on the frequency or energy of any photon travelling between them.*
This fact is most obvious in the Hafele-Kerting experiments[@2] in which no photon was used. The readins of the clocks $A$ and $B$ were directly compared in the earth surface, before and after experiments. During experiments of $48$ hours, the clocks $B$ were flying at $9$ km over the earth surface. Thus the differences of the readings of the clocks did not depend on the frequency of any photon travelling between such clocks[^2].Then the differences of time intervals observed in the GTD experiments made up with clocks can only be due to real differences of the relative frequencies of the standard clock B, compared with A[^3] This means that , during the flight, some fundamental physical change had occurred to every part of the clock $B$.
Then it is expected that *the current relations between quantities measured by observers located in different G potentials are not strictly homogeneous because their reference clocks are not strictly the same with respect to each other, respectively*. They would be sources of errors in the current literature.
It is also said, without fair demonstration, that the GTD experiments would have verified the theory of GR. On the contrary, from them, the photons emitted by the NL clock $B$ would starttheir trips with an initial frequency shift $z$ with respect to the local clock $A$. From the fact that the final redshift of the photons is just equal to $z$ it is concluded that: *during the trip* $BA$*, the relative frequency of the photons, with respect to the observer* $A$*, remains constant*. (Relative frequency conservation law for photons.).
Then *the G redshift of photons is not due to real frequency changes of the free photons. It is due to differences of the natural frequencies of atoms and clocks of observers located at rest in different distances from the field source*. The same conclusion was obtained by Vera in1981 from direct application of wave continuity [@4].
The non local form of the Equivalence Principle {#the-non-local-form-of-the-equivalence-principle .unnumbered}
------------------------------------------------
From above it is concluded that: to relate quantities measured in different G potentials, they must be transformed to some well-defined reference standard in a well-defined G position of the field. Here, the fixed position of such observer (A) is stated by means of a subscript $a$.
From Lorenz equations and GTD experiments, it is inferred that the “relative” quantities can depend on the velocity and distances of the body and of the observer with respect to the field sources.(Vera 1981). On the other hand, from the EP, when an observer moves altogether with his clocks he finds that the local ratios between frequencies ($\nu $) masses ($%
m $) and lengths ($\lambda $) are constants that do not change after a change of position of the measuring system with respect to the G field sources. The opposite comes true when the observer $A$ remains in a fixed position $a$. He finds, from GTD experiments, that the frequency of the clock at rest at $B $ is a function on its position ($r$) in the field, say $\nu _{a}(0,r)$. The first results can be consistent with the last ones only if: “*the relative values of the frequencies, masses and lengths of any well-defined part of the system have changed in just the same proportion after the same change of relative position with respect to the G field sources*”. Only in this way all of the local ratios can remain unchanged. This may be called the NL form of the EP, or NL EP.
The field equations fixed by experimental facts
===============================================
A short cut can be done from results of experiments and applications of the NL EP.
For example, assume that the observer A throws a clock upward with some energy $\Delta E_{a}(0.r)$. From results of free fall experiments, the clock would stop at some NL radius $r=a+\Delta r$ given by the three first members of (1). From results of the GTD experiments made up by the observer $A$, the clock at rest at $B$ would have a frequency $\nu _{a}(0,r)$ given by the third and fourth member of (1)[^4]. The last member of (1) comes from the NL EP applied to any of the frequencies, masses, lengths and wavelengths, of any particle or standing wave of the same system.
$$\Delta \phi (r)=\frac{\Delta E_{a}(0,r)}{m_{a}(0,r)}=\frac{GM}{a}\frac{%
\Delta r}{r}=\frac{\Delta \nu _{a}(0,r)}{\nu _{a}(0,r)}=\frac{\Delta
m_{a}(0,r)}{m_{a}(0,r)}=\frac{\Delta \lambda _{a}(0,r)}{\lambda _{a}(0,r)}=%
\frac{1}{2}\frac{\Delta c_{a}(r)}{c_{a}(r)} \tag{1}$$
The last member results from the application of this equation to any standing wave of the system. In it, $c_{a}(r)$ is the relative (NL) speed of light at $B$ with respect to the observer $A$. This one is the product of its relative frequency $\nu _{a}(0,r)$, and its relative wavelength $\lambda
_{a}(0,r)$.
In a previous article, in (1981), it is proved, step by step, that this equation is consistent with the current tests for G theories[@4].
The relative mass-energy conservation
---------------------------------------
From the 2$^{nd}$ and 5$^{th}$ members of (1), the relative mass of the clock $B$, with respect to the observer $A$, depends on its relative position:
$$m_{a}(0,r)=m_{a}(0,a)+\Delta E_{a}(0,r) \tag{2}$$
The energy $\Delta E_{a}(0,r)$ given up to the clock is not given up to the G field: it remains ”in the clock” as an additional mass. Vice versa, during a free fall from $r$, its relative initial rest mass is $m_{a}(0,r)$. From (2) and special relativity, its final mass passing by $A$ is. $$m_{a}(V,a)=m_{a}(0,a)+\Delta E_{a}(0,r)=m_{a}(0,r) \tag{3}$$
*During the free fall, the relative mass of the clock, with respect to the observer A, remains constant*. This is consistent with the relative frequency conservation law for photons derived above. Then it may be concluded that* there is not a true exchange of energy between bodies or photons and the G field*. This result is in clear contradiction with *energy of the G field assumed by Einstein*[@4; @5].
The linear black hole
----------------------
The theoretical properties of the “linear black hole” (LBH) derived from (1) *are radically different from the conventional ones[@4] F*or $%
2GM>>r$, the gradient of the relative speed of light would produce dielectric reflections preventing the escape of photons and nucleons[@4]. On the other hand it has a larger cross-section for photon capture. Thus the average NL mass-energy of its nucleons increases with the time. When it gets higher than the one in free state, the LBH can explode. Their neutrons would decay into new hydrogen.
The theoretical field equation from the NL form of the EP {#the-theoretical-field-equation-from-the-nl-form-of-the-ep .unnumbered}
---------------------------------------------------------
From the NL EP, a *particle model* made up or radiation in stationary state between any two parts of a system *must obey the same inertial and gravitational laws of the particles in it*. This fact has been verified by Vera (1981,1997) with a full consistency with special relativity, quantum mechanics and equation (1) [@4; @5].When particle models emulate all of the uncharged particles of the universe it is found that, according to the Huygen principle, the particles are the result of constructive interference of wavelets crossing the space. *The properties of the empty space in some position* $r^{i}$* can depend only on the actual perturbation frequency of the space produced by all of the wavelets with random phases crossing it*. Each wavelet contribution must be proportional to the product of its frequency and of its amplitude. After taking into account the cosmological red shift, in which $%
d\nu /\nu $ *=* $dr/R$, the average perturbation frequency of the space, called $w(r^{i})$, must be proportional to:
$$w(r^{i})\propto \sum_{j=1}^{\infty }\frac{\nu ^{j}}{r^{ij}}\exp \left[ \frac{%
r^{ij}}{R}\right] =\sum_{j=1}^{\infty }\frac{m^{j}}{r^{ij}}\exp \left[ \frac{%
r^{ij}}{R}\right] \cong 4\pi \rho R^{2} \tag{4}$$
The average density of the universe, in $joules/m^{3}$ is $\rho $. The Hubble radius is $R$.
The best fit of (4) with (1) occurs for particles in equilibrium with the space:
$$\lambda _{a}(0,r)w_{a}(r)=\text{ Constant} \tag{5}$$
$$\Delta \phi (r)=\frac{\Delta \lambda _{a}(0,r)}{\lambda _{a}(0,r)}=-\frac{%
\Delta w_{a}(r)}{w_{a}(r)}\text{ \ \ \ In which \ }G(r)=-\frac{1}{w_{a}(r)}
\tag{6}$$
From (6), the universe density is about 30 times the average density of luminous matter. This is consistent with dark matter estimated in some *clusters*.
Matter expansion due to universe expansion
------------------------------------------
Assume, as a hypothesis to be tested, that after a time *dt* the distances between the galaxies $i$ and $k$ have increased the proportion,
$$\frac{dr^{ik}}{r^{ik}}=Hdt \tag{7}$$
From (4), after using (6) and (7), it is found that the increase of the G potential produces a gravitational expansion of any standard rod of length $%
\lambda $ given by:
$$d\phi (r)=\frac{d\lambda }{\lambda }=-\frac{dw}{w}=\frac{dr^{ik}}{r^{ik}}=%
\frac{dR}{R}=Hdt\text{ ;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{\lambda }{R}%
=\text{ Constant\ \ } \tag{8}$$
The new cosmological scenario fixed by experimental facts
=========================================================
*From equation* (8)* it is concluded that: in the average, the relative distances and the average density of the universe cannot change with the time.* The universe age must be rather infinite.
*The evolution of the celestial bodies must be occurring according to rather closed cycles between luminous (hot) and non-luminous (cool) states.*
1\) The hydrogen atoms, after stellar evolution, must be evolving, indefinitely, in closed cycles between states of gas and linear black hole (LBH), and vice versa.
2\) The explosions of massive LBHs would fix the initial period of a luminous star cluster or a galaxy. The new stars would be normally formed from condensation of gas over older bodies that existed before the explosion.
3\) A galaxy must also be running, almost indefinitely, in rather closed cycles between luminous and dark states, and vice versa. Something similar may hold for clusters.
Statistically, all of the evolution stages of the galaxies should be present in the proportion fixed by their evolution periods. Since the energy-recovering period of dark galaxies must by of a higher order of magnitude that the luminous period of galaxies, then *most of the universe must be in the state of black galaxy cooled down by LBHs and the rest of the universe*. They should account for most of the higher velocities of galaxies in clusters and for the radiations coming from the intergalactic space, mainly gamma, cosmic and low temperature blackbody radiation (CMB).
Galaxies should start their luminous period with new gas, free of heavy metals, with a high density of randomly oriented angular momentum generated during the LBH explosion. This should correspond with “elliptical galaxies”.
During the luminous period of a galaxy, the luminous volume would decrease with the time. The last luminosity should come from a small region located in its center, in the strong fields of massive bodies. They should correspond with the true (radio noisy) *quasars* of relatively variable luminosity. Most of their red shifts would be gravitational one. They should be not confused with QSOs of large Hubble red shift.
Most of the energy released in a matter cycle, from the state of gas up to LBH, is gravitational. Most of it must be transformed into other kind of energy around neutron stars. Thus the true role of the G energy in the interpretation of the celestial phenomena is most important[@6].
Conclusions
===========
The classical hypothesis on the invariability of the bodies after a change of relative position with respect to other bodies is not consistent with the experimental facts. The true changes occurring to the bodies, after changes of position in the G field must be described by using a position-dependent formalism with respect to some well-defined observer that does not change of position in the G field. In this way the fundamental errors derived from this wrong hypothesis can be eliminated.
The new field relations derived from the EP and the G tests are strictly linear ones. They rule out the presumed energy of the G field. The G field does not exchange energy either with photons or with bodies. The G energy comes from the bodies, not from the field.
The new approach based on particle models made up of photons in stationary states provides more exact relationships for long range interactions and a for unified understanding on the properties of bodies and their G fields. The new relations reveal that, in the average, the relative density of the university cannot change with the time. The average relative distances must remain constant, indefinitely, i.e.,the universe age should be rather infinite.
In the new scenario, the H gas must be evolving in rather closed cycles between the states of gas and LBH and vice versa. A LBH, after absorbing energy from the space, would explode. The new gas, condensed over other bodies, would regenerate star clusters or galaxies. Galaxies would be evolving, rather indefinitely, in rather closed cycles between hot and cool states. Most of the matter of the universe must be in the black galaxies that would be absorbing energy from the rest of the universe. They must account for the black body radiation coming from the intergalactic space, observed in the CMB. They must also account for the anomalous velocities of galaxies in clusters.
The LBH explosions should account for the clean H and high densities of angular momentum with random orientations observed in some galaxies. They would be testimonies of the ”small bangs” that occurred rather recently.
*References*
------------
[9]{} Misner Ch. W., Thorne K. S. and Wheeler J. A., *Gravitation* (Freeman, San Francisco, 1973), p 386.
Hafele R., Keating E., *Science*.(1972), Vol. 177, PP 168-170.
Vessot R. F., Levine M., *J.Gen. Rel. and Grav*.(1979), Vol. 10, pp.181-204.
Vera R. A., *Int. J. Theor. Phys*, (Plenum Publ. Corp, 1981), 20, pp 19-50.
Vera R. A., *The New Universe Fixed by the Equivalence Principle and Properties of Light*, ed. Rafael Vera, Galvarino 482, Concepcion, Chile, 1997)
Vera R. A., in *Inside of the Stars*, IAU. Coll 137, ed. by W. Weiss and A Balgin.(Astr. Soc. of the Pacific, San Francisco, 1993), 40, p. 798-800
[^1]: [email protected]
[^2]: The same fact holds for other experiments in which the time interval between the initial and final readings of the clock $B$ are obtained from electromagnetic signals coming from such clock[@3]. Such time intervals are differences of times in which the time of flight of the initial signals is cancelled out by the one of the last signal. Thus such measurements do not depend at all on the frequency of the photons.
[^3]: An observer at $B$ cannot detect such change because, according to the equivalence principle[@1], all of the natural frequencies of the local bodies have changed in the same proportion after a change of rest position with respect to the earth center.
[^4]: The common unit of mass and energy used here is $1\left[ joule\right] $
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We observed the neutron star X-ray transient 2S1803$-$245 in quiescence with the X-ray satellite XMM-Newton, but did not detect it. An analysis of the X-ray bursts observed during the 1998 outburst of 2S1803$-$245 gives an upper-limit to the distance of $\le$7.3 kpc, leading to an upper-limit on the quiescent 0.5-10 keV X-ray luminosity of $\le$2.8$\times$10$^{32}$ erg s$^{-1}$ (3$\sigma$). Since the expected orbital period of 2S1803$-$245 is several hrs, this limit is not much higher than those observed for the quiescent black hole transients with similar orbital periods.'
author:
- |
R. Cornelisse$^{1,2}$[^1], Wijnands, R.$^{3}$, Homan, J.$^{4}$\
$^{1}$Instituto de Astrofisica de Canarias, Via Lactea, La Laguna E-38200, Santa Cruz de Tenerife, Spain\
$^{2}$School of Physics and Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, UK\
$^{3}$Astronomical institute “Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ, the Netherlands\
$^{4}$MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Avenue, Cambridge, MA 02139, USA\
date: 'Accepted Received ; in original form '
title: 'An XMM-Newton observation of the neutron star X-ray transient 2S1803$-$245 in quiescence'
---
\[firstpage\]
accretion, accretion disks – stars: neutron – stars:individual: 2S1803$-$245 – X-rays:binaries.
Introduction
============
Low mass X-ray binaries (LMXBs) are compact binaries in which the primary is a compact object (a black hole or neutron star) that accretes matter from a low mass ($\le$1$M_\odot$) secondary. An important sub-class of the LMXBs are the soft X-ray transients. These systems spend most of their time in a quiescent state in which little (or no) accretion is thought to take place and X-ray luminosities are $\le$10$^{34}$ erg s$^{-1}$. Only occasionally do these transients show outbursts that can last for weeks upto months and reaching X-ray luminosities of 10$^{36-38}$ erg s$^{-1}$.
Over a dozen neutron star transients have been observed when they were in the quiescent state, and in many cases their spectra show a soft thermal component that is dominant below $\simeq$1 keV. This is thought to be due to the cooling of the neutron star that has been heated during the previous outbursts (e.g. Verbunt et al. 1994; Brown et al. 1998; Campana et al. 1998). Other mechanisms have also been suggested to explain the quiescent emission for neutron stars, such as residual accretion onto the neutron star (e.g. Campana et al. 1998; Campana & Stella et al. 2000).
Apart from a soft component also a hard power-law component that dominates the spectrum above a few keV has been observed in several systems. This component can contribute a significant fraction to the total X-ray flux, and especially in SAXJ1808.4$-$3658 and EXO1745$-$248 this power-law component was the main source of the X-ray flux with no significant contribution from the soft thermal component (Campana et al. 2002; Heinke et al. 2007; Wijnands et al. 2005). The origin of this power-law component is still unclear, although at low luminosities ($L_X$$<$10$^{33}$ erg s$^{-1}$) there appears to be an anti-correlation between the fractional power-law contribution to the luminosity and the source luminosity (e.g. Jonker et al. 2004).
One of the distinct differences between black hole transients and neutron star transients is the difference in quiescent luminosities, with the black hole transients being systematically fainter (e.g. Narayan et al. 1997; Menou et al. 1999; Garcia et al. 2001; Lasota 2007). This has been interpreted as evidence for the presence of an event horizon in black holes. Since the energy is radiated away very inefficiently for such very low accretion rates, this will not happen before the matter has crossed the event horizon for black holes and can therefore not be observed, while in neutron stars this should be emitted at the moment the matter falls on the surface and can be detected (e.g. Narayan et al. 1997). However, alternative explanations for this difference in luminosity have been suggested, such as a transition to a jet-dominated regime for black hole transients that carries away most of the material that would otherwise be accreted (Fender et al. 2003).
2S1803$-$245 (=XTEJ1806$-$246) is a neutron star transient that was first detected with the SAS-3 satellite in 1976 at a maximum intensity of $\simeq$1 Crab (Jernigan 1976), and again during a second outburst in 1998 that lasted for $\simeq$3 months (see Fig.\[asm\]) and that also reached a peak intensity of $\simeq$1 Crab (Marshall et al. 1998). At the beginning of the second outburst the BeppoSAX satellite detected thermonuclear X-ray bursts from this source, establishing its neutron star nature (Muller et al. 1998). A radio counterpart was detected (0.8 mJy) that provided an accurate position ($\alpha$=18h06m50.72s $\delta$=-24$^\circ$35$'$28.6$"$ J2000) and optical follow-up observations showed a weak (V$\simeq$22) counterpart at the position of the radio source (Hjellming et al. 1998; Hynes et al. 1998). During the peak of the outburst 2S1803$-$245 showed some spectral and timing properties of the Z-sources (Wijnands & van der Klis 1999), suggesting that it reached accretion rates comparable to the Eddington rate (Hasinger & van der Klis 1989). Other observations during the decay of the outburst showed that 2S1803$-$245 had the spectral and timing characteristics of Atoll sources.
In this paper we report on an XMM-Newton observation of 2S1803$-$245 in quiescence, made $\simeq$7 years after its last outburst. Thus far, most of the neutron star transients that have been studied in quiescence showed sub-Eddington outbursts, with 2S1803$-$245 being one of the few that reached the Eddington limit. This makes it an interesting source to study how its quiescent properties compare to the other systems. However, in Sect.2.1 we will show that 2S1803$-$245 was not detected during our observation. Combined with the analysis of the X-ray bursts that were detected during its outburst (Sect.2.2) we determine an upper-limit to the distance, and thereby an upper-limit to its luminosity. In Sect.3 we will discuss the implications of our findings.
Observations and Data Reduction
===============================
Quiescent observations
----------------------
We made a 24 ks observation on 2S1803$-$245 using the X-ray satellite XMM-Newton from April 5 2005 (UT 22:23:52) until April 6 2005 (UT 05:04:07). We analysed the data from the three EPIC cameras (PN, MOS1, MOS2) that were observing in full window mode and with a thin filter. The data were processed using the Standard Analysis Software (SAS) version 7.0.0. In order to identify periods of high particle background we extracted high energy ($\ge$10 keV) lightcurves for all cameras. We chose to keep all data where the countrate was less than 0.8 counts s$^{-1}$ for the PN and 0.2 counts s$^{-1}$ for the MOS. This left a net observing time of 14.8 ks for the PN and 20.3 ks for the MOS cameras.
We created images for each individual camera for several energy ranges (0.5-10, 0.5-2, 5-10 keV) but there was no detection of 2S1803$-$245 at the position of the radio source detected by Hjellming et al. (1998). Although we think it is unlikely, since its radio flux (0.8$\pm$0.3 mJy at 4.86 Ghz) is similar to that of the bright neutron star X-ray transients AqlX-1 and XTE1701$-$462 (both 0.5 mJy at 4.8 GHz; Fender & Kuulkers 2001, Fender et al. 2006), it cannot be completely ruled that the radio source is not related to 2S1803$-$245. We therefore checked the region inside the RXTE error-circle, but no source was present. In order to increase sensitivity we also merged all 3 cameras and again created images in different energy ranges. Still no source is present at the position of the radio source, or even inside the RXTE error-circle (see Fig.\[optical\]). We therefore conclude that we have not detected 2S1803$-$245 in quiescence.
In order to determine an upper-limit on the X-ray flux of 2S1803$-$245 we extracted a spectrum for all 3 cameras using a circle with a radius of 20 arcsec around the position of the radio source. This lead to spectra with 9 counts for the PN and 4 counts for each MOS detector. For different spectral models, using the absorption column determined in Sect.2.2 and combining all cameras, we estimated a 3$\sigma$ upper-limit to the 0.5-10 keV unabsorbed X-ray flux in Table\[upperlimits\]. We have also compared these limits with a source located closest to the radio position (see Fig.\[optical\]). This source was detected at 3.8$\sigma$ above the background, and using the same spectral models as in Table\[upperlimits\] gave comparable flux levels as determined for 2S1803$-$245. This makes us confident that the upper-limit on 2S1803$-$245 is correct. Using the upper-limit to the distance determined from the X-ray bursts we also show the corresponding luminosity in Table\[upperlimits\].
Distance estimate
-----------------
2S1803$-$245 was in the field of view of the Wide Field Cameras (WFCs; Jager et al. 1997) onboard the BeppoSAX satellite (Boella et al. 1997) during its campaigns on the Galactic centre region. During the campaign in the first half of 1998 three X-ray bursts were detected from a position coincident with 2S1803$-$245. Using the publicly available data from the All Sky Monitor (ASM) onboard the RXTE satellite we created a lightcurve of the outburst of 2S1803$-$245. In Fig.\[asm\] we show its outburst, and have also indicated the time that the bursts observed by the WFCs occurred. We note that all bursts occurred during the beginning of the outburst, and assuming that the peak of the outburst was at the Eddington-limit the X-ray luminosity must have been $\simeq$10$^{37}$ erg s$^{-1}$ (see below). Since X-ray bursts are most commonly observed when a source is at X-ray luminosities between 0.5-2$\times$10$^{37}$ erg s$^{-1}$, but tend to be suppressed at higher luminosities (e.g. Cornelisse et al. 2003), we can be confident that they originated from 2S1803$-$245.
The X-ray bursts occurred between April 2 and 10 1998, and in Fig.\[bursts\] we show their 3 lightcurves in two different energy-bands. The shapes of the bursts can be described by a fast rise and exponential decay (with e-folding times between 10.1 and 13.5 s), as is characteristic of a thermonuclear X-ray burst. Furthermore, we have also calculated the hardness ratio (8-26 keV/2-8 keV) of the bursts to show that spectral softening occurs during the burst. Finally, we created a spectrum of the peak of the first (and brightest) burst in order to estimate the corresponding flux. The spectrum could be well described by an absorbed black-body with a temperature of 2.6$\pm$0.4 keV (and taking the absorption column fixed at the value determined below), as is typically observed for thermonuclear X-ray bursts. This translates into an unabsorbed bolometric peak flux of 3.1$\pm$0.7$\times$10$^{-8}$ erg cm$^{-2}$ s$^{-1}$.
---------------- -------------- -------------------------- --------------------------
Spectral Model parameter $F_{0.5-10}$ $L_{0.5-10}$
(erg cm$^{-2}$ s$^{-1}$) (erg s$^{-1}$)
black-body kT=0.2 keV $<$4.4$\times$10$^{-14}$ $<$2.8$\times$10$^{32}$
black-body kT=0.5 keV $<$1.5$\times$10$^{-14}$ $<$0.96$\times$10$^{32}$
Powerlaw $\gamma$=1.5 $<$2.0$\times$10$^{-14}$ $<$1.3$\times$10$^{32}$
Powerlaw $\gamma$=2.0 $<$2.1$\times$10$^{-14}$ $<$1.3$\times$10$^{32}$
---------------- -------------- -------------------------- --------------------------
: 3$\sigma$ upper-limits to the 0.5-10 keV unabsorbed X-ray flux of 2S1803$-$245 in quiescence for different spectral models. The temperature (for the black-body model) and photon index, $\gamma$, (for the power-law model) are fixed at the indicated values, while for all models the absorption column is fixed at 1.47$\times$10$^{22}$ cm$^{-2}$. Furthermore, we have indicated the corresponding luminosity for a distance of 7.3 kpc. \[upperlimits\]
Since the timing properties of 2S1803$-$245 suggested Z-source like behaviour, and hence imply near-Eddington luminosities (Wijnands & van der Klis 1999), its persistent flux during the peak of the outburst should be close to the peak flux of the X-ray bursts. Although the quality of the data is not good enough to determine if the X-ray bursts show radius-expansion, a clear indication that they reached the Eddington-limit, we can still test if the persistent flux reached a similar level. From the log by Wijnands & van der Klis (1999) we selected the observation on May 3 1998 with the Proportional Counter Array onboard the RXTE satellite (Jahoda et al. 1996). This observation showed the highest count rate, and also corresponds more or less with the peak of the outburst according the ASM lightcurve in Fig.\[asm\]. The spectrum could be well fitted by a combination of an absorbed black-body and absorbed disc black body models, with an absorption column of 1.47$\times$10$^{22}$ cm$^{-2}$. The unabsorbed 0.5-50 keV flux corresponds to 2.5$\times$10$^{-8}$ erg cm$^{-2}$ s$^{-1}$, which is close to peak flux of the X-ray burst, suggesting that the outburst reached luminosities very close to the Eddington limit.
Discussion
==========
We have observed 2S1803$-$245 during its outburst in 1998 with BeppoSAX and again in a $\simeq$20 ks observation in order to determine its quiescent properties. We did not detect the source during the XMM-Newton observations, and were only able to determine an upper-limit on its quiescent flux of $<$4.4$\times$10$^{-14}$ erg cm$^{-2}$ s$^{-1}$ (3$\sigma$). However, to compare this with other neutron star transients in quiescence and the different cooling models for neutron stars, we first need to determine the luminosity and time-averaged mass transfer of 2S1803$-$245.
In order to determine its luminosity we presented the analysis of the three X-ray bursts that were observed during the outburst of 2S1803$-$245. Since their peak flux was comparable to the continuum flux during the peak of the outburst we can assume that they reached the Eddington limit, which allows us to determine an upper-limit on the distance. However, we must make several assumptions on the neutron star properties in order to determine its Eddington limit. Since all X-ray bursts showed an e-folding time of $\ge$10 s, indicative for the presence of hydrogen during the burst (e.g. Fujimoto et al. 1981; Cornelisse et al. 2003), we assume that 2S1803$-$245 has solar metallicity. Note that we can therefore not use the emperical determined value of 3.8$\times$10$^{38}$ erg s$^{-1}$ by Kuulkers et al. (2003), since this is only valid for hydrogen-poor material. Instead, we assume the canonical properties for the neutron star parameters (i.e. radius of 10 km, mass of 1.4$M\odot$), leading to an Eddington limit of 2$\times$10$^{38}$ erg s$^{-1}$. This leads to a maximum distance of 7.3$\pm$0.7 kpc for 2S1803$-$245. Although the formal error on the distance is only 10%, due to the uncertainties in the Eddington limit it will be larger. The largest uncertainty, as suggested by the Eddington value determined by Kuulkers et al. (2003), is that the actual Eddington luminosity could be $\simeq$2 times larger than we used, leading to a distance that is at most $\simeq$1.5 times larger than we estimated. The other uncertainty is that the bursts do not show a clear indication of radius-expansion, suggesting that they did not reach the Eddington limit. However, this suggests that the Eddington flux for 2S1803$-$245 must be higher, and therefore its distance lower than the upper-limit we determined above. Despite these uncertainties we have used the distance value of 7.3 kpc to determine the upper-limit on the 0.5-10 keV luminosity given in Table\[upperlimits\].
Following Tomsick et al. (2004) we can estimate the time-averaged mass transfer rate for 2S1803$-$245, $\dot M$, by assuming that $\dot M$$=$$s$$L_{\rm peak}$$N$. Here $L_{peak}$ is the peak luminosity, $N$ is the number of outbursts and $s$$=$1.1$\times$10$^{-23}$ s$^2$ cm$^{-2}$ symbolising a value to estimate the average accretion rate over a period of 33 years for a source that has a similar outburst profile and duration as XTEJ2123$-$058 (see Tomsick et al. 2004 for its outburst lightcurve). Since the outburst duration and the profile of 2S1803$-$245 is very similar to that of XTEJ2123$-$058 we can use this value of $s$. Given that 2S1803$-$245 has at least 2 outburst over the last 33 years, and that it reached the Eddington luminosity, we estimate an average mass accretion rate of $\dot
M$$=$7$\times$10$^{-11}$ $M_\odot$ yr$^{-1}$. Obviously, there are many uncertainties in this value. For example, it assumes that we have observed all outbursts of 2S1803$-$245 that occurred in the last 33 years, that all these outbursts were similar, that these 33 years reflects the real time-averaged mass transfer rate. However, since it is comparable to other estimates for the mass transfer rate, such as using the time interval of the ASM lightcurve as done by Heinke et al. (2007), we think this value is currently the best we can derive.
We can compare the quiescent luminosity and average mass transfer rate of 2S1803$-$245 with the predictions of the different cooling models. Heinke et al. (2007) did this for most other neutron star transients that have been observed in quiescence (their Fig.2). As has already been observed for many other systems (for overviews see e.g. Cackett et al. 2006, Heinke et al. 2007), the quiescent luminosity is too low to be explained by standard cooling models for a low-mass neutron star as calculated by Yakovlev & Pethick (2004). This model predicts a luminosity that is at least an order of magnitude higher than the upper-limit determined for 2S1803$-$245. Only the models for more massive neutron stars, where the central density is high enough to have more rapid direct Urca or Urca-like processes, are consistent with our observations. However, we must note that increasing the neutron star mass does increase its Eddington-limit and thereby our estimate for the distance and consequently increases both the upper-limit on the quiescent luminosity and average mass transfer rate. Therefore, we cannot rule out any of the other cooling models at the moment.
Although 2S1803$-$245 is fainter than expected for standard cooling models, it is still an order of magnitude brighter than the currently faintest neutron star transient 1H1905$+$000 (Jonker et al. 2006). At an upper-limit of 1.8$\times$10$^{31}$ erg s$^{-1}$ the luminosity of 1H1905$+$000 is rivalling that of black hole transients in quiescence (Jonker et al. 2006). This system could challenge the idea that black hole systems should have lower luminosities than neutron star systems in quiescence (e.g. Narayan et al. 1997). However, Menou et al. (1999) predicted that this should only be the case for systems with a similar orbital period. Since there is a strong indication that 1H1905$+$000 is an ultra-compact binary (Jonker et al. 2006), it should be able to reach luminosities lower than the average block hole system (but not as low as a black hole transient with a similar period). The orbital period of 2S1803$-$245 is currently unknown, but Lasota (2007) gives a relation between the maximum outburst luminosity and orbital period for an hydrogen dominated disk (his formula 3). Using the maximum observed X-ray luminosity for 2S1803$-$245, we found that this would result in an orbital period of 9 hrs. Although this is only a rough estimate, it strongly indicates that 2S1803$-$245 is not an ultra-compact object. Comparing the quiescent luminosity of 2S1803$-$245 with neutron star and black hole transients which have orbital periods around 9 hrs (see Garcia et al. 2001), we note that it is located at the bottom of the region where the neutron stars are located. More interestingly, the current upper-limit is not that much higher than the luminosity of the black holes. This makes 2S1803$-$245 an excellent candidate for deep observations with the Chandra telescope to determine its quiescent flux, and find out if it reaches X-ray luminosities comparable to the black hole transients.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge Jean in ’t Zand for providing the BeppoSAX Wide Field Cameras data. We would like to thank the RXTE/ASM teams at MIT and GSFC for provision of the on-line ASM data. We acknowledge the use of the Digitized Sky Survey produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. RC acknowledges financial support from a European Union Marie Curie Intra-European Fellowship (MEIFT-CT-2005-024685).
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[^1]: E-mail: [email protected]
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[**Branes, U-folds and hyperelliptic fibrations**]{}
[ *Luca Martucci[^1], Jose Francisco Morales[^2] & Daniel Ricci Pacifici [^3]*]{}
*I.N.F.N. Sezione di Roma “TorVergata” &\
Dipartimento di Fisica, Università di Roma “TorVergata",\
Via della Ricerca ScientiÞca, 00133 Roma, Italy*\
**Abstract**
Introduction and Summary
========================
F-theory [@Vafa:1996xn] provides an elegant framework where fully non-pertubative solutions of type IIB supergravity are described in purely geometric terms. Solutions are characterized by a non-trivial profile of the axio-dilaton field $\tau$. In the simplest set up, one can think of the field $\tau$ as the complex structure of an auxiliary torus fibered over a complex plane with punctures at the points where the torus fiber degenerates. Moving around a puncture, $\tau$ undergoes a non-trivial monodromy in the U-duality group ${\rm SL}(2,\mathbb{Z})$ indicating the presence of a 7-brane charge. The resulting vacua are non-perturbative in nature but in appropriate limits they describe systems of D7-branes and O7-planes non-perturbatively completed by D-instantons [@sen1; @sen2]. In particular, an O7-plane is described in this framework as a composite object, a pair of ($p,q$) 7-branes colliding at weak coupling. In this paper we study an extension of this picture which includes 3-branes as well.
The general philosophy is the same as the one adopted in F-theory, with the difference that now we start from type IIB theory on K3 and consider solutions with non-trivial profiles on a complex plane for a set of scalar fields in the six-dimensional effective theory. The effective six-dimensional supergravity includes 105 scalars, which are rotated by an ${\rm O}(5,21;\mathbb{Z})$ U-duality group. By allowing a subset of these scalars to vary over a complex plane with non-trivial U-duality monodromies, we will construct supersymmetric solutions of the six-dimensional supergravity, [*U-folds*]{}, that incorporate within the same framework 3- and 7-branes [^4]. The solutions generically describe (from the ten-dimensional perspective) non-geometric string vacua which patch together mutually non-local systems of 3- and 7-branes. In analogy with the F-theory case, the U-folds we consider here can be interpreted, in some appropriate limits, in terms of systems of D3,D7-branes and O3,O7-planes complemented by D(-1) and ED3 instantons.
We focus on solutions preserving $\caln=2$ four-dimensional supersymmetry. This allows us to make contact with field-theoretic results based on the Seiberg-Witten analysis [@Seiberg:1994rs] and their M-theory engineering [@Witten:1997sc]. In [@cremonesi], the supergravity vacuum associated with a system of fractional D3-branes at a ${\mathbb{C}}^2/\mathbb{Z}_2$ singularity was obtained by reduction of the M5 brane solution along a two-dimensional curve. More recently, in [@lerdaetnoi], this solution was derived directly from string amplitudes computing the rate of emission of twisted fields from fractional D3-branes and D-instanton sources at the singularity. In particular, the profile for the twisted field on the plane orthogonal to the singularity was related to certain chiral correlators in the dual gauge theory (see also [@Billo:2010mg; @Billo:2011uc; @Fucito:2011kb] for similar results in the elliptic F-theory context). To understand, the gravity counterpart of these results was one of the initial motivations of this work. Here we develop a unifying framework for supergravity solutions describing general systems (geometric or not) of 3- and 7-branes, in which brane instanton corrections are codified in simple geometrical terms. Although we confine ourselves to ${\cal N}=2$ supersymmetric vacua, we believe our techniques can be adapted to less supersymmetric and phenomenologically motivated settings, as for instance those of [@Kachru:2003aw; @Balasubramanian:2005zx].
Let us now briefly discuss our approach and the structure of the paper. In section \[sechol\], we review the construction of ten-dimensional holomorphic vacua of type IIB supergravity. We show that after reduction on K3, the conditions of ten-dimensional supersymmetry translate into the requirement of holomorphicity for a set of six-dimensional fields $(\tau,\sigma,\beta^a)$. $\tau$ is the axio-dilaton field, $\sigma$ characterizes the warp factor and the R-R four-fom and $\beta^a$ correspond to the reduction of the NS-NS/R-R two-form $C_2+\tau B$ along a set of $n$ vanishing exceptional cycles $\calc_a$ at a singularity of K3. These fields transform under an ${\rm O}(2,2+n;\mathbb{Z})$ subgroup of the complete U-duality group. The six-dimensional viewpoint is developed in section \[sec:IIBcomp\], where the ten-dimensional equations (integrated over K3) are reproduced directly from the six-dimensional effective theory, along the lines of [@cstring]. This provides a framework in which the global properties of the vacua, characterized by non-trivial monodromies in the U-duality group, can be addressed.
In section \[sec:examples1\], \[sec:examples2\] we provide some explicit realizations of our general results. The case with no three-form fluxes ($n=0$) is described by a double elliptic fibration over a complex plane, a double copy of the well understood F-theory elliptic geometries. Already for $n=1$ (associated with a ${\mathbb{C}}^2/{\mathbb{Z}}_2$ singularity) one obtains a much richer situation. The U-duality group in this case is ${\rm O}(2,3;\mathbb{Z})\simeq {\rm Sp}(4,{\mathbb{Z}})$ which is nothing but the modular group of a genus two Riemann surface. Moreover, one can see that $\tau,\sigma,\beta$ transform under this group as the three entries of the period matrix ${\bf \Omega}$ of a genus two surface. This suggests that the general solution in this case is described by the fibration of a genus two surface over ${\mathbb{C}}$ such that ${\bf \Omega}(z)$ varies holomorphically on $z$. Locations of branes are associated with points in the $z$-plane where the fiber degenerates. Circling these points, the matrix ${\bf \Omega}(z)$ undergoes non-trivial U-duality monodromies specifying the type of brane at the puncture. We discuss some simple examples and their brane interpretation. A more systematic analysis, and extensions to cases with $n>1$, is left to future investigation. Finally, this paper contains extensive appendices including technical details and background material.
Holomorphic solutions of type IIB supergravity {#sechol}
==============================================
We are interested in describing type IIB vacua preserving $\caln=2$ four-dimensional supersymmetry and characterized by the presence of D3 and D7-branes. We start with a ten-dimensional background of the form $\mathbb{R}^{1,3}\times \mathbb{C}\times X$, where $X$ is a (four-dimensional Ricci-flat) K3 space. We consider D-branes with world-volumes sharing the four-dimensional flat space $\mathbb{R}^{1,3}$ and sitting at certain points in $\mathbb{C}$. Hence, D7’s wrap the entire internal $X$ while D3’s sit at points in $X$. Furthermore D3-branes can be either regular or fractional in the case K3 is singular. These configurations preserve $\caln=2$ four-dimensional supersymmetry and locally admit a specific ten-dimensional supergravity description. Indeed, they can be seen as special sub-cases of the warped Calabi-Yau/F-theory backgrounds discussed in [@GKP], which considers warped flux vacua on space-times of the form $\mathbb{R}^{1,3}\times Y$, where $Y$ is either a Calabi-Yau or a F-theory Kähler space. In our case we locally have $Y=\mathbb{C}\times X$. The general backgrounds of [@GKP] preserve $\caln=1$ four-dimensional supersymmery, which is enhanced to $\caln=2$ for this choice. These kinds of configurations have been previously considered in the literature, as for instance in [@GP].
Supersymmetric vacua
--------------------
In this section we review the construction in [@GP] of holomorphic vacua of type IIB supergravity on a singular K3 describing the local geometry generated by a systems of D3 and D7 branes. In Appendix \[app:10d\] we present a self-contained re-derivation of these solutions by using the generalized complex geometry formalism. Here we just quote the results. The ten-dimensional metric in the Einstein frame is given by \[10dmetric\] s\^2\_[E]{} = e\^[2 A]{}x\^x\_+ e\^[-2A]{} s\^2\_Y with \[ymetric\] s\^2\_Y= e\^[-]{} |h(z)|\^2 z |z +s\^2\_[X]{} Here $\d s^2_X$ is the Ricci flat K3 metric, $h(z)$ is a holomorphic function and the dilaton $\phi$ is constant along $X$. In addition, two-form potentials are taken to be self-dual with respect to the K3 metric defined by hyperkähler structure associated with the triplet of anti-self dual two-forms $(\Re\omega,\Im\omega,j)$, where $j$ is the Kähler form and $\omega$ the holomorphic (2,0) form on $X$. By introducing a set $\chi_a\in H^2(X;\mathbb{Z}) $ of integer self-dual two-forms on $X$, with positive definite non-degenerate pairing \[deltab\] \_[ab]{}=\_X\_a\_b we set C\_2+B=\^a\_a We remark that self-duality of $\chi_a$ implies $j\wedge\chi_a=\omega\wedge\chi_a=\bar\omega\wedge\chi_a=0$ and therefore the two-cycles $\calc_a$ Poincaré dual to the forms $\chi_a=[\calc_a ]$ should have vanishing volume[^5]. Indeed, as we will more precisely discuss in section \[sec:Dbranes\], non-trivial two-form fluxes $\beta^a$ are allowed only for a singular K3 and signal for D5 branes wrapping the vanishing exceptional cycles $ \calc_a$ or equivalently fractional D3-branes. The functions $\beta^a$ are taken to be constant on $X$ and varying over the $z$-plane.
Under these assumptions, the background supersymmetry conditions reduce drastically and can be written in the compact form (see Appendix \[app:10d\] for details) | + |[T]{}=0 \[susytau\] with $\bar\partial=\bar\partial_{{\mathbb{C}}}+\bar\partial_{X}=\d\bar u^\alpha \frac{\partial}{\partial \bar u^\alpha}$ (where $u^\alpha=(z,u^1,u^2)$, are local complex coordinates on $Y$) and $\cal T$ a polyform =e\^B \[tpoly\] packing the RR potentials $C=C_0+C_2+C_4$ and the NS-NS data. In particular J=j- e\^[-]{} |h(z)|\^2 z|z is just the Kähler form associated with the metric (\[ymetric\]). Writing $\calt$ in components
[T]{}\_0 &= C\_0+ e\^[-]{}\
[T]{}\_2 &= C\_2 + B =\^a\_a\
[T]{}\_4 &= C\_4- e\^[-4A]{}JJ +C\_2B + 12 BB \[t024\]
one can immediately see that the first two conditions encoded in (\[susytau\]) just require that we must take $\tau$ and $\beta^a$ to be holomorphic: $\tau=\tau(z)$ and $\beta^a=\beta^a(z)$.
The last equation in (\[susytau\]) requires more care. This can be seen from the integrability condition $\partial \bar\partial {\rm Re}{\cal T}_4=\partial \bar\partial {\rm Im}{\cal T}_4=0$. For instance the second equation implies (e\^[-4A]{}) JJ= \[lapwarp\] with G\_3=C\_2+B= (\^a-)\_a Equation (\[lapwarp\]) determines the warp factor $e^{-4A}$ in terms of the three form flux $G_3$. Notice that the right hand side of this equation is localized at the singularity, which implies that $e^{-4A}$ should depend on $X$ in order to match this behavior. A similar equation and conclusion can be drawn for $C_4$. Still, one can define a holomorphic field on $\mathbb{C}$ out of ${\cal T}_4$. Indeed, by integrating (\[susytau\]) over $X$ one gets (|\_ )=0 \_X [T]{}\_4 which implies that $\sigma(z)$ is holomorphic on ${\mathbb{C}}$.
We can summarize these results by saying that the effective six-dimensional fields which are obtained by integrating $\calt$ along the internal 0-, 4- and 2-cycles
&= [T]{}\_0C\_0+e\^[-]{}\
&= \_[X]{} [T]{}\_4 = \_[X]{} (C\_4- e\^[-4A]{} JJ+BC\_2+12 B B )\
\_a &= \_[\_a]{} [T]{}\_2= \_[\_a]{} (C\_2+B) -\_[ab]{}\^b \[sols0\]
depend holomorphically on $z$, i.e. \[holcond0\] =0 ,=0 ,\^a=0 This result will be confirmed by purely effective six-dimensional arguments in section \[sec:IIBcomp\].
D-branes and monodromies {#sec:Dbranes}
------------------------
In presence of D-branes, the equations (\[holcond0\]) are modified by delta-like functions centered at the brane positions. Let us consider D$p$-branes, $p=3,5,7$, filling $\mathbb{R}^{1,3}$, sitting at a point $0\in{\mathbb{C}}$ and wrapping a $(p-3)$-cycle $\Sigma_{p-3}$ in $X$. The elementary brane couples to the R-R fields via the CS term $\int_{\mathbb{R}^{1,3}\times \Sigma_{p-3}}C_{p+1}$. This coupling generates a source term of the internal R-R field strength (e\^BF)=\_\^[2]{}(0)\_[7-p]{} where $F=F_1+F_3+F_5$, with $F_k$ the R-R field strengths and $\alpha_{7-p}$ stands for the delta-like form which is Poincare dual in $X$ to $\Sigma_{p-3}$. Integrating this equation over $D_2\times [\alpha_{7-p}]$, where $D_2$ is a disk surrounding $0\in{\mathbb{C}}$, we get 1=\_\_[\[\_[7-p]{}\]]{} e\^BF =\_[\[\_[7-p]{}\]]{} (e\^BC) |\^[z e\^[2i]{}]{}\_[z]{} \[uno\] where $\gamma=\partial D_2$ is a curve surrounding $0\in{\mathbb{C}}$. In deriving the last equality we have used the R-R Bianchi identity $\d_H F=0$ to write locally $e^B\wedge F=\d(e^B\wedge C)$. Specifying to $p=3,5,7$, we notice that the quantities in the right hand side of (\[uno\]) are nothing but the real parts of $\sigma,\beta^a$ and $\tau$ respectively as defined in (\[sols0\]). Then, equation (\[sols0\]) implies that the presence of D$p$-branes induces the monodromies [^6] :&&+1\
:&&\^b\^b+\^b\_a \[monD3\]\
:&&+1 A similar analysis can be done for the holonomies associated with O-planes. Still, from the experience of F-theory it is known that O-planes are not elementary objects and at the non-perturbative level they are resolved into more elementary ones. Therefore the solutions will be characterized entirely in terms of the D-branes discussed above and their U-duals, which provide the elementary constituents of our background.
Finally, let us observe that $\tau,\sigma$ and $\beta^a$ can be also interpreted as the tree level complexified gauge couplings $\tau_{\rm YM}=\frac{\theta_{\rm YM}}{2\pi}+\frac{4\pi\ii}{g^2_{\rm YM}}$ appearing in the the four-dimensional effective theories supported by the different branes probing these backgrounds. Indeed, by dimensionally reducing the DBI+CS action, one gets the identifications $\tau^{\rm D3}_{\rm YM}= \tau$, $\tau^{\text{D7-D3}}_{\rm YM}=\sigma$, $\tau^{{\rm D5}_a}_{\rm YM}=\beta_a\equiv -\Delta_{ab}\beta^b$. The non-trivial profiles in the ${\mathbb{C}}$-plane for $(\sigma,\beta^a,\tau)$ in the gravity solutions we will construct describe then the running of these gauge couplings in the dual gauge theories.
U-dualities
------------
So far, we have only considered elementary D-branes but one could consider other branes which are related to the above ones by duality transformations. We are interested in duality transformations whose action closes on fields $\tau,\sigma,\beta^a$ which characterize our vacua.
Already at the level of ten-dimensional supergravity, one has the perturbative dualities which correspond to integral shifts of the R-R and NS-NS gauge potentials, as well as the non-perturbative type IIB S-duality. In addition, in the compactified theory one has an additional duality, the so called Fourier-Mukai transform that we denote by $R$. This action does not has a counterpart in ten-dimensional supergravity. In our context, it can be seen as a sort of ‘T-duality’ (along all four directions of the K3 space $X$) which exchanges regular D3 and D7-branes, leaving fractional D3-branes untouched. More precisely, the $R$-duality exchanges $\tau\leftrightarrow \sigma$ [^7]. We can now combine the Fourier-Mukai tranform $R$ with S-duality and shift dualities obtaining the following minimal set of duality tranformations acting on the fields $\tau,\sigma,\beta^a$ as follows: \[duality\]
generator non-trivial action
----------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------
$S$ $\tau\rightarrow -\frac{1}{\tau} \qquad~ \sigma\rightarrow \sigma-\frac{1}{2\tau}\Delta_{ab}\,\beta^a\beta^b \qquad~ \beta^a\rightarrow \frac{1}{\tau}\beta^a$
$T$ $\tau\rightarrow \tau+1$
$W_{a}$ $\quad \beta^b\rightarrow \beta^b+\delta^b_a $
$ R $ $\tau\leftrightarrow \sigma$
The elements $S$ and $T$ generate the SL(2,$\mathbb{Z})_\tau$ S-duality group of type IIB theory, $R$ is the Fourier-Mukai transform and $W_a$ corresponds to the axionic shift $C_2\rightarrow C_2+\chi_a$ [^8]. The transformations (\[duality\]) generate the U-duality group of our system which will be denoted by ${\rm O}(\Gamma_{2,2+n})$ and is isomorphic to ${\rm O}(2,2+n;\mathbb{Z})$. Here $\Gamma_{2,2+n} \simeq \Gamma_{2,2}\oplus \Gamma_n$ with $\Gamma_{2,2}$ the unique four-dimensional self-dual even lattice of signature $(2,2)$ while $ \Gamma_n$ is a sub-lattice of $H^{2}(X;\mathbb{Z})$ generated by $n$ integer self-dual forms $\chi_a\equiv[\calc_a]$ equipped with the positive definite pairing $\Delta_{ab}$ defined in (\[deltab\]).
U-folds: the six-dimensional perspective {#sec:IIBcomp}
========================================
\[sec:topcond\]
The six-dimensional effective theory describing the dynamics of type IIB supergravity on K3 can be obtained by dimensional reduction. First, we split the ten-dimensional space in $M_6\times X$ and we parametrize the ten-dimensional Einstein frame metric as \[six+four\] s\^2\_[E]{}=e\^[2A]{}s\^2\_[6]{} + e\^[-2A]{}s\^2\_X The warp factor $A$ is taken approximately constant along $X$. After reduction to six-dimensions, the ten-dimensional Einstein-Hilbert term reduces to \^[10]{}xR\_[(10)]{}= M\^4\_[P]{} \^6x R\_6+…\[10d6d\] with \[MP\] M\^4\_[P]{} -12\_X jj the K3 volume computed by the metric $\d s^2_X$. We use units where $2\pi \sqrt{\alpha'}=1$ and denote by $M_{\rm P}$ the six-dimensional Planck mass. On the other hand, the effective dynamical volume of the internal space $X$ is given by =- \_X e\^[-4A-]{} jj = -12 where $\Im\beta\cdot \Im\beta\equiv \Delta_{ab}\Im\beta^a\Im\beta^b$. In addition to $\cal V$, the moduli space of metrics on K3 contains other 57 moduli that specify a hyperkähler structure, i.e. a choice of a triplet of anti-self-dual two-forms $(\Re\omega,\Im\omega,j)$ inside the space ${\mathbb R}^{3,19}$ of two forms on K3. The remaining moduli come from the dilaton $\phi$, the axion $C_0$, one scalar associated with $C_4\in H^4(X;\mathbb{R})$ and 44 scalars coming from $B,C_2\in H^2(X;\mathbb{R})$, since $\dim H^2(X;\mathbb{R})=22$. Altogether, these fields parametrize the moduli space \[modulispaceR\] \_=[O]{}(\_[5,21]{})\\[ [O]{}(5,21;)(5;) (21;)]{} The discrete group ${\rm O}(\Gamma_{5,21})\equiv{\rm O}(5,21;\mathbb{Z}$) is the U-duality group of the effective six-dimensional theory. A more detailed description of this moduli space is presented in Appendix \[app:trunc\].
The reduced moduli space
------------------------
The fields $\tau,\sigma,\beta^a$ characterizing the supersymmetric vacua under study here parametrize the reduced moduli space \[truncmsn\] =[O]{}(\_[2,2+n]{})\\[ [O]{}(2,2+n;)(2;) (2+n;)]{} In Appendix \[app:trunc\] we show that the reduction to (\[truncmsn\]) defines a consistent truncation of the moduli space (\[modulispaceR\]) of type IIB supergravity on K3. Acting with the U-duality group ${\rm O}(\Gamma_{2,2+n})$ on the elementary branes discussed in section \[sechol\] one can generate the monodromies associated with a general system of $(p,q)$ 3-, 5- and 7- branes. We will be interested in describing systems in which different branes are contemporarily present. These are characterized by solutions where the holomorphic fields $\tau,\sigma,\beta^a$ are allowed to jump under the U-duality group of the effective six-dimensional theory. This is the six-dimensional analogue of the more familiar F-theory elliptic backgrounds, in which the U-duality group is just SL(2,$\mathbb{Z})$.
The space (\[truncmsn\]) has the important property of being a Kählerian coset manifold. The fact that it is complex is already evident from its parametrization provided by the fields introduced in (\[sols0\]). It is then possible to show that the standard coset metric can be written as a Kähler metric, with Kähler potential (see Appendix \[app:trunc\]) \[kpot\] K=-with = -12 \[calv\] It is important to notice that the fields $\tau,\sigma,\beta^a$ parametrizing (\[truncmsn\]) must satisfy the conditions \[podefcond\] ,,> 0 and therefore $K$ is real as expected. $K$ defines a well defined metric on the orbifolded coset space $\calm$ as well. Moreover, one can easily check that $\calv$ is invariant under $T$, $W_{a}$ and $R$ in (\[duality\]), while it transforms as $\calv\rightarrow \calv/|\tau|^2$ under $S$. Correspondingly, S :K K--|which is a Kähler transformation. Since this happens for the generators (\[duality\]), the same is true for any element of ${\rm O}(\Gamma_{2,2+n})$.
Six-dimensional equations
--------------------------
We are interested on solutions of this six-dimensional supergravity involving non-trivial backgrounds for the metric and the scalar fields $\varphi^I=(\tau,\sigma,\beta^a)$ spanning the Kählerian coset submanifold (\[truncmsn\]). The setting is completely analogous to the one considered in [@cstring]. The relevant terms of the effective action are: \[effact\] S\_[eff]{}= \^6 x where $K_{I\bar J}\equiv \frac{\del^2 K}{\del\varphi^I\del\bar\varphi^{\bar J}}$ is the Kähler metric associated with the Kähler potential (\[kpot\]). In general, assuming that the complex scalars $\varphi^I$ depend just on one complex coordinate $(z,\bar z)$, for any Kählerian target space the scalars equations of motion reduce to \[scaleq\] \_z\_[|z]{}\^I+\^I\_[JL]{}\_z\^J\_[|z]{}\^L=0 where $\Gamma^I{}_{JK}$ are the Christoffel symbols of the Kähler metric, which have crucially only purely holomorphic or anti-holomorphic indices.
The equations (\[scaleq\]) are easily solved by choosing the scalar fields to be holomorphic, namely \[holcond\] =0 ,=0 ,\^a=0 We see that this purely six-dimensional description reproduces the conditions (\[holcond0\]) obtained by direct integration of the ten-dimensional supersymmetry equations.
On the other hand, one can take an ansatz for the six-simensional metric of the form $\d s^2=\d x^\mu\d x_\mu+e^{\rho(z,\bar z)}\d z\d \bar z$. By using once again the Kähler structure defining the effective action (\[effact\]), the Einstein equations reduce to $\del\delbar(\rho+K)=0$, where $K$ is the Kähler potential (\[kpot\]). Hence, the Einstein equations are solved by the $6$-dimensional metric s\^2\_6=x\^x\_+ M\_P\^[-4]{} [V ]{} |h(z)|\^2 z |z \[g6\] where $\calv$ is defined in (\[calv\]) and the constant factor $M_P^{-4}$ is included for matching the 10-dimensional metric (\[10dmetric\]).
We notice that the invariance of the metric (\[g6\]) under the U-duality group implies that the function $h(z)$ should transform under $S$ in such a way to keep $\calv |h(z)|^2$ invariant. This requirement is satisfied if we choose h(z) =( [\_D((z),(z),\^a(z))\_[i]{} (z-z\_i) ]{} )\^[1D]{} \[hd\] where $\chi_D$ is a generalized ${\rm O}(2,2+n,{\mathbb{Z}})$ modular form of weight $D$, i.e. a function of $\varphi^I=(\tau,\sigma,\beta^a)$ which is invariant under the generators $T,W_a,R$ in (\[duality\]) and transforms under $S$ as \[holmodular\] \_D (S\^I)=\^D \_D (\^I) Finally $z_i$ are simple zeros of $\chi_D(\tau(z),\sigma(z),\beta^a(z))$, with the denominator $\prod_{i} (z-z_i) $ in (\[hd\]) included in order to cancel the zeroes of $\chi_D(\varphi^I(z))$ in ${\mathbb{C}}$, leaving a no-where vanishing function $h(z)$. We will see that the choice (\[hd\]) is compatible with supersymmetry requirements too.
Supersymmetry and topological conditions {#sec:topcond}
----------------------------------------
In order to show that the six-dimensional backgrounds described in this section are supersymmetric, one has to show the existence of Killing spinors under which the supersymmetric variations of the six-dimensional gravitino and matter fermions vanish. The supersymmetry conditions follow from those of ${\cal N}=(2,0)$ supergravity [@romans] after reduction to the truncated moduli space (\[truncmsn\]). This problem is somewhat technical and for this reason its discussion is detailed in Appendix \[app:susy\]. Here we quote the main results.
As discussed in Appendix \[app:susy\], one can in fact write down an ansatz for eight independent Killing spinors (hence providing four-dimensional $\caln$=2 supersymmetry) written just in terms of a single two-dimensional chiral spinor $\eta$ defined on the $z$-plane. The supersymmetry variations of the matter fermions are vanishing once the fields $\varphi^I=(\tau,\sigma,\beta^a)$ are holomorphic, as in (\[holcond\]). On the other hand, the vanishing of the gravitino variation reduces to the following two-dimensional equations \[2dsusy6\] D\_m (\_m-\_[m]{} )=0 where the index $m=1,2$ runs over coordinates of the complex plane. $\nabla_m$ is the covariant derivative associated with the ordinary spin connection, which must be computed by using the two-dimensional metric $\calv |h(z)|^2\d z\d \bar z$ appearing in (\[g6\]). $ \calq_{m}$ are the components of the ${\rm SO}(2)\sim {\rm U}(1)$-connection: \[calq\] =\_[z]{}z+\_[|z]{}|z=(z) It is easy to see that the $\calv$-dependent contribution to the spin connection in (\[2dsusy6\]) cancels against $\calq_m$ leading to the Killing spinor solution \[globeta0\] =()\^[14]{}\_0 with constant $\eta_0$ which satisfies the appropriate projection conditions – see appendix \[app:susy\].
It is well known that codimension-two configurations generically produce deficit angles at large distances [@Deser:1983tn] and this puts severe consistency constraints. If we assume to have a configuration in which the holomorphic fields $\varphi^I(z)$ are asymptotically constant for $|z|\rightarrow \infty$, the deficit angle at infinity is given by $\Delta\theta=\int \calr_{(2)}$, where $\calr_{(2)}$ is the two-dimensional ${\rm SO(2)}\simeq {\rm U}(1)$ curvature. On the other hand, the integrability of (\[2dsusy6\]) requires that $[D_m,D_n]\eta=0$ and then $\calr_{(2)}=F_\calq$, where $F_\calq =\d\calq$ is the curvature associated with the U(1) connection $\calq$. Hence, supersymmetry requires the deficit angle to be given by $\Delta\theta=\int \calr_{(2)}=\int F_\calq$, consistently with the results of section 5 of [@cstring].
In particular, the transverse space closes up to a sphere $\mathbb{P}^1$ when $\Delta\theta=4\pi$. In this case the holomorphic tangent bundle $\calt_{\mathbb{P}^1}$ is isomorphic to $\calo_{\mathbb{P}^1}(2)$, which is the line bundle whose sections are homogeneous polynomials of degree-two in the projective coordinates $[z_0:z_1]$. Indeed, from $\calr_{(2)}=2\pi\,c_1(\calt_{\mathbb{P}^1})$ one gets $\Delta \theta=2\pi \int_{\mathbb{P}^1}c_1(\calo_{\mathbb{P}^1}(2))=4\pi$ as required. On the other hand, the integrability condition $\calr_{(2)}=F_\calq$ implies that the holomorphic line bundle $\call_Q$ associated with the connection $\calq$ is isomorphic to $\calo_{\mathbb{P}^1}(2)$. Noticing that the pull-back of a modular form $\chi_D(\varphi^I)$ of weight $D$ can be regarded as a section of $\call^D_Q$, we see that $\chi_D(\varphi(z)^I)$ must be given by a homogeneous polynomial of degree $2D$ in the projective coordinates $[z_0:z_1]$.
This implies in particular that if $\chi_D(\varphi^I(z))$ appearing in (\[hd\]) has $2D$ zeros in the $z$-plane, the plane compactifies to a ${\mathbb P}^1$. Consistently the metric (\[g6\]) at large $z$ is regular as can be seen from the asymptotic behaviour |h(z)|\^2 z |z \_0|z|\^[-4]{} z |z = \_0w|w with $w=1/z$ the coordinate on the second chart of ${\mathbb P}^1$.
U-fold solutions without 3-form fluxes {#sec:examples1}
======================================
In this and the next section we present some simple examples of U-folds. We start by considering the case $n=0$, in which the K3 space is smooth and there are no three-form fluxes. The restricted moduli space becomes \[truncms22\] \_[n=0]{}=[O]{}(\_[2,2]{})\\[ [O]{}(2,2;)(2;) (2;)]{} \_2\\( [O]{}(\_[1,1]{})\\[ [Sl]{}(2;)(1)]{} )\^2 where $ \mathbb{Z}_2$ refers to the $R$-duality. Being the moduli space factorized, this case can be considered as a doubled elliptic fibration, a double copy of the well known F-theory elliptic geometries [@Vafa:1996xn], where the torus fiber is replaced by a factorized product of two tori with complex structures $\tau$ and $\sigma$ respectively. In analogy with the standard F-theory elliptic geometry [@sen1], the solution describes now a system of regular D7 and D3 branes with O7 and O3 planes non-perturbatively resolved in terms of $(p,q)$-branes. The interpretation in our setting is completely analogous to the ordinary F-theory case. For this reason we will be rather sketchy.
Consider first a non-trival $\tau(z)$. The details of the geometry are encoded in an elliptic curve which can be generally written into the form y\^2 = \_[i=1]{}\^3 (x-e\_i(z) )=x\^3+f\_2(z) x+f\_3(z) \[ellip\] This solution describes systems of $(p,q)$ 7-branes, related to the elementary D7 branes via $SL(2,{\mathbb{Z}})$-duality. In particular for a choice of $f_2$, $f_3$ where (\[ellip\]) matches the Seiberg-Witten curve of a ${\cal N}=2$ SU(2) gauge theory with four fundamentals the solution describes the non-perturbative resolution of a system of four D7-branes and one O7-plane [@sen1] probed by an elementary D3-brane [@Banks:1996nj]. The axio-dilaton profile can be extracted from the standard elliptic formula = [\_2\^4\_3\^4]{}() \[zeta1\] relating the harmonic ratio of the roots to the complex structure $\tau$ of the torus. Here $\theta_{s}$ are the genus one even theta constants (see (\[genus1theta\]) and (\[thetastand\]) for the definition). The positions of D7-branes correspond to points in the $z$-plane where $e_1\to e_2$, i.e. $\tau\to \ii \infty$. Going around this point the axio-dilaton field undergoes the monodromy $\tau\to \tau+1$. Finally the O7 plane corresponds to a pair of degeneration points with overall monodromy $\tau\to \tau-4$. The effects of instantons resolve this plane into a pairs of $(p,q)$-branes which locally look like D7-branes (in a given SL(2,${\mathbb{Z}}$) frame). Encircling the two $(p,q)$ 7-branes one finds the monodromy reproducing the O7-plane charge [@sen1].
The story for the $\sigma(z)$ field follows [*mutatis mutandis*]{} that of $\tau$. Again the details of the geometry are encoded in the elliptic data = [\_2\^4\_3\^4]{}() \[zeta2\] with $\tilde{e}_i$ the roots of an elliptic curve of type (\[ellip\]). The fibration describes now the background of $(p,q)$ 3-branes, related to elementary D3-branes via ${\rm SL(2;\mathbb{Z})}$ duality. The elliptic fibration now corresponds to the Seiberg-Witten curve describing the dynamics of the gauge theories in D7 brane probes of the D3 brane geometry after reduction to four-dimensions.
Summarizing, the U-fold solution with no three-form fluxes is specified by the choice of two elliptic curves fibered over ${\mathbb{C}}$ with punctures signaling the presence of 3 and 7-branes. Notice that in this case there is a natural candidate for the holomorphic function $h(z)$ entering the metric of the solution, which follows from the doubling of the solution of [@cstring]. This is given by h(z)= \[heta\] where $u_i$ and $v_j$ are the points where the elliptic fibrations defining $\tau$ and $\sigma$ respectively degenerate. These are the points at which the discriminant $\Delta(z)=4f^3_2(z)+27f^2_3(z)$ of one of the two elliptic curves vanishes and they signal in general the presence of $(p,q)$-branes. This is consistent with the general form of $h(z)$ given in (\[hd\]) with $D=12$ and $\chi_{12}(\tau,\sigma)=\eta(\tau)^{24}\eta(\sigma)^{24}$. As explained in section \[sec:topcond\], for fibrations chosen such that there are 24 degeneration points in total, the complex plane compactifies to the sphere $\mathbb{P}^1$.
U-folds from hyperelliptic fibrations {#sec:examples2}
=====================================
In this section we discuss in some more details the case $n=1$, i.e. the case in which the K3 develops a local $\mathbb{C}^2/\mathbb{Z}_2$ singularity with a single exceptional cycle $\calc_1\equiv \calc$. An analogous discussion for the case with $n>1$ exceptional cycles is left to the future.
The solutions in the case $n=1$ involve three active scalar fields $(\tau,\sigma,\beta)$, where $\beta\equiv \beta^1$ according to the general notation used in the previous sections. These scalars follow from the reduction of the axio-dilaton field, the warp factor, the RR four-form and the NSNS and RR two-form potentials along $ \calc$. The three complex scalars span the coset \[truncms\] =[O]{}(\_[2,3]{})\\[ [O]{}(2,3;)(2;) (3;)]{} The exceptional cycle $\calc$ sitting at the singularity is a two-sphere with self-intersection $\calc\cdot\calc=-2$ and then $\Delta_{11}=2$. Hence, in this case, the generators of the U-duality group (\[duality\]) reduce to \[duality2\]
&T:+1 R: W:+1\
&S:- -\^2
We observe that the U-duality group ${\rm O}(\Gamma_{2,3})$ generated by (\[duality2\]) is isomorphic to the modular group Sp$(4,{\mathbb{Z}})$ of a genus two hyperelliptic Riemann surface. Moreover, if we organize the three complex scalars into a $2\times 2$ matrix =(
[cc]{} &\
&
) one can see that ${\bf \Omega}$ transforms under the U-duality transformations (\[duality2\]) as the period matrix of a genus two Riemann surface (see appendix \[sgtwo\]) (A[****]{}+B)(C[****]{}+D)\^[-1]{} with $A,B,C,D$ $2\times 2$ matrices defining a matrix of Sp$(4,{\mathbb{Z}})$ via[^9] \[spmatrix\] M= (
[cc]{} A & B\
C & D\
) M (
[cc]{} 0 & [$\mathbbm{1}$]{}\
-[$\mathbbm{1}$]{}& 0\
) M\^T= (
[cc]{} 0 & [$\mathbbm{1}$]{}\
-[$\mathbbm{1}$]{}& 0\
) Furthermore, the quantity $\calv$ defined in (\[calv\]) reduces in this case to =-()\^2 Hence, the consistency condition $\calv>0$ translates into the condition that $\Im{\bf \Omega}$ is a positive definite matrix, as required for ${\bf \Omega}$ being the period matrix of a Riemann surface. The identification of ${\bf \Omega}$ with the period matrix of a genus two Riemann surface suggests that the U-fold solution can be viewed as a holomorphic fibration of a genus two Riemann surface over the complex plane ${\mathbb{C}}$. The period matrix ${\bf \Omega}(z)$ describes the variations of scalar fields over the complex plane ${\mathbb{C}}$. U-duality holonomies around brane locations are encoded in the non-trivial modular group transformations that the cycles of the Riemann surface undergo around a point where the fiber degenerates.
The geometry of genus two fibrations over a complex plane has been extensively studied in the mathematical literature and one can resort to this powerful apparatus to explore the physics of U-folds in this sector. Here we will not attempt an analysis of the general case but rather we focus on some explicit choices of fibrations illustrating few relevant features of the general solution. We start by describing the geometry of the genus two curve, the period matrix ${\bf \Omega}(z)$, its degenerations, holonomies and brane interpretation.
The genus two fibration
-----------------------
We start by describing the geometry of the hyperelliptic fibration. We refer the reader to Appendix \[sgtwo\] for further details. A Riemann surface of genus two can be always described by a hyperelliptic curve (a sextic or a quintic) y\^2 =\_[i=1]{}\^6 (x-e\_i(z) ) =x\^6+f\_2(z) x\^4+ f\_3(z) x\^3+…+f\_6(z) \[sucurve0\] At each point $z$ equation (\[sucurve0\]) specifies a genus two curve. The period matrix ${\bf \Omega}(z)$ of the genus two fiber at $z$ is computed by integrals around the non-trivial cycles in the complex $x$-plane with three cuts pairing the six roots $e_i$ (see Appendix \[AppCycles\] for details).
Alternatively, the hyperelliptic curve can be written directly in terms of the theta functions of the genus two Riemann surface $\theta[^a_b ] =\theta[^a_b] (0|{\bf \Omega})$ with half-characteristics $[^a_b]$. Indeed after using SL$(2,{\mathbb R})$ invariance to map, let us say, points $e_1,e_3,e_5$ to $0,1,\infty$ the curve can be brought to the quintic form \[xipar\] y\^2=x(x-1)(x-\_2)(x-\_4)(x-\_6) with
\_2([****]{}) &=&[ e\_[21]{} e\_[35]{}e\_[25]{} e\_[31]{} ]{} =[ \^2([****]{}) \^2 ([****]{}) \^2([****]{}) \^2([****]{})]{}\
\_4([****]{}) &=& [ e\_[41]{} e\_[35]{}e\_[45]{} e\_[31]{} ]{} =[ \^2([****]{}) \^2([****]{})\^2([****]{}) \^2([****]{})]{}\
\_6([****]{}) &=& [ e\_[61]{} e\_[35]{}e\_[65]{} e\_[31]{} ]{} =[ \^2([****]{})\^2([****]{})\^2([****]{}) \^2 ([****]{})]{} \[xi2340\]
The genus two surface degenerates whenever two roots $e_i$ collide signaling for the presence of a brane. In general, a degeneration shows up in the vanishing of the discriminant of the curve, that we denoted by $I_{10}$ and which is defined by I\_[10]{} = \_[1i<j 6]{} e\_[ij]{}\^2 For $I_{10} \neq 0$ the Riemann surface is smooth. At a point $z=z_0$ where the discriminant vanishes the genus two curve degenerates. Going around $z_0$, the period matrix ${\bf \Omega}(z)$ undergoes non-trivial monodromies. Given a hyperelliptic fibration, the brane content of the system is specified by these monodromies and the full non-perturbative dynamics is coded in the details of the fibration.
There are various basic ways in which a genus two Riemann surface can degenerate, see figure \[fig:deg\]. First, when one of the two handles is pinched, the Riemann surface degenerates to a torus with a double point. This happens if $\tau\rightarrow \ii\infty$ or $\sigma\rightarrow \ii\infty$, signaling the presence of 7- and 3-branes respectively. A genus two surface can also degenerates into two genus one surfaces when $\beta\to 0$. This degeneration will show up, for example, in the solution representing flux-dissolved fractional 3-branes in section \[sec:fracD3\]. On the other hand, a localized D5 brane charge would be signalled by a degeneration $\beta\to \infty$ which, however, can never come alone since $ \Im\tau\Im\sigma-(\Im\beta)^2>0$ for a genus two Riemann surface.
![The basic degenerations of the genus two fiber.[]{data-label="fig:deg"}](hyperfibr.pdf)
Finally, let us recall that in order to completely specify the background, one has to specify the modular form $\chi_D({\bf \Omega})$ entering in the metric. In analogy to F-theory elliptic solutions we expect $\chi_D({\bf \Omega})$ to vanish at the positions of 3- and 7-branes. This suggests that $\chi_D({\bf \Omega})$ is not only a modular form but a cusp form. The ring of cusps forms is generated by the three functions $\chi_{10}({\bf \Omega})$, $\chi_{12}({\bf \Omega})$ and $\chi_{35}({\bf \Omega})$ whose definition is provided in Appendix \[sgtwo\]. It is not clear to us whether any choice of the cusp form defines an admissible solution or if there is a privileged one. We postpone this interesting question to future investigations.
Fractional D3-branes {#sec:fracD3}
--------------------
In this section we describe the solution associated to a system of fractional D3 branes at a $\mathbb{C}^2/\mathbb{Z}_2$ singularity. This corresponds to the limit of large volume $\sigma \to \ii \infty$ and weak coupling $\tau \to \ii \infty$ of the hyperelliptic fibration. We will first show how the solution can be entirely reconstructed from the knowledge of its monodromies (the brane content). The results will be then matched with those coming from the hyperelliptic description.
A system of fractional D3 branes at $\mathbb{C}^2/\mathbb{Z}_2$ is characterized by two integers $N_0,N_1$ counting the number of fractional branes of each type. The brane system realizes a ${\cal N}=2$ quiver gauge theory with gauge group $U(N_0)\times U(N_1)$ and bifundamental matter. On the other hand fractional branes are source for the twisted field $\beta$ describing the NS-NS/R-R two-form fluxes along the exceptional cycle $\calc$ of the singularity. Since fractional D3-branes can be thought as D5-branes wrapping the exceptional cycle $\calc$ with opposite orientation, one expects a monodromy $\beta\rightarrow\beta+N_0-N_1$ for a turn around the location of the brane system in the ${\mathbb{C}}$-plane. One would then be tempted to locally describe them by $\beta\simeq \pm\frac{1}{2\pi\ii}\log(z-z_{\rm D5})$. However, this creates a problem. Around the D5-brane, $\tau$ and $\sigma$ are expected to be approximatively constant and therefore at some point the condition $\calv=\Im\tau\Im\sigma-(\Im\beta)^2 > 0$ would be violated, which would be inconsistent with the formulation based on the hyperelliptic fibration.
In order to address this problem, let us consider a local description of the system of fractional D3-branes. Practically, this can be obtained by taking a very large K3 volume. According to (\[MP\]), this corresponds to the limit of large Planck mass $M_{\rm P}\gg 1$, where then the six dimensional gravity effectively decouples. Furthermore, we take $\Im\sigma$, $\Im\tau$ large in order to keep the coupling of the string small and the combination $M_P^{-4} \,\calv$ appearing in the six-dimensional metric $(\ref{g6})$ finite. Under these conditions, the only allowed U-duality transformations are those leaving $\tau$ and $\sigma$ invariant. These transformations are generated by the two elements W : +1 S\^2:- A holomorphic function $\beta(z)$ with holonomies only of this kind can be written as (z) = \[betapq\] with $P_0(z), P_1(z)$ polynomials in $z$ of order $N_0$ and $N_1$ respectively. The function $\beta(z)$ displays indeed the following monodromies z\_0=: &&+N\_0-N\_1\
{ z\_0 ; P\_1(z\_0)=0}: && -1\
{ z\_0 ; P\_0(z)\^2=P\_1(z\_0)\^2}: && - We stress that the solution (\[betapq\]) is valid in the patch of ${\mathbb{C}}$ where $(\Im\beta)^2<\Im\sigma\, \Im \tau$. Hence, for $z$ large enough or near the zeros of $P_1(z)$, the local approximation breaks down and one should resort to the complete hyperelliptic description. In particular for $N_1=0$ (the pure gauge theory) $\beta(z)$ given in (\[betapq\]) is locally completely finite, with no logarithmic singularities at finite $z$, but is rather characterized by $2N$ points with monodromy $\beta\rightarrow -\beta$. In particular, there are no localized D5-brane sources in this case but a purely flux solution.[^10]
Formula (\[betapq\]) agrees with that one proposed in [@cremonesi], motivated by the M-theory lift of the D-brane system, and recently derived in [@lerdaetnoi] from a direct string computation [^11]. We refer to these references for a more detailed discussion on the physics and holographic interpretation of this solution.
### The hyperelliptic description
Let us now show how the previous results fit (and generalize) into the general hyperelliptic description of U-folds. We consider a hyperelliptic fibration in the limit $\Im\sigma\rightarrow\infty$, while keeping $\tau$ approximately constant but [*finite*]{}. This case covers a class of backgrounds which are dual to one of the M-theory ‘elliptic models’ of [@Witten:1997sc], see also [@Ennes:1999fb; @Petrini:2001fk]. These models can be used to extract $\beta(z)$ from the dual M5 geometry, as in [@cremonesi].
Using the parametrization (\[xipar\]), the genus two curve is specified by the three harmonic ratios $\xi_2,\xi_4,\xi_6$. Their dependence on the period matrix ${\bf \Omega}$ can be expressed in terms of the theta functions by means of the relations (\[xi2340\]). One finds that in the degenerate limit $\sigma\to\ii \infty$ the three harmonic ratios become (see Appendix \[spinching\]) \_2 =-[\_2\^2()\_4\^2()]{}[\_1\^2(|)\_3\^2(|)]{} \_4=1 \_6=-[\_4\^2()\_2\^2()]{}[\_1\^2(|)\_3\^2(|)]{} \[oneh2\] with the genus one theta functions given by (\[thetastand\]). The fact that $\xi_4=1$ means that the genus two curve has degenerated to a two-torus, as already mentioned. The requirement that $\tau$ remains constant over the complex plane translates into the condition that the ratio ${\xi_2(z) \over \xi_6(z)}$ is constant over ${\mathbb{C}}$. Inverting the last relation in (\[oneh2\]) one can then find $\beta(z)$ in terms of the harmonic ratio $\xi_{6}(z)$.
If we also perform the limit $\tau\to\ii \infty$ the three harmonic ratios further degenerate (see Appendix \[twopinch\] for details) \_2 =0 \_4=1 \_6=-\^2()\[oneh0\] This implies that the combination f(z)=2(z)=2(2\_6(z)+1)\^2-1 \[ff\] is a well defined function on the ${\mathbb{C}}$-plane (invariant under $\beta\to -\beta$ or $\beta \to \beta+1$). Inverting (\[ff\]) one finds \_+(z)=-\_-(z)= \[betapq0\] We immediately see that the result (\[betapq\]) is obtained by setting $ \beta(z)=\beta_+(z)$ and $f(z)=P_0(z)/P_1(z)$.
An example with fractional and regular branes
---------------------------------------------
Now we consider a case with both regular and fractional D-branes. A strategy for providing a concrete example, followed for instance in [@sen1] for the case of F-theory, is to exploit the experience coming from Seiberg-Witten (SW) theory. Hence, we are led to consider the hyperelliptic curve describing the ${\cal N}=2$ gauge theory with gauge group SU(3) and six fundamental hypermultiplets. For simplicity we take the SU(3) theory in the so called special vacua [@Argyres:1999ty], parametrized by three parameters, the cubic gauge invariant $z={\rm tr} \Phi^3$, a mass $m$ and the gauge coupling $g$. For this choice the hyperelliptic curve takes the simple form y\^2=\_[i=1]{}\^6 (x-e\_i(z) )=(x\^3-z)\^2-g\^2 (x\^3-m\^3)\^2 \[su3\] The six branch points are given by e\_[2k]{}(z)=w\^[k-1]{} ([ z+ g m\^31+ g]{} )\^[13]{} e\_[2k-1]{}(z)=w\^[k-1]{} ([ z- g m\^31- g]{} )\^[13]{} with $ w=e^{2\pi \ii \over 3}$ and $k=1,2,3$. Let us consider first the asymptotic geometry. At $z$ infinity one finds e\_[2k]{}= [w\^[k-1]{} z\^[13]{} (1+g)\^[1/3]{}]{} e\_[2k-1]{}= [w\^[k-1]{} z\^[13]{} (1-g)\^[1/3]{}]{} Plugging this into (\[xi2340\]), one finds that the harmonic ratios $\xi_{2,4,6}(z)$, and therefore ${\bf \Omega}(z)$ go to a constant and finite value for $z\rightarrow \infty$. Furthermore, in the limit $g\to 0 $, $e_{12},e_{56}\sim g\, z^{1/3}$, and all others $e_{ij}$ go like $z^{1/3}$. For the harmonic ratios $\xi_{2,4,6}(z)$ we have \_2 \~g \_4 \~1 \_6\~g\^[-1]{} \[xig\] To see what this implies for ${\bf \Omega}$, we consider the expansion for very large $\Im\tau,\Im\sigma$ of the theta functions in the right hand side of (\[xi2340\]). By using (\[qqy\]) and (\[thetaexp\]), one finds \_2 \~e\^[(-)]{} \_4 \~e\^[(-)]{} \_6 \~e\^[-]{} \[xizm2\] Comparing with (\[xig\]) we conclude that asymptotically for $z\gg 1$ and in the limit $g\ll 1$ we have the following limiting values of $\tau,\sigma,\beta$: \_0\_0 12 \_0g\^4 In other words the weak coupling of the auxiliary gauge theory corresponds to the limit where the imaginary parts of all entries of the period matrix are very large, which indeed defines the weak coupling of the string theory description.
Now let us consider the points where the genus two fiber degenerates. From the point of view of the auxiliary gauge theory they correspond to points in the Coulomb branch of the moduli space where BPS (in general dyonic) states become massless. The discriminant of the curve is given by I\_[10]{}=\_[i<j]{} e\_[ij]{}\^2=[(6g)\^6(1-g\^2)\^8]{} (z-m\^3)\^6(z\^2-g\^2 m\^6)\^2\[i10\] The zeros of $I_{10}$ signal the collision of some of the branch points. For the present case, there are three degeneration points (punctures) in the $z$-plane z=m\^3 && e\_[2k]{}=e\_[2k-1]{}=w\^[k-1]{} m\
z=g m\^3 && e\_[1]{}=e\_3=e\_5=0\
z=-g m\^3 && e\_[2]{}=e\_4=e\_6=0 for $k=1,2,3$. Going around these points the period matrix ${\bf \Omega}(z)$ undergoes non-trivial monodromies which characterize the brane content. To compute them, we use again the representation (\[xi2340\]) to compute the period matrix ${\bf \Omega}(z)$ in the nearby of the singularities.
Let us first consider the geometry near the degeneration point $z= m^3$. For $z\simeq m^3$, $e_{2k-1,2k} \sim (z-m^3)$ implying \_[2]{} \~(z-m\^3) , \_4\~1 , \_6\~(z-m\^3)\^[-1]{} Following the same arguments as before we conclude that close to $z= m^3$ we have ,\~ (z-m\^3)\^4 \~ (z-m\^3)\^2 We conclude that all the three entries of the period matrix behave logarithmically as $z\simeq m^3$, leading to the monodromies +4 +4 +2 This indicates the presence of 4 D7 branes, 4 D3 branes and 2 fractional D3 branes at $z=m^3$. Notice that in this case the fractional D3-branes are superimposed on D7-branes and D3-branes. This allows, in contrast with the solution discused in section \[sec:fracD3\], for the presence of a fractional brane at a finite distance since $\Im \beta$ can diverge without violating the consistency condition $\calv >0$.
The monodromies around $z=\pm g m^3$ can be studied in a similar way. Since at infinity there is no net monodromy, the total monodromy around these two points should be such that it compensates for those coming from the D-branes at $z=m^3$, i.e. \[Omon\] -4 -4 -2 We notice that in the perturbative limit $g\ll 1$ the two degeneration points $z=\pm g m^3$ become very close and separated by a distance $\Delta z\sim m^3 e^{-\frac\pi2\Im\tau_0}\sim m^3 e^{-\frac\pi2\Im\sigma_0}$, which is exponentially suppressed in this weak coupling limit. Hence, by analogy with the F-theory case discussed in [@sen1; @sen2], it is natural to regard the solution around the degeneration points $z=\pm g m^3$ as a system of mutually non-local branes which provide the non-perturbative resolution of a system of O-planes, with total charges given by the monodromies (\[Omon\]). This is the analogue of the resolution of O7 planes into a pair of $(p,q)$ 7-branes [@sen1], which has been explicitly shown to derive from ED(-1) non-perturbative corrections in [@Billo:2011uc]. In the present case we expect that not only ED(-1), but also fractional ED(-1) and ED3 instantons conspire to resolve the composite O-planes at $z=0$.
In Appendix \[AppPerios\] we present an alternative derivation of the monodromies from a direct evaluation of ${\bf \Omega}(z)$ from the period integrals.
**Acknowledgments**
We would like to thank M. Bianchi, M. Billó, G. Bonelli, S. Cremonesi, F. Fucito, H. Samtleben and A. Uranga for useful discussions. L.M. would like to thank the Physics Department of Università di Parma for kind hospitality during the course of this work. This work is partially supported by the ERC Advanced Grant n.226455 “Superfields", by the Italian MIUR-PRIN contract 20075ATT78, by the NATO grant PST.CLG.978785.
Ten-dimensional supersymmetric solutions {#app:10d}
========================================
In this Appendix we re-derive the supersymmetry conditions presented in [@GKP; @GP], which describe general flux F-theory vacua, hence characterized by seven-branes, regular or fractional D3-branes as well as compatible bulk fluxes. These conditions apply to the local ten-dimensional description of our vacua configurations, which admit a global description only at the level of the effective six-dimensional theory after non-trivial U-duality holonomies are allowed. Our aim is to make the paper self-consistent, to clarify the relation between the six- and ten-dimensional description, and facilitate the potential application of our results and their comparison with others. Here we use the generalized geometry framework, in which the supersymmetry conditions can be expressed in a compact geometrical form [@gmpt] and acquire a clear interpretation from the viewpoint of D-brane physics [@luca1; @luca2].
Take a general type II background preserving four-dimensional Poincaré invariance. The ten-dimensional space-time splits as $\mathbb{R}^{1,3}\times Y$ and the string-frame metric can be written as s\^2\_[st]{}=e\^[2D]{}x\^x\_+s\^2\_6 Four-dimensional $\caln=1$ supersymmetry requires the existence of type II Killing Weyl spinor $\epsilon=\epsilon_1+\ii\epsilon_2$, which in our case takes the form $\epsilon=\zeta\otimes \eta$, where $\zeta$ is a constant chiral spinor on $\mathbb{R}^{1,3}$ and $\eta$ is a chiral spinor on $Y$. We can use the internal spinor $\eta$ to construct the following forms on $Y$: \_[st]{}=\^T\_[mnp]{} y\^[m]{}y\^ny\^p ,J\_[st]{}=\^\_[mn]{} y\^[m]{}y\^n where $|a|^2\equiv \eta^\dagger\eta$ and $y^m$ are some coordinates on $Y$. The normalization is taken such that J\_[st]{}J\_[st]{}J\_[st]{}=- \_[st]{} |\_[st]{} =[dvol]{}\_6 \[norm\] In turn, the information in $J_{\rm st}$ is equivalently encoded in the polyform (alias pure spinor, in generalized geometry language) e\^[J\_[st]{} ]{} In [@gmpt] it is was proved that the ordinary Killing spinor conditions are equivalent to imposing the following equations on $Y$:[^12] \[susycond\] $$\begin{aligned}
\d_H(e^{3D-\phi} \Omega_{\rm st} )&=0 \label{susycond1}\\
\d_H(e^{2D-\phi}\Im \Psi)&=0\label{susycond2}\\
\d_H(e^{4D-\phi}\Re\Psi)&=e^{4D}*_6\lambda(F)\label{susycond3}\end{aligned}$$ where $H$ and $F=F_1+F_3+F_5$ are the field strengths along $Y$ and && \_H=+H (F)=F\_1-F\_3+F\_5 Away from localized sources, the fields $H$ and $F$ satisfy the Bianchi identities $\d H=0$ and $\d_H F=0$ which can be locally solved by setting $H=\d B$ and $F=\d_H C$ with $C=C_0+C_2+C_4$.
By defining e\^[3D-]{}\_[st]{} , Je\^[2D-]{} J\_[st]{} the first two supersymmetry equations (\[susycond1\],\[susycond2\]) can be written as 0== J = H= HJ \[domega\] which are equivalent to requiring that the $(\Omega,J)$ define integrable complex and Kähler structures respectively. Furthermore, the condition $J\wedge H=0$ imply that $H$ is primitive. A solution to (\[domega\]) is given by taking =h(z)z , J=j- e\^[-]{} |h(z)|\^2 z|z where $\omega$ and $j$ are the anti-self dual [^13] closed two-forms on $X$ for $Y=\mathbb{C}\times X$. In addition $B$ is taken self-dual to ensure $H\wedge J=0$. The forms $J$ and $\Omega$ define the Kahler and complex structures of the six-dimensional metric s\^2\_Y=e\^[2D-]{} s\^2\_6= (s\^2\_X+e\^[-]{} |h(z)|\^2 z |z) \[dsy\] Finally let us consider the remaining equation (\[susycond3\]) and specialize to the backgrounds we are interested in. Then $Y=\mathbb{C}\times X$, where $X$ is a K3-space. We use complex coordinates $u^\alpha=(u^1,u^2,z)$ on $Y$ and write the differentials as \[ddc\] &=&+| =u\^\_\
\^c &=& -(-|) |=|u\^\_ The six-dimensional Hodge dual can be computed with the help of the formulas \_6 f&=& J\_[st]{} J\_[st]{} \^c f \_6 (12 J\_[st]{} J\_[st]{} f) = \^c f \_6 \_a f = - \_a \^c f valid for any function $f$ and any self-dual two-form $\chi_a$ on $X$. Two form will be always expanded in the basis of self-dual two forms $\chi_a= [\calc_a]$ associated to a set of exceptional cycles $\calc_a$ at a singularity of K3.
Using this equations (\[susycond3\]) can be written as F\_1 &=& 12 e\^[-4D]{} \*\_6 ( e\^ JJ) = -\^c e\^[-]{}\
F\_3 &=& e\^[-]{} \*\_6 B =-e\^[-]{} \^c B\
F\_5 &=& -e\^[-4D]{} \*\_6 (e\^[4D-]{} ) =12 JJ\^c e\^[-4D]{} \[eqsa0\] Writing $F=\d (e^{B}\wedge C)$ and collecting $\partial$, $\bar\partial $-components the three equations can be written in the compact form ( | )=0 \[dtau\] with[^14] =e\^B (C+ e\^[-]{} [Re ]{}) \[bardt\] In components \_0 &=& C\_0+ e\^[-]{}\
[T]{}\_2 &=& C\_2 + B \^a\_a\
[T]{}\_4 &=& C\_4- e\^[-4A]{}JJ +C\_2B + 12 BB \[t024fin\] after setting $e^{2D}=e^{2A+\frac{\phi}{2}}$. One can then easily see that the first two equations in (\[dtau\]) can be solved by taking $\tau$ and $\beta^a$ some holomorphic functions on $\mathbb{C}$: $\tau=\tau(z)$ and $\beta^a=\beta^a(z)$. The holomorphicity of $\beta^a$ ensures the famous ISD condition on the three form field strength $*_YG_3=\ii G_3$, with $G_3= F_3+\ii e^{-\phi} H$, where $*_Y$ is the Hodge operator associated with the metric (\[dsy\]).
The last equation in (\[dtau\]) requires more care. Indeed, (\[dtau\]) necessarily implies that $ \del\delbar\Im\calt_4= \del\delbar\Re\calt_4=0$, which provides the following equations for the warping and the four-form field \[warpingf4\] && |C\_4= - |( C\_2 B+12 C\_0 BB )\
&& (e\^[-4A]{}) JJ= ( e\^[-]{} B B )
We write the ten-dimensional Einstein metric as s\^2\_[E]{} = e\^[-]{} s\^2\_[st]{}=e\^[2A]{} x\^x\^+e\^[-2A]{} s\^2\_Y which is somewhat more natural in this context and is then used in the main text.
Finally we remark that the structure of the solutions here is preserved under any SU(2)$_R$ transformation which rotates the three two-forms $(\Re \omega,\Im \omega,j)$. This implies that the four-dimensional supersymmetry is enhanced to $\caln=2$.
BPS solutions of the six-dimensional supergravity {#app:trunc}
=================================================
In this paper we construct supersymmetric vacua of ${\cal N}=(2,0)$ six-dimensional supergravity in which a subset of fields vary over a complex plane with non-trivial holonomies under a subgroup of the U-duality group. In this Appendix we discuss the details of the moduli space and the truncations we use.
Moduli space of type IIB supergravity on K3 {#app:trunc1}
-------------------------------------------
Let us first summarize some properties of type IIB compactifications on K3 surfaces, referring to [@aspin] for more details. The complete set of 105 moduli describing a compactification of type IIB supergravity on a K3 surface spans the orbifolded coset moduli space \_=[O]{}(\_[5,21]{})\\[O]{}(5,21;)/ ([O]{}(5;) (21;) \[modulispace\] A point in this space can be thought of as a choice of a time-like five-plane $\Pi\subset \mathbb{R}_{5,21}$ with ${\rm O}(5;\mathbb{R})$ and ${\rm O}(21;\mathbb{R})$ acting as rotations along and perpendicular to $\Pi$ respectively. On the other hand ${\rm O}(\Gamma_{5,21})\simeq{\rm O}(5,21;\mathbb{Z})$ is the U-duality group which acts by rotations in $\mathbb{R}_{5,21}$ preserving an even self-dual lattice $\Gamma_{5,21}$ with a non-degenerate pairing $\cali$ of signature $(5,21)$. One can choose a basis of elements \[latbasis\] \_={\^+\_i,\^-\_[r]{},\_A} i,r=1,2 A=1,…,22 in which the pairing $\cali$ takes the form \[genmetric\] [I]{}\_=(
[ccc]{} 0 & [$\mathbbm{1}$]{}\_2 & 0\
[$\mathbbm{1}$]{}\_2 & 0 & 0\
0 & 0 & \_[AB]{}
) The elements $\chi_A$ span the even self-dual lattice $\Gamma_{3,19}\simeq H^2(X;\mathbb{Z})$ of integer closed two-forms, with natural pairing given by $\cali_{AB}=\int_{X}\chi_A\wedge \chi_B$[^15]. In turn, one may choose a basis in which $\cali_{AB}=H^3\oplus \hat \cali$, where $H\equiv {\tiny \left(\begin{array}{cc} 0 & 1\\ 1 & 0\end{array}\right)}$ and $\hat\cali$ is the (positive definite) Cartan matrix of $E_8\times E_8$.
The moduli space (\[modulispace\]) can be parametrized in terms of a vielbein $U_{{\underline}\Sigma}=U_{{\underline}\Sigma}{}^{\Lambda}\chi_\Lambda$ for $\mathbb{R}_{5,21}$, where ‘flat’ indices are denoted by ${\underline}\Sigma,{\underline}\Lambda,\ldots$ and ‘curved’ indices by $\Sigma,\Lambda,\ldots$. The $26\times 26$ matrix $U_{{\underline}\Sigma}{}^{\Lambda}$ satisfies U U\^T=where $\eta= (-{\ensuremath{\mathbbm{1}}}_5,{\ensuremath{\mathbbm{1}}}_{21})$. In particular, the first five rows of the matrix $U$ span the five-plane $\Pi\subset \mathbb{R}_{5,21}$ which contains the physical information. The vielbein $U$ is defined up to ‘gauge’ rotations $U\to O\, U$ with $O\in {\rm O}(5;\mathbb{R})\times {\rm O}(21;\mathbb{R})$. In addition, the vielbeins $U$ and $ U\, {\cal O} $ have to be identified for $ {\cal O}$ an element of the U-duality group ${\cal O}\in {\rm O}(5,21;\mathbb{Z})$, i.e. $\calo \, \cali \, \calo^T=\cali$.
One can parametrize the gauge invariant degrees of freedom by using the matrix MU\^T U By construction $M$ is gauge invariant, symmetric $M^T=M$ and satisfies $M\cali M\cali={\ensuremath{\mathbbm{1}}}$. Furthermore, it transforms as $M\rightarrow \calo^T M\calo$ under $\calo\in {\rm O}(5,21;\mathbb{Z})$. The matrix $M$ allows to define the sigma model which characterizes the six-dimensional effective action \[sigmamodel\] -\^6x [tr]{} ( \^N M \_N M)
In order to describe an explicit parametrization of the vielbein $U$, it is useful to introduce another $26\times 26$ matrix $V$ related to $U$ by U=A V with \[diagmatrix\] A=(
[ccc]{} -[$\mathbbm{1}$]{}\_5 & [$\mathbbm{1}$]{}\_5 & 0\
[$\mathbbm{1}$]{}\_5 & [$\mathbbm{1}$]{}\_5 & 0\
0 & 0 &
) and ${\cal E}$ a vielbein for the $E_8\times E_8$ pairing $\hat\cali={\cal E}^T {\cal E}$. Then an explicit gauge-fixed parametrization of matrix $V$ is given by \[vv\] V &=&(
[ccc]{} E & -EC & -EY\^T\
0 & (E\^[-1]{})\^T & 0\
0 & Y & [$\mathbbm{1}$]{}
) with C= B+12 Y\^T Y B=-B\^T G=(E\^T E)\^[-1]{} Here $G,B,C,E$ are $5\times 5$ matrices, Y is a $16\times 5$ block, while $\hat\cali$ is the $16\times 16$ matrix providing the $E_8\times E_8$ pairing. A similar gauge-fixed form of the vielbein of the moduli coset space appears for example in [@schw92], in the context of toroidal compactifications of the heterotic string on $T^5$ with $G$, $B$ and $Y$ corresponding to the metric, $B$-field and Wilson lines respectively.
IIB identification of the moduli {#app:10Dmoduli}
--------------------------------
In order to clarify the ten-dimensional interpretation of the description of the moduli space given above, it is convenient to denote the first five (time-like) elements of the vielbein $U_{{\underline}\Sigma}$ by $(U_{{\underline}i},U_{{\underline}\alpha})$, ${\underline}i=1,2$, ${\underline}\alpha=1,2,3$. Then, by using the lattice basis (\[latbasis\]) one can choose $U_{{\underline}\alpha}$ to be of the form $U_{{\underline}\alpha}=U_{{\underline}\alpha}{}^i\zeta^+_i+U_{{\underline}\alpha}{}^A\chi_A$. In this gauge, the metric moduli, up to the overall volume of K3, are encoded in the three elements J\_U\_\^A\_AH\^2(X;) which generate a three-dimensional time-like plane $\Sigma\subset H^2(X;\mathbb{R})$. More explicitly, the elements $J_{{\underline}\alpha}$ can be identified, up to the overall rescaling, with the triplet of real anti-self-dual harmonic two-forms J\_= (,,j) which define the hyperkähler structure of the K3 surface, cf. footnote \[foot:hyper\]. Hence the moduli space of metrics (up to the overall volume) in K3 is spanned by the $57$-dimensional Grassmanian submanifold of $\calm_{\text{K3}}$ given by \_=[O]{}(\_[3,19]{})\\[O]{}(3,19;)/ ([O]{}(3;) (19;) where ${\rm O}(\Gamma_{3,19})$ is the geometrical duality subgroup, which acts on $H^2(X;\mathbb{R})$ while preserving the metric $\cali_{AB}$ of the lattice $H^2(X;\mathbb{Z})$.
On the other hand, we can write $U_{{\underline}i}=U_{{\underline}i}{}^j(\zeta^+_j+\calb_j{}^{A}\chi_A)+U_{{\underline}i}{}^r\zeta^-_r$. Then the $2\times 22$ matrix $\calb_j{}^{A} $ parametrize the $44$ components of the NSNS and RR two-forms $B$ and $C_2$ respectively, while the matrices $U_{{\underline}i}{}^j$ and $U_{{\underline}i}{}^r$ encode the information on $\phi$, $C_0$, $C_4$ and the warp factor. Below we will provide an explicit parametrization of these fields for the cases of interest for this paper.
Truncation of the moduli space
------------------------------
Let us first consider the simplest truncation in which we set $B=C_2=0$. This corresponds to setting $U_{{\underline}i}{}^A=0$ which reduces the matrix $U_{{\underline}\Sigma}{}^\Lambda$ into a block diagonal form with blocks of dimensions $4$ and $22$. We can then truncate the dynamical fields by keeping fixed the metric components $U_{{\underline}\alpha}=U_{{\underline}\alpha}{}^A\chi_A$, while allowing a dynamical $U_{{\underline}i}=U_{{\underline}i}{}^I\zeta_I$, where we have introduced $\zeta_I=(\zeta^+_i,\zeta^-_r)$, $I=1,\ldots, 4$. The two vectors $U_{{\underline}i}$ span a two-dimensional time-like subplane of $\mathbb{R}^{2,2}$ and the restricted moduli space is then given by \[trunc1\] =[O]{}(\_[2,2]{})\\[O]{}(2,2;)/[O]{}(2;)(2;) We notice that this space is invariant under rotations in the orthogonal subgroup ${\rm O}(3,19;\mathbb{R})$ and therefore defines a consistent truncation of the scalar moduli space.
We would like now to extend the truncation (\[trunc1\]) by including non trivial NSNS and RR fluxes. More precisely, we would like to allow for more general dynamical $U_{{\underline}i}$’s, while still keeping the $U_{{\underline}\alpha}$’s fixed. This can be achieved as follows. We first take a set of $n$ space-like integer closed forms $\chi_a\in H^2(X;\mathbb{Z})$, $a=1,...n$, with a non-degenerate positive definite pairing $\Delta_{ab}=\int \chi_a\wedge \chi_b$. Let us also assume that there are other $22-n$ integer forms $\tilde\chi_{\tilde a}$ which are orthogonal to the $\chi_a$’s, i.e. $\int_X\chi_a\wedge \tilde\chi_{\tilde b}=0$. Together $(\chi_a,\tilde\chi_{\tilde a})$ span $H^2(X;\mathbb{R})$ but in general generate a sublattice of $H^2(X;\mathbb{Z})$.
Then, the truncation is specified by restricting $U_{{\underline}\alpha}=U_{{\underline}\alpha}{}^{\tilde a}\tilde \chi_{\tilde a}$ and $U_{{\underline}i}=U_{{\underline}i}{}^{j}\zeta^+_j+U_{{\underline}i}{}^r\zeta^-_r+U_{{\underline}i}{}^{a}\chi_a$ and allowing only $U_{{\underline}i}$ to be dynamical, while $U_{{\underline}\alpha}$ is kept fixed. In other words we focus on the scalar fields which specify a two-dimensional plane $\Pi_2$ spanned by the two vectors $U_{{\underline}i}$ in the space $\mathbb{R}^{2,2+n}\simeq \Gamma_{2,2+n}\otimes \mathbb{R}$, where $\Gamma_{2,2+n}=\Gamma_{2,2}\oplus \Gamma_n$ is the lattice spanned by $(\zeta^+_i,\zeta^-_r,\chi_a)$. Our truncation is then given by a block-diagonal complete vielbein $U_{{\underline}\Sigma}{}^\Lambda$, with dynamical blocks of dimensions $(4+n)$ and a constant block of dimension $(22-n)$.
Clearly, we can use the description of the dynamical moduli in terms of the coset \[redmoduli\] =[O]{}(\_[2,2+n]{})\\[O]{}(2,2+n;)/[O]{}(2;)(2+n;) where ${\rm O}(\Gamma_{2,2+n})$ is the subgroup of ${\rm O}(\Gamma_{5,21})$ which acts only on the basis $(\zeta^+_i,\zeta^-_r,\chi_a)$ and leaves $\tilde \chi_{\tilde a}$ untouched.
Notice that the above ansatz requires that $\int J_{{\underline}\alpha}\wedge \chi_a=0$. According to the discussion provided in section \[app:10Dmoduli\], this means that \[ortcond\] \_[\_a]{}j=\_[\_a]{}=\_[\_a]{}=0 where $\calc_a$ are the cycles which are dual to the integer closed forms $\chi_a\in H^2(X;\mathbb{Z})$. The condition (\[ortcond\]) implies that the cycles $\calc_a$ dual to $\chi_a$ must have vanishing volume. Indeed, it is known that a K3 surface develops an orbifold singularity if an only if the three-plane $\Sigma$ is orthogonal to some points of the lattice $H^2(X;\mathbb{Z})$, see for instance [@aspin]. The cycles $\calc_a$ are just the exceptional cycles associated with the orbifold singularity.[^16] As a concrete example, one can consider $X$ to be $T^4/\mathbb{Z}_2$ blown-up at $16-n$ points. We take $\chi_a$ to be Poincaré dual to the $n$ unresolved exceptional cycles, while the remaining $22-n$ two-forms $\tilde\chi_{\tilde a}$ can be taken to be the Poincaré duals to the remaining $16-n$ (blown-up) exceptional cycles and of the $6$ toroidal cycles inherited from the underlying $T^4$.[^17]
Holomorphic parametrization {#app:gaugefix}
---------------------------
Let us now give an explict parametrization of the $(4+n)\times (4+n)$ vielbein $U$ describing the reduced moduli space coset (\[redmoduli\]). We use the same strategy used for the general case in section \[app:trunc1\]. Namely, we introduce a matrix $V$ related to $U$ by U=A V with \[diagmatrix2\] A=(
[ccc]{} -[$\mathbbm{1}$]{}\_2 & [$\mathbbm{1}$]{}\_2 & 0\
[$\mathbbm{1}$]{}\_2 & [$\mathbbm{1}$]{}\_2 & 0\
0 & 0 &
) where ${\cal E}$ is now a vielbein for the positive definite $\Delta_{ab}$: $\Delta={\cal E}^T {\cal E}$. We can then use a gauge-fixed parametrization on $V$ analogous to (\[vv\]), with $\hat\cali$ substituted by $\Delta$, with $E,B,C$ now $2\times 2$ matrices and $Y$ a $n\times 2$ block. We can then introduce the complex fields $\tau,\sigma,\beta^a$ defined by \[vhol\]
E &=(
[cc]{} & 0\
-& 1
)\
\^a&=-Y\_2\^a+Y\_1\^b\
&=B\_[12]{}+1[2]{}\
where = -12\
and we have used a notation in which, for instance, $\Im\beta\cdot\Im\beta=\Delta_{ab}\Im\beta^a\Im\beta^b$.
One can now compute the coset ‘metric’ (\[genmetric\]) and using it in the sigma model (\[sigmamodel\]) one finds -14 [tr]{} ( \^N M \_N M) =-2 K\_[I|J]{}()\_M\^I\^M|\^[|J]{} with $\varphi^I=(\tau,\sigma,\beta^a)$, $K_{I\bar J}={\partial^2 \over \partial \varphi^I \varphi^{\bar J}} K$ and $K$ the Kähler potential \[kah2\] K=-Namely, the effective action for the truncated scalar sector coupled to gravity takes the form (\[effact\]).
Supersymmetry analysis {#app:susy}
----------------------
In this section we explicitly study the supersymmetry properties of the vacua considered in this paper, from the effective six-dimensional perspective.
In general, a supersymmetric bosonic vacuum must satisfy the Killing spinor equations, obtained by imposing the vanishing of the supersymmetry variations of the fermionic fields. In the case of a compactification of IIB supergravity on K3, the effective six-dimensional theory is given by an $\caln=(2,0)$ supergravity coupled to $21$ self-dual tensor multiplets. The general structure of $\caln=(2,0)$ supergravities has been determined in [@romans] and another useful reference, whose conventions we follow, is provided by [@gutperle]. The fermionic fields are given by the gravitino $\psi_M$ and the 21 tensor multiplet fermions $\rho_{{\underline}{\hat r}}$, ${{\underline}{ \hat r}}=1,\ldots,21$, which carry the (four-dimensional) spin representation of the ${\rm SO}(5;\mathbb{R})$ R-symmetry group. Furthermore, the index ${{\underline}{\hat r}}$ of $\rho_{{\underline}{\hat r}}$ transforms in the fundamental representation of ${\rm SO}(21;\mathbb{R})$. The fermions $\psi_M$ and $\rho_{{\underline}{\hat r}}$ have opposite chirality \_7=- \_7\_[[ r]{}]{}=\_[[ r]{}]{} with $\Gamma_7=\Gamma^{012345}$, and satisfy the symplectic-Majorana conditions $\psi_M=\calc\hat\calc(\psi_M)^*$ and $\rho_{{\underline}r}=\calc\hat\calc(\rho_{{\underline}{\hat r}})^*$, where $\calc$ and $\hat\calc$ are complex conjugation matrices for the spin representations of the space-times SO$(1,5;\mathbb{R})$ and the R-symmetry SO$(5;\mathbb{R})$, respectively. More explicitly, denoting the R-symmetry SO$(5;\mathbb{R})$ gamma matrices by $\gamma^{{\underline}r}$, $\calc$ and $\hat\calc$ are defined by \_M\^[-1]{}=(\_M)\^\* \^[r]{}\^[-1]{}=(\^[r]{})\^\*
We are interested in six-dimensional backgrounds with non-trivial scalars but vanishing three-form fluxes. Hence, the relevant supersymmetry transformations reduce to \[gensusy\]
\_M&=\
\_[[r]{}]{}&=1\^M (P\_M)\_[[r s]{}]{}\^[[s]{}]{}
where $\epsilon$ has the same spinorial property as $\psi_M$: \[epsiloncond\] \_7=- , =\^\* In (\[gensusy\]) the matrices $Q_M$ and $P_M$ are given by the formula \_M U U\^[-1]{}=(
[cc]{} Q\_M & P\_M\
P\^T\_M & S\_M
) where $\{U_{{\underline}\Sigma}\}=\{U_{{\underline}r},U_{{\underline}{\hat r}}\}$ is the coset vielbein introduced in Appendix \[app:trunc1\]. Notice that the indices ${{\underline}r}$ and ${\underline}{\hat r}$, being flat, can be raised and lowered with no problems. In this sense, $Q^T_M=-Q_M$.
We can now evaluate $Q_M$ and $P_M$ for scalars belonging to the truncated moduli space (\[redmoduli\]), by using the gauge-fixed vielbein provided in Appendix \[app:gaugefix\], and plugging them in the supersymmetry transformations (\[gensusy\]).
Let us consider more in detail the R-symmetry connection $Q_M$, which appears in the gravitino supersymmetry condition. Clearly, the only non vanishing components of $Q_M$ are $Q_{M{\underline}{ij}}=\calq_{M}\epsilon_{{\underline}{ij}}$, which can be read from \_[M]{}x\^M=( \_M\^I) \[QM\] where $K$ is defined in (\[kah2\]). We notice that $\calq_{M} $ is the pull back of the of the U(1) connection on $\calm$ associated with the holomorphic line bundle whose sections transform as modular forms of weight one.
The gravitino supersymmetry transformations can then be written as \_M=
### Supersymmetric vacua
Let us now focus on our vacua, which are characterized by the six-dimensional metric s\^2=x\^x\_+M\_P\^[-4]{} |h(z)|\^2z|z and complex scalars $\varphi^I=(\tau,\sigma,\beta^a)$ depending holomorphically just on $z$: $\delbar\varphi^I(z)=0$.
In order to prove that these vacua preserve four-dimensional $\caln=2$ supersymmetry[^18], we can use the following representation of the SO$(1,5;\mathbb{R})$ gamma matrices: \^=\^[$\mathbbm{1}$]{} , \^[5]{}=\_5\_1 , \^[6]{}=\_5\_2 where $\hat\gamma^\mu$ are four-dimensional gamma-matrices, which we choose to be real, and $\hat\gamma_5\equiv -\ii\hat\gamma^{0123}$ is the associated chiral operator. In this representation $\Gamma_7=-\hat\gamma_5\otimes\sigma_3$ and we can take $\calc={\ensuremath{\mathbbm{1}}}\otimes\sigma_2$. Analogously, we can take the following explicit (four-dimensional) representation of the SO$(5;\mathbb{R})_{\rm R}$ R-symmetry (in this section we introduce the suffix $_R$ for clarity) gamma matrices $\gamma^{{\underline}r}=(\gamma^{{\underline}i},\gamma^{{\underline}\alpha})$, adapted to the decomposition ${\rm SO}(5;\mathbb{R})_{\rm R}\rightarrow {\rm SO}(2;\mathbb{R})_{\rm R}\times {\rm SO}(3;\mathbb{R})_{\rm R}$:
\^[i]{}&=(\_1[$\mathbbm{1}$]{},\_2[$\mathbbm{1}$]{})\
\^&=(\_3\_1,\_3\_2,\_3\_3)
where the $\hat\sigma_i$ are Pauli matrices acting on the spin representation of the the R-symmetry group, with associated charge conjugation matrix $\hat\calc=\hat\sigma_1\otimes\hat\sigma_2$.
Then, one can take the following spinorial ansatz for the six-dimensional spinor, which automatically satisfies (\[epsiloncond\]): \[6dsusy\] =+(\_1\^\*)(\_2\_2\^\*) Here $\zeta$ is an arbitrary four-dimensional constant chiral spinor ($\hat\gamma_5\zeta=\zeta$) which transforms as a spin-doublet under the ${\rm SO}(3)_{\rm R}\simeq{\rm SU}(2)_{\rm R}$ R-symmetry sub-group. Hence, it has eight independent components, which correspond to the eight $\caln=2$ four-dimensional supercharges. On the other hand, in (\[6dsusy\]) $\eta$ is a two-dimensional chiral spinor ($\sigma_3\eta=\eta$) which is chiral under $ {\rm SO}(2)_{\rm R}$ too: $\hat\sigma_3\eta=\eta$.
Under these conditions, the gravitino supersymmetry condition reduces to an equation on the internal two-dimensional space: \[2dsusy\] (\_m- \_[m]{} )=0 where $m$ runs over coordinates of the transversal complex plane. In (\[2dsusy\]) $\nabla_m$ must be computed by using the two-dimensional metric $\calv |h(z)|^2\d z\d \bar z$. Hence, by taking into account (\[QM\]), it is not difficult to see that in (\[2dsusy\]) the $\cal V$-dependent terms cancel between the spin-connection and U(1) connections[^19] and one is left with a simple equation satisfied by =()\^[14]{}\_0 with constant $\eta_0=\sigma_3\eta_0=\hat\sigma_3\eta_0$.
The remaining supersymmetry conditions $\delta\rho_{{\underline}i}=\delta\rho_{{\underline}a}=0$ can be analyzed along the same lines. Namely, one can compute $P_M$ from the truncated vielbein, express the result in terms of the complex fields $(\tau,\sigma,\beta^a)$, use the above explicit representations of the space-time and R-symmetry gamma matrices, and evaluate $\delta\rho_{{\underline}i},\delta\rho_{{\underline}a}$ for the spinorial ansatz described around (\[6dsusy\]). The result is that, indeed, $\delta\rho_{{\underline}i}=\delta\rho_{{\underline}a}=0$ once the complex fields $\tau,\sigma,\beta^a$ are chosen to depend holomorphically on $z$.
Genus two curves, modular forms and theta functions {#sgtwo}
===================================================
In this Appendix we summarize some useful facts about hyperelliptic curves of genus two. Although the discussion will be necessarily incomplete, we will try to be self-consistent in treating the results useful for the present paper. For more details the reader should consult, for instance, [@munford; @farkas; @bertola; @Minahan:1995er; @D'Hoker:2001qp][^20].
Genus two hyperelliptic curves {#AppCycles}
------------------------------
A genus two surface $\Sigma$ can be described by a hyperelliptic curve in ${\mathbb{C}}^2$ given by y\^2=a\_0\_[i=1]{}\^6 (x-e\_i) \[sucurve\] The curve can be interpreted in terms of a two-sheet covering of the $x$-complex plane with three cuts pairing the $e_i$’s, let us say along $[e_{2i-1},e_{2i}]$ with $i=1,2,3$. This description maps a point $p\in \Sigma$ to a point $x(p)\in \mathbb{C} $.
The first homology class of $\Sigma$ has dimension $b_1(\Sigma)=4$ and one can choose a symplectic basis of one-cycles $\{\gamma^a,\tilde\gamma_b\}_{a,b=1,2}$, which have intersection numbers $\gamma^a\cdot\tilde\gamma_b=\delta^a_b$. In the double-sheet description provided by (\[sucurve\]), one can make the following choice. The cycle $\gamma^a$ encircles clockwise the cuts $[e_{2a-1},e_{2a}]$ in one sheet, while $\tilde\gamma_a$ goes along one sheet from $[e_{2a-1},e_{2a}]$ towards $[e_{5},e_{6}]$ and comes back along the second sheet, see figure \[cycles\]. It is easy to check that indeed $\gamma^a\cdot \tilde\gamma_b=\delta^a_{b}$ for this choice.
The curve $\Sigma$ is charactered by its [*period matrix*]{} =(
[cc]{} &\
&
) defined as follows. Take a basis of holomorphic one-forms $\lambda_a$, $a=1,2$, on $\Sigma$. Then, \[MN\] [****]{}\_[ab]{}=(N M\^[-1]{})\_[ab]{} M\^a\_[b]{} =\_[\^a]{} \_b N\_[ab]{} =\_[\_a]{} \_b Alternatively, one can introduce the normalized holomorphic differentials $\omega_a=\lambda_b(M^{-1})^b{}_a$, $\oint_{\gamma^a}\omega_b=\delta^a_b$ and define ${\bf \Omega}_{ab}=\oint_{\tilde\gamma_a}\omega_b$. Notice that by construction ${\bf \Omega}_{ab}$ is symmetric and has positive definite imaginary part.
In our setting, we can choose the basis $\lambda_a=\frac{x^{a-1} {\rm d}x}{y }$ and, by using the one-cycles described above, the period integrals reduce to line integrals with the identifications \_[\^a]{} \_b= 2\_[e\_[2a-1]{}]{}\^[e\_[2a]{}]{} \_b , \_[\_a]{} \_b=2\_[p=a]{}\^2 \_[e\_[2p]{}]{}\^[e\_[2p+1]{}]{} \_b \[periods\]
The definition of $\Omega$ depends on the choice of the symplectic basis $\{\gamma^a,\tilde\gamma_b\}$ and, clearly, any re-shuffling of such one-cycles should produce an equivalent period matrix. The most general redefinition of symplectic basis corresponds to an element of ${\rm Sp}(4,\mathbb{Z})$, the [*modular group*]{}, which is parametrized as in (\[spmatrix\]). More explicitly, it is defined by four $2\times 2$ matrices $(A_a{}^b,B_{ab}, C^{ab}, D^a{}_b)$, which take values in $\mathbb{Z}$ and satisfy the constraints A\^T C=C\^T A B\^T D=D\^T B A\^T D-C\^T B=D\^T A-B\^T C=[$\mathbbm{1}$]{} The modular group acts on the period matrix ${\bf\Omega}_{ab}$ by (A[****]{}+B)(C[****]{}+D)\^[-1]{} As a simple example, consider the basis change $\gamma^a\rightarrow \gamma^a$, $\tilde\gamma_a\rightarrow \tilde\gamma_a+n_{ab}\gamma^b$, with $n_{ab}=n_{ba}\in\mathbb{Z}$. This generates the shifts ${\bf\Omega}_{ab}\rightarrow {\bf \Omega}_{ab}+n_{ab}$.
Modular forms
-------------
In our discussions, an important role is played by the [*modular forms*]{}, defined as follows. A modular form $f({\bf \Omega})$ of weight $k$ is a function transforming as f()=[det]{} (C [****]{}+D)\^[k]{} f([****]{}) where $\tilde{\bf\Omega}=(A{\bf\Omega}+B)(C{\bf\Omega}+D)^{-1}$. The ring of modular forms is generated by the Eisenstein series defined as \_k([****]{}) = \_[C,D]{} [det]{} (C [****]{}+D)\^[-k]{} where the sum is taken over the set of bottom halves $(C,D)$ of elements of the ${\rm Sp}(4,{\mathbb{Z}})$ group.
Any intrinsic property of the curve should be invariant under modular transformations. Indeed, one can characterize the genus two surface by the so called [*absolute Igusa invariants*]{} j\_1=[I\_2\^5I\_[10]{} ]{} j\_2=[I\_2\^3 I\_[4]{} I\_[10]{}]{} j\_3=[I\_4\^2 I\_[2]{}I\_[10]{}]{} given in terms of the polynomials, known as the [*homogeneous Igusa-Clebsch invariants*]{}
&I\_2 =a\_0\^2 \_[15 [perms]{}]{} e\_[12]{}\^2 e\_[34]{}\^2 e\_[56]{}\^2\
&I\_[4]{} =a\_0\^4 \_[10 [perms]{}]{} e\_[12]{}\^2 e\_[23]{}\^2 e\_[31]{}\^2 e\_[45]{}\^2 e\_[56]{}\^2 e\_[64]{}\^2\
&I\_[6]{} = a\_0\^6\_[60 [perms]{}]{} e\_[12]{}\^2 e\_[23]{}\^2 e\_[31]{}\^2 e\_[45]{}\^2 e\_[56]{}\^2 e\_[64]{}\^2 e\_[14]{}\^2 e\_[25]{}\^2 e\_[36]{}\^2\
&I\_[10]{} =a\_0\^[10]{} \_[1i<j 6]{} e\_[ij]{}\^2\
&I\_[15]{} = \_[15 [perms]{}]{} [det]{} (
[ccc]{} 1 & e\_1+e\_2 & e\_1 e\_2\
1 & e\_3+e\_4 & e\_3 e\_4\
1 & e\_5+e\_6 & e\_5 e\_6\
)
\[iis\] generating the ring of projective invariants. $I_{10}$ is the discriminant of the curve and for $I_{10}\neq 0$ the Riemann surface is smooth[^21] . The sums in (\[iis\]) run over the $15$ partitions into three groups of two elements, $10$ partitions into two groups of three elements and $60=10\times 6 $ matching between two groups of three elements (10 choices for the partition into two groups and six matching between the two chosen groups).\
The $I_k$ polynomials defined above are invariant under a generic SL(2,${\mathbb R}$) transformation in the $x$ plane x= y= \[sl2r\] that maps the roots $e_i$ and $a_0$ according to e\_i = a\_0= a\_0\_[i=1]{}\^6(c e\_i+d) In terms of the invariants (\[iis\]) one can write the basic Siegel modular forms \_4=14 I\_[4]{} \_6=18(I\_[2]{} I\_[4]{}-3 I\_[6]{} ) \_[10]{}=- I\_[10]{} \_[12]{}= I\_2 I\_[10]{} \_[35]{}=5\^3 I\_[10]{}\^2 I\_[15]{}\
\[siegelf\] with $\psi_{4}, \psi_{6}$ Eisenstein series of weight $4,6$ and $\chi_{10}, \chi_{12},\chi_{35}$ cusp forms of weight $10$, $12$ and $35$ respectively. A cusp form $\chi_{k}$ is a modular form of modular weight $k$ which satisfies the condition [@geer] \_[ , =0 ]{} \_[k]{}([****]{})=0 The modular forms defined in (\[siegelf\]) generate the graded ring of classical Siegel modular forms of genus two.
The Abel map and Jacobi variety
-------------------------------
Introduce the following vectors \[C2basis\] v\^1(1,0)\^2 ,v\^2(0,1)\^2 and consider the map from $\Sigma$ to $\mathbb{C}^2$ defined by \[abel\] (P)=(\^[x\_P]{}\_[e\_1]{}\_a) v\^a where we denote by $x_P \in {\mathbb{C}}$ the projection of the point $P\in \Sigma$ on the $x$-plane, the double-sheet description provided by (\[sucurve\]). If we shift $P$ along a general one-cycle $m_a\gamma^a+n^a\tilde\gamma_a$, then $\phi(p)\rightarrow \phi(p)+(m_a+{\bf \Omega}_{ab}n^b)v^a$. We then see that (\[abel\]) defines a well defined map $\phi:\Sigma\rightarrow \mathbb{C}^2/\Lambda$, the [*Abel map*]{}, where $\Lambda$ is the $\mathbb{Z}^2\subset \mathbb{C}^2$ lattice generated by the vectors $v^a$ and $\tilde v_a\equiv \Omega_{ab}v^b$: ={m\_av\^a+n\^bv\_b| m\_a,n\^b} , v\_a\_[ab]{}v\^b The Abel map takes values into $\mathbb{C}^2/\Lambda$, which is the so-called [*Jacobian variety*]{}.
The elements of the lattice $\Lambda$ are called ‘periods’. We introduce the following notation for the [*half-periods*]{}: \[halfper\] (
[c]{} n\
m
)12(m\_av\^a+n\^bv\_b) As elements of the Jacobian variety $\mathbb{C}^2/\Lambda$, the half-periods are $2\times 2$ matrices with entries 0 or 1. Then, in this notation, by using (\[periods\]) one can express the value of the Abel map at the branch points $P_i\in \Sigma$ as follows \[branchabel\]
&(P\_1)=(
[cc]{} 0 & 0\
0 & 0
)(P\_2)=(
[cc]{} 0 & 0\
1 & 0
) (P\_3)= (
[cc]{} 1 & 1\
1 & 0
)\
&(P\_4)= (
[cc]{} 1 & 1\
1 & 1
)(P\_5)=(
[cc]{} 1 & 0\
1 & 1
) (P\_6)= (
[cc]{} 1 & 0\
0 & 0
)
Theta functions
---------------
The theta functions associated with a genus two Riemann surface with period matrix ${\bf \Omega}$ are defined as (Z|[****]{})=\_[n\^2]{} e\^[ ]{} Here $Z\equiv Z_a v^a\in \mathbb{C}^2$ \[recall (\[C2basis\]) for the definition of $v^a$\], $n\equiv (n^1,n^2)\in {\mathbb{Z}}^2$, $a\equiv (a^1,a^2)\in \mathbb{Z}^2$ and $b\equiv (b_1,b_2)\in \mathbb{Z}^2$. The matrix $[^a_b]$ is called [*half-characteristics*]{}. Furthermore, we have used a notation in which, for instance, $n{\bf \Omega}n^T=n^a{\bf \Omega}_{ab}n^b$ and $nZ^T=n^aZ_a$. The half-characteristics $[^a_b]$ is called even/odd if $ab^T$ is even/odd respectively.
The theta functions obey the following important properties:
$$\begin{aligned}
\theta[^{a}_{b} ](-Z|{\bf \Omega}) &= (-)^{ab^T} \,\theta[^{a}_{b} ](Z|{\bf \Omega}) \label{eotheta}\\
\theta[^{a+2n}_{b+2m} ](Z|{\bf \Omega}) &= (-)^{am^T} \,\theta[^{a}_{b} ](Z|{\bf \Omega}) \label{shift1}\\
\theta[^{a}_{b} ]\left(Z+(^n_m)|{\bf \Omega}\right) &= e^{\pi\ii(am^T-nb^T-2nZ^T-n{\bf \Omega}n^T)} \,\theta[^{a+n}_{b+m} ](Z|{\bf \Omega}) \label{shift2}
\end{aligned}$$
where $(^n_m)$ denotes an half-period as defined in (\[halfper\]). Eq. (\[eotheta\]) is telling us that $\theta[^{a}_{b} ](Z|{\bf \Omega})$ is even/odd in $Z$ if the half-characteristics $[^a_b]$ is even/odd respectively. We will often use the short-hand notation ([****]{})(0|[****]{}) Clearly, by (\[eotheta\]), $\theta[^{a}_{b} ]\equiv 0$ if $[^a_b]$ is odd.
Eq. (\[shift1\]) implies that, up to a sign, we can reduce the half-characteristic matrix to take values $0$ or $1$. We denote by $\nu_i$, $i=1,\ldots, 6$, the odd half-characteristics \_1 = \_2 = \_3 = \_4 = \_5 = \_6 = Even half-characteristics can be obtained as sums mod 2 of three odd half-characteristics and therefore they can be labeled by triplets $\{i,j,k\}$. Hence, we can introduce the following shorthand notation for the theta functions with even half-characteristics evaluated at $Z=0$: \_[ijk]{} where the sum is understood mod 2. More explicitly
& \_[123]{}= \_[124]{}= \_[125]{}= \_[126]{}= \_[134]{}=\
&\_[135]{}= \_[136]{}= \_[145]{} = \_[146]{} = \_[156]{} =
Notice that a triplet of integers and its complementary lead to the same $ \theta_{ijk}$, [*e.g.*]{} $\theta_{123}=\theta_{456}$.
Theta functions evaluated at $Z=0$ transform nicely under modular tranformations: \[thetamodular\] ()=e\^ [det]{} (C[****]{}+D)\^[12]{} ([****]{}) where $\tilde{\bf\Omega}=(A{\bf\Omega}+B)(C{\bf\Omega}+D)^{-1}$ and (
[c]{} \^T\
\^T\
) = (
[cc]{} D & -C\
-B & A\
) (
[c]{} a\^T\
b\^T\
)+ 12 [diag]{} (
[c]{} CD\^T\
AB\^T\
) In (\[thetamodular\]), the phase factor obeys $e^{8\ii\varphi}=1$ and depends on $[^a_b]$ and on the modular transformation.
In terms of theta functions one can write [@streng] \_4 &=& 14 \_[T ]{} \^8\
\_6 &=& 14 \_[C\_3]{} (C\_3) \_[C\_3 ]{} \^4\
\_[10]{} &=& - \_[T]{} \^2\
\_[12]{} &=& \_[C\_4]{} \_[C\_4]{} \^4\
\_[35]{} &=& \_[T]{} \_[C\_3’]{} (C\_3’) \_[C\_3’ ]{} \^[20]{} \[hs\] with $T$ the set of even characteristics, $C_4$ is the set of quartets of even characteristics defined as C\_4={ (\_1,\_2,\_3,\_4) \_[i=1]{}\^4 \_i=(\^[00]{}\_[00]{}) } There are 15 of such quartets. Finally $C_3$ ($C_3'$) is the set of triplets contained in any element of $C_4$, whose sum is even (odd). There are 60 choices for both cases. The signs $\epsilon(C_3)$ and $\epsilon(C_3')$ in the formulae (\[hs\]) for $\psi_6$ and $\chi_{35}$ are fixed by modular invariance.
It can be useful to introduce the alternative parameters \[qqy\] q\_1=e\^[2]{} q\_2=e\^[2]{} y=e\^[2]{} and consider the expansions of theta functions evaluated at $Z=0$ for small values of $q_1,q_2$ \[thetaexp\] ([****]{})= q\_1\^[a\_1\^28 ]{} q\_2\^[ a\_2\^28]{} y\^[a\_1 a\_24 ]{} e\^[a\_i b\_i2]{} +q\_1\^[(1-[a\_12]{})\^2]{} q\_2\^[(1-[a\_12]{})\^2]{} y\^[(1-[a\_12]{}) (1-[a\_22]{}) ]{} e\^[(1-[a\_i 2]{}) b\_i ]{}+… In particular for the cusp forms $\chi_{10}$ and $\chi_{12}$ one finds the expansions \_[10]{} &=& [(1-y)\^24y]{} q\_1 q\_2+…\
\_[12]{} &=& [(1+10 y+y\^2)12y]{} q\_1 q\_2+…\
### The curve in terms of theta functions
One can use theta functions to provide a useful parametrization of the hyperelliptic curve in which the dependence on the period matrix ${\bf \Omega}$ is manifested.
First of all, the definition (\[sucurve\]) of the curve depends on the six branch points $e_i$ while ${\bf \Omega}$ depends on just the three independent parameters $\tau,\sigma,\beta$. One can remove the redundancy of the description (\[sucurve\]) by performing the following transformation (P) = ()( ) with $x(P)$ the projection of the point $P\in \Sigma$ on the $x$-plane. Three of the new branch points, denoted by $\xi_i=\hat{x}(P_i)$, have now fixed values $\xi_1=0$, $\xi_3=1$ and $\xi_5=\infty$, while the other three $\xi_2,\xi_4,\xi_6$ provide non-degenerate information on the curve.
With a proper choice of $a_0$ in (\[sucurve\]) the curve can be written as \[altcurve\] y\^2= (-1)(-\_2)(-\_4)(-\_6) By using (\[abel\]), one can construct the functions on the curve $\Sigma$ \[auxfunct\] f\_1(P)=( )\^2 f\_2(P)=()\^2 which have both a double zero and a double pole at the branch points $P_1$ and $P_5$ respectively, since $\phi(P_i)\sim \sqrt{x-e_i}$. On the other hand, the map $P\in\Sigma\mapsto \hat{x}(P)\in \mathbb{P}^1$ provided by the double-sheeted description (\[altcurve\]) have exactly the same zero/pole structure of $f_1(P)$ and $f_2(P)$. This can be seen by using $y\sim \sqrt{x-e_i}$ as a local coordinate around the branch points. This implies that the three functions must coincide up to a multiplicative constant. The multiplicative constant can be fixed by requiring that $f_i(P_3)=1$: \[xP\] (P)=\^2= \^2 where the normalization is fixed by using $\phi(P_3)$ given in (\[branchabel\]) and the property (\[shift2\]) of theta functions.
By repeatedly using (\[branchabel\]) and (\[xP\]), one can compute the remaining $\xi_i$ as functions of the period matrix
\_2 ([****]{})&=[ e\_[21]{} e\_[35]{}e\_[25]{} e\_[31]{} ]{} =[ \^2 \^2 \^2 \^2]{} =[ \_[126]{}\^2 \_[124]{}\^2 \_[134]{}\^2 \_[136]{}\^2 ]{}\
\_4([****]{}) &= [ e\_[41]{} e\_[35]{}e\_[45]{} e\_[31]{} ]{} =[ \^2 \^2\^2 \^2]{} =[ \_[124]{}\^2 \_[146]{}\^2\_[136]{}\^2 \_[123]{}\^2 ]{}\
\_6 ([****]{})&= [ e\_[61]{} e\_[35]{}e\_[65]{} e\_[31]{} ]{} =[ \^2\^2\^2 \^2 ]{}=[ \_[146]{}\^2 \_[126]{}\^2 \_[123]{}\^2 \_[134]{}\^2 ]{} \[xi234\]
Alternatively, by rescaling $\hat{x}$ and $y$ in (\[altcurve\]), we can rewrite it as \[modcurve\] y\^2=x(x-\^2\_[136]{}\^2\_[123]{}\^2\_[134]{})(x-\^2\_[123]{}\^2\_[126]{}\^2\_[124]{})(x-\^2\_[134]{}\^2\_[124]{}\^2\_[146]{})(x-\^2\_[136]{}\^2\_[146]{}\^2\_[126]{}) This form makes explicit the modular properties of the coefficient of quintic. Indeed shifting $x$ in order to eliminate the $x^4$ term, (\[modcurve\]) can be rewriten as \[modcurve2\] y\^2=x\^5+f\_6([****]{})x\^3+f\_9[([****]{})]{}x\^2+f\_[12]{}([****]{})x+f\_[15]{}([****]{}) where $f_{k}({\bf\Omega})$ are some modular forms of weight $k$.
Degenerations of the Riemann surface
------------------------------------
There are two types of degenerations of a genus two curve depending one squeezes a cycle homologous to zero or not. Squeezing a cycle non homologous to zero the Riemann surface degenerates to a genus one surface with a double point. In this limit $\tau \to \ii \infty$ or $\sigma\to \ii \infty$. We refer to this degeneration class simply as “pinching a handle". Squeezing a cycle homologous to zero the Riemann surface degenerates into two genus one surfaces linked by a long tube and $\beta \to 0$. We refer to this degeneration as “splitting into two genus one surfaces". A complete analytic classification of singular fibers of genus two Riemann surfaces and the definition of their homological monodromies was made by Namikawa and Ueno [@Ueno].
The theta functions of a degenerated surface can be always written in terms of elliptic functions. For the case in which a handle is pinched by sending $\sigma\rightarrow\ii\infty$ one finds
([****]{})&() \
([****]{})&e\^[ i 4]{}(e\^+e\^[a\_1b\_1]{}e\^[-]{}) ( |)\[deg0\]
where ( z |)= \_[n]{} e\^[ (n+[a2]{} )\^2 +2(n+[a2]{} )(z+[b2]{}) ]{} \[genus1theta\] are the genus one theta functions. Finally, we recall the following alternative standard notation for theta functions \_1(z|) (z|), \_2(z|) (z|), \_3(z|) (z|), \_4(z|) (z|) \[thetastand\] In the limit $\tau\rightarrow\ii\infty$ one gets a completely analogous formula:
([****]{})&() \
([****]{})&e\^[ i 4]{}(e\^+e\^[a\_2b\_2]{}e\^[-]{}) ( |)\[deg1\]
For the case where the Riemann surface splits into two genus one surfaces one finds \_[0]{} = +[2]{} \_z (z| ) \_z (z| )|\_[z=0]{}+… \[beta0\] In the following we describe the details of the two basic degenerations of the genus two curve. We refer the reader to the Appendix of [@Bonelli:2010gk] for a clear exposition of these two kinds of degenerations.
### Pinching a handle {#spinching}
Plugging (\[deg0\]) into (\[xi234\]) one finds that the harmonic ratios entering in the hyperelliptic curve at $\sigma\to \ii \infty$ (or $e_{34}\to 0$) reduce to \_2 =-[\_2\^2()\_4\^2()]{}[\_1\^2(|)\_3\^2(|)]{} \_4=1 \_6=-[\_4\^2()\_2\^2()]{}[\_1\^2(|)\_3\^2(|)]{} \[oneh\] where we used the standard notation (\[thetastand\]) for the genus one theta functions. The curve reduces to the form y\^2=x(x-\_2)(x- \_6) y=[y(x-1)]{} which corresponds to a genus one curve with a double point at $x=1$ and harmonic ratio =[\_2\^4()\_4\^4()]{} Other handle degenerations are related to this by the action of the modular group.
### Pinching two handles {#twopinch}
The limit where both handles are pinched can be found from (\[oneh\]) sending $\tau\to \ii \infty$. This leads to $\xi_2\to 0$ and degenerates the curve to an irreducible rational curve with two ordinary double points (in $x=0,1$) y\^2=(x- \_6) y=[yx(x-1)]{} with \_6= - \^2 [2]{}
### Spitting into two genus one surfaces {#splitting}
In the limit $\beta\to 0$, using (\[xi234\]) and (\[beta0\]) one finds the harmonic ratios \[degratios\] \_2= \_4()= [\_3\^4\_4\^4]{}() \_6= with a\_2 &=& - a\_6 =- obtained by using the relation $\partial_{z}\theta_1(z|\tau)|_{z=0}=\pi \,\theta_2 \theta_3\theta_4 (\tau)$. Away from $x= 0$ the Riemann surface is then described by the elliptic curve y\_1\^2 &=& x (x-1)(x- [\_3\^4\_4\^4]{}() ) y=x y\_1 \[y1\] On the other hand, near $x= 0$, one can write x=[\^2x ]{} y= to bring the curve into the elliptic form y\_2\^2 = x ( x-a\_2)( x -a\_6) with harmonic ratio =[\_2\^4()\_4\^4()]{} Summarizing, at $\beta\to 0$, the Riemann surface splits into two genus one curves (near and far away from $x= 0$) with complex structure parameters given in terms of $\sigma$ and $\tau$ respectively. We notice that the limit $\beta\to 0$ corresponds to sending $e_1,e_2,e_6$ together as follows from $\xi_2=\xi_6=0$.
An example of hyperelliptic fibration {#AppPerios}
-------------------------------------
In this section we determine the degenerations and holonomies of the fiber period matrix for a simple choice of hyperelliptic curve. We take y\^2=(x\^3-z)\^2-g\^2 (x\^3-m\^3)\^2=\_[i=1]{}\^6 (x-e\_i(z) ) \[su30\] with $g,m$ some constants. The six branch points are given by e\_[2k]{}(z)=w\^[k-1]{} (z+ g m\^31+ g )\^[13]{} e\_[2k-1]{}(z)=w\^[k-1]{} ( z- g m\^31- g )\^[13]{} with $ w=e^{2\pi i \over 3}$, $k=1,..3$. The curve (\[su30\]) defines a genus two surface at each point $z$. The fiber degenerates at the points z=m\^3 && e\_[2k]{}=e\_[2k-1]{}=w\^[k-1]{} m\
z=g m\^3 && e\_[2k-1]{}=0\
z=-g m\^3 && e\_[2k]{}=0 where some of the branch points collide. The monodromies around these points can be derived from the period matrix in the nearby of the degeneration point. Let us consider for example the curve near $z=m^3$. The period matrix ${\bf \Omega}(z)$ of the Riemann surface is given by (\[MN\]).
At $z= m^3$ the $\gamma$-cycles shrink to zero and the corresponding integrals boil down to a residue M\^a\_[b]{} =[12i ]{} \_[\^b]{} [ x\^[a-1]{} dx (x\^3-m\^3) ]{}= -[w\^[a (b-1)]{} m\^[a-3]{}3 ]{} On the other hand for the $\tilde{\gamma}$-integrals one finds N\^a\_[1]{} &=& [1i ]{} (\_[e\_2]{}\^[e\_3]{} +\_[e\_4]{}\^[e\_5]{}) [ x\^[a-1]{} dx (x\^3-z) ]{} (1-[zm\^3]{} )\
N\^a\_[2]{} &=& [1i ]{} \_[e\_4]{}\^[e\_5]{} [ x\^[a-1]{} dx (x\^3-z) ]{} (1-[z m\^3]{} ) resulting into (z) (1-[zm\^3]{} ) (
[cc]{} 4&2\
2 & 4\
) near $z= m^3$. Going around $z=m^3$ one finds the monodromies +4 +4 +2 A similar analysis can be done for the holonomies at the degeneration points $z=\pm g m^3$. In particular for $g\to 0$ when the two points come together one finds the monodromies associated with O-planes at $z=0$ with charges compensating for D-branes at $z=m^3$.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: See for instance [@Kumar:1996zx; @Liu:1997mb; @Hellerman:2002ax; @Flournoy:2004vn; @Gray:2005ea; @Vegh:2008jn; @McOrist:2010jw] for previous work discussing similar extensions of F-theory.
[^5]: \[foot:Pduality\]We use the following definition of Poincaré duality: a two-form $[\calc_a]$ is Poincaré dual to a two-cycle $\calc_a$ if $\int_{\calc_a}\alpha=-\int_X\alpha\wedge [\calc_a]$, for any two-form $\alpha$. We have $\int_X [\calc_a]\wedge [\calc_b]=-\calc_a\cdot \calc_b$, with $\calc_a\cdot \calc_b$ being the ordinary intersection number of cycles.
[^6]: \[footind\] Curvature corrections for D7-brane wrapping a K3 surface induces a -1 unit of D3-brane charge [@Bershadsky:1995sp]. What we refer here as a D7-brane is better thought as a D3D7 bound state wrapping K3 with zero net D3-brane charge [@hori97].
[^7]: We recall that the shift monodromy in $\tau$ is associated to a D3-D7 bound state wrapping K3 with zero net D3 brane charge, see footnote \[footind\]. The fact that the bound state is dual to a D3-brane on K3 is supported by the analysis in [@hori97], where the two moduli spaces were matched.
[^8]: \[Bshift\]Alternatively, instead of $W_a$, one can use as generators the conjugates $\tilde W_a=S^3\, W_{a} \,S$, acting as shifts of the $B$-field: $\tau\rightarrow \tau$, $\sigma\rightarrow\sigma+\Delta_{ab}\beta^b+\frac12\tau\Delta_{aa}$, $\beta^b\rightarrow\beta^b+\tau\delta_a^b$.
[^9]: Explictly $
{\tiny S=\left(\begin{array}{cccc}
0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)}\, , \,
{\tiny T=\left(\begin{array}{cccc}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)}\, , \, {\tiny R=\left(\begin{array}{cccc}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{array}\right)}\, , \,{\tiny W=\left(\begin{array}{cccc}
1 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)}$
[^10]: Notice that, in our hyperelliptic formulation, the zeroes of $\beta$ correspond to the points where the fiber degenerates into to two genus one surfaces.
[^11]: Our field $\beta$ is related to the fields $\gamma$ and $t$ in [@cremonesi; @lerdaetnoi] via the identifications $\gamma={{\textstyle{\frac{1}{2}}}} t = \beta_1=-\Delta_{11}\beta^1= -2\beta$.
[^12]: In an oriented vielbein $e^a$, the Hodge star is defined by $ *_d \omega_p = \frac{1}{p!(d-p)!}\epsilon_{a_1\ldots a_d}\,\omega^{a_{d+1-p}\ldots a_d} \,
e^{a_1}\wedge\ldots\wedge e^{a_{d-p}}$, where $\epsilon_{12\ldots d}=1$.
[^13]: \[foot:hyper\]In a local oriented vielbein $e^a$, we can write $\omega=(e^1+\ii e^3)\wedge (e^2+\ii e^4)$ and $j=e^{1}\wedge e^3+e^2\wedge e^4 $. In these conventions ${\rm vol}_4=-\frac12 j\wedge j=-\frac14\omega\wedge \bar\omega$, $*_{X}j=-j$ and $*_{X}\omega=-\omega$.
[^14]: The polyform defined in (\[bardt\]) is a specialized version of the polyform $\calt$ used in [@eff1; @eff2] to discuss general warped flux compactifications.
[^15]: We use an orientation convention in which $\cali_{AB}$ has signature $(3,19)$, differently from the one used for instance in [@aspin].
[^16]: Notice however that $\Pi$ itself is generically not orthogonal to any element of the lattice and then the theory is not singular [@aspin].
[^17]: See for instance [@Kumar:2009zc; @Schulz:2012wu] for readable discussions on the structure of $H^2(X;\mathbb{Z})$ and some of its sublattices in the case of Kummer K3 spaces.
[^18]: See for instance [@bergsh; @howe] for analogous discussions on supersymmetric codimension-two configurations.
[^19]: Explicitly $\calq_z= -\frac{\ii}{2} \partial_z K $, $\calq_{\bar z}=\calq_{z}^*$ and $\gamma^{{\underline}{12}} \eta=\ii \eta$. The non trivial components of the spin connection are $w_{z 1 2}={{\textstyle{\frac{\ii}{2}}}}\partial _{z} \log \left( \calv |h|^2 \right)$ and $w_{\bar{z} 12}=-{{\textstyle{\frac{\ii}{2}}}}\partial _{\bar{z}} \log \left( \calv |h|^2 \right)$.
[^20]: We thank M. Billó for notes and explanations contributing to the presentation here.
[^21]: The discriminant of a hyperelliptic curve $y^2=a_0 \prod_{i=1}^{2n} (x-e_i)$ of genus $g=n-1$ is defined as $\Delta=a_0^{4n-2} \prod_{i<j} e_{ij}^2$ and it is invariant under the $SL(2,{\mathbb R})$ transformations (\[sl2r\]) .
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A method is developed to derive simple relations among the reduced matrix elements of the quadrupole operator between low-lying collective states. As an example, the fourth order scalars of $Q$ are considered. The accuracy and validity of the proposed relations is checked for the ECQF Hamiltonian of the IBM–1 in the whole parameter space of the Casten triangle. Furthermore these relations are successfully tested for low-lying collective states in nuclei for which all relevant data is available.'
address:
- '$^1\,$ Institut für Kernphysik, Universität zu Köln, 50937 Köln, Germany'
- '$^2\,$ Bogoliubov Laboratory, Joint Institute for Nuclear Research, 141980 Dubna, Russia'
author:
- 'V. Werner$\,^1$, P. von Brentano$\,^1$ and R.V. Jolos$\,^{1,2}$'
title: 'Simple relations among $E2$ matrix elements of low-lying collective states'
---
Microscopic shell model wave functions of collective nuclear states need a huge configurational space. However, experimental data indicates that there are comparably simple relations between the wave functions of different collective states, including the ground state. The wave functions of some excited states can be described by actions of one-body operators on the ground state wave function with good accuracy. In even-even nuclei, where the ground state is a $0^+$ state, the first $2^+$ state is given by the quadrupole operator $Q$ acting on the ground state. The generalization of this concept has been named the $Q$-phonon approach [@Sie94; @Ots94; @Shi01; @Pie94; @Pie95; @Pie98; @Pal98]. In this approach one describes the low-lying collective positive parity states of even-even nuclei in the basis of multiple $Q$-phonon excitations of the ground state, $|0_1^+\rangle$, $$\label{eq:qexpand}
|L^+,n\rangle = N^{(L,n)} (\underbrace{Q...Q}_{n})^{(L)}
|0_1^+\rangle \ .$$ In the framework of the IBM-1 [@AriIac75; @IacAri87] it has been shown over the whole parameter space of the ECQF Hamiltonian [@WarCas82; @Lip85] that each of the wave vectors of the yrast states can be described by only one multiple $Q$-phonon configuration with good accuracy [@Sie94; @Pie94; @Pie95], which has recently been confirmed by microscopic calculations in [@Shi01].
The $Q$-phonon approximation implies the existence of selection rules for the matrix elements of the quadrupole operator. Thus, one finds that $E2$ transitions between $Q$-phonon configurations, that differ by more than one $Q$-phonon, are weak compared to transitions between those configurations that differ by only one $Q$-phonon. During the last years much data on $\gamma$-soft nuclei has been collected, especially in the A=130 mass region, which support these selection rules, e.g. [@Gad00b; @Wer01].
The $Q$-phonon structure of the low-lying collective states allows one to obtain quadrupole shape invariants [@Kum72; @Cli86; @Jol97; @Wer00] from rather few data. As an example we consider fourth order scalars obtained by coupling the four quadrupole operators in different ways. One obtains several different expressions for the fourth order quadrupole shape invariants in terms of only a few $E2$ matrix elements. These expressions can be used to derive approximate values of various observables, e.g., for the quadrupole moment of the first $2^+$ state or the lifetime of the first excited $0^+$ state, from more easily accessible nuclear data. Such information is desirable for nuclei where complete experimental information about low-lying states is not – or not yet – available, as e.g. nuclei which are produced using rare isotope beams.
There are three possibilities to couple the quadrupole operators to obtain fourth order scalars, $$\begin{aligned}
\label{eq:q40}
q_4^{(0)} & = & \langle 0_1^+| (Q\cdot Q) (Q\cdot Q) |0_1^+\rangle \ ,\\
\label{eq:q42}
q_4^{(2)} & = & \langle 0_1^+| \left[
[QQ]^{(2)}[QQ]^{(2)}\right]^{(0)} |0_1^+\rangle \ ,\\
\label{eq:q44}
q_4^{(4)} & = & \langle 0_1^+| \left[
[QQ]^{(4)}[QQ]^{(4)}\right]^{(0)} |0_1^+\rangle \ ,\end{aligned}$$ The notation $[\ldots]^{(L)}$ abbreviates the tensor coupling of two operators to angular momentum $L$. These three scalars are proportional to each other according to Dobaczewski, Rohoziński and Srebrny in [@Dob87], if the $Q$-operators commute. Then one obtains the relations $$\label{eq:cinf}
q_4^{(0)} = \frac{7\sqrt{5}}{2} q_4^{(2)} = \frac{35}{6} q_4^{(4)} \ .$$ In the IBM-1 the $Q$-operators do not commute. The effect of the noncommutativity of the components of the quadrupole operators scales however with $1/N$ and is therefore neglected in first order. Below, we will check the accuracy of Eq. (\[eq:cinf\]) in the framework of IBM-1.
In order to do this we decompose the scalars into sums over reduced matrix elements, $$\begin{aligned}
q_4^{(0)} & = & \ \sum_{i,j,k} \langle 0_1^+||Q||2_i^+\rangle \langle
2_i^+||Q||0_j^+\rangle \nonumber \\
\label{eq:q40sum}
& & \times \langle 0_j^+||Q||2_k^+\rangle \langle
2_k^+||Q||0_1^+\rangle \ , \\
q_4^{(2)} & = & \frac{1}{5\sqrt{5}} \sum_{i,j,k} \langle
0_1^+||Q||2_i^+\rangle \langle
2_i^+||Q||2_j^+\rangle \nonumber \\
\label{eq:q42sum}
& & \times \langle 2_j^+||Q||2_k^+\rangle \langle
2_k^+||Q||0_1^+\rangle \ , \\
q_4^{(4)} & = & \frac{1}{15} \sum_{i,j,k} \langle
0_1^+||Q||2_i^+\rangle \langle 2_i^+||Q||4_j^+\rangle \nonumber \\
\label{eq:q44sum}
& & \times \langle 4_j^+||Q||2_k^+\rangle \langle
2_k^+||Q||0_1^+\rangle \ .\end{aligned}$$ Using Eqs. (\[eq:q40sum\])-(\[eq:q44sum\]), the quantities (\[eq:q40\])-(\[eq:q44\]) have been calculated gridwise – using the code [Phint]{} [@phint] – for $N$=10 bosons over the whole IBM-1 symmetry space spanned by the ECQF-Hamiltonian [@WarCas82; @Lip85] $$\label{eq:hecqf}
H_{ECQF} = a \ \left[(1-\zeta) \ n_d - \frac{\zeta}{4N} \ Q\cdot
Q\right] \ .$$ The ECQF Hamiltonian interpolates between the symmetry limits of the IBM-1 using two structural parameters, $\zeta$ and $\chi$. Here, $n_d$ is the $d$-boson number operator and $N$ is the total boson number. The parameter $a$ has no structural meaning as it sets an absolute energy scale, and $Q$ is the CQF quadrupole operator, both in the Hamiltonian and the $E2$ transition operator, $$\label{eq:qcqf}
1/e_B \ T(E2)=Q=s^+\tilde{d}+d^+ s +\chi [d^+\tilde{d}]^{(2)} \ ,$$ depending on the structural parameter $\chi$, with $-\sqrt{7}/2 \le
\chi \le 0$; $e_B$ is the effective boson charge. The result of this calculation is a near proportionality of the $q_4^{(0)}$, $q_4^{(2)}$ and $q_4^{(4)}$, in accordance with Eq. (\[eq:cinf\]).
In view of the selection rules of the $Q$-phonon scheme, the sums (\[eq:q40sum\])-(\[eq:q44sum\]) reduce drastically. In the first approximation the set of $E2$ matrix elements necessary for the calculation of $q_4^{(n)}$ ($n=0,2,4$) reduces to the following matrix elements, $$\begin{aligned}
\label{eq:214i}
\langle 2_1^+||Q||4_i^+\rangle & \longrightarrow & i=1 \ , \\
\label{eq:212i}
\langle 2_1^+||Q||2_i^+\rangle & \longrightarrow & i=1,2 \ , \\
\label{eq:210i}
\langle 2_1^+||Q||0_i^+\rangle & \longrightarrow & i=1,2,3 \ .\end{aligned}$$ The first three $0^+$ states are taken into account, because the $0_{2,3}^+$-eigenstates of the ECQF Hamiltonian are mixtures of two- and three-$Q$-phonon $0^+$ configurations. Of course, if there are low-lying non-collective $0^+$ states, the ECQF $0^+_{2,3}$ eigenstates may refer to higher lying physical states. We have introduced the short notation $0^+_{QQ}$ by means of $$\langle 0^+_{QQ}||Q||J\rangle^2 = \langle 0^+_{2}||Q||J\rangle^2 +
\langle 0^+_{3}||Q||J\rangle^2 \ . \\$$
By using only the matrix elements (\[eq:214i\])-(\[eq:210i\]) in (\[eq:q40sum\])-(\[eq:q44sum\]) we see that in each sum a factor $\langle 0_1^+||Q||2_1^+\rangle^2$ appears, which may be dropped, since we are interested in the ratios. Eqs. (\[eq:q40sum\])-(\[eq:q44sum\]) now become: $$\begin{aligned}
\label{eq:t40}
t_4^{(0)} & = & \ \langle 2_1^+||Q||0_1^+\rangle^2 + \langle
2_1^+||Q||0_{QQ}^+\rangle^2 \ , \\
\label{eq:t42}
t_4^{(2)} & = & \frac{1}{5\sqrt{5}} \left(\langle
2_1^+||Q||2_1^+\rangle^2 + \langle 2_1^+||Q||2_2^+\rangle^2\right) \ ,
\\
\label{eq:t44}
t_4^{(4)} & = & \frac{1}{15} \langle 2_1^+||Q||4_1^+\rangle^2 \ ,\end{aligned}$$ where we use $t^{(n)}_4$ to distinguish these quantities from the exact $q_4^{(n)}$ values. These quantities are also approximately proportional to each other, like the quantities $q_4^{(n)}$. For arbitrary values of the boson number $N$ the $t_4^{(n)}$ values are related by factors $c_{0i}^N$ defined by $$\label{eq:cdef}
t_4^{(0)} = \frac{1}{c_{02}^N} t_4^{(2)} = \frac{1}{c_{04}^N}
t_4^{(4)} \ ,$$ which depend on the boson number and the dynamical symmetry character. To obtain the values of the $c_{0i}^N$, we consider the $U(5)$, $SU(3)$ and $O(6)$ dynamical symmetry limits of the IBM-1 at first. The ECQF quadrupole operator (\[eq:qcqf\]) is used, and one obtains $$\begin{aligned}
\label{eq:t40limit}
t_4^{(0)} & = & \left\{ \begin{array}{c@{\qquad}c}
7N-2 & U(5) \\
N(2N+3) & SU(3) \\
N(N+4) & O(6)
\end{array} \right. \ , \\
\label{eq:t42limit}
t_4^{(2)} & = & \left\{ \begin{array}{c@{\qquad}c}
\frac{1}{\sqrt{5}} (2N+\chi^2-2) &
U(5) \\
\frac{1}{28\sqrt{5}} (4N+3)^2 & SU(3)
\\
\frac{2}{7\sqrt{5}} (N-1)(N+5) & O(6)
\end{array} \right. \ , \\
\label{eq:t44limit}
t_4^{(4)} & = & \left\{ \begin{array}{c@{\qquad}c}
\frac{6}{5} (N-1) & U(5) \\
\frac{6}{35} (N-1)(2N+5) & SU(3) \\
\frac{6}{35} (N-1)(N+5) & O(6)
\end{array} \right. \ .\end{aligned}$$ Comparing Eq. (\[eq:cdef\]) and Eqs. (\[eq:t40limit\])-(\[eq:t44limit\]) one obtains proportionality factors for $N\rightarrow\infty$ $$\label{eq:climits}
c^{\infty}_{02} = \frac{2}{7\sqrt{5}} \quad ,
\quad c^{\infty}_{04} = \frac{6}{35} \ ,$$ in agreement with the factors of Eq. (\[eq:cinf\]). These values hold also for finite $N$ in the $O(6)$ and the $SU(3)$ dynamical symmetry limits when one neglects $1/N^2$ terms. Only in the $U(5)$ limit a $1/N$ dependence is left, causing a small deviation from the limiting values. The values of the parameters $c_{0i}^N$ with finite $N$ differ slightly from those with $N\rightarrow\infty$. Using Eqs. (\[eq:cdef\])-(\[eq:t44limit\]) we obtain improved relations in the dynamical symmetry limits including the values of $c_{0i}^N$ for finite $N$. For nuclei far from symmetries one can calculate the exact $c_{0i}^N$ using Eq. (\[eq:cdef\]) and interpolating in the IBM-1. We have done such calculation for the IBM-1 using the ECQF-Hamiltonian (\[eq:hecqf\]). Fig. \[fig:cij\] shows the values of the parameter $$\label{eq:cdev}
1-\frac{c_{0i}^N}{c_{0i}^{\infty}} \quad , \quad (0i)=(02),(04)$$ for $N$=10 bosons. Using the limiting values for $N\rightarrow\infty$ results in a systematical error below 10$\%$. We note that some deviations arise from our use of only the $0^+_2$ state for the $0^+_{QQ}$ configuration in this calculation.
From Eqs. (\[eq:t40\])-(\[eq:cdef\]) one obtains two relations for the quadrupole moment of the $2^+_1$ state: $$\begin{aligned}
\label{eq:q21amat}
\langle 2^+_1||Q||2^+_1\rangle^2 & + & \langle
2^+_1||Q||2^+_2\rangle^2 \nonumber \\
& = &
\left(\frac{c_{02}^N}{c_{02}^{\infty}}\frac{c_{04}^{\infty}}{c_{04}^N}\right)
\cdot \frac{5}{9} \langle 2^+_1||Q||4^+_1\rangle^2 \ ,\end{aligned}$$ $$\begin{aligned}
\label{eq:q21aeb}
Q_{2^+_1}^2 = \frac{32\pi}{35}\left[
\left(\frac{c_{02}^N}{c_{02}^{\infty}}\frac{c_{04}^{\infty}}{c_{04}^N}\right)\right.
& \cdot & B(E2;4^+_1 \rightarrow 2^+_1)
\nonumber \\
& - & \left. B(E2;2^+_2\rightarrow 2^+_1)\right]\end{aligned}$$ and $$\begin{aligned}
\label{eq:q21bmat}
& & \langle 2^+_1||Q||2^+_1\rangle^2 + \langle 2^+_1||Q||2^+_2\rangle^2
\nonumber \\
& & = \frac{10}{7}\cdot \frac{c_{02}^N}{c_{02}^{\infty}} \cdot
(\langle 2^+_1||Q||0^+_1\rangle^2 + \langle
2^+_1||Q||0^+_{QQ}\rangle^2) \ ,\end{aligned}$$ $$\begin{aligned}
\label{eq:q21beb}
Q_{2^+_1}^2 & = & \frac{32\pi}{35} \left[ \frac{2}{7}\cdot
\frac{c_{02}^N}{c_{02}^{\infty}} \cdot [5
B(E2;2^+_1\rightarrow 0^+_1)\right. \nonumber \\
& + & \left. B(E2;0^+_{QQ} \rightarrow 2^+_1)] - B(E2;2^+_2
\rightarrow 2^+_1)\right] \ ,\end{aligned}$$ where Eqs. (\[eq:q21amat\]),(\[eq:q21aeb\]) and (\[eq:q21bmat\]),(\[eq:q21beb\]), respectively, differ only in notation. In a first approximation with $c_{0i}^N/c_{0i}^{\infty}$=1, and if we define $B(E2;2^+_1\rightarrow 2^+_1)\equiv 1/5 \langle
2^+_1||Q||2^+_1\rangle^2$, we can write expression (\[eq:q21amat\]) in an intuitively interesting way: $$B(E2;2^+_1\rightarrow 2^+_1) + B(E2;2^+_2\rightarrow 2^+_1) =
B(E2;4^+_1\rightarrow 2^+_1) \ .$$ A relation similar to (\[eq:q21aeb\]) for $N\rightarrow\infty$ has been obtained in [@Jol96], but was derived in a much less transparent way and was expressed using a rather difficult notation. Rewriting Eq. (\[eq:q21beb\]) we get a relation for $B(E2;0^+_{QQ} \rightarrow
2^+_1)$, and a second relation by inserting (\[eq:q21beb\]) in (\[eq:q21aeb\]).
Extending our previous definitions [@Wer00] of quadrupole shape invariants, we define now not only $K_4$, but $K_4^{(0)}$, $K_4^{(2)}$ and $K_4^{(4)}$, depending on the coupling. We want to obtain values, which characterize the nucleus and do not depend strongly on the coupling scheme. Thus, with $q_2$=$\langle
0^+_1|Q\cdot Q|0^+_1\rangle$, we introduce $$\begin{aligned}
\label{eq:k40}
K_4^{(0)} & = & \frac{q_4^{(0)}}{q_2^2} \ , \\
\label{eq:k42}
K_4^{(2)} & = & \frac{7\sqrt{5}}{2} \ \frac{q_4^{(2)}}{q_2^2} \ , \\
\label{eq:k44}
K_4^{(4)} & = & \frac{35}{6} \ \frac{q_4^{(4)}}{q_2^2} \ .\end{aligned}$$ These quantities are all equal if the quadrupole operators commute. We note that in the large $N$ limit of the IBM-1 the theoretical values for the $K_4^{(n)}$ are 1 for the $SU(3)$ and the $O(6)$, and 1.4 for the $U(5)$ dynamical symmetry limit, distinguishing between $\beta$-rigid and vibrational nuclei, respectively. Applying the above results to the $K_4^{(n)}$ leads to an approximation formula for $K_4^{(0)}$ that has already been obtained for $N\rightarrow\infty$ in [@Jol97], $$\label{eq:k404}
K_4^{(0)}\approx \frac{7}{10}\frac{B(E2;4^+_1\rightarrow
2^+_1)}{B(E2;2^+_1\rightarrow 0^+_1)}\equiv K_4^{\rm appr.} \ .$$ A second approximation is $$\label{eq:k402}
K_4^{(0)}\approx \frac{7}{10} \left[
\frac{\frac{35}{32\pi}Q_{2_1^+}^2 +
B(E2;2^+_2\rightarrow 2^+_1)}{B(E2;2^+_1\rightarrow 0^+_1)}
\right] \ .$$ Due to $K_4^{(0)}\in[1,1.4]$ it emerges from Eq. (\[eq:k402\]) that, e.g., in the transition from $O(6)$ to $SU(3)$, where $K_4^{(0)}$=$1$, the value of $Q_{2^+_1}^2/B(E2;2^+_1\rightarrow
0^+_1)$ rises from zero to 10/7, while the value of $B(E2;2^+_2\rightarrow 2^+_1)/B(E2;2^+_1\rightarrow 0^+_1)$ drops from 10/7 to zero. Thus, these ratios characterize nicely the change of structure.
In order to compare the relations with experimental data we considered nuclei near dynamical symmetries, for which all needed data is available. This data comes mostly from Coulomb excitation experiments by D. Cline and co-workers [@Cli86; @Cli99; @Wu96; @Sve95; @Fa88]. In Tables \[tab:tkerne\],\[tab:qkerne\] the results are given. The Os and Pt nuclei are considered to be $\gamma$-soft [@Wu96; @Bak78; @Cas85] or transitional between $\gamma$-soft and axially deformed nuclei, which is indicated by the large values of the quadrupole moments of the $2^+_1$ states. As examples for vibrational nuclei Cd and Pd nuclei are shown, and Gd and Dy nuclei for the axially deformed case. We used $c_{0i}^N$ values from the appropriate dynamical symmetry.
In Table \[tab:tkerne\] the relations (\[eq:cdef\]) are tested with satisfactory overall agreement. Additionally, the values of $K_4^{\rm appr.}$ are given in Table \[tab:tkerne\].
Table \[tab:qkerne\] shows the experimental values of $Q_{2^+_1}^2$ and $B(E2;0^+_{QQ} \rightarrow 2^+_1)$ for the chosen nuclei, compared to the values obtained by the relations. The values of $Q_{2^+_1}^2$, obtained from the relations (\[eq:q21aeb\]) and (\[eq:q21beb\]), agree with the experimental values within the errors in most cases. A high accuracy of data is necessary for significant results, especially for the vibrator-like and the $\gamma$-soft nuclei, for which the quadrupole moments become very small.
As an example for discrepancies, we consider $^{188}$Os for which the $B(E2;0^+_2\rightarrow 2^+_1)$ value is very small, as expected for an $O(6)$ nucleus. The two-$Q$-phonon content of this state should therefore be very small. However, the values from Eqs. (\[eq:q21aeb\]),(\[eq:q21beb\]) in Table \[tab:qkerne\] may refer to a higher lying $0^+$ state with larger two-$Q$-phonon contribution, for which the lifetime is not known. Thus, with the missing $E2$ strength in $^{188}$Os, the value of $Q_{2^+_1}$ is underestimated by relation (\[eq:q21beb\]), while Eq. (\[eq:q21aeb\]) describes the quadrupole moment well.
One finds significant deviations for other nuclei, too. For example, in $^{194}$Pt the large $B(E2;0^+_4\rightarrow 2^+_1)$ value indicates a two-$Q$-phonon structure for the $0^+_4$ state, in contradiction with the $O(6)$ prediction. Thus, this transition has been included in the calculation of the $B(E2;0^+_{QQ}\rightarrow 2^+_1)$ value. The value of $K_4^{\rm appr.}=0.8$ in $^{192}$Os is considerably smaller than its minimally allowed value: 1. This may be due to the small experimental value of $B(E2;4^+_1\rightarrow 2^+_1)$. Also $K_4^{\rm appr.}$ for $^{108}$Pd is unexpectedly small, which does not support the vibrational character of this nucleus. In the Cd isotopes considered the measured $Q_{2^+_1}$ are smaller than expected from the relations.
To summarize, we propose a simple method to derive sets of relations between the experimentally observable reduced matrix elements of the quadrupole operator. This approach is based on the use of the quadrupole shape invariants, the selection rules of the $Q$–phonon scheme and the fact that corrections from noncommutativity of the components of the quadrupole moment operator in the IBM-1 are small. As an example of the general scheme, fourth order $Q$-invariants of the ground state are given. One can apply the scheme also to higher order invariants, e.g. $q_5$ or $q_6$, or to invariants built on excited states. The accuracy of the derived relations is checked for finite boson number $N$ over the whole parameter space of the ECQF-IBM-1 Hamiltonian and is shown to be rather good. A satisfactory agreement between data and theoretical relations has been obtained in many cases, but some exceptions clearly need further study.
For fruitful discussions we thank A. Dewald, A. Gelberg, J. Jolie, T. Otsuka and Yu.V. Palchikov. One of the authors (P.B.) wants to thank the Institute for Nuclear Theory at the University of Washington for its hospitality during the final stages of this work, and R.J. thanks the Cologne University for support. We thank K. Jessen for the careful reading of this paper. This work was supported by the Deutsche Forschungsgemeinschaft under Contract No. Br 799/10-1.
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-------------------- ----------------------- ---------------- ------------------------------ ------------------- -------------------
data taken $t_4^{(0)}$ $1/c_{02}^N \cdot t_4^{(2)}$ $1/c_{04}^N \cdot $K_4^{\rm appr.}$
t_4^{(4)}$
from $[$e$^2$b$^2]$ $[$e$^2$b$^2]$ $[$e$^2$b$^2]$
$^{186}$Os [@Wu96] $2.84(7)$ $2.79(56)$ $3.06(17)$ $1.06(3)$
\[1mm\] $^{188}$Os [@Wu96] $2.52(3)$ $2.72(48)$ $2.82(8)$ $1.08(1)$
\[1mm\] $^{190}$Os [@Wu96] $2.36(6)$ $1.97(40)$ $2.28(16)$ $0.93(3)$
\[1mm\] $^{192}$Os [@Wu96] $2.12(3)$ $2.20(30)$ $1.84(6)$ $0.82(1)$
\[1mm\] $^{194}$Pt [@Wu96] $1.56(12)$ $1.96(12)$ $1.54(5)$ $1.00(4)$
\[1mm\] $^{196}$Pt [@Bij80] $1.34(6)$ $1.53(25)$ $1.56(9)$ $1.08(7)$
$^{106}$Pd [@Sve95] $0.76(7)$ $0.86(10)$ $0.83(9)$ $1.19(9)$
$^{108}$Pd [@Sve95] $0.92(11)$ $1.10(13)$ $0.86(9)$ $1.04(9)$
$^{112}$Cd [@Ra89; @Ju80] $0.65(5)$ $0.37(6)$ $0.76(7)$ $1.41(14)$
$^{114}$Cd [@Fa88] $0.60(3)$ $0.53(8)$ $0.77(5)$ $1.39(12)$
$^{156}$Gd [@La83; @Go81] $4.67(13)$ $4.58(23)$ $4.66(13)$ $0.98(3)$
$^{158}$Gd [@La83; @Al88; @Go81] $5.03(15)$ $5.01(25)$ $5.20(14)$ $1.02(4)$
$^{160}$Gd [@Klu01; @Ro77] $5.25(5)$ $5.36(26)$ $5.20(13)$ $0.98(2)$
$^{164}$Dy [@Go81] $5.57(8)$ $5.16(100)$ $5.12(27)$ $0.91(5)$
-------------------- ----------------------- ---------------- ------------------------------ ------------------- -------------------
: Values of $t_4^{(0)}$, $1/c_02^N t_4^{(2)}$ and $1/c_04^N
t_4^{(4)}$ for various nuclei. The three values should agree according to Eq. \[eq:cdef\][]{data-label="tab:tkerne"}
------------ --------------------- -------------------- --------------------- ------------------------------------ --------------------- ---------------------
$[$e$^2$b$^2]$ $[$e$^2$b$^2]$ $[$e$^2$b$^2]$ $[$e$^2$b$^2]$ $[$e$^2$b$^2]$ $[$e$^2$b$^2]$
\[2mm\] Eq. (\[eq:q21aeb\]) exp. Eq. (\[eq:q21beb\]) Eqs. (\[eq:q21aeb\],\[eq:q21beb\]) exp. Eq. (\[eq:q21beb\])
$^{186}$Os 1.97$^{+14}_{-29}$ 1.76$^{+26}_{-44}$ 1.80$^{+10}_{-12}$ 0.25$^{+16}_{-17}$ 0.040$^{+24}_{-16}$ $<$ 0.35
$^{188}$Os 1.80$^{+7}_{-21}$ 1.72$^{+10}_{-38}$ 1.56$^{+8}_{-8}$ 0.30$^{+8}_{-8}$ 0.0061$^{+3}_{-3}$ 0.20$^{+15}_{-20}$
$^{190}$Os 1.14$^{+15}_{-30}$ 0.90$^{+19}_{-32}$ 1.20$^{+9}_{-8}$ $<$ 0.10 0.014$^{+2}_{-2}$ 0
$^{192}$Os 0.56$^{+6}_{-19}$ 0.84$^{+24}_{-8}$ 0.78$^{+7}_{-8}$ 0 0.004$^{+1}_{-1}$ 0.08$^{+32}_{-8}$
$^{194}$Pt 0 0.20$^{+2}_{-7}$ $<$ 0.01 0.08$^{+6}_{-8}$ 0.100$^{+6}_{-6}$ 0.50$^{+9}_{-17}$
$^{196}$Pt $0.26(9)$ $0.24(18)$ $0.10(8)$ $0.24(11)$ $0.02(1)$ $0.21(26)$
$^{106}$Pd $0.28^{+8}_{-22}$ $0.30^{+5}_{-6}$ $0.23^{+7}_{-7}$ $0.20^{+11}_{-11}$ $0.14^{+2}_{-2}$ $0.23^{+11}_{-11}$
$^{108}$Pd $0.20^{+9}_{-20}$ $0.38^{+4}_{-8}$ $0.24^{+10}_{-8}$ $0.11^{+11}_{-11}$ $0.16^{+2}_{-2}$ $0.35^{+11}_{-17}$
$^{112}$Cd $0.43(6)$ $0.14(3)$ $0.35(5)$ $0.27(8)$ $0.16(5)$ 0
$^{114}$Cd $0.31(4)$ $0.13(6)$ $0.18(3)$ $0.26(6)$ $0.090(5)$ $0.02(9)$
$^{156}$Gd $3.79(11)$ $3.72(15)$ $3.79(15)$ $< 0.18$ n.a. $< 0.18$
$^{158}$Gd $4.19(11)$ $4.04(16)$ $4.05(18)$ $0.17(20)$ n.a. $< 0.28$
$^{160}$Gd $4.20(10)$ $4.33(17)$ $4.23(14)$ $< 0.09$ n.a. $0.11(26)$
$^{164}$Dy $4.09(22)$ $4.12(81)$ $4.46(15)$ 0 n.a. $< 0.60$
------------ --------------------- -------------------- --------------------- ------------------------------------ --------------------- ---------------------
: Comparison of the quadrupole moments of the $2^+_1$ state and the reduced transition strengths of the $0^+_{QQ}\rightarrow 2^+_1$ transition for various nuclei.[]{data-label="tab:qkerne"}
4.5cm
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Knowledge graphs have become popular over the past years and frequently rely on the Resource Description Framework (RDF) or Property Graphs (PG) as underlying data models. However, the query languages for these two data models – SPARQL for RDF and Gremlin for property graph traversal – are lacking interoperability. We present Gremlinator, a novel SPARQL to Gremlin translator. Gremlinator translates SPARQL queries to Gremlin traversals for executing graph pattern matching queries over graph databases. This allows to access and query a wide variety of Graph Data Management Systems (DMS) using the W3C standardized SPARQL query language and avoid the learning curve of a new Graph Query Language. Gremlin is a system agnostic traversal language covering both OLTP graph database or OLAP graph processors, thus making it a desirable choice for supporting interoperability wrt. querying Graph DMSs. We present a comprehensive empirical evaluation of Gremlinator and demonstrate its validity and applicability by executing SPARQL queries on top of the leading graph stores Neo4J, Sparksee and Apache TinkerGraph and compare the performance with the RDF stores Virtuoso, 4Store and JenaTDB. Our evaluation demonstrates the substantial performance gain obtained by the Gremlin counterparts of the SPARQL queries, especially for star-shaped and complex queries.'
address:
- 'Smart Data Analytics, , '
- 'Department, , '
- ', '
- ', '
- 'TIB Technische Informationsbibliothek & L3S Research Center, , '
author:
- '[^1]'
-
-
-
-
bibliography:
- 'ref.bib'
nocite: '[@*]'
title: 'A Stitch in Time Saves Nine – SPARQL Querying of Property Graphs using Gremlin Traversals'
---
,
and
[^1]: Corresponding author. .
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We discuss the feasibility of predicting, managing and subsequently manipulating, the future evolution of a Complex Adaptive System. Our archetypal system mimics a population of adaptive, interacting objects, such as those arising in the domains of human health and biology (e.g. cells), financial markets (e.g. traders), and mechanical systems (e.g. robots). We show that short-term prediction yields corridors along which the model system will, with high probability, evolve. We show how the widths and average direction of these corridors varies in time as the system passes through regions, or [*pockets*]{}, of enhanced predictability and/or risk. We then show how small amounts of ‘population engineering’ can be undertaken in order to steer the system away from any undesired regimes which have been predicted. Despite the system’s many degrees of freedom and inherent stochasticity, this dynamical, ‘soft’ control over future risk requires only minimal knowledge about the underlying composition of the constituent multi-agent population.'
author:
- |
David M.D. Smith and Neil F. Johnson\
Physics Department, Oxford University, Oxford OX1 3PU, U.K.
title: 'Predictability, Risk and Online Management in a Complex System of Adaptive Agents'
---
Introduction {#sec:intro}
============
Complex Adaptive Systems (CAS) are of great theoretical interest because they comprise large numbers of interacting objects or ‘agents’ which, unlike particles in traditional physics, change their behaviour based on experience [@bocc]. Such adaptation yields complicated feedback processes at the microscopic level, which in turn generate complicated global dynamics at the macroscopic level. CAS also arguably represent the ‘hard’ problem in biology, engineering, computation and sociology[@bocc]. Depending on the application domain, the agents in CAS may be taken as representing species, people, cells, computer hardware or software, and are typically quite numerous, e.g. $10^2-10^3$[@bocc; @Wolp].
There is also great practical interest in the problem of predicting and subsequently controlling a Complex Adaptive System. Consider the enormous task facing a Complex Adaptive Systems ‘manager’ in charge of overseeing some complicated computational, biological, medical, sociological or even economic system. He would certainly like to be able to predict its future evolution with sufficient accuracy that he could foresee the system heading towards any ‘dangerous’ areas. However, prediction is not enough – he also needs to be able to steer the system away from this dangerous regime. Furthermore, the CAS manager needs to be able to achieve this $without$ detailed knowledge of the present state of its thousand different components, nor does he want to have to shut down the system completely. Instead he is seeking for some form of ‘soft’ control which can be applied ‘online’ while the system is still evolving.
Such online management of a Complex System therefore represents a significant theoretical and practical challenge. However, the motivation for pursuing such a goal is equally great given the wide range of important real-world systems that can be regarded as complex – from engineering systems through to human health, social systems and even financial systems.
In purely deterministic systems with only a few degrees of freedom, it is well known that highly complex dynamics such as chaos can arise [@strogatz] making any control very difficult. The ‘butterfly effect’ whereby small perturbations can have huge uncontrollable consequences, comes to mind. One would think that things would be considerably worse in a CAS, given the much larger number of interacting objects. As an additional complication, a CAS may also contain stochastic processes at the microscopic and/or macroscopic levels, thereby adding an inherently random element to the system’s dynamical evolution. The Central Limit Theorem tells us that the combined effect of a large number of stochastic processes tends fairly rapidly to a Gaussian distribution. Hence, one would think that even with reasonably complete knowledge of the present and past states of the system, the evolution would be essentially diffusive and hence difficult to control without imposing substantial global constraints.
In this paper, we address this question of dynamical control for a simplified yet highly non-trivial model of a CAS. We show that a surprising level of prediction and subsequent control can be achieved by introducing small perturbations to the agent heterogeneity, i.e. ‘population engineering’. In particular, the system’s global evolution can be managed and undesired future scenarios avoided. Despite the many degrees of freedom and inherent stochasticity both at the microscopic and macroscopic levels, this global control requires only minimal knowledge on the part of the ‘system manager’. For the somewhat simpler case of Cellular Automata, Israeli and Goldenfeld[@golden] have recently obtained the remarkable result that computationally irreducible physical processes can become computationally reducible at a coarse-grained level of description. Based on our findings, we speculate that similar ideas may hold for a far wider class of system comprising populations of decision-taking, adaptive agents.
It is widely believed (see for example, Ref. [@casti1]) that Arthur’s so-called El Farol Bar Problem [@arthur; @A258] provides a representative toy model of a CAS where objects, components or individuals compete for some limited global resource (e.g. space in an overcrowded area). To make this model more complete in terms of real-world complex systems, the effect of network interconnections has recently been incorporated [@nets1; @nets2; @nets3; @nets4]. The El Farol Bar Problem concerns the collective decision-making of a group of potential bar-goers (i.e. agents) who use limited global information to predict whether they should attend a potentially overcrowded bar on a given night each week. The Statistical Mechanics community has adopted a binary version of this problem, the so-called Minority Game (MG) (see Refs.[@A246] through [@PRE65]), as a new form of Ising model which is worthy of study in its own right because of its highly non-trivial dynamics. Here we consider a general version of such multi-agent games which (a) incorporates a finite time-horizon $H$ over which agents remember their strategies’ past successes, to reflect the fact that the more recent past should have more influence than the distant past, and (b) allows for fluctuations in agent numbers, since agents only participate if they possess a strategy with a sufficiently high success rate[@thmg]. The formalism we employ is applicable to any CAS which can be mapped onto a population of $N$ objects repeatedly taking actions in the form of some global ‘game’.
The paper has several parts. Initially (section \[sec:Dice\]), we discuss a very simple two state ‘game’ to introduce and familiarise the reader to the nature of the mathematics which is explored in further detail in the rest of the paper. We then more formally establish a common framework for describing the spectrum of future paths of the complex adaptive system (section \[sec:formal\]). This framework is general to any complex system which can be mapped onto a general B-A-R (Binary Agent Resource) model in which the system’s future evolution is governed by past history over an arbitrary but finite time window $H$ (the $Time-Horizon$). In fact, this formalism can be applied to any CAS whose internal dynamics are governed by a Markov Process[@thmg], providing the tools whereby we can monitor the future evolution both with and without the perturbations to the population’s composition. In section \[sec:BAR\], we discuss the B-A-R model in more detail, further information is provided in [@BAR]. We emphasize that such B-A-R systems are [*not*]{} limited to the well-known El Farol Bar Problem and Minority Games - instead these two examples are specific limiting cases. Initial investigations of a finite time-horizon version of the Minority Game were first presented in [@thmg]. In section \[sec:natural\], we consider the system’s evolution in the absence of any such perturbations, hence representing the system’s natural evolution. In section \[sec:evolution\], we revisit this evolution in the presence of control, where this control is limited to relatively minor perturbations at the level of the heterogeneity of the population. In section \[sec:reduction\] we revisit the toy model of section \[sec:Dice\] to provide a reduced form of formalism for generating averaged quantities of the future possibilities. In section \[sec:conclusion\] we discuss concluding remarks and possible extensions.
A Tale of Two Dice {#sec:Dice}
==================
In this section we examine a very simple toy model employed to generate a time series (analogue output) and introduce the Future-Cast formalism to describe the model’s properties. This toy model comprises two internal states, ***A*** and ***B***, and two dice also denoted ***A*** and ***B***. We make these dice generic in that we assign their faces values and these are not equal in likelihood. The rules of the model are very simple. When the system is in state ***A***, dice ***A*** is rolled and similarly for dice ***B***. The outcome, $\delta_t$, of the relevant dice is used to increment a time (price) series, whose update can be written $$\label{eq:PriceInc}
S_{t+1} = S_t + \delta_t$$
![ \[fig:DiceTransitions\] The internal state transitions on the De Bruijn graph.](debruijn.eps){width="65.00000%"}
The model employs a very simple rule to govern the transitions between its internal states. If the outcome $\delta_t$ is greater than zero (recall that we have re-assigned the values on the faces) the internal state at time $t+1$ is ***A*** and consequently dice ***A*** will be used at the next step regardless of the dice used at this time step. Conversely, if $\delta_t~<~0$, the internal state at $t+1$ will be ***B*** and dice ***B*** used for the next increment[^1]. These transitions are shown in figure \[fig:DiceTransitions\].
![ \[fig:DiceProbs\] The outcomes and associated probabilities of our two dice.](diceprobs.eps){width="75.00000%"}
Let us prescribe our dice some values and observe the output of the system, namely rolling dice ***A*** could yield values $\{-5,~+3,~ +8 \}$ with probabilities $\{ 0.3,~ 0.5,~ 0.2\}$ and ***B*** yields values $\{-10,~-8,~ +3 \}$ with probabilities $\{ 0.1,~ 0.5,~ 0.4\}$. These are shown in figure \[fig:DiceProbs\]. Let us consider that at some time $t$ the system is in state ***A*** with the value of the time series being $S(t)$ and we wish to investigate the possible output over the next $U$ time-steps ($S(t+U)$). Some examples of the system’s change in output over the next $10$ time-steps are shown in figure \[fig:output\]. The circles represent the system being in state ***A*** and the squares the system in ***B***.
![ \[fig:output\] The toy model’s output over 10 time-steps for 20 realisations where the state at time $t$ is ***A*** for all paths.. The circles represent the system being in state ***A*** and the squares the system in ***B***. ](output.eps){width="75.00000%"}
The stochasticity inherent in the system is evident in figure \[fig:output\]. Many possible paths could be realised even though they all originate from state ***A***, and only a few of them are depicted. If one wanted to know more about system’s output after these $10$ time-steps, one could look at many more runs and look at a histogram of the output for $U~=10$. This Monte-Carlo[@Monte] technique has been carried out in figure \[fig:output2\]. It denotes the possible change in value of the systems analogue output after $10$ time-steps and associated probability derived from $10^6$ runs of the system all with the same initial starting state (***A***) at time $t$. However, this method is a numerical approximation. For accurate analysis of the system over longer time scales, or for more complex systems, it might prove both inaccurate and/or computationally intensive.
![ \[fig:output2\] The change in the system’s output $S(t+U)~-~S(t)$ at $U = 10$, and associated probability as calculated for $10^6$ time-series realisations.](output2.eps){width="75.00000%"}
For a more detailed analysis of our model, we must look at the internal state dynamics and how they map to the output. Let us consider all possible eventualities over $2$ time-steps, again starting in state ***A*** at time $t$. All possible paths are denoted on figure \[fig:2step\]. The resulting possible values of change in the output at time $t+2$ time-steps and their associated probabilities are given explicitly too. These values are exact in that they are calculated from the dice themselves. The $Future-Cast$ frame work which we now introduce will allow us to perform similar analysis over much longer periods.
![ \[fig:2step\] All possible paths after $2$ time-steps and associated probability.](2step.eps){width="75.00000%"}
First consider the possible values of $S(t+2)-S(t)$, which result in the system being in state ***A***. The state transitions that could have occurred for these to arise are ***A*** $\rightarrow$ ***A*** $\rightarrow$ ***A*** or ***A*** $\rightarrow$ ***B*** $\rightarrow$ ***A***. The paths following the former can be considered a convolution (explores all possible paths and probabilities[^2]) of the distribution of possible values of $S(t+1)-S(t)$ in state ***A*** with the distribution corresponding to the ***A*** $\rightarrow$ ***A*** transition. Likewise, the latter a convolution of the distribution of $S(t+1)-S(t)$ in state ***B*** with the distribution corresponding to a ***B*** $\rightarrow$ ***A*** transition. The resultant distribution of possibilities in state ***A*** at time $t+2$ is just the superposition of these convolutions as described in figure \[fig:conv\]. We can set the initial value $S(t)$ to zero because the absolute position has no bearing on the evolution on the system. As such $S(t+U)-S(t)~\equiv S(t+U)$.
![ \[fig:conv\] All possible paths after $2$ time-steps which could result in the system being in state ***A***.](conv.eps){width="95.00000%"}
So we note that if at some time $t+U$ there exists a distribution of values in our output which are in state ***A*** then the convolution of this with the ***B*** $\rightarrow$ ***A*** transition distribution will result in a contribution at $t+U+1$ to our distribution of $S$ which will also be in ***A***. This is evident in the right hand side of \[fig:conv\]. Let us define all these distributions. We denote all possible values in our output $U$ time-steps beyond the present (time $t$) that are in state ***A*** as the function $\varsigma^U_A(S)$ and $\varsigma^U_B(S)$ is the function that describes the values of our output time series that are in state ***B*** (as shown in figure \[fig:conv\]). We can similarly describe the transitions as prescribed by our dice. The possible values allowed and corresponding likelihoods for the ***A*** $\rightarrow$ ***A*** transition are denoted $\Upsilon_{A\to %%@
A}(\delta)$ and similarly $\Upsilon_{A\to B}(\delta)$ for ***A*** $\rightarrow$ ***B***, $\Upsilon_{B\to %%@
A}(\delta)$ for ***B*** $\rightarrow$ ***A*** and $\Upsilon_{B\to B}(\delta)$ for the ***B*** $\rightarrow$ ***B*** state change. These are shown in figure \[fig:dicetrans\].
![ \[fig:dicetrans\] The state transition distributions as prescribed by our dice.](dicetrans.eps){width="75.00000%"}
We can now construct the Future-Cast. We can express the evolution of the output in their corresponding states as the superposition of the required convolutions: $$\begin{aligned}
\label{eqn:fut1}
\varsigma^{U+1}_A(S)&=& \Upsilon_{A\to A}(\delta)~ \otimes~ \varsigma^{U}_A(S) ~+~ \Upsilon_{B\to A}(\delta)\otimes~ %%@
\varsigma^{U}_B(S) \nonumber\\
\varsigma^{U+1}_B(S)&=& \Upsilon_{A\to B}(\delta)~ \otimes~ \varsigma^{U}_B(S) ~+~ \Upsilon_{B\to B}(\delta)\otimes~ %%@
\varsigma^{U}_B(S)\end{aligned}$$ where $\otimes$ is the discrete convolution operator as defined in footnote \[foot:conv\]. Recall that $S(0)$ has been set to zero such that we can consider the possible changes in output and the output itself to be identical. We can write this more concisely: $$\begin{aligned}
\label{eqn:fut2}
\underline{\varsigma^{U+1}}&=& \underline{\underline{\Upsilon}}~\underline{\varsigma^{U}}\end{aligned}$$ Where the element $\Upsilon_{1,1}$ contains the function $\Upsilon_{A\to A}(\delta)$ and the operator $\otimes$ and the element $\varsigma^{U+1}_1$ is the distribution $\varsigma^{U+1}_A(S)$. We note that this matrix of functions and operators is static, so only need computing once. As such we can rewrite equation \[eqn:fut2\] as $$\begin{aligned}
\label{eqn:fut5}
\underline{\varsigma^{U}}&=& \underline{\underline{\Upsilon}}^U~\underline{\varsigma^{0}}\end{aligned}$$ such that $\underline{\varsigma^{0}}$ contains the state-wise information of our starting point (time $t$). This is the Future-Cast process. For a system starting in state ***A*** and with the start value of the time series of zero, the elements of $\underline{\varsigma^{0}}$ are as shown in figure \[fig:start1\].
![ \[fig:start1\] The initial elements of our output distributions vector, $\underline{\varsigma^{0}}$ when starting the system in state ***A*** with initial value of zero in our time series.](start1.eps){width="75.00000%"}
Applying the Future-Cast process, we can look precisely at the systems potential output at any number of time-steps into the future. If we wish to consider the process over 10 steps again, we applying as following: $$\begin{aligned}
\label{eqn:fut3}
\underline{\varsigma^{10}}&=& \underline{\underline{\Upsilon}}^{10}~\underline{\varsigma^{0}}\end{aligned}$$ The resultant distribution of possible outputs (we denote $\Pi^U (S)$) is then just the superposition of the contributions from each state. This distribution of the possible outputs at some time into the future is what we call the $Future-Cast$. $$\begin{aligned}
\label{eqn:fut4}
\Pi^{10}&=& \varsigma^{10}_1 ~+~\varsigma^{10}_2\end{aligned}$$ This leads to distribution shown in figure \[fig:dicefutcast\].
![ \[fig:dicefutcast\] The actual probability distribution $\Pi^{10} (S))$ of the output after 10 time-steps starting the system in state ***A*** with initial value of zero in our time series.](dicefutcast.eps){width="75.00000%"}
Although the exact output as calculated using the Future-Cast process demonstrated in figure \[fig:dicefutcast\] compares well with the brute force numerical results of the Monte-Carlo technique in figure \[fig:output2\], it allows us to perform some more interesting analysis without much more work, let alone computational exhaustion. Consider that we don’t know the initial state of the system or that we want to know characteristic properties of the system. This might include wanting to know what the system does, on average over one time-step increments. We could for example look run the system for a very long time and investigate a histogram of the output movements over single time-steps as shown in figure \[fig:hist1\].
![ \[fig:hist1\] The one step increments of our time-series as run over 10000 steps.](hist1.eps){width="75.00000%"}
However, we can use the formalism to generate this result exactly. Imagine that model has been run for a long time but we don’t know which state it is in. Using the probabilities associated with the transitions between the states, we can infer the likelihood that the system’s internal state is either ***A*** or ***B***. Let the element $\Gamma^t_1$ represent the probability that the system is in state ***A*** at some time $t$ and $\Gamma^t_2$ that the system is in state ***B***. We can express these quantities at time $t+1$ by considering the probabilities of going between the states: $$\begin{aligned}
\label{eqn:trans}
\Gamma^{t+1}_1 & = & T_{A\to A} \Gamma^{t}_1~+~T_{B\to A} \Gamma^{t}_2\nonumber\\
\Gamma^{t+1}_2 & = & T_{A\to B} \Gamma^{t}_2~+~T_{B\to B} \Gamma^{t}_2\end{aligned}$$ Where $T_{A\to A}$ represents the probability that when the system is in state ***A*** it will be in state ***A*** at the next time-step. We can express this more concisely: $$\begin{aligned}
\label{eqn:trans2}
\underline{\Gamma^{t+1}} & = & \underline{\underline{T}}~\underline{\Gamma^{t}}\end{aligned}$$ Such that $T_{1,1}$ is equivalent to $T_{A\to A}$. This is a Markov Chain. From the nature of our dice, we can trivially calculate the elements of $\underline{\underline{T}}$. The value for $T_{A\to A}$ is the sum over all elements in $\Upsilon_{A\to A}(\delta)$ or more precisely: $$\begin{aligned}
\label{eqn:trans3}
T_{A\to A} = \sum_{\delta = -\infty}^{\infty}\Upsilon_{A\to A}(\delta)\end{aligned}$$ The Markov Chain transition matrix $\underline{\underline{T}}$ for our system can thus be trivially written: $$\begin{aligned}
\label{eqn:T}
\underline{\underline{T}} = \left(\begin{array}{c c}
0.7 & 0.4 \\
0.3 & 0.6 \end{array}\right)\end{aligned}$$ The static probabilities of the system being in either of its two states are given by the eigenvector solution to equation \[eqn:eig\] with eigenvalue $1$. This is equivalent to looking at the relative occurrence of the two states if the system were to be run over infinite time. $$\begin{aligned}
\label{eqn:eig}
\underline{\Gamma} & = & \underline{\underline{T}}~\underline{\Gamma}\end{aligned}$$ For our system, the static probability associated with being in state ***A*** is $\frac{4}{7}$ and obviously $\frac{3}{7}$ for state ***B***. To look at the characteristic properties we are going to construct an initial vector similar to $\varsigma^0$ in equation \[eqn:fut1\]) for our Future-Cast formalism to act on but which is related to the static probabilities contained by the solution to equation \[eqn:eig\]. This vector is denoted $\underline{\kappa}$ and its form is described in figure \[fig:kappa\].
![ \[fig:kappa\] Explicit depiction of the elements of vector $\underline{\kappa}$ which is used to analyse characteristic behaviour of the system.](kappa.eps){width="50.00000%"}
We use employ $\underline{\kappa}$ in the Future-Cast over one time-step as in equation \[eqn:fut6\]: $$\begin{aligned}
\label{eqn:fut6}
\underline{\varsigma^{1}}&=& \underline{\underline{\Upsilon}}~\underline{\kappa}\end{aligned}$$ The resulting distribution of possible outputs from superimposing the elements of $\underline{\varsigma}^1$ is the exact representation of the one time-step increments ($S(t+1)~-S(t)$) of the system if it were allowed to be run infinitely. We call this characteristic distribution $\Pi_{char}^1$. Applying the process a number of times will yield the exact distributions $\Pi_{char}^U$ equivalent to looking at all values of $S(t+U)~-S(t)$ which is a rolling window length $U$ over an infinite time series as in figure \[fig:pis\]. This is also equivalent to running the system forward in time $U$ time-steps from unknown initial state, investigating all possible paths. The Markovian nature of the system means that this is not the same as the convolution of the one time-step characteristic Future-Cast convolved with it self $U$ times.
![ \[fig:pis\] The characteristic behaviour of the system for $U = 1, 2, 3$ time-steps into the future from unknown initial state. This is equivalent to looking at the relative frequency of occurrence of changes in output values over 1,2, and 3 time-step rolling windows](pis.eps){width="80.00000%"}
Clearly the characteristic Future-Cast over one time-step in figure \[fig:pis\] compares well with that of figure \[fig:hist1\].
The Evolution of the Complex Adaptive System {#sec:formal}
============================================
Here we provide a general formalism applicable to any Complex System which can be mapped onto a population of $N$ species or ‘agents’ who are repeatedly taking actions in some form of global ‘game’. At each time-step each agent makes a (binary) decision $a_{\mu(t)}$ in response to the global information $\mu(t)$ which may reflect the history of past global outcomes. This global information is of the form of a bitstring of length $m$. For a general game, there exists some winning outcome $w(t)$ based on the aggregate action of the agents. Each agent holds a subset of all possible strategies - by assigning this subset randomly to each agent, we can mimic the effect of large-scale heterogeneity in the population. In other words, we have a simple way of generating a potentially diverse ecology of species, some of which may be similar but others quite different. One can hence investigate a typically-diverse ecology whereby all possible species are represented, as opposed to special cases of ecologies which may themselves generate pathological behaviour due to their lack of diversity.
The aggregate action of the population at each time-step $t$ is represented by $D(t)$, which corresponds to the accumulated decisions of all the agents and hence the (analogue) output variable of the system at that time-step. The goal of the game, and hence the winning decision, could be to favour the minority group (MG), the majority group or indeed any function of the macroscopic or microscopic variables of the system. The individual agents do not themselves need to be conscious of the precise nature of the game, or even the algorithm for deciding how the winning decision is determined. Instead, they just know the global outcome, and hence whether their own strategies predicted the winning action[^3]. The agents then reward the strategies in their possession if the strategy’s predicted action would have been correct if that strategy was implemented. The global history is then updated according to the winning decision. It can be expressed in decimal form as follows: $$\label{eqn:formal1}
\mu(t) = \sum_{i=1}^{m}2^{i-1}[w(t-i)+1]\$$ The system’s dynamics are defined by the rules of the game. We will consider here the class of games whereby each agent uses his highest-scoring strategy at each timestep, and agents only participate if they possess a strategy with a sufficiently high success rate. \[N.B. Both of these assumptions can be relaxed, thereby modifying the actual game being played\]. The following two scenarios might then arise during the system’s evolution:
- An agent has two (or more) strategies which are tied in score and are above the confidence level, and the decisions from them differ.
- The number of agents choosing each of the two actions is equal, hence the winning decision is undecided.
We will consider these cases to be resolved with a fair ‘coin toss’, thereby injecting stochasticity or ‘noise’ into the system’s dynamical evolution. In the first case, each agent will toss his own coin to break the tie, while in the second the Game-master tosses a single coin. To reflect the fact that evolving systems will typically be non-stationary, and hence the more distant past will presumably be perceived as less relevant to the agents, the strategies are rewarded as to whether they would have made correct predictions over the last $H$ time-steps of the game’s running. There is no limit on the size of $H$ other than it is finite and constant. The time-horizon represents a trajectory of length $H$ on the de Bruijn graph in $\mu(t)$ (history) space[@thmg] as shown in figure \[fig:debruijn\]. The stochasticity in the game means that for a given time-horizon $H$ and a given strategy allocation in the population, the output of the system is not always unique. We will denote the set of all possible outputs from the game at some number of time-steps beyond the time-horizon $H$, as the Future-Cast.
![\[fig:debruijn\] A path of time-horizon length $H=5$ (dashed line) superimposed on the de Bruin graph for $m=3$. The 8 global outcome states represent the 8 possible bitstrings for the global information, and correspond to the global outcomes for the past $m=3$ timesteps.](histhor.eps){width="75.00000%"}
It is useful to work in a time-horizon space $\underline{\Gamma_{t}}$ of dimension $2^{m+H}$. An element $\Gamma_{t}$ corresponds to the last $m+H$ elements of the bitstring of global outcomes (or equivalently, the winning actions) produced by the game. This dimension is constant in time whereas for a non-time-horizon game it would grow linearly. For any given time-horizon state, $\Gamma_{t}$, there exists a unique score vector $\underline{G(t)}$ which is the set of scores $G_{R}(t)$ for all the strategies which an agent could possess. As such, for each particular time-horizon state, there exists a unique probability distribution of the aggregate action, $D(t)$. This distribution of possible actions when a specified state is reached will necessarily be the same each time that state is revisited. Thus, it is possible to construct a transition matrix (c.f. Markov Chain[@thmg]) $\underline{\underline{T}}$ of probabilities for the movements between these time-horizon states such that $\underline{P(\Gamma_{t})}$ can be expressed as $$\label{eqn:formal2}
\underline{P(\Gamma_{t})} ~=
~\underline{\underline{T}}
~\underline{P(\Gamma_{t-1})}$$ where $\underline{P(\Gamma_{t})}$ is a vector of dimension $2^{m+H}$ containing the probabilities of being in a given state $\Gamma$ at time $t$
The transition matrix of probabilities is constant in time and necessarily sparse. For each state, there are only two possible winning decisions. The number of non-zero elements in the matrix is thus $\le2^{(m+H+1)}$. We can use the transition matrix in an eigenvector-eigenvalue problem to obtain the stationary state solution of $\underline{P(\Gamma)}
~=\underline{\underline{T}}~\underline{P(\Gamma)} $. This also allows calculation of some time-averaged macroscopic quantities of the game [@thmg][^4].
To generate the Future-Cast, we want to calculate the quantities in output space. To do this, we require;
- The probability distribution of $D(t)$ for a given time-horizon;
- The corresponding winning decisions, $w(t)$, for given $D(t)$;
- An algorithm generating output in terms of $D(t)$.
To implement the Future-Cast, we need to map from the transitions in the state space internal to the system to the macroscopic observables in the output space (often cumulative excess demand). We know that in the transition matrix, the probabilities represent the summation over a distribution of possible aggregate actions which is binomial in the case where the agents are limited to two possible decisions. Using the output generating algorithm, we can construct an ‘adjacency’ matrix $\underline{\underline{\Upsilon}}$ analogous to the transition matrix $\underline{\underline{T}}$, with the same dimensions. The elements of $\underline{\underline{\Upsilon}}$, contain probability distribution functions of change in output corresponding to the non-zero elements of the transition matrix together with the discrete convolution operator $\otimes$ whose form depends on that of the output generating algorithm.
The adjacency matrix of functions and operators can then be applied to a vector, $\underline{\varsigma^{U=0}(S)}$, containing information about the current state of the game and of the same dimension as $\underline{\Gamma_{t}}$ . $\underline{\varsigma^{U=0}(S)}$ not only describes the time-horizon state positionally through its elements but also the current value in the output quantity $S$ within that element. At $U=0$, the state of the system is unique so there is only one non-zero element within $\underline{\varsigma^{U=0}(S)}$. This element corresponds to a probability distribution function of the current output value, its position within the vector corresponding to the current time-horizon state. The probability distribution function is necessarily of value unity at the current value or, for a Future-Cast expressed in terms of change in output from the current value, unity at the origin. The Future-Cast process for $U$ time-steps beyond the present state can then be described by $$\label{eqn:formal3}
\underline{\varsigma^U(S)}~=~ \underline{\underline{\Upsilon}}^{U} \underline{\varsigma^0(S)}$$
The actual Future-Cast, $\Pi(S,U)$, is then computed by superimposing the elements of the output/time-horizon state vector: $$\label{eqn:formal4}
\Pi^U(S) = \sum_{i=1}^{2^{(m+H)}}\varsigma_i^U(S).$$ Thus the Future-Cast, $\Pi^U(S)$, is a probability distribution of the outputs possible at $U$ time-steps in the future.
As a result of the state dependence of the Markov Chain, $\Pi$ is non-Gaussian. As with the steady-state solution of the state space transition matrix, we would like to find a ‘steady-state’ equivalent for the output space[^5] of the form $$\label{eqn:formal5}
\Pi_{char}^1(S) = \big<\Pi^1(S)\big>_\infty$$ where the one-timestep Future-Cast is time-averaged over an infinitely long period. Fortunately, we have the steady state solutions of $\underline{P(\Gamma)}
~=\underline{\underline{T}}~\underline{P(\Gamma)} $ which are the (static) probabilities of being in a given time-horizon state at any time. By representing these probabilities as the appropriate functions, we can construct an ‘initial’ vector, $\underline{\kappa}$, similar in form to $\underline{\varsigma(S,0)}$ in equation \[eqn:formal3\] but equivalent to the eigenvector solution of the Markov Chain. We can then generate the solution of equation \[eqn:formal5\] for the $characteristic$ Future-Cast, $\Pi_{char}^1$, for a given initial set of strategies. An element $\kappa_{i}$ is again a probability distribution which is simply the point (0 , $P_{i}(\Gamma)$), the static probability of being in the time-horizon state denoted by the elements position, $i$. We can then get back to the Future-Cast $$\label{eqn:formal6}
\Pi_{char}^{1}(S)~=~
\sum_{i=1}^{2^{(m+H)}}\varsigma_i^1~~~\textrm{where}~~~\underline{\varsigma^1}~=~\underline{\underline{\Upsilon}}~ %%@
\underline{\kappa}.$$ We can also generate characteristic Future-Casts for any number of time-steps, $U$, by pre-multiplying $\underline{\kappa}$ by $\underline{\underline{\Upsilon}}^U$ $$\label{eqn:formal7}
\Pi_{char}^{U}(S)~=~
\sum_{i=1}^{2^{(m+H)}}\varsigma_i^U~~~\textrm{where}~~~\underline{\varsigma^U}~=~
\underline{\underline{\Upsilon}}^U~\underline{\kappa}\$$ We note that $\Pi_{char}^{U}$ is not equivalent to the convolution of $\Pi_{char}^1$ with itself $U$ times and as such is not necessarily Gaussian. The characteristic Future-Cast over $U$ time-steps is simply the Future-Cast of length $U$ from all the $2^{m+H}$ possible initial states where each contribution is given the appropriate weighting factor. This factor corresponds to the probability of being in that initial state. The characteristic Future-Cast can also be expressed as $$\label{eqn:formal8}
\Pi_{char}^U (S)~~ =~~ \sum_{\Gamma=1}^{2^{(m+H)}}P(\Gamma)~ \Pi^U(S)\mid\Gamma$$
where $\Pi^U(S)\mid\Gamma $ is a normal Future-Cast from an initial time-horizon state $\Gamma$ and $P(\Gamma)$ is the static probability of being in that state at a given time.
The Binary Agent Resource System {#sec:BAR}
================================
The general binary framework of the B-A-R (Binary Agent Resource) system was discussed in section \[sec:formal\]. The global outcome of the ‘game’ is represented as a binary digit which favours either those choosing option $+1$ or option $-1$ (or equivalently $1$ or $0$, A or B etc.). The agents are randomly assigned $s$ strategies at the beginning of the game. Each strategy comprises an action $a_{\mu(t)}^s$ in response to each of the $2^{m}$ possible histories $\mu$, thereby generating a total of $2^{2^{m}}$ strategies in the Full Strategy Space [^6]. At each turn of the game, the agents employ their most successful strategy, being the one with the most virtual points. The agents are thus adaptive if $s>1$.
![\[fig:BARscheme\] Schematic diagram of the Binary Agent Resource (B-A-R) system.](schematic.eps){width="90.00000%"}
We have already extended the B-A-R system by introducing the time-horizon $H$, which determines the number of past time-steps over which virtual points are collected for each strategy. We further extend the system by the introduction of a confidence level. The agents decide whether to participate or not depending on the success of their strategies. As such, the number of active agents $N(t)$ is less than or equal to $N_{tot}$ at any given time-step. This results in a variable number of participants per time-step $V(t)$, and constitutes a ‘Grand Canonical’ game. The threshold, $\tau$, denotes the confidence level: each agent will only participate if he has a strategy with at least $r$ points where $$r = T(2\tau-1).$$ Agents without an active strategy become temporarily inactive.
In keeping with typical biological, ecological, social or computational systems, the Game-master takes into account a finite global resource level when deciding the winning decision at each time-step. For simplicity, we will here consider the specific case[^7] whereby the resource level $L(t) =
\phi$$V(t)$ with $0\le$$\phi$$\le$$1$. We denote the number of agents choosing action $+1$ (or equivalently A) as $N_{+1}(t)$, and those that choose action -1 (or equivalently B) as $N_{-1}(t)$. If $L(t)-N_{+1}(t)>0$ the winning action is $+1$ and vice-versa. We define the winning decision $1$ or $0$ as follows: $$w(t) = {\rm step}[L(t) - N_{+1}(t)]$$ where we define ${\rm step}[x]$ to be $${\rm step}[x] = \left\{\begin{array}{lll} 1 & \textrm{if
}x>0,\\ 0 & \textrm{if }x<0,\\ \textrm{fair coin toss} &
\textrm{if }x
= 0.\end{array}
\right.$$ When $x~=~0$, there is no definite winning option since $N_{+1}(t) = N_{-1}(t)$, hence the Game-master uses a random coin-toss to decide between the two possible outcomes. We use a binary payoff rule for rewarding strategy scores, although more complicated versions can, of course, be used. However, we note that non-binary payoffs (e.g. a proportional payoff scheme) will decrease the probability of tied strategy scores, hence making the system more deterministic. Since we are interested in seeing the extent to which stochasticity can prevent control, we are instead interested in preserving the presence of such stochasticity. The reward function $\chi$ can be written $$\chi[N_{+1}(t),L(t)]
=\left\{\begin{array}{ll} 1 & \textrm{for }
w(t)=1,\\ -1 & \textrm{for } w(t)=0,\end{array}
\right.$$ namely +1 for predicting the correct action and -1 for predicting the incorrect one. For a given strategy, $R$, the virtual points score is given by$$G_{R}(t) =
\sum_{i=t-T}^{t-1}a_{R}^{\mu(i)}\chi[N_{+1}(i),L(i)],$$ where $a_{R}^{\mu(t)}$ is the response of strategy, $R$, to the global information $\mu(t)$ summed over the rolling window of width $H$. The global output signal $D(t) = N_{+1}(t)- N_{-1}(t)$ is calculated at each iteration to generate an output time series.
Looking at the System’s Natural Evolution {#sec:natural}
=========================================
To realize all possible paths within a given game is necessarily computationally expensive. For a Future-Cast $U$ timesteps beyond the current game state, there are necessarily $2^{U}$ winning decisions to be considered. Fortunately, not all winning decisions are realized by a specific game and the numerical generation of the Future-Cast can be made reasonably efficient.
Fortunately we can approach the Future-Cast analytically [*without*]{} having to keep track of the agents’ individual microscopic properties. Instead we group the agents together via the population tensor of rank $s$ given by $\underline{\underline{\Omega}}$, which we will refer to as the Quenched Disorder Matrix (QDM) [@A269]. This matrix is assumed to be constant over the time-scales of interest, and more typically is fixed at the beginning of the game. The entry $\Omega_{R2,R2,\ldots{}}$ represents the number of agents holding the strategies ${R1,R2,\ldots{}}$ such that $$\sum_{R,R',\ldots{}}\underline{\underline{\Omega}}_{R,R',\ldots{}} = N$$ For numerical analysis, it is useful to construct a symmetric version of this population tensor, $\underline{\underline{\Psi}}$ . For the case $s=2$, we will let $\underline{\underline{\Psi}}$ = $\frac{1}{2}(\underline{\underline{\Omega}}$+ $\underline{\underline{\Omega}}^{transpose})$ [@PRL82].
The output variable $D(t)$ can be written in terms of the decided agents $D_d(t)$ who act in a pre-determined way since they have a unique predicted action from their strategies, and the undecided agents $D_{ud}(t)$ who require an additional coin-toss in order to decide which action to take. Hence $$D(t)=D_d(t)+D_{ud}(t).$$ We focus on $s=2$ strategies per agent although the approach can be generalized. The element $\Psi_{R,R'}$ represents the number of agents holding both strategy $R$ and $R'$. We can now write $D_d(t)$ as $$D_d(t) =
\sum_{R=1}^{Q}a_{R}^{\mu(t)}\mathcal{H}[G_{R}(t)-r]\sum_{R'=1}^{Q}(1+\mbox{sgn}[G_{R}(t)-G_{R'}(t)])\Psi_{R,R'}$$ where $Q$ is the size of the strategy space, $\mathcal{H}$ is the Heaviside function and $[x]$ is defined as $$\mbox{sgn}[x] =
\left\{\begin{array}{lll} 1 & \textrm{if }x>0,\\ -1 & \textrm{if }x<0,\\ \textrm{0} &
\textrm{if }x
= 0.\end{array}
\right.$$ The volume $V(t)$ of active agents can be expressed as $$V(t)~=~ \sum_{R,R'}\mathcal{H}[G_{R}(t)-r]\big\{sgn[G_{R}(t)-G_{R'}(t)]+\frac{1}{2}\delta[G_{R}(t)-G_{R'}(t)]
\big\}\Psi_{R,R'}$$ where $\delta$ is the Dirac delta. The number of undecided agents $N_{ud}(t)$ is given by $$N_{ud}(t)~=~ %%@
\sum_{R,R'}\mathcal{H}[G_{R}(t)-r]\delta(G_{R}(t)-G_{R'}(t))[1-\delta(a_{R}^{\mu(t)}-a_{R'}^{\mu(t)})]
\Psi_{R,R'}$$ We note that for $s=2$, because each undecided agent’s contribution to $D(t)$ is an integer, hence the demand of all the undecided agents $D_{ud}(t)$ can be written simply as $$D_{ud}(t)~ \epsilon~
2 ~\mbox{Bin}\bigg(N_{ud}(t),\frac{1}{2}\bigg)-N_{ud}(t)$$ where $(n,p)$ is a sample from a binomial distribution of $n$ trials with probability of success $p$.
For any given time-horizon space-state $\Gamma_{t}$, the score vector $\underline{G(t)}$ (i.e., the set of scores $G_{R}(t)$ for all the strategies in the QDM) is unique. Whenever this state is reached, the quantity $D_d(t)$ will necessarily always be the same, as will the distribution of $D_{ud}(t)$. We can now construct the transition matrix $\underline{\underline{T}}$ giving the probabilities for the movements between these time-horizon states. The element $\underline{\underline{T}}_{\Gamma_{t}\mid\Gamma_{t-1}}$ which corresponds to the transition from state $\Gamma_{t-1}$ to $\Gamma_{t}$, is given for the (generalisable) $s=2$ case by $$\begin{aligned}
\underline{\underline{T}}_{\Gamma_{t}\mid\Gamma_{t-1}} =
\sum_{x=0}^{N_{ud}}\Bigg\{{}^{N_{ud}}C_x(\frac{1}{2})^{N_{ud}}\delta\bigg[Sgn(D_d+2x-N_{ud}
+V(1-2\phi))~+{}\nonumber\\(2\mu_t\% 2-1)\bigg]~+
{}\nonumber\\{}^{N_{ud}}C_x(\frac{1}{2})^{(N_{ud}+1)}\delta\bigg[Sgn(D_d+2x-N_{ud}+V(1-2\phi))~+~0\bigg]
\Bigg\}\end{aligned}$$ where $N_{ud}$, $D_d$ implies $N_{ud}\mid \Gamma_{t-1}$ and $D_d\mid \Gamma_{t-1}$, $V$ implies $V(t-1)$, $\phi$ sets the resource level as described earlier and $\mu_t \% 2$ is the required winning decision to get from state $\Gamma_{t-1}$ to state $\Gamma_{t}$. We use the transition matrix in the eigenvector-eigenvalue problem to obtain the stationary state solution of $\underline{P(\Gamma)}
~=\underline{\underline{T}}~\underline{P(\Gamma)} $. The probabilities in the transition matrix represent the summation over a distribution which is binomial in the $s=2$ case. These distributions are all calculated from the QDM which is fixed from the outset. To transfer to output-space, we require an output generating algorithm. Here we use the equation $$S(t+1) = S(t)+D(t)$$ hence the output value $S(t)$ represents the cumulative value of $D(t)$, while the increment $S(t+1)-S(t)$ is simply $D(t)$. Again, we use the discrete convolution operator $\otimes$ defined as $$(f\otimes g)\mid_{i} ~= \sum_{j=-\infty}^{\infty}
f(i-j)\times g(j).$$ The formalism could be extended for general output algorithms using differently defined convolution operators.
An element in the adjacency matrix for the $s=2$ case can then be expressed as $$\begin{aligned}
\underline{\underline{\Upsilon}}_{\Gamma_{t}\mid\Gamma_{t-1}} =
\Bigg\{\sum_{x=0}^{N_{ud}}\Bigg((D_d+2x-N_{ud}),~~~~~~~~~~~~~~~~~~~~~~~{}\nonumber\\ %%@
{}^{N_{ud}}C_x(\frac{1}{2})^{N_{ud}}\delta\bigg[Sgn(D_d+2x-N_{ud}+V(1-2\phi))+{}
(2\mu_t \% 2-1)\bigg]+{}\nonumber\\
{}^{N_{ud}}C_x(\frac{1}{2})^{(N_{ud}+1)}\delta\bigg[Sgn(D_d+2x-N_{ud}+V(1-2\phi))+0\bigg]
\Bigg)\Bigg\}\otimes~~ \end{aligned}$$ where $N_{ud}$, $D_d$ again implies $N_{ud}\mid \Gamma_{t-1}$ and $D_d \mid\Gamma_{t-1}$, $V$ implies $V(t-1)$, and $\mu_t\% 2$ is the winning decision necessary to move between the required states. The Future-Cast and characteristic Future-Casts ($\Pi^U(S)$, $\Pi_{char}^U $) $U$ time-steps into the future can then be computed for a given initial quenched disorder matrix (QDM).
We now consider an example to illustrate the implementation. In particular, we provide the explicit solution of a Future-Cast in the regime of small $m$ and $H$, given the randomly chosen quenched disorder matrix
$$\underline{\underline{\Omega}} = \left(\begin{array}{cccccccccccccccc}
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 2 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\
0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 1\\
0 & 1 & 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1\\
1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 2 & 0\\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 1 & 0 & 0 & 1\\
0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\
2 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 3 & 1 & 1 & 0 & 1 & 0 & 1\\
0 & 2 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 3 & 2 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\
0 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 &
1\end{array} \right).$$
We consider the full strategy space and the the following game parameters:
---------------------- ----------- ------
Number of agents $N_{tot}$ 101
Memory size $m$ 2
Strategies per agent $s$ 2
Resource level $\phi$ 0.5
Time horizon $H$ 2
Threshold $\tau$ 0.51
---------------------- ----------- ------
The dimension of the transition matrix is thus $2^{H+m} = 16$.
$$\underline{\underline{T}} = \left(\begin{array}{cccccccccccccccc}
0 &0&0&0&0&0&0&0 & 0 &0&0&0&0&0&0&0\\
1 &0&0&0&0&0&0&0 & 1 &0&0&0&0&0&0&0\\
0& 0.5 &0&0&0&0&0&0&0& 0.0625 &0&0&0&0&0&0\\
0& 0.5 &0&0&0&0&0&0&0& 0.9375 &0&0&0&0&0&0\\
0&0& 1 &0&0&0&0&0&0&0& 1 &0&0&0&0&0\\
0&0& 0 &0&0&0&0&0&0&0& 0 &0&0&0&0&0\\
0&0&0& 0.1875 &0&0&0&0&0&0&0& 1 &0&0&0&0\\
0&0&0& 0.8125 &0&0&0&0&0&0&0& 0 &0&0&0&0\\
0&0&0&0& 0.1875 &0&0&0&0&0&0&0& 0.1094 &0&0&0\\
0&0&0&0& 0.8125 &0&0&0&0&0&0&0& 0.8906 &0&0&0\\
0&0&0&0&0& 0 &0&0&0&0&0&0&0& 0.0312 &0&0\\
0&0&0&0&0& 1 &0&0&0&0&0&0&0& 0.9688 &0&0\\
0&0&0&0&0&0& 0.75 &0&0&0&0&0&0&0& 0.5 &0\\
0&0&0&0&0&0& 0.25 &0&0&0&0&0&0&0& 0.5 &0\\
0&0&0&0&0&0&0& 1 &0&0&0&0&0&0&0& 1\\
0&0&0&0&0&0&0& 0 &0&0&0&0&0&0&0& 0
\end{array} \right).$$
Each non-zero element in the transition matrix corresponds to a probability function in the output space in the Future-Cast operator matrix. Consider that the initial state is $\Gamma = 10$ i.e. the last 4 bits are $\{1001\}$ (obtained from running the game prior to the Future-Casting process). The initial probability state vector is the point (0,1) in the element of the vector $\underline{\varsigma}$ corresponding to time-horizon state $10$. We can then generate the Future-Cast for given $U$ (shown in figure \[fig:res1\]).
![\[fig:res1\] The (un-normalized) evolution of a Future-Cast for the given ${\underline{\underline{\Omega}}}$, game parameters and initial state $\Gamma$. The figure shows the last 10 time-steps prior to the Future-Cast, the means of the distributions within the Future-Cast itself, and also an actual realization of the game run forward in time.](res1.ps){width="75.00000%"}
Clearly the probability function in output space becomes smoother as $U$ becomes larger and the number of successive convolutions increases, as highlighted by the probability distribution functions at $U = 15 $ and $U = 25$ (figure \[fig:res2\]).
![\[fig:res2\] The probability distribution function at $U~ =~ 15, 25 $ time-steps beyond the present state.](res3.ps "fig:"){width="45.00000%"} ![\[fig:res2\] The probability distribution function at $U~ =~ 15, 25 $ time-steps beyond the present state.](res2.ps "fig:"){width="45.00000%"}
We note the non-Gaussian form of the probability distribution for the Future-Casts, emphasising the fact that such a Future-Cast approach is essential for understanding the system’s evolution. An assumption of rapid diffusion toward a Gaussian distribution, and hence the future spread in paths increasing as the square-root of time, would clearly be unreliable.
Online Evolution Management via ‘Soft’ Control {#sec:evolution}
==============================================
For less simple parameters, the matrix dimension required for the Future-Cast process become very large very quickly. To generate a Future-Cast appropriate to larger parameters e.g. $m =3$, $H = 10$, it is still however possible to carry out the calculations numerically quite easily. As an example, we generate a random $\underline{\underline{\Omega}}$ (the form of which is given in figure \[fig:figpet\]) and initial time-horizon appropriate to these parameters. This time-horizon is obtained by allowing the system to run prior to the Future-Cast. For visual representation reasons, the Reduced Strategy Space[@A269] is employed. The other game parameters are as previously stated. The game is then instructed to run down every possible winning decision path exhaustively. The spread of output at each step along each path is then convolved with the next spread such that a Future-Cast is built up along each path. Fortunately, not all paths are realized at every time-step since the stochasticity in the winning-decision/state-space results from the condition $N_{ud}\ge D_d $. The Future-Cast as a function of $U$ and $S$, can thus be built up for a randomly chosen initial quenched disorder matrix (QDM) (\[fig:thesis3D\]).
![\[fig:thesis3D\] Evolution of $\Pi^{U}(S)$ for a typical quenched disorder matrix ${\underline{\underline{\Omega}}}$.](thesis_3d.ps){width="75.00000%"}
We now wish to consider the situation where it is required that the system should not behave in a certain manner. For example, it may be desirable that it avoid entering a certain regime characterised by a given value of $S(t)$. Specifically, we consider the case where there is a barrier in the output space that the game should avoid, as shown in figure \[fig:figbarrier1\].
![\[fig:figbarrier1\] The evolution of the Future-Casts, and the barrier to be avoided. For simplicity the barrier is chosen to correspond to a fixed $S(t)$ value of 110, although there is no reason that it couldn’t be made time-dependent. Superimposed on the (un-normalised) distributions, are the means of the Future-Casts, while their variances are shown below.](thesis_figbarrier1.ps){width="75.00000%"}
The evolution of the spread (i.e. standard deviation) of the distributions in time, confirms the non-Gaussian nature of the system’s evolution – we note that this spread can even decrease with time[^8]. In the knowledge that this barrier will be breached by this system, we therefore perturb the quenched disorder at $U = 0$. This perturbation corresponds in physical terms to an adjustment of the composition of the agent population. This could be achieved by ‘re-wiring’ or ‘reprogramming’ individual agents in a situation in which the agents were accessible objects, or introducing some form of communication channel, or even a more ‘evolutionary’ approach whereby a small subset of species are removed from the population and a new subset added in to replace them. Interestingly we note that this ‘evolutionary’ mechanism need neither be completely deterministic (i.e. knowing exactly how the form of the QDM changes) nor completely random (i.e. a random perturbation to the QDM). In this sense, it seems tantalisingly close to some modern ideas of biological evolution, whereby there is some purpose mixed with some randomness.
![\[fig:figpet\]The initial and resulting quenched disorder matrices (QDM), shown in schematic form. The x-y axes are the strategy labels for the two strategies. The absence of a symbol denotes an empty bin (i.e. no agent holding that particular pair of strategies).](thesis_figpet.ps){width="75.00000%"}
Figure \[fig:figbarrier2\] shows the impact of this relatively minor microscopic perturbation on the Future-Cast and global output of the system. In particular, the system has been steered away from the potentially harmful barrier into ‘safer’ territory.
![\[fig:figbarrier2\]The evolution as a result of the microscopic perturbation to the population’s composition (i.e. the QDM).](thesis_figbarrier2.ps){width="75.00000%"}
This set of outputs is specific to the initial state of the system. More typically, we may not know this initial state. Fortunately, we can make use of the characteristic Future-Casts to make some kind of quantitative assessment of the robustness of the quenched disorder perturbation in avoiding the barrier, since this procedure provides a picture of the range of possible future scenarios.
![\[fig:charev\] The characteristic evolution of the initial and perturbed QDMs.](thesis_figbarrier3.ps){width="75.00000%"}
This evolution of the characteristic Future-Casts, for both the initial and perturbed quenched disorder matrices, is shown in figure \[fig:charev\] A quantitative evaluation of the robustness of this barrier avoidance could then be calculated using traditional techniques of risk analysis, based on knowledge of the distribution functions and/or their low-order moments.
A Simplified Implementation of the Future-Cast Formalism {#sec:reduction}
========================================================
We introduced the Future-Cast formalism to map from the internal state space of a complex system to the observable output space. Although the formalism exactly generates the probability distributions of the subsequent output from the system, it’s implementation is far from trivial. This involves keeping track of numerous distributions and performing appropriate convolutions between them. Often, however it is only the lowest order moments which are of immediate concern to the system designer. Here, we show how this information can be generated without the computational exhaustion previously required. We demonstrate this procedure for a very simple two state system, although the formalism is general to a system of any number of states, governed by a Markov chain.
Recall the toy model comprising two dice of section \[sec:Dice\] . We previously broke down the possible outputs of each according to the state transition as shown in figure \[fig:reddicestates2\].
![ \[fig:reddicestates2\] The state transition distributions as prescribed by our dice.](dicetrans.eps){width="75.00000%"}
These distributions were used to construct the matrix $\underline{\underline\Upsilon}$ to form the Future-Cast process as denoted in equation \[eqn:fut2\]. This acted on vector $\underline\varsigma^U$ to generate $\underline\varsigma^{U+1}$. The elements of these vectors contain the partial distribution of outputs which are in the state denoted by the element number at that particular time, so for the two dice model, $\varsigma_{1}^U(S)$ contains the distribution of output values at time $t+U$ (or $U$ time-steps beyond the present) which correspond to the system being in state ***A*** and $\varsigma_{2}^U(S)$ contains those for state ***B***. To reduce the calculation process, we will consider only the moments of each of these individual elements about zero. As such we construct a vector, $^{n}\underline{x}_U$, which takes the form:
$$\begin{aligned}
^{n}\underline{x}_U &=& \left( \begin{array}{c}
\sum_{S = -\infty}^{\infty}\varsigma_{1}^U (S) S^n \\
\sum_{S = -\infty}^{\infty}\varsigma_{2}^U(S) S^n
\end{array}
\right)\end{aligned}$$
The elements are just the $n$th moments about zero of the partial distributions within the appropriate state. For $n = 0$ this vector merely represents the probabilities of being in either state at some time $t+U$. We note that for the $n = 0$ case, $^{0}\underline{x}_{U+1} =\underline{\underline{T}}~ ^{0}\underline{x}_{U} $ where $\underline{\underline{T}}$ is the Markov Chain transition matrix as in equation \[eqn:trans2\]. We also note that the transition matrix $\underline{\underline{T}} =~ %%@
^{0}\underline{\underline{X}} $ where we define the (static) matrix $^{n}\underline{\underline{X}}$ in a similar fashion using the partial distributions (described in figure \[fig:reddicestates2\]) to be
$$^{n}\underline{\underline{X}}~=~
\left( \begin{array}{c c}
\sum_{\delta = -\infty}^{\infty}\Upsilon_{A\to A}(\delta) \delta^n & \sum_{\delta = -\infty}^{\infty}\Upsilon_{B\to A}(\delta) %%@
\delta^n\\
\sum_{\delta = -\infty}^{\infty}\Upsilon_{A\to B}(\delta) \delta^n & \sum_{\delta = -\infty}^{\infty}\Upsilon_{B\to B}(\delta) %%@
\delta^n
\end{array}\right)$$
Again, this contains the moments (about zero) of the partial distributions corresponding to the transitions between states. The evolution of $^{0}\underline{x} $ , the state-wise probabilities with time is trivial as described above. For higher orders, we must consider the effects of superposition and convolution on their values. We know that the for the superposition of two partial distributions, the resulting moments (any order) about zero will be just the sum of the moments of the individual distributions, it is just a summation. The effects of convolution, however must be considered more carefully. The elements of our vector $^{1}\underline{x}_{U} $ are the first order moments of the values associated with either of the two states at time $t+U$. The first element of which corresponds to those values of output in state ***A*** at time $t+U$. Consider that element one step later, $^{1}{x}_{1,U+1} $. This can be written as the superposition of the two required convolutions.
$$\begin{aligned}
\sum_S~\varsigma_{1}^{U+1}~ S & = &
\sum_{\delta}\sum_{S}~\varsigma_{1}^{U}~\Upsilon_{A\to A}~(S~+~\delta)~+~
\sum_{\delta}~\sum_{S}~\varsigma_{2}^{U}~\Upsilon_{B\to A}~(S~+~\delta) \nonumber\\
^{1}{x}_{1,U+1}& = & ^{0}x_{1,U}~^{1}X_{1,1}~ +~ ^{0}X_{1,1}~^{1}x_{1,U}~+~\nonumber\\
{}&{}&~~~~~~~~^{0}x_{2,U}~^{1}X_{1,2}~ + ~^{0}X_{1,2}~^{1}x_{2,U}\end{aligned}$$
In simpler terms, $$^{1}\underline{x}_{U+1} = ~~^{0}\underline{\underline{X}}~^{1}\underline{x}_U~+~
^{1}\underline{\underline{X}}~^{0}\underline{x}_U$$ The $(S~+~\delta)$ term in the expression relates to the nature of series generating algorithm, $S_{t+1} = S_t + \delta_t $. If the series updating algorithm were altered, this would have to be reflected in this convolution.
The overall output of the system is the superposition of the contributions in each state. As such, the resulting first moment about zero (the mean) for the overall output at $U$ time-steps into the future is simply $^{1}\underline{x}_{U+1}\cdot\underline{1}$ where $\underline{1}$ is a vector containing all ones and $\cdot$ is the familiar dot product.
The other moments about zero can be obtained similarly. $$\begin{aligned}
^{0}\underline{x}_{U+1} &=& ~^{0}\underline{\underline{X}}~~^{0}\underline{x}_U \nonumber\\
^{1}\underline{x}_{U+1} &=&~^{0}\underline{\underline{X}}~~^{1}\underline{x}_U~+~
~^{1}\underline{\underline{X}}~^{0}\underline{x}_U\nonumber\\
^{2}\underline{x}_{U+1} &=&~^{0}\underline{\underline{X}}~~^{2}\underline{x}_U~+~
2~^{1}\underline{\underline{X}}~^{1}\underline{x}_U~+~
~^{2}\underline{\underline{X}}~^{0}\underline{x}_U\nonumber\\
^{3}\underline{x}_{U+1} &=&~^{0}\underline{\underline{X}}~~^{3}\underline{x}_U~+~
3~^{1}\underline{\underline{X}}~^{2}\underline{x}_U~+~
3~^{2}\underline{\underline{X}}~^{1}\underline{x}_U~+~
~^{3}\underline{\underline{X}}~^{0}\underline{x}_U~\nonumber\\
\vdots &{}&{}\vdots\end{aligned}$$
More generally $$^{n}\underline{x}_{U+1} ~ =~ \sum_{\gamma=0}^{n} ~^{n}C_\gamma ~^{\gamma}\underline{\underline{X}}~~~^{n-\gamma}\underline{x}_U$$ where $^{n}C_\gamma$ is the conventional $choose$ function.
To calculate time-averaged properties of the system, for example the one-time-step mean or variance, we set the initial vectors such that $$^{0}\underline{x}_{0} =~^{0}\underline{\underline{X}}~ ^{0}\underline{x}_{0}$$
and $^{\beta}\underline{x}_{0} = \underline{0}$ for $\beta>0$. The moments about zero can then be used to calculate the moments about the mean. The mean of the one time-step increments in output averaged over an infinite run will then be $^{1}\underline{x}_{1}\cdot\underline{1}$ and $\sigma^2$ will be $$\sigma^2~=~ ~^{2}\underline{x}_{1}\cdot\underline{1} - (~^{1}\underline{x}_{1}\cdot\underline{1})^2$$ These can be calculated for any size rolling window. The mean of all $U$-step increments, $S(t+U)-S(t))$ or conversely the mean of the Future-Cast $U$ steps into the future from unknown current state is simply $^{1}\underline{x}_{U}\cdot\underline{1}$and $\sigma^2_U$ will be $$\sigma^2_U~=~ ~^{2}\underline{x}_{U}\cdot\underline{1} - (~^{1}\underline{x}_{U}\cdot\underline{1})^2$$ again with initial vectors calculated from $^{0}\underline{x}_{0} =~^{0}\underline{\underline{X}}~ ^{0}\underline{x}_{0}$ and $^{\beta}\underline{x}_{0} = \underline{0}$ for $\beta>0$. Examining this explicitly for our two dice model, the initial vectors are: $$\begin{aligned}
^{0}\underline{x}_0 &=& \left( \begin{array}{c}
\frac{4}{7}\\
\frac{3}{7}
\end{array}\right)\nonumber\\
^{1}\underline{x}_0 &=& \left( \begin{array}{c}
0\\
0
\end{array}\right)\nonumber\\
^{2}\underline{x}_0 &=& \left( \begin{array}{c}
0\\
0
\end{array}\right)\nonumber\\\end{aligned}$$ and the (static) matrices are: $$\begin{aligned}
^{0}\underline{\underline{X}}&=&
\left( \begin{array}{c c}
0.7 & 0.4 \\
0.3 & 0.6
\end{array}\right) \nonumber\\
^{1}\underline{\underline{X}}&=&
\left( \begin{array}{c c}
3.1 & 1.2\\
-1.5 & -5.0
\end{array}\right) \nonumber\\
^{2}\underline{\underline{X}}&=&
\left( \begin{array}{c c}
17.3 & 3.6\\
7.5& 42.
\end{array}\right) \nonumber\\\end{aligned}$$
These are all we require to calculate the means and variances for our system’s potential output at any time in the future. To check that all is well, we employ the Future-Cast to generate the possible future distributions of output up till 10 time-steps, $\Pi^1_{char}$ to $\Pi^{10}_{char}$. The means and variances of these are compared to the reduced Future-Cast formalism and also a numerical simulation. This is a single run of the game over 100000 time-steps. The means and variances are then measured over rolling windows of between 1 and 10 time-steps in length. The comparison is shown in figure \[fig:compare\].
![ \[fig:compare\] The means and variances of the characteristic distributions $\Pi^1_{char}$ to $\Pi^{10}_{char}$ as compared to a numerical evaluation and the reduced Future-Cast.](compare.eps){width="75.00000%"}
Fortunately they all concur. The Reduced Future-Cast formalism and the moments about either the mean or zero from the distributions generated by the Future-Cast formalism are identical. Clearly numerical simulations require progressively longer run times to investigate the properties of distributions further into the future, where the total number of possible paths gets large.
Discussion {#sec:conclusion}
==========
We have presented an analytical formalism for the calculation of the probabilities of outputs from the B-A-R system at a number of time-steps beyond the present state. The construction of the (static) Future-Cast operator matrix allows the evolution of the systems output, and other macroscopic quantities of the system, to be studied without the need to follow the microscopic details of each agent or species. We have demonstrated the technique to investigate the macroscopic effects of population perturbations but it could also be used to explore the effects of exogeneous noise or even news in the context of financial markets. We have concentrated on single realisations of the quenched disorder matrix, since this is appropriate to the behaviour and design of a particular realization of a system in practice. An example could be a financial market model based on the B-A-R system whose derivatives could be analysed quantitatively using expectation values generated with the Future-Casts. We have also shown that through the normalised eigenvector solution of the Markov Chain transition matrix, we can use the Future-Cast operator matrix to generate a characteristic probability function for a given game over a given time period. The formalism is general to any time-horizon game and could, for example, be used to analyse systems (games) where a level of communication between the agents is permitted, or even linked systems (i.e. linked games or ‘markets’). In the context of linked systems, it will then be interesting to pursue the question as to when adding one ‘safe’ complex system to another ‘safe’ complex system, results in an ‘unsafe’ complex system. Or thinking more optimistically, when can we put together two or more ‘unsafe’ systems and get a ‘safe’ one?
We have also presented a simplified and altogether more usable interpretation of the Future-Cast formalism for tracking the evolution of the output variable from a complex system whose internal states can be described as a Markov process. We have illustrated the application of the results for an example case both for the evolution from a known state or when the present state is unknown, to give characteristic information about the output series generated by such a system. The formalism is generalizable to Markov Chains whose state transitions are not limited to just two possibilities and also to systems whose mapping from state transitions to output-space are governed by continuous probability distributions.
Future work will focus on the ‘reverse problem’ of the broad-brush design of multi-agent systems which behave in some particular desired way – or alternatively, ones which will avoid some particular undesirable behaviour. The effects of any perturbation to the system’s heterogeneity could then be pre-engineered in such a system. One possible future application would be to attack the global control problem of discrete actuating controllers[@AIAA]. We will also pursue our goal of tailoring multi-agent model systems to replicate the behaviour of a range of real-world systems, with a particular focus on (1) biological and human health systems such as cancer tumours and the immune system, and (2) financial markets.
[99]{}
See N. Boccara, [*Modeling Complex Systems*]{} (Springer, New York, 2004) and references within, for a thorough discussion.
D.H. Wolpert, K. Wheeler and K. Tumer, Europhys. Lett., $\mathbf{49}$(6) (2000).
S. Strogatz, [*Nonlinear Dynamics and Chaos*]{} (Addison-Wesley,Reading, 1995).
N. Israeli and N. Goldenfeld, Phys. Rev. Lett. [**92**]{}, 074105 (2004).
J.L. Casti, [*Would-be Worlds*]{} (Wiley, New York, 1997).
B. Arthur, Amer. Econ. Rev. [**84**]{}, 406 (1994); Science [**284**]{}, 107(1999).
N.F. Johnson, S. Jarvis, R. Jonson, P. Cheung,Y.R. Kwong, and P.M. Hui, Physica A [**258**]{} 230 (1998).
M. Anghel, Z. Toroczkai, K. E. Bassler, G.Kroniss, Phys. Rev. Lett. [**92**]{}, 058701 (2004).
S. Gourley, S.C. Choe, N.F. Johnson, and P.M.Hui, Europhys. Lett. [**67**]{}, 867 (2004).
S.C. Choe, N.F. Johnson, and P.M. Hui, Phys. Rev. E [**70**]{}, 055101 (2004).
T.S. Lo, H.Y. Chan, P. M. Hui, and N. F.Johnson, Phys. Rev. E [**70**]{}, 056102 (2004).
D. Challet and Y.C. Zhang, Physica A [**246**]{}, 407 (1997);.
D. Challet, M. Marsilli, and G. Ottino, cond-mat/0306445.
N.F. Johnson, S.C. Choe, S. Gourley, T.Jarrett, and P.M. Hui, in [*Advances in Solid State Physics*]{} Vol. [**44**]{} (Springer, Heidelberg, 2004),p. 427.
We originally introduced the finite time-horizon MG, plus its ‘grand canonical’ variable-$N$ and variable-$L$ generalizations, to provide a minimal model for financial markets. See M.L. Hart, P. Jefferies and N.F. Johnson, Physica A **311**, 275(2002); M.L. Hart, D. Lamper and N.F. Johnson, Physica A **316**, 649 (2002); D. Lamper, S. D. Howison, and N. F. Johnson, Phys. Rev. Lett. [**88**]{}, 017902 (2002); N.F. Johnson, P. Jefferies, and P.M. Hui,[*Financial Market Complexity*]{} (Oxford University Press, 2003). See also D. Challet and T. Galla, cond-mat/0404264,which uses this same model.
D. Challet and Y.C. Zhang, Physica A [**256**]{}, 514 (1998).
D. Challet, M. Marsili and R. Zecchina, Phys. Rev. Lett. [**82**]{}, 2203 (1999).
D. Challet, M. Marsili and R. Zecchina, Phys. Rev. Lett. [**85**]{}, 5008 (2000).
See http://www.unifr.ch/econophysics for Minority Game literature.
N.F. Johnson, M. Hart and P.M. Hui, Physica A **269**, 1 (1999).
M. Hart, P. Jefferies, N.F. Johnson and P.M. Hui, Physica A **298**, 537 (2001).
N.F. Johnson, P.M. Hui, Dafang Zheng, and M. Hart, J. Phys. A: Math. Gen. **32**, L427 (1999).
M.L. Hart, P. Jefferies, N.F. Johnson and P.M. Hui, Phys. Rev. E [**63**]{}, 017102 (2001).
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[^1]: This state transition rule is not critical to the formalism. For example, the rule could be to use dice ***A*** if the last increment was odd and ***B*** is even, or indeed any property of the outcomes.
[^2]: \[foot:conv\]We define and use the discrete convolution operator $\otimes$ such that $(f\otimes g)\mid_i~=~\sum_{j = -\infty}^\infty f(i-j)g(i)$.
[^3]: The algorithm used by the ‘Game-master’ to generate the winning decision could also incorporate a stochastic factor.
[^4]: The steady state eigenvector solution is an exact expression equivalent to pre-multiplying the probability state vector $\underline{P(\Gamma_t)}$ by $\underline{\underline{T}}^{\infty}$. This effectively results in a probability state vector which is time-averaged over an infinite time-interval.
[^5]: Note that we can use this framework to generate time-averaged quantities of [*any*]{} of the macroscopic quantities of the system (e.g total number of agents playing) or volatility.
[^6]: We note that many features of the game can be reproduced using a Reduced Strategy Space of $2^{m+1}$ strategies, containing strategies which are either anti-correlated or uncorrelated with each other[@A246]. The framework established in the present paper is general to both the full and reduced strategy spaces, hence the full strategy space will be adopted here.
[^7]: We note that $\phi$ itself could be actually be a stochastic function of the known system parameters.
[^8]: This feature can be understood by appreciating the multi-peaked nature of the distributions in question. The peaks correspond to differing paths travelled in the Future-Cast, the final distribution being a superposition of these. If these individual path distributions mean-revert, the spread of the actual Future-Cast can decrease over short time-scales.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the Supplemental Material, we present details of our analysis of single-vortex solutions in the compact anisotropic KPZ equation. We study these solutions using asymptotic analysis and numerics. Moreover, we calculate the interaction between a vortex and an antivortex in the limit of low mobility of topological defects. In this limit, the interaction can be obtained from a dual electrostatic description. We solve the electrostatic problem perturbatively in the non-linearity of the KPZ equation. Based on this calculation, we derive RG equations that describe the vortex-unbinding crossover in parametrically large systems. Our analysis of the critical RG flow trajectories reveals a peculiar universal divergence of the correlation length as the transition is approached from the disordered side.'
bibliography:
- 'anisotropic\_KPZ.bib'
title: 'Supplemental Material: Topological Defects in Anisotropic Driven Open Systems'
---
A single vortex in the compact anisotropic KPZ equation {#sec:single-vort-comp}
=======================================================
Here we present some details of our analysis of the field generated by a single topological defect in the compact anisotropic KPZ (caKPZ) equation. As explained in the main text, to judge whether the ordered phase is stable in the thermodynamic limit, the crucial question is how the interaction of vortices behaves at asymptotically large distances. This question cannot be addressed within the perturbative treatment of the non-linearity we formulate below in the framework of the electrodynamic duality, since the perturbative expansion breaks down at large distances. We can nevertheless gain some insight by considering the simpler problem of a single vortex: In the isotropic KPZ equation, these vortices emit waves in the radial direction, and the exponential screening of the vortex interaction can be traced back to the shocks which are created when the emitted waves collide. Thus — at least heuristically — we conclude that the interaction is not screened if there are no waves emitted from the vortex cores in the (strongly) anisotropic KPZ equation. (Recall that due to the non-linearity a multi-vortex solution cannot simply be constructed by linear superposition of single vortices.) Below, using a combination of analytical asymptotic analysis and numerics we show that this is indeed the case in the fully anisotropic KPZ equation. We find that in the full weakly anisotropic (WA) regime, topological defects emit (deformed) radial waves and we would expect their interactions to be exponentially screened. Radial waves correspond to the asymptotic behavior $\theta(r, \phi) \sim k_0(\phi) r$ for $r \to \infty$ of the field generated by a topological defect at the origin, where $k_0(\phi)$ is the asymptotic wave number that depends on the polar angle $\phi$. By contrast, in the strongly anisotropic (SA) regime, the leading asymptotic behavior of the far-field of a topological defect is $\theta(r, \phi) \sim b_0 \ln(r/a) + \Phi(\phi)$, and the coefficient $b_0$ vanishes at the fully anisotropic (FA) point ($\lambda_x = - \lambda_y$). Thus, vortices in the fully anisotropic KPZ equation are qualitatively very similar to ordinary vortices in the $XY$ model. Away from the fully anisotropic point, the asymptotics $\theta(r, \phi) \sim b_0 \ln(r/a)$ can be interpreted as a radial wave with a wave number that vanishes as $1/r$. Further studies are required to test whether this behavior leads to sufficient screening to destabilize the ordered phase.
We thus want to find solutions to the caKPZ equation without noise, $$\label{eq:vortex_caKPZ}
\partial_t \theta = D \nabla^2 \theta + \frac{\lambda_x}{2} \left( \partial_x
\theta \right)^2 + \frac{\lambda_y}{2} \left( \partial_y \theta \right)^2,$$ subject to the topological constraint $\oint d \mathbf{l} \cdot \nabla \theta = 2 \pi$, where the line integral encircles the vortex core. As we show below, in the WA regime, such vortex solutions oscillate uniformly, i.e., they take the form $\theta(\mathbf{r}, t) = \theta_0(\mathbf{r}) + \omega_0 t$, where $\omega_0 > 0$ for $\lambda_{x, y} > 0$. We thus find it convenient to rewrite Eq. in a rotating frame by the transformation $\theta
\to \theta - \omega_0 t$, such that $$\label{eq:vortex_caKPZ_rf}
\partial_t \theta = D \nabla^2 \theta + \frac{\lambda_x}{2} \left( \partial_x
\theta \right)^2 + \frac{\lambda_y}{2} \left( \partial_y \theta \right)^2 -
\omega_0 = 0,$$ which is the equation stated in the main text \[Eq. (2)\]. This is a non-linear partial differential equation, and in the absence of rotational symmetry, the solution cannot be separated into parts that depend only on the radial coordinate or polar angle, respectively. As a further complication, the continuum compact KPZ (cKPZ) equation has to be regularized at short distances (e.g., by considering the cKPZ equation as the far-field phase equation derived from the complex Ginzburg-Landau equation (CGLE), or by discretizing the cKPZ equation on a lattice). In particular, the value of the oscillation frequency is determined by the regularization. Nevertheless, some progress can be made if we are modest and consider the asymptotic far-field behavior only. Below, we check our analytical results for the far field with numerics for the full solution of Eq. .
We begin the discussion of the analytical approach by reviewing vortices in the isotropic KPZ equation [@Aranson1998] (see [@Aranson2002] and references therein for vortices in the CGLE). Hence, we set $\lambda_x = \lambda_y = \lambda_+$ in Eq. , $$\partial_t \theta = D \nabla^2 \theta + \frac{\lambda_+}{2} \left( \nabla \theta
\right) - \omega_0 = 0.$$ A vortex sitting at the origin is described by a solution of the form $\theta(r, \phi) = \phi + R(r)$, where due to the rotational symmetry the function $R(r)$ depends only on the radius. For the radial dependence we find the equation ($\alpha_+ = \lambda_+/(2 D)$) $$\label{eq:R}
\left( R'' + \frac{R'}{r} \right) + \alpha_+ \left[ \frac{1}{r^2} + R^{\prime 2}
\right] - \frac{\omega_0}{D} = 0,$$ which can be linearized by means of a Cole-Hopf transformation, $w = e^{\alpha_+ R}$. We note that this implies that $w$ takes values in ${\mathbbm{R}}_{>0}$ since $R \in {\mathbbm{R}}$. The Cole-Hopf transformation brings Eq. to the form of a modified Bessel equation, $$\label{eq:w}
w'' + \frac{w'}{r} + \frac{\alpha_+^2 w}{r^2} = \kappa_0^2 w,$$ where $\kappa_0^2 = \alpha_+ \omega_0/D$. For $\omega_0 = 0$, two linearly independent solutions to this equation are given by $w = r^{\pm i \alpha_+}$, and accordingly real-valued solutions take the form $w = w_0 \cos(\alpha_+ \ln(r/a) + b)$ with $w_0, b \in {\mathbbm{R}}$. However, this oscillating function does not have an inverse Cole-Hopf transformation $\forall r \in {\mathbbm{R}}_{>0}$ and thus it does not yield a valid solution for a vortex. For finite $\omega_0$, the solution to Eq. which is bounded at large $r$ is the modified Bessel function $w(r) = K_{i \alpha_+}(\kappa_0 r)$ At large scales it behaves as $w(r) \sim e^{-\kappa_0 r}/\sqrt{r}$ and it assumes a maximum at $r_0 = e^{-\pi/(2 \alpha_+)}/\kappa_0$, while at shorter scales it starts to oscillate. Hence, the KPZ equation can describe vortices only for $r > r_0$, and some regularization is required at shorter scales. The precise value of $\omega_0$ is determined by the regularization [@Aranson1998]. In the CGLE, one finds $\omega_0 \sim \lambda_+ e^{-\pi D/\lambda_+}/(2 a)$, where $a$ is the vortex core radius [@Aranson1998]. Before we move on to discuss vortices in the anisotropic case, we note that the asymptotic behavior of the Bessel function implies the following asymptotic behavior of $\theta$: $$\label{eq:vortex_asymptotic_isotropic}
\begin{split}
\theta(r, \phi) & = \phi + \frac{1}{\alpha_+} \ln(K_{i \alpha_+}(\kappa_0
r)) \\ & = -k_0 r - \frac{1}{2 \alpha_+} \ln(2 \alpha_+ k_0 r/\pi) + \phi +
O(1/r),
\end{split}$$ where $k_0 = \kappa_0/\alpha_+ = \sqrt{\omega_0/(\alpha_+ D)}$.
Far field of a single anisotropic vortex
----------------------------------------
Next, we consider vortices in the anisotropic KPZ equation, i.e., we seek solutions to $$\label{eq:vortex_caKPZ_rf_rescaled}
\nabla^2 \theta + \alpha_x \left( \partial_x \theta \right)^2 + \alpha_y
\left( \partial_y \theta \right)^2 - \varpi_0 = 0,$$ where $\varpi_0 = \omega_0/D$, $\alpha_{x,y} = \lambda_{x,y}/(2 D)$, and with $\alpha_x \neq \alpha_y$. The asymptotic behavior in the isotropic case motivates the following ansatz for $r \to \infty$: $$\label{eq:vortex_asymptotic}
\theta(r, \phi) = k_0(\phi) r + b(\phi) \ln(r/a) + \Phi(\phi) + O(1/r).$$ Here, $\Phi(\phi)$ contains the topological part, i.e., $\int_0^{2 \pi} d \phi \, \Phi'(\phi) = 2 \pi$, where $\Phi' = d \Phi/d\phi$. We note, that a constant contribution to the vortex field can be added arbitrarily. It is not determined by the caKPZ equation, since the latter contains only derivatives of $\theta$. With the above ansatz, the gradient of the phase behaves at large $r$ as $$\begin{gathered}
\nabla \theta(r, \phi) = {\hat{\mathbf{e}}}_r \left( k_0(\phi) + \frac{b(\phi)}{r}
\right) \\ + \frac{{\hat{\mathbf{e}}}_{\phi}}{r} \left( k_0'(\phi) r + b'(\phi)
\ln(r/a) + \Phi'(\phi) \right) + O(1/r^2),\end{gathered}$$ where ${\hat{\mathbf{e}}}_r = \left( \cos(\phi), \sin(\phi) \right)$, and ${\hat{\mathbf{e}}}_{\phi} = \left( -\sin(\phi), \cos(\phi) \right)$. Inserting this expression in Eq. , we find by matching the leading terms for $r \to \infty$ (for $\alpha_{x,y} > 0$, we take the negative square root to match our numerical findings, see Sec. \[sec:numerics\]; $k(\phi)$ is positive for $\alpha_{x, y} < 0$): $$\label{eq:k_0}
k_0(\phi) = - \sqrt{\frac{\varpi_0}{\alpha_x \cos(\phi)^2 + \alpha_y \sin(\phi)^2}},$$ which describes a vortex emitting a slightly deformed radial wave. Clearly, this solution is well-behaved as a function of $\phi$ as long as both $\alpha_x$ and $\alpha_y$ are positive. As in the isotropic case, we expect that $\varpi_0$ is determined by matching the asymptotic solution to the (regularized) solution in the core region. However, we cannot perform a Cole-Hopf transformation as before, and therefore it is not easily possible to find a solution that is valid at short distances. Crucially, within the WA regime, the structure of a single vortex is qualitatively unchanged, which implies that the vortex interaction is screened and the ordered phase is unstable whenever $\alpha_{x, y}$ have the same sign.
In the SA regime, when $\alpha_x$ and $\alpha_y$ have opposite signs, Eq. indicates that $\varpi_0$ and $k_0(\phi)$ vanish — any non-zero $\varpi_0$ would lead to imaginary values of $k_0(\phi)$ and can be discarded for this reason. This matches our numerical findings, see Sec. \[sec:numerics\]. Dropping the leading term from Eq. , we find by matching terms $O((\ln(r/a)/r)^2)$ in Eq. : $$\left( \alpha_+ - \alpha_- \cos(2 \phi) \right) b'(\phi)^2 = 0,$$ where $\alpha_{\pm} = (\alpha_x \pm \alpha_-)/2$. It follows that $b'(\phi) = 0$ and hence $b(\phi) = b_0$. All terms at $O(\ln(r/a)/r^2)$ vanish for $b'(\phi) = 0$, and the next non-trivial contribution comes at $O(1/r^2)$: $$\begin{gathered}
\label{eq:Phi}
\Phi''(\phi) + \alpha_+ \left( b_0^2 + \Phi'(\phi)^2 \right) + \alpha_- \left(
b_0^2 \cos(2 \phi) \right. \\ \left. - 2 b_0 \Phi'(\phi) \sin(2 \phi) -
\Phi'(\phi)^2 \cos(2 \phi) \right) = 0.\end{gathered}$$ This is an ordinary yet non-linear differential equation, and there is no constructive way to find a solution. However, a vast simplification occurs in the FA limit $\alpha_x = - \alpha_y$. There, the numerical solution shown in Fig. 1 of the main text indicates that $\theta(r, \phi)$ does not depend on $r$ at all, i.e., the ansatz $\theta(r, \phi) = \Phi_0(\phi)$ yields an exact solution. Going back to Eq. , with this ansatz we find $$\begin{gathered}
\nabla^2 \theta + \alpha_- \left[ \left( \partial_x \theta \right)^2 -
\left( \partial_x \theta \right)^2 \right] \\ =
\frac{1}{r^2} \left( \Phi_0'' - \alpha_- \cos(2 \phi) \Phi_0^{\prime 2} \right) = 0,\end{gathered}$$ This equation can be integrated trivially once, $$\begin{split}
\int_0^{\phi} d \phi' \, \frac{\Phi_0''(\phi')}{\Phi_0'(\phi')^2} & = -
\int_0^{\phi} d \phi' \, \frac{d}{d \phi'} \frac{1}{\Phi_0'(\phi')} \\ & = -
\left( \frac{1}{\Phi_0'(\phi)} - \frac{1}{\Phi_0'(0)} \right) \\ & =
\alpha_- \int_0^{\phi} d \phi' \cos(2 \phi') \\ & = \frac{\alpha_-}{2} \sin(2
\phi).
\end{split}$$ We thus find $$\Phi_0'(\phi) = \frac{\Phi_0'(0)}{1 - \frac{\alpha_-}{2} \Phi_0'(0) \sin(2
\phi)},$$ and another integration yields the result $$\begin{gathered}
\label{eq:Phi_0}
\Phi_0(\phi) = 2 \nu \Phi_0'(0) \left( \arctan(\nu \left( 2 \tan(\phi) -
\alpha_- \Phi_0'(0) \right)) \right. \\ \left. + \arctan(\alpha_- \nu
\Phi_0'(0)) \right),\end{gathered}$$ where a constant of integration is added such that $\Phi_0(0) = 0$; in order to obtain a smooth solution we have to choose different branches of the $\arctan$ in the intervals $0 < \phi < \pi/2, \pi/2 < \phi < 3 \pi/2$, and $3 \pi/2 < \phi < 2 \pi$. The constant $\nu$ in Eq. is given by $$\nu = \frac{1}{\sqrt{4 - \left( \alpha_- \Phi_0'(0) \right)^2}}.$$ Finally, $\Phi_0'(0)$ is determined by the condition that for a singly-charged vortex the function $\Phi_0(\phi)$ should wind once around the unit circle, which yields $$\Phi_0'(0) = \frac{2}{\sqrt{4 + \alpha_-^2}}.$$ Vortex solutions for different values of $\alpha_-$ are shown in Fig. \[fig:Phi\_01\](a).
![(a) Angular dependence of the vortex field in the fully anisotropic KPZ equation for $\alpha_- = 0.1, 1, 10, 100$ (blue to light orange). (b) First-order correction to the fully anisotropic vortex for $\alpha_- = 0.5, 1, 10$ (blue to orange).[]{data-label="fig:Phi_01"}](Phi_0 "fig:"){width="0.48\linewidth"} ![(a) Angular dependence of the vortex field in the fully anisotropic KPZ equation for $\alpha_- = 0.1, 1, 10, 100$ (blue to light orange). (b) First-order correction to the fully anisotropic vortex for $\alpha_- = 0.5, 1, 10$ (blue to orange).[]{data-label="fig:Phi_01"}](Phi_1 "fig:"){width="0.48\linewidth"}
For $\alpha_- \to 0$ the solution smoothly deforms into an “ordinary” $XY$-type vortex with $\Phi_0(\phi) = \phi$, while in the opposite limit it approaches a step-like form. In fact, Eq. is analytic in $\alpha_-$ at $\alpha_- = 0$. This is in stark contrast to the isotropic case, where the transition from from the linear to the non-linear problem is highly non-analytic (see the above expression for $\omega_0$). Since turning on the non-linearity in a fully anisotropic system does not alter the *radial dependence* of the far field of a single vortex, we conclude that the interaction of vortices at large distances is not screened as in the isotropic case, and thus the ordered phase can be stable.
Now let’s reinstate $\alpha_+$. First, note that for $b_0 = 0$, Eq. becomes $$\Phi''(\phi) + \left( \alpha_+ - \alpha_- \cos(2 \phi) \right) \Phi'(\phi)^2 = 0.$$ As before, this equation can be integrated and we find $$- \left( \frac{1}{\Phi'(\phi)} - \frac{1}{\Phi'(0)} \right) = - \alpha_+ \phi
+ \frac{\alpha_-}{2} \sin(2 \phi).$$ Since the resulting expression for $\Phi'(\phi)$ is not periodic in $\phi$, evidently this cannot be a valid solution, and we have to allow for a finite value of $b_0$, i.e., away from the FA configuration vortices do have a non-trivial radial dependence. We restrict ourselves to small values of $\alpha_+$ for which we set $$\begin{split}
\Phi(\phi) & = \Phi_0(\phi) + \alpha_+ \Phi_1(\phi) + O(\alpha_+^2), \\
b_0 & = \alpha_+ b_{01} + O(\alpha_+^2),
\end{split}$$ where the zeroth-order solution $\Phi_0(\phi)$ is given by Eq. . Inserting this ansatz in Eq. leads to a linear second-order differential equation for $\Phi_1(\phi)$; $b_{01}$ is determined by the condition that $\Phi_0'(\phi)$ has to be periodic, which yields $b_{01} = -2/\alpha_-^2$. The first constant of integration $\Phi_1'(0)$ has to be chosen such that $\lim_{\phi \searrow 0} \Phi_1(\phi) = \lim_{\phi \nearrow 2 \pi} \Phi_1(\phi)$, while w.l.o.g. we choose the second constant of integration $\Phi_1(0)$ such that $\Phi(0) = 0$. The resulting solution $\Phi_1(\phi)$ is shown in Fig. \[fig:Phi\_01\](b). It is an interesting question for future research how the logarithmic dependence of the vortex field on the distance from the core — corresponding to an emitted wave with wave number $\sim 1/r$ — affects the interaction at asymptotic distances.
Numerics {#sec:numerics}
--------
To confirm the results of the previous section numerically, we discretize Eq. on a lattice with sites $\mathbf{r} = \left( x, y \right)$, i.e., we replace spatial derivatives with finite differences according to (cf. Ref. [@Sieberer2016b]) $$\label{eq:finite_differences}
\begin{split}
\partial_x^2 \theta & \to - \sum_{\sigma = \pm} \sin(\theta_{\mathbf{r}} -
\theta_{\mathbf{r} + \sigma {\hat{\mathbf{x}}}}), \\ \left( \partial_x \theta
\right)^2 & \to - \sum_{\sigma = \pm} \left( \cos(\theta_{\mathbf{r}} -
\theta_{\mathbf{r} + \sigma {\hat{\mathbf{x}}}}) - 1 \right).
\end{split}$$ ${\hat{\mathbf{x}}}$ and ${\hat{\mathbf{y}}}$ are unit vectors, and for convenience we choose the lattice spacing as $a = 1$. With the above prescription, Eq. becomes $$\begin{gathered}
\label{eq:discrete_vortex_caKPZ}
\partial_t \theta_{\mathbf{r}} = - \sum_{\sigma = \pm} \left[ D \left(
\sin(\theta_{\mathbf{r}} - \theta_{\mathbf{r} + \sigma {\hat{\mathbf{x}}}}) +
\sin(\theta_{\mathbf{r}} - \theta_{\mathbf{r} + \sigma {\hat{\mathbf{y}}}})
\right) \vphantom{\frac{\lambda_x}{2}} \right. \\ + \frac{\lambda_x}{2}
\left( \cos(\theta_{\mathbf{r}} - \theta_{\mathbf{r} + \sigma {\hat{\mathbf{x}}}}) -
1 \right) \\ \left. + \frac{\lambda_y}{2} \left( \cos(\theta_{\mathbf{r}} -
\theta_{\mathbf{r} + \sigma {\hat{\mathbf{y}}}}) - 1 \right) \right].\end{gathered}$$ We choose initial conditions corresponding to an ordinary $XY$ vortex that is displaced by half a lattice spacing from the origin, i.e., $\theta_{\mathbf{r}}(0) = \tan((y - 1/2)/(x - 1/2))$, and evolve this configuration in time. For open boundary conditions, we found that the core of the topological defect remains stationary. (For large values of the non-linearities $\lambda_{x, y}$, the vortex starts to move, and new vortices are generated dynamically. In the simulations presented here, we always stay below this instability.) Since the evolution equation is dissipative and the topological charge is conserved, the field configuration converges to a vortex solution of the non-linear problem. For the plots in Fig. 1 of the main text, we evolved Eq. on a lattice of $200 \times 200$ sites.
As discussed in the previous section, vortices in the caKPZ equation oscillate uniformly in the WA regime. From the numerical solution of Eq. , the oscillation frequency can be obtained by fitting the steady linear growth of $\theta_{\mathbf{r}}(t)$ at late times (i.e., when convergence is reached). We find a vanishing oscillation frequency $\varpi_0$ only exactly at the FA point, and small but finite oscillation frequencies throughout the SA region. However, as illustrated in Fig. \[fig:omega0\], this is just a finite-size effect.
{width="0.4\linewidth"} {width="0.4\linewidth"}\
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Interaction of vortices in the compact anisotropic KPZ equation
===============================================================
In linear theories, the superposition of two solutions gives another valid solution. This is no longer true in the presence of a non-linearity as in the case of the caKPZ equation. In particular, the superposition of two single-vortex solutions does not yield a two-vortex solution. This is the main difficulty in trying to find the vortex interaction. Only for very large separations of topological defects, we could gain some insight in the asymptotic behavior of the vortex interaction as discussed in the previous section. Here, we present an alternative approach that works up to parametrically large distances, and is based on a recently developed formulation of the caKPZ equation as non-linear electrodynamics [@Sieberer2016b; @Wachtel2016].
In Sec. \[sec:electr-dual\], we briefly review the electrodynamic duality for the compact KPZ equation [@Sieberer2016b; @Wachtel2016] and its extension to the anisotropic case. Within this framework, we calculate the interaction between a vortex and an antivortex perturbatively. This rather tedious calculation, and some numerical checks of the result, are presented in Sec. \[sec:pert-calc-vort\].
Electrodynamic duality {#sec:electr-dual}
----------------------
In the following, we derive a dual description of the caKPZ equation \[Eq. (1) of the main text\] [@Sieberer2016b; @Wachtel2016], $$\partial_t \theta = \sum_{i = x, y} \left[ D_i \partial_i^2 \theta +
\frac{\lambda_i}{2} \left( \partial_i \theta \right)^2 \right] + \eta.$$ As in the main text, we set $D_x = D_y = D$, which corresponds simply to an anisotropic rescaling of the units of length. We find it convenient to rewrite the non-linear terms in the following way: $$\sum_{i = x, y} \lambda_i \left( \partial_i \theta \right)^2 = \lambda_+
\left( \nabla \theta \right)^2 + \lambda_- \left[ \left( \partial_x \theta
\right)^2 - \left( \partial_y \theta \right)^2 \right],$$ which separates an isotropic contribution that is proportional to $\lambda_+ = \frac{\lambda_x + \lambda_y}{2}$ from a purely anisotropic contribution with coefficient $\lambda_- = \frac{\lambda_x - \lambda_y}{2}$. To keep the notation compact, we define a $\odot$ product of vectors $\mathbf{a} = \left( a_x, a_y \right)$ and $\mathbf{b} = \left( b_x, b_y \right)$ through the relation. $\mathbf{a} \odot \mathbf{b} = a_x b_x - a_y b_y$. We also use the abbreviation $\mathbf{a}^{\odot 2} = \mathbf{a} \odot \mathbf{a} = a_x^2 - a_y^2$. It is straightforward to check that this product is commutative and distributive, i.e., in calculations it can be handled like the usual scalar product. With this notation, the caKPZ equation can be written as $$\label{eq:caKPZ_odot}
\partial_t \theta = D \nabla^2 \theta + \frac{\lambda_+}{2} \left( \nabla \theta
\right)^2 + \frac{\lambda_-}{2} \left( \nabla \theta \right)^{\odot 2} + \eta.$$
To explicitly incorporate vortices in the caKPZ equation, we reformulate it in terms of the electric field, which is defined as $$\label{eq:electric_field}
\mathbf{E} = - {\hat{\mathbf{z}}} \times \nabla \theta.$$ ${\hat{\mathbf{z}}} = \left( 0, 0, 1 \right)$ is a unit vector pointing in the direction perpendicular to the $xy$-plane on which $\theta$ and $\mathbf{E}$ are defined. We note that in the presence of topological defects in the KPZ equation without noise, $\theta$ has a contribution that depends linearly on time and is uniform in space, see Sec. \[sec:single-vort-comp\]. This contribution corresponds to oscillations of vortices in the complex Ginzburg-Landau equation, and drops out if we consider $\mathbf{E}$ instead of $\theta$.
By cyclic permutation of the vectors in the defining relation for $\mathbf{E}$ we obtain $\nabla \theta = {\hat{\mathbf{z}}} \times \mathbf{E}$, which leads to the following expressions for the non-linear terms in the caKPZ equation: $$\left( \nabla \theta \right)^2 = E^2, \qquad \left( \nabla \theta
\right)^{\odot 2} = - \mathbf{E}^{\odot 2}.$$ Thus, Eq. can be written as $$\label{eq:KPZ_E}
\partial_t \mathbf{E} = D \nabla^2 \mathbf{E} - {\hat{\mathbf{z}}} \times \left(
\frac{\lambda_+}{2} E^2 - \frac{\lambda_-}{2} \mathbf{E}^{\odot 2} + \eta
\right).$$ This equation has to be extended to explicitly account for vortices. To this end, we first note that the circulation of the gradient of $\theta$ around a closed loop is determined by the number of enclosed vortices. This statement, recast in differential form, can be written as $$\label{eq:Gauss}
\nabla \cdot \mathbf{E} = 2 \pi n,$$ where $n$ is the vortex density. Second, since vortices are created only in pairs (or at the boundary of the sample), the vortex density $n$ and current $\mathbf{j}$ obey on equation of continuity: $$\label{eq:continuity}
\partial_t n + \nabla \cdot \mathbf{j} = 0.$$ Since Eqs. and can be combined to read $$\nabla \cdot \left( \partial_t \mathbf{E} + 2 \pi \mathbf{j} \right) = 0,$$ we see that $$\label{eq:u}
\partial_t \mathbf{E} = - 2 \pi \mathbf{j} + \nabla \times \mathbf{u},$$ where $\mathbf{u}$ is a vector field that is determined by the condition that Eq. should reproduce the KPZ equation in the absence of vortices, i.e., for $n = \mathbf{j} = 0$. This condition is fulfilled by setting $$\label{eq:156}
\mathbf{u} = - D \nabla \times \mathbf{E} + {\hat{\mathbf{z}}} \left(
\frac{\lambda_+}{2} E^2 - \frac{\lambda_-}{2}
\mathbf{E}^{\odot 2} + \eta \right).$$ The required extension of the KPZ equation to include vortices is thus given by $$\begin{gathered}
\label{eq:dual_KPZ}
\partial_t \mathbf{E} = - D \nabla \times \left( \nabla \times \mathbf{E}
\right) - 2 \pi \mathbf{j} \\ - {\hat{\mathbf{z}}} \times \nabla \left(
\frac{\lambda_+}{2} E^2 - \frac{\lambda_-}{2} \mathbf{E}^{\odot 2} + \eta
\right).\end{gathered}$$ As a final step, as customary in “macroscopic” electrodynamics, we separate the contributions due to free and bound vortices by decomposing the vortex density and current as $$\begin{split}
n & = n_f + n_b, \\
\mathbf{j} & = \mathbf{j}_f + \mathbf{j}_b.
\end{split}$$ Bound vortices lead to polarization of the medium, which can be described by a polarization density $\mathbf{P}$ that satisfies $$\begin{split}
\nabla \cdot \mathbf{P} & = - 2 \pi n_b, \\
\partial_t \mathbf{P} & = 2 \pi \mathbf{j}_b.
\end{split}$$ Adding the polarization to the electric field we obtain the displacement field $$\label{eq:displacement_field}
\mathbf{D} = \mathbf{E} + 2 \pi \mathbf{P} = \left( 1 + 2 \pi \chi \right) \mathbf{E} = \varepsilon \mathbf{E},$$ where we set $\mathbf{P} = \chi \mathbf{E}$ with the susceptibility $\chi$, and the last relation defines the dielectric constant $\varepsilon = 1 + 2 \pi \chi$. We show below the even though we are considering an anisotropic system, to the lowest perturbative order in the KPZ non-linearity it is sufficient to consider a single isotropic dielectric constant instead of a dielectric tensor. With these definitions, Eqs. and can be written as $$\label{eq:macroscopic_Gauss}
\nabla \cdot \mathbf{E} = \frac{2 \pi}{\varepsilon} n_f,$$ and $$\begin{gathered}
\label{eq:macroscopic_dual_KPZ}
\varepsilon \partial_t \mathbf{E} = - D \nabla \times \left( \nabla \times
\mathbf{E} \right) - 2 \pi \mathbf{j}_f \\ - {\hat{\mathbf{z}}} \times \nabla
\left( \frac{\lambda_+}{2} E^2 - \frac{\lambda_-}{2} \mathbf{E}^{\odot 2} +
\eta \right).\end{gathered}$$ In the following, we drop the subscript $f$.
Perturbative calculation of the vortex interaction {#sec:pert-calc-vort}
--------------------------------------------------
As we are interested in the interaction of vortices, which is encoded in the deterministic dynamics, we set the noise to zero, $\eta = 0$. Moreover, we assume that the mobility of vortices is small. This allows us to consider the static limit [@Wachtel2016] in which $\partial_t \mathbf{E} = \mathbf{j} = 0$. Then, Eq. becomes $$\label{eq:16}
0 = D \left( \nabla^2 \mathbf{E} - 2 \pi \nabla n
\right) - {\hat{\mathbf{z}}} \times \nabla \left( \frac{\lambda_+}{2} E^2 - \frac{\lambda_-}{2}
\mathbf{E}^{\odot 2} \right).$$ This equation and Eq. determine the electrostatic field generated by a collection of free vortices with local density $n$. It is convenient to represent the electric field in terms of scalar and vector potentials, $$\label{eq:phi_A}
\mathbf{E} = - \nabla \phi - \mathbf{A}.$$ As usual, this decomposition does not uniquely define the potentials. It is invariant under gauge transformations of the form $\phi' = \phi - \chi$ and $\mathbf{A}' = \mathbf{A} + \nabla \chi$. This redundancy can be eliminated by working in a particular gauge. Here, we impose the Lorenz gauge condition $\partial_t \phi + D \nabla \cdot \mathbf{A} = 0$, which in the static limit reduces to $\nabla \cdot \mathbf{A} = 0$. Then, the scalar potential encodes the longitudinal part of the electric field, and the vector potential the transverse part. It can be written in terms of another potential $\psi$ as $$\label{eq:A_psi}
\mathbf{A} = - {\hat{\mathbf{z}}} \times \nabla \psi.$$ Inserting Eqs. and in Eqs. and , we obtain $$\begin{aligned}
\label{eq:phi_psi_diff_equ}
- \nabla^2 \phi & = \frac{2 \pi}{\varepsilon} n, \\
D \nabla^2 \psi & = \frac{\lambda_+}{2} \left( \nabla \phi - {\hat{\mathbf{z}}} \times
\nabla \psi \right)^2 - \frac{\lambda_-}{2} \left( \nabla \phi - {\hat{\mathbf{z}}} \times
\nabla \psi \right)^{\odot 2}.\end{aligned}$$ These equations can be integrated with the aid of the fundamental solution of the Laplacian (i.e., the electrostatic potential generated by a point charge), $$\label{eq:G}
G(\mathbf{r}) = - \nabla^{-2} \delta(\mathbf{r}) = - \frac{1}{2 \pi}
\ln(r/a).$$ As usual, $a$ is to be understood as a microscopic cutoff. Here and in the following, the potential $G(\mathbf{r})$ should be set to zero for $r < a$. With the aid of the fundamental solution , Eqs. can be rewritten as convolution integrals, $$\label{eq:phi}
\phi(\mathbf{r}) = \frac{2 \pi}{\varepsilon} \int_{\mathbf{r}'} G(\mathbf{r} -
\mathbf{r}') n(\mathbf{r}'),$$ and $$\begin{gathered}
\label{eq:psi}
\psi(\mathbf{r}) = - \frac{1}{2 D} \int_{\mathbf{r}'} G(\mathbf{r} -
\mathbf{r}') \left[ \lambda_+ \left( \nabla \phi(\mathbf{r}') - {\hat{\mathbf{z}}}
\times \nabla \psi(\mathbf{r}') \right)^2 \right. \\ \left. - \lambda_-
\left( \nabla \phi(\mathbf{r}') - {\hat{\mathbf{z}}} \times \nabla
\psi(\mathbf{r}') \right)^{\odot 2} \right],\end{gathered}$$ where we write $\int_{\mathbf{r}} = \int d^2 \mathbf{r}$, and the integration extends over the area occupied by the system. While Eq. fully determines the potential $\phi$ for a given charge distribution $n$, Eq. is an integro-differential equation for $\psi$. By iterating this equation, we obtain a solution in the form of a perturbative expansion in $\lambda_+$ and $\lambda_-$, $$\label{eq:psi_expansion}
\psi = \frac{1}{\varepsilon} \psi^{(0)} + \frac{1}{\varepsilon^2} \psi^{(1)} +
\frac{1}{\varepsilon^3} \psi^{(2)} + \dotsb$$ (we explicitly specify factors of $1/\varepsilon$ to get simpler expressions below). More concretely, setting $\lambda_+ = \lambda_- = 0$ in Eq. , we find that the zeroth-order contribution vanishes, $\psi^{(0)} = 0$. The first-order contribution $\psi^{(1)}$ can be obtained by inserting $\psi = \psi^{(0)} = 0$ on the RHS of Eq. . Another iteration yields the lowest-order terms, $$\label{eq:psi_12}
\begin{split}
\psi^{(1)} & = - \sum_{\sigma = \pm} \sigma
\alpha_{\sigma} \psi_{\sigma}^{(1)}, \\
\psi^{(2)} & = - \sum_{\sigma = \pm} \alpha_{\sigma}^2 \psi_{\sigma}^{(2)} +
\alpha_+ \alpha_- \psi_{+-}^{(2)}.
\end{split}$$ As in the previous section, we denote $\alpha_{\pm} = \lambda_{\pm}/(2 D)$. The explicit expressions for the lowest-order terms read $$\label{eq:psi^1}
\psi_{\sigma}^{(1)}(\mathbf{r}) = \varepsilon^2 \int_{\mathbf{r}'} G(\mathbf{r} - \mathbf{r}')
\left( \nabla \phi(\mathbf{r}') \right)^{\odot 2},$$ (here and in the following, for $\sigma = +$ the $\odot$ product should be replaced by the usual scalar product) $$\label{eq:psi^2_sigma}
\psi_{\sigma}^{(2)}(\mathbf{r}) = 2 \varepsilon \int_{\mathbf{r}'}
G(\mathbf{r} - \mathbf{r}') \left[ \nabla \phi(\mathbf{r}') \odot \left(
{\hat{\mathbf{z}}} \times \nabla \psi_{\sigma}^{(1)}(\mathbf{r}') \right)
\right],$$ and the mixed term is given by $$\begin{gathered}
\label{eq:psi^2_+-_foo}
\psi_{+-}^{(2)}(\mathbf{r}) = 2 \varepsilon \int_{\mathbf{r}'} G(\mathbf{r}
- \mathbf{r}') \left[ \nabla \phi(\mathbf{r}') \cdot \left( {\hat{\mathbf{z}}} \times
\nabla \psi_-^{(1)}(\mathbf{r}') \right) \right. \\ \left. + \nabla
\phi(\mathbf{r}') \odot \left( {\hat{\mathbf{z}}} \times \nabla
\psi_+^{(1)}(\mathbf{r}') \right) \right].\end{gathered}$$ The above expressions are valid for any charge distribution $n$. In the following, we consider a dipole which is described by $$n(\mathbf{r}) = \delta(\mathbf{r} - \mathbf{r}_+) - \delta(\mathbf{r} - \mathbf{r}_-).$$ Inserting this in Eq. yields the scalar electrostatic potential generated by the dipole: $$\phi(\mathbf{r}) = \frac{2 \pi}{\varepsilon} \left( G(\mathbf{r} -
\mathbf{r}_+) - G(\mathbf{r} - \mathbf{r}_-) \right)
= \frac{1}{\varepsilon} \ln \! {\left\lvert \frac{\mathbf{r} -
\mathbf{r}_+}{\mathbf{r} - \mathbf{r}_-} \right\rvert}.$$ Equations , , and require the gradient of the scalar potential. In terms of $$\label{eq:f}
\mathbf{f}(\mathbf{r}) = 2 \pi \nabla G(\mathbf{r}) = -
\frac{\mathbf{r}}{r^2},$$ we find $$\label{eq:grad_phi}
\begin{split}
\nabla \phi(\mathbf{r}) & = \frac{1}{\varepsilon} \left(
\mathbf{f}(\mathbf{r} - \mathbf{r}_+) - \mathbf{f}(\mathbf{r} -
\mathbf{r}_-) \right) \\ & = - \frac{1}{\varepsilon} \left(
\frac{\mathbf{r} - \mathbf{r}_+}{{\left\lvert \mathbf{r} - \mathbf{r}_+ \right\rvert}^2} -
\frac{\mathbf{r} - \mathbf{r}_-}{{\left\lvert \mathbf{r} - \mathbf{r}_- \right\rvert}^2}
\right).
\end{split}$$ Inserting this in Eqs. and yields $$\label{eq:psi_f}
\psi_{\sigma}^{(1)}(\mathbf{r}) = \int_{\mathbf{r}'} G(\mathbf{r} - \mathbf{r}')
\left( \mathbf{f}(\mathbf{r}' - \mathbf{r}_+) -
\mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right)^{\odot 2},$$ and $$\begin{gathered}
\label{eq:psi^2_sigma_f}
\psi_{\sigma}^{(2)}(\mathbf{r}) = 2 \int_{\mathbf{r}'} G(\mathbf{r} -
\mathbf{r}') \left[ \left( \mathbf{f}(\mathbf{r}' - \mathbf{r}_+) -
\mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right) \vphantom{\left(
{\hat{\mathbf{z}}} \times \nabla \psi_{\sigma}^{(1)}(\mathbf{r}') \right)}
\right. \\ \left. \odot \left( {\hat{\mathbf{z}}} \times \nabla
\psi_{\sigma}^{(1)}(\mathbf{r}') \right) \right],\end{gathered}$$ and from Eq. we obtain $$\begin{gathered}
\label{eq:psi^2_+-}
\psi_{+-}^{(2)}(\mathbf{r}) = 2 \int_{\mathbf{r}'} G(\mathbf{r} - \mathbf{r}')
\left( \mathbf{f}(\mathbf{r}' - \mathbf{r}_+) - \mathbf{f}(\mathbf{r}' -
\mathbf{r}_-) \right)^T \\ \left[ \left( {\hat{\mathbf{z}}} \times \nabla
\psi_-^{(1)}(\mathbf{r}') \right) + \sigma_z \left( {\hat{\mathbf{z}}} \times
\nabla \psi_+^{(1)}(\mathbf{r}') \right) \right],\end{gathered}$$ where we used that with the Pauli matrix $\sigma_z$ the $\odot$ product can be written as $\mathbf{a} \odot \mathbf{b} = a_x b_x - a_y b_y = \mathbf{a}^T \sigma_z
\mathbf{b}$.
The integrals in the above expressions for the lowest-order contributions to $\psi$ are divergent in the thermodynamic limit. This complication is resolved upon taking the gradient as required by Eq. . Therefore, in the following we find it convenient to consider $\mathbf{a} = \nabla \psi$. In analogy to Eqs. and we write $$\mathbf{a} = \frac{1}{\varepsilon^2} \mathbf{a}^{(1)} +
\frac{1}{\varepsilon^3} \mathbf{a}^{(2)} + \dotso,$$ (recall that $\psi^{(0)} = 0$) and $$\begin{split}
\mathbf{a}^{(1)} & = - \sum_{\sigma = \pm} \sigma
\alpha_{\sigma} \mathbf{a}_{\sigma}^{(1)}, \\
\mathbf{a}^{(2)} & = - \sum_{\sigma = \pm} \alpha_{\sigma}^2
\mathbf{a}_{\sigma}^{(2)} + \alpha_+ \alpha_- \mathbf{a}_{+-}^{(2)},
\end{split}$$ where $$\label{eq:a^1_sigma}
\mathbf{a}_{\sigma}^{(1)}(\mathbf{r}) = \frac{1}{2 \pi} \int_{\mathbf{r}'} \mathbf{f}(\mathbf{r} - \mathbf{r}')
\left( \mathbf{f}(\mathbf{r}' - \mathbf{r}_+) -
\mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right)^{\odot 2},$$ and $$\begin{gathered}
\label{eq:a^2_sigma}
\mathbf{a}_{\sigma}^{(2)}(\mathbf{r}) = \frac{1}{\pi} \int_{\mathbf{r}'}
\mathbf{f}(\mathbf{r} - \mathbf{r}') \left[ \left( \mathbf{f}(\mathbf{r}' -
\mathbf{r}_+) - \mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right)
\vphantom{\left( {\hat{\mathbf{z}}} \times \mathbf{a}_{\sigma}^{(1)}(\mathbf{r}')
\right)} \right. \\ \left. \odot \left( {\hat{\mathbf{z}}} \times
\mathbf{a}_{\sigma}^{(1)}(\mathbf{r}') \right) \right],\end{gathered}$$ and finally $$\begin{gathered}
\label{eq:a^2_+-}
\mathbf{a}_{+-}^{(2)}(\mathbf{r}) = \frac{1}{\pi} \int_{\mathbf{r}'}
\mathbf{f}(\mathbf{r} - \mathbf{r}') \left( \mathbf{f}(\mathbf{r}' -
\mathbf{r}_+) - \mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right)^T \\ \left[
\left( {\hat{\mathbf{z}}} \times \mathbf{a}_-^{(1)}(\mathbf{r}') \right) + \sigma_z
\left( {\hat{\mathbf{z}}} \times \mathbf{a}_+^{(1)}(\mathbf{r}') \right) \right].\end{gathered}$$
In terms of the quantities defined above, the vector potential is given by $\mathbf{A} = - {\hat{\mathbf{z}}} \times \mathbf{a}$, and up to second order in the KPZ non-linearity the static electric field can be written as $$\label{eq:E_expansion}
\mathbf{E} = \frac{1}{\varepsilon} \mathbf{E}^{(0)} - \sum_{\sigma = \pm}
\left[ \frac{\sigma \alpha_{\sigma}}{\varepsilon^2} \mathbf{E}_{\sigma}^{(1)}
+ \frac{\alpha_{\sigma}^2}{\varepsilon^3} \mathbf{E}_{\sigma}^{(2)} \right]
+ \frac{\alpha_+ \alpha_-}{\varepsilon^3} \mathbf{E}_{+-}^{(2)},$$ where $$\label{eq:E_012}
\begin{split}
\mathbf{E}^{(0)}(\mathbf{r}) & = - \varepsilon \nabla \phi(\mathbf{r}), \\
\mathbf{E}^{(1,2)}_{\sigma}(\mathbf{r}) & = {\hat{\mathbf{z}}} \times
\mathbf{a}^{(1,2)}_{\sigma}(\mathbf{r}), \\
\mathbf{E}^{(2)}_{+-}(\mathbf{r}) & = {\hat{\mathbf{z}}} \times
\mathbf{a}^{(2)}_{+-}(\mathbf{r}).
\end{split}$$
In the following sections, we evaluate the integrals in Eqs. , , and both analytically and — to check the rather lengthy analytical calculations — numerically. For completeness, we also repeat the calculation of the purely isotropic corrections, which can also be found in Ref. [@Wachtel2016]. Actually, we do not need to find the electric field at arbitrary points in space, since the force acting on the charges is determined by the electric field at the position of one of the charges. Therefore, we evaluate the second-order corrections and only at $\mathbf{r} = \mathbf{r}_+$. The first-order correction , however, has to be calculated for any $\mathbf{r}$, since it is required in and .
Equations , , and are integrals over products of the function $\mathbf{f}(\mathbf{r})$ defined in Eq. with different arguments. The pole of $\mathbf{f}(\mathbf{r}) = -\mathbf{r}/r^2$ at $\mathbf{r} = 0$, which is cut off at the scale $a$, can lead to logarithmic contributions to the integrals in the limit $a \to 0$. In some cases, these singular contributions are lifted by the angular integration. The main theme of the calculation we present in the following is therefore to identify the poles that do give singular contributions. Once these poles have been identified, the integrals can be evaluated by shifting the integration variable such that the “dangerous” poles are at the origin $\mathbf{r} = 0$, and the corresponding integration has to be cut at $r = a$, while the remaining integrals can be extended over the entire plane.
### First order correction {#sec:first-order-corr}
We split the first order correction in Eq. into three contributions $$\label{eq:a_1}
\mathbf{a}_{\sigma}^{(1)}(\mathbf{r}) = \mathbf{a}^{(1)}_{\sigma, +}(\mathbf{r}) - 2
\mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r}) +
\mathbf{a}^{(1)}_{\sigma, -}(\mathbf{r}),$$ where (recall that for $\sigma = +$ the $\odot$ product should be replaced by the usual scalar product) $$\begin{aligned}
\mathbf{a}^{(1)}_{\sigma, \pm}(\mathbf{r})
& = \frac{1}{2 \pi}
\int_{\mathbf{r}'} \mathbf{f}(\mathbf{r} - \mathbf{r}')
\mathbf{f}(\mathbf{r}' - \mathbf{r}_{\pm})^{\odot 2}, \\
\label{eq:a^1_pm}
\mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r})
& = \frac{1}{2 \pi}
\int_{\mathbf{r}'} \mathbf{f}(\mathbf{r} - \mathbf{r}') \left(
\mathbf{f}(\mathbf{r}' - \mathbf{r}_+) \odot \mathbf{f}(\mathbf{r}' -
\mathbf{r}_-) \right).\end{aligned}$$ Let’s consider $\mathbf{a}^{(1)}_{\sigma, \pm}(\mathbf{r})$ first. Shifting the integration variable as $\mathbf{r}' \to \mathbf{r}' + \mathbf{r}_{\pm}$ and denoting $\mathbf{R}_{\pm} = \mathbf{r} - \mathbf{r}_{\pm}$, we obtain $$\mathbf{a}^{(1)}_{\sigma, \pm}(\mathbf{r}) = \frac{1}{2 \pi} \int_{\mathbf{r}'}
\mathbf{f}(\mathbf{R}_{\pm} - \mathbf{r}') \mathbf{f}(\mathbf{r}')^{\odot 2}.$$ This and many of the following integrals are conveniently performed using [ Mathematica]{}, resulting in $$\begin{aligned}
\label{eq:a^1_+_result}
\mathbf{a}^{(1)}_{+, \pm}(\mathbf{r}) & = \mathbf{f}(\mathbf{R}_{\pm})
\ln(R_{\pm}/a), \\
\label{eq:a^1_-_result}
\mathbf{a}^{(1)}_{-, \pm}(\mathbf{r}) & = \mathbf{e}(\mathbf{R}_{\pm}),\end{aligned}$$ where $$\begin{gathered}
\mathbf{e}(\mathbf{r}) = -\frac{\cos(\theta)
\sin(\theta)}{r}
\begin{pmatrix}
- \sin(\theta) \\ \cos(\theta)
\end{pmatrix}
\\ = - \cos(\theta) \sin(\theta) \frac{{\hat{\mathbf{z}}} \times \mathbf{r}}{r^2} = -
x y \frac{{\hat{\mathbf{z}}} \times \mathbf{r}}{r^4}.\end{gathered}$$ Note that these expressions are valid for $R_{\pm} > a$ and should be set to zero below the cutoff $a$.
The calculation of $\mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r})$ is more involved. In Eq. , we replace $\mathbf{r}' \to \mathbf{r}' + \mathbf{r}$ and as above we write $\mathbf{R}_{\pm} = \mathbf{r} - \mathbf{r}_{\pm}$, which yields $$\begin{split}
\mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r}) & = - \frac{1}{2 \pi}
\int_{\mathbf{r}'} \mathbf{f}(\mathbf{r}') \left(
\mathbf{f}(\mathbf{r}' + \mathbf{R}_+) \odot \mathbf{f}(\mathbf{r}' + \mathbf{R}_-) \right)
\\ & = \frac{1}{2 \pi} \int_{\mathbf{r}'} \frac{\mathbf{r}'}{r^{\prime 2}}
\frac{\left( \mathbf{r}' + \mathbf{R}_+ \right) \odot
\left( \mathbf{r}' + \mathbf{R}_- \right)}{{\left\lvert \mathbf{r}' + \mathbf{R}_+ \right\rvert}^2
{\left\lvert \mathbf{r}' + \mathbf{R}_- \right\rvert}^2}.
\end{split}$$ To evaluate these integrals, we switch to polar coordinates for $\mathbf{r}'$ and $\mathbf{R}_{\pm}$: $$\mathbf{r}' = r'
\begin{pmatrix}
\cos(\theta' + \theta_+) \\ \sin(\theta' + \theta_+)
\end{pmatrix}
, \quad
\mathbf{R}_{\pm} = R_{\pm}
\begin{pmatrix}
\cos(\theta_{\pm}) \\ \sin(\theta_{\pm})
\end{pmatrix}
.$$ Moreover, we use the following Fourier-cosine series: $$\begin{gathered}
\frac{1}{{\left\lvert \mathbf{r}' + \mathbf{R}_- \right\rvert}^2} = \frac{1}{{\left\lvert r^{\prime
2} - R_-^2 \right\rvert}} \sum_{n = 0}^{\infty} \left( 2 - \delta_{n,0} \right) \\
\times \left( - \frac{r_{<}}{r_{>}} \right)^n \cos(n (\theta' + \theta_+ -
\theta_-)),\end{gathered}$$ where $r_{<}$ and $r_{>}$ are the lesser and greater, respectively, of $r'$ and $R_-$. Finally, we set $$\frac{1}{{\left\lvert \mathbf{r}' + \mathbf{R}_+ \right\rvert}^2} = \frac{1}{r^{\prime 2} + R_+^2} \frac{1}{1 + s'
\cos(\theta')}, \quad s' = \frac{2 r' R_+}{r^{\prime 2} + R_+^2}.$$ Then, Eq. becomes $$\begin{gathered}
\mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r}) = \frac{1}{2 \pi} \sum_{n =
0}^{\infty} \left( 2 - \delta_{n, 0} \right) \int_{\mathbf{r}'}
\frac{1}{r^{\prime 2}} \frac{1}{r^{\prime 2} + R_+^2} \\ \times
\frac{1}{{\left\lvert r^{\prime 2} - R_-^2 \right\rvert}} \left( - \frac{r_{<}}{r_{>}} \right)^n
\cos(n (\theta' + \theta_+ - \theta_-)) \\ \times \mathbf{r}' \frac{\left(
\mathbf{r}' + \mathbf{R}_+ \right) \odot \left( \mathbf{r}' + \mathbf{R}_-
\right)}{1 + s' \cos(\theta')}.\end{gathered}$$ After some lengthy but straightforward algebra, the angular integrals can be preformed using the relation [@Gradshteyn2007] $$\label{eq:angular_int}
\int_0^{2 \pi} d \theta \frac{\cos(n \theta)}{1 + s \cos(\theta)} = \frac{2
\pi}{\sqrt{1 - s^2}} \left( \frac{\sqrt{1 - s^2} - 1}{s} \right)^n,$$ which holds for $s^2 < 1$ and $n \geq 0$. In the resulting expression, the summation over $n$ can be carried out, and finally performing the integral over $r'$ yields the result: $$\mathbf{a}^{(1)}_{+, +-}(\mathbf{r}) = - \frac{1}{2} \left(
\mathbf{f}(\mathbf{R}_+) \ln(R/R_-) +
\mathbf{f}(\mathbf{R}_-) \ln(R/R_+)
\right),$$ where $R$ is the magnitude of $\mathbf{R} = \mathbf{r}_+ - \mathbf{r}_-$. We omit the cumbersome expression for $\mathbf{a}^{(1)}_{-, +-}(\mathbf{r})$. Combining these results with Eqs. and gives the first order correction $\mathbf{E}^{(1)}(\mathbf{r})$. Again, we omit the rather lengthy expression.
The calculation of $\mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r})$ is actually much simpler for the special case $\mathbf{r} = \mathbf{r}_+$ that gives the electric field acting on the charge at $\mathbf{r}_+$. Then, shifting $\mathbf{r}' \to \mathbf{r} + \mathbf{r}_+$, Eq. becomes $$\mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r}_+)
= - \frac{1}{2 \pi}
\int_{\mathbf{r}} \mathbf{f}(\mathbf{r}) \left(
\mathbf{f}(\mathbf{r}) \odot \mathbf{f}(\mathbf{r} +
\mathbf{R}) \right),$$ where we used $\mathbf{f}(-\mathbf{r}) = - \mathbf{f}(\mathbf{r})$. Again, <span style="font-variant:small-caps;">Mathematica</span> does the job, and combining the result with Eqs. and we obtain $$\begin{aligned}
\label{eq:a^1_+_r_+}
\mathbf{a}_+^{(1)}(\mathbf{r}_+) & = \frac{1}{2} \mathbf{f}(\mathbf{R}) \left( 4 \ln(R/a) - 1
\right), \\
\label{eq:a^1_-_r_+}
\mathbf{a}_-^{(1)}(\mathbf{r}_+) & = \frac{3}{2} \mathbf{f}(\mathbf{R}) \cos(2
\theta_{\mathbf{R}}) - \frac{1}{R^2}
\begin{pmatrix}
R_x \\ - R_y
\end{pmatrix}
\left( \ln(R/a) - 1 \right),\end{aligned}$$ where we used the polar representation of $\mathbf{R}$, $$\mathbf{R} =
\begin{pmatrix}
R_x \\ R_y
\end{pmatrix}
= R
\begin{pmatrix}
\cos(\theta_{\mathbf{R}}) \\ \sin(\theta_{\mathbf{R}})
\end{pmatrix}.$$
Equations and give the first order corrections to the electric field at the position of the positive charge, $$\mathbf{E}_+^{(1)}(\mathbf{r}_+) = \frac{1}{2} {\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{R}) \left( 4 \ln(R/a) - 1
\right),$$ and $$\begin{gathered}
\mathbf{E}_-^{(1)}(\mathbf{r}_+) = \frac{3}{2} {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{R}) \cos(2 \theta_{\mathbf{R}}) \\ - \frac{1}{R^2}
\begin{pmatrix}
R_y \\ R_x
\end{pmatrix}
\left( \ln(R/a) - 1 \right).\end{gathered}$$
### Second order correction: diagonal terms {#sec:second-order-corr-diag}
For $\mathbf{r} = \mathbf{r}_+$, the second-order correction Eq. becomes $$\mathbf{a}_{\sigma}^{(2)}(\mathbf{r}_+) = - \frac{1}{\pi} \int_{\mathbf{r}} \mathbf{f}(\mathbf{r})
\left[ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r} + \mathbf{R})
\right) \odot \left( {\hat{\mathbf{z}}}
\times \mathbf{a}_{\sigma}^{(1)}(\mathbf{r} + \mathbf{r}_+) \right) \right].$$ We decompose the second order correction in two contributions, $$\mathbf{a}_{\sigma}^{(2)}(\mathbf{r}_+) = \mathbf{a}^{(2)}_{\sigma, 1}(\mathbf{r}_+) +
\mathbf{a}^{(2)}_{\sigma, 2}(\mathbf{r}_+),$$ where (cf. Eq. ) $$\label{eq:a^2_sigma_12}
\begin{split}
\mathbf{a}^{(2)}_{\sigma, 1}(\mathbf{r}_+) & = - \frac{1}{\pi} \int_{\mathbf{r}} \mathbf{f}(\mathbf{r})
\left\{ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r} + \mathbf{R})
\right) \odot \left[ {\hat{\mathbf{z}}}
\times \left( \mathbf{a}_{\sigma, +}^{(1)}(\mathbf{r} +
\mathbf{r}_+) + \mathbf{a}_{\sigma, -}^{(1)}(\mathbf{r} +
\mathbf{r}_+) \right) \right] \right\}, \\
\mathbf{a}^{(2)}_{\sigma, 2}(\mathbf{r}_+) & = \frac{2}{\pi} \int_{\mathbf{r}} \mathbf{f}(\mathbf{r})
\left[ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r} + \mathbf{R})
\right) \odot \left( {\hat{\mathbf{z}}}
\times \mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r} +
\mathbf{r}_+) \right) \right].
\end{split}$$ To proceed with the calculation of $\mathbf{a}_{\sigma, 1}^{(2)}(\mathbf{r}_+)$, we have to specify whether we are dealing with the isotropic or fully anisotropic case. Before going into that, let us simplify the expression for $\mathbf{a}^{(2)}_{\sigma, 2}(\mathbf{r}_+)$. Here, “simplifying” refers to splitting into two parts, $$\mathbf{a}^{(2)}_{\sigma, 2}(\mathbf{r}_+) = \mathbf{a}^{(2)}_{\sigma, 2, 1}(\mathbf{r}_+) +
\mathbf{a}^{(2)}_{\sigma, 2 ,2}(\mathbf{r}_+),$$ where $$\begin{aligned}
\label{eq:a^2_sigma_21_foo}
\mathbf{a}^{(2)}_{\sigma, 2, 1}(\mathbf{r}_+) & = \frac{2}{\pi} \int_{\mathbf{r}} \mathbf{f}(\mathbf{r})
\left[ \mathbf{f}(\mathbf{r}) \odot \left( {\hat{\mathbf{z}}}
\times \mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r} +
\mathbf{r}_+) \right) \right], \\
\label{eq:a^2_sigma_22_foo}
\mathbf{a}^{(2)}_{\sigma, 2, 2}(\mathbf{r}_+) & = - \frac{2}{\pi} \int_{\mathbf{r}} \mathbf{f}(\mathbf{r})
\left[ \mathbf{f}(\mathbf{r} + \mathbf{R})
\odot \left( {\hat{\mathbf{z}}}
\times \mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r} +
\mathbf{r}_+) \right) \right].\end{aligned}$$ Copying from Eq. , we find $$\label{eq:a^1_sigma_+-_rr_+}
\mathbf{a}^{(1)}_{\sigma, +-}(\mathbf{r} + \mathbf{r}_+) = \frac{1}{2 \pi}
\int_{\mathbf{r}'} \mathbf{f}(\mathbf{r} + \mathbf{r}_+ - \mathbf{r}')
\left( \mathbf{f}(\mathbf{r}' - \mathbf{r}_+) \odot \mathbf{f}(\mathbf{r}' -
\mathbf{r}_-) \right) = \frac{1}{2 \pi} \int_{\mathbf{r}'}
\mathbf{f}(\mathbf{r} - \mathbf{r}') \left(
\mathbf{f}(\mathbf{r}') \odot \mathbf{f}(\mathbf{r}' +
\mathbf{R}) \right).$$ Then, we can write Eq. as $$\label{eq:a^2_sigma_21}
\mathbf{a}^{(2)}_{\sigma, 2, 1}(\mathbf{r}_+) = \frac{1}{\pi^2}
\int_{\mathbf{r}, \mathbf{r}'} \mathbf{f}(\mathbf{r}) \left[
\mathbf{f}(\mathbf{r}) \odot \left( {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r} - \mathbf{r}') \right) \right] \left(
\mathbf{f}(\mathbf{r}') \odot \mathbf{f}(\mathbf{r}' + \mathbf{R}) \right)
= \frac{2}{\pi} \int_{\mathbf{r}'} \mathbf{c}_{\sigma}(\mathbf{r}') \left(
\mathbf{f}(\mathbf{r}') \odot \mathbf{f}(\mathbf{r}' + \mathbf{R})
\right),$$ where $$\label{eq:c_sigma}
\mathbf{c}_{\sigma}(\mathbf{r}') = \frac{1}{2 \pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r}) \odot
\left( {\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r} - \mathbf{r}') \right)
\right].$$ Finally, plugging Eq. into Eq. , the latter becomes $$\label{eq:a^2_sigma_22}
\mathbf{a}^{(2)}_{\sigma, 2, 2}(\mathbf{r}_+) = - \frac{1}{\pi^2} \int_{\mathbf{r},
\mathbf{r}'} \mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r} +
\mathbf{R}) \odot \left( {\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r} -
\mathbf{r}') \right) \right] \left( \mathbf{f}(\mathbf{r}')
\odot \mathbf{f}(\mathbf{r}' + \mathbf{R}) \right).$$ So far, we have split the second order correction into three contributions, given by Eqs. , , and . In the following, we calculate those, first for $\sigma = +$ and then for $\sigma = -$.
#### $\sigma = +$ {#sec:isotropic-case}
We start with $\mathbf{a}^{(2)}_{+, 1}(\mathbf{r}_+)$ defined in Eq. . Using Eq. we obtain $$\label{eq:a^1_++}
\mathbf{a}_{+,+}^{(1)}(\mathbf{r} + \mathbf{r}_+) + \mathbf{a}_{+,-}^{(1)}(\mathbf{r}
+ \mathbf{r}_+) = \mathbf{f}(\mathbf{r}) \ln(r/a) + \mathbf{f}(\mathbf{r} +
\mathbf{R}) \ln({\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}/a),$$ and inserting this relation in Eq. leaves us with $$\label{eq:a^2_+1}
\begin{split}
\mathbf{a}^{(2)}_{+, 1}(\mathbf{r}_+) & = \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left\{ {\hat{\mathbf{z}}} \cdot \left[ \left(
\mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r} + \mathbf{R}) \right)
\times \left( \mathbf{f}(\mathbf{r}) \ln(r/a) + \mathbf{f}(\mathbf{r} +
\mathbf{R}) \ln({\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}/a) \right) \right]
\right\} \\ & = \frac{1}{\pi} \int_{\mathbf{r}} \mathbf{f}(\mathbf{r})
\left[ {\hat{\mathbf{z}}} \cdot \left( \mathbf{f}(\mathbf{r}) \times
\mathbf{f}(\mathbf{r} + \mathbf{R}) \right) \right] \ln(r
{\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}/a^2) \\ & = \mathbf{b}_1(\mathbf{R}) +
\mathbf{b}_2(\mathbf{R}),
\end{split}$$ where $$\begin{aligned}
\mathbf{b}_1(\mathbf{R}) & = \frac{1}{\pi} \int_{\mathbf{r}} \mathbf{f}(\mathbf{r})
\left[ {\hat{\mathbf{z}}} \cdot \left( \mathbf{f}(\mathbf{r}) \times
\mathbf{f}(\mathbf{r} + \mathbf{R}) \right) \right]
\ln(r R /a^2), \\
\mathbf{b}_2(\mathbf{R}) & = \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r})
\left[ {\hat{\mathbf{z}}} \cdot \left( \mathbf{f}(\mathbf{r}) \times
\mathbf{f}(\mathbf{r} + \mathbf{R}) \right) \right]
\ln({\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}/R) =
\frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r} - \mathbf{R})
\left[ {\hat{\mathbf{z}}} \cdot \left(
\mathbf{f}(\mathbf{r} - \mathbf{R}) \times
\mathbf{f}(\mathbf{r}) \right) \right]
\ln(r/R).\end{aligned}$$ The results read as follows: $$\begin{aligned}
\mathbf{b}_1(\mathbf{R}) & = - \frac{1}{4} {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{R}) \left( 6 \ln(R/a)^2 + 4
\ln(R/a) + 1 \right), \\
\mathbf{b}_2(\mathbf{R}) & = 0,\end{aligned}$$ and hence we obtain for the first part of the second order correction: $$\mathbf{a}^{(2)}_{+, 1}(\mathbf{r}_+) = \mathbf{b}_1(\mathbf{R}).$$
We move on to calculate $\mathbf{a}^{(2)}_{+, 2, 1}(\mathbf{r}_+)$, given by Eq. , and find $$\mathbf{a}^{(2)}_{+, 2, 1}(\mathbf{r}_+) = - \frac{1}{2} {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{R}) \left( \ln(R/a)^2 - 1 \right).$$
The nastiest part by far is $\mathbf{a}^{(2)}_{+, 2, 2}(\mathbf{r}_+)$, which can be written as $$\begin{split}
\mathbf{a}^{(2)}_{+, 2, 2}(\mathbf{r}_+) & = - \frac{1}{\pi^2}
\int_{\mathbf{r}, \mathbf{r}'} \mathbf{f}(\mathbf{r}) \left[
\mathbf{f}(\mathbf{r} + \mathbf{R}) \cdot \left( {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r} - \mathbf{r}') \right) \right] \left(
\mathbf{f}(\mathbf{r}') \cdot \mathbf{f}(\mathbf{r}' + \mathbf{R}) \right)
\\ & = - \frac{1}{\pi^2} \int_{\mathbf{r}, \mathbf{r}'}
\mathbf{f}(\mathbf{r}) \left[ {\hat{\mathbf{z}}} \cdot \left( \mathbf{f}(\mathbf{r}
- \mathbf{r}') \times \mathbf{f}(\mathbf{r} + \mathbf{R}) \right)
\right] \left( \mathbf{f}(\mathbf{r}') \cdot \mathbf{f}(\mathbf{r}' +
\mathbf{R}) \right) \\ & = \frac{1}{\pi^2} \int_{\mathbf{r},
\mathbf{r}'} \frac{\mathbf{r}}{r^2} \frac{{\hat{\mathbf{z}}} \cdot \left[
\mathbf{r} \times \mathbf{R} - \mathbf{r}' \times \left( \mathbf{r} +
\mathbf{R} \right) \right]}{{\left\lvert \mathbf{r} - \mathbf{r}' \right\rvert}^2
{\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}^2} \frac{\mathbf{r}' \cdot \left(
\mathbf{r}' + \mathbf{R} \right)}{r^{\prime 2} {\left\lvert \mathbf{r}' +
\mathbf{R} \right\rvert}^2}.
\end{split}$$ To evaluate these integrals, we switch to polar coordinates for $\mathbf{r}, \mathbf{r}',$ and $\mathbf{R}$: $$\mathbf{r} = r
\begin{pmatrix}
\cos(\theta + \theta_{\mathbf{R}}) \\ \sin(\theta + \theta_{\mathbf{R}})
\end{pmatrix}
, \quad \mathbf{r}' = r'
\begin{pmatrix}
\cos(\theta' + \theta_{\mathbf{R}}) \\ \sin(\theta' + \theta_{\mathbf{R}})
\end{pmatrix}
, \quad \mathbf{R} = R
\begin{pmatrix}
\cos(\theta_{\mathbf{R}}) \\ \sin(\theta_{\mathbf{R}})
\end{pmatrix}.$$ Moreover, we use the following Fourier-cosine series: $$\label{eq:cos_r_rp}
\frac{1}{{\left\lvert \mathbf{r} - \mathbf{r}' \right\rvert}^2} = \frac{1}{{\left\lvert r^2 - r^{\prime
2} \right\rvert}} \sum_{n = 0}^{\infty} \left( 2 - \delta_{n,0} \right) \left(
\frac{r_{<}}{r_{>}} \right)^n \cos(n (\theta - \theta')),$$ where $r_{<}$ and $r_{>}$ are the lesser and greater, respectively, of $r$ and $r'$. Finally, we write $$\label{eq:s}
\frac{1}{{\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}^2} = \frac{1}{r^2 + R^2} \frac{1}{1 + s
\cos(\theta)}, \qquad s = \frac{2 r R}{r^2 + R^2},$$ and we use an analogous representation with $\mathbf{r}$ replaced by $\mathbf{r}'$. This leads us to $$\begin{gathered}
\mathbf{a}^{(2)}_{+, 2, 2}(\mathbf{r}_+) = \frac{1}{\pi^2} \sum_{n = 0}^{\infty}
\left( 2 - \delta_{n, 0} \right) \int_{\mathbf{r}, \mathbf{r}'} \frac{1}{r^2 +
R^2} \frac{1}{r^{\prime 2} + R^2} \frac{1}{{\left\lvert r^2 - r^{\prime 2} \right\rvert}} \left(
\frac{r_{<}}{r_{>}} \right)^n \cos(n (\theta - \theta')) \\ \times
\frac{\mathbf{r}}{r^2 r^{\prime 2}} \frac{{\hat{\mathbf{z}}} \cdot \left[ \mathbf{r}
\times \mathbf{R} - \mathbf{r}' \times \left( \mathbf{r} + \mathbf{R}
\right) \right]}{1 + s \cos(\theta)} \frac{\mathbf{r}' \cdot \left(
\mathbf{r}' + \mathbf{R} \right)}{1 + s' \cos(\theta')}.\end{gathered}$$ We then proceed to symmetrize the integrand with respect to $\theta \to -\theta$ and $\theta' \to -\theta'$, and to rearrange the trigonometric functions in the numerator such that the angular integrals can be preformed using Eq. . In the result, the summation over $n$ can be carried out straightforwardly. Performing the integrals over $r$ and $r'$ leads us to $$\mathbf{a}^{(2)}_{+, 2, 2}(\mathbf{r}_+) = 0.$$
Hence, the second order correction to the electric field at the position of the positive charge is given by $$\label{eq:E^2_+_r_+}
\mathbf{E}_+^{(2)}(\mathbf{r}_+) = {\hat{\mathbf{z}}} \times \mathbf{a}^{(2)}_{+, 1}(\mathbf{r}_+) +
{\hat{\mathbf{z}}} \times \mathbf{a}^{(2)}_{+, 2, 1}(\mathbf{r}_+) = \frac{1}{4} \mathbf{f}(\mathbf{R}) \left( 8
\ln(R/a)^2 + 4 \ln(R/a) - 1 \right).$$
#### $\sigma = -$ {#sec:fully-anis-case}
According to Eq. , $$\label{eq:z_cross_a^1}
\begin{split}
{\hat{\mathbf{z}}} \times \left( \mathbf{a}_{-,+}^{(1)}(\mathbf{r} + \mathbf{r}_+)
+ \mathbf{a}_{-, -}^{(1)}(\mathbf{r} + \mathbf{r}_+) \right) & =
{\hat{\mathbf{z}}} \times \left( \mathbf{e}(\mathbf{r}) + \mathbf{e}(\mathbf{r} +
\mathbf{R}) \right) \\ & = - \cos(\theta) \sin(\theta)
\mathbf{f}(\mathbf{r}) - \cos(\theta_{\mathbf{r} + \mathbf{R}})
\sin(\theta_{\mathbf{r} + \mathbf{R}}) \mathbf{f}(\mathbf{r} + \mathbf{R}),
\end{split}$$ where we used $${\hat{\mathbf{z}}} \times \mathbf{e}(\mathbf{r}) = \cos(\theta) \sin(\theta)
\frac{\mathbf{r}}{r^2} = - \cos(\theta) \sin(\theta)
\mathbf{f}(\mathbf{r}).$$ Inserting Eq. in Eq. , we obtain $$\begin{split}
\mathbf{a}^{(2)}_{-, 1}(\mathbf{r}_+) & = \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) -
\mathbf{f}(\mathbf{r} + \mathbf{R}) \right) \odot \left( \cos(\theta)
\sin(\theta) \mathbf{f}(\mathbf{r}) + \cos(\theta_{\mathbf{r} +
\mathbf{R}}) \sin(\theta_{\mathbf{r} +
\mathbf{R}}) \mathbf{f}(\mathbf{r} + \mathbf{R}) \right) \right] \\ &
= \mathbf{k}_1(\mathbf{R}) + \mathbf{k}_2(\mathbf{R}) +
\mathbf{k}_3(\mathbf{R}),
\end{split}$$ where $$\begin{aligned}
\mathbf{k}_1(\mathbf{R})
& = \frac{1}{\pi} \int_{\mathbf{r}} \cos(\theta) \sin(\theta)
\mathbf{f}(\mathbf{r}) \left[ \left(
\mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r} + \mathbf{R}) \right)
\odot \mathbf{f}(\mathbf{r}) \right], \\ \mathbf{k}_2(\mathbf{R})
& = - \frac{1}{\pi} \int_{\mathbf{r}} \cos(\theta_{\mathbf{r} + \mathbf{R}})
\sin(\theta_{\mathbf{r} + \mathbf{R}}) \mathbf{f}(\mathbf{r}) \left(
\mathbf{f}(\mathbf{r} + \mathbf{R}) \right)^{\odot 2}
\\
& = - \frac{1}{\pi} \int_{\mathbf{r}} \cos(\theta) \sin(\theta)
\mathbf{f}(\mathbf{r} - \mathbf{R}) \left( \mathbf{f}(\mathbf{r})
\right)^{\odot 2}, \\ \mathbf{k}_3(\mathbf{R})
& = \frac{1}{\pi} \int_{\mathbf{r}} \cos(\theta_{\mathbf{r} +
\mathbf{R}}) \sin(\theta_{\mathbf{r} + \mathbf{R}}) \mathbf{f}(\mathbf{r})
\left( \mathbf{f}(\mathbf{r}) \odot \mathbf{f}(\mathbf{r} + \mathbf{R})
\right).\end{aligned}$$ Both $\mathbf{k}_{1,2}(\mathbf{R})$ can be calculated directly as before; to simplify $\mathbf{k}_3(\mathbf{R})$, we parameterize $\mathbf{r}$ as $$\mathbf{r} = r
\begin{pmatrix}
\cos(\theta + \theta_{\mathbf{R}}) \\ \sin(\theta + \theta_{\mathbf{R}})
\end{pmatrix},$$ and use $$\mathbf{r} + \mathbf{R} = {\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}
\begin{pmatrix}
\cos(\theta_{\mathbf{r} + \mathbf{R}}) \\ \sin(\theta_{\mathbf{r} + \mathbf{R}})
\end{pmatrix}
= r
\begin{pmatrix}
\cos(\theta + \theta_{\mathbf{R}}) \\ \sin(\theta + \theta_{\mathbf{R}})
\end{pmatrix}
+ R
\begin{pmatrix}
\cos(\theta_{\mathbf{R}}) \\ \sin(\theta_{\mathbf{R}})
\end{pmatrix}.$$ Taking the product of the components of the last equation, we find $$\cos(\theta_{\mathbf{r} + \mathbf{R}}) \sin(\theta_{\mathbf{r} + \mathbf{R}})
= \frac{\left( r \cos(\theta + \theta_{\mathbf{R}}) + R
\cos(\theta_{\mathbf{R}}) \right) \left( r \sin(\theta +
\theta_{\mathbf{R}}) + R \sin(\theta_{\mathbf{R}})
\right)}{r^2 + R^2 + 2 r R \cos(\theta)}.$$ Using this relation, also $\mathbf{k}_3(\mathbf{R})$ can be calculated by <span style="font-variant:small-caps;">Mathematica</span>. We omit the cumbersome results for $\mathbf{k}_{1,2,3}$, and also the result for $\mathbf{a}^{(2)}_{+, 2, 1}(\mathbf{r}_+)$, given by Eq. .
It remains to calculate $\mathbf{a}^{(2)}_{-,2,2}(\mathbf{r}_+)$, $$\mathbf{a}^{(2)}_{-, 2, 2}(\mathbf{r}_+) = \frac{1}{\pi^2} \int_{\mathbf{r},
\mathbf{r}'} \frac{\mathbf{r}}{r^2} \frac{\left( \mathbf{r} + \mathbf{R}
\right) \odot \left[ {\hat{\mathbf{z}}} \times \left( \mathbf{r} - \mathbf{r}'
\right) \right]}{{\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}^2 {\left\lvert \mathbf{r} -
\mathbf{r}' \right\rvert}^2} \frac{\mathbf{r}' \odot \left( \mathbf{r}' + \mathbf{R}
\right)}{r^{\prime 2} {\left\lvert \mathbf{r}' + \mathbf{R} \right\rvert}^2}.$$ Using Eqs. and , this can be written as $$\begin{gathered}
\label{eq:a^2_-_22}
\mathbf{a}^{(2)}_{-, 2, 2}(\mathbf{r}_+) = \frac{1}{\pi^2} \sum_{n = 0}^{\infty}
\left( 2 - \delta_{n, 0} \right) \int_{\mathbf{r}, \mathbf{r}'} \frac{1}{r^2 +
R^2} \frac{1}{r^{\prime 2} + R^2} \frac{1}{{\left\lvert r^2 - r^{\prime 2} \right\rvert}} \left(
\frac{r_{<}}{r_{>}} \right)^n \cos(n (\theta - \theta')) \\ \times
\frac{\mathbf{r}}{r^2 r^{\prime 2}} \frac{\left( \mathbf{r} + \mathbf{R}
\right) \odot \left[ {\hat{\mathbf{z}}} \times \left( \mathbf{r} - \mathbf{r}'
\right) \right]}{1 + s \cos(\theta)} \frac{\mathbf{r}' \odot \left(
\mathbf{r}' + \mathbf{R} \right)}{1 + s' \cos(\theta')}.\end{gathered}$$ Repeating similar steps as above, we can get <span style="font-variant:small-caps;">Mathematica</span> to calculate the final result, which is surprisingly simple: $$\mathbf{a}^{(2)}_{-, 2, 2}(\mathbf{r}_+) = \frac{1}{2 R}
\begin{pmatrix}
\sin(3 \theta_{\mathbf{R}}) \\ \cos(3 \theta_{\mathbf{R}})
\end{pmatrix}.$$
For the anisotropic second order correction to the electric field we thus obtain $$\begin{split}
\mathbf{E}_-^{(2)}(\mathbf{r}_+) & = - \frac{1}{16} \left[
\mathbf{f}(\mathbf{R}) \left( 8 \ln(R/a)^2 - 20 \ln(R/a) + 15 - 8 \cos(4
\theta_{\mathbf{R}}) \right) - \frac{6}{R^2}
\begin{pmatrix}
R_x \\ - R_y
\end{pmatrix}
\cos(2 \theta_{\mathbf{R}}) \left( 4 \ln(R/a) - 5 \right) \right] \\ & \sim -
\frac{1}{2} \mathbf{f}(\mathbf{R}) \ln(R/a)^2,
\end{split}$$ where the asymptotic expansion corresponds to $R \to \infty$. Remarkably, the dominant contribution at large distances has the same form as in the isotropic case . In particular, it is central and potential.
### Second order correction: mixed terms {#sec:second-order-corr-mixed}
Setting $\mathbf{r} = \mathbf{r}_+$ in Eq. we obtain $$\label{eq:a^2_+-_foo}
\mathbf{a}_{+-}^{(2)}(\mathbf{r}_+) = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right)^T \left[ \left( {\hat{\mathbf{z}}} \times
\mathbf{a}_-^{(1)}(\mathbf{r} + \mathbf{r}_+) \right) + \sigma_z \left(
{\hat{\mathbf{z}}} \times \mathbf{a}_+^{(1)}(\mathbf{r} + \mathbf{r}_+) \right)
\right] = \mathbf{h}_1(\mathbf{R}) + \mathbf{h}_2(\mathbf{R}).$$ Here, we defined $$\begin{aligned}
\mathbf{h}_1(\mathbf{R}) & = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \cdot \left( {\hat{\mathbf{z}}} \times
\mathbf{a}_-^{(1)}(\mathbf{r} + \mathbf{r}_+)
\right) \right], \\
\mathbf{h}_2(\mathbf{R}) & = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \odot \left(
{\hat{\mathbf{z}}} \times \mathbf{a}_+^{(1)}(\mathbf{r} +
\mathbf{r}_+) \right) \right].\end{aligned}$$ Some further definitions: For $i = 1,2$ we set $$\mathbf{h}_i(\mathbf{R}) = \mathbf{h}_{i, 1} + \mathbf{h}_{i, 2},$$ where $$\begin{aligned}
\mathbf{h}_{1,1}(\mathbf{R}) & = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left\{ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \cdot \left[ {\hat{\mathbf{z}}} \times
\left( \mathbf{a}_{-, +}^{(1)}(\mathbf{r} +
\mathbf{r}_+) + \mathbf{a}_{-, -}^{(1)}(\mathbf{r} + \mathbf{r}_+)
\right) \right] \right\}, \\
\mathbf{h}_{1,2}(\mathbf{R}) & = \frac{2}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \cdot \left( {\hat{\mathbf{z}}} \times
\mathbf{a}_{-, +-}^{(1)}(\mathbf{r} +
\mathbf{r}_+) \right) \right], \\
\mathbf{h}_{2,1}(\mathbf{R}) & = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left\{ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \odot \left[ {\hat{\mathbf{z}}} \times
\left( \mathbf{a}_{+, +}^{(1)}(\mathbf{r} +
\mathbf{r}_+) + \mathbf{a}_{+, -}^{(1)}(\mathbf{r} + \mathbf{r}_+)
\right) \right] \right\}, \\
\mathbf{h}_{2,2}(\mathbf{R}) & = \frac{2}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \odot \left(
{\hat{\mathbf{z}}} \times \mathbf{a}_{+,
+-}^{(1)}(\mathbf{r} + \mathbf{r}_+) \right) \right].\end{aligned}$$
In $\mathbf{h}_{1,1}(\mathbf{R})$, we can use Eq. , which yields $$\begin{split}
\mathbf{h}_{1,1}(\mathbf{R}) & = \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) -
\mathbf{f}(\mathbf{r} + \mathbf{R}) \right) \cdot \left( \cos(\theta) \sin(\theta) \mathbf{f}(\mathbf{r}) +
\cos(\theta_{\mathbf{r} + \mathbf{R}}) \sin(\theta_{\mathbf{r} +
\mathbf{R}}) \mathbf{f}(\mathbf{r} + \mathbf{R}) \right) \right] \\
& = \mathbf{l}_1(\mathbf{R}) + \mathbf{l}_2(\mathbf{R}) + \mathbf{l}_3(\mathbf{R}),
\end{split}$$ where $$\begin{aligned}
\mathbf{l}_1(\mathbf{R}) & = \frac{1}{\pi} \int_{\mathbf{r}} \cos(\theta) \sin(\theta)
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) -
\mathbf{f}(\mathbf{r} + \mathbf{R}) \right) \cdot
\mathbf{f}(\mathbf{r}) \right], \\
\mathbf{l}_2(\mathbf{R}) & = - \frac{1}{\pi} \int_{\mathbf{r}}
\cos(\theta_{\mathbf{r} + \mathbf{R}}) \sin(\theta_{\mathbf{r} +
\mathbf{R}})
\mathbf{f}(\mathbf{r}) \left( \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right)^2 \\
& = - \frac{1}{\pi} \int_{\mathbf{r}}
\cos(\theta) \sin(\theta)
\mathbf{f}(\mathbf{r} - \mathbf{R}) \left( \mathbf{f}(\mathbf{r}) \right)^2, \\
\mathbf{l}_3(\mathbf{R}) & = \frac{1}{\pi} \int_{\mathbf{r}} \cos(\theta_{\mathbf{r} + \mathbf{R}}) \sin(\theta_{\mathbf{r} +
\mathbf{R}})
\mathbf{f}(\mathbf{r}) \left(
\mathbf{f}(\mathbf{r}) \cdot
\mathbf{f}(\mathbf{r} + \mathbf{R}) \right). \\\end{aligned}$$ These quantities are the same as $\mathbf{k}_{1,2,3}$ defined in the previous section up to the replacement of the $\odot$ product with the usual scalar product. The computation of the integrals goes along the same lines as above.
Next, we consider $\mathbf{h}_{1, 2}(\mathbf{R})$, which we split into two components, $$\mathbf{h}_{1,2}(\mathbf{R}) = \mathbf{h}_{1,2,1}(\mathbf{R}) + \mathbf{h}_{1,2,2}(\mathbf{R}),$$ where $$\begin{aligned}
\mathbf{h}_{1,2,1}(\mathbf{R}) & = \frac{2}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[
\mathbf{f}(\mathbf{r}) \cdot \left( {\hat{\mathbf{z}}} \times
\mathbf{a}_{-, +-}^{(1)}(\mathbf{r} +
\mathbf{r}_+) \right) \right], \\
\mathbf{h}_{1,2,2}(\mathbf{R}) & = - \frac{2}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \cdot \left( {\hat{\mathbf{z}}} \times
\mathbf{a}_{-, +-}^{(1)}(\mathbf{r} +
\mathbf{r}_+) \right) \right].\end{aligned}$$ In the first contribution, we use Eq. and find $$\mathbf{h}_{1,2,1}(\mathbf{R}) = \frac{1}{\pi^2} \int_{\mathbf{r},
\mathbf{r}'} \mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r}) \cdot
\left( {\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r} - \mathbf{r}') \right)
\right] \left( \mathbf{f}(\mathbf{r}') \odot \mathbf{f}(\mathbf{r}' +
\mathbf{R}) \right) = \frac{2}{\pi} \int_{\mathbf{r}'}
\mathbf{c}_+(\mathbf{r}') \left( \mathbf{f}(\mathbf{r}') \odot
\mathbf{f}(\mathbf{r}' + \mathbf{R}) \right),$$ where $\mathbf{c}_+(\mathbf{r}')$ is defined in Eq. .
We move on to $\mathbf{h}_{1,2,2}(\mathbf{R})$, given by $$\mathbf{h}_{1,2,2}(\mathbf{R}) = - \frac{1}{\pi^2} \int_{\mathbf{r},
\mathbf{r}'} \mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r} +
\mathbf{R}) \cdot \left( {\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r} -
\mathbf{r}') \right) \right] \left( \mathbf{f}(\mathbf{r}') \odot
\mathbf{f}(\mathbf{r}' + \mathbf{R}) \right).$$ Comparison with Eq. shows that $\mathbf{h}_{1,2,2}(\mathbf{R})$ is given by Eq. with the first of the $\odot$ products replaced by the usual scalar product, which yields $$\mathbf{h}_{1,2,2}(\mathbf{R}) = \frac{1}{2 R}
\begin{pmatrix}
-\sin(3 \theta_{\mathbf{R}}) \\ \cos(3 \theta_{\mathbf{R}})
\end{pmatrix}.$$
The next on the list is $\mathbf{h}_{2,1}(\mathbf{R})$, which we write — using Eq. — as $$\mathbf{h}_{2,1}(\mathbf{R}) = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left\{ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \odot \left[ {\hat{\mathbf{z}}} \times
\left( \mathbf{f}(\mathbf{r}) \ln(r/a) + \mathbf{f}(\mathbf{r} +
\mathbf{R}) \ln({\left\lvert \mathbf{r} + \mathbf{R} \right\rvert}/a)
\right) \right] \right\}.$$ Unfortunately, this can not be simplified as we did above in Eq. because in general $\mathbf{a} \odot \left( {\hat{\mathbf{z}}} \times \mathbf{a} \right) \neq 0$. Hence, we have to invent something new: $$\mathbf{h}_{2,1}(\mathbf{R}) = \mathbf{m}_1(\mathbf{R}) +
\mathbf{m}_2(\mathbf{R}) + \mathbf{m}_3(\mathbf{R}) + \mathbf{m}_4(\mathbf{R}),$$ where $$\begin{aligned}
\mathbf{m}_1(\mathbf{R}) & = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[
\mathbf{f}(\mathbf{r}) \odot \left( {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r}) \right) \right] \ln(r/a), \\
\mathbf{m}_2(\mathbf{R}) & = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \odot \left( {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r} +
\mathbf{R})
\right) \right] \ln({\left\lvert \mathbf{r} +
\mathbf{R} \right\rvert}/R) \\
& = - \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r} - \mathbf{R}) \left[ \left(
\mathbf{f}(\mathbf{r} - \mathbf{R}) - \mathbf{f}(\mathbf{r}) \right) \odot \left( {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r})
\right) \right] \ln(r/R), \\
\label{eq:m3}
\mathbf{m}_3(\mathbf{R}) & = - \frac{1}{\pi} \ln(R/a) \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \left( \mathbf{f}(\mathbf{r}) - \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \right) \odot \left( {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r} +
\mathbf{R})
\right) \right], \\
\mathbf{m}_4(\mathbf{R}) & = \frac{1}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \odot \left( {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r}) \right) \right] \ln(r/a).\end{aligned}$$ $\mathbf{m}_1(\mathbf{R})$ vanishes because the integrand is antisymmetric under reflections $\mathbf{r} \to -\mathbf{r}$.
Now comes $\mathbf{h}_{2,2}(\mathbf{R})$, which again consists of two contributions, $$\mathbf{h}_{2,2}(\mathbf{R}) = \mathbf{h}_{2,2,1}(\mathbf{R}) + \mathbf{h}_{2,2,2}(\mathbf{R}),$$ where $$\begin{aligned}
\mathbf{h}_{2,2,1}(\mathbf{R}) & = \frac{2}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[
\mathbf{f}(\mathbf{r}) \odot \left(
{\hat{\mathbf{z}}} \times \mathbf{a}_{+,
+-}^{(1)}(\mathbf{r} + \mathbf{r}_+) \right)
\right], \\
\mathbf{h}_{2,2,2}(\mathbf{R}) & = - \frac{2}{\pi} \int_{\mathbf{r}}
\mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \odot \left(
{\hat{\mathbf{z}}} \times \mathbf{a}_{+,
+-}^{(1)}(\mathbf{r} + \mathbf{r}_+) \right) \right].\end{aligned}$$ In the first contribution, we use Eq. and find $$\begin{split}
\mathbf{h}_{2,2,1}(\mathbf{R}) & = \frac{1}{\pi^2} \int_{\mathbf{r},
\mathbf{r}'} \mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r}) \odot
\left( {\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r} - \mathbf{r}') \right)
\right] \left( \mathbf{f}(\mathbf{r}') \cdot \mathbf{f}(\mathbf{r}' +
\mathbf{R}) \right) \\ & = \frac{2}{\pi} \int_{\mathbf{r}'}
\mathbf{c}_-(\mathbf{r}') \left( \mathbf{f}(\mathbf{r}') \cdot
\mathbf{f}(\mathbf{r}' + \mathbf{R}) \right),
\end{split}$$ where $\mathbf{c}_-(\mathbf{r}')$ is defined in Eq. .
Finally, we consider $$\mathbf{h}_{2,2,2}(\mathbf{R}) = - \frac{1}{\pi^2} \int_{\mathbf{r}, \mathbf{r}'}
\mathbf{f}(\mathbf{r}) \left[ \mathbf{f}(\mathbf{r}
+ \mathbf{R}) \odot \left(
{\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r} - \mathbf{r}') \right) \right]
\left( \mathbf{f}(\mathbf{r}' \cdot \mathbf{f}(\mathbf{r}' + \mathbf{R})
\right).$$ Comparing this with Eq. shows that $\mathbf{h}_{2,2,2}(\mathbf{R})$ is given by Eq. with the second of the $\odot$ products replaced by the usual scalar product. We find $$\mathbf{h}_{2,2,2}(\mathbf{R}) = 0.$$
Combining all of the above results (many of which we haven’t stated for brevity), the mixed second order correction to the electric field is thus given by $$\mathbf{E}^{(2)}_{+-}(\mathbf{r}_+) = \frac{1}{16} \left[
2 \mathbf{f}(\mathbf{R}) \cos(2 \theta_{\mathbf{R}}) \left( 28 \ln(R/a) + 3 \right) + \frac{1}{R^2}
\begin{pmatrix}
R_x \\ - R_y
\end{pmatrix}
\left( 8 \ln(R/a)^2 + 12 \ln(R/a) - 5 \right)
\right].$$
### Numerical checks {#sec:numerical-checks}
To check the above calculations, we evaluated Eqs. , , and numerically. The results for some sample parameter values are shown in Figs. \[fig:E1\_numerics\], \[fig:E21\_numerics\], and \[fig:E22\_numerics\]. We find agreement between analytics and numerics up to convergence problems of the numerical integration for select values of $\mathbf{r}_+$ (the position of the positive charge). The slight discrepancy between results for $\mathbf{E}^{(2)}_{+-}(\mathbf{r}_+)$ and, consequently, $\mathbf{E}^{(2)}(\mathbf{r}_+)$ shown in Figs. \[fig:E21\_numerics\] and \[fig:E22\_numerics\], respectively, can be traced back to particularly slow convergence of the numerical integration for integrals of the type of Eq. .
{width="\linewidth"} {width="\linewidth"}
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Dynamics of vortices in the compact anisotropic KPZ equation {#sec:dynam-topol-defects}
============================================================
Having found the electric field acting upon the charges (i.e., vortices) constituting a dipole, we proceed to study the motion of the charges under the influence of Markovian noise. In particular, we are interested in the stationary distribution, which forms the basis of the RG treatment below. The noise acting on the charges originates from the one in the original caKPZ equation, i.e, Eq. (1) in the main text. Hence, the strengths of the noise sources in the caKPZ equation and the equation of motion of vortices are related, but will be renormalized differently [@Sieberer2016b; @Wachtel2016].
Equations of motion {#sec:equations-motion}
-------------------
As above, we consider a dipole consisting of a positive charge (vortex) at $\mathbf{r}_+$, and a negative charge (antivortex) at $\mathbf{r}_-$. The equations of motion for the vortices read $$\label{eq:r_pm_eom}
\frac{d \mathbf{r}_{\sigma}}{d t} = \sigma \mu \mathbf{E}(\mathbf{r}_{\sigma}) +
\boldsymbol{\xi}_{\sigma},$$ where $\sigma = \pm$, cf. Eq. (6) in the main text. As discussed there, the noise correlations read $\langle \xi_{\sigma, i}(t) \xi_{\sigma', j}(t') \rangle = 2 \mu T
\delta_{\sigma \sigma'} \delta_{i j} \delta(t - t'),$ where $\sigma, \sigma' = \pm$, and $\mu$ is the vortex mobility. Thus, the relative and “center-of-mass” coordinates, $\mathbf{r} = \mathbf{r}_+ - \mathbf{r}_-$ and $\mathbf{R} = (\mathbf{r}_+ + \mathbf{r}_-)/2$, respectively, obey the following equations of motion: $$\begin{aligned}
\label{eq:r_eom}
\frac{d \mathbf{r}}{d t} & = \mu \left( \mathbf{E}(\mathbf{r}_+) +
\mathbf{E}(\mathbf{r}_-) \right) + \boldsymbol{\xi}, \\
\label{eq:R_eom}
\frac{d \mathbf{R}}{d t} & = \frac{\mu}{2} \left( \mathbf{E}(\mathbf{r}_+) -
\mathbf{E}(\mathbf{r}_-) \right) + \boldsymbol{\Xi}, \end{aligned}$$ where $\boldsymbol{\xi} = \boldsymbol{\xi}_+ - \boldsymbol{\xi}_-$ and $\boldsymbol{\Xi} = (\boldsymbol{\xi}_+ + \boldsymbol{\xi}_-)/2$. (To avoid confusion, we note that in the above calculation of the electric field we denoted the relative coordinate by $\mathbf{R}$.) The correlations of the noise acting on the relative coordinate read $$\langle \xi_i(t) \xi_j(t') \rangle = \sum_{\sigma, \sigma'} \sigma \sigma'
\langle \xi_{\sigma, i}(t) \xi_{\sigma', j}(t') \rangle = 4 \mu T
\delta_{ij} \delta(t - t').$$ As we show in the following, the relative coordinate is only affected by the zeroth and second order contributions to the electric field, whereas the first order corrections induces motion of the center of mass.
To this end, for the bare Coulomb interaction, from Eqs. and we find $\mathbf{E}^{(0)}(\mathbf{r}_+) = \mathbf{f}(\mathbf{r}_+ - \mathbf{r}_-)$ (recall that $\mathbf{f}(0) = 0$ due to the short-distance cutoff) and hence $$\mathbf{E}^{(0)}(\mathbf{r}_-) = -
\mathbf{f}(\mathbf{r}_- - \mathbf{r}_+) =
\mathbf{f}(\mathbf{r}_+ - \mathbf{r}_-) = \mathbf{E}^{(0)}(\mathbf{r}_+),$$ where we used that $\mathbf{f}(-\mathbf{r}) = - \mathbf{f}(\mathbf{r})$ as can be seen from Eq. . Thus, the Coulomb interaction enters Eq. with a factor of two but drops out of the difference in Eq. .
The first order correction to the electric field is given by (cf. Eqs. and ; recall that the $\odot$ product becomes the usual scalar product for $\sigma = +$) $$\mathbf{E}_{\sigma}^{(1)}(\mathbf{r}_-)
= \frac{1}{2 \pi} \int_{\mathbf{r}'} {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r}_- - \mathbf{r}') \left( \mathbf{f}(\mathbf{r}' -
\mathbf{r}_+) - \mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right)^{\odot 2}.$$ With a change of the integration variable according to $\mathbf{r}' \to
-\mathbf{r}' + \mathbf{r}_+ + \mathbf{r}_-$, this can be written as $$\mathbf{E}_{\sigma}^{(1)}(\mathbf{r}_-)
= - \frac{1}{2 \pi} \int_{\mathbf{r}'} {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r}_+ - \mathbf{r}') \left( \mathbf{f}(\mathbf{r}' -
\mathbf{r}_+) - \mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right)^{\odot 2} = -
\mathbf{E}_{\sigma}^{(1)}(\mathbf{r}_+).$$ As stated above, we see the first order correction contributes to Eq. but not to .
Finally, let us consider the second order contributions, and here first the diagonal parts. According to Eqs. and it is given by $$\begin{split}
\mathbf{E}^{(2)}_{\sigma}(\mathbf{r}_-) & = \frac{1}{\pi} \int_{\mathbf{r}'}
{\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r}_- - \mathbf{r}') \left[ \left(
\mathbf{f}(\mathbf{r}' - \mathbf{r}_+) - \mathbf{f}(\mathbf{r}' -
\mathbf{r}_-) \right) \odot \left( {\hat{\mathbf{z}}} \times
\mathbf{a}_{\sigma}^{(1)}(\mathbf{r}') \right) \right] \\
& = - \frac{1}{\pi} \int_{\mathbf{r}'} {\hat{\mathbf{z}}} \times
\mathbf{f}(\mathbf{r}_+ - \mathbf{r}') \left[ \left( \mathbf{f}(\mathbf{r}' -
\mathbf{r}_+) - \mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right) \odot
\left( {\hat{\mathbf{z}}} \times \mathbf{a}_{\sigma}^{(1)}(-\mathbf{r}' +
\mathbf{r}_+ + \mathbf{r}_-) \right)
\right],
\end{split}$$ where in the second equality we performed the same change of the integration variable $\mathbf{r}'$ as above. Then, from Eq. , $$\mathbf{a}^{(1)}_{\sigma}(-\mathbf{r} + \mathbf{r}_+ + \mathbf{r}_-) =
\frac{1}{2 \pi} \int_{\mathbf{r}'} \mathbf{f}(- \mathbf{r} + \mathbf{r}_+ +
\mathbf{r}_- - \mathbf{r}') \left( \mathbf{f}(\mathbf{r}' - \mathbf{r}_+) -
\mathbf{f}(\mathbf{r}' - \mathbf{r}_-) \right)^{\odot 2} = -
\mathbf{a}^{(1)}_{\sigma}(\mathbf{r}),$$ and hence $\mathbf{E}^{(2)}_{\sigma}(\mathbf{r}_-) =
\mathbf{E}^{(2)}_{\sigma}(\mathbf{r}_+)$. Along the same lines, starting from Eqs. and it is straightforward to see that $\mathbf{E}^{(2)}_{+-}(\mathbf{r}_-) =
\mathbf{E}^{(2)}_{+-}(\mathbf{r}_+)$. Thus, we find $$\begin{aligned}
\label{eq:r_eom_1}
\frac{d \mathbf{r}}{d t} & = 2 \mu \left( \frac{1}{\varepsilon} \mathbf{E}^{(0)}(\mathbf{r}_+) +
\frac{1}{\varepsilon^3} \mathbf{E}^{(2)}(\mathbf{r}_+) \right) +
\boldsymbol{\xi}, \\ \label{eq:R_eom_1}
\frac{d \mathbf{R}}{d t} & = \frac{\mu}{\varepsilon^2}
\mathbf{E}^{(1)}(\mathbf{r}_+) + \boldsymbol{\Xi},\end{aligned}$$ where $$\label{eq:E^1_E^2}
\mathbf{E}^{(1)}(\mathbf{r}) = - \sum_{\sigma = \pm} \sigma \alpha_{\sigma}
\mathbf{E}_{\sigma}^{(1)}, \qquad
\mathbf{E}^{(2)} = - \sum_{\sigma = \pm} \alpha_{\sigma}^2 \mathbf{E}_{\sigma}^{(2)} + \alpha_+
\alpha_- \mathbf{E}_{+-}^{(2)}.$$ For convenience, we list again the various contributions to the electric field obtained in the previous section: $$\begin{split}
\mathbf{E}^{(0)}(\mathbf{r}_+) & = \mathbf{f}(\mathbf{r}) = -
\frac{\mathbf{r}}{r^2}, \\
\mathbf{E}_+^{(1)}(\mathbf{r}_+) & = \frac{1}{2} {\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r}) \left( 4 \ln(r/a) - 1
\right), \\
\mathbf{E}_-^{(1)}(\mathbf{r}_+) & = \frac{3}{2} {\hat{\mathbf{z}}} \times \mathbf{f}(\mathbf{r}) \cos(2
\theta) - \frac{1}{r^2}
\begin{pmatrix}
y \\ x
\end{pmatrix}
\left( \ln(r/a) - 1 \right), \\
\mathbf{E}_+^{(2)}(\mathbf{r}_+) & = \frac{1}{4} \mathbf{f}(\mathbf{r}) \left( 8
\ln(r/a)^2 + 4 \ln(r/a) - 1 \right), \\
\mathbf{E}_-^{(2)}(\mathbf{r}_+) & = - \frac{1}{16} \left[
\mathbf{f}(\mathbf{r}) \left( 8 \ln(r/a)^2 - 20 \ln(r/a) + 15 - 8 \cos(4
\theta) \right) - \frac{6}{r^2}
\begin{pmatrix}
x \\ - y
\end{pmatrix}
\cos(2 \theta) \left( 4 \ln(r/a) - 5 \right) \right], \\
\mathbf{E}^{(2)}_{+-}(\mathbf{r}_+) & = \frac{1}{16} \left[
2 \mathbf{f}(\mathbf{r}) \cos(2 \theta) \left( 28 \ln(r/a) + 3 \right) + \frac{1}{r^2}
\begin{pmatrix}
x \\ - y
\end{pmatrix}
\left( 8 \ln(r/a)^2 + 12 \ln(r/a) - 5 \right)
\right],
\end{split}$$
where $\mathbf{r} = \mathbf{r}_+ - \mathbf{r}_- = \left( x, y \right) = r \left(
\cos(\theta), \sin(\theta) \right)$ is the dipole moment.
Let’s consider first the “isotropic” corrections, $\mathbf{E}^{(1)}_+$ and $\mathbf{E}^{(2)}_{++}$, which were previously obtained in Ref. [@Wachtel2016]. We note that the first-order correction is perpendicular to the dipole moment and causes motion of the center of mass in this direction, while the second-order term is a central force and adds to the Coulomb force $\mathbf{E}^{(0)}$ affecting the relative motion. Both $\mathbf{E}^{(0)}$ and $\mathbf{E}^{(2)}_{++}$ can be derived from a potential by taking the derivative with respect to the relative coordinate $\mathbf{r}$. The “anisotropic” and “mixed” second order corrections, $\mathbf{E}^{(2)}_{--}$ and $\mathbf{E}^{(2)}_{+-}$, on the other hand, cannot be derived from a potential. In addition to the central contributions $\propto \mathbf{r}$, they include terms $\propto \left( x, - y \right)$ that favor alignment of the dipole along the $x$ or $y$-axis (depending on the signs of $\alpha_+$ and $\alpha_-$). As mentioned in the main text, this is in line with numerical simulations of the anisotropic complex Ginzburg-Landau equation [@Faller1998; @ROLANDFALLERandLORENZKRAMER1999].
All types of contributions have the common structure of being power series in logarithms. For this reason, perturbation theory is valid up to the scale $L_v \sim a e^{1/\alpha_{\mathrm{max}}}$ where $\alpha_{\mathrm{max}} = \max \{ {\left\lvert \alpha_{\pm} \right\rvert} \}$. For distances $r$ which are much larger than the microscopic cutoff but below $L_v$, $a \ll r \ll L_v$, the second order corrections are dominated by the leading powers of logarithms, $\ln(r/a)^2$. Remarkably, for both $\mathbf{E}^{(2)}_{++}$ and $\mathbf{E}^{(2)}_{--}$ these contributions take the same form, i.e., they are centrally symmetric and potential — in spite of $\mathbf{E}^{(2)}_{--}$ originating from a fully anisotropic non-linearity in the caKPZ equation. The crucial difference between the “isotropic” and “anisotropic” (according to their origin) second order contributions is that the former gives a repulsive correction to the Coulomb force, while the latter gives an *attractive* one.
Stationary distribution of a dipole {#sec:stat-distr}
-----------------------------------
We proceed to derive the stationary distribution of a dipole subject to the Langevin equation . The associated Fokker-Planck equation reads [@Kamenev2011] $$\label{eq:FP}
\partial_t \mathcal{P} = - 2 \mu \nabla \cdot \left( \mathbf{F}
\mathcal{P} - T \nabla \mathcal{P} \right),$$ with the drift generated by the electric field: $$\mathbf{F}(\mathbf{r}) = \mathbf{F}^{(0)}(\mathbf{r}) +
\mathbf{F}^{(2)}(\mathbf{r}) = \frac{1}{\varepsilon}
\mathbf{E}^{(0)}(\mathbf{r}_+) + \frac{1}{\varepsilon^3}
\mathbf{E}^{(2)}(\mathbf{r}_+).$$ All the physics below the microscopic cutoff scale $a$ is contained in a single phenomenological parameter, the vortex fugacity $y$, which quantifies the probability of finding a dipole at the separation $a$ and thus sets the boundary condition for the stationary distribution of the dipole, $\mathcal{P}(\mathbf{r}) = y^2$ for $r = a$. We seek a steady-state solution of Eq. in the form $$\label{eq:P}
\mathcal{P}(\mathbf{r}) \sim y^2 e^{- \Phi(\mathbf{r})/T}.$$ This is the exact form of the solution in thermal equilibrium, when $\nabla \Phi = \mathbf{F}^{(0)}$ and hence $$\Phi(\mathbf{r}) = \Phi^{(0)}(\mathbf{r}) = \left( 1/\varepsilon \right)
\ln(r/a).$$ Out of equilibrium, the ansatz yields the leading behavior at low noise strengths [@Kamenev2011]. Inserting this ansatz in the Fokker-Planck equation and expanding the potential as $\Phi = \Phi^{(0)} + \Phi^{(2)}$, where $\Phi^{(0)}$ is the equilibrium solution, results for $T \ll 1$ in $$\mathbf{F}^{(0)} \cdot \left( \mathbf{F}^{(2)} + \nabla \Phi^{(2)} \right) = 0.$$ In order to solve this partial differential equation for $\Phi^{(2)}$ we apply the method of characteristics, which yields the following system of ordinary differential equations: $$\begin{split}
\frac{d \mathbf{r}}{d t} & = \mathbf{F}^{(0)}, \\
\frac{d \Phi^{(2)}}{d t} & = - \mathbf{F}^{(0)} \cdot \mathbf{F}^{(2)}.
\end{split}$$ The integral curves of the first equation flow upstream against the equilibrium part of the drift field, i.e., they are the activation trajectories of the unperturbed equilibrium problem. Integrating the second equation and inserting the first one for $\mathbf{F}^{(0)}$ we get $$\Phi^{(2)}(\mathbf{r}) = - \int_{t_0}^t d t' \frac{d \mathbf{r}}{d t} \cdot \mathbf{F}^{(2)} = -
\int_{a {\hat{\mathbf{r}}}}^{\mathbf{r}} d \mathbf{s} \cdot \mathbf{F}^{(2)},$$ where the line integral has to be taken along the activation trajectory of the equilibrium problem that connects $\mathbf{r}(t_0) = a {\hat{\mathbf{r}}}$ and $\mathbf{r}$. The initial condition at $t_0$ is chosen to ensure $\Phi^{(2)}(\mathbf{r}) = 0$ and thus $\mathcal{P}(\mathbf{r}) = y^2$ for $r = a$. We thus find
$$\begin{gathered}
\label{eq:Phi^2}
\Phi^{(2)}(\mathbf{r}) = - \frac{1}{\varepsilon^3} \left\{ \frac{1}{3} \left(
2 \alpha_+^2 - \frac{\alpha_-^2}{2} + \frac{\alpha_+ \alpha_-}{2} \cos(2
\theta) \right) \ln(r/a)^3 \right. \\ + \frac{1}{2} \left[ \alpha_+^2 +
\frac{\alpha_-^2}{2} \left( 1 - \frac{3}{2} \cos(4 \theta) \right) -
\frac{11 \alpha_+ \alpha_-}{4} \cos(2 \theta) \right] \ln(r/a)^2 \\ \left. -
\frac{1}{4} \left( \alpha_+^2 - \frac{23 \alpha_-^2}{4} \cos(4 \theta) +
\frac{11 \alpha_+ \alpha_-}{4} \cos(2 \theta) \right) \ln(r/a) \right\}.\end{gathered}$$
RG flow {#sec:rg-flow}
=======
The “macroscopic” electrodynamics of the previous sections captures the screening of the electric field due to bound vortex-antivortex pairs by introducing the dielectric constant $\varepsilon$. The latter describes the response of the dielectric medium of bound pairs to an electric field. This definition as a response leads to an implicit equation for $\varepsilon$, since the polarization of a single test dipole due to an external electric field is determined by the balance between the external field and the screened Coulomb interaction between the charges constituting the dipole — with the screening in turn determined by $\varepsilon$. The resulting implicit equation for $\varepsilon$ can be solved by a renormalization group (RG) approach described in the following. Section \[sec:derivation-rg-flow\] describes the derivation of RG flow equations, which generalizes the one given in Ref. [@Wachtel2016] for the isotropic case. We study the phases and fixed points of the RG flow in Sec. \[sec:phases-fixed-point\]. The rather peculiar divergence of the correlation length at the critical point is discussed in Sec. \[sec:asympt-analys-RG\].
Derivation of the RG flow equations {#sec:derivation-rg-flow}
-----------------------------------
As outlined above, in the following we derive an implicit equation for the dielectric constant $\varepsilon$ by calculating the polarization of a single test dipole that is induced in linear order by an external electric field. This implicit equation is the starting point from which we obtain a system of RG flow equations.
Adding an external electric field $\mathbf{E}_{\mathrm{eff}}$ in the equations of motion modifies the potential as $\Phi(\mathbf{r}) \to \Phi(\mathbf{r}) - \mathbf{E}_{\mathrm{ext}} \cdot
\mathbf{r}$. To first order in the external field, the resulting average polarization of our test dipole is given by
$$\label{eq:polarization}
\left\langle \mathbf{P} \right\rangle = \frac{1}{L^2} \int \frac{d^2
\mathbf{R}}{a^2} \frac{d^2 \mathbf{r}}{a^2} \mathbf{r}
\mathcal{P}(\mathbf{r}) \left( 1 + \frac{1}{T} \mathbf{E}_{\mathrm{ext}}
\cdot \mathbf{r} \right) = \frac{1}{T} \int \frac{d^2
\mathbf{r}}{a^2} \mathcal{P}(\mathbf{r}) \frac{\mathbf{r}
\mathbf{r}^T}{a^2} \mathbf{E}_{\mathrm{ext}} = \chi
\mathbf{E}_{\mathrm{ext}},$$
which defines the scusceptibility tensor $\chi$. In an isotropic system, when $\mathcal{P}(\mathbf{r})$ does not depend on the direction, the susceptibility $\chi \propto \int d^2 \mathbf{r} \, \mathcal{P}(\mathbf{r}) \mathbf{r}
\mathbf{r}^T$ is proportional to the identity matrix, which can be seen by noting that $\mathcal{P}(\mathbf{r})$ is symmetric under (each of the transformations) $x \to -x, y \to -y,$ and $x \leftrightarrow y$. As can be seen in Eq. , the anisotropy we consider here leaves the reflection symmetries under $x \to -x$ and $y \to -y$ intact, but breaks the symmetry under the exchange $x \leftrightarrow y$ (note that $\cos(2 \theta) = (x^2 - y^2)/r^2$). Hence, the susceptibility tensor $\chi$ is still diagonal, $$\label{eq:chi_tensor}
\chi =
\begin{pmatrix}
\chi_x & 0 \\
0 & \chi_y
\end{pmatrix},$$ but in general its eigenvalues are distinct, $\chi_x \neq \chi_y$. More specifically, we find $$\label{eq:chi_xy}
\begin{pmatrix}
\chi_x \\ \chi_y
\end{pmatrix}
= \frac{y^2}{T} \int_a^{\infty} \frac{d r}{a} \frac{r^3}{a^3} e^{-\Phi^{(0)}(r)/T} \int_0^{2 \pi} d \theta
\begin{pmatrix}
\cos(\theta)^2 \\
\sin(\theta)^2
\end{pmatrix}
e^{-\Phi^{(2)}(\mathbf{r})/T}.$$ We note that strictly speaking the integral over $r$ should be cut at the scale $L_v$ at which the perturbative expansion of the vortex interaction and hence the potential $\Phi(\mathbf{r})$ breaks down. This is assumed implicitly in the following. To make progress with the expression for $\chi$, we expand the last exponential in Eq. . This is justified up to parametrically large distances, for which $$\label{eq:L_T}
\frac{\alpha_{\sigma} \alpha_{\sigma'}}{T} \ln(r/a)^3 \ll 1 \quad
\Longleftrightarrow \quad r \ll L_T = a e^{\left[ T/(\alpha_{\sigma}
\alpha_{\sigma'}) \right]^{1/3}}.$$ where we use the bare value $\varepsilon = 1$ to estimate $L_T$. We note that for distances below $L_T$, the temperature is always “high” with regard to the terms $\propto \alpha_{\sigma} \alpha_{\sigma'}$, which means that by noise-induced fluctuations the test dipole can explore all possible orientations. Only at much larger distances $r \gg L_T$ the test dipole is essentially restricted to the direction that minimizes $\Phi^{(2)}(\mathbf{r})$ with strongly suppressed fluctuations around this direction. Thus, we find ($i = x,y$) $$\label{eq:chi_i}
\chi_i = \frac{\pi y^2}{T} \int_a^{\infty} \frac{d r}{a} \frac{r^3}{a^3}
e^{-\Phi^{(0)}(r)/T} \left( 1 - \frac{1}{\varepsilon^3 T} \sum_{n = 1}^3
\beta_{i, n} \ln(r/a)^n \right),$$ with the coefficients $$\label{eq:beta}
\begin{aligned}
\beta_{x, 1} & = \frac{1}{32} \alpha_+ \left( 8 \alpha_+ + 11 \alpha_-
\right), & \beta_{x, 2} & = - \frac{1}{16} \left( 8 \alpha_+^2 - 11 \alpha_+
\alpha_- + 4 \alpha_-^2 \right),
& \beta_{x, 3} & = - \frac{1}{12} \left( 8 \alpha_+^2 + \alpha_+ \alpha_- - 2 \alpha_-^2 \right), \\
\beta_{y, 1} & = \frac{1}{32} \alpha_+ \left(8 \alpha_+ -11 \alpha_-
\right), & \beta_{y, 2} & = - \frac{1}{16} \left( 8 \alpha_+^2 + 11 \alpha_+
\alpha_- + 4 \alpha_-^2 \right), & \beta_{y, 3} & = - \frac{1}{12} \left(
8 \alpha_+^2 - \alpha_+ \alpha_- - 2 \alpha_-^2 \right).
\end{aligned}$$ Despite the noise-induced angular averaging, the coefficients $\beta_{x,n} \neq \beta_{y,n}$ for $n = 1,2,3$ if both $\alpha_+, \alpha_- \neq 0$. In consequence, the eigenvalues of the susceptibility tensor are distinct, $\chi_x \neq \chi_y$, and a single dielectric constant $\varepsilon$ — as we have assumed in our derivation — is insufficient to describe the resulting anisotropic screening. Remarkably, this complication does not arise in the fully anisotropic configuration in which $\alpha_+ = 0$ (and, of course, also in an isotropic system with $\alpha_- = 0$ a single dielectric constant suffices). In the following, we focus on this case. Then, we find $\chi_x = \chi_y = \chi$, where $$\label{eq:chi}
\chi = \frac{\pi y^2}{T} \int_a^{\infty} \frac{dr}{a} \left( \frac{r}{a}
\right)^{3 - \frac{1}{\varepsilon T}} \left[ 1 - \frac{\alpha_-^2}{6
\varepsilon^3 T} \left( \ln(r/a)^3 - c \ln(r/a)^2 \right) \right].$$ Anticipating renormalization of the coefficient of the last term in brackets, we introduced a coupling $c$ with microscopic value $c = 3/2$. The crucial difference between the above expression and the corresponding result in the isotropic case [@Wachtel2016] is the sign of the leading term $\propto \ln(r/a)^3$. This difference brings about major qualitative changes in the behavior of vortices. According to Eq. , the renormalized dielectric constant is then $$\label{eq:epsilon_R}
\varepsilon_R = \varepsilon + 2 \pi \chi = \varepsilon + \frac{2
\pi^2 y^2}{T} \int_a^\infty \frac{dr}{a} \left(
\frac{r}{a}\right)^{3-\frac{1}{\varepsilon T}} \left[ 1 -
\frac{\alpha_-^2}{6 \varepsilon^3 T} \left( \ln(r/a)^3 - c \ln(r/a)^2
\right) \right].$$ In the integral on the RHS of this relation, the dielectric constant should be interpreted as the renormalized, scale-dependent value. Thus, Eq. is an implicit integral equation for $\varepsilon_R$. It can be solved by breaking the integral into small steps and absorbing the contribution of each of them progressively in renormalized coefficients. To derive RG differential equations that describe this procedure in the limit of infinitesimal steps, we separate the integral into two parts, $$\int_a^\infty=\int_a^{a \left( 1 + d \ell \right)} + \int_{a \left( 1 + d \ell \right)}^\infty.$$ The first part is used to redefine $\varepsilon$ on a slightly larger cutoff scale $a \left( 1 + d \ell \right)$, $$\varepsilon' = \varepsilon + \frac{2 \pi^2 y^2}{T} d \ell.$$ In the remaining integral, we rescale $r$ to restore the lower limit of integration to $a$, $$\begin{split}
\varepsilon_R & = \varepsilon' + \frac{2 \pi^2 y^2}{T} \int_{a \left( 1 + d
\ell \right)}^\infty \frac{dr}{a} \left(
\frac{r}{a}\right)^{3-\frac{1}{\varepsilon T}} \left[ 1 -
\frac{\alpha_-^2}{6 \varepsilon^3 T} \left( \ln(r/a)^3 - c \ln(r/a)^2
\right) \right] \\ & = \varepsilon' + \frac{2 \pi^2 y^2}{T}
\int_{a}^\infty \frac{dr}{a} \left(
\frac{r}{a}\right)^{3-\frac{1}{\varepsilon T}} \left( 1 + d \ell
\right)^{4 - \frac{1}{\varepsilon T}} \left[ 1 - \frac{\alpha_-^2}{6
\varepsilon^3 T} \left( \ln \! \left( \frac{r \left( 1 + d \ell
\right)}{a} \right)^3 - c \ln \! \left( \frac{r \left( 1 + d \ell
\right)}{a} \right)^2 \right) \right].
\end{split}$$ Expanding in $d \ell$ we find $$\begin{split}
\left( 1 + d \ell \right)^{4 - \frac{1}{\varepsilon T}} & = 1 + \left( 4 -
\frac{1}{\varepsilon T} \right) d \ell + O(d \ell^2), \\
\left( \ln(r/a) + \ln(1 + d \ell) \right)^n & = \ln(r/a)^n + n \ln(r/a)^{n -
1} d \ell + O(d \ell^2). \\
\end{split}$$ A redefinition of the other coupling constants is required to bring the expression for $\varepsilon_R$ to its original form, Eq. , $$\label{eq:epsilon_R_rescaled}
\begin{split}
\varepsilon_R & = \varepsilon' + \frac{2 \pi^2 y^2}{T} \left[ 1 + \left( 4 -
\frac{1}{\varepsilon T} \right) d \ell \right] \int_a^\infty
\frac{dr}{a} \left(\frac{r}{a}\right)^{3 - \frac{1}{\varepsilon T} \left( 1
- \frac{c \alpha_-^2}{3 \varepsilon^2} d \ell \right)} \left\{ 1 -
\frac{\alpha_-^2}{6 \varepsilon^3 T} \left[ \ln(r/a)^3 - \left( c - 3 d
\ell \right) \ln(r/a)^2 \right] \right\} \\ & = \varepsilon' + \frac{2
\pi^2 y^{\prime 2}}{T'} \int_a^\infty \frac{dr}{a} \left( \frac{r}{a}
\right)^{3 - \frac{1}{\varepsilon T'}} \left[ 1 - \frac{\alpha_-^2}{6
\varepsilon^3 T} \left( \ln(r/a)^3 - c' \ln(r/a)^2 \right) \right].
\end{split}$$ In the last line, we identified the following renormalized coupling constants: $$\begin{aligned}
\frac{y'^2}{T'} & = \frac{y^2}{T} \left[ 1 + \left( 4 - \frac{1}{\varepsilon
T} \right) d \ell \right]
& \Rightarrow &
& \frac{d}{d \ell} \left(
\frac{y^2}{T} \right)
& = \left( 4 - \frac{1}{\varepsilon T} \right) \frac{y^2}{T}, \\
\frac{1}{T'} & = \frac{1}{T} \left( 1 - \frac{c \alpha_-^2}{3 \varepsilon^2}
d \ell \right)
& \Rightarrow & & \frac{d}{d \ell} \frac{1}{T}
& = - \frac{c \alpha_-^2}{3 \varepsilon^2 T}, \\
c' & = c - 3 d \ell
& \Rightarrow &
& \frac{d c}{d \ell} & = - 3.\end{aligned}$$
We note that at the given order of $y^2$ and $\alpha_-^2$, all couplings on the RHS of Eq. can be replaced by the renormalized values. The last line can be integrated trivially and yields $$\label{eq:c}
c = \frac{3}{2} \left( 1 - 2 \ell \right),$$ where the bare value at $\ell = 0$ is $c = 3/2$ as indicated below Eq. . The remaining flow equations read $$\label{eq:RG_flow}
\begin{split}
\frac{d \varepsilon}{d \ell} & = \frac{2 \pi^2 y^2}{T}, \\
\frac{d y}{d \ell} & = \frac{1}{2} \left( 4 - \frac{1}{\varepsilon T} +
\frac{c \alpha_-^2}{3 \varepsilon^2} \right) y, \\
\frac{d T}{d \ell} & = \frac{c \alpha_-^2 T}{3 \varepsilon^2}.
\end{split}$$ This is the form reported in the main text. The appearance of the logarithmic scale $\ell$ in the flow equations upon inserting Eq. reflects that our perturbative treatment of the non-linearity does not yield the true large-distance behavior of the vortex interaction. For this reason, the RG flow has to be cut when the perturbative corrections become large, i.e., at the scale $L_v$ (or, if it is smaller, at $L_T$ given in Eq. where the angular averaging becomes invalid).
Phases and fixed point of the RG flow {#sec:phases-fixed-point}
-------------------------------------
To get a feeling for the RG flow described by Eqs. , we disregard for the moment that they are valid only up to parametrically large distances, and integrate the flow for a sample of microscopic values. As can be seen in Fig. \[fig:anisotropic\_RG\_flow\_3D\] (and also Fig. 2 of the main text), there is a critical temperature $T_c$ separating two phases with distinct flow patterns.
![RG flow of $\varepsilon$, $y$, and $T$ with $\alpha_-^2 = 0.1$ as described by Eqs. (red). Two phases are clearly distinguishable: At low temperatures $T < T_c \approx 0.13$, $y, T \to 0$ and $\varepsilon \to \mathrm{const.}$, while at $T > T_c$, $y, \varepsilon \to \infty$ and $T \to \mathrm{const.}$ For comparison we show the equilibrium KT flow with $\alpha_-^2 = 0$ (blue). Here, the value of $T$ is conserved in the RG flow. In this figure, the microscopic value of the fugacity is $y = 0.1$, and the temperature is varied from $T = 0.1$ to $0.2$.[]{data-label="fig:anisotropic_RG_flow_3D"}](anisotropic_RG_flow_3D){width="\linewidth"}
In the low-temperature phase, the dielectric constant approaches a constant value, while the temperature and the fugacity flow to zero. This is in stark contrast to the corresponding phase in the equilibrium KT case, where the temperature is conserved in the RG flow. In fully anisotropic non-equilibrium systems, such a scale-invariant temperature is encountered only asymptotically in the high-temperature phase, when $\varepsilon \sim e^{4 \ell}, y \sim e^{2 \ell}$, and $T \to T_{\infty} = \mathrm{const.}$ At large scales, the flow equations simplify as $$\frac{d \varepsilon}{d \ell} \sim \frac{2 \pi^2 y^2}{T_{\infty}}, \qquad
\frac{d y}{d \ell} \sim 2 y, \qquad
\frac{d T}{d \ell} \to 0.$$ This is just the usual KT flow with a renormalized temperature.
The existence of two distinct phases in the RG flow suggests there is a fixed point separating these phases and controlling critical behavior at the transition. In stark contrast to the usual case encountered in continuous phase transitions, the flow equations cannot have a true fixed point since $c$ always grows logarithmically with the running cutoff (i.e., linearly in $\ell$). However, as we show in the following, there is nevertheless a fixed point of the flow of a reduced set of logarithmically rescaled variables. To this end, it is convenient to regard the couplings $\varepsilon$, $y$, and $T$ as functions of $x = -c \in [-3/2, \infty)$ instead of $\ell$, $$\begin{split}
\frac{d \varepsilon}{d x} & = \frac{2 \pi^2 y^2}{3 T}, \\ \frac{d y}{d x} &
= \frac{1}{6} \left( 4 - \frac{1}{\varepsilon T} -
\frac{x \alpha_-^2}{3 \varepsilon^2} \right) y, \\
\frac{d T}{d x} & = - \frac{x \alpha_-^2 T}{9 \varepsilon^2}.
\end{split}$$ To find the fixed point of these equations, we effect another change of variables: $$\label{eq:rescaled_couplings}
\tilde{\varepsilon} = \varepsilon/x, \qquad \tilde{y} = \sqrt{x} y, \qquad
\tilde{T} = x T.$$ Strictly speaking, the rescaled variables are ill-defined at the beginning of the flow when $x < 0$. However, here we are concerned with the behavior of the solutions to the flow equations for $x \to \infty$. The flow equations are then recast as $$\begin{split}
\frac{d \tilde{\varepsilon}}{d x} & = \frac{1}{x} \left( \frac{2 \pi^2
\tilde{y}^2}{3 \tilde{T}} - \tilde{\varepsilon} \right), \\
\frac{d \tilde{y}}{d x} & = \frac{1}{6} \left[ 4 -
\frac{1}{\tilde{\varepsilon} \tilde{T}} + \frac{1}{x} \left( 3 - \frac{\alpha_-^2}{3
\tilde{\varepsilon}^2} \right) \right] \tilde{y}, \\
\frac{d \tilde{T}}{d x} & = \frac{1}{3 x} \left( 3 - \frac{\alpha_-^2}{3
\tilde{\varepsilon}^2} \right) \tilde{T}.
\end{split}$$ In this form, it is straightforward to see there is a fixed point at $\tilde{\varepsilon}_{*}$, $\tilde{y}_{*}$, and $\tilde{T}_{*}$ determined by $$\frac{2 \pi^2 \tilde{y}_{*}^2}{3 \tilde{T}_{*}} - \tilde{\varepsilon}_{*} = 0, \qquad 4 -
\frac{1}{\tilde{\varepsilon}_{*} \tilde{T}_{*}} = 0, \qquad 3 - \frac{\alpha_-^2}{3
\tilde{\varepsilon}_{*}^2} = 0.$$ We find $$\label{eq:critical_point}
\tilde{\varepsilon}_{*} = \frac{{\left\lvert \alpha_- \right\rvert}}{3}, \qquad \tilde{y}_{*} =
\sqrt{\frac{3}{8}} \frac{1}{\pi}, \qquad \tilde{T}_{*} = \frac{3}{4 {\left\lvert \alpha_- \right\rvert}}.$$ Flow trajectories close to criticality are shown in Fig. \[fig:close-to-crit\_flow\], both for the original and rescaled couplings (see panels (a-c) and (d-f), respectively). The rescaled couplings are close to their fixed-point values in the range $30 \lesssim \ell \lesssim 90$, during which the original ones evolve according to Eq. . This logarithmic flow is the origin of the peculiar singularity of the correlation length $\xi$ at the critical point which is distinct from both the algebraic scaling at conventional second order phase transitions, and the essential singularity at the equilibrium KT transition. In the next section, we discuss how the singularity of the correlation length can be inferred from the a linearization of the flow around the fixed point .
{width="\linewidth"}
Asymptotic analysis of the linearized flow equations {#sec:asympt-analys-RG}
----------------------------------------------------
As usual, we define the correlation length $\xi$ as the scale on which the renormalized fugacity reaches the value $y_1 = 1$. We fix the microscopic value $y_0$ and regard the temperature $T$ as the tuning parameter through the transition. The origin of the singularity of $\xi$ as $T \to T_c$ is apparent from panels (b) and (e) in Fig. \[fig:close-to-crit\_flow\] which show the flow of $y$ and $\tilde{y}$, respectively. This flow can be divided into three stages: (i) for $0 \leq \ell < \ell_0$ the rescaled fugacity $\tilde{y}$ approaches its fixed-point value $\tilde{y}_{*}$ and (ii) stays close to this value for $\ell_0 \leq \ell < \ell_1$; eventually, (iii) $\tilde{y}$ flows away from the fixed point and $y$ grows strongly until it reaches $y_1 = 1$ for $\ell_1 \leq \ell \leq \ell_2$. When $T \to T_c$, $\ell_1 \to \infty$, while $\ell_0$ and $\ell_2 - \ell_1$ remain finite, and therefore $\xi \sim a e^{\ell_2}$. Consequently, to determine the singularity of $\xi$, it is sufficient to consider stage (ii) of the flow in the vicinity of the fixed point where we can linearize the flow equations.
We collect the deviations from the fixed point in the variable $\mathbf{a} = ( \delta \tilde{\varepsilon}, \delta \tilde{y}, \delta \tilde{T} )
= ( \tilde{\varepsilon} - \tilde{\varepsilon}_{*}, \tilde{y} - \tilde{y}_{*},
\tilde{T} - \tilde{T}_{*} )$. The linearized flow equations read $$\label{eq:linearized_flow}
\frac{d \mathbf{a}}{d x} = A \mathbf{a}, \qquad A = A_0 + \frac{A_1}{x},$$ where $$\begin{split}
A_0 & =
\begin{pmatrix}
0 & 0 & 0 \\
- \sqrt{\frac{3}{2}} \frac{1}{\pi \alpha_-} & 0 & - \sqrt{\frac{2}{3}}
\frac{2 \alpha_-}{3 \pi} \\
0 & 0 & 0
\end{pmatrix},
\\ A_1 & =
\begin{pmatrix}
-1 & \sqrt{\frac{2}{3}} \frac{4 \pi \alpha_-}{3} & - \frac{4 \alpha_-^2}{9}
\\
\sqrt{\frac{3}{2}} \frac{3}{2 \pi \alpha_-} & 0 & 0 \\
\frac{9}{2 \alpha_-^2} & 0 & 0
\end{pmatrix}.
\end{split}$$ Note that these equations still depend on $x$ and thus cannot be solved straightforwardly. However, since $\ell_2 \gg 1$ when $t = (T - T_c)/T_c \ll 1$, we only need to know the asymptotic behavior of the solution for $x \sim 3 \ell \to \infty$. To solve this problem, we closely follow the method described in Ref. [@Wasow2002].
What makes finding an asymptotic expansion of the solution to Eq. slightly complicated is first that $x = \infty$ is an irregular singular point of this equation and second that $A_0$, the leading matrix for $x \to \infty$, has only one eigenvalue. In the following, we apply a series of transformations to bring Eq. to a form in which the leading matrix has three distinct eigenvalues. Then, the leading matrix can be diagonalized, which results in three decoupled equations that can be integrated straightforwardly.
The first step is to bring $A_0$ to Jordan normal form by means of a transformation $P_1$, $$\mathbf{a}_1 = P_1^{-1} \mathbf{a}, \quad \frac{d \mathbf{a}_1}{d x} = A_1
\mathbf{a}_1, \quad A_1 = P_1^{-1} A P_1 = A_{10} + \frac{A_{11}}{x},$$ where $$\begin{gathered}
P_1 =
\begin{pmatrix}
-\frac{4 \alpha_-^2}{9} & 0 & \sqrt{\frac{2}{3}} \pi \alpha_- \\
0 & 1 & 0 \\
1 & 0 & 0 \\
\end{pmatrix},
\qquad A_{10} =
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
\end{pmatrix},
\\ A_{11} =
\begin{pmatrix}
-2 & 0 & \sqrt{\frac{3}{2}} \frac{3 \pi}{\alpha_-} \\
\sqrt{\frac{2}{3}} \frac{\alpha_-}{\pi } & 0 & \frac{3}{2} \\
-\sqrt{\frac{2}{3}} \frac{4 \alpha_-}{3 \pi} & \frac{4}{3} & 1 \\
\end{pmatrix}.\end{gathered}$$ The matrix $A_{10} = H_1 \oplus H_2$ is the direct sum of two shifting matrices, which are matrices with ones on the superdiagonal and zeroes elsewhere, $H_1 = 0$ and $H_2 =
\begin{psmallmatrix}
0 & 1 \\
0 & 0
\end{psmallmatrix}
$. We next apply a transformation $P_2$ to bring the sub-leading matrix $A_{11}$ to a form in which the only non-zero entries occur in the rows corresponding to the last rows of the blocks $H_{1,2}$, i.e., $$\label{eq:a_2}
\mathbf{a}_2 = P_2^{-1} \mathbf{a}_1, \quad \frac{d \mathbf{a}_2}{d x} =
A_2 \mathbf{a}_2, \quad A_2 = P_2^{-1} A_1 P_2 - P_2^{-1} \frac{d P_2}{d x},$$ where the matrix $A_2$ has the structure $$\label{eq:A2}
A_2 =
\begin{pmatrix}
A_{211} & A_{212} & A_{213} \\
0 & 0 & 1 \\
A_{231} & A_{232} & A_{233}
\end{pmatrix}.$$ Inserting in Eq. the asymptotic ansätze $P_2 \sim \sum_{r = 0}^{\infty} P_{2r}/x^r$ and $A_2 \sim \sum_{r = 0}^{\infty} A_{2r}/x^r$ and identifying coefficients of the same powers of $x$, we obtain $A_{10} P_{20} - P_{20} A_{20} = 0$ and $$\begin{gathered}
\label{eq:A2_recursion}
A_{10} P_{2r} - P_{2r} A_{20} \\ =
\sum_{s = 0}^{r - 1} \left( P_{2s} A_{2, r - s} - A_{1, r - s} P_{2s} \right)
- \left( r - 1 \right) P_{2, r - 1}.\end{gathered}$$ The first relation can be solved by setting $A_{20} = A_{10}$ and $P_{20} =
{\mathbbm{1}}$; the second relation determines $A_{2r}$ and $P_{2r}$ for $r \geq 1$ recursively. We obtain the desired transformation by restricting the form of $P_{2r}$ with $r \geq 1$ as $$\label{eq:P2r}
P_{2r} =
\begin{pmatrix}
0 & 0 & 0 \\
0 & P_{222r} & 0 \\
P_{231r} & P_{232r} & 0
\end{pmatrix}.$$ Insertion of Eqs. and in Eq. yields a sequence of linear equations for the elements of $A_2$ and $P_2$ that can be solved straightforwardly to any desired order. We omit the explicit expressions. Next, we apply a first shearing transformation, $$\mathbf{a}_3 = P_3^{-1} \mathbf{a}_2, \quad \frac{d \mathbf{a}_3}{d x} =
A_3 \mathbf{a}_3,$$ where $$\begin{split}
A_3 & = P_3^{-1} A_2 P_3 - P_3^{-1} \frac{d P_3}{d x}, \\ P_3 & =
{\mathop{\mathrm{diag}}}(1, x^{-g_1}, x^{-2 g_1}).
\end{split}$$ We choose $g_1 = 1/3$ (this choice is, of course, not arbitrary, but is determined by a well-defined procedure [@Wasow2002]), and to bring the equation back to a form that involves only integer powers of the variables, we switch to $x = \alpha_1 y^{p_1}$, where $\alpha_1 = p_1^{1/(g_1 - 1)}$ and $p_1 = 3$. This yields $$\frac{1}{y} \frac{d \mathbf{a}_3}{d y} = \tilde{A}_3 \mathbf{a}_3, \qquad \tilde{A}_3
= x^{g_1} A_3.$$ The leading matrix in the last equation, $\tilde{A}_{30} = \lim_{y \to \infty} \tilde{A}_3$, still has only one distinct eigenvalue, and it seems as if we would not have gained anything. However, we must not despair. Instead, we bring $\tilde{A}_{30}$ again to Jordan normal form, $$\mathbf{a}_4 = P_4^{-1} \mathbf{a}_3, \quad \frac{1}{y} \frac{d \mathbf{a}_4}{d y} = A_4
\mathbf{a}_4, \quad A_4 = P_4^{-1} \tilde{A}_3 P_4,$$ with $$P_4 =
\begin{pmatrix}
0 & 0 & - \sqrt{\frac{3}{2}} \frac{3 \pi}{4 \alpha_-} \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{pmatrix},$$ and perform a second shearing transformation, $$\mathbf{a}_5 = P_5^{-1} \mathbf{a}_4, \quad \frac{1}{y} \frac{d \mathbf{a}_5}{d y} =
A_5 \mathbf{a}_5,$$ where $$\begin{split}
A_5 & = P_5^{-1} A_4 P_5 - \frac{1}{y} P_5^{-1} \frac{d P_5}{d y}, \\ P_5
& = {\mathop{\mathrm{diag}}}(1, y^{-g_2}, y^{-2 g_2}).
\end{split}$$ This time, we choose $g_2 = 1/2$, and another change of variables $y = \alpha_2 z^{p_2}$ with $\alpha_2 = p_2^{1/(g_2 - 2)}$ and $p_2 = 2$ brings us to $$\frac{1}{z^2} \frac{d \mathbf{a}_5}{d z} = \tilde{A}_5 \mathbf{a}_5, \qquad
\tilde{A}_5 = y^{g_2} A_5.$$ Miraculously, $\tilde{A}_{50} = \lim_{z \to \infty} \tilde{A}_5$ has three distinct eigenvalues, $0$ and $\pm 2/3^{1/4}$. Hence, we can now go ahead and diagonalize $\tilde{A}_5$ order by order in $1/z$. The first step is to diagonalize the leading matrix $\tilde{A}_{50}$, $$\mathbf{a}_6 = P_6^{-1} \mathbf{a}_5, \quad \frac{1}{z^2} \frac{d \mathbf{a}_6}{d z}
= A_6 \mathbf{a}_6, \quad A_6 = P_6^{-1} \tilde{A}_5 P_6,$$ where $$\label{eq:P6}
P_6 =
\begin{pmatrix}
-\frac{3^{1/4}}{2} & \frac{3^{1/4}}{2} & -\frac{\sqrt{3}}{4} \\
1 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}.$$ We move on to diagonalize the sub-leading parts of $A_6$ in two steps, $$\begin{gathered}
\mathbf{a}_7 = P_7^{-1} \mathbf{a}_6, \quad \frac{1}{z^2} \frac{d \mathbf{a}_7}{d z} =
A_7 \mathbf{a}_7, \\ A_7 = P_7^{-1} A_6 P_7 - \frac{1}{z^2} P_7^{-1} \frac{d P_7}{d z},\end{gathered}$$ and $$\begin{gathered}
\label{eq:linearized_flow_8}
\mathbf{a}_8 = P_8^{-1} \mathbf{a}_7, \quad \frac{1}{z^2} \frac{d \mathbf{a}_8}{d z} =
A_8 \mathbf{a}_8, \\ A_8 = P_8^{-1} A_7 P_8 - \frac{1}{z^2} P_8^{-1} \frac{d P_8}{d z}.\end{gathered}$$ The matrices $A_{7,8}$ and $P_{7,8}$ can be found order by order in $1/z$ by recursion relations similar to Eq. . They take the forms $$\begin{gathered}
A_7 =
\begin{pmatrix}
A_{711} & 0 & 0 \\
0 & A_{722} & A_{723} \\
0 & A_{732} & A_{733} \\
\end{pmatrix}, \quad
P_7 =
\begin{pmatrix}
0 & P_{712} & P_{713} \\
P_{721} & 0 & 0 \\
P_{731} & 0 &
\end{pmatrix}, \\
P_8 =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & P_{823} \\
0 & P_{832} & 0
\end{pmatrix},\end{gathered}$$ and $A_8$ is diagonal. The general solution to Eq. is thus $$\label{eq:asymptotic_solution_8}
\mathbf{a}_8(z) = e^{\int_{z_0}^z d z' \, z^{\prime 2} A_8(z')} \mathbf{a}_{80}
\sim e^{\int_{z_0}^z d z' \, z^{\prime 2} A_8^{\infty}(z')} D_8^{\infty} \mathbf{a}_{80}.$$ In the last equation, in $A_8^{\infty}$ we keep terms in the asymptotic expansion of $A_8$ up to order $O(1/z^3)$ so that the lowest order term in the exponent is $O(\ln(z))$ — this is the order to which we have to perform all the above transformations to find the leading asymptotic behavior. $D_8^{\infty}$ is a constant diagonal matrix which could only be found be carrying out the above analysis *exactly* (not just asymptotically) since every order of $1/z$ in $A_8$ contributes to $D_8^{\infty}$. However, the precise value of $D_8^{\infty}$ is not important for our purposes. From Eq. , we can reconstruct $\mathbf{a}(x)$ by undoing all transformations: $$\begin{split}
\mathbf{a}(x) & = P_1 P_2 P_3 P_4 P_5 P_6 P_7 P_8 \mathbf{a}_8(z) \\ & \sim
P_1 P_2 P_3 P_4 P_5 P_6 P_7 P_8 e^{\int_{z_0}^z d z' \, z^{\prime 2}
A_8^{\infty}(z')} D_8^{\infty} \mathbf{a}_{80},
\end{split}$$ where $$z = \left[ \frac{1}{\alpha_2} \left( \frac{x}{\alpha_1} \right)^{1/p_1} \right]^{1/p_2}.$$ We find, for $x \to \infty$, $$\label{eq:linearized_flow_asymptotic_x}
\begin{split}
\delta \tilde{\varepsilon} & \sim x^{1/4} e^{4 \sqrt{x/3}}, \\ \delta
\tilde{y} & \sim x^{3/4} e^{4 \sqrt{x/3}}, \\ \delta \tilde{T} & \sim x^{-1/4}
e^{4 \sqrt{x/3}}.
\end{split}$$ As we demonstrate in Fig. \[fig:asymptotic\_expansion\], these asymptotic expressions give an excellent approximation to the exact solution of the linearized flow equation even at relatively small values of $x$.
{width="\linewidth"}
The corresponding asymptotic behavior of the original couplings follows from Eq. and $x = 3/2 \left( 2 \ell - 1 \right) \sim 3 \ell$, $$\label{eq:asymtotic_expansion}
\begin{split}
\varepsilon & \sim \delta \varepsilon_{\infty} = \delta \varepsilon_{\infty,0} \ell^{5/4} e^{4
\sqrt{\ell}}, \\ y & \sim \delta y_{\infty} = \delta y_{\infty,0} \ell^{1/4} e^{4
\sqrt{\ell}}, \\ T & \sim \delta T_{\infty} = \delta T_{\infty,0} \ell^{-5/4} e^{4
\sqrt{\ell}}.
\end{split}$$ At $T = T_c$, the couplings flow to the fixed point, and $\delta \varepsilon_{\infty, 0} = \delta y_{\infty, 0} = \delta T_{\infty, 0} =
0$. Close to criticality, when $t = (T - T_c)/T_c \ll 1$, we expect that the amplitudes $\varepsilon_{\infty, 0}$, $\delta y_{\infty, 0}$, and $\delta T_{\infty, 0}$ in Eq. are proportional to $t$. In particular, setting $\delta y_{\infty} \propto t$, the correlation length is $\xi = a e^{\ell}$ where $$\begin{gathered}
\label{eq:xi_t}
4 \sqrt{\ell} + \frac{1}{4} \ln(\ell) \sim - \ln(t) \\ \Rightarrow \quad
\ln(\xi/a) \sim \frac{1}{16} \ln(t) \left( \ln(t) + \ln( {\left\lvert \ln(t) \right\rvert}) \right).\end{gathered}$$ This type of singularity is between true scaling behavior encountered at a second order phase transition and the essential singularity of $\xi$ at the equilibrium KT transition: $$\begin{aligned}
& \text{true scaling:} & \frac{1}{\nu} \ell & \sim - \ln(t) & \Rightarrow & &
\xi/a & \sim
t^{-\nu}, \\
& \text{equilibrium KT:} & 2 \ln(\ell) & \sim - \ln(t) & \Rightarrow & &
\xi/a & \sim e^{C/\sqrt{t}}.
\end{aligned}$$ A comparison showing the good agreement between our analytical prediction and the correlation length obtained from a numerical integration of the flow equations is shown in Fig. \[fig:corr\_length\].
![Divergence of the correlation length at the critical temperature for $\alpha_-^2 = 0.1, 0.2, 0.5$ (top to bottom). The vertical axis is rescaled as $4 \sqrt{\ln(\xi/a)} + \ln(\ln(\xi/a))/4$ (cf. Eq. ) so that the curves approach straight lines with slope $-1$ as $T \to T_c$ (for comparison shown as black dashed line).[]{data-label="fig:corr_length"}](xit){width="\linewidth"}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex combination. More specifically, given two persistence diagrams and a choice of metric, one obtains a bijection realizing the distance between the diagrams, and uses this bijection to linearly interpolate from one diagram to another. We prove that for several families of metrics, every geodesic in persistence diagram space arises as such a convex combination. For certain other choices of metrics, we explicitly construct infinite families of geodesics that cannot have this form.'
address: 'Department of Mathematics, The Ohio State University.'
author:
- Samir Chowdhury
bibliography:
- '../biblio.bib'
title: Geodesics in persistence diagram space
---
Introduction and problem statement
==================================
A *persistence diagram* or *barcode* is a countable multiset of above-diagonal points in ${\mathbb{R}}^2$ along with the diagonal, which is counted with countably infinite multiplicity. We denote the collection of all possible diagrams by ${{\mathcal}{D}}$. Persistence diagrams were originally formulated as shape descriptors arising from applying *persistent homology* to point cloud or metric datasets. In recent years, they have been generalized to the point where they can be studied as algebraic objects in their own right, without necessarily arising as a shape descriptor for a dataset. Relevant to this paper is the development of a variety of metrics on persistence diagrams with the overarching goal of defining Fréchet means and related generalizations [@mileyko2011probability; @turner2013means; @tmmh; @munch2015probabilistic].
Persistence diagrams are typically compared using the *bottleneck distance*, which is an $l^\infty$ matching distance where the matching cost is computed using an $l^\infty$ ground metric. In the aforementioned papers, the objects of study were variants of the bottleneck distance. Specifically, [@mileyko2011probability] considered $l^p$ matching for $p \in [1,\infty)$ with the $l^\infty$ ground metric, [@tmmh; @munch2015probabilistic] considered $l^2$ matching with the $l^2$ (Euclidean) ground metric, and [@turner2013means] considered $l^p$ matching with an $l^p$ ground metric for $p \in [1,\infty]$.
By an overload of notation, let ${\varnothing}$ denote the *empty diagram* consisting of just the diagonal with countably infinite multiplicity. In [@tmmh], the authors defined a type of $l^2$ metric on ${{\mathcal}{D}}$ (denoted $d_2[l^2]$) and studied the space ${\mathcal}{D}_2[l^2] {\overset{\tiny\operatorname{def}}{=}}\{X \in {{\mathcal}{D}}: d_2[l^2](X,{\varnothing}) < \infty\}$. On this space, they characterized Fréchet means and gave a procedure for computing these means. A necessary step for their constructions was a result showing that $({\mathcal}{D}_2[l^2],d_2[l^2])$ is an *Alexandrov space* with nonnegative curvature [@tmmh Theorem 2.5]. The proof of [@tmmh Theorem 2.5] in turn requires one to show that all geodesics in this space are of a convex combination form. Indeed, we show in Section \[sec:branch\] that for certain other choices of metrics on ${{\mathcal}{D}}$, there exist geodesics which are not given by a convex combination form, and moreover there exist *branching geodesics* which preclude a space from having nonnegative curvature in the sense of Alexandrov. Finally in Section \[sec:geod-char\], we show that for certain families of metrics, including the important case $p=q=2$, all geodesics are indeed of a convex combination form.
Our proof of this characterization result follows the strategy used by Sturm in proving an analogous result about geodesics in the space of metric measure spaces [@sturm2012space]. The existence results about branching geodesics and geodesics not given by a convex combination form are related to constructions we previously investigated in [@dgh-era].
#### Contributions
Following [@munch2015probabilistic], we study persistence diagram metrics ${d_p[l^q]}$ which involve an $l^p$ matching metric over an $l^q$ ground metric. For certain families of metrics, we show that ${{\mathcal}{D}}$ has geodesics that can be uniquely characterized as convex combinations. We are able to prove our result for the following families:
- $q = 2$, $p \in (1,\infty)$
- $p = q \in [2,\infty)$.
We also provide counterexamples showing that geodesics are *not* uniquely characterized in the cases $p = q =1$ and $p =\infty, \, q\in[1,\infty]$.
Said differently: whereas it is easy to show that any optimal bijection yields a geodesic (via the convex-combination form), here we prove the harder reverse direction, i.e. that any geodesic arises as the convex-combination geodesic of an optimal bijection, at least for certain ranges of $p,q$. Furthermore, for certain other ranges of $p$ and $q$, we show that the negative result holds. So our focus is on the dashed line shown below. $$\begin{aligned}
\centering
\{\text{optimal bijection}\} \xrightarrows[\text{convex combination}] \{\text{geodesic}\} \end{aligned}$$
Definitions and statement of results
====================================
Given sets $X,Y$ and an element $z \in X\times Y$, we write $\pi_X(z), \pi_Y(z)$ to denote the canonical projections of $z$ into $X$ and $Y$, respectively. The diagonal in ${\mathbb{R}}^2$ is denoted $\Delta := \{(x,x) \in {\mathbb{R}}^2 : x \in {\mathbb{R}}\}$. We also define $\Delta_{\mathbb{Q}}:= \{(x,x) \in {\mathbb{R}}^2 : x \in {\mathbb{Q}}\}$, i.e. the rational points on the diagonal. We write $\Delta^\infty$ or $\Delta_{\mathbb{Q}}^\infty$ to denote these sets counted with countably infinite multiplicity. The part of the plane above the diagonal is denoted ${\mathbb{R}}^2_>$, and the part of the plane above and including the diagonal is denoted ${\mathbb{R}}^2_{\geq}$. The $p$-norm in ${\mathbb{R}}^2$, for $p \in [1,\infty]$, is denoted ${\left\|}\cdot {\right\|}_p$. Given an above-diagonal point $x \in {\mathbb{R}}^2_>$, we write ${\left\|}x - \Delta {\right\|}_p$ to denote the perpendicular distance (in $p$-norm) between $x$ and the diagonal. We also write $\pi_{\Delta}(x)$ to denote the projection of $x$ onto the diagonal. When we suppress notation and write ${\left\|}\cdot {\right\|}$, we mean the Euclidean norm in ${\mathbb{R}}^2$. We will occasionally use the canonical identification between ${\mathbb{R}}^2$ and ${\mathbb{C}}$.
The transpose of a vector $[v_1,v_2,\ldots,v_n]$ will be denoted $[v_1,v_2,\ldots,v_n]^T$. Given an infinite-dimensional vector $V \in {\mathbb{R}}^{\mathbb{N}}$ and a function $f$ defined on each element of $V$, we will write $f(V)$ to denote $(f(v_1),f(v_2),\ldots ).$
\[def:pdgm\] A *persistence diagram* is a countable subset of ${\mathbb{R}}^2_> \times {\mathbb{N}}$ along with countably infinite copies of $\Delta$. This naming convention differs slightly from that of the standard persistence diagram (cf. [@tmmh]), which involves multisets in ${\mathbb{R}}^2$. However, we introduce the ${\mathbb{N}}$ coordinate so that different copies of the same point can be defined to occupy different entries in ${\mathbb{N}}$. We refer to the ${\mathbb{N}}$ component as the indexing component, and the ${\mathbb{R}}^2$ component as the geometric component. For a persistence diagram $X$, we let $X_>$ denote the above-diagonal portion of the diagram. The collection of all persistence diagrams is denoted ${{\mathcal}{D}}$. For any $x \in X$, the cardinality of $(\pi_{{\mathbb{R}}^2}){^{-1}}\circ \pi_{{\mathbb{R}}^2}(x)$ is the multiplicity of $\pi_{{\mathbb{R}}^2}(x)$. We write $m(x)$ to denote the multiplicity of $x$.
Note that persistence diagrams are typically formulated as multisets, i.e. as a subset $Z\subseteq {\mathbb{R}}^2_{\geq}$ along with a multiplicity function $m:Z {\rightarrow}{\mathbb{N}}$. This multiset formulation can be recovered from the ${\mathbb{R}}^2_\geq \times {\mathbb{N}}$ formulation given above; the advantage of the above formulation is that it enables some of our later arguments involving convergence of sequences.
Crucially, given persistence diagrams $X,Y$ and points $x \in X,\, y\in Y$, we write ${\left\|x - y \right\|}_p$ to mean ${\left\|{\pi_{{\mathbb{R}}^2}}(x) - {\pi_{{\mathbb{R}}^2}}(y) \right\|}_p$. In other words, when computing distances between points in persistence diagrams, only the geometric component of each point is considered.
\[ex:pers-dgm\] Let $X = \{(0,1,1)\} \cup \Delta^\infty$, $Y = \{(0,1,1), (0,1,2), (1,3,3)\} \cup \Delta^\infty$, and $Z= \{(0,1,2)\} \cup \Delta^\infty$. All three are persistence diagrams. Each of $X$ and $Z$ has a single off-diagonal point at $(0,1)$. $Y$ has an off-diagonal with multiplicity two at $(0,1)$, and another point with multiplicity one at $(1,3)$. For any of the metrics we later define, the distance between $X$ and $Z$ is zero. This is because their off-diagonal points only differ in the ${\mathbb{N}}$ coordinate, which is not relevant for the distances we consider.
Let $X, Y \in {{\mathcal}{D}}$ be two persistence diagrams. We can always obtain bijections between $X$ and $Y$, matching points to the diagonal if needed. Next we introduce a family of $l^p$ matching distances which compute the expected cost of an optimal matching between $X$ and $Y$, where optimality is with respect to an $l^q$ ground metric. Given $p \in [1,\infty)$, $q \in [1,\infty]$, the *$l^p[l^q]$ matching distance* between persistence diagrams is the function ${d_p[l^q]}: {{\mathcal}{D}}\times {{\mathcal}{D}}{\rightarrow}[0,\infty]$ given by writing $${d_p[l^q]}(X,Y) {\overset{\tiny\operatorname{def}}{=}}\inf {\left\{{\left(}\sum_{x\in X} {\left\|}x - {\varphi}(x){\right\|}_q^p {\right)}^{1/p}
: {\varphi}: X {\rightarrow}Y \text{ a bijection} \right\}} \text{ for any } X,Y \in {{\mathcal}{D}}.$$ For $p = \infty$, we have $${d_p[l^q]}(X,Y) {\overset{\tiny\operatorname{def}}{=}}\inf {\left\{\sup_{x\in X} {\left\|}x - {\varphi}(x){\right\|}_q
: {\varphi}: X {\rightarrow}Y \text{ a bijection} \right\}} \text{ for any } X,Y \in {{\mathcal}{D}}.$$ A bijection ${\varphi}$ for which the infimum above is attained is said to be *optimal*.
Here are special cases of the preceding definition.
- The *bottleneck distance* corresponds to $p = \infty, \, q= \infty$.
- The case $p \in [1,\infty)$ and $q = \infty$ was considered in [@mileyko2011probability].
- Both [@tmmh; @munch2015probabilistic] considered the case $p=2, \, q=2$.
- [@turner2013means] considered the case $p = q \in [1,\infty]$.
For $p,q \in [1,\infty]$, the set $\{X \in {{\mathcal}{D}}: {d_p[l^q]}(X,{\varnothing}) < \infty \}$ is denoted by ${{{\mathcal}{D}}_{p}[l^q]}$. Note that if $X \in {{{\mathcal}{D}}_{p}[l^q]}$ and $p < \infty$, then any open ball $U \subseteq {\mathbb{R}}^2_>$ separated from the diagonal by some ${\varepsilon}> 0$ can contain only finitely many points of ${\pi_{{\mathbb{R}}^2}}(X)$.
We make some simple but important remarks to guide the reader:
- Typically the persistence diagram is defined to be a multiset of points in the extended plane (including $\infty$). Note that our definition only allows for points on the plane, which is in keeping with the definition in [@tmmh].
- In [@tmmh; @munch2015probabilistic], the distance ${d_p[l^q]}$ above is called the $l^q$-Wasserstein metric; we avoid this terminology because Wasserstein distances typically refer to distances between probability measures.
- A priori, ${d_p[l^q]}$ is only a pseudometric on ${{\mathcal}{D}}$. To see this, let $A, B $ be two countable dense subsets of $[0,1]$ that are not equal. Write $X:= \{ (0,a,1) : a \in A\} \cup \Delta^\infty$ and $Y:= \{(0,b,1) : b\in B\} \cup \Delta^\infty$. Then ${d_p[l^q]}(X,Y) = 0$, even though $X \neq Y$.
A *curve* in $({{{\mathcal}{D}}_{p}[l^q]},{d_p[l^q]})$ is a continuous map ${\gamma}:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$. Such a curve is called a *geodesic* [@bridson2011metric Section I.1] if for any $s, t \in [0,1]$, $${d_p[l^q]}({\gamma}(s),{\gamma}(t)) = |t-s|\cdot {d_p[l^q]}({\gamma}(0),{\gamma}(1)).$$ In [@tmmh], the authors gave a constructive proof showing that $({{{\mathcal}{D}}_{2}[l^2]},{d_2[l^2]})$ is a *geodesic space*, i.e. that for any $X, Y \in {{{\mathcal}{D}}_{2}[l^2]}$, there exists a geodesic from $X$ to $Y$. As a precursor to the construction, they first proved the following result showing that between any $X, Y \in {{{\mathcal}{D}}_{2}[l^2]}$, there exists an optimal bijection ${\varphi}$ realizing the infimum in the definition of ${d_2[l^2]}(X,Y)$:
\[thm:existence\] Let $X, Y \in {{{\mathcal}{D}}_{2}[l^2]}$. Then there exists a bijection ${\varphi}: X {\rightarrow}Y$ such that $\sum_{x \in X} {\left\|x - {\varphi}(x) \right\|}^2 = {d_2[l^2]}(X,Y)^2$.
The construction of the geodesics is as follows: given any $X, Y \in {{{\mathcal}{D}}_{p}[l^q]}$, let ${\varphi}$ be an optimal bijection. For the time being, we ignore the ${\mathbb{N}}$ coordinates of the persistence diagrams. Write ${\gamma}(0) {\overset{\tiny\operatorname{def}}{=}}X$, ${\gamma}(1) {\overset{\tiny\operatorname{def}}{=}}Y$, and for any $t \in (0,1)$, $${\gamma}(t) {\overset{\tiny\operatorname{def}}{=}}{\left\{ (1-t)x + t{\varphi}(x) : x \in X\right\}}.$$
Regardless of the choice of $p,q \in [1,\infty]$, such a curve defines a geodesic (cf. Corollary \[cor:geod-exist\]). Note that different choices of $p,q$ may lead to different bijections ${\varphi}$ being optimal. We call any geodesic of this form a *convex-combination geodesic*. Conversely, we refer to geodesics *not* of this form as *deviant* geodesics. Returning to the question of dealing with the indexing coordinate ${\mathbb{N}}$: recall that ${d_p[l^q]}$ is blind to this coordinate, so we can define the convex-combination geodesic $\gamma$ in the following manner and still maintain continuity: $${\gamma}(t) {\overset{\tiny\operatorname{def}}{=}}{\left\{ [(1-t)x_1 + t{\varphi}(x)_1, (1-t)x_2 + t{\varphi}(x)_2, x_3]^T : x=[x_1,x_2,x_3]^T \in X\right\}} \text{ for } t\in [0,1),$$ and ${\gamma}(1) {\overset{\tiny\operatorname{def}}{=}}Y$. In other words, the indexing coordinate stays constant for $t \in [0,1)$, and switches to the appropriate coordinate at $t =1$.
We will occasionally discuss *branching geodesics*. A geodesic ${\gamma}:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ *branches* at $t_0 \in (0,1)$ if there exists a geodesic $\widetilde{{\gamma}}: [0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ such that $\widetilde{{\gamma}}$ agrees with ${\gamma}$ on $[0,t_0]$, and is distinct from ${\gamma}$ on $(t_0,t_0+{\varepsilon}]$ for some ${\varepsilon}> 0$.
With this terminology, we now pose the main question motivating this paper.
\[q:convex-comb\] For which pairs $(p,q)$ can we say that all geodesics in ${{{\mathcal}{D}}_{p}[l^q]}$ have the form of convex-combination geodesics?
Our first result shows that setting $p=\infty$ simultaneously produces branching and deviant geodesics in ${{{\mathcal}{D}}_{p}[l^q]}$. In particular, the existence of branching geodesics implies that $({{{\mathcal}{D}}_{p}[l^q]},{d_p[l^q]})$ for $q \in [1,\infty]$, $p = \infty$ cannot have nonnegative Alexandrov curvature ([@burago Chapter 10]).
\[thm:branch-dev-infty\] Let $p = \infty, \, q \in [1,\infty]$. There exist infinite families of both branching and deviant geodesics in $({{{\mathcal}{D}}_{p}[l^q]},{d_p[l^q]})$.
[@turner2013means] showed—via a direct examination of an inequality characterizing Alexandrov curvature—that in the case $p = q \in [1,2) \cup (2,\infty]$, ${{{\mathcal}{D}}_{p}[l^q]}$ does not have nonnegative Alexandrov curvature.
We collect another related result for the case $p =q =1$:
\[thm:branch-dev-one\] Let $q = p =1$. There exist infinite families of branching and deviant geodesics in $({{{\mathcal}{D}}_{p}[l^q]},{d_p[l^q]})$.
Theorems \[thm:branch-dev-infty\] and \[thm:branch-dev-one\] serve to make Question \[q:convex-comb\] more interesting. The next result is the finite version of our answer to Question \[q:convex-comb\].
\[thm:char-I\] Fix $p,q$ in the following ranges:
- $p = q \in [2,\infty),$
- $q = 2$, $p \in (1,\infty).$
Let $X, Y \in {{{\mathcal}{D}}_{p}[l^q]}$ be diagrams having finitely many points outside the diagonal, and let $\mu:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ be a geodesic from $X$ to $Y$. Then there exists a convex-combination geodesic ${\gamma}:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ from $X$ to $Y$ such that for each $t \in [0,1]$, we have ${d_p[l^q]}({\gamma}(t),\mu(t)) = 0$.
This result in fact generalizes to the setting of countably-many off-diagonal points.
\[thm:char-II\] Fix $p,q$ in the following ranges:
- $p = q \in [2,\infty),$
- $q = 2$, $p \in (1,\infty).$
Let $X, Y \in {{{\mathcal}{D}}_{p}[l^q]}$, and let $\mu:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ be a geodesic from $X$ to $Y$. Then there exists a convex-combination geodesic ${\gamma}:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ from $X$ to $Y$ such that for each $t \in [0,1]$, we have ${d_p[l^q]}({\gamma}(t),\mu(t)) = 0$.
These characterization theorems have the following interpretation. Suppose we are given $X, Y \in {{{\mathcal}{D}}_{p}[l^q]}$ for the specified choices of $p$ and $q$ and a geodesic $\mu$ from $X$ to $Y$. Then $\mu$ is a convex-combination geodesic for some optimal bijection ${\varphi}:X {\rightarrow}Y$. Furthermore, for each $x \in X$, we obtain a straight-line path $\gamma_x$ from $x$ to ${\varphi}(x)$. By the construction of convex-combination geodesics, any optimal bijection produces a geodesic; these theorems assert the *inverse* result that any geodesic comes from an optimal bijection, at least for the prescribed choices of $p$ and $q$.
Recasting diagram metrics as $l^p$ norms and OT problems {#sec:OT}
--------------------------------------------------------
A key property of persistence diagrams is that the diagonal is counted with infinite multiplicity; this geometric trick ensures that bijections are always possible, and hence the ${d_p[l^q]}$-type distances are always defined. While a persistence diagram contains uncountably many points according to Definition \[def:pdgm\], only countably many points are actually ever involved in computing a ${d_p[l^q]}$-type distance. Specifically, we can view ${d_p[l^2]}$ as an $l^p$ norm. To see this, let $X,Y \in {{{\mathcal}{D}}_{p}[l^2]}$. Recall that $X_>, Y_>$ consists of the (countably many) off-diagonal points of $X$ and $Y$. Define the $^*$ operation as the following: $$\begin{aligned}
X^* := X_> \cup \{ \pi_{\Delta}(y) : y \in Y_>\} \cup \Delta_{\mathbb{Q}}^\infty, \qquad \text{for } X \in {{{\mathcal}{D}}_{p}[l^q]}.\end{aligned}$$ The multiset $X^*$ consists of the off-diagonal points of $X$, a copy of the diagonal projection for each off-diagonal point of $Y$, and the rational diagonal points counted with countably infinite multiplicity. To ease the notation, we did not specify the indexing coordinates for the points in $\{ \pi_{\Delta}(y) : y \in Y_>\}$, but it is to be understood that the indices are chosen such that multiple off-diagonal points with the same diagonal projection are mapped to different slots in ${\mathbb{N}}$. The idea of including the rationals on the diagonal is the following: the set in the middle contains redundancies, so when obtaining matchings, it may be the case that the redundant diagonal points in $X_> \cup \{ \pi_{\Delta}(y) : y \in Y_>\}$ have to get matched to diagonal points in $Y_> \cup \{ \pi_{\Delta}(x) : x \in X_>\}$. By including rational points on the diagonal, we ensure (by the density of the rationals) that this matching of diagonal points contributes zero cost.
In particular, $X^*$ is a countable set (perhaps invoking the axiom of countable choice as necessary). Fix an enumeration $X^*= \{x^1,x^2,\ldots \}$. Then we think of $X^*$ as the map $X^*:{\mathbb{N}}{\rightarrow}{\mathbb{C}}$ given by $i \mapsto x_i \mapsto {\pi_{{\mathbb{R}}^2}}(x_i)$. Next define $Y^*$ analogously, and consider any bijection ${\varphi}:X^* {\rightarrow}Y^*$. We again treat ${\varphi}(X^*)$ as an infinite-dimensional vector, i.e, a map ${\varphi}(x): {\mathbb{N}}{\rightarrow}{\mathbb{C}}$ given by $i\mapsto {\varphi}(x_i) \mapsto {\pi_{{\mathbb{R}}^2}}({\varphi}(x_i))$. Here we are using the canonical identification of ${\mathbb{C}}$ with ${\mathbb{R}}^2$.
Next we introduce some cost functions. Let $p,q \in [1,\infty]$, and let $X,Y \in {{{\mathcal}{D}}_{p}[l^q]}$. Define the following functional for a bijection ${\varphi}:X {\rightarrow}Y$: $$\begin{aligned}
C_p[l^q]({\varphi}):= \begin{cases}
{\left(}\sum_{x\in X} {\left\|x - {\varphi}(x) \right\|}_q^p {\right)}^{1/p} &: p \in [1,\infty)\\
\sup_{x\in X} {\left\|x - {\varphi}(x) \right\|}_q &: p = \infty.
\end{cases}
\label{eq:Cp-defn}\end{aligned}$$ When $q = 2$, we reduce notation and simply write $C_p$ instead of $C_p[l^2]$.
For the next few definitions, we fix $q=2$ and consider $X,Y \in {{{\mathcal}{D}}_{p}[l^2]}$. Now for $p \in [1,\infty)$, consider the functional $$\begin{aligned}
J_p({\varphi}) := {\left\| X^* - {\varphi}(X^*) \right\|}_{l^p},
\label{eq:dgm-norm}\end{aligned}$$ where the $l^p$-norm is given as $$\begin{aligned}
{\left\|X^* - {\varphi}(X^*) \right\|}_{l^p} := {\big(}\sum_{i \in {\mathbb{N}}} | x_i - {\varphi}(x_i) |^p {\big)}^{1/p} = {\big(}\sum_{i \in {\mathbb{N}}} {\left\| x_i - {\varphi}(x_i) \right\|}_2^p {\big)}^{1/p}\end{aligned}$$ if the sum converges, and as $\infty$ otherwise. Note that by our choice of $X,Y \in {{{\mathcal}{D}}_{p}[l^2]}$, there always exists ${\varphi}$ such that the preceding sum converges. For such ${\varphi}$, the vector $X^* - {\varphi}(X^*)$ belongs to $l^p$. Here also recall from Definition \[def:pdgm\] that ${\left\|x - y \right\|}_p = {\left\| {\pi_{{\mathbb{R}}^2}}(x) - {\pi_{{\mathbb{R}}^2}}(y) \right\|}_p$. Each summand is an absolute value, i.e. a Euclidean norm, that is raised to the $p$th power. The $l^\infty$ norm is likewise defined as $${\left\|x - {\varphi}(x) \right\|}_{l^\infty} : = \sup_{i\in {\mathbb{N}}} {\left\|x_i - {\varphi}(x_i) \right\|}_2.$$
For any bijection ${\varphi}':X {\rightarrow}Y$, define $\Lambda_{{\varphi}'}$ to be the collection of bijections ${\varphi}:X^* {\rightarrow}Y^*$ agreeing with ${\varphi}'$ on off-diagonal points of $X$ and $Y$.
By the construction of $X^*, Y^*$ and the density of the rationals, we have $$\begin{aligned}
C_p({\varphi}') = {\big(}\sum_{x \in X} {\left\|x - {\varphi}'(x) \right\|}_2^p {\big)}^{1/p} = \inf \{ J_p({\varphi}) : {\varphi}\in \Lambda_{{\varphi}'} \} \, \text{for any bijection ${\varphi}':X {\rightarrow}Y$}.
\label{eq:bijection-approx}\end{aligned}$$ In particular, the matching cost of ${\varphi}\in \Lambda_{{\varphi}'}$ differs from that of ${\varphi}'$ only in how it produces a matching among the “redundant" points on the diagonal.
Finally we observe that for all $p \in [1,\infty]$, $${d_p[l^2]}(X,Y) = \inf \{C_p({\varphi}) : {\varphi}:X {\rightarrow}Y \text{ a bijection} \} = \inf \{J_p({\varphi}) : {\varphi}:X^* {\rightarrow}Y^* \text{ a bijection} \}.$$ Here we are using the following observations: (1) any ${\varphi}$ infimizing $C_p$ does not move diagonal points unnecessarily, and (2) any ${\varphi}$ infimizing $J_p$ agrees with a $C_p$ infimizer on off-diagonal points and incurs zero cost for infinitesimally “sliding" points along the diagonal.
The distinction between $J_p$ and $C_p$ is that $J_p$ is an $l^p$ norm. This reformulation allows us to use powerful $l^p$ space inequalities to produce results for ${d_p[l^2]}$. It is not clear to us if this approach can be extended to ${d_p[l^q]}$ for $q \neq 2$; attempting to prove one of the inequalities we need (Clarkson’s inequality, Lemma \[lem:clarkson\]) with $q \neq 2$ leads to some difficulty.
At least in the case of diagrams having finitely many off-diagonal points, one could similarly reformulate a ${d_p[l^2]}$ distance as an *optimal transportation* (OT) problem. This idea is used below, where we describe a method ([@lacombe2018large]) for recasting the computation of a diagram metric as an OT problem.
Given appropriately defined measures $\mu, \nu$ on measure spaces $X$ and $Y$, we write ${\mathscr{C}}(\mu,\nu)$ to denote the collection of all *coupling measures*, i.e. measures ${\gamma}$ on $X\times Y$ with marginals $\mu$ and $\nu$.
Following [@lacombe2018large], we let ${\Delta^{\bullet}}$ denote a virtual point representing the diagonal. We also use the notation ${\mathbb{R}}^{2\bullet}:= {\mathbb{R}}^2 \cup \{{\Delta^{\bullet}}\}$ (and resp. ${\mathbb{R}}^{2\bullet}_>:= {\mathbb{R}}^2_> \cup \{{\Delta^{\bullet}}\}$). For $x \in {\mathbb{R}}^2$, we use the notation ${\left\|}x - {\Delta^{\bullet}}{\right\|}_p$ to denote ${\left\|}x - \pi_{\Delta}(x) {\right\|}_p$. We also set ${\left\|}{\Delta^{\bullet}}- {\Delta^{\bullet}}{\right\|}_p = 0$.
Let $X, Y \in {{{\mathcal}{D}}_{p}[l^2]}$, $p \in [1,\infty]$, be diagrams having finitely many off-diagonal points. Let $n_X := |X_>|, \, n_Y:= |Y_>|$, and set $n:= n_X + n_Y$. Then we define: $$\begin{aligned}
X^\bullet &:= X_> \cup \{ ({\Delta^{\bullet}}, j) : 1\leq j \leq n_Y \} \subseteq {\mathbb{R}}^{2\bullet}_> \times {\mathbb{N}}, \\
Y^\bullet &:= Y_> \cup \{ ({\Delta^{\bullet}}, j) : 1\leq j \leq n_X \} \subseteq {\mathbb{R}}^{2\bullet}_> \times {\mathbb{N}}. \end{aligned}$$
Then $n = |X^\bullet| = |Y^\bullet|$. Given arbitrary measures $\mu_{X^\bullet},\mu_{Y^\bullet}$ on $X^\bullet$ and $Y^\bullet$, respectively, and a coupling measure ${\gamma}\in {\mathscr{C}}(\mu_{X^\bullet},\mu_{Y^\bullet})$ (i.e. a transport plan), the $L^p[l^2]$ transport cost is defined as: $$\begin{aligned}
T_p({\gamma}) := {\left\|}\pi_{X^\bullet} - \pi_{Y^\bullet} {\right\|}_{L^p({\gamma})}
&= {\left(}\int_{X^\bullet\times Y^\bullet} | \pi_{X^\bullet}(x,y) - \pi_{Y^\bullet}(x,y) |^p \,d{\gamma}(x,y) {\right)}^{1/p}\\
&= {\left(}\sum_{i,j} {\left\|}x_i - y_j {\right\|}_2^p {\gamma}(x_i,y_j) {\right)}^{1/p}.\end{aligned}$$ Here $\pi_{X^\bullet} : X^\bullet\times Y^\bullet {\rightarrow}X^\bullet , \pi_{Y^\bullet}:X^\bullet\times Y^\bullet {\rightarrow}Y^\bullet$ are the canonical projection maps. More specifically, by taking the canonical identification of ${\mathbb{R}}^2$ with ${\mathbb{C}}$, these are maps $X^\bullet \times Y^\bullet {\rightarrow}{\mathbb{C}}$, so we are able to view them as maps in the $L^p$ space of complex-valued measurable functions. Measurability holds because these maps, being defined on discrete spaces, are trivially continuous. The absolute value in the integrand is taken for complex numbers, i.e. it corresponds to the Euclidean norm. The $l^2$ ground norm is the canonical choice when working over an $L^p$ space.
Next let $\mu_{X^\bullet}:= \sum_{i=1}^n {\delta}_{x_i}$ denote the uniform measure on $X^\bullet$, and similarly let $\mu_{Y^\bullet}$ denote the uniform measure on $Y^\bullet$. Then we have (see also [@lacombe2018large Proposition 1]):
\[prop:OT\] Given $X,Y, X^\bullet,Y^\bullet$ as above, we have the following identity: $${d_p[l^2]}(X,Y) = \inf_{{\gamma}\in {\mathscr{C}}(\mu_{X^\bullet},\mu_{Y^\bullet})} T_p({\gamma}).$$
It is well-known as a consequence of Birkhoff’s theorem (see [@villani2003topics §0.1]) that the OT cost between measures on $n$-point spaces giving equal mass to all points is realized by a coupling that can be represented as an $n\times n$ permutation matrix. This permutation ${\sigma}$ provides the bijection in the definition of ${d_p[l^2]}(X,Y)$.
The preceding OT formulation appears to work only in the case of diagrams with finitely many off-diagonal points. It would be interesting to clarify if a ${d_p[l^2]}$ distance between diagrams having countably many off-diagonal points can be formulated as an OT problem. The difficulty arises from ensuring that the optimal transportation plans correspond to permutation matrices, as required for the bijections in the definition of ${d_p[l^2]}$.
Branching and deviant geodesics {#sec:branch}
===============================
We now proceed to the proofs of Theorems \[thm:branch-dev-infty\] and \[thm:branch-dev-one\].
\[node distance = 0pt, X/.style=[circle,draw=purple,fill=purple,thick,inner sep = 0pt,minimum size = 6pt]{}, Y/.style=[rectangle,draw=NavyBlue,fill=NavyBlue,thick,inner sep = 0pt,minimum size = 6pt]{}\] (0,0)–(5,0) ; (0,0)–(0,5) ; (0,0)–(4.5,4.5); (x) at (0,4.5) ; (x’) at (0,2.5) ; (mu) at (1,4) [$\mu$]{}; (0,2.5)–(1.25,1.25); (0,4.5)–(2.25,2.25); ; ;
\[node distance = 0pt, X/.style=[circle,draw=purple,fill=purple,thick,inner sep = 0pt,minimum size = 6pt]{}, Y/.style=[rectangle,draw=NavyBlue,fill=NavyBlue,thick,inner sep = 0pt,minimum size = 6pt]{}\] (0,0)–(5,0) ; (0,0)–(0,5) ; (0,0)–(4.5,4.5); (y) at (0,4.5) ; (y’) at (0,1) ; (nu) at (1,4) [$\nu$]{}; (0,1)–(0.5,0.5); (0,4.5)–(2.25,2.25); ; ;
We begin with the proof of branching geodesics. Let $X = {\varnothing}$, $Y = \{(0,k),(0,j)\}$, and $Z = \{(0,k),(0,l)\}$ for $0 < l < j < 3j < k$. Now we define two curves $\mu, \nu: [0,1] {\rightarrow}{{{\mathcal}{D}}_{\infty}[l^q]}$ as follows: $$\begin{aligned}
\mu(t) &{\overset{\tiny\operatorname{def}}{=}}{
\begin{cases}
{\left\{{\left(}\frac{k}{2}t, \frac{k}{2}(2-t){\right)}, {\left(}\frac{j}{2}(3t), \frac{j}{2}(2-3t) {\right)}\right\}} & 0 \leq t \leq \frac{1}{3}\\
{\left\{ {\left(}\frac{k}{2}t, \frac{k}{2}(2-t) {\right)}\right\}} & \frac{1}{3} \leq t \leq 1.
\end{cases}
}\\
\nu(t) &{\overset{\tiny\operatorname{def}}{=}}{
\begin{cases}
{\left\{{\left(}\frac{k}{2}t, \frac{k}{2}(2-t){\right)}, {\left(}\frac{l}{2}(3t), \frac{l}{2}(2-3t) {\right)}\right\}} & 0 \leq t \leq \frac{1}{3}\\
{\left\{ {\left(}\frac{k}{2}t, \frac{k}{2}(2-t) {\right)}\right\}} & \frac{1}{3} \leq t \leq 1.
\end{cases}
}\end{aligned}$$
Thus $\mu, \nu$ are curves from $Y, Z$ to $X$. For convenience, define $$\mathbf{k}(t){\overset{\tiny\operatorname{def}}{=}}{\left(}\tfrac{k}{2}t, \tfrac{k}{2}(2-t){\right)}\text{ for }t \in [0,1], \qquad \mathbf{j}(t) {\overset{\tiny\operatorname{def}}{=}}{\left(}\tfrac{j}{2}(3t), \tfrac{j}{2}(2-3t) {\right)}\text{ for } t \in [0,\tfrac{1}{3}].$$
We check that $\mu,\nu$ are geodesics. It suffices to show this for $\mu$. First we see that ${d_{\infty}[l^q]}(X,Y)$ is the perpendicular $q$-norm distance from $(0,k)$ to the diagonal; this is just $2^{(1/q) - 1}k$.
Let $s,t \in [\frac{1}{3},1]$. We observe that an optimal bijection matches $\mathbf{k}(s)$ and $\mathbf{k}(t)$; hence we have: $${d_{\infty}[l^q]}(\mu(s),\mu(t)) = {\left\|}{\left(}\tfrac{k}{2}t, \tfrac{k}{2}(2-t){\right)}- {\left(}\tfrac{k}{2}s, \tfrac{k}{2}(2-s){\right)}{\right\|}_q = 2^{(1/q)-1}k{\left|}t - s {\right|}= {\left|}t- s {\right|}{d_{\infty}[l^q]}(X,Y).$$
Let $s,t \in [0,\frac{1}{3}]$. First we claim that ${d_{\infty}[l^q]}(\mu(s),\mu(t))$ is realized by the $q$-norm distance between $\mathbf{k}(s)$ and $\mathbf{k}(t)$. By the previous work, this is just $2^{(1/q)-1}k{\left|}t - s {\right|}$. We compare this to ${\left\|}\mathbf{j}(s) - \mathbf{j}(t){\right\|}_q$: $${\left\|}{\left(}\tfrac{j}{2}(3t), \tfrac{j}{2}(2-3t) {\right)}-
{\left(}\tfrac{j}{2}(3s), \tfrac{j}{2}(2-3s) {\right)}{\right\|}_q
=
(3j)|t-s|2^{(1/q)-1}
< k|t-s|2^{(1/q)-1},$$ where the last inequality holds because $3j < k$ by assumption. Thus ${d_{\infty}[l^q]}(\mu(s),\mu(t)) = {\left|}t- s {\right|}{d_{\infty}[l^q]}(X,Y).$ Notice that in this computation, it was implicit that an optimal matching would match $\mathbf{k}(s)$ to $\mathbf{k}(t)$ and $\mathbf{j}(s)$ to $\mathbf{j}(t)$; a cross-matching would not be optimal due to the greater distance that would need to be traversed.
Finally let $s \in [0,\frac{1}{3}], t \in (\frac{1}{3},1]$. Again we claim that ${d_{\infty}[l^q]}(\mu(s),\mu(t))$ is realized by ${\left\|}\mathbf{k}(s) - \mathbf{k}(t) {\right\|}_q$. The previous work shows that ${\left\|}\mathbf{k}(s) - \mathbf{k}(t) {\right\|}_q > {\big\|}\mathbf{j}(s) - (\tfrac{j}{2},\tfrac{j}{2}) {\big\|}_q$. It follows that ${d_{\infty}[l^q]}(\mu(s),\mu(t)) = {\left|}t- s {\right|}{d_{\infty}[l^q]}(X,Y).$
This shows that $\mu$ is a geodesic. The proof for $\nu$ is analogous. So $\mu, \nu$ are geodesics which are equal on $[\tfrac{1}{3},1]$, but clearly they branch at $t= \tfrac{1}{3}$ since ${d_{\infty}[l^q]}(Y,Z) > 0$. Since $l < j < 3j < k$ were arbitrary, there are in fact infinitely many such branching geodesics. This concludes the first part of the proof. $\blacksquare$
Notice that $\mu, \nu$ are not convex-combination geodesics; the points at $(0,j)$ and $(0,l)$ move too fast for the geodesics to be convex-combination, but slow enough that the geodesic property still holds. Even though these are deviant geodesics, there still seem to be bijections providing straight lines for the points to interpolate through. However, this need not be the case, and deviant geodesics may exist even when there is no supporting bijection. We see such a construction next.
Let $W = \{(0,k)\}$. Now we define a curve ${\omega}:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ as follows: $$\begin{aligned}
{\omega}(t) &{\overset{\tiny\operatorname{def}}{=}}{
\begin{cases}
{\left\{{\left(}\frac{k}{2}t, \frac{k}{2}(2-t){\right)}, {\left(}j(\frac{1}{2} - t) , j(\frac{1}{2} + t) {\right)}\right\}} & 0 \leq t \leq \frac{1}{2}\\
{\left\{ {\left(}\frac{k}{2}t, \frac{k}{2}(2-t) {\right)}, {\left(}j(t - \frac{1}{2}) , j(\frac{3}{2} - t) {\right)}\right\}} & \frac{1}{2} \leq t \leq 1.
\end{cases}
}\end{aligned}$$ Then ${\omega}$ is a curve from $W$ to $X$. Note that ${\omega}(0)$ contains one off-diagonal point $(0,k)$, and this point linearly moves to the diagonal as $t \uparrow 1$. However, starting at $t = 0$, a point emerges from the diagonal at $(j/2,j/2)$ and moves linearly to $(0,j)$ as $t\uparrow 1/2$, which then returns to the diagonal as $t \uparrow 1$. Calculations such as the ones carried out above show that ${\omega}$ is a geodesic; for the reader’s convenience, we note that the point moving back and forth between $(0,j)$ and the diagonal has speed $j < k/3,$ so the $l^\infty$ matching only sees the $q$-norm distance between $\mathbf{k}(s)$ and $\mathbf{k}(t)$. This is the reason ${\omega}$ is a geodesic. However, ${\omega}$ is not a convex-combination geodesic from $W$ to $X$. Moreover, for different choices of $j$, we get infinitely many geodesics from $W$ to $X$, all of which are mutually distinct. This concludes the proof.
Next we proceed to the proof of Theorem \[thm:branch-dev-one\].
Fix $k \gg 0$ so that we do not have to consider situations where points are matched to the diagonal. Let $X = \{(0,k), (1,k-1)\}$ and $Y = \{(1,k+1),(2,k)\}$. This configuration is illustrated in Figure \[fig:deviant11\]. Define a curve $\mu:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ as follows:
$$\begin{aligned}
\mu(t) &{\overset{\tiny\operatorname{def}}{=}}{
\begin{cases}
{\left\{{\left(}2t,k {\right)}, {\left(}1, k-1+2t {\right)}\right\}} & 0 \leq t \leq \frac{1}{2}\\
{\left\{ {\left(}1,k+2(t- \frac{1}{2}) {\right)}, {\left(}1 + 2(t-\frac{1}{2}), k {\right)}\right\}} & \frac{1}{2} \leq t \leq 1.
\end{cases}
}\end{aligned}$$
This curve corresponds to the lefmost configuration in Figure \[fig:deviant11\]. The points $x,x'$ come together at the center, then bend and travel to $y,y'$, respectively. Next we verify that $\mu$ is a geodesic. First note that $d_1[l^1](X,Y) = 4$. Next let $s\leq t \in [0,\frac{1}{2}].$ The optimal matching between $\mu(s)$ and $\mu(t)$ happens in the simple way: points along the dashed line get matched, and points along the solid line get matched (here we are referring to Figure \[fig:deviant11\]). The cost of this matching is as follows: $$\begin{aligned}
d_1[l^1](\mu(s),\mu(t)) &= {\left\|}(2s,k) - (2t,k) {\right\|}_1 + {\left\|}(1,k-1+2s) - (1,k-1+2t) {\right\|}_1\\
&= 2(t-s) + 2(t-s) = |t-s| \, d_1[l^1](X,Y).\end{aligned}$$
The verification for $s,t \in [\frac{1}{2},1]$ is analogous. An interesting case is $s \in [1,\frac{1}{2}),\, t \in [\frac{1}{2},1]$. By virtue of using the $l^1$ ground metric, there are two optimal bijections: matching the points according to the dashed/solid lines, and cross-matching points on the dashed and solid lines. Using the first of these bijections, we calculate: $$\begin{aligned}
d_1[l^1](\mu(s),\mu(t)) &= {\big\|}(2s,k) - (1,k+2(t-\tfrac{1}{2})) {\big\|}_1 + {\big\|}(1,k-1+2s) - (1+2(t-\tfrac{1}{2}),k) {\big\|}_1\\
&= 2(t-s) + 2(t-s) = |t-s| \, d_1[l^1](X,Y).\end{aligned}$$ Thus $\mu$ is a geodesic. Note that it is different from the convex-combination geodesic illustrated at the right of Figure \[fig:deviant11\].
Moreover, note that curves with corners, as illustrated in the middle of Figure \[fig:deviant11\], would also be geodesics by virtue of the ground metric being $l^1$. There is an infinite choice of positions for these corners, and so we get an infinite family of deviant geodesics which are all distinct from each other. $\blacksquare$
Now we proceed to the proof of branching geodesics. We refer the reader to Figure \[fig:branch11\]. Starting with $X = \{(0,k), (1,k-1)\}$ as before and a fixed $r \in [0,1]$, consider the curve $\nu_r$ which: (1) transports the points $x = (0,k)$ and $x' = (1,k-1)$ to $(1,k)$ at constant speed over the interval $t \in [0,\frac{1}{2}]$, and (2) moves $x,x'$ jointly to $(1,k+r)$ and then to $(1 + (1-r), k+r)$, all at constant speed over the interval $t \in [\frac{1}{2},1]$. The cases $r = 1, 0, 0.5$ are illustrated from left to right, respectively, in Figure \[fig:branch11\]. Calculations analogous to the ones carried out above show that these curves are all geodesics, and by construction, they branch at $t = \frac{1}{2}$. Thus $\{\nu_r : r \in [0,1]$ is an infinite family of branching geodesics in ${{{\mathcal}{D}}_{p}[l^q]}$.
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\[node distance = 0pt, X/.style=[circle,draw=purple,fill=purple,thick,inner sep = 0pt,minimum size = 6pt]{}, Y/.style=[rectangle,draw=NavyBlue,fill=NavyBlue,thick,inner sep = 0pt,minimum size = 6pt]{}\] (x) at (-1,0) ; (x’) at (0,-1) ; (y) at (0,1) ; (y’) at (1,0) ; (y’) at (0,0) ; (-1,0)–(0,0); (0,-1)–(0,0); (0,0)–(0,1); ; ;
\[node distance = 0pt, X/.style=[circle,draw=purple,fill=purple,thick,inner sep = 0pt,minimum size = 6pt]{}, Y/.style=[rectangle,draw=NavyBlue,fill=NavyBlue,thick,inner sep = 0pt,minimum size = 6pt]{}\] (x) at (-1,0) ; (x’) at (0,-1) ; (y) at (0,1) ; (y’) at (1,0) ; (y’) at (0,0) ; (-1,0)–(0,0); (0,-1)–(0,0); (0,0)–(1,0); ; ;
\[node distance = 0pt, X/.style=[circle,draw=purple,fill=purple,thick,inner sep = 0pt,minimum size = 6pt]{}, Y/.style=[rectangle,draw=NavyBlue,fill=NavyBlue,thick,inner sep = 0pt,minimum size = 6pt]{}\] (x) at (-1,0) ; (x’) at (0,-1) ; (y) at (0.5,0.5) ; (y’) at (0,0) ; (-1,0)–(0,0); (0,-1)–(0,0); (0,0)–(1,0); (0,0)–(0,1); (1,0)–(0,1); (0,0.5)–(0.5,0.5); (0,0)–(0,0.5); ; ;
Characterization of geodesics {#sec:geod-char}
=============================
A preliminary result about limiting bijections
----------------------------------------------
We now collect a lemma (Lemma \[lem:limiting-bijection\]) showing how, given a sequence of bijections between persistence diagrams, we can pick out a subsequence of bijections that converges pointwise to a limiting bijection. This lemma is used directly in proving Theorem \[thm:char-II\] from Theorem \[thm:char-I\], and as a corollary we also obtain the existence of optimal bijections between diagrams, which is used throughout the proof of Theorem \[thm:char-I\]. The main proof technique is a standard diagonal argument with some additional consideration for the multiset nature of persistence diagrams, and a similar proof appeared in [@turner2013means Proposition 1].
Our reason for viewing persistence diagrams in ${\mathbb{R}}^2 \times {\mathbb{N}}$ becomes apparent in this section. We view ${\mathbb{R}}^2 \times {\mathbb{N}}$ as a subset of ${\mathbb{R}}^3$ endowed with the subspace topology. This allows us to invoke the Bolzano-Weierstrass theorem to obtain convergent sequences.
#### Notation
Below we will write $\lim$ to denote limits with respect to the usual topology in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$. This is different from a ${d_p[l^q]}$ limit, which uses only the geometric component of a point in a persistence diagram and ignores the indexing coordinate. Also, we interchangeably write $X_>$ or $X\setminus \Delta$ to denote the off-diagonal points of a persistence diagram $X$, depending on which notation better preserves typography. To emphasize that each point in a persistence diagram is a vector, we use boldface notation, e.g. ${\mathbf{x}}$ or ${\mathbf{y}}$.
For a point $a$ in a persistence diagram, we write $\pi_{\Delta}(a) \in {\mathbb{R}}^2$ to denote its projection onto the diagonal, ignoring the indexing coordinate ${\mathbb{N}}$. In other words, it is the shorthand notation for projecting a point to its geometric component in ${\mathbb{R}}^2$, and then further projecting the resulting point to the diagonal.
\[lem:bw-r3\] Fix $p,q \in [1,\infty]$. Let $X, Y \in {{{\mathcal}{D}}_{p}[l^q]}$, and let ${\mathbf{y}}\in Y\setminus \Delta$. Then,
1. There exists ${\varepsilon}>0 $ such that $B_{{\mathbb{R}}^3}({\mathbf{y}},{\varepsilon}) \cap Y = \{{\mathbf{y}}\}$.
Suppose also that $\Phi_k:X {\rightarrow}Y$ is a sequence of bijections and ${\mathbf{x}}\in X$ is such that $\lim_k \Phi_k({\mathbf{x}}) = {\mathbf{y}}$.
1. Then there exists $k_0 \in {\mathbb{N}}$ such that for all $k \geq k_0$, we have $\Phi_k({\mathbf{x}}) = {\mathbf{y}}$.
Finally, suppose ${\mathbf{y'}}\in \Delta$ and ${\mathbf{x'}}\in X\setminus \Delta$ is such that $\lim_k \Phi_k({\mathbf{x'}}) = {\mathbf{y'}}$. Suppose also that $C_p[l^q](\Phi_k) {\rightarrow}{d_p[l^q]}(X,Y)$ as $k {\rightarrow}\infty$. Then,
1. $\pi_{{\mathbb{R}}^2}({\mathbf{y'}}) = \pi_{\Delta}({\mathbf{x'}})$ (i.e. optimal bijections map ${\mathbf{x'}}$ to the diagonal via orthogonal projection).
Let ${\varepsilon}>0$ be small enough so that $B_{{\mathbb{R}}^3}({\mathbf{y}},2{\varepsilon}) \cap \Delta = {\varnothing}$, and define $U:=B_{{\mathbb{R}}^3}({\mathbf{y}},{\varepsilon}) \cap Y \cap \Delta = {\varnothing}$. Then $U$ has a strictly positive distance to the diagonal. Since $Y \in {{{\mathcal}{D}}_{p}[l^q]}$, there can only be finitely many points, including multiplicity, in $U$. Different copies of ${\mathbf{y}}$ in $U$ have the same ${\mathbb{R}}^2$ coordinates, but differ on the ${\mathbb{N}}$ coordinate by at least 1. Thus ${\varepsilon}$ can be made sufficiently small so that $B_{{\mathbb{R}}^3}({\mathbf{y}},{\varepsilon}) \cap Y = \{{\mathbf{y}}\}$. This proves the first assertion. The second assertion follows immediately.
The third assertion is also easy to see, and we provide a few lines of proof. Suppose toward a contradiction that ${\mathbf{y'}}\in \Delta$ and $\lim_k \Phi_k({\mathbf{x'}}) = {\mathbf{y'}}$, but $\pi_{{\mathbb{R}}^2}({\mathbf{y'}}) \neq \pi_{\Delta}({\mathbf{x'}})$, i.e. ${\mathbf{y'}}$ is not the diagonal projection of ${\mathbf{x'}}$. Then there exists ${\varepsilon}> 0$ and $\eta_{\varepsilon}>0$ such that the distance from $\pi_{{\mathbb{R}}^2}({\mathbf{x'}})$ to $B_{{\mathbb{R}}^2}(\pi_{{\mathbb{R}}^2}({\mathbf{y'}}),{\varepsilon})$ is at least ${\left\|\pi_{{\mathbb{R}}^2}({\mathbf{x'}}) - \pi_{\Delta}({\mathbf{x'}}) \right\|}_{q} + \eta_{\varepsilon}$. Thus there exists $k_0 \in {\mathbb{N}}$ such that for all $k \geq k_0$, ${\left\|\pi_{{\mathbb{R}}^2}({\mathbf{x'}}) - \pi_{{\mathbb{R}}^2}(\Phi_k({\mathbf{x'}})) \right\|}_{q} > {\left\|\pi_{{\mathbb{R}}^2}({\mathbf{x'}}) - \pi_{\Delta}({\mathbf{x'}}) \right\|}_{q} + \eta_{\varepsilon}$. But then $C_p[l^q](\Phi_k) \geq {d_p[l^q]}(X,Y) + \eta_{\varepsilon}$ for all $k \geq k_0$. This is a contradiction.
Here is the main result of this section.
\[lem:limiting-bijection\] Let $p,q \in [1,\infty]$. Let $X,Y \in {{{\mathcal}{D}}_{p}[l^q]}$, and let $\Phi_k:X {\rightarrow}Y$ be a sequence of bijections such that $C_p[l^q](\Phi_k) {\rightarrow}{d_p[l^q]}(X,Y)$. Then there exists a subsequence indexed by $L \subseteq {\mathbb{N}}$ and a limiting bijection $\Phi_*$ such that $\Phi_k \xrightarrow{k \in L,\, k {\rightarrow}\infty} \Phi_*$ pointwise and $C_p[l^q](\Phi_*) = {d_p[l^q]}(X,Y)$.
For each $\Phi_k$, we let ${\Psi^{XY}}_k:X {\rightarrow}Y|_{{\mathbb{R}}^2}$ denote the geometric part (i.e. the ${\mathbb{R}}^2$ component) of $\Phi_k$. We also write ${\Psi^{YX}}_k: Y {\rightarrow}X|_{{\mathbb{R}}^2}$ to denote the geometric component of the inverse map $\Phi_k{^{-1}}$. Recall that only the geometric component is involved in ${d_p[l^q]}$ computations (cf. Definition \[def:pdgm\]).
Define $Y_0:= (Y\setminus \Delta) \cup
\{{\mathbf{y}}\in \Delta : \pi_{{\mathbb{R}}^2}({\mathbf{y}}) = \pi_{\Delta}({\mathbf{x}}),\; {\mathbf{x}}\in X\setminus \Delta\}$. Then $Y_0$ denotes the union of the countably many off-diagonal points of $Y$ with the countably many copies of diagonal points that are projections of off-diagonal points in $X$. This is a countable set. Fix an enumeration $Y_0 = \{{\mathbf{y}}^{(n)}\}_{n \in {\mathbb{N}}}$.
Since $X,Y \in {{{\mathcal}{D}}_p[l^q]}$, ${d_p[l^q]}(X,Y) < \infty$, and $C_p[l^q](\Phi_k) {\rightarrow}{d_p[l^q]}(X,Y)$, we know ${\left(}{\Psi^{YX}}_k({\mathbf{y}}{^{(i)}}){\right)}_k$ is a bounded sequence in ${\mathbb{R}}^2$ for each $i \in {\mathbb{N}}$. By a diagonal argument and the Bolzano-Weierstrass theorem, we obtain a diagonal subsequence indexed by $J \subseteq {\mathbb{N}}$ such that $({\Psi^{YX}}_k)_{k \in J}$ converges pointwise on $Y_0$. Define ${\Psi^{YX}}_*$ on $Y_0$ by setting ${\Psi^{YX}}_*({\mathbf{y}}) := \lim_{k {\rightarrow}\infty, \, k\in J}{\Psi^{YX}}_k({\mathbf{y}})$ for each ${\mathbf{y}}\in Y_0$. Note that if ${\Psi^{YX}}_*({\mathbf{y}}{^{(i)}}) \in \Delta$ for some ${\mathbf{y}}{^{(i)}}\in Y_0 \setminus \Delta$, then by an argument analogous to that of Lemma \[lem:bw-r3\], we have ${\Psi^{YX}}_*({\mathbf{y}}{^{(i)}}) = \pi_{\Delta}({\mathbf{y}}{^{(i)}}) \in X$.
Next define: $$\begin{aligned}
Q:= \{( \pi_{\Delta}({\mathbf{y}}{^{(i)}}) , i ) : {\mathbf{y}}{^{(i)}}\in Y_0 \setminus \Delta, \; {\Psi^{YX}}_*({\mathbf{y}}{^{(i)}}) \in \Delta \}, && X_1:= (X\setminus \Delta) \sqcup
Q.\end{aligned}$$
$Q$ contains all the diagonals of $X$ matched to off-diagonals in $Y$. $X_1$ is countable, so another application of a diagonal argument and the Bolzano-Weierstrass theorem gives a subsequence indexed by $K\subseteq J$ such that $({\Psi^{XY}}_k)_{k \in K}$ converges pointwise on $X_1$. Define ${\Psi^{XY}}_*({\mathbf{x}}):= \lim_{k {\rightarrow}\infty, \, k \in K}({\Psi^{XY}}_k({\mathbf{x}}))$ for each ${\mathbf{x}}\in X_1$.
Next write $Y_0^{{\triangle}}$ and $Y_0^{{\blacktriangle}}$ to denote the off-diagonal and on-diagonal points of $Y_0$, respectively. Define $A:= \{{\mathbf{x}}\in (X\setminus \Delta) : {\Psi^{XY}}_*({\mathbf{x}}) \in Y_0^{{\triangle}}|_{{\mathbb{R}}^2}\}$ and $B:= (X\setminus \Delta) \setminus A$. Fix an enumeration $\{{\mathbf{x}}{^{(n)}}\}_{n \in {\mathbb{N}}}$ on $B$. Note the following descriptions of the sets $A$ and $B$ in terms of how they should be matched by the limiting bijection: $A$ contains all the off-diagonal points of $X$ that are matched to off-diagonals in $Y$, and $B$ contains all the off-diagonals of $X$ matched to diagonals in $Y$. In particular, $X_1 = A \sqcup B \sqcup Q$.
Each point in $Y_0^{{\triangle}}$ has finite multiplicity, because otherwise we would have $Y \not\in{{{\mathcal}{D}}_p[l^q]}$. Thus for any ${\mathbf{x}}\in A$, ${\left(}\Phi_{k}({\mathbf{x}}) {\right)}_{k\in K}$ is a bounded sequence in ${\mathbb{R}}^2 \times {\mathbb{N}}$ by Lemma \[lem:bw-r3\]. By the Bolzano-Weierstrass theorem and a diagonal argument as above, we get a subsequence ${\left(}\Phi_{k}{\right)}_{k\in L}$ indexed by $L\subseteq K$ converging pointwise on $A$. Since $L\subseteq K$, we have $\lim_{k{\rightarrow}\infty,\, k\in L} \pi_{{\mathbb{R}}^2}{\left(}\Phi_{k}({\mathbf{x}}){\right)}= {\Psi^{XY}}_*({\mathbf{x}})$ for each ${\mathbf{x}}\in A$.
Define $\Phi_*:X_1 {\rightarrow}Y$ by writing the following for each ${\mathbf{x}}\in X_1$: $$\Phi_* ({\mathbf{x}}):=
\begin{cases}
\lim_{k {\rightarrow}\infty,\, k \in L} \Phi_{k}({\mathbf{x}}) &: {\mathbf{x}}\in A\\
{\mathbf{y}}{^{(i)}}&: {\mathbf{x}}\in Q,\, {\mathbf{x}}= (\pi_{\Delta}({\mathbf{y}}{^{(i)}}),i) \\
(\pi_{\Delta}({\mathbf{x}}{^{(i)}}),i) &: {\mathbf{x}}\in B,\, {\mathbf{x}}= {\mathbf{x}}{^{(i)}}.
\end{cases}$$
**Claim: $\Phi_*|_A: A {\rightarrow}Y_0^{{\triangle}}$ is injective.** Let ${\mathbf{x}},{\mathbf{x'}}\in A$ be such that $\Phi_*({\mathbf{x}}) = \Phi_*({\mathbf{x'}})$. Write ${\mathbf{y}}:= \Phi_*({\mathbf{x}})$. By Lemma \[lem:bw-r3\], we obtain ${\varepsilon}> 0$ and $k_0 \in {\mathbb{N}}$ such that $\Phi_{k}({\mathbf{x}}), \Phi_{k}({\mathbf{x'}}) \in B_{{\mathbb{R}}^3}({\mathbf{y}},{\varepsilon})$ for all $ k \geq k_0$, $k \in L$. Thus for such $k$, ${\mathbf{x}}= \Phi_{k}^{-1}({\mathbf{y}}) = {\mathbf{x'}}$.
**Claim: $\Phi_*|_Q: Q {\rightarrow}Y_0^{{\triangle}}$ is injective.** Let ${\mathbf{x}},{\mathbf{x'}}\in Q$ be such that $\Phi_*({\mathbf{x}}) = \Phi_*({\mathbf{x'}}) = {\mathbf{y}}{^{(i)}}$. Then ${\mathbf{x}}= (\pi_{\Delta}({\mathbf{y}}{^{(i)}}),i) = {\mathbf{x'}}$ by the definition of $\Phi_*$ on $Q$.
**Claim: $\Phi_*|_{A\cup Q}: A\cup Q {\rightarrow}Y_0^{{\triangle}}$ is injective.** We have already dealt with the cases ${\mathbf{x}},{\mathbf{x'}}\in A$ and ${\mathbf{x}},{\mathbf{x'}}\in Q$. Now we deal with the remaining case. Let ${\mathbf{x}}\in A$, ${\mathbf{x'}}\in Q$ be such that $\Phi_*({\mathbf{x}}) = {\mathbf{y}}{^{(i)}}= \Phi_*({\mathbf{x'}})$ for some ${\mathbf{y}}{^{(i)}}\in Y\setminus \Delta$. By Lemma \[lem:bw-r3\], there exists $k_0$ such that for all $k\geq k_0$, $\Phi_k({\mathbf{x}}) = {\mathbf{y}}{^{(i)}}$. Then for all such $k$, $\Psi^{YX}_k({\mathbf{y}}{^{(i)}}) = \pi_{{\mathbb{R}}^2}({\mathbf{x}})$, which is bounded away from $\Delta$. On the other hand, since $\Phi_*({\mathbf{x'}}) = {\mathbf{y}}{^{(i)}}$, we know that $\lim_{k {\rightarrow}\infty, \, k \in L}\Psi^{YX}_k({\mathbf{y}}{^{(i)}}) = \pi_{{\mathbb{R}}^2}({\mathbf{x'}}) = \pi_{\Delta}({\mathbf{y}}{^{(i)}}) \in \Delta$. This is a contradiction.
**Claim: $\Phi_*|_{A\cup Q}: A \cup Q {\rightarrow}Y_0^{{\triangle}}$ is surjective.** Let ${\mathbf{y}}{^{(i)}}\in Y_0^{{\triangle}}$, and consider ${\Psi^{YX}}_*({\mathbf{y}}{^{(i)}})$. There are two cases: ${\Psi^{YX}}_*({\mathbf{y}}{^{(i)}})$ is either off-diagonal or on-diagonal. Suppose first that ${\Psi^{YX}}_*({\mathbf{y}}{^{(i)}})$ is off-diagonal. Then by an argument similar to that of Lemma \[lem:bw-r3\], we obtain $k_0 \in {\mathbb{N}}$ such that for all $k \geq k_0$, $k \in L$, ${\Psi^{YX}}_k({\mathbf{y}}{^{(i)}}) = {\Psi^{YX}}_*({\mathbf{y}}{^{(i)}})$. Since $X \in {{{\mathcal}{D}}_p[l^q]}$, there are only finitely many ${\mathbf{x}}\in X$ such that $\pi_{{\mathbb{R}}^2}({\mathbf{x}}) = {\Psi^{YX}}_*({\mathbf{y}}{^{(i)}})$. Let $X_{{\mathbf{y}}{^{(i)}}}:= \{{\mathbf{x}}_1,{\mathbf{x}}_2,\ldots, {\mathbf{x}}_n\}$ denote this collection.
We know that $\Phi{^{-1}}_k({\mathbf{y}}{^{(i)}}) \in X_{{\mathbf{y}}{^{(i)}}}$ for all $k \geq k_0$, $k \in L$. By the pigeonhole principle, choose a subsequence indexed by $M \subseteq L$ such that $(\Phi{^{-1}}_k({\mathbf{y}}{^{(i)}}))_{k \in M,\, k \geq k_0}$ is constant. Let ${\mathbf{x}}\in X_{{\mathbf{y}}{^{(i)}}}$ denote the value of this constant sequence. Then for all $k \geq k_0,\, k \in M$, we have $\Phi_k({\mathbf{x}}) = {\mathbf{y}}{^{(i)}}$. Since ${\mathbf{x}}\in A$ and $(\Phi_k)_{k \in L}$ converges pointwise on $A$, we know furthermore that $\Phi_*({\mathbf{x}}) = {\mathbf{y}}{^{(i)}}$.
Suppose next that ${\Psi^{YX}}_*({\mathbf{y}}{^{(i)}})$ is on-diagonal. Then by what we have observed before, ${\Psi^{YX}}_*({\mathbf{y}}{^{(i)}}) = \pi_{\Delta}({\mathbf{y}}{^{(i)}}) \in \Delta$. By definition, $Q$ contains $(\pi_{\Delta}({\mathbf{y}}{^{(i)}}), i)$, and $\Phi_*$ maps this to ${\mathbf{y}}{^{(i)}}$.
**Claim: $\Phi_*|_B: B {\rightarrow}Y_0^{{\blacktriangle}}$ is injective.**
First note that the codomain of $\Phi_*|_B$ is not $Y_0^{{\blacktriangle}}$ a priori, because the points in ${\operatorname{im}}(\Phi_*|_B)$ and $Y_0^{{\blacktriangle}}$ may differ on the ${\mathbb{N}}$ coordinate. But this is simply a matter of choosing representatives from the ${\mathbb{N}}$-indexed diagonal points, and we may relabel the ${\mathbb{N}}$-coordinates of points in $Y_0^{\blacktriangle}$ to have ${\operatorname{im}}(\Phi_*|_B) \subseteq Y_0^{{\blacktriangle}}$.
To see injectivity, suppose ${\mathbf{x}}{^{(i)}}, {\mathbf{x}}{^{(j)}}\in B$ are such that $\Phi_*({\mathbf{x}}{^{(i)}}) = \Phi_*({\mathbf{x}}{^{(j)}})$. Then $(\Psi_*({\mathbf{x}}{^{(i)}}),i) = (\Psi_*({\mathbf{x}}{^{(j)}}),j)$, so $i = j$ and hence ${\mathbf{x}}{^{(i)}}= {\mathbf{x}}{^{(j)}}$.
Finally we extend $\Phi_*$ to a bijection from $X$ to $Y$ by matching the points of $X\setminus X_1$ to the points of $Y\setminus Y_0$ and $Y_0^{{\blacktriangle}} \setminus {\operatorname{im}}(\Phi_*|_B)$, all of which are diagonal. We continue writing $\Phi_*:X {\rightarrow}Y$ to denote this bijection. It follows from the construction that $\Phi_*$ satisfies the statement of the theorem. This concludes the proof.
As a corollary of this lemma, we see that ${{{\mathcal}{D}}_{p}[l^q]}$ is a geodesic space. This result was already implicit in [@turner2013means Proposition 1], where it was stated in the case $p = q \in [1,\infty]$. See also [@tmmh] for a different argument in the case $p=q=2$. In addition to using the result about existence of geodesics throughout this paper, we specifically use Lemma \[lem:limiting-bijection\] to prove Theorem \[thm:char-II\] via Theorem \[thm:char-I\].
\[cor:geod-exist\] Fix $p,q \in [1,\infty]$. Let $X,Y \in {{{\mathcal}{D}}_{p}[l^q]}$. Then there exists a bijection $\Phi:X {\rightarrow}Y$ such that $C_p[l^q](\Phi) = {d_p[l^q]}(X,Y)$. Thus we immediately have a convex-combination geodesic from $X$ to $Y$.
Lemma \[lem:limiting-bijection\] yields an optimal bijection $\Phi$. Let ${\gamma}$ denote the associated convex combination curve. To conclude, we need to show that ${\gamma}$ is geodesic. Let $s,t \in [0,1]$. To compare ${\gamma}(s)$ and ${\gamma}(t)$, consider the bijection associating $(1-t)x+ t\Phi(x)$ with $(1-s)x + s\Phi(s)$. Then we have: $$\begin{aligned}
d_p[l^q]({\gamma}(s),{\gamma}(t)) &\leq {\left(}\sum_{x \in X} {\left\|}(1-t)x+ t\Phi(x) - (1-s)x- s\Phi(x) {\right\|}_q^p {\right)}^{1/p} \\
&= {\left(}\sum_{x \in X} {\left\|}(s-t) (x - \Phi(x) ) {\right\|}_q^p {\right)}^{1/p} \\
&= |t-s| {\left(}\sum_{x\in X} {\left\|}x - \Phi(x) {\right\|}_q^p {\right)}^{1/p} = |t-s| d_p[l^q](X,Y).\end{aligned}$$ By a property of geodesics, showing the inequality is sufficient to guarantee equality (cf. [@dgh-era Lemma 1.3]). This concludes the proof.
Lemmas related to the characterization of geodesics
---------------------------------------------------
The proof of Theorem \[thm:char-I\] will follow the strategy used by Sturm in proving an analogous result about geodesics in the space of metric measure spaces [@sturm2012space]. We first present a sequence of lemmas.
\[lem:clarkson\] Let $p \in [2,\infty)$, and let $v,w \in l^p$. Then, $$\begin{aligned}
{\left\| v + w \right\|}_p^p + {\left\|v-w \right\|}^p_p & \leq 2^{p-1}{\left(}{\left\|v \right\|}_p^p + {\left\|w \right\|}_p^p{\right)}.
$$
\[ineq:sturm-2\] Let $t\in (0,1)$, $p \in [2,\infty)$. Then there exists a constant $C>0$ depending on $p$ and $t$ such that for any $v, w \in l^p$, we have $${\left\|tv + (1-t)w \right\|}_p^p \leq t{\left\|v \right\|}_p^p +(1-t){\left\|w \right\|}_p^p - t(1-t)C{\left\|v-w \right\|}_p^p.$$
\[lem:bcl\] Let $p \in (1,2]$, and let $v,w \in l^p$. Then, $${\left\| v + w \right\|}_p^2 + {\left\|v-w \right\|}_p^2 \geq 2{\left\|v \right\|}_p^2 + 2(p-1){\left\|w \right\|}_p^2.$$
\[ineq:sturm-3\] Let $t\in (0,1)$, $p \in (1,2]$. Then for any vectors $v, w \in l^p$, we have $${\left\|tv + (1-t)w \right\|}_p^2 \leq t{\left\|v \right\|}_p^2 +(1-t){\left\|w \right\|}_p^2 - (p-1)t(1-t){\left\|v-w \right\|}_p^2.$$
\[ineq:sturm-1\] Let $a_1,\ldots, a_n \in [0,\infty)$, $t_0 < t_1 < \ldots < t_n \in {\mathbb{R}}$, and let $p \in (1,\infty)$. Then, $$\frac{1}{(t_n - t_0)^{p-1}} {\left(}\sum_{i=1}^na_i {\right)}^p \leq \sum_{i=1}^n\frac{a_i^p}{(t_i-t_{i-1})^{p-1}}.$$
For each $1\leq i \leq n$, write $\lambda_i = \tfrac{t_i - t_{i-1}}{t_n - t_0}$ and $x_i = \tfrac{a_i}{t_i-t_{i-1}}$. Notice that $\sum_{i=1}^n \lambda_i =\sum_{i =1}^n\tfrac{t_i - t_{i-1}}{t_n - t_0} = 1$.
By Jensen’s inequality, $$\begin{aligned}
{\left(}\sum_{i=1}^n \lambda_ix_i{\right)}^p
&= {\left(}\sum_{i=1}^n\frac{a_i}{t_i - t_{i-1}}\cdot \frac{t_i-t_{i-1}}{t_n-t_0} {\right)}^p
= \frac{1}{(t_n-t_0)^p}{\left(}\sum_{i=1}^na_i {\right)}^p \\
& \leq
\sum_{i=1}^n {\left(}\frac{a_i}{t_i - t_{i-1}}{\right)}^p \cdot \frac{t_i - t_{i-1}}{t_n - t_0}
= \frac{1}{t_n - t_0}\sum_{i=1}^n\frac{a_i^p}{(t_i - t_{i-1})^{p-1}}.\end{aligned}$$
This verifies Lemma (\[ineq:sturm-1\]).
It suffices to show the inequality for dyadic rationals in the unit interval, and then invoke the density of the dyadic rationals. We consider dyadic rationals of the form $t=a/2^b$, for integers $b\geq 0$ and $0\leq a \leq 2^b$. The proof is by induction, based on the following inductive hypothesis: for each $b \in {\mathbb{N}}$, there exists a constant $C>0$ depending on $b$ and $p$ such that: $${\left\|}(\tfrac{a}{2^{b}})v + (1-\tfrac{a}{2^{b}})w {\right\|}_p^p \leq (1- \tfrac{a}{2^{b}}){\left\|}w {\right\|}^p_p + (\tfrac{a}{2^{b}}){\left\|}v {\right\|}^p_p - C(1- \tfrac{a}{2^{b}})(\tfrac{a}{2^{b}}){\left\|}w-v {\right\|}^p_p.$$
For the base case, we use Lemma \[lem:clarkson\] (a sharper result is obtained via the parallelogram law for $p=2$):
$$\begin{aligned}
{\left\|{\tfrac{v}{2}}+ {\tfrac{w}{2}}\right\|}^p_p \leq 2^{p-1}{\left\|{\tfrac{v}{2}}\right\|}^p_p + 2^{p-1}{\left\|{\tfrac{w}{2}}\right\|}^p_p - {\left\|{\tfrac{v}{2}}- {\tfrac{w}{2}}\right\|}^p_p
&= {\tfrac{1}{2}}{\left\|v \right\|}^p_p + {\tfrac{1}{2}}{\left\|w \right\|}^p_p - \tfrac{1}{2^p} {\left\|v - w \right\|}^p_p\\
&= {\tfrac{1}{2}}{\left\|v \right\|}^p_p + {\tfrac{1}{2}}{\left\|w \right\|}^p_p - C \cdot {\tfrac{1}{2}}\cdot {\tfrac{1}{2}}{\left\|v - w \right\|}^p_p.\end{aligned}$$
Suppose the inductive hypothesis is true up to some $b \in {\mathbb{N}}$ and all $0\leq a \leq 2^b$. This means that at the $(b+1)$th step, the inductive hypothesis is true for all $2a/2^{b+1}$, where $0 \leq a \leq 2^b$. Fix $a$ in this range, and consider the midpoint $(2a+1)/2^{b+1}$ of $2a/2^{b+1}$ and $(2a+2)/2^{b+1}$. For the inductive step, we have: $$\begin{aligned}
{\left\| {\tfrac{2a+1}{2^{b+1}}}v + {\left(}1 - {\tfrac{2a+1}{2^{b+1}}}{\right)}w \right\|}^p_p
&= {\left\| {\tfrac{1}{2}}{\left(}{\tfrac{2a}{2^{b+1}}}v + {\tfrac{2a+2}{2^{b+1}}}v {\right)}+ {\tfrac{1}{2}}{\left(}{\left(}1 - {\tfrac{2a}{2^{b+1}}}{\right)}w + {\left(}1 - {\tfrac{2a+2}{2^{b+1}}}{\right)}w {\right)}\right\|}^p_p\\
&= {\left\| {\tfrac{1}{2}}{\left(}{\tfrac{2a}{2^{b+1}}}v + {\left(}1-{\tfrac{2a}{2^{b+1}}}{\right)}w {\right)}+ {\tfrac{1}{2}}{\left(}{\tfrac{2a+2}{2^{b+1}}}v + {\left(}1 - {\tfrac{2a+2}{2^{b+1}}}{\right)}w {\right)}\right\|}_p^p \\
&\leq 2^{p-1}{\left\| {\tfrac{1}{2}}{\left(}{\tfrac{2a}{2^{b+1}}}v + {\left(}1 - {\tfrac{2a}{2^{b+1}}}{\right)}w {\right)}\right\|}^p_p + 2^{p-1}{\left\| {\tfrac{1}{2}}{\left(}{\tfrac{2a+2}{2^{b+1}}}v + {\left(}1 - {\tfrac{2a+2}{2^{b+1}}}{\right)}w {\right)}\right\|}^p_p\\
& \phantom{\hspace{0.5 in}} - {\left\| {\tfrac{1}{2^{b+1}}}w - {\tfrac{1}{2^{b+1}}}v \right\|}^p_p.\end{aligned}$$
Here we used Lemma \[lem:clarkson\] for the inequality, and showed the final term after simplification. The inductive hypothesis can now be applied to the first two terms. Either by hand or a computer algebra package, we see that the inductive step holds for $(b+1)$. This completes the proof of Lemma \[ineq:sturm-2\].
We again proceed by showing the inequality for dyadic rationals. Writing $t = a/2^b$ as before, the inductive hypothesis now becomes: $${\left\|}(\tfrac{a}{2^{b}})v + (1-\tfrac{a}{2^{b}})w {\right\|}_p^2 \leq (1- \tfrac{a}{2^{b}}){\left\|}w {\right\|}^2_p + (\tfrac{a}{2^{b}}){\left\|}v {\right\|}^2_p - (p-1)(1- \tfrac{a}{2^{b}})(\tfrac{a}{2^{b}}){\left\|}w-v {\right\|}^2_p.$$
For the base case, we use Lemma \[lem:bcl\]. Set $c = (v+w)/2$ and $d = (v-w)/2$. Then $c + d =v$ and $c-d = w$, and we have:
$$\begin{aligned}
{\left\|c+d \right\|}^2_p + {\left\|c-d \right\|}^2_p &\geq 2{\left\|c \right\|}^2_p + 2(p-1){\left\|d \right\|}^2_p, \text{ so }\\
{\left\|v \right\|}^2_p + {\left\|w \right\|}^2_p &\geq 2{\left\|{\tfrac{v}{2}}+ {\tfrac{w}{2}}\right\|}^2_p + 2(p-1){\left\|{\tfrac{v}{2}}- {\tfrac{w}{2}}\right\|}^2_p, \text{ and hence}\\
{\left\|{\tfrac{v}{2}}+ {\tfrac{w}{2}}\right\|}^2_p &\leq {\tfrac{1}{2}}{\left\|v \right\|}^2_p + {\tfrac{1}{2}}{\left\|w \right\|}^2_p - (p-1)({\tfrac{1}{2}})^2{\left\|v-w \right\|}^2_p.\end{aligned}$$
The inductive step then applies as in the proof of Lemma \[ineq:sturm-2\].
Geometric intuition for the proof of Theorem \[thm:char-I\] {#sec:geom-idea}
-----------------------------------------------------------
The proof of Theorem \[thm:char-I\] proceeds via a geometric argument about $p$-norms. We abstract away these arguments and present the intuition here.
The setup of Theorem \[thm:char-I\] involves two persistence diagrams $X$ and $Y$ with finitely many off-diagonal points and a geodesic $\mu:[0,1] {\rightarrow}{{{\mathcal}{D}}_{p}[l^q]}$ from $X$ to $Y$. Let $\Phi$ be a bijection between $X$ and $Y$ (we will specify this bijection in the actual proof), and let $\gamma$ be the convex-combination curve in ${{{\mathcal}{D}}_{p}[l^q]}$ from $X$ to $Y$ induced by $\Phi$. For each $t \in [0,1]$, let $x_t := (1-t)x + t \Phi(x) \in \gamma(t)$. Fix $t \in (0,1)$. Suppose also that we have a bijection between $\gamma(t)$ and $\mu(t)$ (this will be specified in the proof). Let $\psi(x_t) \in \mu(t)$ denote the image of $x_t$ under this bijection.
In what follows, we will write norms in some $l^q$ space raised to some power $p>1$ (i.e. terms of the form ${\left\|\cdot \right\|}_q^p$) without specifying the elements of the normed space, but this will be written explicitly in the proof. Our goal is to prove that $$\sum_{x \in X} {\left\|}x_t - \psi(x_t) {\right\|}_q^p = 0,$$ which would show that ${d_p[l^q]}(\gamma(t),\mu(t)) = 0$. To approach this, consider the following quantities: $${\left\|}x - \psi(x_t){\right\|}_q^p, \qquad {\left\|}\psi(x_t) - \Phi(x) {\right\|}_q^p, \qquad {\left\|}x - x_t {\right\|}_q^p, \qquad {\left\|}x_t - \Phi(x) {\right\|}_q^p.$$
\[node distance = 0pt, X/.style=[circle,draw=purple,fill=purple,thick,inner sep = 0pt,minimum size = 6pt]{}, Y/.style=[rectangle,draw=NavyBlue,fill=NavyBlue,thick,inner sep = 0pt,minimum size = 6pt]{}\] (x) at (0,0) ; (phix) at (10,0) ; (psix) at (4,2) ; (xt) at (2.5,0) ;
(phix)–(x); (phix)–(psix) node\[pos=0.65, above right,rotate = -20\][$\psi(x_t) - \Phi(x)$]{} ; (psix)–(x) node\[midway, above, rotate=30\][$x - \psi(x_t)$]{}; (psix)–(xt); ; ; ; ;
These are of course related geometrically, as suggested by Figure \[fig:geom-intuit\]. More specifically, write $Q(x):= (x - \psi(x_t))/t$ and $R(x):= (\psi(x_t) - \Phi(x))/(1-t)$. Then the following is true: $$\begin{aligned}
{\left\|}Q(x) - R(x) {\right\|}_q^p &= {\left\|}\frac{ (1-t)(x - \psi(x_t)) - t(\psi(x_t) - \Phi(x))}{t(1-t)} {\right\|}_q^p\\
&= \frac{1}{t^p(1-t)^p} {\left\|}(1-t)x + t\Phi(x) - \psi(x_t) {\right\|}_q^p \\
&=\frac{1}{t^p(1-t)^p} {\left\|}x_t - \psi(x_t) {\right\|}_q^p.\end{aligned}$$
So to show $\sum_{x \in X} {\left\|}x_t - \psi(x_t) {\right\|}_q^p = 0$, it suffices to show $$\sum_{x \in X} {\left\|}Q(x) - R(x) {\right\|}_q^p = 0.$$
To obtain $Q(x) - R(x)$, one computes: $$\begin{aligned}
{d_p[l^q]}(X,Y)^p &\leq \sum_{x\in X} {\left\|}x - \Phi(x) {\right\|}_q^p \\
&= \sum_{x \in X} {\left\|}x - \psi(x_t) + \psi(x_t) - \Phi(x) {\right\|}_q^p\\
&= \sum_{x \in X} {\left\|}tQ(x) + (1-t)R(x) {\right\|}_q^p.\end{aligned}$$
Suppose also that one can bound: $${\left\|}tQ(x) + (1-t)R(x) {\right\|}_q^p \leq t {\left\|}Q(x) {\right\|}_q^p + (1-t) {\left\|}R(x) {\right\|}_q^p - C{\left\|}Q(x) - R(x) {\right\|}_q^p
\label{eq:geom-1}$$ for some positive constant $C$. If one can show that $$\begin{aligned}
\sum_{x\in X} t {\left\|}Q(x) {\right\|}_q^p + (1-t) {\left\|}R(x) {\right\|}_q^p = {d_p[l^q]}(X,Y)^p,\end{aligned}$$ then by summing over $x\in X$ in Inequality (\[eq:geom-1\]), we necessarily have $\sum_{x \in X} {\left\|}Q(x) - R(x) {\right\|}_q^p = 0$, which is what we need.
Notice that if $\psi(x_t) = x_t$ for each $x \in X$, i.e. if $\gamma(t) = \mu(t)$, then $${\left\|}x - \psi(x_t) {\right\|}_q = t{\left\|}x - \Phi(x) {\right\|}_q \text{ and }
{\left\|}\psi(x_t) - \Phi(x) {\right\|}_q = (1-t){\left\|}x - \Phi(x) {\right\|}_q, \text{ so}$$ $${\left\|}Q(x) {\right\|}_q = {\left\|}x - \Phi(x) {\right\|}_q = {\left\|}R(x) {\right\|}_q.$$ Thus we have $$\begin{aligned}
\sum_{x\in X} t {\left\|}Q(x) {\right\|}_q^p + (1-t) {\left\|}R(x) {\right\|}_q^p
= \sum_{x\in X} t {\left\|}x - \Phi(x) {\right\|}_q^p + (1-t) {\left\|}x - \Phi(x) {\right\|}_q^p
= {d_p[l^q]}(X,Y)^p.\end{aligned}$$
Now we proceed to the main result.
Proof of the characterization result
------------------------------------
We split the proof into two parts: first we construct bijections $\Phi_{k}$ that induce geodesics ${\gamma}_{k}$ which agree with $\mu$ at all $i2^{-k}$, for integers $0 < i < 2^k$. Then we will use the sequence ${\left(}\Phi_{k} {\right)}_k$ to construct a “limiting bijection" that induces a geodesic satisfying the statement of the theorem.
We also alert the reader to certain notational choices we will make in this proof. We will occasionally deal with infinite-dimensional vectors $V \in {\mathbb{R}}^{\mathbb{N}}$. Recall that when there is a function $f$ defined on each element of $V$, we will write $f(V)$ to denote $(f(v_1),f(v_2),\ldots )$. Whenever we use ${\left\|\cdot \right\|}_{l^p}$ notation, we assert that the vector in the argument does indeed belong to $l^p$. Typically this will be easy to see, and we will remind the reader to this effect.
#### Part I {#part-i .unnumbered}
For this proof, we will use an argument about dyadic rationals. Fix $k \in {\mathbb{N}}$. For each $0\leq i \leq 2^k-1$, fix optimal bijections ${\varphi}_i: \mu(i2^{-k}) {\rightarrow}\mu((i+1)2^{-k})$ (Corollary \[cor:geod-exist\]). Composing these bijections together gives a bijection $\Phi_k: X {\rightarrow}Y$. Let ${\gamma}_k$ be the convex-combination curve induced by $\Phi_k$. Also for each $0\leq i \leq 2^k-1$, let $\psi_i:{\gamma}_k(i2^{-k}) {\rightarrow}\mu(i2^{-k})$ denote the bijection induced by the $\Phi_k$ and ${\varphi}_i$ terms. There is a choice here: one can pass to $X$ and then use the maps ${\varphi}_0, {\varphi}_1, {\varphi}_2,\ldots$ or pass to $Y$ and then use the inverses of the ${\varphi}_i$ maps. This choice will not matter, so for convenience, suppose we make the former choice. We wish to show ${d_p[l^q]}({\gamma}_k(i2^{-k}), \mu(i2^{-k})) = 0$ for each $i$. For notational convenience, define $t_i := i2^{-k}$ for each $0\leq i \leq 2^k$. Also for each $x\in X$ and each $t \in [0,1]$, define $x_t := (1-t)x + t \Phi_k(x)$. Note that each $x_{t_i}$ belongs to ${\gamma}_k(t_i)$.
Let $i \in \{1, \ldots, 2^k-1\}$. For each $x \in X$, define $Q(x):=\frac{x - \psi_i(x_{t_i})}{t_i - 0}$ and $R(x):= \frac{\psi_i(x_{t_i}) - \Phi_k(x)}{1 - t_i}$ (recall the idea described in §\[sec:geom-idea\]).
#### The case $p=q \in [2,\infty)$.
First we consider the case $p=q \in [2,\infty)$. Then we have: $$\begin{aligned}
{d_p[l^p]}(X,Y)^p &\leq \sum_{x\in X} {\left\| x - \Phi_k(x) \right\|}_p^p \nonumber \\
&= \sum_{x\in X} {\left\|x - \psi_i(x_{t_i}) + \psi_i(x_{t_i}) - \Phi_k(x) \right\|}_p^p \nonumber \\
&= \sum_{x\in X} {\left\|t_iQ(x) + (1-t_i)R(x) \right\|}_p^p \label{eq:QR} \\
&\leq \sum_{x\in X}\left[ t_i{\left\|Q(x) \right\|}_p^p + (1-t_i){\left\|R(x) \right\|}_p^p - Ct_i(1-t_i){\left\|Q(x) - R(x) \right\|}_p^p \right] \label{eq:pp-clarkson}\end{aligned}$$ For the last inequality, we have used Lemma \[ineq:sturm-2\]. Specifically, $Q(x)$ and $R(x)$ are both vectors in ${\mathbb{C}}$, and we regard them as elements in $l^p$ when applying Lemma \[ineq:sturm-2\]. Notice also that this is the step described in Inequality (\[eq:geom-1\]). We split the last term into a positive part $P(p=q\in [2,\infty)):= \sum_{x\in X} \left[t_i{\left\|Q(x) \right\|}_p^p + (1-t_i){\left\|R(x) \right\|}_p^p\right]$ and a negative part $N(p=q\in [2,\infty)):= \sum_{x\in X} t_i(1-t_i) {\left\|Q(x) - R(x) \right\|}_p^p$ (the constant $C>0$ will not be important in the sequel). As described in §\[sec:geom-idea\], our goal would be show that $N(p=q\in [2,\infty)) = 0$.
#### The case $q = 2, \, p \in [2,\infty)$.
We now consider the case $q = 2, \, p \in [2,\infty)$. Recall the construction of $X^*$ and $Y^*$ in §\[sec:OT\], as well as the functional $J_p$ defined in Equation (\[eq:dgm-norm\]). By the observation in Equation (\[eq:bijection-approx\]), we can approximate $\Phi_k$ arbitrarily well by a bijection ${\sigma}: X^* {\rightarrow}Y^*$ that agrees with $\Phi_k$ on off-diagonal points of both $X$ and $Y$. Recall also from §\[sec:OT\] that we regard $X^* = (x^1,x^2,x^3,\ldots )$ and ${\sigma}(X^*) = ({\sigma}(x^1),{\sigma}(x^2),{\sigma}(x^3),\ldots )$ as infinite-dimensional vectors.
Let ${\varepsilon}> 0$, and let ${\sigma}:X^* {\rightarrow}Y^*$ be a bijection in $\Lambda_{\Phi_k}$ such that $|C_p(\Phi_k)^p - J_p({\sigma})^p| < {\varepsilon}$. By this choice we know $X^* - {\sigma}(X^*) \in l^p$. Combining these observations, we have: $$\begin{aligned}
{d_p[l^2]}(X,Y)^p &\leq J_p({\sigma})^p + {\varepsilon}= {\left\|}X^* - {\sigma}(X^*) {\right\|}_{l^p}^p + {\varepsilon}.\end{aligned}$$
Now let $X^*_{t_i}$ denote the vector $(x^1_{t_i}, x^2_{t_i},\ldots )$. This is just $\gamma_k(i2^{-k})^*$ with a particular ordering on the elements that is consistent with the ordering initially placed on $X^*$.
Once again, by the observation in Equation (\[eq:bijection-approx\]), we can approximate $\psi_i: {\gamma}_k(i2^{-k}) {\rightarrow}\mu(i2^{-k})$ arbitrarily well by a bijection $\rho_i: X^*_{t_i} {\rightarrow}\mu(i2^{-k})^*$ such that $\rho_i \in \Lambda_{\psi_i}$. Let $\rho_i \in \Lambda_{\psi_i}$ be such that $|C_p(\psi_i)^p - J_p(\rho_i)^p| < {\varepsilon}$. Next define $$\widetilde{Q}(X^*):= {\left(}\frac{x^1 - \rho_i(x^1_{t_i})}{t_i - 0}, \frac{x^2 - \rho_i(x^2_{t_i})}{t_i - 0}, \ldots {\right)}, \,
\widetilde{R}(X^*):= {\left(}\frac{\rho_i(x^1_{t_i}) - {\sigma}(x^1)}{1- t_i }, \frac{\rho_i(x^2_{t_i}) - {\sigma}(x^2)}{ 1- t_i }, \ldots {\right)}.$$ The choice of $\rho_i$ ensures that $\widetilde{Q}(X^*)$ and $\widetilde{R}(X^*)$ are both in $l^p$. Then we have: $$\begin{aligned}
{\left\|}X^* - {\sigma}(X^*) {\right\|}_{l^p}^p + {\varepsilon}&= {\left\|}X^* - \rho_i(X^*_{t_i}) + \rho_i(X^*_{t_i}) - {\sigma}(X^*) {\right\|}_{l^p}^p + {\varepsilon}\\
&= {\big\|}t_i \widetilde{Q}(X^*) + (1-t_i)\widetilde{R}(X^*) {\big\|}_{l^p}^p + {\varepsilon}\\
&\leq t_i {\big\|}\widetilde{Q}(X^*) {\big\|}_{l^p}^p
+ (1-t_i){\big\|}\widetilde{R}(X^*) {\big\|}_{l^p}^p
- Ct_i(1-t_i) {\big\|}\widetilde{Q}(X^*) - \widetilde{R}(X^*) {\big\|}_{l^p}^p + {\varepsilon}.\end{aligned}$$ where the last inequality is obtained via Lemma \[ineq:sturm-2\]. Notice that this application uses the full strength of Lemma \[ineq:sturm-2\], in the sense that is used as an inequality between norms of truly infinite-dimensional vectors, as opposed to being used as an inequality between norms in ${\mathbb{R}}^2$ (cf. Inequality (\[eq:pp-clarkson\])).
Next we compare ${\big\|}\widetilde{Q}(X^*){\big\|}_{l^p}^p$ with $\sum_{x\in X}{\left\|Q(x) \right\|}_2^p$. Define the bijection ${\alpha}: X {\rightarrow}\mu(i2^{-k})$ by $x \mapsto \psi_i(x_{t_i})$. Define another bijection $\beta: X^* {\rightarrow}\mu(i2^{-k})^*$ by $x^j \mapsto \rho_i(x^j_{t_i})$. By our choices of ${\sigma}$ and $\rho_i$, we know that ${\alpha}$ and ${\beta}$ agree on off-diagonal elements of $X$ and $\mu(i2^{-k})$. Furthermore, ${\alpha}$ is the identity on diagonal points that are not matched to off-diagonal points, and ${\beta}$ incurs a total cost bounded by a function of ${\varepsilon}$ from moving such points infinitesimally along the diagonal. Repeating this argument for the other terms, we conclude in particular that $${d_p[l^2]}(X,Y)^p \leq \sum_{x\in X}\left[ t_i{\left\|Q(x) \right\|}_2^p + (1-t_i){\left\|R(x) \right\|}_2^p - Ct_i(1-t_i){\left\|Q(x) - R(x) \right\|}_2^p \right] + f({\varepsilon}),$$ where $f({\varepsilon})$ is some positive function of ${\varepsilon}$ that tends to zero as ${\varepsilon}{\rightarrow}0$. As before, we define $$\begin{aligned}
P(p\in[2,\infty),q=2)&:= \sum_{x\in X} \left[t_i{\left\|Q(x) \right\|}_2^p + (1-t_i){\left\|R(x) \right\|}_2^p\right] \\
N(p\in [2,\infty),q=2)&:= \sum_{x\in X} t_i(1-t_i) {\left\|Q(x) - R(x) \right\|}_2^p.\end{aligned}$$
We now show how to obtain similar quantities in the final remaining case.
#### The case $q = 2, \, p \in (1,2)$.
Let ${\varepsilon}> 0$. Now let ${\sigma}\in \Lambda_{\Phi_k}$ be such that $|C_p(\Phi_k)^2 - J_p({\sigma})^2| < {\varepsilon}$, and let $\rho_i \in \Lambda_{\psi_i}$ be such that $|C_p(\psi_i)^2 - J_p(\rho_i)^2| < {\varepsilon}$. As before, we have: $$\begin{aligned}
{d_p[l^2]}(X,Y)^2&\leq J_p({\sigma})^2 + {\varepsilon}\\
&= {\big\|}t_i \widetilde{Q}(X^*) + (1-t_i)\widetilde{R}(X^*) {\big\|}_{l^p}^2 + {\varepsilon}\\
&\leq t_i {\big\|}\widetilde{Q}(X^*) {\big\|}_{l^p}^2
+ (1-t_i){\big\|}\widetilde{R}(X^*){\big\|}_{l^p}^2
- (p-1)t_i(1-t_i) {\big\|}\widetilde{Q}(X^*) - \widetilde{R}(X^*) {\big\|}_{l^p}^2 + {\varepsilon},\end{aligned}$$ where the last inequality holds via Lemma \[ineq:sturm-3\]. In this case, the argument deviates from that of the preceding cases. We define: $$\begin{aligned}
P(p\in (1,2),q=2)&:= t_i {\big\|}\widetilde{Q}(X^*) {\big\|}_{l^p}^2
+ (1-t_i){\big\|}\widetilde{R}(X^*){\big\|}_{l^p}^2 + {\varepsilon}\\
N(p\in (1,2),q=2)&:= {\big\|}\widetilde{Q}(X^*) - \widetilde{R}(X^*) {\big\|}_{l^p}^2.\end{aligned}$$
In each of these three cases, we have obtained an inequality of the form ${d_p[l^q]}(X,Y)^k \leq P - CN$, where $C>0$ is some constant.
\[cl:positive-part\] We claim that in each case presented above, $P \leq {d_p[l^q]}(X,Y)^k$ for the appropriate $k$. In the first two cases, $k = p$, and in the third case, $k = 2$.
As explained in §\[sec:geom-idea\], this shows—at least in the first case—that $N = 0$, and so ${d_p[l^q]}(\mu(t_i),{\gamma}_k(t_i)) = 0$. In the second and third cases, we will obtain an additional positive term $f({\varepsilon})$ which is a function of ${\varepsilon}$ that tends to zero as ${\varepsilon}{\rightarrow}0$, so we will drop this term and again obtain $N=0$. So assuming Claim \[cl:positive-part\], and because $i$ was arbitrary, we now have ${d_p[l^q]}(\mu(i2^{-k}),{\gamma}_k(i2^{-k})) = 0$ for each $i \in \{1,\ldots, 2^k-1\}$. Additionally, this argument shows that $\Phi_k$ is indeed an optimal bijection.
The proof of this claim comprises the rest of Part I.
#### Proof of Claim \[cl:positive-part\] in Part I {#proof-of-claim-clpositive-part-in-part-i .unnumbered}
First we deal with the case $P(p=q\in [2,\infty))$. Here we have: $$\begin{aligned}
P &= \sum_{x\in X} t_i{\left\|Q(x) \right\|}^p_p + (1-t_i){\left\|R(x) \right\|}^p_p \label{step-begin}\\
&= \sum_{x\in X} \frac{1}{t_i^{p-1}} {\left\|x - \psi_i(x_{t_i}) \right\|}^p_p + \frac{1}{(1-t_i)^{p-1}}{\left\|\psi_i(x_{t_i}) - \Phi_k(x) \right\|}^p_p \nonumber \\
&= \sum_{x\in X} \frac{1}{(t_i - t_0)^{p-1}} {\left\|\psi_0(x_0) - \psi_i(x_{t_i}) \right\|}^p_p + \frac{1}{(t_{2^k}-t_i)^{p-1}}{\left\|\psi_i(x_{t_i}) - \psi_1(x_1) \right\|}^p_p \nonumber \\
&= \sum_{x\in X} \frac{1}{(t_i - t_0)^{p-1}} {\big\|}\sum_{j=1}^i \psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) {\big\|}^p_p + \frac{1}{(t_{2^k}-t_i)^{p-1}}{\big\|}\sum_{j=i+1}^{2^k} \psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) {\big\|}^p_p \nonumber \\
&\leq \sum_{x\in X} \frac{1}{(t_i - t_0)^{p-1}} {\big(}\sum_{j=1}^i {\left\|\psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) \right\|}_p {\big)}^p + \frac{1}{(t_{2^k}-t_i)^{p-1}} {\big(}\sum_{j=i+1}^{2^k} {\left\|\psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) \right\|}_p {\big)}^p
\label{step-triangle-ineq}
\\
&\leq \sum_{x\in X} \sum_{j=1}^i \frac{{\left\|\psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) \right\|}_p^p}{(t_j - t_{j-1})^{p-1}} + \sum_{j=i+1}^{2^k} \frac{{\left\|\psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) \right\|}^p_p}{(t_j - t_{j-1})^{p-1}}
\label{step-jensen-ineq}
\\
& = \sum_{x\in X} \sum_{j=1}^{2^k} \frac{{\left\|\psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) \right\|}^p_p}{(t_j - t_{j-1})^{p-1}} \nonumber \\
& = 2^{k(p-1)} \sum_{x\in X} \sum_{j=1}^{2^k} {\left\|\psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) \right\|}^p_p \nonumber \\
& = 2^{k(p-1)} \sum_{j=1}^{2^k} \sum_{x\in X} {\left\|\psi_{j-1}(x_{t_{j-1}}) - \psi_{j}(x_{t_j}) \right\|}^p_p \nonumber \\
& = 2^{k(p-1)} \sum_{j=1}^{2^k} {d_p[l^p]}(\mu(t_{j-1}),\mu(t_j))^p
\label{step-optimal-bij}
\\
&= 2^{k(p-1)} \sum_{j=1}^{2^k} {\left(}\frac{{d_p[l^p]}(X,Y)}{2^k}{\right)}^p = {d_p[l^p]}(X,Y)^p.\end{aligned}$$
Step (\[step-triangle-ineq\]) follows from the triangle inequality, (\[step-jensen-ineq\]) follows from Lemma \[ineq:sturm-1\], and (\[step-optimal-bij\]) follows because the $\psi_j$ maps are constructed using the ${\varphi}_j$ maps, which are optimal by assumption.
The case $P(p\in[2,\infty),q=2)$ follows almost immediately, as it is exactly analogous to the preceding case with ${\left\|\cdot \right\|}_p^p$ terms replaced by ${\left\|\cdot \right\|}_2^p$. Specifically, we get $${d_p[l^2]}(X,Y)^p \leq P - CN + f({\varepsilon}) \leq {d_p[l^2]}(X,Y)^p - CN + f({\varepsilon}).$$ But ${\varepsilon}>0$ was arbitrary, and $f({\varepsilon}) {\rightarrow}0$ as ${\varepsilon}{\rightarrow}0$. So we have ${d_p[l^2]}(X,Y)^p \leq {d_p[l^2]}(X,Y)^p - CN$, and so $N = 0$.
Finally we handle the case $P(p\in (1,2),q=2)$. Here we have: $$\begin{aligned}
P -{\varepsilon}&= (t_i - t_0) {\big\|}\widetilde{Q}(X^*) {\big\|}_{l^p}^2 + (1-t_i){\big\|}\widetilde{R}(X^*) {\big\|}_{l^p}^2 \nonumber \\
&= \frac{t_i - t_0}{(t_i - t_0)^2} {\big\|}\rho_0(X^*_{t_0}) - \rho_i(X^*_{t_i}) {\big\|}_{l^p}^2 +
\frac{1 - t_i }{(1 - t_i)^2} {\big\|}\rho_i(X^*_{t_i}) - \rho_{2^k}(X^*_{t_{2^k}}) {\big\|}_{l^p}^2 \nonumber \\
&= \frac{1}{t_i - t_0} {\big\|}\sum_{j=1}^i \rho_{j-1}(X^*_{t_{j-1}}) - \rho_j(X^*_{t_j}) {\big\|}_{l^p}^2 +
\frac{1}{1 - t_i} {\big\|}\sum_{j=i+1}^{2^k} \rho_{j-1}(X^*_{t_{j-1}}) - \rho_{j}(X^*_{t_{j}}) {\big\|}_{l^p}^2 \nonumber \\
&\leq \frac{1}{t_i - t_0} {\left(}\sum_{j=1}^i {\big\|}\rho_{j-1}(X^*_{t_{j-1}}) - \rho_j(X^*_{t_j}) {\big\|}_{l^p} {\right)}^2 +
\frac{1}{1 - t_i} {\left(}\sum_{j=i+1}^{2^k} {\big\|}\rho_{j-1}(X^*_{t_{j-1}}) - \rho_{j}(X^*_{t_{j}}) {\big\|}_{l^p} {\right)}^2 \label{step:case3-minkowski} \\
&\leq \sum_{j=1}^i \frac{ {\big\|}\rho_{j-1}(X^*_{t_{j-1}}) - \rho_j(X^*_{t_j}) {\big\|}_{l^p}^2}{t_j - t_{j-1}} +
\sum_{j=i+1}^{2^k} \frac{{\big\|}\rho_{j-1}(X^*_{t_{j-1}}) - \rho_{j}(X^*_{t_{j}}) {\big\|}_{l^p}^2}{t_j - t_{j-1}} \label{step:case3-jensen} \\
&= \sum_{j=1}^{2^k} \frac{ {\big\|}\rho_{j-1}(X^*_{t_{j-1}}) - \rho_j(X^*_{t_j}) {\big\|}_{l^p}^2}{t_j - t_{j-1}} \nonumber \\
&\leq 2^k \sum_{j=1}^{2^k} \frac{{d_p[l^2]}(X,Y)^2}{2^{2k}} + f({\varepsilon}) \label{step:case3-optimal} \\
&= {d_p[l^2]}(X,Y)^2 + f({\varepsilon}). \end{aligned}$$ Adjusting $f$ as needed, we write $P \leq {d_p[l^2]}(X,Y)^2 + f({\varepsilon})$. Here $f({\varepsilon}) {\rightarrow}0$ as ${\varepsilon}{\rightarrow}0$. Since ${\varepsilon}> 0$ was arbitrary, it follows that $P \leq {d_p[l^2]}(X,Y)^2$. Here Steps (\[step:case3-minkowski\]), (\[step:case3-jensen\]), and (\[step:case3-optimal\]) hold by the Minkowski inequality for $l^p$ norms, by Lemma \[ineq:sturm-1\], and by the optimality of the $\phi_j$ maps, respectively. Note that the ${\varepsilon}$ error term comes from using the $\rho$ maps, which agree with the ${\varphi}$ maps on off-diagonal points and incur an infinitesimal error from moving diagonal points. This concludes the proof of the claim.
By the discussion following the statement of Claim \[cl:positive-part\], and the discussion in §\[sec:geom-idea\], we immediately obtain in the first two cases that ${d_p[l^q]}({\gamma}(t_i),\mu(t_i)) = 0$ for each $i \in {\left\{1,\ldots, 2^k - 1\right\}}$. In the third case, we obtain ${\big\|}\widetilde{Q}(X^*) - \widetilde{R}(X^*) {\big\|}_{l^p} = 0$ for a given $t_i$. This in turn implies that for each $ j \in {\mathbb{N}}$, we have $\rho_i(x^j_{t_i}) = (1-t_i)x^j + t{\sigma}(x^j)$. In the cases where $x^j$ or ${\sigma}(x^j)$ is off-diagonal, we then have $\psi_i(x^j_{t_i}) = \rho_i(x^j_{t_i})$. On the diagonal points of $X$ that get matched to diagonal points of $Y$, $\psi$ is the identity by definition. Thus we again have $\sum_{x \in X} {\left\|x_{t_i} - \psi_i(x_{t_i}) \right\|}_2^p = 0$, which shows ${d_p[l^q]}({\gamma}(t_i),\mu(t_i)) = 0$ in the case $q = 2$, $p \in (1,2)$.
#### Part II {#part-ii .unnumbered}
We begin with an observation. Let $\Phi: A {\rightarrow}B$ be any optimal bijection between diagrams $A,B \in {{{\mathcal}{D}}_p[l^q]}$ that have finitely many off-diagonal points. Any off-diagonal point of $A$ is mapped either to an off-diagonal point of $B$ or to a copy of its projection onto the diagonal in $B$. In particular, we know by optimality that $\Phi$ is the identity on each point on the diagonal of $A$ that is not the diagonal projection of a point in $B$. Since $A$ and $B$ both have finitely many off-diagonal points and hence finitely many diagonal projections, we know that $\Phi$ is the identity on all but finitely many points of $A$.
Now consider the sequence ${\left(}\Phi_{k} {\right)}_k$ of bijections $X{\rightarrow}Y$ chosen at the beginning of the proof. Let $X_1 \subseteq X$ denote the union of the finitely many off-diagonal points of $X$ with the finitely many copies of diagonal points that could possibly be matched to an off-diagonal point of $Y$ by projection. Define $Y_1 \subseteq Y$ similarly. We showed above that each $\Phi_k$ is optimal, so we know by the preceding observation that each $\Phi_k$ is the identity on $X\setminus X_1$. Thus we view each $\Phi_k$ as a map $X_1 {\rightarrow}Y$.
Write $X_1 = \{x_1,x_2,\ldots, x_n\}$. Choose a subsequence of ${\left(}\Phi_k {\right)}_k$ that is constant on $x_1$. Such a subsequence must exist by finiteness of $Y_1$. By choosing further subsequences, we obtain a subsequence that is constant on $X_1$. Let $\Phi_*:X {\rightarrow}Y$ denote the bijection given by this subsequence, and let ${\gamma}_*$ denote its induced geodesic. Fix $p,q$ in the prescribed ranges. Then we have ${d_p[l^q]}(\mu(i2^{-k}),{\gamma}_*(i2^{-k})) = 0$ for each $i \in \{1,\ldots, 2^k-1\}$, for arbitrarily large $k$. Continuity of $\mu$ and ${\gamma}_*$ now shows that ${d_p[l^q]}(\mu(t),{\gamma}_*(t)) = 0$ for each $t \in [0,1]$.
Finally we supply the proof of Theorem \[thm:char-II\].
Part I of Theorem \[thm:char-I\] is independent of any finiteness assumption, so it holds in this setting as well. Assume we are in the setup obtained from Part I of Theorem \[thm:char-I\], i.e. we have a sequence of optimal bijections $\Phi_k: X {\rightarrow}Y$. In particular, we have $C_p[l^q](\Phi_k) = {d_p[l^q]}(X,Y)$ for each $k$. By Lemma \[lem:limiting-bijection\], we obtain a subsequence indexed by $L \subseteq {\mathbb{N}}$ and a limiting bijection $\Phi_*:X {\rightarrow}Y$ such that $\Phi_k \xrightarrow{k \in L, \, k {\rightarrow}\infty} \Phi_*$ pointwise and $C_p[l^q](\Phi_*) = {d_p[l^q]}(X,Y)$.
Let ${\gamma}_*$ denote the geodesic induced by $\Phi_*$. Then we have ${d_p[l^q]}(\mu(i2^{-k}),{\gamma}_*(i2^{-k})) = 0$ for each $i \in \{1,\ldots, 2^k-1\}$, for arbitarily large $k$. Continuity of $\mu$ and ${\gamma}_*$ now shows that ${d_p[l^q]}(\mu(t),{\gamma}_*(t)) = 0$ for each $t \in [0,1]$. This proves the theorem.
Discussion
==========
We have proved that in persistence diagram space equipped with several families of $l^p[l^q]$ metrics, every geodesic can be represented as a convex combination. The most interesting special case of this result is when $p =q=2$. The convex combination structure of geodesics in this case can be applied to obtain a variety of important geometric consequences, as shown in [@tmmh]. Several other cases remain open, e.g. the cases $p=q \in (1,2)$ and $p \in (1,\infty),\, q \in (1,2)\cup (2,\infty)$.
#### Acknowledgements
We thank Facundo Mémoli and Katharine Turner for many useful comments and suggestions.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Network modeling is a critical component for building self-driving Software-Defined Networks, particularly to find optimal routing schemes that meet the goals set by administrators. However, existing modeling techniques do not meet the requirements to provide accurate estimations of relevant performance metrics such as delay and jitter. In this paper we propose a novel Graph Neural Network (GNN) model able to understand the complex relationship between topology, routing and input traffic to produce accurate estimates of the per-source/destination pair mean delay and jitter. GNN are tailored to learn and model information structured as graphs and as a result, our model is able to generalize over arbitrary topologies, routing schemes and variable traffic intensity. In the paper we show that our model provides accurate estimates of delay and jitter (worst case $R^2=0.86$) when testing against topologies, routing and traffic not seen during training. In addition, we present the potential of the model for network operation by presenting several use-cases that show its effective use in per-source/destination pair delay/jitter routing optimization and its generalization capabilities by reasoning in topologies and routing schemes not seen during training.'
author:
- Krzysztof Rusek
- 'José Suárez-Varela'
- Albert Mestres
- 'Pere Barlet-Ros'
- 'Albert Cabellos-Aparicio'
bibliography:
- 'bibliography.bib'
title: |
Unveiling the potential of Graph Neural Networks\
for network modeling and optimization in SDN
---
Introduction
============
Motivation
----------
Network optimization is a well-known and established topic with the fundamental goal of operating networks efficiently. In the context of the SDN paradigm, network optimization is achieved by incorporating two components to the SDN controller: (i) a network model and (ii) an optimization algorithm (e.g, [@TEakyildiz]). Typically, the network administrator configures the network policy (goals) in the optimization algorithm that uses the network model to obtain the configuration that meets the goals.
In this traditional and well-known architecture the model is responsible for predicting the performance (e.g, per-link utilization) of the network (e.,g topology) for a particular configuration (e.g, routing). Then the optimization algorithm is tasked to explore the configurations to find one that meets the goals of the network administrator. An example of this is Traffic Engineering, where the goal is finding a routing configuration that keeps the per-link utilization below the per-link capacity. Since the dimensionality of the configuration is typically very large, efficient optimization strategies reduce them by using expert knowledge. The networking community has developed over decades a large set of network models and optimization strategies [@rexfordOptimization].
One of the fundamental characteristics of network optimization is that *we can only optimize what we can model*. For example, in order to optimize the jitter of the packets traversing the network we need a model able to understand how jitter relates to other network characteristics. In the field of fixed networks many accurate network models have been developed in the past, particularly using Queuing Theory [@queuingModels]. However, such models make some simplifications like assuming some non-realistic properties of real-world networks (e.g., generation of traffic with Poisson distribution, probabilistic routing). Moreover, they do not work well for networking problems involving multi-hop routing (i.e., multi-point to multi-point queueing) and estimation of end-to-end performance metrics [@experienceDriven]. As a result, they are not accurate for large networks with realistic routing configurations and as such, delay, jitter and losses remain as critical performance metrics for which no practical model exists.
Recent advances in Artificial Intelligence (AI) [@googleNature] have led to a new era of Machine Learning (ML) techniques such as Deep Learning [@deepLearning]. This has attracted the interest of the networking community to try to take advantage of these novel techniques to develop a new breed of models, particularly focused on complex network behavior and/or metrics.
In this context, relevant research efforts are being devoted into this new field. Researchers are using neural networks to model computer networks [@wangMachineLearning] and using such models for network optimization [@IntelligentRouting], in some cases in combination with advanced strategies based on Deep Reinforcement Learning [@learningRouting; @experienceDriven; @deepRMSA].
Such proposals [@deepQ; @mestresModeling] typically use well-known Neural Networks (NN) architectures like fully-connected Neural Networks, Convolutional Neural Networks (extensively used for image processing), Recurrent Neural Networks (used for text processing) or Variational Auto-Encoders. However, computer networks are fundamentally represented as graphs, and such types of NN are *not designed to learn information structured as graphs*. As a result, the models trained are strongly limited: they provide limited accuracy and are unable to generalize in terms of topologies or routing configurations. This is one of the main reasons why ML-based network optimization techniques have -at the time of this writing- failed to meet its expectations and clearly outperform traditional techniques.
Objectives
----------
In this paper we aim to address these issues and we present RouteNet, a pioneering network model based on Graph Neural Networks (GNN) [@graphNetworks]. Our model is able to understand the complex relationship between topology, routing and input traffic to produce accurate estimates of the per-source/des-tination pair mean delay and jitter. GNN are tailored to learn and model information structured as graphs and as a result our model is able to generalize over arbitrary topologies, routing schemes and variable traffic intensity. To the best of our knowledge, this is the first work to address such fundamental networking problem using ML-based techniques able to learn and *generalize*.
Graph Neural Network (GNN) models have grown in popularity in recent years and are particularly designed to operate on graphs with the aim of achieving relational reasoning and combinatorial generalization. In other words, GNNs facilitate the learning of relations between entities in a graph and the rules for composing them (i.e., they have a strong *relational inductive bias* [@relationalInductiveBias]). Specifically, our model is inspired by Message-passing Neural Networks [@MPNN], such models are already used in chemistry to develop new compounds. With this framework we design a new model that captures *meaningfully* traffic routing over network topologies. This is achieved by modeling the relationships of the links in topologies with the source-destination paths resulting from the routing schemes and the traffic flowing through them.
Contributions
-------------
In order to test the accuracy of our model we train it with a dataset generated using a per-packet simulator (Omnet++ [@omnet]) resulting in high estimation accuracy of delay and jitter (worst case $R^2=0.86$) when testing against topologies, routing and traffic not seen during training. More importantly, we verify that our model is able to generalize and for instance, when training the model with samples of a 14-node topology the model is able to provide accurate estimates in a never seen 24-node network.
Finally, and in order to showcase the potential of our model we present a series of use-cases applicable to a SDN architecture:
1. **Routing Optimization:** We first show that our model can be used to find routing schemes that minimize per-source/destination average delay and/or jitter. We benchmark it against traditional utilization-aware models (e.g., OSPF) achieving improvements up to 43.5%. We show that this model can be also used for SLA optimization, where delay or jitter SLA is maintained for a set of source-destination pairs even when the overall traffic volume increases.
2. **Link failures:** In order to show the generalization capabilities of our model we show that it is able to produce estimates of delay and jitter in the presence of link-failures. That is, changes in the topology and the routing.
3. **What-if Scenarios:** Finally we show that the model can be used to reason in what-if scenarios answering the following questions: What will happen to the delay/jitter of the network if a new user is added? And, how should I upgrade the network to significantly reduce the overall delay and jitter?
Network architecture
====================
Network modeling enables the control plane to further exploit the potential of SDN to perform fine-grained management. This permits to evaluate the resulting performance of what-if scenarios without the necessity to modify the state of the data plane. It may be profitable for a number of network management applications such as optimization, planning or fast failure recovery. For instance, in Fig. \[fig:scenario\] we show an architecture of a use-case that performs network optimization within the context of the *knowledge-Defined Networking* (KDN) paradigm [@kdn]. In this case, we assume that the control plane receives timely updates of the network state (e.g., traffic matrix, delay measurements). This can be achieved by means of “conventional” SDN-based measurement techniques (e.g., OpenFlow [@openflow], OpenSketch [@opensketch]) or more novel telemetry proposals such as INT for P4 [@int-p4] or iOAM [@ioam]. Likewise, in the knowledge plane there is an optimizer whose behavior is defined by a given target policy. This policy, in line with intent-based networking, may be defined by a declarative language such as NEMO [@nemo] and finally being translated to a (multi-objective) network optimization problem. In this point, an accurate network model can play a crucial role in the optimization process by leveraging it to run optimization algorithms (e.g., hill-climbing) that iteratively explore the performance of candidate solutions in order to find the optimal configuration. We intentionally leave out of the scope of this architecture the training phase.
![Architecture for network optimization[]{data-label="fig:scenario"}](figures/network_architecture.pdf){width="1.0\linewidth"}
{width="1.0\linewidth"}
\[fig:DL-model\]
To be successful in scenarios like the one proposed above, the network model should meet two main requirements: $(i)$ providing accurate results, and $(ii)$ having a low computational cost to allow network optimizers to operate in short time scales. Moreover, it is essential for optimizers to have enough flexibility to simulate what-if scenarios involving different routing schemes, changes in the topology and variations in the traffic matrix. To this end, we rely on the capability of Graph Neural Network (GNN) models to efficiently operate and generalize over environments represented as graphs. Our GNN-based model, RouteNet, inspired by the Message-Passing Neural Network [@MPNN] used in the chemistry field, is able to propagate any routing scheme throughout a network topology and abstract meaningful information of the current network state. Fig. \[fig:DL-model\] shows a schematic representation of the model. More in detail, RouteNet takes as input $(i)$ a given topology, $(ii)$ a source-destination routing scheme (i.e., relations between end-to-end paths and links) and $(iii)$ a traffic matrix (defined as the bandwidth between each pair of nodes in the network), and finally produces performance metrics according to the current network state (e.g., per-path delays or jitter). To achieve it, RouteNet uses fixed-dimension vectors that encode the states of paths and links and propagate the information among them according to the routing scheme. In Section \[sec:use-cases\], we provide some relevant use-cases with experiments that exhibit how we can benefit from this GNN model in different network-related problems.
Network modeling with GNN {#sec:modeling-gnn}
=========================
In this section, we provide a detailed mathematical description of RouteNet, the GNN-based model proposed in this paper and designed specifically to operate in networking scenarios.
Notation
--------
A computer network can be represented by a set of links $\mathcal{N}=\{l_i\},\quad i \in (0,1,\ldots,n_l)$, and the routing scheme in the network by a set of paths $\mathcal{R}=\{p_k\} \quad k \in (0,1,\ldots,n_p) $. Each path is defined as a sequence of links $p_k=(l_{k(0)},\ldots, l_{k(|p_k|)})$, where $k(i)$ is the index of the $i$-th link in the path $k$. The properties (features) of both links and paths are denoted by $\mathbf x_{l_i}$ and $\mathbf x_{p_i}$.
Message Passing on Paths
------------------------
Let us consider the delay on path $p_k=(l_{k(0)},l_{k(1)},l_{k(2)}\ldots)$. The state of every link in this path and consequently, the associate delays, depend on all the traffic traversing these links. If packet loss is negligible, the order of links in the path does not matter. Then, the delay could be computed as $\sum_i d(l_{k(i)})$, where $d(l_j)$ represents the delay on the $j$-th link. However, the presence of links with losses introduces sequential dependence between the link states.
Let the state of a link be described by $\mathbf{h}_{l_i}$, which is an unknown hidden vector. Similarly, the state of a path is defined by $\mathbf{h}_{p_i}$. We expect the link state vector to contain some information about the link delay, packet loss rate, link utilization, etc. Likewise, the path state is expected to contain information about end-to-end metrics such as delays or total losses. Considering these assumptions, we can state the following principles:
1. The state of a path depends on the states of all the links in the path.
2. The state of a link depends on the states of all the paths including the link.
These principles can be matematically formulated with the following expressions: $$\begin{gathered}
\mathbf{h}_{l_i} = f(\mathbf{h}_{p_1},\ldots, \mathbf{h}_{p_j}), \quad l_i \in p_k,\space k=1, \ldots, j \label{eq:hl}\\
\mathbf{h}_{p_k} = g(\mathbf{h}_{l_{k(0)}},\ldots, \mathbf{h}_{l_{k(|p_k|)}}) \label{eq:hp}\end{gathered}$$ where $f$ and $g$ are some unknown functions.
It is well-known that neural networks can work as universal function approximators. However, a direct approximation of functions $f$ and $g$ is not possible in this case given that: $(i)$ Equations and define an implicit function (a nonlinear system of equations with the states being hidden variables), $(ii)$ these functions depend on the input routing scheme, and $(iii)$ the dimensionality of each function is very large. This would require a vast set of training samples.
Our goal is to achieve a structure for $f$ and $g$ being invariant for the routing scheme but still being aware of it. For this purpose, we propose RouteNet, a Graph Neural Network architecture based on *message-passing neural networks* (MPNN) [@MPNN], which were already successfully applied to a quantum chemistry problem.
Algorithm \[alg:mpnn\] describes the forward propagation (and the internal architecture) of the network. In this process, RouteNet receives as input the initial path and link features $\mathbf x_p$, $\mathbf x_l$ and the routing description $\mathcal{R}$, and outputs inferred per-path metrics ($\hat{\mathbf y}_p$). Note that we simplified the notation by dropping sub-indexes of paths and links.
\[lin:in\] \[lin:hp0\] \[lin:hl0\] $\hat{\mathbf y}_p \leftarrow F_p(\mathbf h_p)$\[lin:read\]
RouteNet’s architecture enables dealing with the circular dependencies described in equations and , and supporting arbitrary routing schemes (which are inherently represented within the architecture). This is all thanks to the ability of GNNs to address problems represented as graphs and solve circular dependencies by making an iterative approximation to fixed point solutions.
In order to address the circular dependencies, RouteNet repeats the same operations over the state vectors $T$ times (loop from line \[lin:forT\]). These steps represent the convergence process to the fixed point of a function from the initial states $\mathbf{h}_{p}^{0}$ and $\mathbf{h}_{l}^{0}$.
Regarding the issue of routing invariance (more generically known as topology invariance in the context of graph-related problems). This requires the use of a structure that is able to represent graphs of different topologies and sizes. In our case, we aim at representing different routing schemes in a uniform way. One state-of-the-art solution for this problem [@rusek2018message] proposes using neural message passing architectures that combine both: a representation of the topology as a graph, and vectors to encode the link states. In this context, RouteNet can be interpreted as an extension of a vanilla message passing neural network that is specifically suited to represent the dependencies among links and paths given a routing scheme (Equations and ).
In Algorithm \[alg:mpnn\], the loop from line \[lin:mp\] and the line \[lin:ml\] represent the *message-passing* operations that exchange mutually the information encoded (hidden states) among links and paths. Likewise, lines \[lin:up\] and \[lin:ul\] are *update* functions that encode the new collected information into the hidden states. The path update (line \[lin:up\]) is a simple assignment, while the link update (line \[lin:ul\]) is a trainable neural network. In general, the path update could be also a trainable neural network.
From a computational point of view, the loops over links and paths are the most expensive parts of the algorithm. An upper bound estimate of complexity is $O(n^3)$, where $n$ is the number of nodes in the network. This assumes the worst case scenario, where all the paths have length $n$. Typically, the expected diameter in real networks is around $\log(n)$ (e.g., Erd˝os–R´enyi random graphs), thereby the complexity would decrease to $O(n^2\log(n))$.
This architecture provides flexibility to represent any source-destination routing scheme. This is achieved by the direct mapping of $\mathcal{R}$ (i.e., the set of end-to-end paths) to specific message passing operations among link and path entities that define the architecture of RouteNet. Thus, each path collects messages from all the links included in it (loop from line \[lin:mp\]) and, similarly, each link receives messages from all the paths containing it (line \[lin:ml\]). Given that the order of paths traversing the same link does not matter, we used a simple summation for the path-level message aggregation. However, in the case of links, the presence of packet losses may imply sequential dependence in the links that form every path. Consequently, we use a Recurrent Neural Network (RNN) for the link-level message aggregation. Note that RNNs are well suited to capture dependence in sequences of variable size (e.g., text processing). This allows us to model losses in links and propagate this information through all the paths. For an input sequence $\mathbf i_1,\mathbf i_2, \ldots$ and an initial hidden state $\mathbf s_0$, the output of a RNN is defined as:
$$(\mathbf o_{t}, \mathbf s_{t}) = RNN(\mathbf s_{t-1},\mathbf i_{t}).$$
In our case, we use a simple version of a RNN, where $o_t=s_t$.
Moreover, the use of these message aggregation functions (RNN and summation) enables to significantly limit the dimensionality of the problem. The purpose of these functions is to collect an arbitrary number of messages received in every (link or path) entity, and compress this information into fixed-dimension arrays (i.e., hidden states). Note that the size of the hidden states of links and paths are configurable parameters. In the end, all the hidden states in RouteNet represent an explicit function containing information of the link and path states. This enables to leverage them to infer various features at the same time. Given a set of hidden states $\mathbf h_p^T$ and $\mathbf h_l^T$, it is possible connect readout neural networks to estimate some path and/or link-level metrics. This can be typically achieved by using ordinary fully-connected networks with some layers and proper activation functions. In Algorithm \[alg:mpnn\], the function $F_p$ (line \[lin:read\]) represents a readout function that predicts some path-level features ($\hat{\mathbf y}_p$) using as input the path hidden states $\mathbf h_p$. Similarly, it would be possible to infer some link-level features ($\hat{\mathbf y}_l$) using information from the link hidden states $\mathbf h_l$.
Delay model {#subsec:delay-model}
-----------
RouteNet is a general neural architecture capable of modeling various network performance metrics. In order to apply it to particular problems, the following design decisions may be considered:
The size of the hidden states for both paths ($\mathbf h_p$) and links ($\mathbf h_l$).
The number of message passing iterations ($T$).
The neural network architectures for $RNN$, $U$, and $Fp$.
The last decision may be the most complex one, since there are multiple types of neural networks and possible configurations. In our particular case, where we use RouteNet to model per-path delays, we decided to use Gated Recurrent Units (GRU) for both $U$ and $RNN$. The reason behind this, is that GRUs are simpler than LSTM networks (i.e., there is no output gate) and *a priori* can achieve comparable performance [@Chung14a]. GRUs are recurrent units that have an internal structure that by design reuses weights (i.e., weight tying), which considerably simplifies the model.
We modeled the readout function ($Fp$) with a fully-con-nected neural network and use SELU activation functions in order to achieve desirable scaling properties [@Klambauer2017]. These hidden layers are interleaved with two dropout layers.
The dropout layers play two important roles in the model. During training, they help to avoid overfitting, and during the inference, they can be used for Bayesian posterior approximation [@Gal2015]. This allows us to asses the confidence of the network predictions and avoid the issue of adversarial examples [@Goodfellow2014ExplainingAH]. Typically, when a neural network is optimized to minimize the error for a particular output, the solution may be too optimistic. Thus, repeating an inference multiple times with random dropout can provide a probabilistic distribution of results, and this distribution (e.g., the spread) can be used to estimate the confidence of the predictions.
Jitter model {#subsec:jitter-model}
------------
In order to assess the ability of RouteNet to generalize to different network metrics, we took a model in an early training stage that produces delay estimates and trained it to produce per-path jitter estimates. The main difference between these two metrics is in the scaling factor, since they are closely related but the jitter spans on different range than the delay.
Evaluation of the accuracy of the GNN model
===========================================
In this section, we evaluate the accuracy of RouteNet (Sec. \[sec:modeling-gnn\]) to estimate the per-source/destination mean delays and jitter in different network topologies and routing schemes.
Simulation setup {#sec:simulation-setup}
----------------
In order to build a ground truth to train and evaluate the GNN model, we implemented a custom-built packet-level simulator with queues using OMNeT++ (version 4.6) [@omnet]. In this simulator, the delay and jitter modeled in each queue are related to the bandwidth capacity of the corresponding egress links. For each simulation, we measure the average end-to-end delay and jitter experienced during 16k units of time by every pair of nodes. We model the traffic matrix ($\mathcal{TM}$) for each S-D pair in the network as:
$\mathcal{TM}(S_i,D_j)~=~\mathcal{U}(0.1, 1)*TI/(N-1)\quad \forall~i,j \in nodes, i\neq j$
Where $\mathcal{U}(0.1, 1)$ represents a uniform distribution in the range \[0.1, 1\], *TI* represents a parameter to tune the overall traffic intensity in the network scenario and *N* is the number of nodes in the network.
To train and evaluate the model, we used the 14-node and 21-link NSF network topology [@nsfnet]. Moreover, we use the 24-node Geant2 topology [@geant2] and the 17-node German Backbone Network (GBN) [@gbn] only for evaluation purposes. For the sake of simplicity, we consider the same capacity for all the links in these networks and vary the traffic intensity in each scenario.
Training and Evaluation
-----------------------
We implemented the RouteNet models of delay and jitter in TensorFlow. The source code and all the training/evaluation datasets used in this paper are publicly available at [@kdngit].
We trained both models (delay and jitter) on a collection with 260,000 training samples from the NSF network generated with our packet-level simulator. Despite this dataset only contains samples from single topology, it includes around 100 different routing schemes and a wide variety of traffic matrices with different traffic intensity. For the evaluation, we use 30,000 samples.
In our experiments, we select a size of 32 for the path’s hidden states ($\mathbf h_p$) and 16 for the link’s hidden states ($\mathbf h_l$). The initial path features ($\mathbf x_p$) are defined by the bandwidth that each source-destination path carries (extracted from the traffic matrix). In this case, we do not add initial link features ($\mathbf x_l$). Note that, for larger networks, it might be necessary to use larger sizes for the hidden states. Moreover, every forward propagation we execute $T$=8 iterations. The Dropout rate is equal to 0.5, this means that each training step we randomly deactivate half of neurons in the readout neural network. This also allows us to make a probabilistic sampling of results and infer the confidence of the estimates.
During the training we minimized the mean squared error ($\mathit{MSE}$) between the prediction of RouteNet and the ground truth plus the $L2$ regularization loss ($\lambda =0.1$). The loss function is minimized using an Adam optimizer with an initial learning rate of 0.001. This rate is decreased to 0.0003 after 60,000 training steps approximately.
\[fig:loss\]
We executed the training over 300,000 batches of 32 samples randomly selected from the training set. In our testbed with a GPU Nvidia Tesla K40 XL, this took around 96 hours ($\approx$ 27 samples per second). Figure \[fig:loss\] shows the loss during the training process. Here, we observe that the training is stable and the loss drops quickly.
Table \[tab:eval\] shows a summary of all the experiments we made in the three different network topologies. We report two statistics: $(i)$ the Pearson correlation $\rho$ and $(ii)$ the percentage of variance explained by the model ($R^2$). For the Geant2 and GBN topologies, we tested the accuracy over a dataset with 100,000 samples. For the NSF network, we utilize the same 30,000 samples used for evaluation during the training process. To calculate the statistics in Table \[tab:eval\], we compute for each sample in the evaluation dataset 50 independent predictions using random dropout and take the median value. Note that *the Geant2 and GBN networks were never included in the training.* The model was only trained with samples from the NSF network (14 nodes). The high accuracy on Geant2 (24 nodes) and GBN (17 nodes) networks reveals the ability of RouteNet to well generalize even to larger networks.
-- --------------------------------------------------------------- -- -- -- -- --
**Delay & **Jitter & **Delay & **Jitter & **Delay & **Jitter\
$R^2$ & 0.99 & 0.98 & 0.97 & 0.86 & 0.99 & 0.97\
$\rho$ & 0.998 & 0.993 & 0.991 & 0.942 & 0.997 & 0.987\
************
-- --------------------------------------------------------------- -- -- -- -- --
: Summary of the obtained evaluation results[]{data-label="tab:eval"}
This generalization capability is partly thanks to the Bayesi-an nature of the network (i.e., the use of layers with random dropout). Figure \[fig:regplot\] shows an example of the probabilistic prediction for a single sample of the dataset of Geant2 (Fig. \[fig:regplot\_geant\]) and GBN (Fig. \[fig:regplotgbn\]). The dots represent the median value of the predictions of RouteNet, while gray lines show the range containing 95% of the results.
![Regression plots with uncertainty.[]{data-label="fig:regplot"}](figures/regplot_geant2.pdf){width="0.97\linewidth"}
![Regression plots with uncertainty.[]{data-label="fig:regplot"}](figures/regplot_gbn.pdf){width="0.97\linewidth"}
Statistics like $\rho$ or $R^2$ provide a good picture of the general accuracy of the model. However, there are more elaborated methods that offer a more detailed description of the model behavior. One alternative to gain insight into prediction models is to provide regression plots with the evaluation results. Thus, in Figure \[fig:regplot\] we present relevant regression plots for specific scenarios of the evaluation in Geant2 (Fig. \[fig:regplot\_geant\]) and GBN (Fig. \[fig:regplotgbn\]). However, it is not functional showing a regression plot with all the points predicted by RouteNet in all the evaluation scenarios (dozens of millions of points). Hence, we focus on the distribution of residuals (i.e., the error of the model). Particularly, we present a CDF of the relative error (Fig. \[fig:eval\_all\]) over all the evaluation samples. This allows us to provide a comprehensive view of the whole evaluation in a single plot. In these results, we can observe that the prediction error in general is considerably low. Moreover, we see that the jitter model is more biased compared to the delay model. This can be explained by the fact that this model was trained from a model previously trained for the delay (Sec. \[subsec:jitter-model\]), while the delay model was optimized from the beginning to predict delays. Nevertheless, this shows the feasibility to adapt pre-trained models optimized for a specific metric to predict different network performance metrics (i.e., transfer learning [@tranferLearning]).
Generalization Capabilities
---------------------------
This section discusses the generalization capabilities and limitations of RouteNet. As in all ML-based solutions, RouteNet is expected to provide more accurate inference as the distribution of the input data is closer to the distribution of training samples. In our case, it involves topologies with similar number of nodes and distribution of connectivity, routing schemes with similar patterns (e.g., variations of shortest path) and similar ranges of traffic intensities. We experimentally observe the capability of RouteNet to generalize from a 14-node network up to a 24-node network while still providing accurate estimates. Finally, and in order to expand the generalization capabilities of RouteNet, an extended training set must be used including a wider range of distributions of the input elements.
RouteNet’s architecture is built to estimate path-level metrics using information from the output path-level hidden states. However, it is relatively easy to change the architecture and use information encoded in the link-level hidden states to produce link-related metrics inference (e.g., congestion probability on links). In addition, the implementation of RouteNet at the time of this writing does not support topologies with different link capacity. Similarly, we made experiments involving different link utilization, which implies comparable complexity. In order to support various per-link capacity, this should be encoded in the initial hidden states of links.
Use-cases {#sec:use-cases}
=========
In this section, we present some use-cases to illustrate the potential of RouteNet (Section \[sec:modeling-gnn\]) to be used in relevant network optimization tasks. In these use-cases, we use the delay (Section \[subsec:delay-model\]) and jitter (Section \[subsec:jitter-model\]) models of RouteNet to evaluate the resulting performance after applying different network configurations. Particularly, we limit the optimization problem to evaluate a given set of candidate configurations (e.g., routing schemes) and select the one that results in better performance according to a given target policy. We compare the performance achieved by our optimizer using the delay/jitter estimates of RouteNet to the results obtained by the same optimizer using measurements of the links’ utilization. As a reference, we also provide the results obtained when applying a traditional Shortest Path routing policy. Note that more elaborated state-of-the-art optimization strategies (e.g., [@defo]) could (and most likely will) result in better performance than these baselines and also could be combined with RouteNet to further improve the resulting performance. However, we leave the analysis of such techniques out of the scope of this paper, since the purpose of this section is to show the added value of using the lightweight and accurate delay/jitter models provided by RouteNet to perform delay/jitter-aware network optimization.
[1.0]{} ![Delay-based optimization[]{data-label="fig:delay"}](figures/optMeanDelay.pdf "fig:"){width="1.0\linewidth"}
[1.0]{} ![Delay-based optimization[]{data-label="fig:delay"}](figures/optMaxDelay.pdf "fig:"){width="1.0\linewidth"}
In this context, using state-of-the-art delay/jitter models to perform online network optimization is typically unfeasible since these models often result into inaccurate estimation (e.g., theoretical models) and/or prohibitive processing cost (e.g., packet-level simulators). All the evaluation in this section is carried in the NSF network topology.
Delay/jitter-based routing optimization {#sec:use-cases:1}
---------------------------------------
In this use-case, the objective is to optimize multiple Key Performance Indicators (KPI) of the network. In particular, we made different experiments where the optimizer aims to: $(i)$ minimize the mean end-to-end delay and jitter, and $(ii)$ minimize the maximum delay and jitter experienced among all the source-destination pairs.
We compare the optimal routing policy obtained by Route-Net with two traditional approaches: $(i)$ Shortest Path (SP) routing, where we compute different SP schemes using the Dijkstra algorithm, and $(ii)$ a more elaborated routing optimizer whose objective is to minimize the bandwidth utilization. This latter strategy represents an upper-bound of the results that could be obtained by traditional routing optimizers based on links’ utilization. In particular, for the case of minimizing the mean delay/jitter, we select the routing scheme with minimum mean utilization. In the case of minimizing the maximum delay/jitter, we select the scheme that minimizes the utilization of the link more loaded.
We evaluated the performance obtained by all the routing approaches varying the traffic intensity. Moreover, for a fair comparison, all of them perform the optimization over the same set with 100 different routing schemes randomly generated.
Fig. \[fig:mean\_delay\] shows the minimum mean delay obtained w.r.t. the traffic intensity. Note that traffic intensities (in the x-axis) are in TI units according to the expression in Section \[sec:simulation-setup\] to generate traffic matrices ($\mathcal{T}$). Moreover, for each traffic intensity, we randomly generated 100 different traffic matrices (TMs) with various per-source/destination traffic distributions. The lines show the average results over the experiments (with those 100 TMs) and the error bars represent the 20/80 percentiles. Likewise, in Fig. \[fig:max\_delay\] we show the results for the use-case where all routing techniques aim at minimizing the maximum end-to-end delay.
[1.0]{} ![Jitter-based optimization[]{data-label="fig:jitter"}](figures/optMeanJitter.pdf "fig:"){width="1.0\linewidth"}
[1.0]{} ![Jitter-based optimization[]{data-label="fig:jitter"}](figures/optMaxJitter.pdf "fig:"){width="1.0\linewidth"}
The same experiments were made to evaluate the results optimizing the mean (Fig. \[fig:mean\_jitter\]) and the maximum (Fig. \[fig:max\_jitter\]) jitter experienced by the source-destination pairs in the network.
Considering these results, we can see that, as expected, the performance achieved by the different routing techniques does not differ with low traffic intensity (TI<9). However, the optimizer based on RouteNet delay estimations begins to achieve better performance with medium traffic intensity (TI=10-13) and, for high traffic intensity (TI=13-15), it achieves considerable higher performance. Particularly, with TI=15, it obtains the following results:
- When optimizing the mean delay/jitter, the RouteNet-based optimizer achieves 20.87%/35.27% lower delay/jit-ter than the SP policy, and 12.18%/27.21% lower delay/jitter than the utilization-based optimizer.
- When optimizing the maximum delay/jitter, the Route-Net-based optimizer achieves 40.08%/48.09% lower delay/jitter than the SP policy, and 8.11%/43.53% lower delay/jitter than the utilization-based optimizer.
SLA optimization
----------------
This use-case represents a network scenario where the routing optimizer must comply a Service Level Agreement (SLA) for some specific clients, while minimizing the impact on the performance of the rest of users in the network. In particular, we consider 4 source-destination pairs to have specific delay requirements. We made the experiments in the NSF network and selected the following source-destination pairs (S-D pairs) that must comply a certain delay requirement: (0,3) (3,4) (3,5) (3,6). Then, the objective is to optimize the routing configuration to guarantee that the traffic among those sources and destinations is below the target delay.
[1.0]{} ![Delay optimization under SLA guarantees[]{data-label="fig:sla"}](figures/optSLA_mean.pdf "fig:"){width="1.0\linewidth"}
[1.0]{} ![Delay optimization under SLA guarantees[]{data-label="fig:sla"}](figures/optSLA_max.pdf "fig:"){width="1.0\linewidth"}
Fig. \[fig:mean\_SLA\] shows the results in the case that the RouteNet-based optimizer aims to optimize the mean delay experienced in the network, while Fig. \[fig:max\_SLA\] shows the results for the case of minimizing the maximum delay for all the source-destination pairs. In these figures, the dashed line (labeled as “Non SLA scenario”) represents the results if the optimizer does not distinguish between different traffic classes, and the solid lines represent the results after applying the optimal routing scheme that complies the SLA of the 4 S-D pairs. The dotted line represents the delay requirement of the S-D pairs with SLA, which is an input parameter of the optimizer. Then, we can observe that for the optimization case that considers the SLAs, the delay experienced by the 4 S-D pairs with SLA (labeled as “SLA S-D pairs”) fulfills the delay requirements (dotted line) even with high traffic intensities (TI=13-16). Moreover, we observe that the rest of S-D pairs without SLA requirements (labeled as “Rest of S-D pairs”) do not experience a great increase in the mean/maximum delays compared to the “non SLA scenario”. For instance, with high traffic intensity (TI=15), in the case of optimizing the mean delay, the rest of the traffic only experiences an increase of 9.9% in the average delay (14.8% in the case of optimizing maximum delay).
Robustness against links failures
---------------------------------
In this use-case, we show how our model is able to generalize in the presence of link failures. When a certain link fails, it is necessary to find a new routing that avoids this link to reroute the traffic. As the number of links failures increases, less paths are available and the network becomes more saturated.
[1.0]{} ![Delay optimization under the presence of different link failures[]{data-label="fig:failures"}](figures/optMeanFailures.pdf "fig:"){width="1.0\linewidth"}
[1.0]{} ![Delay optimization under the presence of different link failures[]{data-label="fig:failures"}](figures/optMaxFailures.pdf "fig:"){width="1.0\linewidth"}
We evaluate the performance of the aforementioned methods under the presence of link failures following the same methodology than in the first use-case (see Sect. \[sec:use-cases:1\]). The initial network state is a low traffic intensity scenario (TI=8).
Fig. \[fig:failures\] shows the optimized mean delay (Fig. \[fig:mean\_failures\]) and the optimized max delay (Fig. \[fig:max\_failures\]). Each point in the graph corresponds to the optimal delay obtained under 10 random possible links failures. We observe that, as shown in the first use-case, the mean and the maximum delays increase as the network is more congested and, in these scenarios, the RouteNet-based optimization mechanism outperforms traditional approaches.
What-if scenarios
-----------------
One application of interest of network modeling is that network operators can simulate hypothetical what-if scenarios to evaluate the resulting performance before making strategic decisions. These decisions, for instance, include making agreements to route a considerable bulk of traffic from other network (e.g., BGP peering agreements) or finding a network upgrade that results more beneficial given a limited budget.
[1.0]{} ![Delay optimization as a function of the number of new users[]{data-label="fig:users"}](figures/optMeanUsers.pdf "fig:"){width="1.0\linewidth"}
[1.0]{} ![Delay optimization as a function of the number of new users[]{data-label="fig:users"}](figures/optMaxUsers.pdf "fig:"){width="1.0\linewidth"}
**Adding new users**
The objective of this use-case is to evaluate the performance of the network under the presence of potential new users. Each new user in the network increases the amount of traffic that it has to support, and consequently the average and the maximum delay are increased.
Specifically, we explore when certain delay requirements cannot be fulfilled as the number of users with high bandwidth requirements increases. We model these new users as follows: each user multiplies by 2.5 the existing bandwidth demand in a certain node, the first user is connected to node 10, the second one to node 2, the third one to node 8, the fourth one to node 5, the fifth one to node 12, the sixth one to node 1, the seventh one to node 7 and the last one to node 0. We repeat this process under 3 different traffic matrices with initial low traffic intensity (TI=8).
Fig. \[fig:users\] shows the mean and maximum delay as new users are subscribed to the network. The dotted line represents the delay requirement, whereas the other lines represent the delay obtained with these different traffic matrices. We observe that the RouteNet model is able to predict the future performance of the network and to know “a priori” when the delay requirements will not be accomplished. For example, we observe that a network operating with TM$_1$ will require an update before than the networks operating under the other traffic matrices.
**Budget-constrained network upgrade**
In this final use-case, we address a common problem in networking, how to optimally upgrade the network by adding a new link between two nodes. For this, we take advantage of the RouteNet-based model to explore different options to place this new link to select the one that minimizes the mean delay.
Table \[tab:newLink\] shows the optimal new placement in the NSF network topology under 10 different traffic matrices with high traffic intensity (TI=15). For each, we also show the average delay before placing the link, the obtained delay with the new optimal link and the delay reduction achieved. We observe that we can achieve an important reduction on the average delay by properly choosing between which nodes this new link is deployed. Note that the optimal placement for the new link depends on the traffic conditions in the network.
\[tab:newLink\]
Related work
============
Network modeling with deep neural networks is a recent topic proposed in the literature [@wangMachineLearning; @kdn] with few pioneering attempts. The closest works to our contribution are first Deep-Q [@deepQ], where the authors infer the QoS of a network using the traffic matrix as an input using Deep Generative Models. And second [@mestresModeling], where a fully-connected feed-forward neural network is used to model the mean delay of a set of networks using as input the traffic matrix, the main goal of the authors is to understand how fundamental network characteristics (such as traffic intensity) relate with basic neural network parameters (depth of the neural network). RouteNet is also able to produce accurate estimates of performance metrics -delay and jitter-, but it does not assume a fixed topology and/or routing, rather it is able to produce such estimates with arbitrary topologies and routing schemes not seen during training. This enables RouteNet to be used for network operation, optimization and what-if scenarios.
Finally, an early attempt to use Graph Neural Networks for computer networks can be found in [@geyer2018learning]. In this case the authors use a GNN to learn shortest-path routing and max-min routing using supervised learning. While this approach is able to generalize to different topologies it cannot generalize to different routing schemes beyond the ones for which has been specifically trained. In addition the focus of the paper is not to estimate the performance of such routing schemes.
Conclusions
===========
SDN has brought an unprecedented degree of flexibility to network management, which allows the network controller to configure the network behavior up to the flow-level granularity. This flexibility combined with the information provided by network telemetry opens many possibilities for online network optimization.
However, existing network modeling techniques based on analytic models cannot handle this huge complexity. As a result, current optimization approaches are limited to improve a global performance metric, such as network utilization. Although Deep Learning (DL) is a promising solution to handle such complexity and to exploit the full potential of the SDN paradigm, previous attempts to apply DL to networking problems resulted in tailor-made solutions that failed to generalize to other network scenarios.
In this paper, we presented RouteNet, a new type of Graph Neural Network (GNN) that is specifically designed for modeling computer networks. RouteNet is inspired by the Message-Passing Neural Network (MPNN) previously proposed in the field of quantum chemistry. The main innovation behind RouteNet is a novel message-passing protocol that allows the GNN to capture the complex relationships between the paths and links that form a network topology and the network traffic.
We used RouteNet to model the per-source/destination delay and jitter of a network. Our results show that RouteNet is able to generalize to other network topologies, routing configurations and traffic matrices that were not present in the training set. We finally presented some illustrative use-cases that show the potential of RouteNet to be applied for network optimization in SDN. In particular, we showed that an SDN controller can use RouteNet to optimize multiple KPI and to guarantee the SLA of a particular set of flows, as well as to analyze different what-if scenarios.
This work was supported by AGH University of Science and Technology grant, under contract no. 15.11.230.400, the Spanish MINECO under contract TEC2017-90034-C2-1-R (ALLIANCE) and the Catalan Institution for Research and Advanced Studies (ICREA). The research was also supported in part by PL-Grid Infrastructure.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $R$ be a regular local ring, containing an infinite field. Let ${\mathbf G}$ be a reductive group scheme over $R$. We prove that a principal ${\mathbf G}$-bundle over $R$ is trivial, if it is trivial over the fraction field of $R$. In other words, if $K$ is the fraction field of $R$, then the map of non-abelian cohomology pointed sets $$H^1_{\text{\''et}}(R,{\mathbf G})\to H^1_{\text{\''et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, has a trivial kernel.'
address:
- 'Mathematics Department, 138 Cardwell Hall, Manhattan, KS 66506, USA'
- 'Steklov Institute of Mathematics at St.-Petersburg, Fontanka 27, St.-Petersburg 191023, Russia'
author:
- Roman Fedorov
- Ivan Panin
bibliography:
- 'GrSerre.bib'
title: 'A proof of Grothendieck-Serre conjecture on principal bundles over regular local rings containing infinite fields'
---
Introduction
============
Assume that $U$ is a regular scheme, ${\mathbf G}$ is a reductive $U$-group scheme. Recall that a $U$-scheme ${\mathcal G}$ with an action of ${\mathbf G}$ is called *a principal ${\mathbf G}$-bundle over $U$*, if ${\mathcal G}$ is faithfully flat over $U$ and the action is simple transitive, that is, the natural morphism ${\mathbf G}\times_U{\mathcal G}\to{\mathcal G}\times_U{\mathcal G}$ is an isomorphism, see [@FGA Section 6]. It is well known that such a bundle is trivial locally in étale topology but in general not in Zariski topology. Grothendieck and Serre conjectured that ${\mathcal G}$ is trivial locally in Zariski topology, if it is trivial generically. More precisely
Let $R$ be a regular local ring, let $K$ be its field of fractions. Let ${\mathbf G}$ be a reductive group scheme over $U:=\operatorname{Spec}R$, let ${\mathcal G}$ be a principal ${\mathbf G}$-bundle. If ${\mathcal G}$ is trivial over $\operatorname{Spec}K$, then it is trivial. Equivalently, the map of non-abelian cohomology pointed sets $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, has a trivial kernel.
The main result of this paper is a proof of this conjecture for regular local rings $R$, containing infinite fields. Our proof was inspired by the theory of affine Grassmannians. It also uses significantly the geometric part of the paper [@PSV] by the second author with A. Stavrova and N. Vavilov.
Our result implies that two principal ${\mathbf G}$-bundles over $U$ are isomorphic, if they are isomorphic over $\operatorname{Spec}K$ as proved in the next section. This result is new even for constant group schemes (that is, for group schemes coming from the ground field).
Recall that a part of the Gersten conjecture asserts that the natural homomorphism of $\mathrm K$-groups $\mathrm K_q(R)\to\mathrm K_q(K)$ is injective. Very roughly speaking, the Grothendieck–Serre conjecture is a non-abelian version of this part of the Gersten conjecture.
History of the topic
--------------------
Here is a list of known results in the same vein, corroborating the Grothendieck–Serre conjecture.
$\bullet$ The case, where the group scheme ${\mathbf G}$ comes from an infinite ground field, is completely solved by J.-L. Colliot-Thélène, M. Ojanguren, and M. S. Raghunatan in [@C-TO] and [@R1; @R2]; O. Gabber announced a proof for group schemes coming from arbitrary ground fields.
$\bullet$ The case of an arbitrary reductive group scheme over a discrete valuation ring or over a henselian ring is completely solved by Y. Nisnevich in [@Ni1]. He also proved the conjecture for two-dimensional local rings in the case, when ${\mathbf G}$ is quasi-split in [@Ni2].
$\bullet$ The case, where ${\mathbf G}$ is an arbitrary torus over a regular local ring, was settled by J.-L. Colliot-Thélène and J.-J. Sansuc in [@C-T-S].
$\bullet$ For some simple group schemes of classical series the conjecture is solved in works of the second author, A. Suslin, M. Ojanguren, and K. Zainoulline; see [@Oj1], [@Oj2], [@PS], [@OP2], [@Z], [@OPZ].
$\bullet$ Under an isotropy condition on ${\mathbf G}$ the conjecture is proved in a series of preprints [@PSV] and [@Pa2].
$\bullet$ The case of strongly inner simple adjoint group schemes of the types $E_6$ and $E_7$ is done by the second author, V. Petrov, and A. Stavrova in [@PPS]. No isotropy condition is imposed there.
$\bullet$ The case, when ${\mathbf G}$ is of the type $F_4$ with trivial $g_3$-invariant and the field is of characteristic zero, is settled by V. Chernousov in [@Chernous]; the case, when ${\mathbf G}$ is of the type $F_4$ with trivial $f_3$-invariant and the field is infinite and perfect, is settled by V. Petrov and A. Stavrova in [@PetrovStavrova].
Acknowledgments
---------------
The authors collaborated during Summer schools “Contemporary Mathematics”, organized by Moscow Center for Continuous Mathematical Education. They would like to thank the organizers of the school for the perfect working atmosphere. The authors benefited from the talks with A. Braverman. They also thank J.-L. Colliot-Th[é]{}l[è]{}ne and P. Gille for their interest to the topic.
A part of the work was done, while the first author was a member of Max Planck Institute for Mathematics in Bonn. The first author also benefited from talks with D. Arinkin.
The second author is very grateful to A. Stavrova and N. Vavilov for a stimulating interest to the topic. He also thanks the RFBR-grant 13–01–00429–a for the support.
Main results {#Introduction}
============
Let $R$ be a commutative unital ring. Recall that an $R$-group scheme ${\mathbf G}$ is called *reductive*, if it is affine and smooth as an $R$-scheme and if, moreover, for each algebraically closed field $\Omega$ and for each ring homomorphism $R\to\Omega$ the scalar extension ${\mathbf G}_\Omega$ is a connected reductive algebraic group over $\Omega$. This definition of a reductive $R$-group scheme coincides with [@SGA3-3 Exp. XIX, Definition 2.7]. A well-known conjecture due to J.-P. Serre and A. Grothendieck (see [@Se Remarque, p.31], [@Gr1 Remarque 3, p.26-27], and [@Gr2 Remarque 1.11.a]) asserts that given a regular local ring $R$ and its field of fractions $K$ and given a reductive group scheme ${\mathbf G}$ over $R$, the map $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, has a trivial kernel. The following theorem, which is the main result of the present paper, asserts that this conjecture holds, provided that $R$ contains an infinite field.
\[MainThm1\] Let $R$ be a regular semi-local domain containing an infinite field, and let $K$ be its field of fractions. Let ${\mathbf G}$ be a reductive group scheme over $R$. Then the map $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, has a trivial kernel. In other words, under the above assumptions on $R$ and ${\mathbf G}$, each principal ${\mathbf G}$-bundle over $R$ having a $K$-rational point is trivial.
Theorem \[MainThm1\] has the following
Under the hypothesis of Theorem \[MainThm1\], the map $$H^1_{\text{\'et}}(R,{\mathbf G})\to H^1_{\text{\'et}}(K,{\mathbf G}),$$ induced by the inclusion of $R$ into $K$, is injective. Equivalently, if ${\mathcal G}_1$ and ${\mathcal G}_2$ are two principal bundles isomorphic over $\operatorname{Spec}K$, then they are isomorphic.
Let ${\mathcal G}_1$ and ${\mathcal G}_2$ be two principal ${\mathbf G}$-bundles isomorphic over $\operatorname{Spec}K$. Let $\operatorname{Iso}({\mathcal G}_1,{\mathcal G}_2)$ be the scheme of isomorphisms. This scheme is a principal $\operatorname{Aut}{\mathcal G}_2$-bundle. By Theorem \[MainThm1\] it is trivial, and we see that ${\mathcal G}_1\cong{\mathcal G}_2$.
Note that, while Theorem \[MainThm1\] was previously known for reductive group schemes ${\mathbf G}$ coming from the ground field, in certain cases the corollary is a new result even for such group schemes.
For a scheme $U$ we denote by ${\mathbb A}^1_U$ the affine line over $U$ and by ${\mathbb P}^1_U$ the projective line over $U$. Let $T$ be a $U$-scheme. By a principal ${\mathbf G}$-bundle over $T$ we understand a principal ${\mathbf G}\times_UT$-bundle.
In Section \[sect:redtopsv\] we deduce Theorem \[MainThm1\] from the following result of independent interest (cf. [@PSV Theorem 1.3]).
\[th:psv\] Let $R$ be the semi-local ring of finitely many closed points on an irreducible smooth affine variety over an infinite field $k$, set $U=\operatorname{Spec}R$. Let ${\mathbf G}$ be a simple simply-connected group scheme over $U$ (see [@SGA3-3 Exp. XXIV, Sect. 5.3] for the definition). Let ${\mathcal E}_t$ be a principal ${\mathbf G}$-bundle over the affine line ${\mathbb A}^1_U=\operatorname{Spec}R[t]$, and let $h(t)\in R[t]$ be a monic polynomial. Denote by $({\mathbb A}^1_U)_h$ the open subscheme in ${\mathbb A}^1_U$ given by $h(t)\ne0$ and assume that the restriction of ${\mathcal E}_t$ to $({\mathbb A}^1_U)_h$ is a trivial principal ${\mathbf G}$-bundle. Then for each section $s:U\to{\mathbb A}^1_U$ of the projection ${\mathbb A}^1_U\to U$ the ${\mathbf G}$-bundle $s^*{\mathcal E}_t$ over $U$ is trivial.
The derivation of Theorem \[MainThm1\] from Theorem \[th:psv\] is based on results of the second author, A. Stavrova, and N. Vavilov, namely, on [@Pa2] and [@PSV Theorem 1.2].
Let $Y$ be a semi-local scheme. We will call a simple $Y$-group scheme isotropic, if its restriction to each connected component of $Y$ contains a proper parabolic subgroup scheme. (Note that by [@SGA3-3 Exp. XXVI, Cor. 6.14] this is equivalent to the usual definition, that is to the requirement that the group scheme contains a torus isomorphic to $\mathbb G_{m,Y}$.) Theorem \[th:psv\] is, in turn, derived from the following statement.
\[MainThm2\] Let $R$, $U$, and ${\mathbf G}$ be as in Theorem \[th:psv\].
Let $Z\subset{\mathbb P}^1_U$ be a closed subscheme finite over $U$. Let $Y\subset{\mathbb P}^1_U$ be a closed subscheme étale over $U$. Assume that $Y\cap Z={\varnothing}$, and ${\mathbf G}_Y:={\mathbf G}\times_UY$ is isotropic. Assume also that for every closed point $u\in U$ such that the algebraic group ${\mathbf G}_u:={\mathbf G}|_u$ is isotropic, there is a $k(u)$-rational point in $Y_u:={\mathbb P}^1_u\cap Y$. (Here $k(u)$ is the residue field of $u$.)
Let ${\mathcal G}$ be a principal ${\mathbf G}$-bundle over ${\mathbb P}^1_U$ such that its restriction to ${\mathbb P}^1_U-Z$ is trivial. Then the restriction of ${\mathcal G}$ to ${\mathbb P}^1_U-Y$ is also trivial.
The proof of this result was inspired by the theory of affine Grassmannians (see Section \[sect:loops\]).
1\. Assume that for every closed point $u\in U$ the algebraic group ${\mathbf G}_u$ is anisotropic. Then we can take $Y={\varnothing}$.
2\. It is not necessary to assume that $Y\cap Z={\varnothing}$. Indeed, let $Y$ satisfy the conditions of the theorem except that it may intersect $Z$. Since $U$ is semi-local, applying a projective transformation of ${\mathbb P}^1_U$ to $Y$, we can find $Y'\subset{\mathbb P}^1_U$ satisfying the same conditions as $Y$ but such that $Y'\cap Y=Y'\cap Z={\varnothing}$. By the above theorem the restriction of ${\mathcal G}$ to ${\mathbb P}^1_U-Y'$ is trivial. Now we can apply the theorem again with $Z=Y'$ to show that the restriction of ${\mathcal G}$ to ${\mathbb P}^1_U-Y$ is trivial.
3\. In the situation of Theorem \[MainThm2\], let ${\mathbf G}$ be isotropic. Then it follows from the theorem that one can take $Y=\{\infty\}\times U\subset{\mathbb P}^1_U$, that is, the restriction of ${\mathcal G}$ to ${\mathbb A}^1_U$ is trivial. In fact, this is a partial case of [@PSV Theorem 1.3]. On the other hand, if ${\mathbf G}$ is anisotropic, this restriction is not in general trivial. For an example see [@FedorovExotic].
Organization of the paper
-------------------------
In Section \[sect:redtopsv\], we reduce Theorem \[MainThm1\] to Theorem \[th:psv\]. In Section \[sect:reducing\], we reduce Theorem \[th:psv\] to Theorem \[MainThm2\]. This reduction is based on [@Pa2], [@PSV Theorem 1.2], and a theorem of D. Popescu [@P].
In Section \[sect:proof2\] we prove Theorem \[MainThm2\]. The main idea is to modify the principal bundle ${\mathcal G}$ in a neighborhood of $Y$ so that ${\mathcal G}$ becomes trivial. We give an outline of the proof in Section \[sect:outline\]. We use the technique of henselization. An essentially equivalent proof based on formal loops is outlined in Section \[sect:loops\].
In Section \[sect:Grassm\] we explain the relation to affine Grassmannians, hinting at the circle of ideas that led to the proof.
In Section \[sect:application\] we give an application of Theorem \[MainThm1\].
In the Appendix we recall the definition of henselization from [@Gabber Section 0].
Reducing Theorem \[MainThm1\] to Theorem \[th:psv\] {#sect:redtopsv}
===================================================
In what follows “${\mathbf G}$-bundle” always means “principal ${\mathbf G}$-bundle”. Now we assume that Theorem \[th:psv\] holds. We start with the following particular case of Theorem \[MainThm1\].
\[pr:geometric\] Let $R$, $U=\operatorname{Spec}R$, and ${\mathbf G}$ be as in Theorem \[th:psv\]. Let ${\mathcal E}$ be a principal ${\mathbf G}$-bundle over $U$, trivial at the generic point of $U$. Then ${\mathcal E}$ is trivial.
Under the hypothesis of the proposition, the following data are constructed in [@PSV Theorem 1.2]:\
[[)]{} ]{}a principal ${\mathbf G}$-bundle ${\mathcal E}_t$ over ${\mathbb A}^1_U$;\
[[)]{} ]{}a monic polynomial $h(t)\in R[t]$.\
Moreover these data satisfies the following conditions:\
(1) the restriction of ${\mathcal E}_t$ to $({\mathbb A}^1_U)_h$ is a trivial principal ${\mathbf G}$-bundle;\
(2) there is a section $s:U\to{\mathbb A}^1_U$ such that $s^*{\mathcal E}_t={\mathcal E}$.
Now it follows from Theorem \[th:psv\] that ${\mathcal E}$ is trivial.
\[pr:reductivegeometric\] Let $U$ be as in Theorem \[th:psv\]. Let ${\mathbf G}$ be a reductive group scheme over $U$. Let ${\mathcal E}$ be a principal ${\mathbf G}$-bundle over $U$ trivial at the generic point of $U$. Then ${\mathcal E}$ is trivial.
Firstly, using [@Pa2 Th. 1.0.1], we can assume that ${\mathbf G}$ is semi-simple and simply-connected. Secondly, standard arguments (see for instance [@PSV Section 9]) show that we can assume that ${\mathbf G}$ is simple and simply-connected. (Note that for this reduction it is necessary to work with semi-local rings.) Now the proposition is reduced to Proposition \[pr:geometric\].
Let us prove a general statement first. Let $k'$ be an infinite field, $X$ be a $k'$-smooth irreducible affine variety, ${\mathbf H}$ be a reductive group scheme over $X$. Denote by $k'[X]$ the ring of regular functions on $X$ and by $k'(X)$ the field of rational functions on $X$. Let ${\mathcal H}$ be a principal ${\mathbf H}$-bundle over $X$ trivial over $k'(X)$. Let $\mathfrak p_1,\dots,\mathfrak p_n$ be prime ideals in $k'[X]$, and let ${\mathcal O}_{\mathfrak p_1,\dots,\mathfrak p_n}$ be the corresponding semi-local ring.
\[lm:primemax\] The principal ${\mathbf H}$-bundle ${\mathcal H}$ is trivial over ${\mathcal O}_{\mathfrak p_1,\dots,\mathfrak p_n}$.
For each $i=1,2,\ldots,n$ choose a maximal ideal $\mathfrak m_i\subset k'[X]$ containing $\mathfrak p_i$. One has inclusions of $k'$-algebras $${\mathcal O}_{\mathfrak m_1,\dots,\mathfrak m_n}\subset{\mathcal O}_{\mathfrak p_1,\dots,\mathfrak p_n}\subset k'(X).$$ By Proposition \[pr:reductivegeometric\] the principal ${\mathbf H}$-bundle ${\mathcal H}$ is trivial over ${\mathcal O}_{\mathfrak m_1,\dots,\mathfrak m_n}$. Thus it is trivial over ${\mathcal O}_{\mathfrak p_1,\dots,\mathfrak p_n}$.
Let us return to our situation. Let ${\mathfrak m}_1,\ldots,{\mathfrak m}_n$ be all the maximal ideals of $R$. Let ${\mathcal E}$ be a ${\mathbf G}$-bundle over $R$ trivial over the fraction field of $R$. Clearly, there is a non-zero $f\in R$ such that ${\mathcal E}$ is trivial over $R_f$. Let $k'$ be the algebraic closure of the prime field of $R$ in $k$. Note that $k'$ is perfect. It follows from Popescu’s theorem ([@P; @Sw]) that $R$ is a filtered inductive limit of smooth $k'$-algebras $R_\alpha$. Modifying the inductive system $R_\alpha$ if necessary, we can assume that each $R_\alpha$ is integral. There are an index $\alpha$, a reductive group scheme ${\mathbf G}_{\alpha}$ over $R_{\alpha}$, a principal ${\mathbf G}_{\alpha}$-bundle ${\mathcal E}_{\alpha}$ over $R_{\alpha}$, and an element $f_{\alpha }\in R_{\alpha}$ such that ${\mathbf G}={\mathbf G}_{\alpha}\times_{\operatorname{Spec}R_{\alpha}}\operatorname{Spec}R$, ${\mathcal E}$ is isomorphic to ${\mathcal E}_{\alpha}\times_{\operatorname{Spec}R_{\alpha}}\operatorname{Spec}R$ as principal ${\mathbf G}$-bundle, $f$ is the image of $f_{\alpha}$ under the homomorphism ${\varphi}_{\alpha}: R_{\alpha}\to R$, ${\mathcal E}_{\alpha}$ is trivial over $(R_{\alpha})_{f_{\alpha}}$.
If the field $k'$ is infinite, then for each maximal ideal $\mathfrak m_i$ in $R$ ($i=1,\dots, n$) set $\mathfrak p_i={\varphi}_{\alpha}^{-1}(\mathfrak m_i)$. The homomorphism ${\varphi}_\alpha$ induces a homomorphism of semi-local rings $(R_{\alpha})_{\mathfrak p_1,\dots,\mathfrak p_n}\to R$. By Lemma \[lm:primemax\] the principal ${\mathbf G}_{\alpha}$-bundle ${\mathcal E}_{\alpha}$ is trivial over $(R_{\alpha})_{\mathfrak p_1,\dots,\mathfrak p_n}$. Whence the ${\mathbf G}$-bundle ${\mathcal E}$ is trivial over $R$.
If the field $k'$ is finite, then $k$ contains an element $t$ transcendental over $k'$. Thus $R$ contains the subfield $k'(t)$ of rational functions in the variable $t$. So, if $R'_{\alpha}:= R_{\alpha}\otimes_{k'} k'(t)$, then ${\varphi}_{\alpha}$ can be decomposed as follows $$R_{\alpha}\to R_{\alpha}\otimes_{k'}k'(t)=R'_{\alpha}\xrightarrow{\psi_{\alpha}}R.$$ Let ${\mathbf G}'_{\alpha}={\mathbf G}_{\alpha}\times_{\operatorname{Spec}R_{\alpha}}\operatorname{Spec}R'_{\alpha}$, ${\mathcal E}'_{\alpha}={\mathcal E}_{\alpha}\times_{\operatorname{Spec}R_{\alpha}}\operatorname{Spec}R'_{\alpha}$, $f'_{\alpha}=f_{\alpha}\otimes 1\in R'_{\alpha}$, then the ${\mathbf G}'_{\alpha}$-bundle ${\mathcal E}'_{\alpha}$ is trivial over $(R'_{\alpha})_{f'_{\alpha}}$.
Let $\mathfrak q_i=\psi_{\alpha}^{-1}(\mathfrak m_i)$ for $i=1,\dots, n$. The ring $R'_{\alpha}$ is a $k'(t)$-smooth algebra over the infinite field $k'(t)$, and the ${\mathbf G}'_{\alpha}$-bundle ${\mathcal E}'_{\alpha}$ is trivial over $(R'_{\alpha})_{f'_{\alpha}}$. By Lemma \[lm:primemax\] the ${\mathbf G}'_{\alpha}$-bundle ${\mathcal E}'_{\alpha}$ is trivial over $(R'_{\alpha})_{\mathfrak q_{1},\dots,\mathfrak q_{n}}$. The homomorphism $\psi_{\alpha}$ can be factored as $$R'_{\alpha}\to (R'_{\alpha})_{\mathfrak q_{1},\dots,\mathfrak q_{n}}\to R.$$ Thus the ${\mathbf G}$-bundle ${\mathcal E}$ is trivial over $R$.
If $k$ is perfect, we can use it instead of $k'$, and the above proof simplifies.
Reducing Theorem \[th:psv\] to Theorem \[MainThm2\] {#sect:reducing}
===================================================
Now we assume that Theorem \[MainThm2\] is true. Let $U$ and ${\mathbf G}$ be as in Theorem \[th:psv\]. Let $u_1,\ldots,u_n$ be all the closed points of $U$. Let $k(u_i)$ be the residue field of $u_i$. Consider the reduced closed subscheme ${\mathbf u}$ of $U$, whose points are $u_1$, …, $u_n$. Thus $${\mathbf u}\cong\coprod_i\operatorname{Spec}k(u_i).$$ Set ${\mathbf G}_{\mathbf u}={\mathbf G}\times_U{\mathbf u}$. By ${\mathbf G}_{u_i}$ we denote the fiber of ${\mathbf G}$ over $u_i$; it is a simple simply-connected algebraic group over $k(u_i)$. Let ${\mathbf u}'\subset{\mathbf u}$ be the subscheme of all closed points $u_i$ such that the group ${\mathbf G}_{u_i}$ is isotropic. Set ${\mathbf u}''={\mathbf u}-{\mathbf u}'$. It is possible that ${\mathbf u}'$ or ${\mathbf u}''$ is empty.
There is a closed subscheme $Y\subset{\mathbb P}^1_U$ such that $Y$ is étale over $U$, ${\mathbf G}_Y={\mathbf G}\times_UY$ is isotropic, and for all $u_i\in{\mathbf u}'$ there is a $k(u_i)$-rational point $y_i\in Y$ lying over $u_i$.
If ${\mathbf u}'$ is empty, we just take $Y={\varnothing}$.
Otherwise, for every $u_i$ in ${\mathbf u}'$ choose a proper parabolic subgroup ${\mathbf P}_{u_i}$ in ${\mathbf G}_{u_i}$. Let ${\mathcal P}_i$ be the $U$-scheme of parabolic subgroup schemes of ${\mathbf G}$ of the same type as ${\mathbf P}_{u_i}$. It is a smooth projective $U$-scheme (see [@SGA3-3 Cor. 3.5, Exp. XXVI]). The subgroup ${\mathbf P}_{u_i}$ in ${\mathbf G}_{u_i}$ is a $k(u_i)$-rational point $p_i$ in the fibre of ${\mathcal P}_i$ over the point $u_i$. Using a variant of Bertini theorem, we can find a closed subscheme $Y_i$ of ${\mathcal P}_i$ such that $Y_i$ is étale over $U$ and $p_i\in Y_i$ (take an embedding of ${\mathcal P}_i$ into a projective space ${\mathbb P}^N_U$ and intersect ${\mathcal P}_i$ with appropriately chosen family of quadrics containing the point $p_i$. Arguing as in the proof of [@OP2 Lemma 7.2], we get the desired scheme $Y_i$ finite and étale over $U$.)
Now consider $Y_i$ just as a $U$-scheme and set $Y=\coprod_{u_i\in{\mathbf u}'}Y_i$. Next, ${\mathbf G}_{Y_i}$ is isotropic by the choice of $Y_i$. Thus ${\mathbf G}_Y$ is isotropic as well. Since the field $k$ is infinite and $Y$ is finite étale over $U$, we can choose a closed $U$-embedding of $Y$ in ${\mathbb A}^1_U$. We will identify $Y$ with the image of this closed embedding. Since $Y$ is finite over $U$, it is closed in ${\mathbb P}_U^1$.
Set $Z:=\{h=0\}\cup s(U)\subset{\mathbb A}^1_U$. Note that $\{h=0\}$ is closed in ${\mathbb P}^1_U$ and finite over $U$ because $h$ is monic. Further, $s(U)$ is also closed in ${\mathbb P}^1_U$ and finite over $U$ because it is a zero set of a degree one monic polynomial. Thus $Z\subset{\mathbb P}^1_U$ is closed and finite over $U$.
Further, $U$ is semi-local, so we may assume that $Z\cap Y={\varnothing}$ (use the fact that $k$ is infinite and apply to $Y$ an appropriative affine $U$-transformation of ${\mathbb A}^1_U$).
Since the principal ${\mathbf G}$-bundle ${\mathcal E}_t$ is trivial over $({\mathbb A}^1_U)_h$, and ${\mathbf G}$-bundles can be glued in Zariski topology, there exists a principal ${\mathbf G}$-bundle ${\mathcal G}$ over ${\mathbb P}^1_U$ such that\
(i) its restriction to ${\mathbb A}^1_U$ coincides with ${\mathcal E}_t$;\
(ii) its restriction to ${\mathbb P}^1_U-Z$ is trivial.
Applying Theorem \[MainThm2\] with the above choice of $Y$ and $Z$, we see that the restriction of ${\mathcal G}$ to ${\mathbb P}^1_U-Y$ is a trivial ${\mathbf G}$-bundle. Since $s(U)$ is in $({\mathbb P}^1_U-Y)\cap{\mathbb A}^1_U$, and ${\mathcal G}|_{{\mathbb A}^1_U}$ coincides with ${\mathcal E}_t$, we conclude that $s^*{\mathcal E}_t$ is a trivial principal ${\mathbf G}$-bundle over $U$.
Proof of Theorem \[MainThm2\] {#sect:proof2}
=============================
We will be using notation from Theorem \[MainThm2\]. Let ${\mathbf u}$, ${\mathbf u}'$, and ${\mathbf u}''$ be as in Section \[sect:reducing\]. For $u\in{\mathbf u}$ set ${\mathbf G}_u={\mathbf G}|_u$.
\[pr:trivclsdfbr\] Let ${\mathcal E}$ be a ${\mathbf G}$-bundle over ${\mathbb P}^1_U$ such that ${\mathcal E}|_{{\mathbb P}^1_u}$ is a trivial ${\mathbf G}_u$-bundle for all $u\in{\mathbf u}$. Assume that there exists a closed subscheme $T$ of ${\mathbb P}^1_U$ finite over $U$ such that the restriction of ${\mathcal E}$ to ${\mathbb P}^1_U-T$ is trivial. Then ${\mathcal E}$ is trivial.
This follows from Theorem 7.6 of [@PSV].
An outline of a proof of Theorem \[MainThm2\] {#sect:outline}
---------------------------------------------
A detailed proof will be given in the present text below. Firstly, we give an outline of the proof.
Denote by $Y^h$ the henselization of the pair $({\mathbb A}_U^1,Y)$, it is a scheme over ${\mathbb A}_U^1$. Let $s:Y\to Y^h$ be the canonical closed embedding, see Section \[sect:distinguishedLimit\] for more details. Set $\dot Y^h:=Y^h-s(Y)$. Let ${\mathcal G}'$ be a ${\mathbf G}$-bundle over ${\mathbb P}^1_U-Y$. Denote by $\operatorname{Gl}({\mathcal G}',{\varphi})$ the ${\mathbf G}$-bundle over ${\mathbb P}_U^1$ obtained by gluing ${\mathcal G}'$ with the trivial ${\mathbf G}$-bundle ${\mathbf G}\times_U Y^h$ via a ${\mathbf G}$-bundle isomorphism ${\varphi}:{\mathbf G}\times_U\dot Y^h\to{\mathcal G}'|_{\dot Y^h}$.
Similarly, set $Y_{\mathbf u}:=Y\times_U{\mathbf u}$ and denote by $Y_{\mathbf u}^h$ the henselization of the pair $({\mathbb A}_{\mathbf u}^1,Y_{\mathbf u})$, let $s_{\mathbf u}:Y_{\mathbf u}\to Y_{\mathbf u}^h$ be the closed embedding. Set $\dot Y_{\mathbf u}^h:=Y_{\mathbf u}^h-s_{\mathbf u}(Y_{\mathbf u})$. Let ${\mathcal G}'_{\mathbf u}$ be a ${\mathbf G}_{\mathbf u}$-bundle over ${\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}$, where ${\mathbf G}_{\mathbf u}:={\mathbf G}\times_U{\mathbf u}$. Denote by $\operatorname{Gl}_{\mathbf u}({\mathcal G}'_{\mathbf u},{\varphi}_{\mathbf u})$ the ${\mathbf G}_{\mathbf u}$-bundle over ${\mathbb P}_{\mathbf u}^1$ obtained by gluing ${\mathcal G}'_{\mathbf u}$ with the trivial bundle ${\mathbf G}_{\mathbf u}\times_{\mathbf u}Y_{\mathbf u}^h$ via a ${\mathbf G}_{\mathbf u}$-bundle isomorphism ${\varphi}_{\mathbf u}:{\mathbf G}_{\mathbf u}\times_{\mathbf u}\dot Y_{\mathbf u}^h\to{\mathcal G}'_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$.
Note that the ${\mathbf G}$-bundle ${\mathcal G}$ can be presented in the form $\operatorname{Gl}({\mathcal G}',{\varphi})$, where ${\mathcal G}'={\mathcal G}|_{{\mathbb P}^1_U-Y}$. The idea is to show that
($*$) **
If we find $\alpha$ satisfying condition ($*$), then Proposition \[pr:trivclsdfbr\], applied to $T=Y\cup Z$, shows that the ${\mathbf G}$-bundle $\operatorname{Gl}({\mathcal G}',{\varphi}\circ\alpha)$ is trivial over ${\mathbb P}^1_U$. On the other hand, its restriction to ${\mathbb P}^1_U-Y$ coincides with the ${\mathbf G}$-bundle ${\mathcal G}'={\mathcal G}|_{{\mathbb P}^1_U-Y}$. *Thus ${\mathcal G}|_{{\mathbb P}^1_U-Y}$ is a trivial ${\mathbf G}$-bundle*.
To prove ($*$), one should show that\
(i) the bundle ${\mathcal G}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}$ is trivial;\
(ii) each element $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ can be written in the form $$\alpha|_{\dot Y_{\mathbf u}^h}\cdot\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$$ for certain elements $\alpha\in{\mathbf G}(\dot Y^h)$ and $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$.
If we succeed to show (i) and (ii), then we proceed as follows. Present the ${\mathbf G}$-bundle ${\mathcal G}$ in the form $\operatorname{Gl}({\mathcal G}',{\varphi})$ as above. Observe that $$\operatorname{Gl}({\mathcal G}',{\varphi})|_{{\mathbb P}^1_{\mathbf u}}\cong\operatorname{Gl}_{\mathbf u}({\mathcal G}'_{\mathbf u},{\varphi}_{\mathbf u}),$$ where ${\mathcal G}'_{\mathbf u}:={\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}$, ${\varphi}_{\mathbf u}:={\varphi}|_{{\mathbf G}_{\mathbf u}\times_{\mathbf u}\dot Y_{\mathbf u}^h}$.
Using property (i), find an element $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ such that the ${\mathbf G}_{\mathbf u}$-bundle $\operatorname{Gl}_{\mathbf u}({\mathcal G}'_{\mathbf u},{\varphi}_{\mathbf u}\circ\gamma_{\mathbf u})$ is trivial. For this $\gamma_{\mathbf u}$ find elements $\alpha$ and $\beta_{\mathbf u}$ as in (ii). Finally take the ${\mathbf G}$-bundle $\operatorname{Gl}({\mathcal G}',{\varphi}\circ\alpha)$. Then its restriction to ${\mathbb P}_{\mathbf u}^1$ is trivial. Indeed, one has a chain of ${\mathbf G}_{\mathbf u}$-bundle isomorphisms $$\begin{gathered}
\operatorname{Gl}({\mathcal G}',{\varphi}\circ \alpha)|_{{\mathbb P}^1_{\mathbf u}}\cong\operatorname{Gl}_{\mathbf u}({\mathcal G}'_{\mathbf u},{\varphi}_{\mathbf u}\circ \alpha|_{\dot Y_{\mathbf u}^h})
\cong\\
\operatorname{Gl}_{\mathbf u}({\mathcal G}_{\mathbf u}',{\varphi}_{\mathbf u}\circ
\alpha|_{\dot Y_{\mathbf u}^h}\circ \beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h})=
\operatorname{Gl}_{\mathbf u}({\mathcal G}_{\mathbf u}', {\varphi}_{\mathbf u}\circ \gamma_{\mathbf u}),\end{gathered}$$ which is trivial by the very choice of $\gamma_{\mathbf u}$. Thus, ($*$) will be achieved.
Let us prove (i) and (ii). If $u\in{\mathbf u}'$, then there is a $k(u)$-rational point in $Y_u:={\mathbb P}^1_u\cap Y$. Hence the ${\mathbf G}_u$-bundle ${\mathcal G}_u:={\mathcal G}|_{{\mathbb P}_u^1}$ is trivial over ${\mathbb P}^1_u-Y_u$ (see [@GilleTorseurs Corollary 3.10(a)]). If $u\in{\mathbf u}''$, then ${\mathbf G}_u$ is anisotropic and ${\mathcal G}_u$ is trivial even over ${\mathbb P}^1_u$ (again, by [@GilleTorseurs Corollary 3.10(a)]). Thus ${\mathcal G}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}$ is trivial. So, (i) is achieved.
To achieve (ii) recall that for a domain $A$, its fraction field $L$, and a simple group scheme ${\mathbf H}$ over $A$, having a parabolic subgroup scheme ${\mathbf P}$, one can form a subgroup ${\mathbf E}(L)$ of “elementary matrices” in ${\mathbf H}(L)$. It is known (see [@Gille:BourbakiTalk Fait 4.3, Lemma 4.5]) that if $A$ is a henselian discrete valuation ring and ${\mathbf H}$ is simply-connected, then every element $\gamma\in{\mathbf H}(L)$ can be written in the form $\gamma=\alpha\cdot\beta$, where $\alpha\in{\mathbf E}(L)$ and $\beta\in{\mathbf H}(A)$. Applying this observation in our context, we see that $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ can be written in the form $\gamma_{\mathbf u}=\alpha_{\mathbf u}\cdot\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$, where $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$ and $\alpha_{\mathbf u}\in{\mathbf E}(\dot Y_{\mathbf u}^h)$. It remains to observe that the natural homomorphism ${\mathbf E}(\dot Y^h)\to{\mathbf E}(\dot Y_{\mathbf u}^h)$ is surjective, since $\dot Y_{\mathbf u}^h$ is a closed subscheme of the affine scheme $\dot Y^h$, and so (ii) is achieved.
A realization of this plan in details is given below in the paper.
Henselization of affine pairs {#sect:distinguishedLimit}
-----------------------------
We will use the theory of henselian pairs and, in particular, a notion of a henselization $A^h_I$ of a commutative ring $A$ at an ideal $I$ (see Appendix and [@Gabber Section 0]). The geometric counterpart is this. Let $S=\operatorname{Spec}A$ be a scheme and $T=\operatorname{Spec}(A/I)$ be a closed subscheme. Then we define a category ${\widetilde\operatorname{Neib}}(S,T)$ whose objects are triples $(W',\pi':W'\to S,s': T\to W')$ satisfying the following conditions:\
(i) $W'$ is affine;\
(ii) $\pi'$ is an étale morphism;\
(iii) $\pi'\circ s'$ coincides with the inclusion $T\hookrightarrow S$ (thus $s'$ is a closed embedding).
A morphism from $(W',\pi',s')$ to $(W'',\pi'',s'')$ in this category is a morphism $\rho:W'\to W''$ such that $\pi''\circ\rho=\pi'$ and $\rho\circ s'=s''$. Note that such $\rho$ is automatically étale by [@LNM146 Ch. 6, Prop. 4.7].
Consider the functor from ${\widetilde\operatorname{Neib}}(S,T)$ to the category of $S$-schemes, sending $(W',\pi',s')$ to $(W',\pi')$. It is easy to see (and follows from [@Gabber Section 0]) that the category ${\widetilde\operatorname{Neib}}(S,T)$ is co-filtered, thus this functor has a projective limit $(T^h,\pi)$. We also get a closed embedding $s:T\to T^h$. Clearly, it is a morphism of $S$-schemes, thus $\pi\circ s$ coincides with the inclusion $T\hookrightarrow S$. We call $(T^h,\pi,s)$ *the henselization of the pair $(S,T)$* (cf. Definition \[def:henzelisation\]). Note that the pair $(T^h,s(T))$ is henselian, which means that for any affine étale morphism $\pi:Z\to T^h$, any section $\sigma$ of $\pi$ over $s(T)$ uniquely extends to a section of $\pi$ over $T^h$.
In the notation of [@Gabber Section 0] we have $T^h=\operatorname{Spec}A^h_I$, $\pi:T^h\to S$ is induced by the structure of $A$-algebra on $A^h_I$.
Denote by $\operatorname{Neib}(S,T)$ the full subcategory of ${\widetilde\operatorname{Neib}}(S,T)$ consisting of triples $(W',\pi',s')$ such that\
(iv) the schemes $(\pi')^{-1}(T)$ and $s'(T)$ coincide.
Let $(W',\pi',s')$ and $(W'',\pi'',s'')$ be objects of $\operatorname{Neib}(S,T)$. Let $\rho:W'\to W''$ be a morphism such that $\pi''\circ\rho=\pi'$. Then it is easy to see that $\rho\circ s'=s''$ so that $\rho$ is a morphism in $\operatorname{Neib}(S,T)$. (Again, $\rho$ is automatically étale.)
$\operatorname{Neib}(S,T)$ is co-final in ${\widetilde\operatorname{Neib}}(S,T)$.
Let $(W',\pi',s')$ be an object of $\widetilde{\operatorname{Neib}}(S,T)$. Let $\pi'_T:(\pi')^{-1}(T)\to T$ be the base-changed morphism; it is étale. It follows from (iii) that $s'$ is a section of $\pi'_T$. A section of an étale morphism is étale by [@LNM146 Ch. 6, Prop. 4.7]. Thus $s'$ is both an open and a closed embedding, and we have a disjoint union decomposition $(\pi')^{-1}(T)=s'(T)\coprod T_0$ for a scheme $T_0$. All our schemes are affine, so there is a regular function $f$ on $W'$ such that $f=0$ on $T_0$ and $f=1$ on $s'(T)$.
Set $W''=W'-\{f=0\}$, $\pi''=\pi'|_{W''}$, $s''=s'$. Then $W''$ is affine; thus $(W'',\pi'',s'')\in\operatorname{Neib}(S,T)$, and we have an obvious morphism $(W'',\pi'',s'')\to(W',\pi',s')$.
We see that the category $\operatorname{Neib}(S,T)$ is co-filtered, and the henselization can be computed by taking the limit over $\operatorname{Neib}(S,T)$, instead of $\widetilde{\operatorname{Neib}}(S,T)$. It is now easy to check that $\pi^{-1}(T)=s(T)$.
Note two properties of henselization of affine pairs\
(i) Let $T$ be a semi-local scheme. Then the henselization commutes with restriction to closed subschemes. In more details, if $S'\subset S$ is a closed subscheme, then we get a base change functor ${\widetilde\operatorname{Neib}}(S,T)\to{\widetilde\operatorname{Neib}}(S',T\times_SS')$. This gives a morphism $(T\times_SS')^h\to T^h\times_SS'$. This morphism is an isomorphism and the canonical section $s':T\times_SS'\to(T\times_SS')^h$ coincides under this identification with $$s\times_S\operatorname{Id}_{S'}:T\times_SS'\to T^h\times_SS'.$$
Let us construct a morphism in the opposite direction. Since $T$ is semi-local, $T^h$ is also semi-local (the proof is straightforward). Therefore by Lemma \[basechhens\] the pair $(T^h\times_SS',s(T)\times_SS')$ is henselian.
Let $(W',\pi',s')\in{\widetilde\operatorname{Neib}}(S',T\times_SS')$. From $\pi'$ by a base change we get an étale morphism $\tilde\pi:(T^h\times_SS')\times_{S'}W'\to T^h\times_SS'$. This morphism has an obvious section over $s(T)\times_SS'$. Since the pair $(T^h\times_SS',s(T)\times_SS')$ is henselian, this section extends uniquely to a section of $\tilde\pi$ over $T^h\times_SS'$, which, in turn, gives a morphism $T^h\times_SS'\to W'$. These morphisms give the desired morphism $T^h\times_SS'\to(T\times_SS')^h$.
\(ii) If $T=\coprod_i T_i$ is a disjoint union, then $T^h=\coprod_i T_i^h$. This follows from the observation that the functor from $\prod_i{\widetilde\operatorname{Neib}}(S,T_i)$ to ${\widetilde\operatorname{Neib}}(S,T)$, sending a collection of schemes to their disjoint union, is co-final.
Gluing principal ${\mathbf G}$-bundles
--------------------------------------
Recall that $U=\operatorname{Spec}R$, where $R$ is the semi-local ring of finitely many closed points on an irreducible smooth affine variety over an infinite field $k$. Also, ${\mathbf G}$ is a simple simply-connected group scheme over $U$, and $Y$ is a closed subscheme of ${\mathbb P}_U^1$ étale over $U$. We may and will assume that $Y\subset{\mathbb A}_U^1$ (otherwise, just change the coordinate on ${\mathbb P}_U^1$). Specifically for our context, we take $S={\mathbb A}_U^1$, $T=Y$.
\[lm:affine\] If $(W',\pi',s')\in\operatorname{Neib}({\mathbb A}_U^1,Y)$, then $s'(Y)$ is a principal divisor in $W'$ and therefore $W'-s'(Y)$ is affine.
Since $U$ is a regular semi-local ring, $Y$ is a principal divisor in ${\mathbb A}_U^1$. Thus $s'(Y)=(\pi')^{-1}(Y)$ is also a principal divisor in the affine scheme $W'$.
Let us make a general remark. Let ${\mathcal F}$ be a ${\mathbf G}$-bundle over a $U$-scheme $T$. By definition, a trivialization of ${\mathcal F}$ is a ${\mathbf G}$-equivariant isomorphism ${\mathbf G}\times_UT\to{\mathcal F}$. Equivalently, it is a section of the projection ${\mathcal F}\to T$. If ${\varphi}$ is such a trivialization and $f:T'\to T$ is a $U$-morphism, we get a trivialization $f^*{\varphi}$ of $f^*{\mathcal F}$. Sometimes we denote this trivialization by ${\varphi}|_{T'}$. We also sometimes call a trivialization of $f^*{\mathcal F}$ *a trivialization of ${\mathcal F}$ on $T'$*.
For each object $(W',\pi',s')$ in $\operatorname{Neib}({\mathbb A}_U^1,Y)$ there is an elementary distinguished square (see [@VoevodskyCongress Definition 2.1]) $$\label{eq:distinguished}
\begin{CD}
W'-s'(Y) @>>>W'\\
@VVV @VV in\circ\pi'V\\
{\mathbb P}_U^1 - Y @>>>{\mathbb P}^1_U,
\end{CD}$$ where $in:{\mathbb A}^1_U\hookrightarrow{\mathbb P}^1_U$ is the inclusion. It is used here that $Y$ is closed in ${\mathbb P}^1_U$.
The elementary distinguished square (\[eq:distinguished\]) can be used like a Zariski cover to construct principal ${\mathbf G}$-bundles over ${\mathbb P}_U^1$. A partial case of this procedure is this. Let ${\mathcal A}(W',\pi',s')$ be the category of pairs $({\mathcal E},{\varphi})$, where ${\mathcal E}$ is a ${\mathbf G}$-bundle on ${\mathbb P}_U^1$, ${\varphi}$ is a trivialization of ${\mathcal E}|_{W'}:=(in\circ\pi')^*{\mathcal E}$. A morphism between $({\mathcal E},{\varphi})$ and $({\mathcal E}',{\varphi}')$ is an isomorphism ${\mathcal E}\to{\mathcal E}'$ compatible with trivializations.
Similarly, let ${\mathcal B}(W',\pi',s')$ be the category of pairs $({\mathcal E},{\varphi})$, where ${\mathcal E}$ is a ${\mathbf G}$-bundle on ${\mathbb P}_U^1-Y$, ${\varphi}$ is a trivialization of ${\mathcal E}|_{W'-s'(Y)}$.
Consider the restriction functor $\Phi:{\mathcal A}(W',\pi',s')\to{\mathcal B}(W',\pi',s')$. The following proposition is a version of Nisnevich descent.
\[pr:gluing\] The functor $\Phi$ is an equivalence of categories.
This follows from Lemma \[lm:affine\] and [@C-TO Prop. 2.6(iv)].
The main cartesian square we will work with is $$\begin{CD}\label{eq:distinguishedLimit}
\dot Y^h @>>> Y^h\\
@VVV @VV{in\circ\pi}V\\
{\mathbb P}^1_U - Y @>>>{\mathbb P}^1_U.
\end{CD}$$
\[pr:affine\][[)]{} ]{}\[b\] $\dot Y^h$ is the projective limit of $W'-s'(Y)$ over $\operatorname{Neib}({\mathbb A}_U^1,Y)$.\
[[)]{} ]{}\[c\] $\dot Y^h$ is an affine scheme.
Part follows from the definition of projective limit and the equality $s(Y)=\pi^{-1}(Y)$. Part follows from Lemma \[lm:affine\] and part .
Let ${\mathcal A}$ be the category of pairs $({\mathcal E},\psi)$, where ${\mathcal E}$ is a ${\mathbf G}$-bundle on ${\mathbb P}_U^1$, $\psi$ is a trivialization of ${\mathcal E}|_{Y^h}:=(in\circ\pi)^*{\mathcal E}$. A morphism between $({\mathcal E},\psi)$ and $({\mathcal E}',\psi')$ is an isomorphism ${\mathcal E}\to{\mathcal E}'$ compatible with trivializations.
Similarly, let ${\mathcal B}$ be the category of pairs $({\mathcal E},\psi)$, where ${\mathcal E}$ is a ${\mathbf G}$-bundle on ${\mathbb P}_U^1-Y$, $\psi$ is a trivialization of ${\mathcal E}|_{\dot Y^h}$.
It is easy to see that the category ${\mathcal A}(W',\pi',s')$ is a groupoid whose objects have no non-trivial automorphisms. The same is true for ${\mathcal B}(W',\pi',s')$, ${\mathcal A}$, and ${\mathcal B}$. We are not going to use this explicitly.
Consider the restriction functor $\Psi:{\mathcal A}\to{\mathcal B}$.
\[pr:gluing2\] The functor $\Psi$ is an equivalence of categories.
Let us prove that $\Psi$ is essentially surjective; let $({\mathcal E},\psi)\in{\mathcal B}$. Using Lemma \[lm:affine\] and Proposition \[pr:affine\], we can find $(W',\pi',s')\in\operatorname{Neib}({\mathbb A}_U^1,Y)$ and a trivialization ${\varphi}$ of ${\mathcal E}$ on $W'-s'(Y)$ such that ${\varphi}|_{\dot Y^h}=\psi$. By proposition \[pr:gluing\] there is $(\tilde{\mathcal E},\tilde{\varphi})\in{\mathcal A}(W',\pi',s')$ such that $\Phi(\tilde{\mathcal E},\tilde{\varphi})\cong({\mathcal E},{\varphi})$. Then $\Psi(\tilde{\mathcal E},\tilde{\varphi}|_{Y^h})\cong({\mathcal E},\psi)$.
We leave it to the reader to prove that $\Psi$ is full and faithful.
\[equally\_well\] By Proposition \[pr:gluing2\] we can choose a functor quasi-inverse to $\Psi$. Fix such a functor $\Theta$. Let $\Lambda$ be the forgetful functor from ${\mathcal A}$ to the category of ${\mathbf G}$-bundles over ${\mathbb P}_U^1$. For $({\mathcal E},\psi)\in{\mathcal B}$ set $$\operatorname{Gl}({\mathcal E},\psi)=\Lambda(\Theta({\mathcal E},\psi)).$$ Note that $\operatorname{Gl}({\mathcal E},\psi)$ comes with a canonical trivialization over $Y^h$.
Conversely, if ${\mathcal E}$ is a principal ${\mathbf G}$-bundle over ${\mathbb P}_U^1$ such that its restriction to $Y^h$ is trivial, then ${\mathcal E}$ can be represented as $\operatorname{Gl}({\mathcal E}',\psi)$, where ${\mathcal E}'={\mathcal E}|_{{\mathbb P}_U^1-Y}$, $\psi$ is a trivialization of ${\mathcal E}'$ on $\dot Y^h$.
Let ${\mathbf u}$ be as in Section \[sect:reducing\], $Y_{\mathbf u}:=Y\times_U{\mathbf u}$. Let $(Y_{\mathbf u}^h,\pi_{\mathbf u},s_{\mathbf u})$ be the henselization of $({\mathbb A}^1_{\mathbf u},Y_{\mathbf u})$. Using property (i) of henselization, we get $Y_{\mathbf u}^h=Y^h\times_U{\mathbf u}$. Thus we have a natural closed embedding $Y_{\mathbf u}^h\to Y^h$. Set $\dot Y_{\mathbf u}^h=Y_{\mathbf u}^h-s_{\mathbf u}(Y_{\mathbf u})$. We get a closed embedding $\dot Y_{\mathbf u}^h\to\dot Y^h$. Thus the pull-back of the cartesian square (\[eq:distinguishedLimit\]) by means of the closed embedding ${\mathbf u}\hookrightarrow U$ has the form $$\begin{CD}
\dot Y_{\mathbf u}^h @>>>Y_{\mathbf u}^h\\
@VVV @VV in_{\mathbf u}\circ\pi_{\mathbf u}V\\
{\mathbb P}^1_{\mathbf u}- Y_{\mathbf u}@>>>{\mathbb P}^1_{\mathbf u},
\end{CD}$$ where $in_{\mathbf u}:{\mathbb A}_{\mathbf u}^1\to{\mathbb P}_{\mathbf u}^1$. Similarly to the above, we can define categories ${\mathcal A}_{\mathbf u}$ and ${\mathcal B}_{\mathbf u}$ and an equivalence of categories $\Psi_{\mathbf u}:{\mathcal A}_{\mathbf u}\to{\mathcal B}_{\mathbf u}$. Note that we have a diagram, where $R_{\mathcal A}$ and $R_{\mathcal B}$ are restriction functors. $$\label{CD:pullback}
\begin{CD}
{\mathcal A}@>R_{\mathcal A}>> {\mathcal A}_{\mathbf u}\\
@V \Psi VV @VV \Psi_{\mathbf u}V\\
{\mathcal B}@>R_{\mathcal B}>> {\mathcal B}_{\mathbf u}.
\end{CD}$$ This diagram commutes in the sense that the functors $\Psi_{\mathbf u}\circ R_{\mathcal A}$ and $R_{\mathcal B}\circ\Psi$ are isomorphic.
Let $\Theta_{\mathbf u}$ be a functor quasi-inverse to $\Psi_{\mathbf u}$ and $\Lambda_{\mathbf u}$ be the forgetful functor from ${\mathcal A}_{\mathbf u}$ to the category of ${\mathbf G}_{\mathbf u}$-bundles over ${\mathbb P}_{\mathbf u}^1$. Let ${\mathcal E}_{\mathbf u}$ be a principal ${\mathbf G}_{\mathbf u}$-bundle over ${\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}$ and $\psi_{\mathbf u}$ be a trivialization of ${\mathbf G}_{\mathbf u}$ on $\dot Y_{\mathbf u}^h$. Set $\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u})=\Lambda_{\mathbf u}(\Theta_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u}))$.
\[basechange\_limit\] Let $({\mathcal E},\psi)\in{\mathcal B}$, and let $\operatorname{Gl}({\mathcal E},\psi)$ be the ${\mathbf G}$-bundle obtained by Construction \[equally\_well\]. Then $$\operatorname{Gl}_{\mathbf u}({\mathcal E}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},\psi|_{\dot Y_{\mathbf u}^h}) \ \text{and} \ \operatorname{Gl}({\mathcal E},\psi)|_{{\mathbb P}^1_{\mathbf u}}$$ are isomorphic as ${\mathbf G}_{\mathbf u}$-bundles over ${\mathbb P}^1_{\mathbf u}$.
By definition of $\operatorname{Gl}$ we have $$\Theta({\mathcal E},\psi)=(\operatorname{Gl}({\mathcal E},\psi),\sigma),$$ where $\sigma$ is the canonical trivialization of $\operatorname{Gl}({\mathcal E},\psi)$ on $Y^h$. Similarly, $$\Theta_{\mathbf u}({\mathcal E}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},\psi|_{\dot Y_{\mathbf u}^h})=(\operatorname{Gl}_{\mathbf u}({\mathcal E}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},\psi|_{\dot Y_{\mathbf u}^h}),\sigma_{\mathbf u}),$$ where $\sigma_{\mathbf u}$ is the canonical trivialization of $\operatorname{Gl}_{\mathbf u}({\mathcal E}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},\psi|_{\dot Y_{\mathbf u}^h})$ on $Y_{\mathbf u}^h$. Thus (since $\Psi_{\mathbf u}$ is an equivalence of categories) it is enough to check that $$\Psi_{\mathbf u}\bigl(R_{\mathcal A}(\Theta({\mathcal E},\psi))\bigr)\cong\Psi_{\mathbf u}(\Theta_{\mathbf u}({\mathcal E}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},\psi|_{\dot Y_{\mathbf u}^h})).$$ In fact, both sides are isomorphic to $({\mathcal E}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},\psi|_{\dot Y_{\mathbf u}^h})$ because diagram is commutative.
\[coboundary\_limit\] For any $({\mathcal E}_{\mathbf u},\psi_{\mathbf u})\in{\mathcal B}_{\mathbf u}$ and any $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$ the ${\mathbf G}_{\mathbf u}$-bundles $$\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u})\ \text{and} \ \operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u}\circ\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h})$$ are isomorphic (here $\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$ is regarded as an automorphism of the ${\mathbf G}_{\mathbf u}$-bundle ${\mathbf G}_{\mathbf u}\times_{\mathbf u}\dot Y_{\mathbf u}^h$ given by the right translation by $\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$).
Let $\sigma_{\mathbf u}, \tau_{\mathbf u}$ be the canonical trivializations of $\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u}, \psi_{\mathbf u})$ and $\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u}\circ \beta_{\mathbf u}|_{\dot Y^h_{\mathbf u}})$ respectively. It is straightforward to check that $({\mathcal E}_{\mathbf u},\psi_{\mathbf u})$ is isomorphic in ${\mathcal B}_{\mathbf u}$ to both $\Psi_{\mathbf u}(\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u}),\sigma_{\mathbf u})$ and $\Psi_{\mathbf u}(\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u}\circ\beta_{\mathbf u}|_{\dot Y^h_{\mathbf u}}),\tau_{\mathbf u}\circ\beta_{\mathbf u}^{-1})$.
Since $\Psi_{\mathbf u}$ is an equivalence of categories, we conclude that $(\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u}, \psi_{\mathbf u}), \sigma_{\mathbf u})$ and $(\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u}\circ\beta_{\mathbf u}|_{\dot Y^h_{\mathbf u}}),\tau_{\mathbf u}\circ\beta_{\mathbf u}^{-1})$ are isomorphic in ${\mathcal A}_{\mathbf u}$. Applying the functor $\Lambda_{\mathbf u}$, we see that the ${\mathbf G}_{\mathbf u}$-bundles $\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u}, \psi_{\mathbf u})$ and $\operatorname{Gl}_{\mathbf u}({\mathcal E}_{\mathbf u},\psi_{\mathbf u}\circ \beta_{\mathbf u}|_{\dot Y^h_{\mathbf u}})$ are isomorphic.
Proof of Theorem \[MainThm2\]: presentation of ${\mathcal G}$ in the form $\operatorname{Gl}({\mathcal G}',{\varphi})$ {#sect:presentation}
----------------------------------------------------------------------------------------------------------------------
Let $U$, ${\mathbf G}$, $Z$, and ${\mathcal G}$ be as in Theorem \[MainThm2\]. We may and will assume that $Z\subset{\mathbb A}_U^1$.
\[presentation\] The ${\mathbf G}$-bundle ${\mathcal G}$ over ${\mathbb P}^1_U$ is of the form $\operatorname{Gl}({\mathcal G}',{\varphi})$ for the ${\mathbf G}$-bundle ${\mathcal G}':={\mathcal G}|_{{\mathbb P}^1_U-Y}$ and a trivialization ${\varphi}$ of ${\mathcal G}'$ over $\dot Y^h$.
In view of Construction \[equally\_well\], it is enough to prove that the restriction of the principal ${\mathbf G}$-bundle ${\mathcal G}$ to $Y^h$ is trivial. Let us choose a closed subscheme $Z'\subset{\mathbb A}^1_U$ such that $Z'$ contains $Z$, $Z'\cap Y={\varnothing}$, and ${\mathbb A}^1_U-Z'$ is affine. Then ${\mathbb A}^1_U-Z'$ is an affine neighborhood of $Y$. Thus the henselization of the pair $({\mathbb A}^1_U-Z',Y)$ coincides with the henselization of the pair $({\mathbb A}^1_U,Y)$. Since ${\mathcal G}$ is trivial over ${\mathbb A}^1_U-Z'$, its pull-back to $Y^h$ is trivial too. The proposition is proved.
*Our aim is to modify the trivialization ${\varphi}$ via an element $$\alpha\in{\mathbf G}(\dot Y^h)$$ so that the ${\mathbf G}$-bundle $\operatorname{Gl}({\mathcal G}',{\varphi}\circ\alpha)$ becomes trivial over* ${\mathbb P}^1_U$.
Proof of Theorem \[MainThm2\]: proof of property (i) from the outline {#sect:properties_i}
---------------------------------------------------------------------
Now we are able to prove property (i) from the outline of the proof. In fact, we will prove the following
\[podpravka\] Let $\operatorname{Gl}({\mathcal G}',{\varphi})$ be the presentation of the ${\mathbf G}$-bundle ${\mathcal G}$ over ${\mathbb P}^1_U$ given in Proposition \[presentation\]. Set ${\varphi}_{\mathbf u}:={\varphi}|_{\dot Y_{\mathbf u}^h}$. Then there is $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ such that the ${\mathbf G}_{\mathbf u}$-bundle $\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},{\varphi}_{\mathbf u}\circ\gamma_{\mathbf u})$ is trivial.
We show first that ${\mathcal G}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}$ is trivial. One has $${\mathbb P}^1_{\mathbf u}= \left(\coprod_{u\in{\mathbf u}'}{\mathbb P}^1_u\right)\coprod\left(\coprod_{u\in{\mathbf u}''}{\mathbb P}^1_u\right).$$ For $u\in{\mathbf u}$ set $Y_u:=Y\times_Uu$, ${\mathbf G}_u:={\mathbf G}\times_Uu$, and ${\mathcal G}_u:={\mathcal G}\times_Uu$.
For $u\in{\mathbf u}''$ the algebraic $k(u)$-group ${\mathbf G}_u$ is anisotropic. Since ${\mathcal G}_u$ is trivial over an open subset of ${\mathbb P}^1_u$, the exact sequence from [@GilleTorseurs Corollary 3.10(a)] shows that ${\mathcal G}_u$ is locally trivial in Zariski topology. Now the second part of [@GilleTorseurs Corollary 3.10(a)] shows that ${\mathcal G}_u$ is trivial. Thus ${\mathcal G}|_{{\mathbb P}^1_u-Y_u}$ is trivial.
Take $u\in{\mathbf u}'$. By our assumption on $Y$, there is a $k(u)$-rational point $p_u\in Y_u$. Set ${\mathbb A}_u^1={\mathbb P}_u^1-p_u$. Then we can write $Y_u=p_u\coprod T_u$ and ${\mathbb P}^1_u-Y_u\cong{\mathbb A}^1_u-T_u$. The ${\mathbf G}_u$-bundle ${\mathcal G}_u$ is trivial over ${\mathbb A}^1_u-Z$. Thus, again by [@GilleTorseurs Corollary 3.10(a)], it is trivial over ${\mathbb A}^1_u$. Whence it is trivial over ${\mathbb P}^1_u-Y_u$.
We see that ${\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}={\mathcal G}|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}$ is trivial. Choosing a trivialization, we may identify ${\varphi}_{\mathbf u}$ with an element of ${\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$. Set $\gamma_{\mathbf u}={\varphi}_{\mathbf u}^{-1}$. By the very choice of $\gamma_{\mathbf u}$ the ${\mathbf G}_{\mathbf u}$-bundle $\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},{\varphi}_{\mathbf u}\circ\gamma_{\mathbf u})$ is trivial.
Note that the exact sequence from [@GilleTorseurs Corollary 3.10(a)] is a particular case of Grothendieck-Serre conjecture.
Proof of Theorem \[MainThm2\]: reduction to property (ii) from the outline {#sect:properties_ii}
--------------------------------------------------------------------------
The aim of this section is to deduce Theorem \[MainThm2\] from the following
\[alpha\] Each element $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ can be written in the form $$\alpha|_{\dot Y_{\mathbf u}^h}\cdot\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h}$$ for certain elements $\alpha\in{\mathbf G}(\dot Y^h)$ and $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$.
Let $\operatorname{Gl}({\mathcal G}',{\varphi})$ be the presentation of the ${\mathbf G}$-bundle ${\mathcal G}$ from Proposition \[presentation\]. Let $\gamma_{\mathbf u}\in{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)$ be the element from Lemma \[podpravka\]. Let $\alpha\in{\mathbf G}(\dot Y^h)$ and $\beta_{\mathbf u}\in{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)$ be the elements from Proposition \[alpha\]. Set $${\mathcal G}^{new}=\operatorname{Gl}({\mathcal G}',{\varphi}\circ\alpha).$$ *Claim.* The ${\mathbf G}$-bundle ${\mathcal G}^{new}$ is trivial over ${\mathbb P}^1_U$.\
Indeed, by Lemmas \[basechange\_limit\] and \[coboundary\_limit\] one has a chain of isomorphisms of ${\mathbf G}_{\mathbf u}$-bundles $$\begin{gathered}
{\mathcal G}^{new}|_{{\mathbb P}^1_{\mathbf u}}\cong
\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},{\varphi}_{\mathbf u}\circ \alpha|_{\dot Y_{\mathbf u}^h})
\cong\\
\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}},{\varphi}_{\mathbf u}\circ
\alpha|_{\dot Y_{\mathbf u}^h}\circ\beta_{\mathbf u}|_{\dot Y_{\mathbf u}^h})=
\operatorname{Gl}_{\mathbf u}({\mathcal G}'|_{{\mathbb P}^1_{\mathbf u}-Y_{\mathbf u}}, {\varphi}_{\mathbf u}\circ \gamma_{\mathbf u}),\end{gathered}$$ which is trivial by the choice of $\gamma_{\mathbf u}$. The ${\mathbf G}$-bundles ${\mathcal G}|_{{\mathbb P}^1_U-Y}$ and ${\mathcal G}^{new}|_{{\mathbb P}^1_U-Y}$ coincide by the very construction of ${\mathcal G}^{new}$. By Proposition \[pr:trivclsdfbr\], applied to $T=Z\cup Y$, the ${\mathbf G}$-bundle ${\mathcal G}^{new}$ is trivial. Whence the claim.
The claim above implies that the ${\mathbf G}$-bundle ${\mathcal G}|_{{\mathbb P}^1_U-Y}$ is trivial. Theorem \[MainThm2\] is proved.
End of proof of Theorem \[MainThm2\]: proof of property (ii) from the outline
-----------------------------------------------------------------------------
*In the remaining part of Section \[sect:proof2\] we will prove Proposition \[alpha\]. This will complete the proof of Theorem \[MainThm2\]*.
By our assumption on $Y$, the group scheme ${\mathbf G}_Y={\mathbf G}\times_UY$ is isotropic. Thus we can and will choose a parabolic subgroup scheme ${\mathbf P}^+$ in ${\mathbf G}$ such that the restriction of ${\mathbf P}^+$ to each connected component of $Y$ is a proper subgroup scheme in the restriction of ${\mathbf G}$ to this component of $Y$.
Since $Y$ is an affine scheme, by [@SGA3-3 Exp. XXVI, Cor. 2.3, Th 4.3.2(a)] there is an opposite to ${\mathbf P}^+$ parabolic subgroup scheme ${\mathbf P}^-$ in ${\mathbf G}_Y$. Let ${\mathbf U}^+$ be the unipotent radical of ${\mathbf P}^+$, and let ${\mathbf U}^-$ be the unipotent radical of ${\mathbf P}^-$.
\[EYi\] We will write ${\mathbf E}$ for the functor, sending a $Y$-scheme $T$ to the subgroup ${\mathbf E}(T)$ of the group ${\mathbf G}_Y(T)={\mathbf G}(T)$ generated by the subgroups ${\mathbf U}^+(T)$ and ${\mathbf U}^-(T)$ of the group ${\mathbf G}_Y(T)={\mathbf G}(T)$.
\[lm:surjectivity\] The functor ${\mathbf E}$ has the property that for every closed subscheme $S$ in an affine $Y$-scheme $T$ the induced map ${\mathbf E}(T)\to{\mathbf E}(S)$ is surjective.
The restriction maps ${\mathbf U}^\pm(T)\to{\mathbf U}^\pm(S)$ are surjective, since ${\mathbf U}^\pm$ are isomorphic to vector bundles as $Y$-schemes (see [@SGA3-3 Exp. XXVI, Cor. 2.5]).
Recall that $(Y^h,\pi,s)$ is the henselization of the pair $({\mathbb A}_U^1,Y)$. Also, $in:{\mathbb A}_U^1\to{\mathbb P}_U^1$ is the embedding. Denote the projection ${\mathbb A}_U^1\to U$ by $pr$ and the projection ${\mathbb A}_Y^1\to Y$ by $pr_Y$.
\[lm:retraction\] There is a morphism $r:Y^h\to Y$ making the following diagram commutative $$\label{eq:retraction}
\begin{CD}
Y^h @>r>> Y\\
@V{in\circ\pi}VV @VV{pr|_{Y}}V\\
{\mathbb P}^1_U @>pr>> U
\end{CD}$$ and such that $r\circ s=\operatorname{Id}_Y$.
As usual, we may assume that $Y\subset{\mathbb A}_U^1$. Note that the morphism $$\pi':=\operatorname{Id}\times(pr|_Y):{\mathbb A}^1_Y\to{\mathbb A}^1_U$$ is étale. Let $s':Y\to{\mathbb A}^1_U\times_UY={\mathbb A}^1_Y$ be the morphism induced by the embedding $Y\to{\mathbb A}^1_U$ and $\operatorname{Id}_Y$. Then $({\mathbb A}_Y^1,\pi',s')\in{\widetilde\operatorname{Neib}}({\mathbb A}_U^1,Y)$. Thus there is a canonical morphism $can:Y^h\to{\mathbb A}^1_Y$ such that $(\operatorname{Id}\times (pr|_Y))\circ can =\pi$. Set $$r:= pr_Y\circ can: Y^h\to Y.$$ With this $r$ diagram commutes, and $r\circ s=\operatorname{Id}_Y$.
We view $Y^h$ as a $Y$-scheme via $r$. Thus various subschemes of $Y^h$ also become $Y$-schemes. In particular, $\dot Y^h$ and $\dot Y_{\mathbf u}^h$ are $Y$-schemes, and we can consider $${\mathbf E}(\dot Y^h)\subset{\mathbf G}(\dot Y^h) \ \text{ and } \ {\mathbf E}(\dot Y_{\mathbf u}^h)\subset{\mathbf G}(\dot Y_{\mathbf u}^h)={\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h).$$
\[nastia\] $${\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)={\mathbf E}(\dot Y_{\mathbf u}^h){\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h).$$
Firstly, one has $Y_{\mathbf u}=\coprod_{u\in{\mathbf u}}\coprod_{y\in Y_u}y$. (Note that $Y_u$ is a finite scheme.) Thus by property (ii) of henselization, we have $$Y_{\mathbf u}^h=\coprod_{u\in{\mathbf u}}\coprod_{y\in Y_u}y^h,\qquad
\dot Y_{\mathbf u}^h=\coprod_{u\in{\mathbf u}}\coprod_{y\in Y_u}\dot y^h,$$ where $(y^h,\pi_y,s_y)$ is the henselization of the pair $({\mathbb A}_{\mathbf u}^1,y)$, $\dot y^h:=y^h-s_y(y)$. We see that $y^h$ and $\dot y^h$ are subschemes of $Y^h$, so we can view them as $Y$-schemes, and ${\mathbf G}_{y^h}:={\mathbf G}_Y\times_Y{y^h}$ is isotropic. Also, ${\mathbf E}(\dot y^h)$ makes sense as a subgroup of ${\mathbf G}(\dot y^h)={\mathbf G}_u(\dot y^h)={\mathbf G}_{y^h}(\dot y^h)$.
One has $$\begin{split}
{\mathbf G}_{\mathbf u}(\dot Y_{\mathbf u}^h)&=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf G}_u(\dot y^h)=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf G}_{y^h}(\dot y^h),\\
{\mathbf E}(\dot Y_{\mathbf u}^h)&=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf E}(\dot y^h),\\
{\mathbf G}_{\mathbf u}(Y_{\mathbf u}^h)&=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf G}_u(y^h)=\prod_{u\in{\mathbf u}}\prod_{y\in Y_u}{\mathbf G}_{y^h}(y^h).
\end{split}$$ Thus it suffices for each $u\in{\mathbf u}$ and each $y\in Y_u$ to check the equality $${\mathbf G}_{y^h}(\dot y^h)={\mathbf E}(\dot y^h){\mathbf G}_{y^h}(y^h).$$ This equality holds by Fait 4.3 and Lemma 4.5 of [@Gille:BourbakiTalk]. In fact, $y^h=\operatorname{Spec}\mathcal O$, where $\mathcal O=k(u)[t]_{\mathfrak m_y}^h$ is a henselian discrete valuation ring, and $\mathfrak m_y\subset k(u)[t]$ is the maximal ideal defining the point $y\in{\mathbb A}^1_u$. (Without loss of generality we can assume that $y$ is not the infinite point of ${\mathbb P}^1_u$.) Further, $\dot y^h=\operatorname{Spec}L$, where $L$ is the fraction field of $\mathcal O$. The lemma is proved.
By Lemma \[lm:surjectivity\] and Proposition \[pr:affine\] the restriction map ${\mathbf E}(\dot Y^h)\to{\mathbf E}(\dot Y_{\mathbf u}^h)$ is surjective. Since ${\mathbf E}(\dot Y^h)\subset{\mathbf G}(\dot Y^h)$, the proposition follows. *This completes the proof of Theorem \[MainThm2\]*.
Formal loops and affine Grassmannians {#sect:loops}
=====================================
Here we sketch another proof of Theorem \[MainThm2\] using formal loops rather than henselization. Recall that $k\subset R$ is an infinite field. Let $k[[t]]$ be the ring of formal series, $k((t))=k[[t]][t^{-1}]$ be the field of formal Laurent series. We start with the following diagram, analogous to the square : $$\begin{CD}
\operatorname{Spec}k((t))\times Y @>>> \operatorname{Spec}k[[t]]\times Y \\
@VVV @VVV \\
{\mathbb P}^1_U-Y @>>> {\mathbb P}^1_U.
\end{CD}$$ (Here $\times$ is the product in the category of $k$-schemes.) Naively, the right vertical morphism is thought as taking $({\varepsilon},y)$ to $y+{\varepsilon}$. Formally, it is defined as the composition $$\operatorname{Spec}k[[t]]\times Y\to{\mathbb A}^1_Y\to{\mathbb A}^1_U\hookrightarrow{\mathbb P}^1_U.$$ Here the first morphism is induced by the inclusion $k[{\mathbb A}^1_k]=k[t]\to k[[t]]$. The second morphism is the restriction of the group scheme multiplication morphism ${\mathbb A}^1_U\times_U{\mathbb A}^1_U\to{\mathbb A}^1_U$ to ${\mathbb A}^1_Y={\mathbb A}^1_U\times_UY$.
The following is similar to Construction \[equally\_well\].
\[formalglue\] Given a principal ${\mathbf G}$-bundle ${\mathcal E}$ over ${\mathbb P}_U^1-Y$, and a trivialization ${\varphi}$ of ${\mathcal E}$ on $\bigl(\operatorname{Spec}k((t))\bigr)\times Y$, we get a principal ${\mathbf G}$-bundle $\operatorname{Gl}'({\mathcal E},{\varphi})$ over ${\mathbb P}_U^1$. This follows from results of [@BeauvilleLaszlo] (see also [@FedorovExotic]).
Clearly, the ${\mathbf G}$-bundle $\operatorname{Gl}'({\mathcal E},{\varphi})$ has the following properties:\
(a) its restriction to ${\mathbb P}_U^1-Y$ is isomorphic to the ${\mathbf G}$-bundle ${\mathcal E}$,\
(b) its pull-back to $\operatorname{Spec}k[[t]]\times Y$ is trivial and even trivialized.
Conversely, if ${\mathcal E}$ is a principal ${\mathbf G}$-bundle over ${\mathbb P}_U^1$ such that its restriction to $\operatorname{Spec}k[[t]]\times Y$ is trivial, then ${\mathcal E}$ can be represented as $\operatorname{Gl}'({\mathcal E}',{\varphi})$, where ${\mathcal E}'={\mathcal E}|_{{\mathbb P}_U^1-Y}$, ${\varphi}$ is a trivialization of ${\mathcal E}'$ on $\operatorname{Spec}k((t))\times Y$.
One can prove Theorem \[MainThm2\], using the above construction instead of Construction \[equally\_well\]. The proof is almost identical, the only significant difference is in the proof of (the analogue of) Lemma \[nastia\], where, instead of Fait 4.3 and Lemma 4.5 of [@Gille:BourbakiTalk], one should use
Let $k$ be an infinite field, $G$ be a simple simply-connected $k$-group, $P^\pm\subset G$ be opposite parabolic subgroups of $G$ (defined over $k$). Let $U^\pm$ be unipotent radicals of $P^\pm$. For a $k$-scheme $T$ denote by $E(T)$ the subgroup of $G(T)$ generated by $U^\pm(T)$. We have $$G\bigl(k((t))\bigr)=E\bigl(k((t))\bigr)G\bigl(k[[t]]\bigr).$$
Let $\alpha\in G\bigl(k((t))\bigr)$. By [@GilleTorseurs Theorem 3.4] we can write $\alpha=\beta\gamma$, where $\beta\in G(k(t))$, $\gamma\in G\bigl(k[[t]]\bigr)$. By Propositions 6.5 and 6.7 of [@PSV] we can write $\beta$ as $\beta'\beta''$, where $\beta'\in E(k(t))\subset E\bigl(k((t))\bigr)$, $\beta''\in G\subset G\bigl(k[[t]]\bigr)$.
Relation to affine Grassmannians {#sect:Grassm}
--------------------------------
Here we give a hint on how Theorem \[MainThm2\] was initially proved in the case $k={\mathbb C}$. Thus we assume that $k$ is the field of complex numbers. Assume also for simplicity that $R$ is local and that the Dynkin diagram of ${\mathbf G}$ has no non-trivial automorphisms. Let ${\mathcal G}$, $Z$, and $Y$ be as in Theorem \[MainThm2\], and choose a trivialization of ${\mathcal G}$ on ${\mathbb P}^1_U-Z$. By a *modification of ${\mathcal G}$ at $Y$* we mean a ${\mathbf G}$-bundle ${\mathcal F}$ over ${\mathbb P}_U^1$ together with an isomorphism $${\mathcal F}|_{{\mathbb P}^1_U-Y}\xrightarrow{\cong}{\mathcal G}|_{{\mathbb P}^1_U-Y}.$$ To prove the theorem it suffices to choose a modification of ${\mathcal G}$ at $Y$ such that ${\mathcal F}$ is trivial.
Let us give a description of modifications in terms of affine Grassmannians. The fiber $G:={\mathbf G}_u$ at the closed point $u$ of $U$ is a simple complex group. Let $X$ be a smooth, not necessarily projective, complex curve, let $x\in X$ be a point. Choose a formal coordinate at $x$. Then the $G$-bundles over $X$ with a trivialization off a point $x$ are classified by the affine Grassmannian $Gr_G:=G\bigl(k((t))\bigr)/G\bigl(k[[t]]\bigr)$ (see, e.g. [@Sorger Proposition 5.3.2]).
Let ${\mathcal E}$ be a $G$-bundle over ${\mathbb P}^1_k$ trivialized away from a point $y$. Then its modifications at a point $x\ne y$ are classified by $Gr_G$. Indeed, viewing a point of $Gr_G$ as a $G$-bundle ${\mathcal E}'$ over ${\mathbb P}^1_k-y$ trivialized on ${\mathbb P}^1_k-\{x,y\}$, we construct ${\mathcal F}$ by gluing ${\mathcal E}$ and ${\mathcal E}'$ over ${\mathbb P}^1_k-\{x,y\}$.
One can also define a ${\mathbf G}$-twisted Grassmannian $Gr_{{\mathbf G}}\to U$ as follows. First of all, since ${\mathbf G}$ is coming from the ground field $k$ locally in étale topology, it corresponds to a class in $H^1_{\text{\'et}}(U,\operatorname{Aut}G)$, that is, to a right $\operatorname{Aut}G$-torsor ${\mathcal T}$. On the other hand, $\operatorname{Aut}G$ acts on $Gr_G$, and we set $$Gr_{\mathbf G}:={\mathcal T}\times_{\operatorname{Aut}G}Gr_G.$$ It is easy to see that the modifications of ${\mathcal G}$ at $Y$ are classified by $U$-morphisms $Y\to Gr_{\mathbf G}$.
Further, recall that the orbits of $G\bigl(k[[t]]\bigr)$ on $Gr_G$ are numerated by dominant coweights $\lambda$ of $G$ (see for example [@BravermanFinkelberg Section 1.2 and Section 2.1]); denote such an orbit by $Gr_G^\lambda$. This induces a similar decomposition of the twisted Grassmannian; denote the corresponding subspace by $Gr_{\mathbf G}^\lambda\subset Gr_{\mathbf G}$. Let $\succ$ be the usual order on the coweight lattice. Recall that a coweight $\lambda$ is called regular, if it does not belong to any coroot hyperplane.
Recall also that there is a natural bijection between isomorphism classes of principal $G$-bundles over ${\mathbb P}^1_k$ and dominant coweights. In this situation one can prove the following statement, similar to Theorem \[MainThm2\].
*Let $U$, ${\mathbf G}$, and ${\mathcal G}$ be as in Theorem \[MainThm2\]. Let ${\mathcal G}|_{{\mathbb P}^1_u}$ correspond to $\lambda_0$ under the bijection described above. Then there is a closed subscheme $Y$ of ${\mathbb P}_U^1$ such that the following conditions are satisfied\
(1) $Y$ is étale over $U$;\
(2) $Y\cap Z={\varnothing}$;\
(3) for all regular coweights $\lambda\succ\lambda_0$ there is a modification $Y\to Gr_{\mathbf G}^\lambda$, making the ${\mathbf G}$-bundle ${\mathcal G}$ trivial.*
One can choose a maximal torus ${\mathbf T}\subset {\mathbf G}$. Let ${\mathcal B}$ be the $U$-scheme of Borel subgroup schemes in ${\mathbf G}$. Let $Y\subset{\mathcal B}$ be the closed subscheme of Borel subgroup schemes containing ${\mathbf T}$. Then $Y$ is a $W$-cover of $U$, where $W$ is the Weyl group of $G$.
For a regular coweight $\lambda$ there is a morphism from $Gr_G^\lambda$ to the variety of Borel subgroups in $G$. The fibers of this morphism are isomorphic to vector spaces. Similarly, we get a morphism $Gr_{\mathbf G}^\lambda\to{\mathcal B}$. Since its fibers are vector spaces, the embedding $Y\to{\mathcal B}$ can be lifted to a $U$-morphism $Y\to Gr_{\mathbf G}^\lambda$. With a little work one can choose this lift so that the corresponding modification renders ${\mathcal G}$ trivial.
An application {#sect:application}
==============
The following result is a straightforward consequence of Theorem \[MainThm1\] and an exact sequence for étale cohomology. Recall that by our definition a reductive group scheme has geometrically connected fibres.
\[Norms\] Let $R$ be as in Theorem \[MainThm1\] and ${\mathbf G}$ be a reductive $R$-group scheme. Let $\mu:{\mathbf G}\to{\mathbf T}$ be a group scheme morphism to an $R$-torus ${\mathbf T}$ such that $\mu$ is locally in the étale topology on $\operatorname{Spec}R$ surjective. Assume further that the $R$-group scheme ${\mathbf H}:=\operatorname{Ker}(\mu)$ is reductive. Let $K$ be the fraction field of $R$. Then the group homomorphism $${\mathbf T}(R)/\mu({\mathbf G}(R))\to{\mathbf T}(K)/\mu({\mathbf G}(K))$$ is injective.
This theorem extends all the known results of this form proved in [@C-TO], [@PS], [@Z], [@OPZ]. Theorem \[Norms\] has the following
Under the hypothesis of Theorem \[Norms\] let additionally the $K$-algebraic group ${\mathbf G}_K$ be $K$-rational as a $K$-variety and let the ring $R$ be of characteristic $0$. Then the norm principle holds for all finite flat $R$-domains $S\supset R$. That is, if $S\supset R$ is such a domain, and $a\in{\mathbf T}(S)$ belongs to $\mu({\mathbf G}(S))$, then the element $N_{S/R}(a)\in{\mathbf T}(R)$ belongs to $\mu({\mathbf G}(R))$.
Let $L$ be the fraction field of $S$. Let $\alpha\in{\mathbf G}(S)$ be such that $\mu(\alpha)=a\in{\mathbf T}(S)$. Then $\mu(\alpha_L)=a_L\in{\mathbf T}(L)$, where $\alpha_L$ is the image of $\alpha$ in ${\mathbf G}(L)$, $a_L$ is the image of $a$ in ${\mathbf T}(L)$. The hypothesis on the algebraic $K$-group ${\mathbf G}_K$ implies that there exists an element $\beta\in{\mathbf G}(K)$ such that $\mu(\beta)=N_{L/K}(a_L)\in{\mathbf T}(K)$ (see [@Merk]). Note that $N_{L/K}(a_L)=(N_{S/R}(a))_K\in{\mathbf T}(K)$. By Theorem \[Norms\] there exists an element $\gamma\in{\mathbf G}(R)$ such that $\mu(\gamma)=N_{S/R}(a)\in{\mathbf T}(R)$. Whence the Corollary.
For a commutative ring $A$ we denote by $\operatorname{Rad}(A)$ its Jacobson ideal. The following definition one can find in [@Gabber Section 0].
If $I$ is an ideal in a commutative ring $A$, then the pair $(A,I)$ is called *henselian*, if $I\subset\operatorname{Rad}(A)$ and for every two relatively prime monic polynomials $\bar g, \bar h\in\bar A[t]$, where $\bar A=A/I$, and monic lifting $f\in A[t]$ of $\bar g\bar h$, there exist monic liftings $g,h\in A[t]$ such that $f=gh$. (Two polynomials are called relatively prime, if they generate the unit ideal.)
\[basechhens\] Let $(A,I)$ be a henselian pair with a semi-local ring $A$ and $J\subset A$ be an ideal. Then the pair $(A/J,(I+J)/J)$ is henselian.
Clearly $(J+I)/J\subset\operatorname{Rad}(A/J)$. Now let $\bar g,\bar h\in (A/(J+I))[t]$ be two relatively prime monic polynomials and let $f\in(A/J)[t]$ be a monic polynomial such that $f\bmod(J+I)/J=\bar g\bar h\in (A/(J+I))[t]$.
We claim that there exist relatively prime monic liftings of $\bar g$ and $\bar h$ to $(A/I)[t]$. Indeed, let ${\mathfrak m}_1$, …, ${\mathfrak m}_n$ be all the maximal ideals of $A/I$ not containing $(J+I)/I$ (recall that $A$ is semi-local). By the Chinese remainder theorem we can find monic $\bar G,\bar H\in(A/I)[t]$ such that $$\begin{split}
\bar G\bmod (J+I)/I=\bar g, &\quad \bar G\bmod{\mathfrak m}_i=t^{\deg\bar g}\mbox{ for } i=1,\ldots,n,\\
\bar H\bmod (J+I)/I=\bar h, &\quad \bar H\bmod{\mathfrak m}_i=t^{\deg\bar h}-1\mbox{ for } i=1,\ldots,n.
\end{split}$$ Then $\bar G$ and $\bar H$ are relatively prime. The ring homomorphism $$A\to(A/I)\times_{A/(J+I)}(A/J)$$ is surjective. Thus there exists a monic polynomial $F\in A[t]$ such that $F\bmod I=\bar G\bar H$ and $F\bmod J=f$.
The pair $(A,I)$ is henselian. Thus there exist monic liftings $G,H\in A[t]$ of $\bar G,\bar H$ such that $F=GH$. Let $g=G\bmod J\in(A/J)[t]$ and $h=H\bmod J\in(A/J)[t]$. Clearly, $g$ and $h$ are monic polynomials in $(A/J)[t]$, $f=gh\in(A/J)[t]$. And finally, $g\bmod(J+I)=\bar g$, $h\bmod(J+I)=\bar h$ in $(A/(J+I))[t]$. Whence the Lemma.
The following definition one can find in [@Gabber Section 0].
\[def:henzelisation\] The henselization of any pair $(A,I)$ is the pair $(A_I^h,I^h)$ (over $(A,I)$) defined as follows $$(A_I^h,I^h):=\text{the filtered inductive limit over the category $\mathcal N$ of }(A',\operatorname{Ker}(\sigma)),$$ where $\mathcal N$ is the filtered category of pairs $(A',\sigma)$ such that $A'$ is an étale $A$-algebra and $\sigma\in\text{Hom}_{A-alg}(A',A/I)$.
|
{
"pile_set_name": "ArXiv"
}
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abstract: 'Canonical formulation of quantum field theory on the Light Front (LF) is reviewed. The problem of constructing the LF Hamiltonian which gives the theory equivalent to original Lorentz and gauge invariant one is considered. We describe possible ways of solving this problem: (a) the limiting transition from the equal-time Hamiltonian in a fast moving Lorentz frame to LF Hamiltonian, (b) the direct comparison of LF perturbation theory in coupling constant and usual Lorentz-covariant Feynman perturbation theory. The results of the application of method (b) to QED-1+1 and QCD-3+1 are given. Gauge invariant regularization of LF Hamiltonian via introducing a lattice in transverse coordinates and imposing periodic boundary conditions in LF coordinate $x^-$ for gauge fields on the interval $|x^-|<L$ is also considered.'
author:
- |
V. A. Franke, Yu. V. Novozhilov, S. A. Paston, E. V. Prokhvatilov\
[*St.-Petersburg State University, Russia*]{}
title: Quantization of Field Theory on the Light Front
---
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Introduction
============
A possible approach to solving the strong interaction field theory is the canonical quantization on the Light-Front (LF) with the application of corresponding Schroedinger equation. To realize such program one introduces LF coordinates [@dirak] $x^{\pm}=(x^0\pm x^3)/\sqrt{2}, x^1, x^2,$ where $x^0, x^1, x^2, x^3$ are Lorentz coordinates. The $x^+$ plays the role of time, and canonical quantization is carried out on a hypersurface $x^+=const$. The advantage of this scheme is connected with the positivity of the momentum $P_-$ (translation operator along $x^-$ axis), which becomes quadratic in fields on the LF. As a consequence the lowest eigenstate of the operator $P_-$ is both physical vacuum and the “mathematical” vacuum of perturbation theory [@leut]. Using Fock space [@fock] over this vacuum one can solve stationary Schroedinger equation with Hamiltonian $P_+$ (translation operator along $x^+$ axis) to find the spectrum of bound states. The problem of describing the physical vacuum, very complicated in usual formulation with Lorentz coordinates, does not appear here. Such approach is called LF Hamiltonian approach. It attracts attention for a long time as a possible mean for solving Quantum Field Theory problems.
While giving essential advantages, the application of LF coordinates in Quantum Field Theory leads to some difficulties. The hyperplane $x^+=\mbox{const}$ is a characteristic surface for relativistic differential field equations. It is not evident without additional investigation that quantization on such hypersurface generates a theory equivalent to one quantized in the usual way in Lorentz coordinates [@chang1; @chang2; @yan; @pred1; @pred2; @pirner; @burk1; @tmf97; @lenc]. This is in particular essential because of the special divergences at $p_-=0$ appearing in LF quantization scheme. Beside of usual ultraviolet regularization one has to apply special regularization of such divergences. We will consider the following simplest prescription of such regularization:
\(a) cutoff of momenta $p_-$ [$$\displaylines{\refstepcounter{equation} \label{1.1}\hskip 1em minus 1em |p_- |\ge{\varepsilon},\quad{\varepsilon}>0;{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
\(b) cutoff of the $x^-$ [$$\displaylines{\refstepcounter{equation} \label{1.2}\hskip 1em minus 1em -L\le x^-\le L.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} with periodic boundary conditions in $x^-$ for all fields.
The regularization (b) discretizes the spectrum of the operator $P_-$ ($p_-=\pi n/L$, where $n$ is an integer). This formulation is called sometime “Discretized LF Quantization” [@brodsk]. Fourier components of fields, corresponding to $p_-=0$ (and usually called “zero modes”) turn out to be dependent variables and must be expressed in terms of nonzero modes via solving constraint equations (constraints). These constraints are usually very complicated, and solving of them is a difficult problem.
The prescriptions of regularization of divergences at $p_-=0$ described above are convenient for Hamiltonian approach, but both of them break Lorentz invariance and the prescription (a) breaks also the gauge invariance. Therefore the equivalence of LF and original Lorentz (and gauge) invariant formulation can be broken even in the limit of removed cutoff. To avoid this nonequivalence some modification of usual renormalization procedure may be necessary, see for example [@tmf97; @tmf99] and [@kniga]. The problem of constructing the LF Hamiltonian which gives a theory equivalent to original Lorentz and gauge invariant one turned out to be rather difficult. We will describe possible approaches to this problem.
In sect. 2 we give basic relations of quantum field theory in LF coordinates. In sect. 3 we consider the limiting transition from fast moving Lorentz frame to the LF. This transition relates Hamiltonian formulations in Lorentz and LF coordinates and firstly was presented in [@pred1; @pred2]. In sect. 4 we investigate the relation between LF perturbation theory in coupling constant and usual Lorentz-covariant Feynman perturbation theory. With the help of this investigation we show how to construct LF Hamiltonian giving a theory perturbatively equivalent to original one for Yukawa model, for QCD in four-dimensional space-time, and for QED in two-dimensional space-time (originally this was considered in [@tmf97; @tmf99; @shw2; @tmf02; @rasch]. In sect. 5 we carry out gauge invariant ultraviolet regularization of LF Hamiltonian introducing a lattice in transverse coordinates $x^1, x^2$ and using, instead of transverse components of gauge fields, complex matrices in color space as independent gauge variables on the links of the lattice [@heplat; @tmf04]. We find that LF canonical formalism for gauge theories with this regularization avoid usual most complicated constraints connecting zero and nonzero modes. However, the quantization leads to Lorentz-noninvariant results.
Formal canonical quantization of Field Theory [\
]{}on the Light Front and the problem of bound states
=====================================================
In order to find the bound state spectrum in some field theory quantized on the LF the following system of equations is usually solved: [$$\displaylines{\refstepcounter{equation} \label{2.1a}\hskip 1em minus 1em
P_+ |\Psi\rangle=P'_+ |\Psi\rangle,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} -8mm [$$\displaylines{\refstepcounter{equation} \label{2.1b}\hskip 1em minus 1em
P_- |\Psi\rangle=P'_- |\Psi\rangle,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} -8mm [$$\displaylines{\refstepcounter{equation} \label{2.1v}\hskip 1em minus 1em P_{\bot}|\Psi\rangle =0,{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $P_\bot =\{P_1,P_2\}$. The mass of bound state is equal to [$$\displaylines{\refstepcounter{equation} \label{2.2}\hskip 1em minus 1em m = \sqrt{2P'_+P'_-}.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
It was taken into account that nonzero components of metric tensor in LF coordinates are [$$\displaylines{\refstepcounter{equation} \label{2.3}\hskip 1em minus 1em
g_{-+}=g_{+-}=1,\quad g_{11}=g_{22}=-1.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The operators $P_-$, $P_\bot $ are quadratic in fields, and the solution of equations (\[2.1b\]), (\[2.1v\]) is trivial. The problem is in solving the Schroedinger equation (\[2.1a\]). Physical vacuum $|\Omega\rangle $ is lowest eigenstate of the operator $P_- $, and must satisfy the equations: [$$\displaylines{\refstepcounter{equation} \label{2.4a}\hskip 1em minus 1em
P_+ |\Omega\rangle =0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} -8mm [$$\displaylines{\refstepcounter{equation} \label{2.4b}\hskip 1em minus 1em
P_- |\Omega\rangle =0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} -8mm [$$\displaylines{\refstepcounter{equation} \label{2.4v}\hskip 1em minus 1em
P_{\bot}|\Omega\rangle =0.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} To fulfil equations (\[2.4a\]), (\[2.4b\]) one should subtract , if it is necessary, corresponding renormalizing constants from the operators $P_+$, $P_-$. The $|\Omega\rangle $ plays simultaneously the role of mathematical vacuum of LF Fock space. A solution $|\Psi\rangle$ of Schrodinger equation (\[2.1a\]) belongs to this space.
Expressions for the operators $P_+$, $P_-$ can be obtained by canonical quantization on the LF. Let us describe the procedure of such quantization via some examples, without analyzing so far the question about the equivalence of appearing theory and original Lorentz-covariant one. It is assumed that in addition to explicit regularization of divergences at $p_-=0$ also ultraviolet regularization is implied.
Scalar selfinteracting field in (1+1)-dimensional space-time.
-------------------------------------------------------------
Peculiarities of LF quantization are well seen even in this simple example. We have only LF coordinates $x^+$, $x^-$. The Lagrangian is equal to [$$\displaylines{\refstepcounter{equation} \label{2.5}\hskip 1em minus 1em
L=\int dx^-\,{\left(}{\partial}_\mu{\varphi}{\partial}^\mu{\varphi}-
\frac{1}{2}m^2{\varphi}^2-{\lambda}{\varphi}^4{\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} or [$$\displaylines{\refstepcounter{equation} \label{2.6}\hskip 1em minus 1em
L=\int dx^-\,{\left(}\frac{1}{2}{\partial}_+{\varphi}{\partial}_-{\varphi}-
\frac{1}{2}m^2{\varphi}^2-{\lambda}{\varphi}^4{\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The “time” derivative ${\partial}_+{\varphi}$ enters into this Lagrangian only linearly. For the transition to canonical theory it is sufficient to rewrite the expression $\int dx^-\,\frac{1}{2}{\partial}_+{\varphi}{\partial}_-{\varphi}$ in standard canonical form. To achieve this let us take the Fourier decomposition [$$\displaylines{\refstepcounter{equation} \label{2.7}\hskip 1em minus 1em
{\varphi}(x^-)=(2\pi)^{-\frac{1}{2}}{\int\limits}_0^\infty dk\,|2k|^{-\frac{1}{2}}
{\left(}a(k)\exp(-ikx^-)+a^+(k)\exp(ikx^-){\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $k\equiv k_-,$ ${\varphi}(x^-)\equiv{\varphi}(x^+,x^-),$ $a(k)\equiv a(x^+,k)$. The Lagrangian (\[2.6\]) takes the form [$$\displaylines{\refstepcounter{equation} \label{2.8}\hskip 1em minus 1em
L={\int\limits}_0^\infty dk\,{\left(}\frac{a^+(k)\dot a(k)-a(k)\dot a^+(k)}{2i}
{\right)}-H,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $\dot a\equiv{\partial}a/{\partial}x^+$ and [$$\displaylines{\refstepcounter{equation} \label{2.9}\hskip 1em minus 1em
H=\int dx^-\,{\left(}\frac{1}{2}m^2{\varphi}^2+{\lambda}{\varphi}^4{\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Here we have used the equality [$$\displaylines{\refstepcounter{equation} \label{2.10}\hskip 1em minus 1em
{\int\limits}_0^\infty dk\,{\int\limits}_0^\infty dk'\,{\delta}(k+k')k'{\left(}a(k)\dot a(k')-a^+(k)\dot a^+(k'){\right)}=0.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} It is implied that the function ${\varphi}(x)$ in (\[2.9\]) is expressed in terms of $a^+(k)$ and $a(k)$ with the help of formulae (\[2.6\]). “Time” derivatives $\dot a(k),$ $\dot a^+(k)$ enter into Lagrangian $L$ in a form standard for canonical theory. Therefore one can interpret after quantization the $a^+(k)$ and $a(k)$ as creation and annihilation operators satisfying the following commutation relations at fixed $x^+$ and $k>0,$ $k'>0$: [$$\displaylines{\refstepcounter{equation} \label{2.11}\hskip 1em minus 1em
[a(k),a^+(k')]={\delta}(k-k'),\quad [a(k),a(k')]=0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} It is also seen that the $H$ is LF Hamiltonian, i.e. $H=P_+$.
We have also the formulae [$$\displaylines{\refstepcounter{equation} \label{2.12}\hskip 1em minus 1em P_\mu =\int dx^-\,T_{-\mu},{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the energy-momentum tensor $T_{\nu\mu}$ is equal to [$$\displaylines{\refstepcounter{equation} \label{2.13}\hskip 1em minus 1em T_{\nu\mu}={\partial}_\nu{\varphi}{\partial}_\mu{\varphi}-g_{\mu\nu}{\cal L}.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Via this relation one can reproduce the expression (\[2.9\]) for $P_+\equiv H$ , and obtain the equality [$$\displaylines{\refstepcounter{equation} \label{2.14}\hskip 1em minus 1em P_-=\int dx^-\,({\partial}_-{\varphi})^2=\frac{1}{2}{\int\limits}_0^\infty dk\, k{\left(}a^+(k)a(k)+a(k)a^+(k){\right)}.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
The lowest eigenstate of the operator $P_-$ is the physical vacuum $|\Omega\rangle$ for which [$$\displaylines{\refstepcounter{equation} \label{2.15}\hskip 1em minus 1em a(k)|\Omega\rangle =0{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} at any $k$. It is seen that vacuum expectation values $\langle\Omega|P_-|
\Omega\rangle$, $\langle\Omega|P_+|\Omega\rangle$ are infinite. The renormalization can be got by taking normal ordered forms $:\!P_+\!:$, $:\!P_-\!:$ with respect to the operators $a^+$, $a$ (the symbol $::$ means as usual that operators $a^+$ stand everywhere before of operators $a$). Normal ordering of the $\lambda{\varphi}^4$ term in the Hamiltonian $P_+$ leads also to renormalization of the mass. In the following, writing $P_+$, $P_-$, we mean the expressions $:\!P_+\!:$, $:\!P_-\!:$, satisfying conditions (\[2.4a\]), (\[2.4b\]). Normal ordering of the operators $P_+$, $P_-$ allows to avoid all ultraviolet divergences in this simple model.
Theory of interacting scalar and fermion fields[\
]{}in (3+1)-dimensional space-time (Yukawa model)
-------------------------------------------------
The Lagrangian of the model is [$$\displaylines{\refstepcounter{equation} \label{2.16}\hskip 1em minus 1em L=\int d^2x^\bot dx^-{\left(}\overline{\psi}{\left(}i\gamma^\mu{\partial}_\mu-M
{\right)}\psi +\frac{1}{2}{\partial}_\mu{\varphi}{\partial}^\mu{\varphi}-\frac{1}{2}m^2{\varphi}^2-g
\overline{\psi}\psi {\varphi}-{\lambda}'{\varphi}^3-{\lambda}{\varphi}^4{\right)},{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $M$ is the fermion mass, $m$ is the boson mass, $\overline{\psi}=\psi^+
\gamma^0$, ${\varphi}={\varphi}^+$; $g$, ${\lambda}$, ${\lambda}'$ are coupling constants. Here and so on $\mu,\nu,\ldots=+,-,1,2$; $i,k,\ldots = 1,2$; $x^\bot\equiv{\left(}x^1,x^2{\right)}.$ For Dirac’s $\gamma$-matrices we use: [$$\displaylines{\refstepcounter{equation} \label{2.17}\hskip 1em minus 1em \gamma^0 = {\left(}\begin{array}{cc}0 & -iI\\iI & 0\end{array}{\right)},\qquad
\gamma^3 ={\left(}\begin{array}{cc}0 & iI\\iI & 0\end{array}{\right)},\qquad
\gamma^\bot ={\left(}\begin{array}{cc}-i\sigma^\bot & 0\\0 & i\sigma^\bot
\end{array}{\right)},{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $I$ is a unit $2\times 2$ matrix, $\sigma^\bot\equiv\{\sigma^1,\sigma^2
\}$, $\sigma^i$ are Pauli matrices.
We introduce $2$-component spinors $\chi$, $\xi$, writing [$$\displaylines{\refstepcounter{equation} \label{nnn1}\hskip 1em minus 1em \psi={\left(}\begin{array}{c}\chi\\\xi\end{array}{\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The Lagrangian $L$ can be written in the form [$$\displaylines{\refstepcounter{equation} \label{2.18}\hskip 1em minus 1em
L=\int d^2x^\bot dx^-\,\Bigl( i\sqrt{2}\chi^+{\partial}_+\chi+
i\sqrt{2}\xi^+{\partial}_-\xi+{\left(}i\xi^+{\left(}\sigma^i{\partial}_i-M{\right)}\chi+\mbox{h.c.}
{\right)}+{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}+{\partial}_+{\varphi}{\partial}_-{\varphi}-\frac{1}{2}{\partial}_i{\varphi}{\partial}_i{\varphi}-\frac{1}{2}m^2{\varphi}^2-ig{\varphi}{\left(}\xi^+\chi-\chi^+\xi{\right)}-{\lambda}'{\varphi}^3-{\lambda}{\varphi}^4
\Bigr),
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where h.c. means Hermitian conjugation. The variation of this Lagrangian with respect to $\chi^+$ leads to the equation [$$\displaylines{\refstepcounter{equation} \label{2.19}\hskip 1em minus 1em \sqrt{2}{\partial}_-\xi=-{\left(}{\sigma}^i{\partial}_i-M{\right)}\chi+g{\varphi}\chi.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} This equation does not contain derivatives in $x^+$ and therefore is a constraint. One should solve it with respect to $\xi$ and substitute the result into the Lagrangian. In doing this we must invert the operator ${\partial}_-$ which becomes an operator of multiplication $ik_-$ after Fourier transformation. Inverse operator $(ik_-)^{-1}$ has singularity at $k_-=0$. To avoid this singularity we introduce the regularization (\[1.1\]). For any function $f(x^-)\equiv f(x^+,x^-,x^\bot)$ we define Fourier transform [$$\displaylines{\refstepcounter{equation} \label{2.20}\hskip 1em minus 1em f(x^-)=\frac{1}{\sqrt{2\pi}}\int dk_-\,e^{ik_-x^-}\tilde f(k_-),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $\tilde f(k_-)\equiv\tilde f(x^+,k_-,x^\bot)$, and put [$$\displaylines{\refstepcounter{equation} \label{2.21}\hskip 1em minus 1em [f(x^-)]\equiv\frac{1}{\sqrt{2\pi}}\int dk_-\,e^{ik_-x^-}\tilde f(k_-),
\qquad |k_-|\ge{\varepsilon}>0.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
We insert into the Lagrangian (\[2.18\]) the variables $[\chi]$, $[\chi^+]$, $[\xi]$, $[\xi^+]$, $[{\varphi}]$ instead of $\chi$, $\chi^+$, $\xi$, $\xi^+$, ${\varphi}$ and obtain the constraint equation [$$\displaylines{\refstepcounter{equation} \label{2.22}\hskip 1em minus 1em \sqrt{2}{\partial}_-[\xi]=-{\left(}{\sigma}^i{\partial}_i-M{\right)}[\chi]+g\big[[{\varphi}][\chi]\big]
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} instead of (\[2.19\]). Its solution is [$$\displaylines{\refstepcounter{equation} \label{2.23}\hskip 1em minus 1em [\xi]=\frac{1}{\sqrt{2}}{\partial}^{-1}_-\Big(-{\left(}{\sigma}^i{\partial}_i-M{\right)}[\chi]+g\big[[{\varphi}]
[\chi]\big]\Big),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the operator ${\partial}_-^{-1}$ is completely defined by the condition [$$\displaylines{\refstepcounter{equation} \label{2.24}\hskip 1em minus 1em {\partial}_-^{-1}[f]={\left[}{\partial}_-^{-1}[f]{\right]}.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} After Fourier transformation the operator ${\partial}_-^{-1}$ is replaced by $(ik_-)^{-1}$.
Substituting the expression (\[2.23\]) into the Lagrangian (where all fields $\chi$, $\chi^+,\ldots$ are replaced with $[\chi]$, $[\chi^+],\ldots$) we come to the result: [$$\displaylines{\refstepcounter{equation} \label{2.25}\hskip 1em minus 1em
L=\int d^2x^\bot dx^-\,\bigg( i\sqrt{2}{\left[}\chi^+{\right]}{\partial}_+{\left[}\chi{\right]}+
{\partial}_-[{\varphi}]{\partial}_+[{\varphi}]+{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}}+\frac{1}{\sqrt{2}}\Big({\left(}\sigma^i{\partial}_i-M{\right)}[\chi]-
g\big[[{\varphi}][\chi]\big]\Big)^+\!{\left(}-i{\partial}_-{\right)}^{-1}\!\Big({\left(}\sigma^k
{\partial}_k-M{\right)}[\chi]-g\big[ [{\varphi}][\chi]\big]\Big)-{{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}-\frac{1}{2}{\partial}_i[{\varphi}]{\partial}_i
[{\varphi}]-\frac{1}{2}m^2[{\varphi}]^2-{\lambda}'[{\varphi}]^3-{\lambda}[{\varphi}]^4\bigg),
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} As in sect. 2a time derivatives ${\partial}_+[\chi]$, ${\partial}_+[{\varphi}]$ enter into the Lagrangian (\[2.25\]) linearly. Therefore to go to canonical formalism it is sufficient to find a standard canonical form for the expression [$$\displaylines{\refstepcounter{equation} \label{nnn2}\hskip 1em minus 1em
i\sqrt{2}[\chi^+]{\partial}_+[\chi]+{\partial}_-[{\varphi}^+]{\partial}_+[{\varphi}]
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} (before a quantization the quantities $\chi^+$, $\chi$ are elements of Grassman algebra). We write [$$\displaylines{\refstepcounter{equation} \label{2.26a}\hskip 1em minus 1em \big[{\varphi}(x^-)\big]=(2\pi)^{-1/2}{\int\limits}_{\varepsilon}^\infty dk_-\,(2k_-)^{-1/2}
\big(a(k_-)\exp(-ik_-x^-)+a^+(k_-)\exp(ik_-x^-)\big),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} -8mm [$$\displaylines{\refstepcounter{equation} \label{2.26b}\hskip 1em minus 1em \big[\chi_r(x^-)\big]=(2\pi)^{-1/2}2^{-1/4}{\int\limits}_{\varepsilon}^\infty dk_-
\big(b_r(k_-)\exp(-ik_-x^-)+c^+_r(k_-)\exp(ik_-x^-)\big),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} -8mm [$$\displaylines{\refstepcounter{equation} \label{2.26v}\hskip 1em minus 1em \big[\chi_r^+(x^-)\big]=(2\pi)^{-1/2}2^{-1/4}{\int\limits}_{\varepsilon}^\infty dk_-
\big(c_r(k_-)\exp(-ik_-x^-)+b^+_r(k_-)\exp(ik_-x^-)\big),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where [$$\displaylines{\refstepcounter{equation} \label{nnn3}\hskip 1em minus 1em
\big[{\varphi}(x^-)\big]\equiv\big[{\varphi}(x^+,x^-,x^\bot)\big],\qquad
a(k_-)\equiv a(x^+,k_-,x^\bot)
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} et cetera, $r=1,2$. The Lagrangian (\[2.25\]) takes the form [$$\displaylines{\refstepcounter{equation} \label{2.27}\hskip 1em minus 1em L=\int d^2x^\bot{\int\limits}_{\varepsilon}^\infty dk_-\,\big(a(k_-)\dot a^+(k_-)
-a^+(k_-)\dot a(k_-)-{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}-b_r(k_-)\dot b_r^+(k_-)-b_r^+(k_-)\dot b_r(k_-)-
c_r(k_-)\dot c_r^+(k_-)-c_r^+(k_-)\dot c_r(k_-)\big)-H,{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where [$$\displaylines{\refstepcounter{equation} \label{2.28}\hskip 1em minus 1em H=\bigg(-\frac{1}{\sqrt{2}}\Big({\left(}\sigma^i{\partial}_i-M{\right)}[\chi]-
g\big[[{\varphi}][\chi]\big]\Big)^+\!{\left(}-i{\partial}_-{\right)}^{-1}\!\Big({\left(}\sigma^k
{\partial}_k-M{\right)}[\chi]-g\big[ [{\varphi}][\chi]\big]\Big)+{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}+
\frac{1}{2}{\partial}_i[{\varphi}]{\partial}_i
[{\varphi}]+\frac{1}{2}m^2[{\varphi}]^2+{\lambda}'[{\varphi}]^3+{\lambda}[{\varphi}]^4\bigg).{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} It is assumed that the quantities $[\chi]$, $[\chi^+]$, $[{\varphi}]$ in the formulae (\[2.28\]) are expressed in terms of $b$, $b^+$, $c$, $c^+$, $a$, $a^+$ with the help of (\[2.26a\]), (\[2.26b\]), (\[2.26v\]).
It follows from (\[2.27\]) that $a^+$, $a$, $b^+$, $b$, $c^+$, $c$ play a role of creation and annihilation operators. After quantization they satisfy the commutation relations (at $x^+=$const): [$$\displaylines{\refstepcounter{equation} \label{2.29a}\hskip 1em minus 1em [a(k_-,x^\bot),a^+(k'_-,x'^\bot)]_-={\delta}(k_--k'_-){\delta}^2(x^\bot-x'^\bot),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{2.29b}\hskip 1em minus 1em [b(k_-,x^\bot),b^+(k'_-,x'^\bot)]_+={\delta}(k_--k'_-){\delta}^2(x^\bot-x'^\bot),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{2.29v}\hskip 1em minus 1em [c(k_-,x^\bot),c^+(k'_-,x'^\bot)]_+={\delta}(k_--k'_-){\delta}^2(x^\bot-x'^\bot),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $[x,y]_\pm=xy\pm yx$. The remaining (anti)commutators are equal to zero. It is seen that the quantity $H$ is LF Hamiltonian ($H=P_+$). The operator of the momentum $P_-$ is equal to [$$\displaylines{\refstepcounter{equation} \label{2.30}\hskip 1em minus 1em
P_-=\int d^2xdx^-\,T_{--}=\frac{1}{2}\int d^2x^\bot{\int\limits}_{\varepsilon}^\infty
dk_-\,k_-\big(a^+(k_-)a(k_-)
+a(k_-)a^+(k_-)+{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}+b_r^+(k_-)b_r(k_-)-b_r(k_-)b_r^+(k_-)+
c_r^+(k_-)c_r(k_-)-c_r(k_-)c_r^+(k_-)\big).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
The quantities $P_+\equiv H$ and $P_-$ should be normally ordered with respect to creation and annihilation operators. The lowest eigenstate of the momentum $P_-$ is the physical vacuum. It is defined by conditions [$$\displaylines{\refstepcounter{equation} \label{2.31}\hskip 1em minus 1em
a(k_-,x^\bot)|\Omega\rangle =0,\quad
b(k_-,x^\bot)|\Omega\rangle =0,\quad
c(k_-,x^\bot)|\Omega\rangle =0,\quad\forall\;x^\bot ,\, k_-\ge{\varepsilon}>0.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The equalities (\[2.4a\]), (\[2.4b\]) become true after normal ordering of the operators $P_+$ and $P_-$. For the $P_-$ it is seen from the formulae (\[2.30\]), and for the $P_+$ it follows from the fact that every term of $P_+$ contains a ${\delta}$-function of difference between the sum of momenta $k_-$ of creation operators and the sum of momenta $k_-$ of annihilation operators. Due to the positivity of all momenta $k_-$, in our regularization scheme every term of the $P_+$ contains at least one annihilation operator. Therefore for normally ordered $P_+$ we have $P_+|\Omega\rangle=0$.
The model under consideration requires ultraviolet regularization. It can be done in different ways. We consider this question together with the renormalization problem in sect. 3 and 4.
The U(N)-theory of pure gauge fields
------------------------------------
We consider the $U(N)$ rather than the $SU(N)$ theory because it is technically more simple. The transition to the $SU(N)$ can be done easily. Gauge field is described by Hermitian matrices [$$\displaylines{\refstepcounter{equation} \label{2.31a}\hskip 1em minus 1em A_\mu(x)=A^+_\mu (x)\equiv\left\{A^{ij}_\mu(x)\right\},{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $\mu=+,-,1,2$; $i,j=1,2,\ldots ,N$. Let us assume, that for the indexes $i,j$ and analogous the usual rule of summation on repeated indexes is not used, and where it is necessary the sign of a sum is indicated. Field strengths tensor is [$$\displaylines{\refstepcounter{equation} \label{2.32}\hskip 1em minus 1em F_{\mu\nu}={\partial}_\mu A_\nu -{\partial}_\nu A_\mu -ig[A_\mu ,A_\nu],{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and gauge transformation has the form [$$\displaylines{\refstepcounter{equation} \label{2.33}\hskip 1em minus 1em A_\mu\to A'_\mu=U^+A_\mu U+\frac{i}{g}U^+{\partial}_\mu U,\quad U^+U=I.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} To escape a breakdown of gauge invariance we apply the regularization of the type (\[1.2\]) with periodic boundary conditions [$$\displaylines{\refstepcounter{equation} \label{2.34}\hskip 1em minus 1em A_\mu(x^+,-L,x^\bot)=A_\mu(x^+,L,x^\bot),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} on the interval $-L\le x^-\le L$ (this exact periodicity can be always achieved starting from gauge invariant one, i. e. the periodicity up to a gauge transformation). All Fourier modes of $A_\mu(x)$ in $x^-$ must be kept, including zero modes (at $k_-=0$).
The Lagrangian has the form [$$\displaylines{\refstepcounter{equation} \label{2.35}\hskip 1em minus 1em L=-\frac{1}{2}\int d^2x^\bot{\int\limits}_{-L}^Ldx^-{\rm Tr}{\left(}F_{\mu\nu}
F^{\mu\nu}{\right)},{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} or [$$\displaylines{\refstepcounter{equation} \label{2.36}\hskip 1em minus 1em L=\int d^2x^\bot{\int\limits}_{-L}^Ldx^-{\rm Tr}{\left(}F^2_{+-}+2F_{-k}F_{+k}-
\frac{1}{2}F_{kk'}F_{kk'}{\right)},{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $k,k'=1,2$. Time derivatives ${\partial}_+A_k$ are present only in the term [$$\displaylines{\refstepcounter{equation} \label{nnn4}\hskip 1em minus 1em
{\rm Tr}{\left(}2({\partial}_-A_k-{\partial}_kA_--ig[A_-,A_+]){\partial}_+A_k{\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} This expression can be rewritten in standard canonical form only after fixing the gauge in a special form of the type $A_-=0$ because then the term above becomes similar to that of scalar field theory ${\rm Tr}{\left(}2({\partial}_-A_k){\partial}_+A_k{\right)}$. However not every field, periodic in $x^-$, can be transformed to the $A_-=0$ gauge. Indeed, the loop integral [$$\displaylines{\refstepcounter{equation} \label{2.37}\hskip 1em minus 1em \Gamma(x^+,x^\bot)={\rm Tr}\left\{{\rm P}\exp{\left(}i{\int\limits}_{-L}^Ldx^-\,A_-
(x^+,x^-,x^\bot){\right)}\right\},{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the symbol ’P’ means the ordering of operators along the $x^-$, is a gauge invariant quantity. If this integral is not equal to $N$ for some field then the gauge $A_-=0$ is not possible for this field. Therefore we choose more weak gauge condition [$$\displaylines{\refstepcounter{equation} \label{2.38}\hskip 1em minus 1em A^{ij}_-= 0,\quad i\ne j,\qquad {\partial}_-A^{ii}_-=0,{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and put [$$\displaylines{\refstepcounter{equation} \label{2.39}\hskip 1em minus 1em A_-^{ii}(x)=v^i(x^+,x^\bot).{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The gauge (\[2.38\]) breaks not only the local gauge invariance but, unlike the $A_-=0$ gauge, also global gauge invariance (it remains only the abelian subgroup of gauge transformations not depending on $x^-$). This has some technical inconvenience but now any periodic field can be described in the gauge (\[2.38\]). Furthermore, if we restrict the class of possible periodic fields by the condition $A_-=0$, disregarding the described consideration, we come to canonical theory with even more complicate constraints if zero modes are taken into account [@nov1; @nov2; @nov3].
From the point of view of LF canonical formalism the variables $A_-^{ij}$ are “coordinates”. Therefore one can restrict their values by the condition (\[2.38\]) directly in the Lagrangian without loosing any equations of motion. Let us introduce the denotations [$$\displaylines{\refstepcounter{equation} \label{2.40a}\hskip 1em minus 1em D_-A^{ij}_+=({\partial}_--ig(v^i-v^j))A^{ij}_+,{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{2.40b}\hskip 1em minus 1em D_-A^{ij}_k=({\partial}_--ig(v^i-v^j))A^{ij}_k,{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where obviously [$$\displaylines{\refstepcounter{equation} \label{2.40v}\hskip 1em minus 1em D_-A^{ii}_k={\partial}_-A^{ii}_k.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Also for any function $f(x^-)\equiv f(x^+,x^-,x^\bot)$ periodic in $x^-$ we denote [$$\displaylines{\refstepcounter{equation} \label{2.41a}\hskip 1em minus 1em f_{(0)}=\frac{1}{2L}{\int\limits}_{-L}^Ldx^-\,f(x^-),{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{2.41b}\hskip 1em minus 1em [f(x^-)]=f(x^-)-f_{(0)}.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Obviously, [$$\displaylines{\refstepcounter{equation} \label{2.41v}\hskip 1em minus 1em {\int\limits}_{-L}^Ldx^-\,[f(x^-)]=0.{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
After gauge fixing (\[2.38\]) the Lagrangian (\[2.36\]) takes the form [$$\displaylines{\refstepcounter{equation} \label{2.42}\hskip 1em minus 1em L=\int d^2x{\int\limits}_{-L}^Ldx^-\Bigg\{2\sum_i{\left(}{\partial}_-{\left[}A^{ii}_k{\right]}{\right)}{\partial}_+{\left[}A^{ii}_k{\right]}+2\sum_{i,j,\, i\ne j}{\left(}D_-
A^{ij}_k{\right)}{\partial}^+A^{ji}_k+\sum_i{\left(}{\partial}_-{\left[}A^{ii}_+{\right]}{\right)}^2+{\hfill\cr\hfil}+\!\!\!\sum_{i,j,\, i\ne j}\!\!{\left(}D_-
A^{ij}_+{\right)}\! D_-A^{ji}_++2\sum_i\!{\left[}A^{ii}_+{\right]}\!\!{\left[}{\partial}_k{\partial}_-
{\left[}A^{ii}_k{\right]}-ig\!\!\sum_{j', \,j'\ne i}\!
{\left(}A^{ij'}_kD_-A^{j'i}_k-{\left(}D_-A^{ij'}_k{\right)}A^{j'i}_k{\right)}{\right]}+\!{\hfill\cr\hfil}+2\sum_{i,j,\, i\ne j}A^{ij}_+{\left(}{\partial}_kD_-A^{ji}_k-ig\sum_{j'}
{\left(}A^{jj'}_kD_-A^{j'i}_k-{\left(}D_-A^{jj'}_k{\right)}A^{j'i}_k{\right)}{\right)}-\frac{1}{2}
\sum_{i,j}F^{\phantom{kl}ij}_{kl}F^{klji}\Bigg\}+{\hfill\cr\hfil}+\int d^2x\Bigg\{ 2L\sum_i({\partial}_+v^i)^2-4L\sum_i{\left(}{\partial}_k A^{ii}_{k(0)}
{\right)}{\partial}_+v^i-2\sum_iA^{ii}_{+(0)}\bigg(2L{\partial}_k{\partial}_kv^i+{{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}+ig{\int\limits}_{-L}^L
dx^-\sum_{j', \,j'\ne i}{\left(}A^{ij'}_kD_-A^{j'i}_k-{\left(}D_-A^{ij'}_k{\right)}A^{j'i}_k
{\right)}\bigg)\Bigg\}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Here we have ignored some unessential surface terms.
Variation of the Lagrangian in $[A_+^{ii}]$, $A_+^{ij}$ at $i\ne j$ leads to constraints, the solution of which can be written in the form [$$\displaylines{\refstepcounter{equation} \label{2.43}\hskip 1em minus 1em
[A_+^{ii}]={\partial}_-^{-2}{\left[}{\partial}_k{\partial}_-[A_k^{ii}]-ig\sum_{j',\, j'\ne i}{\left(}A_k^{ij'}D_-A_k^{j'i}-
(D_-A_k^{ij'})A_k^{j'i}{\right)}{\right]},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{2.44}\hskip 1em minus 1em
A_+^{ij}\Bigr|_{i\ne j}=
D_-^{-2}{\left(}{\partial}_kD_-A_k^{ij}-ig\sum_{j'}
{\left(}A_k^{ij'}D_-A_k^{j'j}-(D_-A_k^{ij'})A_k^{j'j}{\right)}{\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The operator ${\partial}_-^{-1}$ is completely defined, as before, by the condition (\[2.24\]) being well defined on functions $[f(x)]$. The operator $D_-^{-1}$ after Fourier transformation in $x^-$ is reduced to the multiplication by ${\left(}i(k_- -g(v^i- v^j)){\right)}^{-1}$. Therefore it has, in general, no singularities for any $k_-=n(\pi /L)$ with integer n.
Substituting the expressions (\[2.43\]), (\[2.44\]) into the Lagrangian (\[2.42\]), we exclude from it the quantities $[A_+^{ii}]$ and $A_+^{ij}$ at $i\ne j$. The variation of the Lagrangian in $A_{+(0)}^{ii}$ leads to the constraints [$$\displaylines{\refstepcounter{equation} \label{2.45}\hskip 1em minus 1em
Q^{ii}(x^+,x^{\bot})\equiv -2{\left(}2L{\partial}_k{\partial}_k v^i+ig{\int\limits}_{-L}^Ldx^-\sum_{j',\, j'\ne i}
{\left(}A_k^{ij'}D_-A_k^{j'i}-(D_-A_k^{ij'})A_k^{j'i}{\right)}{\right)}=0.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} They are first class constraints which can be posed on physical state vectors after quantization. Therefore we can keep the term with this constraint in the Lagrangian.
Now we must put in the standard canonical form the terms of the Lagrangian [$$\displaylines{\refstepcounter{equation} \label{2.46}\hskip 1em minus 1em
\int d^2x{\int\limits}_{-L}^L dx^-\left\{2\sum_i({\partial}_-[A_k^{ii}]){\partial}_+[A_k^{ii}]+
2\sum_{i,j,\, i\ne j}(D_-A_k^{ij}){\partial}_+A_k^{ji}\right\}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} It can be reached by going to Fourier transform [$$\displaylines{\refstepcounter{equation} \label{2.47}\hskip 1em minus 1em
[A_k^{ii}(x^-)]=\frac{1}{2\sqrt{2L}}\sum_{k_-=\pi/L}^\infty
k_-^{-1/2}\left\{ a_k^i(k_-)\exp(-ik_-x^-)+{a_k^i}^+(k_-)\exp(ik_-x^-)\right\},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where we sum over $k_-=n\pi /L$, $n=1,2,\dots$, and [$$\displaylines{\refstepcounter{equation} \label{2.48}\hskip 1em minus 1em
A_k^{ij}(x^-)\Bigr|_{i\ne j}=
\frac{1}{2\sqrt{2L}}\left\{
\sum_{k_->g(v^i-v^j)}{\left(}k_--g(v^i-v^j){\right)}^{-1/2}{a_k^{ij}}^+(k_-)
\exp(ik_-x^-)+\right.{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}+\left.\sum_{k_->g(v^j-v^i)}{\left(}k_--g(v^j-v^i){\right)}^{-1/2}a_k^{ji}(k_-)
\exp(-ik_-x^-)\right\},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where we sum over all $k_-=n\pi /L$ satisfying corresponding inequalities. The expression (\[2.46\]) takes the form [$$\displaylines{\refstepcounter{equation} \label{2.49}\hskip 1em minus 1em
(2i)^{-1}\int d^2x^{\bot}\left\{
\sum_i\sum_{k_-=\pi/L}^\infty {\left(}a_k^i(k_-){\partial}_+(a_k^i)^+(k_-)-
{a_k^i}^+(k_-){\partial}_+a_k^i(k_-){\right)}+\right.{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\left.+\sum_{i,j,\, i\ne j}\sum_{k_->g(v^i-v^j)}
{\left(}a_k^{ij}(k_-){\partial}_+{a_k^{ij}}^+(k_-)-
{a_k^{ij}}^+(k_-){\partial}_+a_k^{ij}(k_-){\right)}\right\}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Further, in the Lagrangian (\[2.42\]) there is a part [$$\displaylines{\refstepcounter{equation} \label{2.50}\hskip 1em minus 1em
L_v=\int d^2x^{\bot}\left\{
2L\sum_i({\partial}_+v^i)^2-4L\sum_i({\partial}_kA_{k(0)}^{ii}){\partial}_+v^i\right\}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The “momentum” conjugated to $v^i$ is [$$\displaylines{\refstepcounter{equation} \label{2.51}\hskip 1em minus 1em
{\cal P}^i=\frac{{\delta}L_v}{{\delta}({\partial}_+v^i)}=4L{\left(}{\partial}_+v^i-{\partial}_kA_{k(0)}^{ii}{\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Hence, [$$\displaylines{\refstepcounter{equation} \label{2.52}\hskip 1em minus 1em
{\partial}_+v^i=\frac{1}{4L}{\cal P}^i+{\partial}_kA_{k(0)}^{ii}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The corresponding part of the Hamiltonian equals to [$$\displaylines{\refstepcounter{equation} \label{2.53}\hskip 1em minus 1em
H_v=\int d^2x^{\bot}\sum_i{\left(}{\cal P}^i{\partial}_+v^i{\right)}-L_v=
\int d^2x^{\bot}2L\sum_i{\left(}\frac{{\cal P}^i}{4L}+{\partial}_kA_{k(0)}^{ii}{\right)}^2,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and the corresponding part of canonical Lagrangian is [$$\displaylines{\refstepcounter{equation} \label{2.54}\hskip 1em minus 1em
L_v=\int d^2x^{\bot}\sum_i{\left(}{\cal P}^i{\partial}_+ v^i{\right)}-H_v.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
Excluding from the Lagrangian (\[2.42\]) the quantities $[A_+^{ii}]$ and $A_+^{ij}$ (at $i\ne j$) via the equations (\[2.47\]), (\[2.48\]), replacing the terms (\[2.46\]) by the expression (\[2.49\]) and the part (\[2.50\]) by the expression (\[2.54\]), we obtain the result [$$\displaylines{\refstepcounter{equation} \label{2.55}\hskip 1em minus 1em
L=(2i)^{-1}\int d^2x^{\bot}\left\{\sum_i\sum_{k_-=\pi/L}^\infty{\left(}a_k^i(k_-){\partial}_+{a_k^i}^+(k_-)-{a_k^i}^+(k_-){\partial}_+a_k^i(k_-){\right)}\right.+{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}}+\sum_{i,j,\, i\ne j}\sum_{k_->g(v^i-v^j)}{\left(}a_k^{ij}(k_-){\partial}_+{a_k^{ij}}^+(k_-)-{a_k^{ij}}^+(k_-){\partial}_+a_k^{ij}(k_-){\right)}+{{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}+\left.\sum_i {\cal P}^i{\partial}_+v^i+\sum_iA_{+(0)}^{ii}Q^{ii}\right\}-H,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $Q^{ii}$ are defined by (\[2.45\]), and the Hamiltonian $H=P_+$ is equal to [$$\displaylines{\refstepcounter{equation} \label{2.56}\hskip 1em minus 1em
H=\int d^2x\int\limits_{-L}^Ldx^-\left\{
\sum_i{\left(}{\partial}_-{\left[}A_+^{ii}{\right]}{\right)}^2+
\sum_{ij,\, i\ne j}{\left(}D_-A_+^{ij}{\right)}D_-A_+^{ji}-
\frac{1}{2}\sum_{i,j}F_{kl}^{\phantom{kl}ij}F^{klji}\right\}+{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}+2L\int d^2x^{\bot}\sum_i{\left(}\frac{{\cal P}_i}{4L}+{\partial}_kA_{k(0)}^{ii}{\right)}^2.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
It is implied that instead of the quantities $[A_+^{ii}]$ and $A_+^{ij}$ (at $i\ne j$) one uses the expressions (\[2.43\]), (\[2.44\]) and the $A_k^{ij}$ are expressed in terms of $a_k^i$, ${a_k^i}^+$, $a_k^{ij}$, ${a_k^{ij}}^+$, (at $i\ne j$) and of $A_{k(0)}^{ii}$ with the help of equations (\[2.47\]), (\[2.48\]) and [$$\displaylines{\refstepcounter{equation} \label{2.57}\hskip 1em minus 1em
A_k^{ii}={\left[}A_k^{ii}{\right]}+A_{k(0)}^{ii}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
It is seen from the formulae (\[2.55\]) that ${a_k^i}^+$, $a_k^i$, ${a_k^{ij}}^+$, $a_k^{ij}$ play the role of creation and annihilation operators. After quantization they satisfy the following commutation relations (at $x^+=const$): [$$\displaylines{\refstepcounter{equation} \label{2.58a}\hskip 1em minus 1em
{\left[}a_k^i(k_-,x^{\bot}),{a_l^j}^+(k'_-,x'^{\bot}){\right]}_-={\delta}^{ij}{\delta}_{kl}
{\delta}_{k_-,k'_-}{\delta}^2(x^{\bot}-x'^{\bot}),
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{2.58b}\hskip 1em minus 1em
{\left[}a_k^{ij}(k_-,x^{\bot}),{a_l^{i'j'}}^+(k'_-,x'^{\bot}){\right]}_-={\delta}^{ii'}
{\delta}^{jj'}{\delta}_{kl}{\delta}_{k_-,k'_-}{\delta}^2(x^{\bot}-x'^{\bot}),\quad i\ne j,\;\; i'\ne j'.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Also we have [$$\displaylines{\refstepcounter{equation} \label{2.58c}\hskip 1em minus 1em
{\left[}{\cal P}^i(x^{\bot}),v^j(x'^{\bot}){\right]}_-=-i{\delta}^{ij}{\delta}^2(x^{\bot}-x'^{\bot}).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Remaining commutators are equal to zero.
The operator of the momentum $P_-$, defined by [$$\displaylines{\refstepcounter{equation} \label{2.59}\hskip 1em minus 1em
P_-=\int d^2x\int\limits_{-L}^Ldx^-T_{--},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} acts on physical states $|\Psi{\rangle}$, satisfying the condition [$$\displaylines{\refstepcounter{equation} \label{2.60}\hskip 1em minus 1em
Q^{ii}(x^{\bot})|\Psi{\rangle}=0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} equivalent to the canonical operator [$$\displaylines{\refstepcounter{equation} \label{2.61}\hskip 1em minus 1em
P_-^{can}=\!\int\! d^2x\!{\left(}\sum_i\sum_{k_-=\pi/L}^\infty\! k_-{a_k^i}^+(k_-)a_k^i(k_-)+
\sum_{i,j,\, i\ne j}\sum_{k_->g(v^i-v^j)}\! k_-
{a_k^{ij}}^+(k_-)a_k^{ij}(k_-){\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the normal ordering was made.
Physical vacuum $|\Omega{\rangle}$ satisfies the relations [$$\displaylines{\refstepcounter{equation} \label{2.62a}\hskip 1em minus 1em
a_k^i(k_-,x^{\bot})|\Omega{\rangle}=0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{2.62b}\hskip 1em minus 1em
a_k^{ij}(k_-,x^{\bot})|\Omega{\rangle}=0,\qquad i\ne j,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and the condition (\[2.60\]).
This scheme is connected with the following essential difficulty. The zero modes $A_{k(0)}^{ii}(x^{\bot}\!)$ are present in the Lagrangian (\[2.55\]) and in the Hamiltonian (\[2.56\]) but the derivatives ${\partial}_+ A_{k(0)}^{ii}$ are absent there. Therefore new constraints arise [$$\displaylines{\refstepcounter{equation} \label{2.63}\hskip 1em minus 1em
\frac{{\delta}H}{{\delta}A_{k(0)}^{ii}(x^{\bot})}=0.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} These constraints are of the 2nd class and they must be solved with respect to $A_{k(0)}^{ii}$ and then the $A_{k(0)}^{ii}$ have to be excluded from the Hamiltonian. The constraints (\[2.63\]) are very complicated and explicit resolution of them is practically impossible. The application of Dirac brackets does not simplify this.
Due to this difficulty a practical calculation usually ignores all zero modes from the beginning. It makes the approximation worse. It is interesting that in the framework of lattice regularization it is possible to overcome the difficulties caused by the constraints (\[2.63\]) [@heplat; @tmf04]. This question will be considered in sect. 5.
Limiting transition from the theory in Lorentz[\
]{}coordinates to the theory on the Light Front
================================================
To clarify the connection between the theory in Lorentz coordinates in Hamiltonian form and analogous theory on the LF we perform the limiting transition from one to the other. Here this transition is considered in the fixed frame of Lorentz coordinates by introducing states that move at a speed close to the speed of light in the direction of the $x^3$ axis. Constructing the matrix elements of the Hamiltonian between such states and studying the limiting transition to the speed of light (an infinite momentum), we can derive information about the Hamiltonian in the light-like coordinates. This information also takes into account the contribution from intermediate states with finite momenta. Here, we illustrate the results of such an investigation using $(1+1)$-dimensional theory of scalar field with the $\lambda \varphi^4$-interaction. Instead of $x^3$ we denote analogous space coordinate by $x^1$. The generalization of the method to $(3+1)$-dimensional Yukawa model is discussed briefly at the end of this section. The limiting transition studied here is accomplished approximately by subjecting the momenta $p_1$ to an auxiliary cutoff that separates fast modes of the fields (with high $p_1$ values) from slow modes (with finite $p_1$ values). This cutoff is parameterized in terms of the quantities ${\Lambda}$, ${\Lambda}_1$, and ${\delta}$ and the limiting-transition parameter ${\eta}$ (${\eta}>0$, ${\eta}\to 0$): we have ${\eta}^{-1}{\Lambda}_1\ge |p_1|\ge {\eta}^{-1}{\delta}$ for the fast modes and $p_1\le {\Lambda}$ for the slow modes (${\Lambda}\gg {\delta}$). For ${\eta}\to 0$, the inequality ${\eta}^{-1}{\delta}>{\Lambda}$ holds, so that the above momentum intervals are separated. The field modes with the momenta ${\eta}^{-1}{\delta}>|p_1|>{\Lambda}$ are discarded. This procedure is justified by the fact that the resulting Hamiltonian in the limit ${\eta}\to 0$ reproduces the canonical LF Hamiltonian (without zero modes) when only the fast modes are taken into account and is consistent with conventional Feynman perturbation theory for ${\delta}\to 0$. Therefore, even an approximate inclusion of the other (slow) modes may provide a description of nonperturbative effects, such as vacuum condensates. The effective LF Hamiltonian obtained here for the model under consideration differs from the canonical Hamiltonian only by the presence of the vacuum expectation value of the scalar field and by an additional renormalization of the mass of this field. The renormalized mass involves the vacuum expectation value of the squared slow part of the field. Masses of bound states can be found by solving Schrodinger equation [$$\displaylines{\refstepcounter{equation} \label{3.1}\hskip 1em minus 1em
P_+|\Psi\rangle=\frac{m^2}{2p_-}|\Psi\rangle,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} with obtained Hamiltonian $P_+$.
We start from the standard expression for the Hamiltonian of scalar field $\varphi (x)$ in $(1+1)$-dimensional space-time in Lorentz coordinates $x^{\mu}=(x^0,x^1)$, at $x^0=0$: [$$\displaylines{\refstepcounter{equation} \label{3.2}\hskip 1em minus 1em
H=:\int d^1x{\left(}\frac{1}{2}\Pi^2+\frac{1}{2}
({\partial}_1{\varphi})^2+\frac{m^2}{2}{\varphi}^2+{\lambda}{\varphi}^4{\right)}:,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $\Pi(x^1)$ are the variables that are canonically conjugate to ${\varphi}(x^1)\equiv{\varphi}(x^0=0,x^1)$, and the symbol $:\quad :$ of the normal ordering refers to the creation and annihilation operators $a$ and $a^+$ that diagonalize the free part of the Hamiltonian in the Fock space over the corresponding vacuum $|0{\rangle}$. These operators are given by [$$\displaylines{\refstepcounter{equation} \label{3.3}\hskip 1em minus 1em
{\varphi}(x^1)=\frac{1}{\sqrt{4\pi}}\int dp_1(m^2+p^2_1)^{-1/4}
{\left[}a(p_1)\exp (-ip_1x^1)+h.c.{\right]},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{3.4}\hskip 1em minus 1em
\Pi(x^1)=\frac{-i}{\sqrt{4\pi}}\int dp_1(m^2+p^2_1)^{-1/4}
{\left[}a(p_1)\exp (-ip_1x^1)-h.c.{\right]},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $a(p_1)|0{\rangle}=0$.
To investigate the limiting transition to the LF Hamiltonian (defined at $x^+= 0$), it is more convenient to go over from the Hamiltonian (\[3.2\]) to the operator $H+P_1=\sqrt{2}P_+$, where the momentum $P_1$ has the form [$$\displaylines{\refstepcounter{equation} \label{3.5}\hskip 1em minus 1em
P_1=\int dp_1 a^+(p_1)a(p_1)p_1.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
Applying the above parametrization of high momenta in terms of ${\eta},{\eta}\to 0$, to $p_1$ we can then consider the transition to an infinitely high momentum of states as a limit of the corresponding Lorentz transformation with parameter ${\eta}$. To be more specific, we have $p_1\to (-{\eta}\sqrt{2})^{-1}q_-$, where $q_-$ is a finite momentum in the light-like coordinates, and [$$\displaylines{\refstepcounter{equation} \label{3.6}\hskip 1em minus 1em
\lim_{{\eta}\to 0}{\left(}({\eta}\sqrt{2})^{-1}\langle p'_1|(H+P_1)_{x^0=0}|p_1{\rangle}{\right)}=
\langle q'_-|(P_+)_{x^+=0}|q_-{\rangle}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} It follows that the eigenvalues $E_+$ of the operator $(P_+)_{x^+=0}$ that correspond to the momentum $q_-$ are obtained as the corresponding limit of the eigenvalues $E({\eta})$ of the operator $(H + P_1)_{x^0=0}$ at momentum $p_1$: [$$\displaylines{\refstepcounter{equation} \label{3.7}\hskip 1em minus 1em
E_+=\lim_{{\eta}\to 0}({\eta}\sqrt{2})^{-1}E({\eta}).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} In the following, we consider this limiting transition as a part of the eigenvalue problem for the operator $H+P_1$, using perturbation theory in the parameter ${\eta}$. Separating the Fourier modes of the field into fast and slow ones, as is indicated above, and neglecting the region of intermediate momenta (${\eta}^{-1}{\delta}\le |p_1|\le {\eta}^{-1}{\Lambda}_1$ is the region of the fast modes, and $|p_1|\le {\Lambda}$ is the region of the slow modes), we can substantially simplify this perturbation theory. The ${\eta}$ dependence of the field operators and Hamiltonian can then be determined by making, in the region of fast momenta, the change of the variables as [$$\displaylines{\refstepcounter{equation} \label{3.8}\hskip 1em minus 1em
p_1={\eta}^{-1}k,\quad a(p_1)=\sqrt{{\eta}}\tilde a(k),\quad
{\delta}\le |k|\le {\Lambda}_1,\quad [\tilde a(k),{\tilde a}^+(k')]={\delta}(k-k').
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The fast part $[{\varphi}(x^1)]_f$ of the operator ${\varphi}(x^1)$ is estimated as [$$\displaylines{\refstepcounter{equation} \label{3.9}\hskip 1em minus 1em
[{\varphi}(x^1)]_f=\tilde {\varphi}(y)+O({\eta}^2),\quad y={\eta}^{-1}x^1,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{3.10}\hskip 1em minus 1em
\tilde{\varphi}(y)=(4\pi)^{-1/2}\int dk|k|^{-1/2}[\tilde a(k)\exp (-iky)+h.c.].
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} We denote the slow part of the field ${\varphi}$ by $\breve{\varphi}$ (${\varphi}=[{\varphi}]_f+\breve{\varphi}$). Substituting formulas (\[3.8\])-(\[3.10\]) into Hamiltonian (\[3.2\]), we obtain [$$\displaylines{\refstepcounter{equation} \label{3.11}\hskip 1em minus 1em
H+P_1={\eta}^{-1}{\left(}h_0+{\eta}h_1+{\eta}^2 h_2+\dots{\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{3.12}\hskip 1em minus 1em
h_0=2\int\limits_{\delta}^{{\Lambda}_1}dk\tilde a^+(k)\tilde a(k)k,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{3.13}\hskip 1em minus 1em
h_1=(H+P_1)_{{\varphi}=\breve{\varphi},\Pi=\breve\Pi}\equiv (\breve H+\breve P_1),
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{3.14}\hskip 1em minus 1em
h_2=\int\limits_{{\delta}\le |k|\le {\Lambda}_1}\hskip -3mm dk{\left(}\frac{m^2}{2|k|}{\right)}\tilde a^+(k)\tilde a(k)+:{\lambda}\int dy{\left[}\tilde{\varphi}^4(y)+4\breve {\varphi}(0)\tilde{\varphi}^3(y)+
6\breve{\varphi}^2(0)\tilde{\varphi}^2(y){\right]}:.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Prior to performing integration with respect to $y$, we formally expanded the operators $\breve{\varphi}(x^1)=\breve{\varphi}({\eta}y)$ in Taylor series in the variable ${\eta}y$ and estimated their orders in the parameter ${\eta}$ at fixed $y$. Such estimates can be justified at least in Feynman perturbation theory. The operator $(\breve H + \breve P_1)$ in (\[3.13\]) is defined in such a way that its minimum eigenvalue is zero. Let us consider perturbation theory in the parameter ${\eta}$ for the equation [$$\displaylines{\refstepcounter{equation} \label{3.15}\hskip 1em minus 1em
(H+P_1)|f{\rangle}=E|f{\rangle}{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} under the condition that the states $|f{\rangle}$ have, as in formula (\[3.6\]), a negative value of $P_1$ proportional to ${\eta}^{-1}$ for ${\eta}\to 0$ and also describe the states with a finite mass. The expansions of the quantity $E$ and the vector $|f{\rangle}$ in power series in ${\eta}$ can then be written as [$$\displaylines{\refstepcounter{equation} \label{3.16}\hskip 1em minus 1em
E={\eta}^{-1}\sum_{n=2}^\infty {\eta}^nE_n,\qquad |f{\rangle}=\sum_{n=0}^\infty {\eta}^n|f_n{\rangle}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} We arrive at the system of equations [$$\displaylines{\refstepcounter{equation} \label{3.17}\hskip 1em minus 1em
h_0|f_0{\rangle}=0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{3.18}\hskip 1em minus 1em
h_0|f_1{\rangle}+h_1|f_0{\rangle}=0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{3.19}\hskip 1em minus 1em
h_0|f_2{\rangle}+h_1|f_1{\rangle}+(h_2-E_2)|f_0{\rangle}=0,\quad \dots.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} To describe solutions to these equations, we use the basis generated by the fast-field operators $\tilde a^+(k)$ over the vacuum $|0{\rangle}$ and the slow-field operators $\breve{\varphi}$ and $\breve\Pi$ over the vacuum $|v{\rangle}$ that corresponds to the Hamiltonian $(\breve H +\breve P_1)$. The vectors of this basis can be symbolically represented as [$$\displaylines{\refstepcounter{equation} \label{3.20}\hskip 1em minus 1em
\tilde a^+\dots\tilde a^+|0{\rangle}\breve{\varphi}\dots\breve{\varphi}\breve\Pi\dots\breve\Pi |v{\rangle}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} By virtue of (\[3.12\]), the manifold of solutions $|f_0{\rangle}$ to equation (\[3.17\]) is reduced to the set of vectors (\[3.20\]), which do not contain the operators $\tilde a^+(k)$ with $k\ge{\delta}$. Let ${\cal P}_0$ be the projection operator onto this set. According to equation (\[3.18\]), we then have [$$\displaylines{\refstepcounter{equation} \label{3.21}\hskip 1em minus 1em
{\cal P}_0h_0|f_1{\rangle}=(\breve H+\breve P_1)|f_0{\rangle}=0.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Equation (\[3.21\]) requires that the vectors $|f_0{\rangle}$ be the lowest eigenstates of the operator $(\breve H+\breve P_1)$, that is, linear combinations of basis vectors (\[3.20\]) including neither the operators $\tilde a^+(k)$ with $k\ge{\delta}$ nor the operators $\breve{\varphi}$ and $\breve \Pi$. We denote the projection operator on this set of vectors (\[3.20\]) by ${\cal P}'_0$. To determine the quantity $E_2$ which we are interested in, it is sufficient to consider the ${\cal P}'_0$-projection of equation (\[3.19\]). Taking into account (\[3.17\]), (\[3.20\]), and (\[3.21\]), we find that $E_2$ appears as a solution to the eigenvalue problem [$$\displaylines{\refstepcounter{equation} \label{3.22}\hskip 1em minus 1em
{\cal P}'_0h_2|f_0{\rangle}=E_2|f_0{\rangle}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Thus, in accordance with (\[3.6\]) and (\[3.7\]), the operator ${\cal P}'_0h_2{\cal P}'_0$ plays the role of the effective LF Hamiltonian $P_+^{eff}$. Substituting formula (\[3.14\]) for the operator $h_2$ into the expression for $P_+^{eff}$, we take into account that, between the projection operators ${\cal P}'_0$, the contribution of the field modes with positive momenta ($k\ge{\delta}$) vanishes and that the products of the operators of the slow part of the field can be replaced with their expectation values for the vacuum $|v{\rangle}$. In addition, we note that, under the Lorentz transformation corresponding to the limiting transition ${\eta}\to 0$ in formula (\[3.6\]), the variable $y$ goes over into the light-like coordinate $y^-=-y/\sqrt{2}$, the momenta $k$ go over into the light-like momenta $q_-=-\sqrt{2}k$, and the corresponding coordinate $y^+$ vanishes at $x^0=0$ (for finite values of $y^-$). Going over to the operators $A(q_-)=2^{-1/4}\tilde a(-k)$ for $k\le-{\delta}$ ($q_-\ge{\delta}\sqrt{2}$) and to the corresponding field $\Phi(y^+=0,y^-)=\tilde{\varphi}_-(y)$, where $\tilde{\varphi}_-$ is the part of the $\tilde{\varphi}$ containing only the modes with negative momenta ($k\le-{\delta}$), we obtain the effective Hamiltonian $P_+^{eff}$ in the form [$$\displaylines{\refstepcounter{equation} \label{3.23}\hskip 1em minus 1em
P_+^{eff}=:\int dy^-\left\{\frac{1}{2}{\left[}m^2+12{\lambda}\langle :\breve{\varphi}^2:{\rangle}_v
{\right]}\Phi^2+4{\lambda}\langle \breve{\varphi}{\rangle}_v\Phi^3+{\lambda}\Phi^4\right\}:,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $\langle\dots{\rangle}_v$ is the expectation value for the vacuum $|v{\rangle}$. Expression (\[3.23\]) coincides with the canonical effective Hamiltonian if, in the latter, we take into account the shift of the field by the constant $\langle\breve{\varphi}{\rangle}_v$ and the change in the mass squared by $12{\lambda}{\left[}\langle :\breve{\varphi}^2:{\rangle}_v-\langle\breve{\varphi}{\rangle}^2_v{\right]}$.
Analogous results were obtained for Yukawa model in $(3+1)$-dimensional space-time [@pred2] in regularization of Pauli-Villars type, introducing a number of nonphysical fields with very large masses. The absence of essential difference between the Hamiltonian obtained via limiting transition and canonical LF Hamiltonian is connected with this choice of regularization. Other regularizations can lead to more complicated results.
This method of limiting transition can not be directly expanded to gauge theories, because the approximations used for nongauge theories are not justified.
Comparison of Light Front perturbation theory[\
]{} with the theory in Lorentz coordinates
===============================================
As is already known, canonical quantization in LF, i.e., on the $x^+=const$ hypersurface, can result in a theory not quite equivalent to the Lorentz-invariant theory (i.e., to the standard Feynman formalism). This is due, first of all, to strong singularities at zero values of the “light-like” momentum variables . To restore the equivalence with a Lorentz-covariant theory, one has to add unusual counter-terms to the formal canonical Hamiltonian for the LF, $H=P_+$. These counter-terms can be found by comparing the perturbation theory based on the canonical LF formalism with Lorentz-covariant perturbation theory [@tmf97]. This is done in the present section. The LF Hamiltonian thus obtained can then be used in nonperturbative calculations. It is possible, however, that perturbation theory does not provide all of the necessary additions to the canonical Hamiltonian, as some of these additions can be nonperturbative. In spite of this, it seems necessary to examine this problem within the framework of perturbation theory first.
For practical purposes a stationary noncovariant LF perturbation theory, which is similar to the one applied in nonrelativistic quantum mechanics, is widely used. It was found [@bur1; @har; @lay] that the “light-front” Dyson formalism allows this theory to be transformed into an equivalent LF Feynman theory (under an appropriate regularization). Then, by re-summing the integrands of the Feynman integrals, one can recast their form so that they become the same as in the Lorentz-covariant theory. (This is not the case for diagrams without external lines, which we do not consider here.) Then, the difference between the LF and Lorentz-covariant approaches that persists is only due to the different regularizations and different methods of calculating the Feynman integrals (which is important because of the possible absence of their absolute convergence in pseudo-Euclidean space). In the present section, we concentrate on the analysis of this difference.
A LF theory needs not only the standard UV regularization, but also a special regularization of the singularities $Q_-=0$. In our approach, this regularization (by method (\[1.1\])) eliminates the creation operators $a^+(Q)$ and annihilation operators $a(Q)$ with $|Q_-^i|<{\varepsilon}$ from the Fourier expansion of the field operators in the field representation. As a result, the integration w.r.t. the corresponding momentum $Q_-$ over the range $(-\infty,-{\varepsilon})\cup({\varepsilon},\infty)$ is associated with each line before removing the ${\delta}$-functions. Different propagators are regularized independently, which allows the described re-arrangement of the perturbation theory series. On the other hand, this regularization is convenient for further nonperturbative numerical calculations with the LF Hamiltonian, to which the necessary counter-terms are added (the “effective” Hamiltonian). We require that this Hamiltonian generate a theory equivalent to the Lorentz-covariant theory when the regularization is removed. Note that Lorentz-invariant methods of regularization (e.g., Pauli-Villars regularization) are far less convenient for numerical calculations and we shall only briefly mention them.
The specific properties of the LF Feynman formalism manifest themselves only in the integration over the variables $Q_{\pm}={1\over\sqrt{2}}(Q_0\pm Q_3)$, while integration over the transverse momenta $Q_{{\bot}}\equiv \{Q_1, Q_2\}$ is the same in the LF and the Lorentz coordinates (though it might be nontrivial because it requires regularization and renormalization). Therefore, we concentrate on a comparison of diagrams for fixed transverse momenta (which is equivalent to a two-dimensional problem).
In this section we propose a method that allows one to find the difference (in the limit ${\varepsilon}\to 0$) between any LF Feynman integral and the corresponding Lorentz-covariant integral without having to calculate them completely. Based on this method, a procedure is elaborated for constructing an effective LF Hamiltonian correct to any order of perturbation theory. This procedure can be applied to nongauge field theories as well as to Abelian and non-Abelian gauge theories in the gauge $A_-=0$ with the gauge vector field propagator chosen according to the Mandelstam-Leibbrandt prescription [@man; @lei]. The question of whether the additions to the Hamiltonian, that arise under that procedure, can be combined into a finite number of counter-terms must be considered separately in each particular case.
We will consider the application of this formalism to Yukawa model, to QCD in four-dimensional space-time and to QED in two-dimensional space-time.
Reduction of Light Front and Lorentz-covariant Feynman[\
]{}integrals to a form convenient for comparison {#integ}
--------------------------------------------------------
Let us examine an arbitrary one particle irreducible Feynman diagram. We fix all external momenta and all transverse momenta of integration, and integrate only over $Q_+$ and $Q_-$: [$$\displaylines{\refstepcounter{equation} \label{4.1}\hskip 1em minus 1em
F=\lim_{{\char'32}\to 0}\int{{\prod_i d^2Q^i \quad f(Q^i,p^k)}
\over {\prod_i(2Q_+^iQ_-^i-M^2_i+i{\char'32})}}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} We assume that all vertices are polynomial and that the propagator has the form [$$\displaylines{\refstepcounter{equation} \label{4.1.2}\hskip 1em minus 1em
{{z(Q)}\over {Q^2-m^2+i{\char'32}}},\qquad{\rm or}\qquad {{z(Q)\; Q_+}
\over{(Q^2-m^2+i{\char'32})(2Q_+Q_-+i{\char'32})}},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $z(Q)$ is a polynomial. A propagator of the second type in (\[4.1.2\]) arises in gauge theories in the gauge $A_-=0$ if the Mandelstam-Leibbrandt formalism [@man; @lei] with the vector field propagator [$$\displaylines{\refstepcounter{equation} \label{nnn5}\hskip 1em minus 1em
\frac{1}{Q^2+i{\char'32}}{\left(}g_{\mu\nu}-
\frac{Q_\mu{\delta}_\nu^++Q_\nu{\delta}_\mu^+}{2Q_+Q_-+i{\char'32}}2Q_+{\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} is used. In equation (\[4.1\]) either $M^2_i=m^2_i+{Q^{i}_{{\bot}}}^2\ne 0$, where $m_i$ is the particle mass, or $M^2_i=0$.
The function $f$ involves the numerators of all propagators and all vertices with the necessary ${\delta}$-functions, that include the external momenta $p^k$ (the same expression without the ${\delta}$-functions is a polynomial, which we denote by $\tilde f$). We assume for the diagram $F$ and for all of its subdiagrams that the conditions [$$\displaylines{\refstepcounter{equation} \label{4.1.1}\hskip 1em minus 1em
{\omega}_{{\scriptscriptstyle \|}}<0, \qquad {\omega}_+<0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} hold, where ${\omega}_+$ is the index of divergence w.r.t. $Q_+$ at $Q_-^i\ne 0\ \forall i$, and ${\omega}_{{\scriptscriptstyle \|}}$ is the index of divergence in $Q_+$ and $Q_-$ (simultaneously); . The diagrams that do not meet these conditions should be examined separately for each particular theory (their number is usually finite). We seek the difference between the value of integral (\[4.1\]) obtained by the Lorentz-covariant calculation and its value calculated in LF coordinates (LF calculation).
In the LF calculation, one introduces and then removes the LF cutoff $|Q_-|\ge{\varepsilon}>0$: [$$\displaylines{\refstepcounter{equation} \label{nnn6}\hskip 1em minus 1em
F_{\rm lf}=\lim_{{\varepsilon}\to 0}\lim_{{\char'32}\to 0}
\int\limits_{V_{\scriptstyle {\varepsilon}}}\prod_i dQ_-^i \int
\prod_i dQ_+^i
{{f(Q^i,p^k)}\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $V_{{\varepsilon}}=\prod_i{\left(}{\left(}-\infty,-{\varepsilon}{\right)}\cup{\left(}{\varepsilon},\infty{\right)}{\right)}$. Here (and in the diagram configurations to be defined below) we take the limit w.r.t. ${\varepsilon}$, but, generally speaking, this limit may not exist. In this case, we assume that we do not take the limit, but take the sum of all nonpositive power terms of the Laurent series in ${\varepsilon}$ at the zero point. If conditions (\[4.1.1\]) are satisfied, Statement 2 from Appendix I can be used. This results in the equality [$$\displaylines{\refstepcounter{equation} \label{4.5}\hskip 1em minus 1em
F_{\rm lf}=\lim_{{\varepsilon}\to 0}\lim_{{\char'32}\to 0}\int\prod_k dq_+^k
\int\limits_{V_{\scriptstyle {\varepsilon}}\cap B_L} \prod_k dq_-^k
{{\tilde f(Q^i,p^s)}
\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} From here on, the momenta of the lines $Q^i$ are assumed to be expressed in terms of the loop momenta $q^k$, $B_L$ is a sphere of a radius $L$ in the $q_-^k$-space, and $L$ depends on the external momenta. Now, using Statement 2 from Appendix I, we obtain [$$\displaylines{\refstepcounter{equation} \label{4.7}\hskip 1em minus 1em
F_{\rm lf}=\lim_{{\varepsilon}\to 0}\lim_{{\char'32}\to 0}
\lim_{{\beta}\to 0}\lim_{{\gamma}\to 0}
\int\prod_k dq_+^k \int\limits_{V_{\scriptstyle {\varepsilon}}}
\prod_k dq_-^k
{{\tilde f(Q^i,p^s) \; e^{-{\gamma}\sum_i {Q_+^i}^2-{\beta}\sum_i{Q_-^i}^2}
}\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} To reduce the covariant Feynman integral to a form similar to (\[4.5\]), we introduce a quantity $\hat F$: [$$\displaylines{\refstepcounter{equation} \label{4.8}\hskip 1em minus 1em
\hat F=\lim_{{\char'32}\to 0}\lim_{{\beta}\to 0}\lim_{{\gamma}\to 0}
\int \prod_k d^2q^k {{\tilde f(Q^i,p^s) \;
e^{-{\gamma}\sum_i {Q_+^i}^2-{\beta}\sum_i{Q_-^i}^2}}\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Let us prove that this quantity coincides with the result of the Lorentz-covariant calculation $F_{\rm cov}$. To this end, we introduce the in the Minkowski space of the propagator [$$\displaylines{\refstepcounter{equation} \label{4.9}\hskip 1em minus 1em
{{z(Q^i)}\over{2Q_+^iQ_-^i-M^2_i+i{\char'32}}}
=-iz{\left(}-i{{{\partial}}\over{{\partial}y_i}}{\right)}\int\limits_0^{\infty}
e^{i{\alpha}_i(2Q_+^iQ_-^i-M^2_i+i{\char'32})+i(Q^i_+y_i^++Q^i_-y_i^-)}
d{\alpha}_i \Bigr|_{y_i=0}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Then we substitute (\[4.9\]) into (\[4.8\]). Due to the exponentials that cut off $q_+^k$, $q_-^k$ and ${\alpha}^i$ the integral over these variables is absolutely convergent. Therefore, one can interchange the integrations over $q_+^k$, $q_-^k$ and ${\alpha}^i$. As a result, we obtain the equality [$$\displaylines{\refstepcounter{equation} \label{4.18}\hskip 1em minus 1em
\hat F=\lim_{{\char'32}\to 0}\lim_{{\beta}\to 0}\lim_{{\gamma}\to 0}
\int\limits_0^\infty \prod_n d{\alpha}_i \;\hat {\varphi}({\alpha}_i,p^s,{\gamma},{\beta})
\; e^{-{\char'32}\sum_i{\alpha}_i},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where [$$\displaylines{\refstepcounter{equation} \label{4.19}\hskip 1em minus 1em
\hat {\varphi}({\alpha}_i,p^s,{\gamma},{\beta})=(-i)^n
\tilde f{\left(}-i{{{\partial}}\over{{\partial}y_i}}{\right)}\times {\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\times \int \prod_k d^2q^k \;
e^{\sum_i {\left[}i{\alpha}_i(2Q_+^iQ_-^i-M^2_i)+
i(Q^i_+y_i^++Q^i_-y_i^-)-{\gamma}{Q_+^i}^2-{\beta}{Q_-^i}^2{\right]}}
\Bigr|_{y_i=0}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} For the Lorentz-covariant calculation in the satisfying conditions (\[4.1.1\]), there is a known expression [@bo] [$$\displaylines{\refstepcounter{equation} \label{4.17}\hskip 1em minus 1em
F_{\rm cov}=\lim_{{\char'32}\to 0}
\int\limits_0^\infty \prod_n d{\alpha}_i \;{\varphi}_{\rm cov}({\alpha}_i,p^s)
\; e^{-{\char'32}\sum_i{\alpha}_i},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where [$$\displaylines{\refstepcounter{equation} \label{4.16}\hskip 1em minus 1em
{\varphi}_{\rm cov}({\alpha}_i,p^s)=(-i)^n \tilde f{\left(}-i{{{\partial}}\over
{{\partial}y_i}}{\right)}\times{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\times \lim_{{\gamma},{\beta}\to 0} \int \prod_k d^2q^k
\; e^{\sum_i {\left[}i{\alpha}_i(2Q_+^iQ_-^i-M^2_i)+
i(Q^i_+y_i^++Q^i_-y_i^-)-{\gamma}{Q_+^i}^2-{\beta}{Q_-^i}^2{\right]}}
\Bigr|_{y_i=0}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} In Appendix 2, it is shown that in (\[4.18\]) the limits in ${\gamma}$ and ${\beta}$ can be interchanged, in turn, with the integration over $\{{\alpha}_i\}$, and then with $\tilde f{\left(}-i{{{\partial}}\over{{\partial}y_i}}{\right)}$. After that, a comparison of relations (\[4.18\]), (\[4.19\]) and (\[4.17\]), (\[4.16\]), clearly shows that $\hat F=F_{\rm cov}$. Considering (\[4.8\]) and using Statement 1 from Appendix 1, we obtain the equality [$$\displaylines{\refstepcounter{equation} \label{4.20}\hskip 1em minus 1em
F_{\rm cov}=\lim_{{\char'32}\to 0}\int\prod_k dq_+^k
\int\limits_{B_L}
\prod_k dq_-^k {{\tilde f(Q^i,p^s)}
\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Expression (\[4.20\]) differs from (\[4.5\]) only by the range of the integration over $q_-^k$.
Reduction of the difference between the Light Front and[\
]{}Lorentz-covariant Feynman integrals to a sum of configurations {#polos}
-----------------------------------------------------------------
Let us introduce a partition for each line, [$$\displaylines{\refstepcounter{equation} \label{4.22}\hskip 1em minus 1em
{\left(}\int\limits_{-\infty}^{-{\varepsilon}}
dQ_- + \int\limits_{{\varepsilon}}^{\infty} dQ_- {\right)}=
{\left[}\int dQ_-+(-1)\int\limits_{-{\varepsilon}}^{{\varepsilon}} dQ_-{\right]}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} We call a line with integration w.r.t. the momentum $Q_-^i$ in the range $(-{\varepsilon},{\varepsilon})$ (before removing the ${\delta}$-functions) a type-1 line, a line with integration in the range $(-\infty,-{\varepsilon})\cup({\varepsilon},\infty)$ a type-2 line, and a line with integration over the whole range $(-\infty,\infty)$ a full line. In the diagrams, they are denoted as shown in Figs. la, b, and c, respectively.
fig1.pic
Let us substitute partition (\[4.22\]) into expression (\[4.5\]) for $F_{\rm lf}$ and open the brackets. Among the resulting terms, there is $F_{\rm cov}$ (expression (\[4.20\])). We call the remaining terms “diagram configurations” and denote them by $F_j$. Then we arrive at the relation $F_{\rm lf}-F_{\rm cov}=\sum\limits_jF_j$, where [$$\displaylines{\refstepcounter{equation} \label{4.24}\hskip 1em minus 1em
F_j=\lim_{{\varepsilon}\to 0}\lim_{{\char'32}\to 0}\int\prod_k dq_+^k
\int\limits_{V^j_{\scriptstyle {\varepsilon}}\cap B_L} \prod_k dq_-^k
{{\tilde f(Q^i,p^s)}
\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and $V^j_{{\varepsilon}}$ is the region corresponding to the arrangement of full lines and type-1 lines in the given configuration.
Note that before taking the limit in ${\varepsilon}$, equations (\[4.20\]) and (\[4.24\]) can be used successfully: first, they are applied to a subdiagram and, then, are substituted into the formula for the entire diagram. This is admissible because, after the deformation of the contours described in the proof of Statement 1 from Appendix 1, the integral over the loop momenta $\{q_+^k\}$ of the subdiagram converges (after integration over the variables $\{q_-^k\}$ of this subdiagram) absolutely and uniformly with respect to the remaining loop momenta $\{q_-^{k'}\}$. Therefore, one can interchange the integrals over $\{q_+^k\}$ and $\{q_-^{k'}\}$.
Thus, the difference between the LF and Lorentz-covariant calculations of the diagram is given by the sum of all of its configurations. A configuration of a diagram is the same diagram, but where each line is labeled as a full or type-1 line, provided that at least one type-1 line exists.
Behavior of the configuration as ${\varepsilon}\to 0$ {#epsil}
-----------------------------------------------------
We assume that all external momenta $p^s$ are fixed for the diagram in question and [$$\displaylines{\refstepcounter{equation} \label{4.D0}\hskip 1em minus 1em
p_-^s\ne 0, \quad \sum_{s'}p_-^{s'}\ne 0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the summation is taken over any subset of external momenta; all of these momenta are assumed to be directed inward.
Let us consider an arbitrary configuration. We apply the term “${\varepsilon}$-line” to all type-1 lines and those full lines for which integration over $Q_-$ actually does not expand outside the domain $(-r{\varepsilon},r{\varepsilon})$, where $r$ is a finite number (below, we explain when these lines appear). The remaining full lines are called $\Pi$-lines. In the diagrams, the ${\varepsilon}$-lines and $\Pi$-lines are denoted as shown in Figs. 1d and e, respectively. Note that the diagram can be drawn with lines “a” and “c” from Fig. 1 (this defines the configuration unambiguously), or with lines “d” and “e” (then the configuration is not uniquely defined).
If among the lines arriving at the vertex only one is full and the others are type-1 lines, this full line is an ${\varepsilon}$-line by virtue of the momentum conservation at the vertex. The remaining full lines form a subdiagram (probably unconnected). By virtue of conditions (\[4.D0\]), there is a connected part to which all of the external lines are attached. All of the external lines of the remaining connected parts are ${\varepsilon}$-lines. Consequently, using Statement 1 from Appendix 1, we can see that integration over the internal momenta of these connected parts can be carried out in a domain of order ${\varepsilon}$ in size, i.e., all of their internal lines are ${\varepsilon}$-lines. Thus, an arbitrary configuration can be drawn as in Fig. 2 and integral (\[4.24\]), with the corresponding integration domain, is associated with it.
fig2.pic
Let us investigate the behavior of the configuration as ${\varepsilon}\to 0$. From here on, it is convenient to represent the propagator as [$$\displaylines{\refstepcounter{equation} \label{4.D1}\hskip 1em minus 1em
{{\tilde z(Q)}\over {Q^2-m^2+i{\char'32}}},\quad{\rm where}\quad
\tilde z(Q)=z(Q)\; \; {\rm or}\; \; \tilde z(Q)={{z(Q)}\over
{2Q_-+i{\char'32}/Q_+}}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} rather than as (\[4.1.2\]). Then, in (\[4.1\]), $M^2_i=m^2_i+{Q^{i}_{{\bot}}}^2\ne 0$ and the function $\tilde f$ is no longer a polynomial. If the numerator of the integrand consists of several terms, we consider each term separately (except when the terms arise from expressing the propagator momentum $Q_-^i$ in terms of loop and external momenta).
We denote the loop momenta of subdiagram $\Pi$ in Fig. 2 by $q^l$ and the others by $k^m$. We make following change of integration variables in (\[4.24\]): [$$\displaylines{\refstepcounter{equation} \label{4.D10}\hskip 1em minus 1em
k_-^m \to {\varepsilon}\; k_-^m.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Then, the integration over $k_-^m$ goes within finite limits independent of ${\varepsilon}$. We denote the power of ${\varepsilon}$ in the common factor by ${\tau}$ (it stems from the volume elements and the numerators when the transformation (\[4.D10\]) is made). The contribution to ${\tau}$ from the expression $1/(2Q_-+i{\char'32}/Q_+)$ (equation (\[4.D1\])), which is related to the ${\varepsilon}$-line, is equal to -1. We divide the domain of integration over $k_+^m$ and $q_+^l$ into sectors such that the momenta of all full lines $Q_+^i$ have the same sign within one sector.
In Statement 1 of Appendix 1, it is shown that for each sector, the contours of integration over $q_-^l$ and $k_-^m$ can be bent in such a way that absolute convergence in $q_+^l$, $k_+^m$, $q_-^l$ and $k_-^m$ takes place. Since, in this case, the momenta $Q_-^i$ of $\Pi$-lines are separated from zero by an ${\varepsilon}$-independent constant, the corresponding $\Pi$-line-related propagators and factors from the vertices can be expanded in a series in ${\varepsilon}$. This expansion commutes with integration.
It is also clear that the denominators of the propagators allow the following estimates under an infinite increase in $|Q_+|$: $$\displaylines{
\hfill
\left|{1\over{2Q_+ Q_--M^2+i{\char'32}}}\right|\le \cases{
{\displaystyle {1\over{c \; |Q_+|}}} & for $\Pi$-lines,
\refstepcounter{equation} \label{4.D11} \hskip 30mm \hfill (\theequation)\cr
{\displaystyle {1\over{\tilde c\; {\varepsilon}\; |Q_+|}}} & for ${\varepsilon}$-lines,
\refstepcounter{equation} \label{4.D12} \hfill (\theequation)\cr}
}$$ Here $c$ and $\tilde c$ are ${\varepsilon}$-independent constants. Note that for fixed finite $Q_+$, the estimated expressions are bounded as ${\varepsilon}\to 0$. After transformation (\[4.D10\]) and release of the factor ${1\over{{\varepsilon}}}$ (in accordance with what was said about the contribution to ${\tau}$), the ${\varepsilon}$-line-related expression from (\[4.D1\]) becomes [$$\displaylines{\refstepcounter{equation} \label{nnn7}\hskip 1em minus 1em
\left| {1\over{2Q_-+i{\char'32}/Q_+}}\right| \to
\left| {1\over{2Q_-+i{\char'32}/(Q_+{\varepsilon})}}\right| \le
{1\over{2|Q_-|}},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where a $Q_+$-independent quantity was used for the estimate (this quantity is meaningful and does not depend on ${\varepsilon}$ because the value of $Q_-$ is separated from zero by an ${\varepsilon}$-independent constant).
We integrate first over $q_+^l$, $k_+^m$ within one sector and then over $q_-^l$, $k_-^m$ (the latter integral converges uniformly in ${\varepsilon}$). Let us examine the convergence of the integral over $q_+^l$, $k_+^m$ with canceled denominators of the ${\varepsilon}$-lines (which is equivalent to estimating the expressions (\[4.D12\]) by a constant). If it converges, then the initial integral is obviously independent of ${\varepsilon}$ and the contribution from this sector to the configuration is proportional to ${\varepsilon}^{\tau}$.
Let us show that if it diverges with a degree of divergence ${\alpha}$, the contribution to the initial integral is proportional to ${\varepsilon}^{{\tau}-{\alpha}}$ up to logarithmic corrections. To this end, we divide the domain of integration over $q_+^l$, $k_+^m$ into two regions: $U_1$, which lies inside a sphere of radius ${\Lambda}/{\varepsilon}$ (${\Lambda}$ is fixed), and $U_2$, which lies outside this sphere (recall that in our reasoning, we deal with each sector separately). Now we estimate (\[4.D11\]) (like (\[4.D12\])) in terms of ${\displaystyle{1\over{\hat c{\varepsilon}|Q_+|}}}$ (which is admissible) and change the integration variables as follows: [$$\displaylines{\refstepcounter{equation} \label{4.D13}\hskip 1em minus 1em
q_+^l\to {1\over{{\varepsilon}}}\; q_+^l,\quad k_+^m\to {1\over{{\varepsilon}}}\; k_+^m.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} After ${\varepsilon}$ is factored out of the numerator and the volume element, the integrand becomes independent of ${\varepsilon}$. Thus, the integral converges.
One can choose such ${\Lambda}$ (independent of ${\varepsilon}$) that the contribution from the domain $U_2$ is smaller in absolute value than the contribution from the domain $U_1$. Consequently, the whole integral can he estimated via the integral over the finite domain $U_1$. Now we make an inverse replacement in (\[4.D13\]) and estimate (\[4.D12\]) by a constant (as above). Since the size of the integration domain is ${\Lambda}/{\varepsilon}$ and the degree of divergence is ${\alpha}$, the integral behaves as ${\varepsilon}^{-{\alpha}}$ (up to logarithmic corrections), q.e.d. This reasoning is valid for each sector and, thus, for the configuration as a whole. Obviously, [$$\displaylines{\refstepcounter{equation} \label{4.D13.1}\hskip 1em minus 1em
{\alpha}=\max\limits_{r}{\alpha}_r,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where ${\alpha}_r$ is the subdiagram divergence index and the maximum is taken over all subdiagrams $D_r$ (including unconnected subdiagrams for which ${\alpha}_r$ is the sum of the divergence indices of their connected parts). In the case under consideration, ${\alpha}_r={\omega}_+^r+{\nu}^r$, where ${\nu}^r$ is the number of internal ${\varepsilon}$-lines in the subdiagram $D_r$. The quantities ${\omega}_{\pm}^r$ are the UV divergence indices of the subdiagram $D_r$ w.r.t. $Q_{\pm}$.
Above, we introduced a quantity ${\tau}$, which is equal to the power of ${\varepsilon}$ that stems from the numerators and volume elements of the entire configuration. We can write ${\tau}={\omega}_-^r-{\mu}^r+{\nu}^r+\eta^r$, where ${\mu}^r$ is the index of the UV divergence in $Q_-$ of a smaller subdiagram (probably, a tree subdiagram or a nonconnected one) consisting of $\Pi$-lines entering $D_r$. The term $\eta^r$ is the power of ${\varepsilon}$ in the common factor, which, during transformation (\[4.D10\]), stems from the volume elements and numerators of the lines that did not enter $D_r$. (It is implied that the integration momenta are chosen in the same way as when calculating the divergence indices of $D_r$.) Then, up to logarithmic corrections, we have [$$\displaylines{\refstepcounter{equation} \label{4.13.4}\hskip 1em minus 1em
F_j\sim {\varepsilon}^{{\sigma}},\quad {\sigma}=\min_r({\tau},{\omega}_-^r-{\omega}_+^r-{\mu}^r+\eta^r).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Consequently, for ${\varepsilon}\to 0$, the configuration is equal to zero if ${\sigma}>0$. Relation (\[4.13.4\]) allows all essential configurations to be distinguished.
Correction procedure and analysis of counter-terms {#ispra}
--------------------------------------------------
We want to build a corrected LF Hamiltonian $H_{\rm lf}^{\rm cor}$ with the cutoff $|Q_-^i|>{\varepsilon}$, which would generate Green’s functions that coincide in the limit ${\varepsilon}\to 0$ with covariant Green’s functions within the perturbation theory. We begin with a usual canonical Hamiltonian in the LF coordinates $H_{\rm lf}$ with the cutoff $|Q_-^i|>{\varepsilon}$. We imply that the integrands of the Feynman diagrams derived from this LF Hamiltonian coincide with the covariant integrands after some resummation [@bur1; @har; @lay]. However, a difference may arise due to the various methods of doing the integration, e.g., due to different auxiliary regularizations. As shown in Sec. \[polos\], this difference (in the limit ${\varepsilon}\to 0$) is equal to the sum of all properly arranged configurations of the diagram. One should add such correcting counter-terms to $H_{\rm lf}$, which generates additional “counter-term” diagrams, that reproduce nonzero (after taking limit w.r.t. ${\varepsilon}$) configurations of all of the diagrams. Were we able to do this, we would obtain the desired $H_{\rm lf}^{\rm cor}$. In fact, we can only show how to seek the $H_{\rm lf}^{\rm cor}$ that generates the Green’s functions coinciding with the covariant ones everywhere except the null set in the external momentum space (defined by condition (\[4.D0\])). However, this restriction is not essential because this possible difference does not affect the physical results.
Our correction procedure is similar to the renormalization procedure. We assume that the perturbation theory parameter is the number of loops. We carry out the correction by steps: first, we find the counterterms to the Hamiltonian that generate all nonzero configurations of the diagrams up to the given order and, then, pass to the next order. We take into account that this step involves the counter-term diagrams that arose from the counter-terms added to the Hamiltonian for lower orders. Thus, at each step, we introduce new correcting counter-terms that generate the difference remaining in this order. Let us show how to successfully look for the correcting counter-terms.
We call a configuration nonzero if it does not vanish as ${\varepsilon}\to 0$. We call a nonzero configuration “primary” if $\Pi$ is a tree subdiagram in it (see Fig. 2). Note that for this configuration, breaking any $\Pi$-line results in a violation of conditions (\[4.D0\]); then, the resulting diagram is not a configuration. We say that the configuration is changed if all of the $\Pi$-lines in the related integral (\[4.24\]) are expanded in series in ${\varepsilon}$ (see the reasoning above equation (\[4.D11\]) in Sec. \[epsil\]) and only those terms that do not vanish in the limit ${\varepsilon}\to 0$ after the integration are retained. As mentioned above, developing this series and integration are interchangeable operations. Thus, in the limit ${\varepsilon}\to 0$, the changed and unchanged configurations coincide. Therefore, we always require that the Hamiltonian counter-terms generate changed configurations, as this simplifies the form of the counter-terms. Using additional terms in the Hamiltonian, one can generate only counter-term diagrams, which are equal to zero for external momenta meeting the condition $|p_-^s|<{\varepsilon}$, because with the cutoff used, the external lines of the diagrams do not carry momenta with $|p_-^s|<{\varepsilon}$. We bear this in mind in what follows.
We seek counter-terms by the induction method. It is clear that, in the first order in the number of loops, all nonzero configurations are primary. We add the counter-terms that generate them to the Hamiltonian. Now, we examine an arbitrary order of perturbation theory. We assume that in lower orders, all nonzero configurations that can be derived from the counter-terms, accounting for the above comment, have already been generated by the Hamiltonian.
Let us proceed to the order in question. First, we examine nonzero configurations with only one loop momentum $k$ and a number of momenta $q$ (see the notation above equation (\[4.D10\])). We break the configuration lines one by one without touching the other lines (so that the ends of the broken lines become external lines). The line break may result in a structure that is not a configuration (if conditions (\[4.D0\]) are violated); a line break may also result in a zero configuration or in a nonzero configuration. If the first case is realized for each broken line, then the initial configuration is primary and it must be generated by the counter-terms of the Hamiltonian in the order under consideration. If breaking of each line results in either the first or the second case, we call the initial configuration real and it must be also generated in this order.
Assume that breaking a line results in the third case. This means that the resulting configuration stems from counter-terms in the lower orders. Then, after restoration of the broken line (i.e., after the appropriate integration), it turns out that the counter-terms of the lower orders have generated the initial configuration (we take into account the comment on successive application of equation (\[4.24\]); see the end of Sec. \[polos\]) with the following distinctions: (i) the broken line (and, probably, some others, if a nonsimply connected diagram arises after breaking the line) is not a $\Pi$-line but a type-2 line, due to the conditions $|p_-^s|>{\varepsilon}$; (ii) if, after restoration of the broken line, the behavior at small ${\varepsilon}$ becomes worse (i.e., ${\sigma}$ decreased), then fewer terms than are necessary for the initial configuration were considered in the above-mentioned series in ${\varepsilon}$. We expand these arising type-2 lines by formula (\[4.22\]) and obtain a term where all of these lines are replaced by $\Pi$-lines or other terms where some (or all) of these lines have become type-1 lines. In the latter case, one of the momenta $q$ becomes the momentum $k$. We call these terms “repeated parts of the configuration” and analyze them together with the configurations that have two momenta $k$. In the former case, we obtain the initial configuration up to distinction (ii). We add a counter-term to the Hamiltonian that compensates this distinction (the counter-term diagrams generated by it are called the compensating diagrams).
If there is only one line for which the third case is realized, it turns out that, in the given order, it is not necessary to generate the initial configuration by the counter-terms, except for the compensating addition and the repeated part that is considered at the next step. If there are several lines for which the third case is realized, the initial configuration is generated in lower orders more than once. For compensation, it should be generated (with the corresponding numerical coefficient and the opposite sign) by the Hamiltonian counter-terms in the given order. We call this configuration a secondary one. Next, we proceed to examine configurations with two momenta $k$ and so on up to configurations with all momenta $k$, which are primary configurations.
Thus, the configurations to be generated by the Hamiltonian counter-terms can be primary (not only the initial primary configurations but also the repeated parts analogous to them, called primary-like), real, compensating, and secondary. If the theory does not produce either the loop consisting only of lines with $Q_+$ in the numerator (accounting for contributions from the vertices) or a line with ${Q_+}^n$ in the numerator for $n>1$, then real configurations are absent because a line without $Q_+$ in the numerator can always be broken without increasing ${\sigma}$ (see equation (\[4.13.4\])). It is not difficult to demonstrate that if each appearing primary, real, and compensating configuration has only two external line, then there are no secondary configurations at all.
The dependence of the primary configuration on external momenta becomes trivial if its degree of divergence ${\alpha}$ is positive, the maximum in formula (\[4.D13.1\]) is reached on the diagram itself, and ${\sigma}=0$. Then, only the first term is taken into account in the above-mentioned series. Thus, not all of the $\Pi$-line-related propagators and vertex factors depend on $k_-^m$ and they can be pulled out of the sign of the integral w.r.t. $\{k_-^m\}$ in (\[4.24\]). We then obtain [$$\displaylines{\refstepcounter{equation} \label{4.D14}\hskip 1em minus 1em
F_j^{\rm prim}=\lim_{{\varepsilon}\to 0}\lim_{{\char'32}\to 0}\int\prod_m dk_+^m
{{\tilde f'(k^m,p^s)}\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}} \times{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\times \int\limits_{V_{\scriptstyle {\varepsilon}}} \prod_m dk_-^m
{{\tilde f''(k^m)} \over{\prod_k (2Q_+^kQ_-^k-M^2_k+i{\char'32})}},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $V_{{\varepsilon}}$ is a domain of order ${\varepsilon}$ in size. Let us carry out transformations (\[4.D10\]) and (\[4.D13\]). For the denominator of the $\Pi$-line, we obtain [$$\displaylines{\refstepcounter{equation} \label{nnn8}\hskip 1em minus 1em
{1\over{2({1\over{{\varepsilon}}}\sum k_+ +\sum p_+)(\sum p_-)-M^2+i{\char'32}}}\to
{{{\varepsilon}}\over{2(\sum k_+)(\sum p_-)}}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Here we neglect terms of order ${\varepsilon}$ in the denominator because the singularity at $k_+^m=0$ is integrable under the given conditions for ${\alpha}$ and everything can be calculated in zero order in ${\varepsilon}$ at ${\sigma}=0$. Thus, the dependence on external momenta can be completely collected into an easily obtained common factor.
Application to the Yukawa model {#jukav}
-------------------------------
The Yukawa model involves diagrams that do not satisfy condition (\[4.1.1\]). These are displayed in Figs. 3a and b. We have ${\omega}_{{\scriptscriptstyle \|}}=0$ for diagram “a” and ${\omega}_+=0$ for diagram “b”.
fig3.pic
Nevertheless, these diagrams can be easily included in the general scheme of reasoning. To this end, one should subtract the divergent part, independent of external momenta, in the integrand of the logarithmically divergent (in two-dimensional space, with fixed internal transverse momenta) diagram “a”. We obtain an expression with ${\omega}_{{\scriptscriptstyle \|}}<0$ (i.e., which converges in two-dimensional space) and ${\omega}_+=0$, as in diagram “b”. This means that the integral over $q_+$ converges only in the sense of the principal value (and it is this value of the integral that should be taken in the LF coordinates to ensure agreement with the stationary noncovariant perturbation theory). This value can be obtained by distinguishing the $q_+$-even part of the integrand.
Two approaches are possible. One is to introduce an appropriate regularization in transverse momenta and to imply integration over them; then, it is convenient to distinguish the part that is even in four-dimensional momenta $q$. The other is to keep all transverse momenta fixed; then, the part that is even in longitudinal momenta $q_{{\scriptscriptstyle \|}}$ can be released. For the Yukawa theory, we use the first approach. For the transverse regularization, we use a “smearing” of vertices, which is equivalent to dividing each propagator by $1+{Q_{{\bot}}^i}^2/{{\Lambda}_{{\bot}}}^2$. In four-dimensional space, diagram “a” diverges quadratically. Under introduction and subsequent removal of the transverse regularization, the divergent part, which was previously subtracted from this diagram, acquires the form $C_1+C_2\> p_{{\bot}}^2$.
After separating the even part of the regularized expression, we fix all of the transverse momenta again. Then it turns out that diagrams “a” and “b” in Fig. 3 meet conditions (\[4.1.1\]) and one can show that after all of the operations mentioned, the exponent ${\sigma}$ (see (\[4.13.4\])) does not decrease for any of their configurations. Hence, they can be included in the general scheme without any additional corrections.
Let us first analyze the primary configurations (see the definition in Sec. \[ispra\]). In the numerators, $k_-$ appears only in the zero or one power and there are no loops where the numerators of all of the lines contain $k_-$. Consequently, one always has ${\tau}>0$, ${\mu}^r\le 0$, and $\eta^r\ge 0$ (see the definitions in Sec. \[epsil\]). Analyzing the properties of the expression ${\omega}_-^r-{\omega}_+^r$ for the Yukawa model diagrams, we conclude from (\[4.13.4\]) that ${\sigma}\ge 0$ always holds. The general form of the nonzero primary configurations with ${\sigma}=0$ is depicted in Fig. 4. Note that they are all configurations with two external line.
fig4.pic
Further, it is clear that there are no nonzero real configurations (see the comment at the end of Sec. \[ispra\]), and it can be shown by induction that there are no nonzero compensating or secondary configurations either (the definitions are given in Sec. \[ispra\] also). Thus, only primary or primary-like configurations can be nonzero and all of them have the form shown in Fig. 4. It can be shown that their degree of divergence ${\alpha}$ is positive and the maximum in formula (\[4.D13.1\]) is reached for the diagram itself. Thus, the reasoning above and below formula (\[4.D14\]) applies to them. Then, denoting the configurations displayed in Figs. 4a-d by , we arrive at the equalities , , and , where the expressions depend only on the masses and transverse momenta, but not on the external longitudinal momenta, and have a finite limit as ${\varepsilon}\to 0$.
Now we assume that $D_{a}$ – $D_{d}$ are not single configurations but are the sums of all configurations of the same form and that integration over the internal transverse momenta has already been carried out, (with the above-described regularization). In four-dimensional space, the diagrams $D_{a}$ and $D_{b}$ diverge linearly while $D_{c}$ and $D_{d}$ diverge quadratically. Therefore, because of the transverse regularization, the coefficients $C_c$ and $C_d$ in the limit of removing this regularization take the form $C_1+C_2\> p_{{\bot}}^2$, where $C_1$ and $C_2$ do not depend on the external momenta (neither do $C_a$, $C_b$)). Thus, to generate all nonzero configurations by the LF Hamiltonian, only the expression [$$\displaylines{\refstepcounter{equation} \label{4.U1}\hskip 1em minus 1em
H_c=\tilde C_1\; {\varphi}^2+\tilde C_2\> p_{{\bot}}^2\; {\varphi}^2+
\tilde C_3\; \bar \psi\; {{{\gamma}^+}\over{p_-}}\; \psi,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} should be added, where ${\varphi}$ and $\psi$ are the boson and fermion fields, respectively, and $\tilde C_i$, are the constant coefficients.
Comparing (\[4.U1\]) with the initial canonical LF Hamiltonian, one can easily see that the found counter-terms are reduced to a renormalization of various terms of the Hamiltonian (in particular the boson mass squared and the fermion mass squared without changing the term linear in fermion mass). The explicit Lorentz invariance is absent but results are Lorentz invariant because the counterterms compensate the violation of Lorentz invariance inherent to chosen LF formalism.
Note that the second approach, mentioned at the beginning of this section, can give the same results. The only difference is that in two-dimensional space, the contributions from the configurations displayed in Fig. 3 would additionally depend on external transverse momenta. However, this dependence disappears after integration over internal transverse momenta with the introduction and subsequent removal of an appropriate regularization.
In the Pauli Villars regularization, it is easy to verify that the expression ${\omega}_-^r-{\omega}_+^r-{\mu}^r+\eta^r$ from (\[4.13.4\]) increases. This is because the number of terms in the numerators of the propagator increases. Then, the contribution from the ${\varepsilon}$-lines does not change, while the $\Pi$-lines belonging to $D_r$ make zero contribution to ${\omega}_-^r-{\omega}_+^r$ and $\eta^r$, but $-1$ contribution to ${\mu}^r$. Since ${\tau}>0$, this regularization makes it possible to meet the condition ${\sigma}>0$ for the configurations that were nonzero (one additional boson field and one additional fermion field are enough). Then it turns out that the canonical LF Hamiltonian need not be corrected at all.
Obtained results agree with the conclusions of the work [@bur1], where a comparison of LF and Lorentz-covariant methods was made for self-energy diagrams in all orders of perturbation theory and for other diagrams in lowest orders.
Application to QCD(3+1) {#kalib}
-----------------------
Applying the LF Hamiltonian approach to gauge theories under regularization (\[1.1\]), where zero modes of fields are thrown out, one has to use the gauge $A_-=0$ (see, for example, sect. 2.3). To carry out successfully renormalization procedure for this scheme it is necessary to take gauge field propagator in accordance with Mandelstam-Leibbrandt prescription [@man; @lei] (which allows to perform Euclidean continuation, see [@bas1; @bas2; @skark]). Such a propagator has the form [$$\displaylines{\refstepcounter{equation} \label{4n1}\hskip 1em minus 1em
\frac{-i{\delta}^{ab}}{Q^2+i{\char'32}}
{\left(}g_{{\mu}{\nu}}-\frac{Q_\mu{\delta}_\nu^++Q_\nu{\delta}_\mu^+}{2Q_+Q_-+i{\char'32}}2Q_+{\right)}=
\frac{-i{\delta}^{ab}}{Q^2+i{\char'32}}
{\left(}g_{{\mu}{\nu}}-\frac{Q_{\mu}n_{\nu}+Q_{\nu}n_{\mu}}{2(Qn^*)(Qn)+i{\char'32}}2(Qn^*){\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $n_+=1$, $n_-=n_{\bot}=0$, $n^*_-=1$, $n^*_+=n^*_{\bot}=0$.
The formalism described in sect. 4.1-4.4 was such that it could be applied to a theory with the propagator (\[4n1\]) (at fixed transverse momenta $Q_{{\bot}}\ne 0$). It turns out that there are nonzero configurations with arbitrarily large numbers of external lines. An example of such a configuration is given in Fig. 5.
fig5.pic
Indeed, using formula (\[4.13.4\]), we can see that for the configuration in Fig. 5, ${\tau}=0$ and, thus, ${\sigma}\le 0$, i.e., this is a nonzero configuration. It is also clear that introduction of the Pauli-Villars regularization does not improve the situation because it does not affect ${\tau}$.
The difficulty is that the distortion of the pole in (\[4n1\]) due to LF cutoff $|Q_-|\ge{\varepsilon}>0$ does not disappear in the limit ${\varepsilon}\to 0$, and infinite number of new counterterms are required to compensate this distortion [@tmf97]. The simplest way to avoid this difficulty is to add small mass-like parameter $\mu^2$ in the denominator: [$$\displaylines{\refstepcounter{equation} \label{4n16}\hskip 1em minus 1em
\frac{1}{2Q_+Q_-+i{\char'32}}\longrightarrow \frac{1}{2Q_+Q_--\mu^2+i{\char'32}},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and take the limit ${\varepsilon}\to 0$ before $\mu\to 0$). To describe this modification with local Lagrangian we need to introduce ghost fields $A'_{\mu}$ in addition to conventional $A_{\mu}$. We write the free part of pure gluon Lagrangian as follows (using higher derivatives and the parameter ${\Lambda}$ for UV regularization): [$$\displaylines{\refstepcounter{equation} \label{4n17}\hskip 1em minus 1em
L_0=-\frac{1}{4}\Biggl( f^{a,\mu{\nu}} {\left(}1+\frac{{\partial}^2}{{\Lambda}^2}{\right)}f_{\mu{\nu}}^a-f'^{a,\mu{\nu}}
{\left(}1+\frac{{\partial}^2}{{\Lambda}^2}+\frac{2{\partial}_+{\partial}_-}{\mu^2}\Biggr)
f'^a_{\mu{\nu}}{\right)},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $f^a_{\mu{\nu}}={\partial}_\mu A^a_{{\nu}}-{\partial}_{\nu}A^a_{\mu}$, $f'^a_{\mu{\nu}}={\partial}_\mu A'^a_{{\nu}}-{\partial}_{\nu}A'^a_{\mu}$ and $A^a_-=A'^a_-=0$. Interaction terms depend only on summary field $\bar A^a_{\mu}=A^a_{\mu}+A'^a_{\mu}$.
At fixed $\mu$ and ${\Lambda}$ we get a theory with broken gauge invariance but with preserved global $SU(3)$-invariance. We put into the Lagrangian all necessary interaction terms (but with unknown coefficients) including those that are needed for UV renormalization: [$$\displaylines{\refstepcounter{equation} \label{4n18}\hskip 1em minus 1em
L=L_0+c_0{\partial}_\mu\bar A^a_{\nu}{\partial}^\mu\bar A^{a,{\nu}}+
c_{01}{\partial}_\mu\bar A^a_{\nu}n^{\mu}n^{*{\alpha}}{\partial}^{\alpha}\bar A^{a,{\nu}}+
c_1{\partial}_\mu\bar A^{a,\mu}\,{\partial}_{\nu}\bar A^{a,{\nu}}+{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}}+
c_{11}n^{\mu}n^{*{\alpha}}{\partial}_\mu\bar A^{a,{\alpha}}\,{\partial}_{\nu}\bar A^{a,{\nu}}+
c_{12}n^{\mu}n^{*{\alpha}}{\partial}_\mu\bar A^{a,{\alpha}}
n^{\nu}n^{*{\beta}}{\partial}_{\nu}\bar A^a_{\beta}+
c_2\bar A^a_\mu\bar A^{a,\mu}+{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}+
c_3 f^{abc}\bar A^a_\mu\bar A^b_{\nu}{\partial}^\mu\bar A^{c,{\nu}}+
c_{31} f^{abc}\bar A^a_\mu\bar A^b_{\nu}n^{\alpha}n^{*{\mu}}{\partial}^{\alpha}\bar A^{c,{\nu}}+{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}+
\bar A^a_\mu\bar A^b_{\nu}\bar A^c_{\gamma}\bar A^d_{\delta}\bigl(c_4 f^{abe}f^{cde} g^{\mu{\gamma}}g^{{\nu}{\delta}}+
{\delta}^{ab}{\delta}^{cd}{\left(}c_5 g^{\mu{\gamma}}g^{{\nu}{\delta}}+
c_6 g^{\mu{\nu}}g^{{\gamma}{\delta}}\bigr){\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} These terms are local and can be taken in Lorentz covariant form due to the restoring of this symmetry in the ${\varepsilon}\to 0$, $\mu \to 0$ limit. However in constructing of these terms both vectors $n^{\mu}$ and $n^{*{\nu}}$, included in the definition of the propagator (\[4n1\]), can participate.
For this theory one can apply the formalism described in sect. 4.1-4.4 to compare (at ${\varepsilon}\to 0$) the LF perturbation theory and that taken in Lorentz coordinates within the same regularization scheme. We find that the difference between mentioned perturbation theories can be compensated by changing of the value of coefficient $c_2$ before the term of gluon mass form $\bar A^a_\mu\bar A^{\mu,a}$ in naive LF Hamiltonian of this theory. After that we can analyse further our regularized theory in Lorentz coordinates and even make Euclidean continuation.
It is possible to prove by induction to all orders of perturbation theory that in the limit $\mu\to 0$, $\Lambda\to\infty$ our theory can be made finite and coinciding with the usual renormalized (dimensionally regularized) theory in Light Cone gauge [@bas1; @bas2] (for all Green functions). This was done in the paper [@tmf99], but there the terms of the Lagrangian, including the vector $n^{*{\mu}}$, were missed. Right expression for LF Hamiltonian should correctly take into account the contribution of all terms, written in the (\[4n18\]).
The values of the unknown coefficients $c_i$ before all counterterms in (\[4n18\]) must be chosen so that the Green functions in each order coincided (after removing the regularization) with those obtained in conventional dimensionally regularized formulation and therefore satisfied Ward identities. Besides, we need to correlate the limits $\mu\to 0$ and ${\Lambda}\to\infty$ to avoid infrared divergencies at $\mu\to 0$. It is sufficient to take $\mu=\mu({\Lambda})$ and to require that $\mu{\Lambda}\to
0$ and $(\log\mu)/{\Lambda}\to 0$.
Our resulting LF Hamiltonian for pure $SU(3)$ gluon fields contains 11 unknown coefficients, including coefficient before gluon mass term that takes also into account the difference between LF and Lorentz coordinate formulations of our regularized theory. The generalization of our scheme for full QCD with fermions is described in [@tmf99]. In this case there are 20 unknown coefficients in the LF Hamiltonian. We hope that it is possible to find an analog of Ward identities relating the coefficients $c_i$ at fixed ${\Lambda}$. This problem seems very important for our approach.
Application to QED(1+1)
-----------------------
The procedure, described in previous section, allows to construct LF Hamiltonian for QCD in four-dimensional space-time, but such LF Hamiltonian contains many additional fields and unknown coefficients. This complicates calculations with this Hamiltonian. Moreover, because only the perturbation theory with respect to the coupling constant was analyzed, there could remain purely nonperturbative effects that are not taken into consideration. It is therefore useful to consider an example of “nonperturbative” (with respect to usual coupling constant) construction of LF Hamiltonian for gauge theory, which is possible for two-dimensional quantum electrodynamics (QED(1+1)), i.e. for “massive” Schwinger model.
The QED(1+1), defined originally by the Lagrangian [$$\displaylines{\refstepcounter{equation} \label{4d1}\hskip 1em minus 1em
L=-\frac{1}{4}F_{\mu{\nu}}F^{\mu{\nu}}+\bar\Psi(i{\gamma}^mD_\mu-M)\Psi,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} can be transformed to its bosonized form [@colm; @naus], described by scalar field Lagrangian [$$\displaylines{\refstepcounter{equation} \label{4d2}\hskip 1em minus 1em
L=\frac{1}{2}{\left(}{\partial}_\mu\Phi{\partial}^\mu\Phi-m^2\Phi^2{\right)}+
\frac{Mme^C}{2\pi}\cos({\theta}+\sqrt{4\pi}\Phi),
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $m=e/\sqrt{\pi}$ is the Schwinger boson mass (the $e$ is original coupling), $C=0.577216\dots$ is the Euler constant, and the ${\theta}$ is the “${\theta}$”-vacuum parameter, which takes into account the nontriviality of QED(1+1) quantum vacuum due to instantons. Here the fermion mass $M$ plays the role of the coupling in bosonized theory so that perturbation theory in this coupling corresponds to chiral perturbation theory in QED(1+1). The nonpolynomial form of scalar field interaction leads in perturbation theory to infinite sums of diagrams in each finite order. It can be proved [@shw2; @tmf03] that some partial sums of these infinite sums are UV divergent in the 2nd order, whereas for full (Lorentz-covariant) Green functions these divergencies cancel (remaining only the divergent vacuum diagrams). Therefore physical quantities are UV finite in this theory. Only at intermediate steps of our analysis we need some UV regularization.
We compare LF and Lorentz-covariant perturbation theories for such bosonized model using an effective resummation of perturbation series in coordinate representation for Feynman diagrams [@shw2; @tmf02] and also using the formalism described in sect. 4.1-4.4. The results of this comparison can be formulated as follows.
The difference between considered perturbation theories can be eliminated in the limit of removing regularizations if we use instead of the naive LF Hamiltonian [$$\displaylines{\refstepcounter{equation} \label{4d3}\hskip 1em minus 1em
H=\int dx^-{\left(}\frac{1}{8\pi}\,m^2:{\varphi}^2:
-\frac{{\gamma}}{2}\,e^{i{\theta}}:e^{i{\varphi}}:-
\frac{{\gamma}}{2}\,e^{-i{\theta}}:e^{-i{\varphi}}:{\right)},{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}{\gamma}=\frac{Mme^C}{2\pi},\quad
{\varphi}=\sqrt{4\pi}\,\Phi,\quad
|p_-|\ge{\varepsilon}>0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} the “corrected” LF Hamiltonian: [$$\displaylines{\refstepcounter{equation} \label{4d4}\hskip 1em minus 1em
H=\int dx^-{\left(}\frac{1}{8\pi}\,m^2:{\varphi}^2:
-B:e^{i{\varphi}}:-B^*:e^{-i{\varphi}}:{\right)}-{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}}-2\pi e^{-2C}\frac{|B|^2}{m^2}
\int dx^-dy^- {\left(}:e^{i{\varphi}(x^-)}e^{-i{\varphi}(y^-)}:-1{\right)}{\theta}(|x^--y^-|-{\alpha})
\frac{v({\varepsilon}(x^--y^-))}{|x^--y^-|}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Here the terms, linear in $B$ and $B^*$ (new coupling constants), are of the same form as in naive Hamiltonian; only the term, containing the $|B|^2$, is of new form (nonlocal in $x^-$). The ${\alpha}$ is the UV regularization parameter, and the $v(z)$ is some arbitrary continuous function rapidly decreasing at the infinity and going to unity as $z\to 0$. The coupling $B$ can be perturbatively written as a series in ${\gamma}$: [$$\displaylines{\refstepcounter{equation} \label{4d5}\hskip 1em minus 1em
B=\frac{{\gamma}}{2}e^{i{\theta}}+\sum_{n=2}^{\infty}{\gamma}^nB_n.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} On the other side, it is related to the sum of all connected “generalized tadpole” diagrams (i.e. diagrams with external lines attached to only one vertex), which is described by the “condensate” parameter $A=\frac{{\gamma}}{2}\langle{\Omega}|:e^{i({\varphi}+{\theta})}:|{\Omega}{\rangle}$ of the Lorentz-covariant formulation (the $|{\Omega}{\rangle}$ is the physical vacuum state in this formulation): [$$\displaylines{\refstepcounter{equation} \label{4d5.1}\hskip 1em minus 1em
B+|B|^2w=A,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{4d6}\hskip 1em minus 1em
w=\frac{2\pi e^{-2C}}{m^2}\int dx^-\frac{{\theta}(|x^-|-{\varepsilon}{\alpha})}{|x^-|}v(x^-).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The equation (\[4d5.1\]) can be solved with respect to the $B$: [$$\displaylines{\refstepcounter{equation} \label{4d6.1}\hskip 1em minus 1em
B=-\frac{1}{2w}+\sqrt{\frac{1}{4w^2}+\frac{A'}{w}-A''^2}+iA'',
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $A=A'+iA''$, and the sign before the root respects the perturbation theory. Within the perturbation theory in ${\gamma}$ one cannot remove UV regularization (i. e. to put ${\alpha}\to 0$ and therefore $w\to\infty$) in this expression due to UV divergencies of the coefficients $B_n$. However, taking into account the validity of the equation (\[4d5.1\]) to all orders in ${\gamma}$, we can consider it beyond the perturbation theory. Then we use the estimation for the $A$ at ${\alpha}\to 0$ [@shw2]: [$$\displaylines{\refstepcounter{equation} \label{4d6.2}\hskip 1em minus 1em
A=\frac{{\gamma}^2}{4}w+const
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and get for the $B$ in ${\alpha}\to 0$ limit UV finite result: [$$\displaylines{\refstepcounter{equation} \label{4d7}\hskip 1em minus 1em
B={\rm sign}(\cos{\theta})\sqrt{\frac{{\gamma}^2}{4}-A''^2}+iA''=
\frac{{\gamma}}{2}e^{i\hat{\theta}},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} so that all information about the condensate is contained in the phase factor $e^{i\hat{\theta}}$: [$$\displaylines{\refstepcounter{equation} \label{4d8}\hskip 1em minus 1em
\sin\hat{\theta}=2\frac{{\rm Im}A}{{\gamma}}=\langle{\Omega}|:\sin({\varphi}+{\theta}):|{\Omega}{\rangle}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
Then we can make a transformation, inverse to the bosonization, but on the LF. Actually we need the expression only for one independent component $\psi_+ $ of the bispinor field ${\left(}\psi_+\atop\psi_-{\right)}$ due to the LF constraint, permitting to write the $\psi_-$ in terms of $\psi_+$. One can use the exact expression for the $\psi_+$ in terms of the ${\varphi}$ obtained in the theory on the interval $|x^-|\le L$ with periodic boundary conditions [@naus; @shw2]. We need only to modify our corrected bosonized theory by using discretized LF momentum $p_-$ instead of continuous one and hence replacing the cutoff parameter ${\varepsilon}$ by $\pi/L$. The necessary formulae for the $\psi_+$ has the following form [@shw2; @naus] (we choose here antiperiodic boundary conditions for the fermion fields): [$$\displaylines{\refstepcounter{equation} \label{4d9}\hskip 1em minus 1em
\psi_+(x)=\frac{1}{\sqrt{2L}}e^{-i {\omega}}e^{-i\frac{\pi}{L}x^- Q}
e^{i\frac{\pi}{2L}x^-}:e^{-i{\varphi}(x)}:.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} The operator ${\omega}$ and the charge operator $Q$ are canonically conjugated so that the $\psi_+$ defined by the equation (\[4d9\]) has proper commutation relation with the charge. On the other side the operator $e^{i{\omega}}$ shifts Fourier modes $\psi_n$ of the field $\psi_+$ [@shw2; @naus]: [$$\displaylines{\refstepcounter{equation} \label{4d10}\hskip 1em minus 1em
e^{i{\omega}}\psi_n e^{-i{\omega}}=\psi_{n+1}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} If we separate the modes related with creation and annihilation operators on the LF putting [$$\displaylines{\refstepcounter{equation} \label{4d11}\hskip 1em minus 1em
\psi_+(x)=\frac{1}{\sqrt{2L}}\Biggl(
\sum_{n\ge 1}b_ne^{-i\frac{\pi}{L}(n-\frac{1}{2})x^-}+
\sum_{n\ge 0}d_n^+ e^{i\frac{\pi}{L}(n+\frac{1}{2})x^-}\Biggr),\quad
b_n|0{\rangle}=d_n|0{\rangle}=0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} we can define the operator $e^{i{\omega}}$ uniquely by specifying its action on the LF vacuum $|0{\rangle}$ as follows: [$$\displaylines{\refstepcounter{equation} \label{4d12}\hskip 1em minus 1em
e^{i {\omega}}|0{\rangle}=b^+_1 |0{\rangle},\qquad
e^{-i {\omega}}|0{\rangle}=d^+_0 |0{\rangle}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} In such sense this operator is similar to a fermion.
We can now rewrite our corrected boson LF Hamiltonian in terms of ${\psi}_+$ and $e^{i{\omega}}$. The result is remarkably simple: [$$\displaylines{\refstepcounter{equation} \label{4d13}\hskip 1em minus 1em
H=\int\limits_{-L}^Ldx^-
\biggl(\frac{e^2}{2}{\left(}{\partial}_-^{-1}
[\psi_+^+ \psi_+]{\right)}^2-\frac{iM^2}{2}
\psi_+^+{\partial}_-^{-1} \psi_+
-{\left(}\frac{Me\:e^C}{4\pi^{3/2}} e^{-i\hat{\theta}}\:e^{i {\omega}}d_0^+
+h.c.{\right)}\biggr).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} This fermionic LF Hamiltonian differs from canonical one (in corresponding DLCQ scheme) only by last term, depending on zero modes and vacuum condensate parameter $\hat{\theta}$ which can be related to chiral condensate by transforming the variables in the equation (\[4d8\]): [$$\displaylines{\refstepcounter{equation} \label{4d14}\hskip 1em minus 1em
\sin\hat{\theta}=-\frac{2\pi^{3/2}}{e\;e^C}
\langle{\Omega}|:\bar\Psi{\gamma}^5\Psi:|{\Omega}{\rangle}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
Let us remark that the presents of linear in $M$ term in LF Hamiltonian (\[4d13\]) can be considered as a results of a modification of the LF constraint, connecting the $\psi_-$ with the $\psi_+$. An analogous modification of this constraint was got in the paper [@mac] where the method of exact operator solution of massless Schwinger model was applied.
The constructed LF Hamiltonian (\[4d13\]) was applied to nonperturbative numerical calculation of mass spectrum of QED(1+1) [@rasch]. The results of this calculation were compared with those of lattice calculations in Lorentz coordinates [@ham1; @ham2].
The calculations were carried out in wide domain of values of fermion mass $M$ for all values of the parameter $\hat{\theta}$, which is a function of the $M/e$ and vacuum parameter ${\theta}$. We do not know exactly this function, but know that it must be zero at ${\theta}=0$ and be equal to $\pi$ at ${\theta}=\pi$.
For ${\theta}= 0$ the obtained spectrum is bounded from below at any values $M$, and is in good agreement with the results of the paper [@ham1] for two bound states of lowest mass.
For the value ${\theta}=\pi$, at which phase transition can take place, the obtained spectrum for the lowest bound state agrees well with the results of the paper [@ham2] for sufficiently small $M$; at greater $M$ we start to see the disagreement with that paper, and then at $M$, greater some critical value, the spectrum becomes unbounded from below. This critical value approximately coincides with the point of phase transition found in [@ham2]. It can be supposed that at $M$ greater than the point of phase transition the calculations with our LF Hamiltonian, constructed via the analysis of perturbation theory in $M$, become incorrect.
Our calculations show that LF Hamiltonian (\[4d13\]) can give good results only in the limited domain of the parameters: from perturbative domain to some values which define the limits of applicability of our LF Hamiltonian. In the domain where our Hamiltonian becomes incorrect we need to take into account nonperturbative (in M/e) effects. Such an investigation can be useful for finding a LF Hamiltonian approach to realistic gauge theories.
The LF Hamiltonian (\[4d13\]) includes the operator ${\omega}$, which has no simple expression in terms of field operators. It is defined only by its properties (\[4d10\]),(\[4d12\]). Due to this fact the expression for the Hamiltonian depends essentially on the form of the regularization, i.e. $|x^-|\le L$ and antiperiodic boundary conditions in $x^-$ for the field $\psi$. Now we have found a possibility to rewrite the expression (\[4d13\]) in such a way that it contains only fermion field operators, and describes in the limit of removing the regularizations the same theory as the Hamiltonian (\[4d13\]). This new expression has at $\hat{\theta}={\theta}=0$ the following form: [$$\displaylines{\refstepcounter{equation} \label{new1}\hskip 1em minus 1em
H=\int\limits_{-L}^Ldx^-{\left(}\frac{e^2}{2}{\left(}{\partial}_-^{-1}
[\psi^+\psi]{\right)}^2
+\frac{eMe^C}{4\pi^{3/2}}{\left(}d_0^+d_0+b_1^+b_1{\right)}-
\frac{iM^2}{2}\psi^+{\partial}_-^{-1}\psi{\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Preliminary calculations of the mass spectrum, produced by this Hamiltonian, show that in the limit of removing the regularization, $L\to\infty$, results indeed coinside, with a good accuracy, with those for the bound state mass spectrum found here for the Hamiltonian (\[4d13\]). Work on studing of the Hamiltonian (\[new1\]) spectrum and also on the constructing the analogous expression for the Hamiltonian at $\hat{\theta}\ne0$ will be continued in future.
Transverse lattice regularization of Gauge Theories[\
]{}on the Light Front
=====================================================
The introduction of space-time lattice for gauge-invariant regularization of nonabelian gauge theories is well known [@wils]. Gauge invariant regularization in continuous space-time is also known [@halp] but it seems not suitable for the LF quantization. For the LF formulation only the lattice in transverse coordinates $x^1, x^2$ is used. In this formulation it is convenient to define variables so as to have the action polynomial in these variables [@barpir1; @barpir2; @heplat; @tmf04]. Such a regularization is not Lorentz invariant, and one can only hope that Lorentz invariance can be restored in continuous space limit. Nevertheless many attempts to apply LF Hamiltonian formulation with periodic boundary conditions in $x^-$, combined with transverse space lattice, are undertaken (for “color dielectric” type models [@dalley1; @dalley4; @pirner1; @mack]). In all of these works zero modes of fields are thrown out, so that, in fact, gauge invariance is violated.
We consider canonical LF formulation of gauge theories, regularized in gauge-invariant way. To achieve this goal we introduce transverse space lattice, discretize the momentum $p_-$ according to the prescription (b) (see introduction, equation (\[1.2\])) with all zero modes of fields included and apply the so called “finite mode” ultraviolet regularization in $p_-$. The last means a cutoff in eigenvalues of covariant derivative operator $D_-$ in the expansion of lattice field variables in eigen functions of this operator. These field variables are lattice modification of transverse components of usual gauge fields. They are described by complex matrices, defined on lattice links. Only such variables admit mentioned above “finite mode” regularization (for fermion fields analogous method was applied in [@konmod1; @konmod2]).
It is interesting that in the framework of this formulation one can avoid complicated canonical 2nd-class constraints, usually present in canonical LF formalism in continuous space. This greatly simplifies canonical quantization procedure. However the absence of explicit Lorentz invariance of the regularization scheme makes the investigation of the connection with conventional Lorentz-covariant formulation difficult. In particular, there is a problem of the description of quantum vacuum as common lowest eigen state of both operators $P_-$ and $P_+$.
Gauge-invariant action on the transverse lattice
------------------------------------------------
At first we introduce particular ultraviolet regularization via a lattice in transverse coordinates $x^1, x^2$ and choose variables so as to have the action, which is polynomial in these variables [@barpir1; @barpir2]. Furthermore, we use the described in the introduction gauge-invariant regularization (b) (see equation (\[1.2\])) of singularities at $p_- \to 0$ and gauge-invariant ultraviolet cutoff in modes of covariant derivative $D_-$ (then ultraviolet regularization of the theory is complete). For simplicity further we consider again the $U(N)$ theory of pure gauge fields because this example is technically more simple than the $SU(N)$ theory.
The components of gauge field along continuous coordinates $x^+$, $x^-$ can be taken without a modification and related to the sites of the lattice. Transverse components are described by complex $N\times N$ matrices $M_k(x)$, $k=1,2$. Each matrix $M_k(x)$ is related to the link directed from the site $x-e_k$ to the site $x$. The transverse vector $e_k$ connects two neighbouring sites on the lattice being directed along the positive axis $x^k$ ($|e_k|\equiv a$), see fig. 6.
fig6.pic
The matrix $M_k^+(x)$ is related to the same link but with opposite direction, see fig. 7.
fig7.pic
In the following the usual rule of summation over repeated indices is not used for the index $k$. If this summation is necessary, the symbol of a sum is indicated.
The elements of these matrices are considered as independent variables. This makes the action polynomial. For any closed directed loop in the lattice we can construct the trace of the product of matrices $M_k(x)$ sitting on the links in the loop and order these matrices from the right to the left along this loop. For example the expression [$$\displaylines{\refstepcounter{equation} \label{3}\hskip 1em minus 1em
{\rm Tr\;}\left\{M_2(x)M_1(x-e_{2})M_2^+(x-e_{1})M_1^+(x)\right\}
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} is related to the loop shown in fig. 8.
fig8.pic
It should be noticed that a product of matrices related to closed loop, consisting of one and the same link passed in both directions, is not identically unity because the matrices $M_k$ are not unitary (see, for example, fig. 9).
fig9.pic
The unitary matrices $U(x)$ of gauge transformations act on the $M$ and $M^+$ in the following way: [$$\displaylines{\refstepcounter{equation} \label{5.1a}\hskip 1em minus 1em
M_k(x)\to M'_k(x)=U(x)M_k(x)U^+(x-e_{k}),
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{1b}\hskip 1em minus 1em
M_k^+(x)\to M'^+_k(x)=U(x-e_{k})M_k^+(x)U^+(x).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} A trace of the product of the matrices, related to closed loop along lattice links, is invariant with respect to these transformations. To relate the matrices $M_k$ with usual gauge fields of continuum theory let us write these matrices in the following form: [$$\displaylines{\refstepcounter{equation} \label{5.2a}\hskip 1em minus 1em
M_k(x)=I+gaB_k(x)+igaA_k(x),\qquad B_k^+=B_k,\quad A_k^+=A_k.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
Then in the $a\to 0$ limit the fields $A_k(x)$ coincide with transverse gauge field components, and the $B_k(x)$ turn out to be extra (nonphysical) fields which should be switched off in the limit. Below we show how to get this.
The analog of the field strength [$$\displaylines{\refstepcounter{equation} \label{5.2b}\hskip 1em minus 1em
F_{\mu\nu}={\partial}_{\mu}A_{\nu}-{\partial}_{\nu}A_{\mu}-ig[A_{\mu},A_{\nu}],
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} multiplied by $i$, can be defined as follows: [$$\displaylines{\refstepcounter{equation} \label{5.n1}\hskip 1em minus 1em
G_{+-}=iF_{+-},\qquad F_{+-}(x)={\partial}_+A_-(x)-{\partial}_-A_+(x)-ig[A_+(x),A_-(x)],{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}G_{\pm,k}(x)=\frac{1}{ga}{\left[}{\partial}_{\pm}M_k(x)-ig{\left(}A_{\pm}(x)M_k(x)-
M_k(x)A_{\pm}(x-e_k){\right)}{\right]},{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}G_{12}(x)=-\frac{1}{ga^2}{\left[}M_1(x)M_2(x-e_1)-M_2(x)M_1(x-e_2){\right]}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Under gauge transformation these quantities transform as follows: [$$\displaylines{\refstepcounter{equation} \label{5.n2}\hskip 1em minus 1em
G_{+-}(x)\to G'_{+-}(x)=U(x)G_{+-}(x)U^+(x),{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}G_{\pm,k}(x)\to G'_{\pm,k}(x)=U(x)G_{\pm,k}(x)U^+(x-e_k),{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}G_{12}(x)\to G'_{12}(x)=U(x)G_{12}(x)U^+(x-e_1-e_2).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
We choose a simplest form of the action having correct naive continuum limit: [$$\displaylines{\refstepcounter{equation} \label{5.n3}\hskip 1em minus 1em
S=a^2\sum_{x^{\bot}}\int\! dx^+\!\int\limits^L_{-L}\! dx^-\;{\rm Tr}
{\left[}G^+_{+-}G_{+-}+\sum_k{\left(}G^+_{+k}G_{-k}+
G^+_{-k}G_{+k}{\right)}-G^+_{12}G_{12}{\right]}+S_m,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the additional term $S_m$ gives an infinite mass to extra fields $B_k$ in the $a\to 0$ limit: [$$\displaylines{\refstepcounter{equation} \label{5.n4}\hskip 1em minus 1em
S_m=-\frac{m^2(a)}{4g^2}\sum_{x_{\bot}}\int dx^+
\int\limits^L_{-L}dx^-\sum_k{\rm Tr}{\left[}{\left(}M_k^+(x)M_k(x)-I{\right)}^2{\right]}{\mathrel{\mathop{\longrightarrow}\limits_{a\to 0}}}{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}{\mathrel{\mathop{\longrightarrow}\limits_{a\to 0}}} -m^2(a)\int d^2x^{\bot}\int dx^+
\int\limits^L_{-L}dx^-\sum_k{\rm Tr} {\left(}B^2_k{\right)},\qquad m(a){\mathrel{\mathop{\longrightarrow}\limits_{a\to 0}}}\infty.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} It is supposed that this leads to necessary decoupling of the fields $B_k$.
Canonical quantization on the Light Front
-----------------------------------------
Let us fix the gauge as follows: [$$\displaylines{\refstepcounter{equation} \label{5.n5}\hskip 1em minus 1em
{\partial}_-A_-=0,\qquad A_-^{ij}(x)={\delta}^{ij}v^j(x^{\bot},x^+).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} For simplicity below we denote the argument of quantities, not depending on the $x^-$, again by $x$. Let us remark that starting with arbitrary field $A_\mu$, periodic in $x^-$, it is not possible to take zero modes of the $A_-$ equal to zero without a violation of the periodicity. But it is possible to make the $A_-$ diagonal as in the equation (\[5.n5\]) [@nov1; @nov2; @nov2a; @nov3].
Then the action (\[5.n3\]) can be written in the form: [$$\displaylines{\refstepcounter{equation} \label{5.n6}\hskip 1em minus 1em
S=a^2\sum_{x^{\bot}}\int dx^+\int\limits^L_{-L}dx^-\Biggl\{
\sum_i{\left[}2F^{ii}_{+-}(x){\partial}_+ v^i(x){\right]}+{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}+
\frac{1}{(ga)^2}\sum_{i,j}\sum_k{\left[}D_-{M_k^{ij}}^+(x){\partial}_+M_k^{ij}(x)
+h.c.{\right]}+
\sum_{i,j} A_+^{ij}(x)Q^{ji}(x)-{\cal H}(x)\Biggr\},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where [$$\displaylines{\refstepcounter{equation} \label{5.n7}\hskip 1em minus 1em
D_-M_k^{ij}(x)\equiv {\left(}{\partial}_--igv^i(x)+igv^j(x-e_k){\right)}M^{ij}_k(x),{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}D_-{M_k^{ij}}^+(x)\equiv {\left(}{\partial}_-+igv^i(x)-igv^j(x-e_k){\right)}{M^{ij}_k}^+(x),{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}D_-F_{+-}^{ij}(x)\equiv {\left(}{\partial}_--igv^i(x)+igv^j(x){\right)}F^{ij}_{+-}(x),
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} the $A_+^{ij}(x)$ play the role of Lagrange multipliers, [$$\displaylines{\refstepcounter{equation} \label{5.n8}\hskip 1em minus 1em
Q^{ji}(x)\equiv 2D_-F_{+-}^{ji}(x)+{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}}+\frac{i}{ga^2}\sum_{j'}\sum_k
\biggl[ {M^{ij'}_k}^+(x) D_-M_k^{jj'}(x)-
{M_k^{j'j}}^+(x+e_k)D_-M_k^{j'i}(x+e_k)-{{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}-
{\left(}D_- {M^{ij'}_k}^+(x){\right)}M_k^{jj'}(x)+
{\left(}D_- {M_k^{j'j}}^+(x+e_k){\right)}M_k^{j'i}(x+e_k)
\biggr]=0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} are gauge constraints and [$$\displaylines{\refstepcounter{equation} \label{5.n81}\hskip 1em minus 1em
{\cal H}=\sum_{ij}{\left(}{F_{+-}^{ij}}^+F_{+-}^{ij} +
{G_{12}^{ij}}^+G_{12}^{ij}{\right)}+{\cal H}_m
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} is Hamiltonian density. The term ${\cal H}_m$ can be obtained from the expression (\[5.n4\]) in standard way.
The constraints can be resolved explicitly by expressing the $F_{+-}^{ij}$ in terms of other variables, but zero mode components $F^{ii}_{+-(0)}$ can not be found from constraint equations and play the role of independent canonical variables. Zero modes $Q^{ii}_{(0)}(x^{{\bot}},x^+)$ of the constraints remain unresolved and are imposed as conditions on physical states: [$$\displaylines{\refstepcounter{equation} \label{5.4.9}\hskip 1em minus 1em
Q^{ii}_{(0)}(x^{\bot},x^+)\left|\Psi_{phys}\right>=0.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} In order to find complete set of independent canonical variables we write Fourier transformation in $x^-$ of fields $M_k^{ij}(x)$ in the following form: [$$\displaylines{\refstepcounter{equation} \label{5.4.10}\hskip 1em minus 1em
M^{ij}_k(x)=\frac{g}{\sqrt{4L}}\sum_{n=-\infty }^{\infty }
\left\{ \Theta {\left(}p_n+gv^i(x)-
gv^j(x-e_k){\right)}M^{ij}_{nk}(x^{\bot},x^+)+\right.{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}}\left. +\Theta {\left(}-p_n-gv^i(x)+
gv^j(x-e_k){\right)}{M^{ij}_{nk}}^+(x^{\bot},x^+) \right \}\times{{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\times \left | p_n+gv^i(x)-
gv^j(x-e_k) \right |^{-1/2} e^{-ip_nx^-},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where [$$\displaylines{\refstepcounter{equation} \label{5.n9}\hskip 1em minus 1em
\Theta (p) = \cases{
$1$, & $p > 0$ \cr
$0$, & $p < 0$ \cr},
\qquad p_n=\frac{\pi}{L}\,n, \quad n\epsilon Z.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Due to the gauge (\[5.n5\]) this Fourier transformation coincides with the expansion in eigen modes of the operator $D_-$. Therefore the ultraviolet cutoff in these modes, which we will apply, reduces to the following condition on the number of terms in the sum (\[5.4.10\]): [$$\displaylines{\refstepcounter{equation} \label{5.15a}\hskip 1em minus 1em
|p_n + gv^i(x) - gv^j(x-e_k) | \le \frac{\pi}{L}\,\bar n,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the $\bar n$ is integer parameter of ultraviolet cutoff. Let us stress that this regularization is gauge invariant.
The action can be rewritten in the following form (up to nonessential surface terms): [$$\displaylines{\refstepcounter{equation} \label{5.4.11}\hskip 1em minus 1em
S=a^2\sum_{x^{{\bot}}} \int dx^+ \left\{ \sum_i 4LF_{+-(0)}^{ii}
{\partial}_+v^i+\right.{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\left. +\frac{i}{a^2}\sum_{i,j}\sum_k{\sum_n}'
{M^{ij}_{nk}}^+{\partial}_+M^{ij}_{nk}+2L \sum_i
A_{+(0)}^{ii}Q_{(0)}^{ii}-\tilde {\cal H}(x) \right\},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the $\sum'_n $ means that the sum is cut off by the condition (\[5.15a\]), and the $\tilde{\cal H}$ is obtained from the ${\cal H}$ via the substitution of the expression [$$\displaylines{\refstepcounter{equation} \label{5.n91}\hskip 1em minus 1em
F_{+-}^{ij}={\left(}F_{+-}^{ij}-{\delta}^{ij}F_{+-(0)}^{ii}{\right)}+
{\delta}^{ij}F_{+-(0)}^{ii},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the $F_{+-}^{ij}-{\delta}^{ij}F_{+-(0)}^{ii}$ are to be written in terms of the $M^{ij}_{nk}$, ${M^{ij}_{nk}}^+$, $v^i$ by solving the constraints (\[5.n8\]) and using the equation (\[5.4.10\]). The $F_{+-(0)}^{ii}$ remain independent. The $G_{12}^{ij}$ are also to be expressed in terms of the $M^{ij}_{nk}$, ${M^{ij}_{nk}}^+$, $v^i$ via the equations (\[5.n1\]), (\[5.4.10\]).
We have the following set of canonically conjugated pairs of independent variables: [$$\displaylines{\refstepcounter{equation} \label{5.4.13}\hskip 1em minus 1em
\left \{ v^i,\quad \Pi^i=4La^2 F_{+-(0)}^{ii}\right \},\qquad
\left \{ M^{ij}_{nk},\quad i{M^{ij}_{nk}}^+ \right \}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} In quantum theory these variables become operators which satisfy usual canonical commutation relations: [$$\displaylines{\refstepcounter{equation} \label{5.n10}\hskip 1em minus 1em
[v^i(x),\Pi^j(x')]_{x^+=0}=i{\delta}^{ij}{\delta}_{x^{\bot},x'^{\bot}},{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}[M_{nk}^{ij}(x),{M_{n'k'}^{i'j'}}^+(x')]_{x^+=0}=
{\delta}^{ii'}{\delta}^{jj'}{\delta}_{nn'}{\delta}_{kk'}{\delta}_{x^{\bot},x'^{\bot}};
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} the other commutators being equal to zero.
Let us remark that the condition (\[5.n5\]) does not fix the gauge completely. In particular, discrete group of gauge transformations, depending on the $x^-$, of the form [$$\displaylines{\refstepcounter{equation} \label{5.n10.1}\hskip 1em minus 1em
U_n^{ij}(x)={\delta}^{ij}\exp\left\{ i\frac{\pi}{L}n^i(x^{{\bot}})x^-\right\},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where $n^i(x^{{\bot}})$ are integers, remains, and, of course, transformations, not depending on the $x^-$, are allowed. Under the transformations (\[5.n10.1\]) canonical variables change as follows: [$$\displaylines{\refstepcounter{equation} \label{5.n10.2}\hskip 1em minus 1em
v^i(x) \longrightarrow v^i(x)-\frac{\pi}{gL}\;n^i(x^{{\bot}}),\qquad
\Pi^i \longrightarrow \Pi^i,{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}M_{nk}^{ij}(x^{{\bot}}) \longrightarrow M_{n'k}^{ij}(x^{{\bot}}),\quad
n'=n+ n^i(x^{{\bot}}) - n^j(x^{{\bot}}- e_k).
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
Let us write the expression for quantum operators $Q_{(0)}^{ii}(x^{{\bot}},x^+)$, which define the physical subspace of states. We fix the order of the operators in such a way as to relate with classical expression $G_{\mu\nu}^{\;+}G^{\,\mu\nu}$ a quantum one of the form: [$$\displaylines{\refstepcounter{equation} \label{5.n10.2a}\hskip 1em minus 1em
\frac{1}{2}{\left(}G_{\mu\nu}^{\;+}G^{\,\mu\nu} + G^{\,\mu\nu}G_{\mu\nu}^{\;+}{\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} We remark that other choices of the ordering do not admit reasonable vacuum solution. Then the operators $Q_{(0)}^{ii}(x^{{\bot}},x^+)$ have the following form in terms of canonical variables: [$$\displaylines{\refstepcounter{equation} \label{5.4.16}\hskip 1em minus 1em
2LQ^{ii}_{(0)}(x^{\bot},x^+)
=-\frac{g}{4a^2}
\sum_j\sum_k{\sum_n}'
{\left[}{\varepsilon}{\left(}p_n+gv^j(x+e_k)-gv^i(x){\right)}\times\right.\hfill{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}}\times{\left(}{M^{ji}_{nk}}^+(x+e_k)M^{ji}_{nk}(x+e_k)+
M^{ji}_{nk}(x+e_k){M^{ji}_{nk}}^+(x+e_k){\right)}-{{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\hfill\left.
-{\varepsilon}{\left(}p_n+gv^i(x)-gv^j(x-e_k){\right)}{\left(}{M^{ij}_{nk}}^+(x)M^{ij}_{nk}(x)+
M^{ij}_{nk}(x){M^{ij}_{nk}}^+(x){\right)}{\right]},
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where [$$\displaylines{\refstepcounter{equation} \label{5.n11}\hskip 1em minus 1em
{\varepsilon}(p) = \cases{
$1$, & $p > 0$ \cr
$-1$, & $p < 0$ \cr}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
One can easily construct canonical operator of translations in the $x^-$: [$$\displaylines{\refstepcounter{equation} \label{5.4.14}\hskip 1em minus 1em
P_-^{can}=\frac{1}{2}\sum_{x^{{\bot}}} \sum_{i,j}\sum_k{\sum_n}'
p_n\,\, {\varepsilon}{\left(}p_n+gv^i(x)-gv^j(x-e_k) {\right)}\times\hfill{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\hfill\times{\left(}{M_{nk}^{ij}}^+(x)M_{nk}^{ij}(x)+M_{nk}^{ij}(x){M_{nk}^{ij}}^+(x){\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
This expression differs from the physical gauge invariant momentum operator $P_-$ by a term proportional to the constraint. The operator $P_-$ is [$$\displaylines{\refstepcounter{equation} \label{5.4.15}\hskip 1em minus 1em
P_-=\frac{a^2}{2}\sum_{x^{{\bot}}} \sum_k\; \int\limits_{-L}^L dx^-\, {\rm Tr}
{\left(}G^+_{-k}G_{-k}+G_{-k}G^+_{-k}{\right)}=
P_-^{can.} + 4La^2 \sum_{x^{{\bot}}} \sum_i
v^iQ^{ii}_{(0)}={\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}=\frac{1}{2}\sum_{x^{{\bot}}} \sum_{i,j}\sum_k{\sum_n}'
\left| p_n+gv^i(x)-gv^j(x-e_k) \right|
\times{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}\times
{\left(}{M^{ij}_{nk}}^+(x)M^{ij}_{nk}(x)+M^{ij}_{nk}(x){M^{ij}_{nk}}^+(x){\right)}={\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}=\sum_{x^{{\bot}}} \sum_{i,j}\sum_k{\sum_n}'
\left| p_n+gv^i(x)-gv^j(x-e_k) \right|{\left(}{M^{ij}_{nk}}^+(x)M^{ij}_{nk}(x)+\frac{1}{2}{\right)}.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
Let us choose a representation of the state space, in which the variables $v^i(x)$ are the multiplication operators. The states are described in this representation by normalizable functionals $F[v]$ of classical functions $v^i(x)$ (in fact by functions, depending on the values of the $v^i$ in different points $x^{\bot}$ due to the discreteness of these $x^{\bot}$). One can define full space of states as direct product of Fock space, in which the ${M^{ij}_{nk}}^+(x)$ and $M^{ij}_{nk}(x)$ play the role of creation and annihilation operators, and the space of functionals $F[v]$. Let us call $M$-vacuum the states of the form $|0{\rangle}\cdot F[v]$, where the $|0{\rangle}$ satisfies the condition [$$\displaylines{\refstepcounter{equation} \label{5.n12}\hskip 1em minus 1em
M^{ij}_{nk}(x)|0{\rangle}=0,
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and the $F$ is some functional. Arbitrary state can be represented in the form of linear combination of vectors $|\{m\};F{\rangle}$ of the form [$$\displaylines{\refstepcounter{equation} \label{5.4.17}\hskip 1em minus 1em
\prod_{x^{{\bot}}}\prod_{i,j}\prod_k{\prod_n}'
{\left(}{M^{ij}_{nk}}^+(x){\right)}^{m^{ij}_{nk}(x)}|0{\rangle}\cdot F[v]
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} with different nonnegative integer functions $m^{ij}_{nk}(x^{\bot})$ and functionals $F$. One can define orthonormalized set of such functionals if necessary.
One can see from (\[5.4.15\]) that the state, corresponding to the absolute minimum of the $P_-$ must satisfy the conditions (\[5.n12\]), i.e. to be a $M$-vacuum. The value of the $P_-$ in this state can be written in the form [$$\displaylines{\refstepcounter{equation} \label{5.4.25}\hskip 1em minus 1em
\langle\left. 0;F\right|P_-\left|\;0;F\right.{\rangle}={\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}=\frac{1}{2}\int \prod_{x^{{\bot}}}\prod_i dv^i(x)
\sum_{x^{{\bot}}}\sum_{i,j}\sum_k{\sum_n}'
\left|p_n+g v^i(x)-g v^j(x-e_k)\right| |F[v]|^2.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} Remind that the ${\sum\limits_n}'$ denotes the sum in $n$ limited by the condition (\[5.15a\]). If one uses this condition and shifts the index $n$ in these sums by integer part of the quantities $(gL(v^i(x)-v^j(x-e_k))/\pi)$, one sees that the dependence on the $v^i$ cancels in sums over $n$ and that the expression (\[5.4.25\]) does not depend on the $F[v]$ if it is normalized to unity. Thus the momentum $P_-$ has the minimum in all $M$-vacua. One can make the value of the $P_-$ in these vacua equal to zero by subtracting corresponding constant from the operator $P_-$.
Let us show that $M$-vacua are the physical states, i. e. satisfy the condition (\[5.4.9\]). Indeed, in $M$-vacua this condition looks as follows: [$$\displaylines{\refstepcounter{equation} \label{5.4.24}\hskip 1em minus 1em
\sum_j\sum_k{\sum_n}'{\left[}{\varepsilon}{\left(}p_n+g v^j(x+e_k)-g v^i(x){\right)}-
{\varepsilon}{\left(}p_n+g v^i(x)-g v^j(x-e_k){\right)}{\right]}F[v]=0
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} and is satisfied for any $F[v]$, because for every link in the sum (\[5.4.24\]) the numbers of positive and negative values of the ${\varepsilon}$-functions are equal. For arbitrary basic states (\[5.4.17\]) analogous conditions have the following form: [$$\displaylines{\refstepcounter{equation} \label{5.4.24a}\hskip 1em minus 1em
\sum_j\sum_k{\sum_n}'{\left[}{\varepsilon}{\left(}p_n+g v^j(x+e_k)-g v^i(x){\right)}m^{ij}_{nk}(x+e_k)-\right.{\hfill{\hfil \hskip 1em minus 1em\phantom{(\theequation)} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}-\left.{\varepsilon}{\left(}p_n+g v^i(x)-g v^j(x-e_k){\right)}m^{ij}_{nk}(x){\right]}F[v]=0.
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{}
The eigen values $p_-$ of the operator $P_-$ can be found from the equation [$$\displaylines{\refstepcounter{equation} \label{5.4.25a}\hskip 1em minus 1em
\sum_{x^{{\bot}}}\sum_{i,j}\sum_k{\sum_n}'
\left|p_n+g v^i(x)-g v^j(x-e_k)\right| m^{ij}_{nk}(x)F[v]=
p_- F[v]
{\hfil\hskip 1em minus 1em (\theequation)}\hfilneg}$$]{} where the functional $F[v]$ is normalized.
To define physical vacuum state correctly one must consider not only states, corresponding to the minimum of the operator $P_-$, but also to the minimum of the operator $P_+$. One can try to do this via minimization of the $P_+$ on the $M$-vacua, i. e. on the set of states with $p_-=0$. The expression $\langle 0|P_+|0{\rangle}$, where $|0{\rangle}$ is the Fock space vacuum w. r. t. the $M_{nk}$, ${M_{nk}}^+$, depends on the functions $v^i(x)$ (which enter, in particular, into the “normal contractions” of the operators $M_k$, $M_k^+$) and on the operators $\Pi^i$, canonically conjugated to the $v^i(x)$. The expectation value of this expression is to be minimized on the set of functionals $F[v]$. The resulting functional $F[v]$ must decrease in the vicinity of those values of the $v^i(x)$, for which the operator $D_-$ has zero eigen value, because the Hamiltonian is singular at these values (it is seen, for example, from the expansion (\[5.4.10\])).
The vacuum state constructed in such a way strongly differs from the usual vacuum in continuous space theory, because for $M$-vacuum we get zero expectation values of the operators $M_k$, but not of the operators $(M_k - I)/ga = B_k + iA_k$, related to usual gauge fields. Beside of this the condition of the unitarity of the matrices $M_k$ in the continuum limit (or equivalent condition of switching off the nonphysical fields $B_k$) cannot be fulfilled in such a vacuum. This disagreement with conventional theory is caused by the absence of explicit Lorentz invariance in our formulation, that leads to different quantum states, corresponding to the minima of the operators $P_-$ and $P_+$. It is not clear whether these states can be made coinciding at least in the limit of continuous space. This requires further investigation.
Nevertheless the method of the quantization of gauge theories on the LF, described here, can be useful for completely gauge-invariant formulation of some effective models, based on analogous formalism (but without complete gauge invariance due to throwing out of all zero modes of fields and due to the absence of gauge-invariant regularization of ultraviolet divergencies in the $p_-$). Such models are described, for example, in papers [@dalley1; @dalley4], where the ideas of papers [@pirner1; @mack] were developed.
Conclusion
==========
The LF Hamiltonian approach to Quantum Field Theory briefly reviewed here is an attempt to apply a beautiful idea of Fock space representation for quantum field nonperturbatively in the framework of canonical formulation on the LF. The problem of describing the physical vacuum state becomes formally trivial in this approach because such vacuum state coincides with mathematical vacuum of LF Fock space.
However the breakdown of Lorentz and gauge symmetries due to regularizations generates difficulties in proving the equivalence of LF formalism and the usual one in Lorentz coordinates. This problem can be solved for nongauge theories but turns out to be very difficult for gauge theories in special (LF) gauge which is needed here. Nevertheless we hope that these difficulties can be overcome by finding a modified form of canonical LF Hamiltonian which generates the perturbation theory equivalent to usual covariant and gauge invariant one.
1em [**Acknowledgments.**]{} The work was supported by the Russian Foundation for Basic Research, grant no. 05-02-17477 (S.A.P. and E.V.P.).
$\protect\vphantom{a}$Appendix 1 {#protectvphantomaappendix-1 .unnumbered}
================================
[**Statement 1.**]{} [*If conditions (\[4.1.1\]) are satisfied, then, for fixed external momenta $p^s$ and $p^s_-\ne 0 \;\forall s$, the equality [$$\displaylines{\refstepcounter{equation} \label{4.A1}\hskip 1em minus 1em
\lim_{{\beta}\to 0}\lim_{{\gamma}\to 0}
\int\prod_k dq_+^k \int\limits_{V_{\scriptstyle {\varepsilon}}}
\prod_k dq_-^k {{\tilde f(Q^i,p^s) e^{-{\gamma}\sum_i
{Q_+^i}^2-{\beta}\sum_i{Q_-^i}^2} }\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}}={\hfill{\hfil \hskip 1em minus 1em\phantom{({A.1.\arabic{equation}})} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}=\int\prod_k dq_+^k \int\limits_{V_{\scriptstyle {\varepsilon}}\cap B_L}
\prod_k dq_-^k
{{\tilde f(Q^i,p^s)}\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}},
{\hfil\hskip 1em minus 1em ({A.1.\arabic{equation}})}\hfilneg}$$]{} holds while the expressions appearing in (\[4.A1\]) exist and the integral over $\{q_+^k\}$ on the right-hand side is absolutely convergent. It is assumed that the momenta of lines $Q^i$ are expressed in terms of loop momenta $q^k$, $V_{{\varepsilon}}$ is the domain corresponding to the presence of full lines, type-1 lines, and type-2 lines (the definitions are given following formula (\[4.22\])), $B_L$ is the sphere of radius $L$, where $L\ge S \: \max\limits_s |p_-^s|$, and $S$ is a number depending on the diagram structure.* ]{}
Let us prove the statement. For each type-1 line in (\[4.A1\]), we perform the following partitioning: [$$\displaylines{\refstepcounter{equation} \label{nnn9}\hskip 1em minus 1em
\int\limits_{-{\varepsilon}}^{{\varepsilon}} dQ_-^i={\left[}\int dQ_-^i+(-1)
{\left(}\int\limits_{-\infty}^{-{\varepsilon}}
dQ_-^i + \int\limits_{{\varepsilon}}^{\infty} dQ_-^i {\right)}{\right]}.
{\hfil\hskip 1em minus 1em ({A.1.\arabic{equation}})}\hfilneg}$$]{} Then both sides of equation (\[4.A1\]) become the sum of expressions of the same form in which, however, the domain $V_{{\varepsilon}}$ corresponds to the presence of only full and type-2 lines. It is clear that by proving the statement for this $V_{{\varepsilon}}$: (which is done below), we prove the original statement as well.
Let $\tilde B$ be a domain such that the surfaces on which are not tangent to the boundary $\tilde B$. First, we prove that in the expression [$$\displaylines{\refstepcounter{equation} \label{4.A2}\hskip 1em minus 1em
\int\prod_k dq_+^k
\int\limits_{V_{\scriptstyle {\varepsilon}}\cap \tilde B} \prod_k dq_-^k
{{\tilde f(Q^i,p^s)\: e^{-{\beta}\sum_i{Q_-^i}^2}}\over{\prod_i
(2Q_+^iQ_-^i-M^2_i+i{\char'32})}}
{\hfil\hskip 1em minus 1em ({A.1.\arabic{equation}})}\hfilneg}$$]{} the integral over $\{q_+^k\}$ is absolutely convergent (here the integral over $\{q_-^k\}$ is finite because x${\char'32}>0$, ${\beta}>0$). This becomes obvious (considering conditions (\[4.1.1\]) and the fact that, in type-2 lines, the momentum $Q_-^i$ is separated from zero) if the contours of the integration over $\{q_-^k\}$ can be deformed in such a way that the momenta $Q_-^i$ of the full lines are separated from zero by a finite quantity (within the domain $V_{{\varepsilon}}\cap \tilde B$). In this case, we can repeat the well-known Weinberg reasoning [@wein]. What can prevent deformation is either a “clamping” of the contour or the point $Q_-^i=0$ falling on the integration boundary.
Let us investigate the first alternative. We divide the domain of integration over $q_+^k$ into sectors such that the momenta of all full lines $Q_+^i$ have a constant sign within one sector. Let us examine one sector. We take a set of full lines whose $Q_-^i$ may simultaneously vanish. In the vicinity of the point where $Q_-^i$ from this set vanish simultaneously, we bend the contours of the integration over $\{q_-^k\}$ such that these contours pass through the points $Q_-^i=iB^i$ and the momenta $Q_-^i$ of the type-2 lines do not change. Let $B^i$ be such that $B^i Q^i_+ \ge 0$ for the lines from the set (for $Q^i_+$ from the sector under consideration). It is easy to check that this bending is possible. (Since the contours of integration over $q_-^k$ are bent and $Q^i_-$ are expressed in terms of $q_-^k$, one should only check that such $b^k$ exist, where the necessary $B^i$ are expressed in the same way, i.e., that $B^i$ obey the conservation laws and flow only along the full lines). With this bending, rather small in relation to the deviation and the size of the deviation region, the contours do not pass through the poles because, for the denominator of each line from the set in question, we have [$$\displaylines{\refstepcounter{equation} \label{nnn10}\hskip 1em minus 1em
{\left(}2Q_+^iQ_-^i-M^2_i+i{\char'32}{\right)}\to
{\left(}2Q_+^i{\left(}Q_-^i+iB^i{\right)}-M^2_i+i{\char'32}{\right)}, \quad Q_+^iB^i\ge 0,
{\hfil\hskip 1em minus 1em ({A.1.\arabic{equation}})}\hfilneg}$$]{} and for the other denominators, the bending takes place in a region separated from the point where the corresponding momenta $Q_-^i$ are equal to $0$. Repeating the reasoning for all sets, we can see that there is no contour “clamping”.
The other alternative is excluded by the above condition for $\tilde B$. To make this clear, one should introduce such coordinates $\xi^{{\alpha}}$ in the $q^k$-space that the boundary of the domain $\tilde B$ is determined by the equation $\xi^1=a=const$ and then argue as above for the coordinates $\xi^{{\alpha}}$ with ${\alpha}\ge 2$.
After bending the contours, integral (\[4.A2\]) is absolutely convergent in $q_+^k$, $q_-^k$ if the integration in $q_+^k$ is carried out within the sector under consideration. On pointing out that the result, of internal integration in (\[4.A2\]) does not depend on the bending, we add the integrals over all sectors and conclude that (\[4.A2\]) converges in $\{q_+^k\}$ absolutely.
Now let us prove that if $\tilde B$ is a quite small, finite vicinity of the point $\{\tilde q_-^k\}$ that lies outside the sphere $B_L$, then expression (\[4.A2\]) is equal to zero. We consider the momentum $Q_-^i$ of one line. Flowing along the diagram, it can ramify or it can merge with other momenta. Clearly, two situations are possible: either it flows away completely through external lines, or, probably, after long wandering, part of it, $\tilde Q_-$, makes a complete loop. The former situation is possible only if $|Q_-^i|\le \sum_r |p_-^r|$, where all external momenta leaving the diagram (but not entering it) are summed. Obviously, $S$ can be chosen such that for $\{q_-^k\}$ from $\tilde B$, a line exists whose momentum violates this condition.
The latter situation results in the existence of a loop, where the inequality $Q_-^i>\tilde Q_-$ holds for all momenta of its lines and the positive direction of the momenta is along the loop. Then the integral over $q_+^k$ of the loop in question can be interchanged with the integrals over $\{q_-^k\}$ (because it is absolutely and uniformly convergent for all $q_-^k$) and the residue formula can be used to perform this integration. Since, for the loop in question, the momenta $Q_-^i$ of the lines of this loop are separated from zero and are of the same sign, the result is zero. This has a simple physical meaning. If we pass to stationary noncovariant perturbation theory, we find that only quanta with positive $Q_-$ can exist. In this case, external particles with positive $p_-$ are incoming and those with negative $p_-$ are outgoing. Then, the momentum conservation law favors the occurrence of the first situation.
The entire outside space for $\tilde B$ can be composed of the above domains $B_L$ (everything converges well at infinity due to the factor $\exp(-{\beta}\sum_i{Q_-^i}^2)$). Thus, on the left-hand side of (\[4.A1\]), one can substitute the integration domain $V_{{\varepsilon}}\cap B_L$ for $V_{{\varepsilon}}$, set the limit in ${\gamma}$ under the sign of integration over $\{q_+^k\}$ because of its absolute convergence, and also set the limit in ${\beta}$ under the integration sign because the domain of the integration over $\{q_-^k\}$ is bounded. Thus, we obtain the right-hand side. The statement is proved.
[**Statement 2.**]{} [*If $V_{{\varepsilon}}$ corresponds to the presence of type-2 lines alone, then, under the same conditions as in Statement 1, the equality [$$\displaylines{\refstepcounter{equation} \label{nnn11}\hskip 1em minus 1em
\int\limits_{V_{\scriptstyle {\varepsilon}}}\prod_k dq_-^k\int\prod_k dq_+^k
{{\tilde f(Q^i,p^s)}\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}}={\hfill{\hfil \hskip 1em minus 1em\phantom{({A.1.\arabic{equation}})} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}=\int\prod_k dq_+^k \int\limits_{V_{\scriptstyle {\varepsilon}}\cap B_L}
\prod_k dq_-^k
{{\tilde f(Q^i,p^s)}\over{\prod_i (2Q_+^iQ_-^i-M^2_i+i{\char'32})}}.
{\hfil\hskip 1em minus 1em ({A.1.\arabic{equation}})}\hfilneg}$$]{} is valid.* ]{}
The proof of this statement is analogous to the second part of the proof of Statement 1.
$\protect\vphantom{a}$Appendix 2 {#protectvphantomaappendix-2 .unnumbered}
================================
[**Statement.**]{} [*If conditions (\[4.1.1\]) are satisfied, the limits in ${\gamma}$ and ${\beta}$ in (\[4.18\]) can be interchanged (in turn) with the sign of the integral over $\{{\alpha}_i\}$ and then with $\tilde f{\left(}-i{{{\partial}}\over{{\partial}y_i}}{\right)}$.* ]{}
To prove this, we define the vectors [$$\displaylines{\refstepcounter{equation} \label{nnn12}\hskip 1em minus 1em
\{q_+^1,q_-^1,\dots, q_+^l,q_-^l\}\equiv S,\qquad
\{Q_+^1,Q_-^1,\dots, Q_+^n,Q_-^n\}\equiv {\mu}S+P,{\hfil \hskip 1em minus 1em\phantom{({A.2.\arabic{equation}})} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\{y_1^+,y_1^-,\dots, y_n^+,y_n^-\}\equiv Y,
{\hfil\hskip 1em minus 1em ({A.2.\arabic{equation}})}\hfilneg}$$]{} where the vector $P$ is built only from external momenta and ${\mu}$ is an $l\times n$ matrix of rank $l$, ${\mu}^{2i}_{2k-1}={\mu}^{2i-1}_{2k}=0$, ${\mu}^{2i}_{2k}={\mu}^{2i-1}_{2k-1}$. Next, we introduce the following notation: [$$\displaylines{\refstepcounter{equation} \label{nnn13}\hskip 1em minus 1em
\tilde {\Lambda}_i={\left(}\begin{array}{cc}
{\gamma}&-i{\alpha}_i \\
-i{\alpha}_i&{\beta}\end{array}{\right)},\quad
{\Lambda}=diag\{\tilde {\Lambda}_1,\dots,\tilde {\Lambda}_n\}, \quad A={\mu}^t{\Lambda}{\mu},{\hfil \hskip 1em minus 1em\phantom{({A.2.\arabic{equation}})} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}B={\mu}^t {\Lambda}P -{1\over 2}i{\mu}^t Y,\quad
C=-P^t{\Lambda}P+iY^t P-i\sum_i{\alpha}_i M^2_i.
{\hfil\hskip 1em minus 1em ({A.2.\arabic{equation}})}\hfilneg}$$]{} Then it follows from (\[4.19\]) that [$$\displaylines{\refstepcounter{equation} \label{4.B4}\hskip 1em minus 1em
\hat {\varphi}({\alpha}_i,p^s,{\gamma},{\beta})=(-i)^n \tilde f{\left(}-i{{{\partial}}\over{{\partial}y_i}}{\right)}\int d^{2l}S \; e^{-S^tAS-2B^tS+C}\Bigr|_{y_i=0}={\hfill{\hfil \hskip 1em minus 1em\phantom{({A.2.\arabic{equation}})} \hfilneg\cr\hfilneg\hskip 1em minus 1em\hfil}\hfill}=(-i)^n \tilde f{\left(}-i{{{\partial}}\over{{\partial}y_i}}{\right)}e^{B^tA^{-1}B+C}{{\pi^l}\over{\sqrt{\det A}}}\biggr|_{y_i=0}.
{\hfil\hskip 1em minus 1em ({A.2.\arabic{equation}})}\hfilneg}$$]{} The function $\tilde f$ is a polynomial and we consider each of its terms separately. Up to a factor, each term has the form ${{{\partial}}\over{{\partial}y_{i_1}}}\dots {{{\partial}}\over{{\partial}y_{i_r}}}$. These derivatives act on $C$ and $B$. The action on $C$ results in the constant factor $iN^tP$, the action on $B$ results in the factor $-(1/2)iN^t{\mu}A^{-1}B$ or $-(1/4)N_1^t{\mu}A^{-1}{\mu}^tN_2$ (the latter is the result of the action of two derivatives; $N$, $N_1$, and $N_2$ are constant vectors).
It is necessary to prove the correctness of the following three procedures: (i) setting the limit in ${\gamma}$ under the integral sign for fixed ${\beta}>0$; (ii) setting the limit in ${\beta}$ for ${\gamma}=0$; (iii) setting the limits in ${\gamma}$ and ${\beta}$ under the signs of differentiation with respect to $Y$. In cases (i) and (ii), one must obtain the bounds [$$\displaylines{\refstepcounter{equation} \label{4.B5}\hskip 1em minus 1em
|\hat {\varphi}({\alpha}_i,p^s,{\gamma},{\beta})|\le {\varphi}' ({\alpha}_i,p^s,{\beta}),
{\hfil\hskip 1em minus 1em ({A.2.\arabic{equation}})}\hfilneg}$$]{} [$$\displaylines{\refstepcounter{equation} \label{4.B6}\hskip 1em minus 1em
|\hat {\varphi}({\alpha}_i,p^s,0,{\beta})|\le {\varphi}'' ({\alpha}_i,p^s),
{\hfil\hskip 1em minus 1em ({A.2.\arabic{equation}})}\hfilneg}$$]{} where ${\varphi}'$ and ${\varphi}''$ are functions integrable (for ${\varphi}'$ if ${\beta}>0$) in any finite domain over ${\alpha}_i$, with ${\alpha}_i\ge 0$. Then, for case (i), we have [$$\displaylines{\refstepcounter{equation} \label{nnn14}\hskip 1em minus 1em
|\hat {\varphi}({\alpha}_i,p^s,{\gamma},{\beta})\; e^{-{\char'32}\sum_i {\alpha}_i}|\le
{\varphi}' ({\alpha}_i,p^s,{\beta})\; e^{-{\char'32}\sum_i {\alpha}_i},
{\hfil\hskip 1em minus 1em ({A.2.\arabic{equation}})}\hfilneg}$$]{} i.e., a limit on the integrated function arises, and, thus, the limit in ${\gamma}$ can be put under the integral sign. The situation is similar for case (ii). It is evident from (\[4.B4\]) that the function $\hat {\varphi}$ can be singular only if the eigenvalues of matrix $A$ become zero. On finding the lower bound of these eigenvalues, one can prove through rather long reasoning that bounds (\[4.B5\]), (\[4.B6\]) exist if condition (\[4.1.1\]) is satisfied.
After the limits in ${\gamma}$ and ${\beta}$ are put under the integral sign, it is not difficult to interchange them with the differentiation with respect to $Y$. One need do it only for ${\alpha}_i>0$ (for each $i$) and, in this case, one can show that the eigenvalues of the matrix $A$ are nonzero and $\hat {\varphi}$ is not singular.
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|
{
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|
---
abstract: 'When a gas in an externally imposed potential field is compressed, temperature gradients appear. This has been called the piezothermal effect. It is possible to analytically calculate the time-dependent behavior of the piezothermal effect using a linearized fluid model. Quantitative differences between the fluid-model results and previous numerical calculations can be explained by the effects of viscosity and heat conductivity. The fluid model casts the piezothermal effect as a spectrum of buoyancy oscillations, which yields new physical insights into the effect.'
address:
- 'Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA'
- 'Lawrence Livermore National Laboratory, Livermore, California, 94550, USA'
- 'Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA'
author:
- 'E. J. Kolmes'
- 'V. I. Geyko'
- 'N. J. Fisch'
bibliography:
- '../../../Master.bib'
title: Fluid Model for the Piezothermal Effect
---
Rotating fluid, piezothermal effect, compression, Brunt-Väisälä oscillations
Introduction
============
Consider a gas at rest in a potential field. If the gas is compressed, it will be heated. Moreover – contrary to the usual intuition about compressional heating – the resulting temperature will be spatially nonuniform, such that regions that are higher in the potential well are hotter. This effect was described by Geyko and Fisch [@Geyko2016] and called the *piezothermal* effect. Intuitively, it results from the fact that particles starting in equilibrium move toward (and further compress) regions of higher potential as they are heated.
In the original paper on the piezothermal effect, Geyko and Fisch observed the phenomenon in particle simulations. Analytically, they used a toy model to explain the scalings and some of the quantitative behavior of the simulations. Their model described the gas as two homogeneous regions separated by a massive movable membrane, so that the two sides of the system could have different temperatures and densities and could exert pressure on one another. For the simulation tools, they used a one-dimensional Monte Carlo code with exact energy and momentum conservation properties and a hard-sphere binary-collision operator. While their models correctly described the essential characteristics of the effect, they left room for discussion and future improvement in a number of respects. This paper analyzes the piezothermal effect by instead using a fluid model. The fluid approach to the piezothermal effect makes it possible to analytically calculate the behavior of the piezothermal effect in a wider range of scenarios, in greater detail, and using fewer simplifying assumptions than was done previously. Numerical fluid simulations confirm the validity of the analytic model and – when compared in detail to the results of the Monte Carlo code used in the original paper – help to explain quantitative discrepancies between the fluid-model results and the previous numerical results.
The piezothermal effect is closely related to the physics to the rotation-dependent heat capacity effect also studied by Geyko and Fisch, in which the energy required to compress a rotating cylinder changes when the gas is spinning [@Geyko2013; @Geyko2017]. That effect has applications in engine design, where it could be used to improve the efficiency of Otto and Diesel cycles [@Geyko2014]. In addition, the piezothermal effect is phenomenologically similar to the behavior observed in Ranque-Hilsch vortex tubes, which also produce radial temperature gradients in a rotating gas [@Ranque1933; @Hilsch1947; @Kassner1948; @Ahlborn1997; @Ahlborn1998; @Ahlborn2000; @Liew2012; @Kolmes2017]. Vortex tubes are used for spot cooling in a variety of industrial applications. In general, the ability to move energy in rotating and compressing systems – either spatially or between degrees of freedom – can be of great practical utility [@Geyko2014; @Davidovits2016]. These effects can also be useful for understanding the natural world. In particular, the fluid treatment of the piezothermal effect makes it clear that there is a strong connection between the piezothermal effect and Brunt-Väisälä oscillations, which are observed in a variety of naturally stratified media [@Brunt1927; @Durran1982; @Emery1984; @Brassard1991].
Linearized Fluid Model for Fast Compression {#sec:fastCompression}
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For simplicity, we consider the potential field to be gravitational, although practical applications are more likely in spinning systems, where centrifugal forces take the role of gravitational forces. Thus, to describe the key effects most simply, consider a gas in a gravitational field, such that all quantities vary only in the direction of the field. Suppose the fluid is compressed in a direction perpendicular to the gravitational field. The behavior of the system depends on four timescales: the collisional timescale $\tau_c$, the compression timescale $\tau_E$, the sound timescale $\tau_s$, and the timescale $\tau_H$ associated with spatial heat conduction. Geyko and Fisch studied the piezothermal effect in a fast-compression scenario and in a slow-compression scenario. In the fast-compression scenario, $\tau_c \ll \tau_E \ll \tau_s \ll \tau_H$. The first part of this inequality implies that the gas is always in local equilibrium. The second inequality means that the input of energy due to compression happens much more quickly than the system can react spatially. The last part of the inequality states that spatial heat conductivity can be neglected.
Because of the very fast collisional timescale, it is appropriate to describe the system with a fluid model (a system with less frequent collisions could behave very differently [@Kolmes2016]). Using an adiabatic equation of state, the fluid density, velocity, and temperature can be modeled by $$\begin{gathered}
\frac{\partial n}{\partial t} + \frac{\partial}{\partial x} \big(n v \big) = 0 \label{eqn:nonlinearN} \\
m n \bigg( \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial x} \bigg) = - \frac{\partial (nT)}{\partial x} - m n g \label{eqn:nonlinearV} \\
\bigg( \frac{\partial}{\partial t} + v \frac{\partial}{\partial x} \bigg)\bigg( \frac{T}{n^{\gamma-1}} \bigg) = 0. \label{eqn:nonlinearT}\end{gathered}$$ Suppose the system is bounded between $x = 0$ and $x = L$. Define equilibrium profiles $$\begin{gathered}
n_0(x) \doteq \bigg(\frac{mg/T_0}{1-e^{-mgL/T_0}}\bigg) \, e^{-mgx/T_0} \\
T_0(x) \doteq T_0 = \text{const} \\
v_0(x) \doteq 0. \end{gathered}$$ Now suppose the system is perturbed so that at $t = 0$, the temperature is (uniformly) changed from $T_0$ to $T_i$. This can occur, for example, by lateral compression as shown in Figure \[fig:compressionCartoon\]. Define $$\begin{gathered}
\delta \doteq \frac{T_i - T_0}{T_0} \end{gathered}$$ and suppose $\delta \ll 1$. $n$, $T$, and $v$ can be expanded about equilibrium so that $$\begin{gathered}
n = n_0 + n_1 + \mathcal{O}(\delta^2) \\
T = T_0 + T_1 + \mathcal{O}(\delta^2) \\
v = v_1 + \mathcal{O}(\delta^2) . \end{gathered}$$ The initial conditions for $n_1$, $T_1$, and $v_1$ are $$\begin{gathered}
n_1 \big|_{t=0} = 0 \label{eqn:initialN1} \\
T_1 \big|_{t=0} = T_0 \delta \label{eqn:initialT1} \\
v_1 \big|_{t=0} = 0. \label{eqn:initialV1}\end{gathered}$$ The initial conditions for their time derivatives can be derived by combining these with the equations of motion. Define the equilibrium scale height $z_0$ by $$\begin{gathered}
z_0 \doteq \frac{T_0}{mg} \, .\end{gathered}$$To first order in $\delta$, the equations of motion can be written as $$\begin{gathered}
\frac{\partial n_1}{\partial t} = \frac{1}{z_0} \, n_0 v_1 - n_0 \frac{\partial v_1}{\partial x} \label{eqn:nLinear} \\
\frac{\partial v_1}{\partial t} = \frac{1}{m z_0} \, T_1 - \frac{1}{m} \frac{\partial T_1}{\partial x} - \frac{T_0}{m n_0} \frac{\partial n_1}{\partial x} - \frac{T_0}{m z_0 n_0} \, n_1 \label{eqn:vLinear} \\
\frac{\partial T_1}{\partial t} = (\gamma-1) \frac{T_0}{n_0} \bigg( \frac{\partial n_1}{\partial t} - \frac{1}{z_0} \, v_1 n_0 \bigg) \label{eqn:TLinear} .\end{gathered}$$ Taking an additional time derivative of Eq. (\[eqn:vLinear\]) and plugging in Eqs. (\[eqn:nLinear\]) and (\[eqn:TLinear\]), $$\begin{gathered}
\frac{\partial^2 v_1}{\partial t^2} = \frac{\gamma T_0}{m} \bigg( \frac{\partial^2 v_1}{\partial x^2} - \frac{1}{z_0} \frac{\partial v_1}{\partial x} \bigg) .\end{gathered}$$ Define $c_s^2 \doteq \gamma T_0 / m$ and $f \doteq v_1 e^{-x/2z_0}$. Then $$\begin{gathered}
\frac{\partial^2 f}{\partial t^2} = c_s^2 \bigg( \frac{\partial^2 f}{\partial x^2} - \frac{1}{4 z_0^2} \, f \bigg) . \label{eqn:fWave}\end{gathered}$$ Applying the boundary conditions at $x = 0$ and $x = L$, $f$ can be written as $$\begin{gathered}
f(t,x) = \sum_{n=1}^\infty \Xi_n(t) \sin \bigg( \frac{\pi n x}{L} \bigg) \, \end{gathered}$$ for some functions $\Xi_n(t)$. Then Eq. (\[eqn:fWave\]) implies $$\begin{gathered}
\ddot{\Xi}_n(t) = - c_s^2 \bigg( \frac{\pi^2 n^2}{L^2} + \frac{1}{4 z_0^2} \bigg) \Xi_n. \end{gathered}$$ The time-dependent coefficients are linear combinations of sine and cosines in time. In order to get $v_1 = 0$ at $t=0$, only the sine terms can survive. As such, $$\begin{gathered}
f(t,x) = \sum_{n=1}^\infty \alpha_n \sin (k_n x) \sin (\omega_n t) \label{eqn:fSeriesUndetermined}\end{gathered}$$ for some constants $\alpha_n$, with $k_n$ and $\omega_n$ defined by $$\begin{gathered}
k_n \doteq \frac{\pi n}{L} \\
\omega_n \doteq c_s \sqrt{ \frac{\pi^2 n^2}{L^2} + \frac{1}{4 z_0^2} } = \omega_0 \sqrt{1 + 4 z_0^2 k_n^2} \, . \label{eqn:omegaN}\end{gathered}$$ Here $\omega_0 = c_s / 2 z_0$. In order to determine the constants $\alpha_n$, consider the initial condition on $\partial v_1 / \partial t$. Combining Eq. (\[eqn:vLinear\]) with Eqs. (\[eqn:initialN1\]), (\[eqn:initialT1\]), and (\[eqn:initialV1\]), $$\begin{gathered}
\frac{\partial v_1}{\partial t} \bigg|_{t=0} = g \delta, \end{gathered}$$ so $$\begin{gathered}
\frac{\partial f}{\partial t} \bigg|_{t=0} = g \delta e^{-x/2z_0} .\end{gathered}$$ The sine series for $e^{-x/\lambda}$ is $$\begin{gathered}
e^{-x/\lambda} = \sum_{n=1}^\infty \frac{2 n \pi \lambda^2}{L^2 + n^2 \pi^2 \lambda^2} \big[ 1 + (-1)^{n+1} e^{-L/\lambda} \big] \sin (k_n x). \end{gathered}$$ Using this, $$\begin{aligned}
&\frac{\partial f}{\partial t} \bigg|_{t=0} = \frac{g z_0 \delta}{L} \nonumber \\
&\hspace{9 pt}\times \sum_{n=1}^\infty \frac{8 k_n z_0}{1 + 4 k_n^2 z_0^2} \big[ 1 + (-1)^{n+1} e^{-L/2 z_0} \big] \sin(k_n x). \end{aligned}$$ Eq. (\[eqn:fSeriesUndetermined\]) implies that $$\begin{aligned}
\frac{\partial f}{\partial t} \bigg|_{t=0} &= \sum_{n=1}^\infty \omega_n \alpha_n \sin(k_n x). \end{aligned}$$ This determines the $\alpha_n$ parameters. $$\begin{aligned}
&f = \frac{2 g z_0^2 \delta}{L c_s} \sum_{n=1}^\infty \bigg[ \frac{8 k_n z_0}{(1 + 4 k_n^2 z_0^2)^{3/2}} \nonumber \\
&\hspace{28 pt}\times \big[ 1 + (-1)^{n+1} e^{-L/2 z_0} \big] \sin(k_n x) \sin(\omega_n t) \bigg].\end{aligned}$$ The governing equation for $T_1$ can be written as $$\begin{aligned}
\frac{\partial T_1}{\partial t} &= - (\gamma - 1) T_0 \bigg( \frac{\partial f}{\partial x} + \frac{f}{2 z_0} \bigg) e^{x/2z_0}, \end{aligned}$$ which is $$\begin{aligned}
&\frac{\partial T_1}{\partial t} = - \frac{\gamma - 1}{\gamma} \frac{2 c_s T_0 \delta}{L} e^{x/2z_0} \nonumber \\
&\hspace{10 pt}\times \sum_{n=1}^\infty \bigg[ \frac{4 k_n z_0}{(1+4 k_n^2 z_0^2)^{3/2}} \big[ 1 + (-1)^{n+1} e^{-L/2z_0} \big] \nonumber \\
&\hspace{30 pt}\times \bigg( \sin(k_n x) + 2 k_n z_0 \cos (k_n x) \bigg) \sin(\omega_n t) \bigg] .\end{aligned}$$ Integrating and applying the initial condition on $T_1$, $$\begin{aligned}
&\frac{T_1}{T_0} = \delta - \frac{\gamma - 1}{\gamma} \frac{4 z_0 \delta}{L} e^{x/2z_0} \nonumber \\
&\hspace{5 pt}\times \sum_{n=1}^\infty \bigg[ \frac{4 k_n z_0}{(1+4 k_n^2 z_0^2)^2} \big[ 1 + (-1)^{n+1} e^{-L/2z_0} \big] \nonumber \\
&\hspace{9 pt} \times \bigg( \sin(k_n x) + 2 z_0 k_n \cos (k_n x) \bigg) [1 - \cos(\omega_n t)] \bigg] . \label{eqn:fastCompressionSolution}\end{aligned}$$ Define the field-strength parameter $G$ as $$\begin{gathered}
G \doteq \frac{L}{z_0} = \frac{m g L}{T_0} \, .\end{gathered}$$ In terms of $G$, $$\begin{aligned}
&\frac{T_1(t,x)}{T_0} = \delta - \frac{\gamma - 1}{\gamma} \big( 8 G \delta \big) e^{(x/L) (G/2)} \nonumber \\
&\hspace{5 pt}\times \sum_{n=1}^\infty \bigg[ \frac{4 \pi n}{(G^2+4 \pi^2 n^2)^2} \big[ 1 + (-1)^{n+1} e^{-G/2} \big] \nonumber \\
&\hspace{9 pt} \times \bigg( G \sin(k_n x) + 2 \pi n \cos (k_n x) \bigg) \sin^2 \bigg( \frac{\omega_n t}{2} \bigg) \bigg] . \label{eqn:fastCompressionSolutionDimensionlessParameters}\end{aligned}$$ Qualitatively, it is clear from Eq. (\[eqn:fastCompressionSolutionDimensionlessParameters\]) that the shape of $T_1(t,x)$ will depend strongly on $G$. Modes other than $n=1$ will contribute significantly when $n \lesssim G / 2 \pi$. When the $n=1$ mode is dominant, the spatial and temporal structure are simple, with a well-defined wavelength and oscillation frequency. As G increases, the spatial structure becomes progressively more complicated.
In the weak-field $G \ll 1$ limit, Eq. (\[eqn:fastCompressionSolutionDimensionlessParameters\]) becomes $$\begin{aligned}
&\lim_{G \rightarrow 0} \frac{T_1(t,x)}{T_0} = \delta - \frac{\gamma - 1}{\gamma} \big( 4 G \delta \big) \nonumber \\
&\hspace{39 pt}\times \sum_{n=1}^\infty \frac{1 + (-1)^{n+1}}{\pi^2 n^2} \cos (k_n x) \sin^2 \bigg( \frac{\omega_n t}{2} \bigg) .\end{aligned}$$ When $G \ll 1$ and $t = L / c_s$, $\sin^2 (\omega_n t / 2) \rightarrow 1+\mathcal{O}(G^2)$ $\forall n \in \mathbb{Z}$. Therefore, the maximal temperature difference between $x = 0$ and $x = L$ is $$\begin{aligned}
\label{eq:kappa_limit}
\lim_{G \rightarrow 0} \frac{T_1(L/c_s, L) - T_1(L/c_s, 0)}{T_0} = \frac{\gamma-1}{\gamma} (2 G \delta). \end{aligned}$$ When $\gamma = 5/3$, this is $0.8 G \delta$. This is precisely the analytic result found by Geyko and Fisch in this limit. However, it disagrees with the results of their simulations, in which $\Delta T_1 / T_0 \approx 0.64 G \delta$.
Simulations of the full nonlinear fluid equations given by Eqs. (\[eqn:nonlinearN\]), (\[eqn:nonlinearV\]), and (\[eqn:nonlinearT\]) were performed using the 1D fluid code SNeuT, which uses components of the SUNDIALS suite [@Hindmarsh2005; @Cohen1996]. Figure \[fig:simulations\] shows these simulations alongside the analytically predicted results from the fluid model; when $\delta$ is small, they are in close agreement, including the coefficient of 0.8. The origin of the discrepancy between these and the original paper’s results is discussed in Section \[sec:two\_codes\]. Now consider the opposite limit, where $G \gg 1$: $$\begin{aligned}
&\lim_{G \rightarrow \infty}\frac{T_1(t,x)}{T_0} = \delta - \frac{\gamma - 1}{\gamma} \big( 8 G \delta \big) e^{(x/L) (G/2)} \nonumber \\
&\hspace{5 pt}\times \sum_{n=1}^\infty \bigg[ \frac{4 \pi n}{(G^2+4 \pi^2 n^2)^2} \bigg( G \sin(k_n x) + 2 \pi n \cos (k_n x) \bigg) \nonumber \\
&\hspace{140 pt}\times \sin^2 \bigg( \frac{\omega_n t}{2} \bigg) \bigg] .\end{aligned}$$ This can be converted to an integral: $$\begin{aligned}
&\lim_{G \rightarrow \infty}\frac{T_1(t,x)}{T_0} = \delta - \frac{\gamma - 1}{\gamma} \frac{8 \delta}{\pi} e^{(x/L) (G/2)} \nonumber \\
&\hspace{0 pt}\times \int_0^\infty \bigg[ \frac{4 y \, {\mathrm{d}}y}{(1+4 y^2)^2} \bigg( \sin\bigg( \frac{G y x}{L} \bigg) + 2 y \cos \bigg( \frac{G y x}{L} \bigg) \bigg) \nonumber \\
&\hspace{100 pt}\times \sin^2 \bigg( \frac{G c_s t}{L} \sqrt{y^2 + \frac{1}{4}} \bigg) \bigg] . \label{eqn:largeGSolution}\end{aligned}$$ When $G$ becomes very large, the fluid becomes strongly rarefied and heated near $x = L$. When calculating the size of the temperature separation across the system, it makes more sense to compare the temperature at $x = 0$ with that at a scaled height $x = z_0 \log 10$. The integral in Eq. (\[eqn:largeGSolution\]) can be evaluated numerically, and the maximal difference between $T_1(t, z_0\log 10) / T_0$ and $T_1(t, 0) / T_0$ is about $0.49 \delta$ when $\gamma = 5/3$ (the minimum is about $-0.53 \delta$). Geyko and Fisch did not make an analytic prediction of this dependence, but they did investigate it numerically, and their simulations found $0.47 \delta$ for the maximum.
![image]("arrowFig".pdf){width="\linewidth"}
Formally, the analytic calculations in this section are done in the limit of small $\delta$. It is natural to wonder how small $\delta$ has to be in order for the calculations to be accurate. The nonlinear fluid simulations shown in Figure \[fig:simulations\] shed some light on this point. When $\delta = 0.01$, the fluid simulations are almost indistinguishable from the analytic results. When $\delta$ is increased to $0.5$, the accuracy of the analytic results depends strongly on $G$.
For $G = 0.1$ and $G = 1$, the $\delta = 0.5$ simulations are qualitatively very similar to the small-$\delta$ analytic results, except that the oscillations appear to take place at a higher frequency. This results from the temperature dependence of the system frequencies $\omega_n$. In Eq. (\[eqn:omegaN\]), these frequencies are written as functions of the pre-compression temperature $T_0$. However, physically, the system’s frequency response after compression should scale with $T_i = (1 + \delta) T_0$ rather than $T_0$ (though the value of $T_0$ will determine which modes are excited). This distinction is not important when $\delta$ is small, but as $\delta$ grows larger it begins to matter. The simulations with $G = 0.1$ and $G = 1$ are dominated by the $n=1$ mode. If the frequency $\omega_1$ is evaluated at $T_i$ rather than $T_0$, $\omega_1$ increases by about 22% when $G = 0.1$ or $1$. This is consistent with the higher-frequency $n=1$ modes observed in the simulations.
However, when $\delta = 0.5$ and $G = 8$, the fluid simulations no longer resemble the small-$\delta$ calculations. This can be explained by the dependence of $T_1$ on $G$. $T_1$ depends nonlinearly on $G$, but in general $T_1$ grows larger as $G$ increases. As such, the $\delta$ that is required to keep $T_1 \ll T_0$ is smaller for larger values of $G$. For the simulations in Figure \[fig:simulations\], $T - T_0 < T_0$ when $G = 0.1$ and $G = 1$, but when $G = 8$ and $\delta = 0.5$, there are regions with $T - T_0 > T_0$ and the perturbative model is no longer valid.
Arbitrary Compression Profiles {#sec:slowCompression}
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The analysis in Section \[sec:fastCompression\] describes fast compression, so that the system starts out of equilibrium at $t = 0$ and is not driven after $t = 0$. It is possible to approach the case of more general heating profiles by instead allowing the system to start at equilibrium and imposing a time-dependent heat source. Suppose, to leading order, the heat source produces a spatially constant change in temperature. Then Eq. (\[eqn:TLinear\]) becomes $$\begin{aligned}
\frac{\partial T_1}{\partial t} = (\gamma-1) \frac{T_0}{n_0} \bigg( \frac{\partial n_1}{\partial t} - \frac{1}{z_0} \, v_1 n_0 \bigg) + \chi(t)\end{aligned}$$ for some heating function $\chi(t)$. $f = v_1 e^{-x/2z_0}$ can be defined the same way, but its governing equation now depends on $\chi$: $$\begin{aligned}
\frac{\partial^2 f}{\partial t^2} = c_s^2 \bigg( \frac{\partial^2 f}{\partial x^2} - \frac{1}{4 z_0^2} f \bigg) + \frac{\chi(t)}{m z_0} \, e^{x/2z_0} \, .\end{aligned}$$ Define $\Phi_n(t)$ by $$\begin{aligned}
&\Phi_n(t) \doteq \frac{\omega_n}{T_0} \int_0^t {\mathrm{d}}t' \sin(\omega_n t') \int_0^{t'} {\mathrm{d}}t'' \, \chi(t'') \cos(\omega_n t'') \nonumber \\
&\hspace{16 pt}- \frac{\omega_n}{T_0} \int_0^t {\mathrm{d}}t' \cos(\omega_n t') \int_0^{t'} {\mathrm{d}}t'' \, \chi(t'') \sin(\omega_n t'') \, . \label{eqn:Phi}\end{aligned}$$ In terms of $\Phi_n(t)$, the solution for $T_1$ is $$\begin{aligned}
&\frac{T_1}{T_0} = \int_0^t \frac{\chi(t') {\mathrm{d}}t'}{T_0} - \frac{\gamma-1}{\gamma} \big( 4 G \big) e^{(x/L)(G/2)} \nonumber \\
&\hspace{0 pt} \times \sum_{n=1}^\infty \bigg[ \frac{4 \pi n}{(G^2+4 \pi^2 n^2)^2} [1+(-1)^{n+1} e^{-G/2}] \nonumber \\
&\hspace{40 pt} \times \bigg( G \sin(k_n x) + 2 \pi n \cos(k_n x) \bigg) \Phi_n(t) \bigg]. \label{eqn:slowCompressionSolution}\end{aligned}$$ Consider the case of steady heating for an interval $\tau$. Set $$\begin{gathered}
\chi(t) = \begin{cases}
\delta T_0 / \tau & 0 \leq t \leq \tau \\
0 & t<0 \text{, } t > \tau.
\end{cases} \label{eqn:steadyPsi}\end{gathered}$$ Here, the parameter $\delta$ is analogous to the corresponding parameter in the fast-compression case. Using this choice of $\chi(t)$, $$\begin{aligned}
&\Phi_n(0 \leq t \leq \tau) = \bigg( \frac{t}{\tau} - \frac{\sin(\omega_n t)}{\omega_n \tau} \bigg) \delta\end{aligned}$$ and $$\begin{aligned}
&\Phi(t > \tau) = \bigg( 1 + \frac{\sin(\omega_n(t-\tau))}{\omega_n \tau} - \frac{\sin(\omega_n t)}{\omega_n \tau} \bigg) \delta .\label{eqn:PhiAfterTau}\end{aligned}$$ In the fast-compression limit where $\tau \rightarrow 0$, Eqs. (\[eqn:slowCompressionSolution\]) and (\[eqn:PhiAfterTau\]) reduce to Eq. (\[eqn:fastCompressionSolutionDimensionlessParameters\]). On the other hand, in the limit of very slow compression, $$\begin{aligned}
&\lim_{\omega_n \tau \rightarrow \infty} \frac{T_1(t>\tau)}{T_0} = \delta - \frac{\gamma-1}{\gamma} \big( 4 G \delta \big) e^{(x/L)(G/2)} \nonumber \\
&\hspace{0 pt} \times \sum_{n=1}^\infty \bigg[ \frac{4 \pi n}{(G^2+4 \pi^2 n^2)^2} [1+(-1)^{n+1} e^{-G/2}] \nonumber \\
&\hspace{40 pt} \times \bigg( G \sin(k_n x) + 2 \pi n \cos(k_n x) \bigg) \bigg]. \label{eqn:verySlowCompressionSolution}\end{aligned}$$ When $\omega_n \tau$ is large, the temperature gradient is not oscillatory. This is consistent with the intuition that a slowly driven system will remain close to force equilibrium. The temperature difference across the system can be written in closed form as $$\begin{aligned}
&\lim_{\omega_n \tau \rightarrow \infty} \frac{T_1(t > \tau, L) - T_1(t > \tau, 0)}{T_0} = \frac{\gamma - 1}{\gamma} \big(G \delta \big).\end{aligned}$$ In the limit where $G \ll 1$, the temperature difference across the system for slow compression will be half of the maximal temperature difference for fast compression. This agrees exactly with the analytic result of Geyko and Fisch in that limit, though their simulations yielded a somewhat smaller coefficient.
Of course, Eqs. (\[eqn:Phi\]) and (\[eqn:slowCompressionSolution\]) make it clear that things can turn out quite differently if $\chi$ has a more complicated time dependence. It was already true in the simple case described by Eq. (\[eqn:steadyPsi\]) that a careful choice of $\tau$ could either suppress or enhance the oscillations associated with a particular mode number. If, for instance, $\chi$ itself were oscillatory, then particular modes could be driven or suppressed even more dramatically. Consider the oscillatory heating function $$\begin{gathered}
\chi(t) = \delta \, \Omega \, T_0 \sin(\Omega t) \end{gathered}$$ where $\Omega$ is some positive frequency. Heating of precisely this form may not necessarily be practically realizable, but it is an informative formal example. For this choice of $\chi$, $$\begin{gathered}
\Phi_n(t) = \frac{[\omega_n^2 - \Omega^2 - \omega_n^2 \cos(\Omega t) + \Omega^2 \cos(\omega_n t) ] \delta}{\omega_n^2-\Omega^2} \, . \end{gathered}$$ When the driving frequency is close to $\omega_n$, there is a secular term. To leading order in $\Omega - \omega_n$, $$\begin{gathered}
\Phi_n(t) \rightarrow \bigg( 1 - \cos(\omega_n t) - \frac{\omega_n t}{2} \sin(\omega_n t) \bigg) \delta. \end{gathered}$$ This holds even for higher-frequency oscillations whose role in the bulk behavior of the system would normally be small. Driving at one of the system’s natural frequencies can produce temperature oscillations that (at least as far as the linear theory is concerned) can grow without bound. If the system is driven at $\omega_n$, the resonant oscillations will be associated with the corresponding spatial wavenumber $k_n$. All of this behavior is intuitive, if the system’s response to $\chi(t)$ is understood in terms of the mode decomposition that comes naturally from the fluid picture.
Comparison of the fluid and Monte Carlo simulations {#sec:two_codes}
===================================================
As pointed out, the numerical results from the original paper on the piezothermal effect [@Geyko2014], obtained via Monte Carlo simulations, are qualitatively similar to the ones obtained in the present work, yet deviate quantitatively in many cases. The main reason for this is the fact that the Monte Carlo code has intrinsic physical and numerical damping built in due to the finite mean free paths of the particles. To get a better understanding of this phenomenon, we briefly review the Monte Carlo code from the original paper.
The object of the simulations is a set of ideal particles that move in a one-dimensional box in a constant gravitational field $\textbf{g} = - g \hat{x}$. The box is considered infinite or periodic in the perpendicular directions $\hat{y}$ and $\hat{z}$, and of the length $L$ in the $\hat{x}$ direction. Particle velocities, however, have all three components ($v_x$, $v_y$, and $v_z$) for the sake of preserving the proper value of the adiabatic gas constant $\gamma=5/3$. A particle’s motion is exactly integrated for every time step $\delta t$, and takes into account the possibility of multiple particle-wall collisions on the box floor.
A non-interacting ensemble of particles does not represent a fluid-like motion. Instead, it will produce complex but uncorrelated behavior, like the density waves described in [@Kolmes2016]. In order to make the system behave like a fluid, particle collisions are added. In the code, only binary elastic collisions are considered, such that energy, momentum, and angular momentum are conserved up to machine precision for each individual collision and, as a result, for the whole system. The main problem of such a collision operator is that any two particles are never located at the same point in space. In principle, a given pair of particles can be tracked and the time of the true collision can be found, yet this is too complicated if all the particles are required to collide every time step. Thus, some nearly located particles are picked for each collision. The domain is divided in the $\hat{x}$ direction into a number of cells, each of the same length $L_c$ for simplicity. Since the particles are not at exactly the same point, the collision should be acting along the direction $\hat{\ell}$ connecting the centers of the two particles, otherwise the angular momentum will not be conserved. One can think about this type of collision as an instantaneous force acting between the two particles, like gravitational attraction. This force should change somehow the projections of particle velocities $v_{1\ell}$ and $v_{2\ell}$ in such a way that the total kinetic energy and momentum are conserved. For identical particles, it is done by exchanging their velocity projections: $v_{1\ell} \to v_{2\ell}$ and $v_{2\ell} \to v_{1\ell}$. Since the two particles are picked at random inside a cell, the distance $d$ between them is of the order of $L_c$. The angle between the direction $\hat{\ell}$ and $\hat{x}$ is $\theta$, and it is picked at random but is typically about $\theta\approx \pi/3$ or similar, because the perpendicular displacement is picked uniformly in both directions from $-L_c$ to $L_c$.
This collision operator exactly conserves energy, momentum and angular momentum, but suffers from numerical heat and momentum transfer due to finite cell size effects. This can be understood in the following way: imagine the cell size is equal to the box height, and a hot population of the particles is sitting at the bottom. In this case, the numerical thermalization would occur instantly, and the particles on the top would get hot even faster than a sound wave can travel across the domain.
To be more specific, consider two particles inside a cell located at coordinates $x_1$ and $x_2$, respectively. For highly collisional gas, which is of interest here, a Maxwellian distribution can be assumed, with temperature $T(x)$, mean velocity $u(x) \hat x$, and density $n(x)$. As a collision occurs, an instantaneous transfer of the momentum from the second particle to the first one can be written as $$\begin{gathered}
\frac{\Delta \textbf{p}}{m} = \int d^3v_1 f_1(\textbf{v}_1, x_1) \int d^3v_2 f_2(\textbf{v}_2, x_2) [\tilde{\textbf{v}}_2 - \tilde{\textbf{v}}_1] ,\end{gathered}$$ where $\tilde{\textbf{v}}$ is a projection of the velocity to the $\hat{\ell}$ direction $\tilde{\textbf{v}}=\hat{\ell}(\hat{\ell}\cdot\textbf{v})$. Integrals with respect to $v_y$ and $v_z$ vanish, because the integrated function is antisymmetric, and the integral with respect to $v_x$ yields $$\begin{gathered}
\label{eq:mom_1}
\Delta \textbf{p} = m \hat{\ell} \ell_x \left( u(x_2)-u(x_1)\right),\end{gathered}$$ where only $\Delta p_x$ is of interest since the other two components vanish, as an averaging over $\hat{\ell}$ is performed, thus, $$\begin{gathered}
\label{eq:mom_2}
\Delta p = \Delta p_x = m \cos^2(\theta)\left( u(x_2)-u(x_1)\right).\end{gathered}$$
For a particle at a given position $\bar{x}$ inside the cell ($\bar{x}=0$ at the center of the cell), the total momentum transfer from all the particles around is found as a mass weighed integral over all the cell of Eq. (\[eq:mom\_2\]), where density and velocity are Taylor expanded around the cell-center point $x_c$. This integral should be also multiplied by a collision rate parameter $R_c$, which is proportional to the number of collisions occurred in the given cell each time step. $$\begin{aligned}
\label{eq:mom_tot}
&\Delta p_\text{tot} = m R_c \int\limits_{-L_c/2}^{L_c/2} \cos^2 \theta \left[\left(n_c + n'\xi + \frac{n''}{2}\xi^2\right) \right. \nonumber \\
&\hspace{70 pt}\left. \cdot \left( u'(\xi-\bar{x}) +\frac{u''}{2}(\xi^2 -\bar{x}^2) \right) \right]d\xi.\end{aligned}$$ The result of expression (\[eq:mom\_tot\]) depends on the value of $\bar{x}$, however for any $\bar{x}$ there always present a term proportional to $m R_c n_c u'' L_c^3$. Notice that $n_c L_c \approx N_p$, where $N_p$ is the number of particles in the cell, and the momentum transfer found in Eq. (\[eq:mom\_tot\]) happens in a time step $\delta t$. Therefore, there is a momentum transfer term with $$\begin{gathered}
\frac{\partial p}{\partial t} \propto \frac{m R_c N_p L_c^2}{\delta t}\frac{\partial^2 u}{\partial x^2}, \end{gathered}$$ and Eq. (\[eqn:nonlinearV\]) then reads as $$\begin{gathered}
m n \bigg( \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial x} \bigg) = - \frac{\partial (nT)}{\partial x} - m n g + \nu m n \frac{\partial^2 u}{\partial x^2},\end{gathered}$$ where $\nu$ is the derived numerical viscosity with $\nu \propto R_c L_c^2/\delta t$. The derivation of numerical heat conductivity is very similar to the one for viscosity, and therefore is omitted here.
![image]("dampingComparison".pdf){width="\linewidth"}
Apart from numerical viscosity and heat conductivity, driven mainly by a finite cell size, there is a physical mechanism of heat conductivity due to finite particle mean free path. The last is determined be the collision rate $R_c$, the time step $\delta t$, and the mean particle velocity $v_t$ and does not depend on the cell size. Indeed, consider a generalized version of Eq. (\[eqn:nonlinearT\]) with heat transfer term included in it $$\begin{gathered}
\frac{n}{\gamma-1}\left(\frac{\partial T}{\partial t} + v\frac{\partial T}{\partial x}\right) + nT\frac{\partial v}{\partial x} = \frac{\partial}{\partial x}\left(\Psi \frac{\partial T}{\partial x}\right).\label{eqn:energy2}\end{gathered}$$ Here, $\Psi$ is the heat conductivity coefficient, given in terms of the mean free path $\lambda_\text{mfp}$ as $\Psi \approx n \lambda_\text{mfp} v_t c_v/3$. Eq. (\[eqn:energy2\]) reduces to Eq. (\[eqn:nonlinearT\]) if $\Psi = 0$. When $\Psi > 0$, heat diffusion leads to wave dissipation and system equilibration.
Notice that the aforementioned arguments are not a rigorous derivation of the numerical viscosity and heat conductivity in the Monte Carlo code. They can only provide some insights on why Monte Carlo simulations sometimes produce different results. However, even such a simplified picture is enough to explain, for example, why the piezothermal coefficient $$\begin{gathered}
\kappa \doteq \frac{T_1(L/c_s,L)-T_1(L/c_s,0)}{G\delta T_0}\end{gathered}$$ was 0.64 instead of 0.8 (see Eq. (\[eq:kappa\_limit\])) in the numerical results from the original paper. In particular, we are interested in how $\kappa$ depends on the length $L_c$, which was described by a parameter $N_c$ in the code, where $N_c L_c = 1$.
Figure \[fig:osc\_NC\] shows how the piezothermal temperature difference evolves as a function of time in a series of simulations using two different codes: one performing Monte Carlo simulations and the other performing fluid simulations. The Monte Carlo simulations, denoted by plus marks, show the temperature difference for four different values of $N_c$, while all other parameters of the code were fixed, namely, $\delta t = 0.001$, $T_0 = 0.3698$, $R_c = 10$ (collisions per particle per cell), $G = 1.352$, $\delta T_0 = 0.0518$. Only for $N_c=240$ the first peak of the oscillations is sufficiently close to the predicted value 0.8, yet the oscillations nevertheless slowly damp in time. For low values of $N_c$ fluid oscillations are very quickly damped, and the system decays to a new equilibrium.
The solid lines in Figure \[fig:osc\_NC\] show a corresponding series of fluid simulations. In these simulations, the field strength parameter $G$ and the heating parameter $\delta$ are chosen to match the values in the Monte Carlo simulations. Each of these fluid simulations includes a spatially constant viscosity $\eta$ and heat conductivity $\Psi$. Of course, discretization error is not a phenomenon unique to Monte Carlo algorithms. Fluid simulations also have finite-grid-size effects. The fluid simulations shown here use sufficiently fine-grained grids that these errors are negligible compared to the corresponding effects in the Monte Carlo code (in this example, the fluid simulations used 128 cells).
Both the Monte Carlo simulations and the fluid simulations show oscillations that are “lopsided," in the sense that they are asymmetric about their extrema. The asymmetry is most apparent in the $N_c = 240$ case. This results from the same nonlinearity discussed at the end of Section \[sec:fastCompression\], in which $\delta$ and $G$ are large enough for the oscillations not to be small perturbations. It is worth noting that these asymmetric oscillations still appear even in fluid simulations without any viscosity or heat conductivity (not shown in Figure \[fig:osc\_NC\]).
In any case, there are two major conclusions to be drawn from the comparison in Figure \[fig:osc\_NC\]. First, the finite-cell-size effects seen in the Monte Carlo simulations appear to be equivalent to an effective viscosity and heat conductivity. Second, the effective viscosity and heat conductivity become small when $N_c$ is large.
Discussion and Conclusions
==========================
Using a fluid model, we have derived analytic expressions for the temperature gradients of the piezothermal effect as they evolve in time. The fluid solutions recover the original analytic model’s predictions for $G \ll 1$ and they make it possible to make predictions when $G$ is not small. Similarly, they recover the original model’s qualitative predictions for very slow and very fast compression while also handling more general compression profiles, including compression that is not constant in time and compression that is neither very fast nor very slow. The analytic solutions to the fluid equations are in very good agreement with fluid simulations performed using the SNeuT fluid code.
There are places where the results from fluid models disagree quantitatively with some of the numerical results from the original paper. The comparison between the present fluid and the original Monte Carlo simulations provides some explanation for why the previous results were different, and what can be done in order to improve them in the Monte Carlo model. In general, a small time step and a very large number of cells are required in order to sufficiently suppress numerical and physical heat diffusion and viscosity in the Monte Carlo simulations. That brings extra complication for the total number of particles in the system, as the number of particles in a cell should be large enough to mitigate statistical noise. However, there is evidence that (in the appropriate limit) the Monte Carlo simulations converge to results that agree with the fluid model.
The fluid model used in this paper makes assumptions. The strict timescale ordering means that viscosity and heat conductivity are neglected (with the exception of the simulations used to produce Figure \[fig:osc\_NC\], which included both), though the calculation in Section \[sec:slowCompression\] makes it possible to relax the requirement for an ordering between the compression timescale $\tau_E$ and the sound timescale $\tau_s$. The analytic calculations presented here use linearized fluid equations; they become invalid when the compression parameter $\delta$ is large. However, these assumptions were also necessary for the model used in the original paper.
The mode structure of the analytic solutions helps to provide intuition for the behavior of the piezothermal effect. The critical dependence of the effect on the field-strength parameter $G$ can be explained by the mode structure: as $G$ increases, modes other than $n=1$ become important when $n \lesssim G / 2 \pi$. When $G$ is small, the piezothermal effect is dominated by a single frequency and a single wavenumber; when $G$ is large, many frequencies and wavenumbers contribute, and the oscillations can become much more complicated.
The characteristic frequencies $\omega_n$ are closely related to the Brunt-Väisälä frequency, which is important in a variety of geophysical, astrophysical, oceanographic, and atmospheric contexts [@Brunt1927; @Durran1982; @Emery1984; @Brassard1991]. Brunt-Väisälä oscillations occur when a fluid element is displaced within a stratified background. For a parcel of air displaced in a dry, isothermal atmosphere, the Brunt-Väisälä frequency can be written as [@Brunt1927] $$\begin{gathered}
\omega_\text{BV} = \sqrt{ \frac{g \Gamma_d}{T} } = \sqrt{ \frac{g^2}{c_p T} } = \frac{2 \omega_0}{\sqrt{\gamma m c_p}} \, ,\end{gathered}$$ where $\Gamma_d$ is the dry adiabatic lapse rate and $c_p$ is the specific heat capacity.
The scenario being considered here is not quite identical to the prototypical Brunt-Väisälä buoyancy oscillation; for one thing, the entire system is displaced, rather than a small fluid element within the system. However, the oscillations associated with the piezothermal effect can be understood as a spectrum of buoyancy oscillations which are closely related to Brunt-Väisälä oscillations.
Acknowledgements {#acknowledgements .unnumbered}
================
VIG was supported under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. EJK and NJF were supported by NSF PHY-1506122 and NNSA 83228-10966 \[Prime No. DOE (NNSA) DE-NA0003764\]. The SNeuT simulation code uses the CVODE package, an open source software package which is part of Lawrence Livermore National Laboratory’s SUNDIALS suite. SNeuT is a fork of the MITNS plasma transport code [@KolmesMITNSarXiv]. Authors are thankful to Eric Emdee, Mike Mlodik, and Jace Waybright for fruitful discussions, and to Ian Ochs for fruitful discussions and for involvement in code development.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Leon Loveridge[^1]'
- 'Paul Busch[^2]'
date: '[**To appear in: Eur. Phys. J. D (2011)**]{}'
title: '‘Measurement of Quantum Mechanical Operators’ Revisited'
---
Introduction {#sec:intro}
============
Quantum mechanical experiments involving the manipulation of individual quantum objects no longer reside only in the minds of a few theoretical physicists, but are a routine occurrence across many physical disciplines such as quantum optics and quantum information. This not only provides new and exciting opportunities for future technologies such as quantum computing, but necessitates a fundamental re-examination of the quantum mechanical formalism itself, and a new understanding of its role in modern applications. With the ever-decreasing size of the components involved in these technologies, it is both interesting from a foundational viewpoint and important in more practical respects to understand any fundamental limitations on the possible size of such microscopic instruments.
One such limitation arises as a consequence of conservation laws for additive quantities that do not commute with the observable to be measured. Whilst considering spin-$\frac 12$ measurements, Wigner [@wigner] discovered that the total angular momentum of the object plus apparatus cannot be conserved in an accurate and repeatable measurement of a particular component. This observation was soon stated in greater generality as a theorem by Araki and Yanase [@way] that has become known as the Wigner-Araki-Yanase (WAY) theorem. Despite the fact that the original papers [@wigner] and [@way] have been widely noted and the WAY theorem has been extended in various respects, its full scope is still unknown.
It is the purpose of this paper to survey the evolution of formulations of WAY-type theorems, elucidate the significance of the underlying assumptions, and clarify the general structure and extent of such theorems. We will also provide some new extensions of known results and propose an answer to a long-standing question concerning the possibility of momentum-conserving measurements of the position of a quantum particle.
In Sec. \[sec:wigner\] we revisit Wigner’s 1952 paper [@wigner]. In particular we scrutinize the final page where Wigner examines the consequences of dropping the assumption that the measurement be repeatable. This is a relaxation which is physically relevant, but is still not appreciated by many practitioners of quantum theory. Wigner notes that in this case the issue arises of the distinguishability of the states of the measuring apparatus, given that the limitation imposed by the conservation law also applies to a measurement of the pointer. The paper [@wigner] is written (in German) with the simplicity and elegance characteristic of Wigner; in order to make it more widely accessible, a translation into English is provided as a concurrent publication [@wigner-en].
In Sec. \[sec:ay\] we proceed to give a modification of the proof of Araki and Yanase [@way] leading to a sharpening and extension of the WAY theorem. They prove for certain classes of observables and conserved quantities that under the assumption of accuracy and repeatability, the observable to be measured must commute with the (object part of) the conserved quantity. Here we show that the same conclusion follows if the repeatability of the measurement is replaced by the assumption that the pointer observable commutes with the conserved quantity. This condition, which following Ozawa [@ozawacon] we shall call *Yanase condition*, was already alluded to in [@wigner] and [@yanase-opt]. In fact, the WAY theorem also precludes accurate and repeatable measurements of the pointer observable (given the conservation law) unless the Yanase condition is fulfilled.
In Sec. \[sec:approx\] we review formulations of WAY-type limitations for approximate measurements. In particular we present and develop an inequality first formulated by Ozawa [@ozawacon] that demonstrates trade-off relations between a measure of error and the “size" of the apparatus (suitably defined). In Sec. \[sec:wayout\] we revisit some model measurement schemes, notably by Ohira and Pearle [@ohira/pearle], and observe that the “ways out" of the WAY limitation sought there always come at the expense of violating the repeatability *and* Yanase conditions. This helps to highlight the fact that the WAY theorem is often paraphrased in a superficial way, ignoring the repeatability property and the relevance of the Yanase condition.
Sec. \[sec:pos\] contains a description of the largely unexplored question of whether position measurements that respect momentum conservation are subject to a WAY-type limitation. Here we adapt Ozawa’s inequality to establish the necessity of a large apparatus for good measurements, provided that the Yanase condition is satisfied. We also formulate a trade-off inequality analogous to Ozawa’s inequality with which one can quantify the degree of repeatability achievable given the size of the apparatus. Finally we provide an affirmative answer, in a certain approximate sense, to a problem posed by Stein and Shimony in 1979 [@Stein-Shimony] concerning the feasibility of repeatable position measurements obeying momentum conservation.
The paper concludes with some remarks on the relavance of the WAY theorem in contemporary quantum physics and quantum information.
We begin with an outline of concepts of quantum measurement theory relevant to our investigation.
Preliminaries {#sec:prelim}
=============
We will apply the standard formulation of quantum measurements (e.g., [@QTM]) where the quantum system and apparatus are represented by complex Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. These come equipped with the usual inner products denoted $\left\langle \cdot|\cdot\right\rangle$. Observables are given as positive operator valued measures (POVMs) $\mathsf{E}:X\mapsto \mathsf{E}(X)$. The operators $\mathsf{E}(X)$ associated with subsets $X$ of $\mathbb R$ are positive operators whose expectation values $\langle\psi |{\mathsf{E}}(X) \psi \rangle$ (for normalized states $\psi\in\mathcal{H}$) represent the measurement outcome probabilities of finding a result in $X$; these operators are called [*effects*]{}. If these effects are projections, the POVM $\mathsf{E}$ is a spectral measure from which one recovers the standard representation of an observable as a self-adjoint operator, namely $\int x\mathsf{E}(dx)\equiv M$.
For a measurement to take place, there must be an interaction between the object system and a macroscopic measuring apparatus, whereby the experimenter can read off the values of the measured observable. The part of this device that actually comes into contact with the quantum system under investigation may only be a small component of the whole apparatus (and could be referred to as a [*probe*]{}). We shall not discuss the process of amplification by which information from the interaction generates macroscopic outcome values.
The composite system-apparatus Hilbert space is described by the tensor product $\mathcal{H}_{T}:=\mathcal{H}\otimes \mathcal{K}$. The time evolution of the system-plus-apparatus is then given by a unitary operator $U$ on $\mathcal{H}_{T}$, which serves to couple the states of the system to those of the apparatus during an interaction period $\tau $. In order that this interaction may be said to lead to a measurement of an observable ${\mathsf{E}}$, some extra elements are still required; these are a self-adjoint operator $Z$ on $\mathcal{K}$, which represents a pointer observable, and a (fixed) initial apparatus state in $\mathcal{K}$, chosen to be the pure (vector) state $\phi $, with $\left\Vert
\phi \right\Vert=1$.
We then define a measurement ${\mathcal{M}}$ of an observable ${\mathsf{E}}$ as the 5-tuple $\langle \mathcal{K},U,\phi ,Z,f\rangle $ satisfying the *probability reproducibility condition*; that the outcome distribution for ${\mathsf{E}}$ in any state $\varphi $ be recovered from the pointer statistics in the final state $\Psi _{\tau }=U(\varphi \otimes \phi )\in
\mathcal{H}_{T}$. This condition can be written symbolically as $$\left\langle \Psi _{\tau }|\boldsymbol{1}\otimes {\mathsf{E}}^{Z}(f^{-1}(X))\Psi _{\tau}
\right\rangle \equiv \left\langle \varphi |{\mathsf{E}}(X)\varphi \right\rangle,
\label{PRC}$$ where $ {\mathsf{E}}^{Z}(f^{-1}(X))$ are spectral projections of $Z$, and holds for all $\varphi$ and $X$. Conversely, given a measurement scheme as described above, this relation determines the measured observable ${\mathsf{E}}$. The scaling function $f$ is used to map the values of the pointer to those of the measured observable.
A measurement is said to be *repeatable* if, upon immediate repetition of the measurement, the same outcome is achieved with certainty. This may be written: $$\left\langle \Psi _{\tau} |{\mathsf{E}}(X)\otimes {\mathsf{E}}^{Z}(f^{-1}(X))\Psi _{\tau}
\right\rangle =\left\langle \varphi |{\mathsf{E}}(X)\varphi \right\rangle.$$ \[rep\] It should be noted that even when holds, it is not guaranteed that is satisfied, and as such the question of repeatability must be treated independently of that of probability reproducibility. Conditions. and can be rephrased in a more concise form as follows: if the outcome of the first measurement is described by a set $X$, the system’s state is obtained by taking a partial trace operation: $$\rho_X={\rm tr}_{\mathcal K}[\boldsymbol{1}\otimes {\mathsf{E}}^{Z}(f^{-1}(X))|\Psi _{\tau}\rangle\langle \Psi _{\tau}|];$$ Now condition reads: $${\rm tr}[\rho_X]=\left\langle \varphi |{\mathsf{E}}(X)\varphi \right\rangle.$$ Writing $\hat\rho_X=\rho_X/{\rm tr}[\rho_X]$ for the normalized state, the repeatability condition then becomes: $${\rm tr}[\hat\rho_X\,{\mathsf{E}}(X)]=1.$$
Although in many textbooks the term *measurement* is understood as comprising the repeatability property, it is important to recognize that most realistic measurements are not repeatable. Furthermore, measurements are rarely accurate and the actually measured observable is appropriately described as a POVM. We will see below that even as early as 1952 Wigner was working with more general measurement models that do not satisfy the repeatability criterion and whose associated observable is a POVM.
Wigner 1952 {#sec:wigner}
===========
Wigner’s example {#subsec:wigner-example}
----------------
Wigner first noticed that repeatable measurements of the $x$-component of the spin of a spin-$\frac{1}{2}$ particle necessarily violate the conservation of the $z$-component of the total angular momentum of the system plus apparatus, written $S_{z}\otimes \boldsymbol{1}+
\boldsymbol{1}\otimes J_{z}$. He also demonstrated the feasibility of recovering arbitrarily accurate and repeatable measurements if the apparatus becomes “large”. This is a significant feature in much of the work following Wigner’s discovery, and we sketch the argument here. We continue with the notation that $\phi\in\mathcal{K}$ represents the initial (normalized) apparatus state, and $\phi _{\pm }\in\mathcal{K}$ orthonormal pointer states, and throughout we shall choose units where $\hbar =1$. The unitary evolution takes the form (with $\varphi _{\pm }$ representing $S_{x}$ eigenstates): $$\begin{aligned}
\varphi _{+}\otimes \phi &\longrightarrow &\varphi _{+}\otimes \phi _{+},
\label{intro1} \\
\varphi _{-}\otimes \phi &\longrightarrow &\varphi _{-}\otimes \phi _{-};
\label{intro2}\end{aligned}$$ the evolution for the eigenstates $\psi _{\pm }=(\varphi _{+}\pm \varphi _{-})/ \sqrt{2}$ of $S_{z}$ is then $$\begin{aligned}
\psi _{+}\otimes \phi &\longrightarrow &\frac{1}{2}\left[ \psi _{+}\otimes
(\phi _{+}+\phi _{-})+\psi _{-}\otimes (\phi _{+}-\phi _{-})\right] ,
\label{contra} \\
\psi _{-}\otimes \phi &\longrightarrow &\frac{1}{2}\left[ \psi _{+}\otimes
(\phi _{+}-\phi _{-})+\psi _{-}\otimes (\phi _{+}+\phi _{-})\right] .
\label{contra 2}\end{aligned}$$This violates angular momentum conservation, since the expectations $\left\langle S_{z}+J_{z}\right\rangle $ agree on the right hand sides of (\[contra\]) and (\[contra 2\]) but differ by one unit on the left hand sides. Since, as Wigner argues, spin component measurements are “practically possible”, he introduces the following modification in order to model an approximate realization of the measurement: $$\begin{aligned}
\varphi _{+}\otimes \phi &\longrightarrow &\varphi _{+}\otimes \phi
_{+}+\varphi _{-}\otimes \eta , \\
\varphi _{-}\otimes \phi &\longrightarrow &\varphi _{-}\otimes \phi
_{-}-\varphi _{+}\otimes \eta ,\end{aligned}$$with $\left\langle \eta ,\phi _{\pm }\right\rangle =0$. There are now three (un-normalized) pointer states, representing a three-outcome measurement, the third (labelled by $\eta $) corresponding to an undetermined spin, representing a situation where the apparatus cannot identify a definite spin. The two definite outcomes are represented by effects $E_{\pm }=(1- \left\Vert
\eta \right\Vert ^2) P[\varphi_{\pm}]$, and the third is represented by a trivial effect $E_0=\left\Vert\eta \right\Vert ^2\mathbf{1}$ (with probability given by $\left\Vert\eta \right\Vert ^2 $). Wigner shows that $\left\Vert\eta \right\Vert ^2$ can be made arbitrarily small given a “large” apparatus. Specifically he shows that if the state $\phi $ has a very large number of components in its expansion in terms of $J_z$-eigenvectors $\phi_\nu$, so that $\phi =\sum\limits_{\nu =1}^{n}\phi _{\nu}$, then with some suitable assumptions and conditions, $\left\Vert\eta \right\Vert ^2=1/(2n-1)$. Thus in the large-$n$ limit, $\left\Vert
\eta \right\Vert \rightarrow 0$ and accurate and repeatable measurements are, to a very good approximation, recovered.
We note that the large size of the apparatus is used here only as a sufficient condition to achieve good measurement accuracy; the argument does not yield it as a necessary one.
Implications of dropping repeatability {#subsec:alternative}
--------------------------------------
Wigner’s consideration in his final page is intriguing although very sketchy and somewhat open-ended; there he discusses a more general measurement scheme in which the repeatability restriction is dropped. We carefully reconstruct his argument in the Appendix; here we provide a more concise and more general calculation, which contains Wigner’s conclusion as a special case. This approach has considerably less cumbersome algebra, and relies on exploiting the condition that the interaction must be a measurement (in the sense of ) from the beginning. We make no assumption on the product form of the final states, and allow the most general (entangled) final state in the system-apparatus Hilbert space.
For notational convenience and following Wigner, when required we shift the spectral values of the observables concerned in order that they are integers; for example the eigenvalues of the object part of the conserved quantity are now $0$ and $1$. In contrast to Wigner, we do not make the assumption that the spectrum of the apparatus’ conserved quantity is bounded below by zero. With $\chi_{k}^{\prime}$, $\chi_{k}^{\prime \prime}$, $\phi_{k}^{\prime}$ and $\phi_{k}^{\prime \prime}$ representing (un-normalized) eigenstates of $J_{z}$ and $\psi _{0}$, $\psi _{1}$ (normalized) $S_{z}$ eigenstates, the unitary evolution $U$ gives: $$\begin{aligned}
(\psi _{0}+\psi _{1})\otimes \sum \phi _{k} &\overset{U}{\longrightarrow }
&\psi_{0}\otimes \sum \phi _{k}^{\prime }+\psi _{1}\otimes \sum \chi _{k}^{\prime}, \qquad \label{wiggen1} \\
(\psi _{0}-\psi _{1})\otimes \sum \phi _{k} &\overset{U}{\longrightarrow }
&\psi_{0}\otimes \sum \phi _{k}^{\prime \prime }+\psi _{1}\otimes \sum \chi_{k}^{\prime \prime } . \label{wiggen2}\end{aligned}$$ In order to exploit the conservation law we take sums and differences of (\[wiggen1\]) and (\[wiggen2\]), and obtain $$\begin{aligned}
2\psi _{0}\otimes \sum \phi _{k}\longrightarrow\psi _{0}\otimes &\sum
(\phi _{k}^{\prime }+\phi _{k}^{\prime \prime }) \nonumber\\
&+\psi _{1}\otimes \sum (\chi_{k}^{\prime }+\chi _{k}^{\prime \prime }) , \\
2\psi _{1}\otimes \sum \phi _{k}\longrightarrow\psi _{0}\otimes &\sum
(\phi _{k}^{\prime }-\phi _{k}^{\prime \prime }) \nonumber \\
&+\psi _{1}\otimes \sum (\chi_{k}^{\prime }-\chi _{k}^{\prime \prime}).\end{aligned}$$The conservation law now entails that for any $k$: $$\begin{aligned}
\label{wigspec1}
2\psi _{0}\otimes \phi _{k} &\longrightarrow
&\psi _{0}\otimes (\phi
_{k}^{\prime }+\phi _{k}^{\prime \prime })+\psi _{1}\otimes (\chi
_{k-1}^{\prime }+\chi _{k-1}^{\prime \prime }),\qquad \\
\label{wigspec2}
2\psi _{1}\otimes \phi _{k} &\longrightarrow
&\psi _{0}\otimes (\phi
_{k+1}^{\prime }-\phi _{k+1}^{\prime \prime })+\psi _{1}\otimes (\chi
_{k}^{\prime }-\chi _{k}^{\prime \prime }).\end{aligned}$$At this point we wish to make contact with Wigner’s work, and so specify that the apparatus carries no units of the conserved quantity. This is implemented by setting $k=0$, and so $\phi=\phi _{0}$. With this stipulation and allowing for the fact that, in general, the final apparatus states may have negative angular momentum values, we combine (\[wigspec1\]) and (\[wigspec2\]) to obtain: $$\begin{aligned}
(\psi _{0}&+\psi _{1})\otimes \phi _{0} \longrightarrow \frac{1}{2}\psi
_{0}\otimes (\phi _{0}^{\prime }+\phi _{1}^{\prime }+\phi _{0}^{\prime
\prime }-\phi _{1}^{\prime \prime })\nonumber\\
&+\frac{1}{2}\psi _{1}\otimes (\chi
_{-1}^{\prime }+\chi _{-1}^{\prime \prime }+\chi _{0}^{\prime }-\chi
_{0}^{\prime \prime }), \\
(\psi _{0}&-\psi _{1})\otimes \phi _{0} \longrightarrow \frac{1}{2}\psi
_{0}\otimes (\phi _{0}^{\prime }-\phi _{1}^{\prime }+\phi _{0}^{\prime
\prime }+\phi _{1}^{\prime \prime })\nonumber\\
&+\frac{1}{2}\psi _{1}\otimes (\chi
_{-1}^{\prime }+\chi _{-1}^{\prime \prime }-\chi _{0}^{\prime }+\chi
_{0}^{\prime \prime }).\end{aligned}$$From here it follows, by comparison with ([wiggen1]{}) and (\[wiggen2\]), that $\phi _{0}^{\prime \prime }=\phi
_{0}^{\prime }$, $\phi _{1}^{\prime \prime }=-\phi _{1}^{\prime }$, $\chi
_{-1}^{\prime \prime }=\chi _{-1}^{\prime }$, $\chi _{0}^{\prime \prime
}=-\chi _{0}^{\prime }.$ Thus$$\begin{aligned}
(\psi _{0}+\psi _{1})\otimes \phi _{0}&\longrightarrow \nonumber \\
&\psi _{0}\otimes (\phi_{0}^{\prime }+\phi _{1}^{\prime })
+\psi _{1}\otimes (\chi _{0}^{\prime}+\chi _{-1}^{\prime }), \label{gen1}\\
(\psi _{0}-\psi _{1})\otimes \phi _{0}&\longrightarrow \nonumber \\
&\psi _{0}\otimes (\phi_{0}^{\prime }-\phi _{1}^{\prime })
+\psi _{1}\otimes (-\chi _{0}^{\prime}+\chi _{-1}^{\prime }). \label{gen2}\end{aligned}$$Taking the partial trace over the system’s Hilbert space in ([gen1]{}) and (\[gen2\]) yields (mixed) reduced probe states $\rho ^{+}$ and $\rho ^{-}$ respectively. With $\{e_i\}$ an arbitrary orthonormal basis in $\mathcal{K}$, $$\rho ^{\pm }:=\mathrm{tr}_{\mathcal{H}}(P[U(\varphi ^{\pm }\otimes \phi _{0}
)])=\sum \left\langle e_{i}|P[U(\varphi ^{\pm }\otimes \phi
)e_{i}\right\rangle, \label{reduced}$$where $P[U(\varphi ^{\pm }\otimes \phi _{0} )]$ are the orthogonal projections onto the final states, and $\varphi ^{\pm }=\frac{1}{\sqrt{2}}(\psi _{0}\pm
\psi _{1})$. For $U$ to yield a measurement in the sense of , it is required that the reduced states corresponding to two orthogonal initial states are unambiguously distinguishable; that is that they are supported on orthogonal subspaces of $\mathcal{K}$. This is equivalent to the statement that $\mathrm{tr}(\rho ^{+}\rho ^{-})$ must vanish, and it readily emerges that$$\begin{gathered}
0=\mathrm{tr}(\rho ^{+}\rho ^{-})=(\left\Vert \phi _{0}^{\prime }\right\Vert
^{2}-\left\Vert \phi _{1}^{\prime }\right\Vert ^{2})^{2}\\+(\left\Vert \chi
_{-1}^{\prime }\right\Vert ^{2}-\left\Vert \chi _{0}^{\prime }\right\Vert
^{2})^{2}+2\left\vert \left\langle \phi _{0}^{\prime }|\chi _{0}^{\prime
}\right\rangle \right\vert ^{2} . \label{trace}\end{gathered}$$Since each term in this sum is non-negative, it follows that they must each vanish individually, and so $\left\Vert \phi _{0}^{\prime }\right\Vert
^{2}=\left\Vert \phi _{1}^{\prime }\right\Vert ^{2}$, $\left\Vert \chi
_{-1}^{\prime }\right\Vert ^{2}=\left\Vert \chi _{0}^{\prime }\right\Vert
^{2}$ and $\left\langle \phi _{0}^{\prime }|\chi _{0}^{\prime }\right\rangle
=0$. Hence (\[trace\]) is only satisfied if either $\phi _{0}^{\prime }=\phi _{1}^{\prime }=0$ or $\chi _{-1}^{\prime }=\chi
_{0}^{\prime }=0$, since $\phi_{0}^{\prime }$ and $\chi_{0}^{\prime }$ are collinear. There are two scenarios to consider: first, where $\chi _{-1}^{\prime }=\chi _{0}^{\prime }=0$ and the measurement takes the form $$\begin{aligned}
\label{mmt1}
(\psi _{0}+\psi _{1})\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes
(\phi _{0}^{\prime }+\phi _{1}^{\prime }),\\
(\psi _{0}-\psi _{1})\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes
(\phi _{0}^{\prime }-\phi _{1}^{\prime })\label{mmt2} .\end{aligned}$$ This is the form that Wigner arrives at on his final page (see our Appendix). The second scenario is given by $\phi _{0}^{\prime }=\phi _{1}^{\prime }=0$ where $$\begin{aligned}
\label{mmt3}
(\psi _{0}+\psi _{1})\otimes \phi _{0} \longrightarrow \psi _{1}^{\prime
}\otimes (\chi _{0}^{\prime }+\chi _{-1}^{\prime }),\\
(\psi _{0}-\psi _{1})\otimes \phi _{0} \longrightarrow \psi _{1}^{\prime
}\otimes (-\chi _{0}^{\prime }+\chi _{-1}^{\prime })\label{mmt4} .\end{aligned}$$
It is now easy to verify the unitarity of the interaction. The measurement property guarantees that $\phi _{0}^{\prime }$ and $\phi _{1}^{\prime }$ have equal (squared) norm, as do $\chi _{0}^{\prime }$ and $\chi _{-1}^{\prime }$, leaving the right hand sides of and orthogonal, and so too and . For both scenarios, the final system state is independent of the initial one, and repeatability is clearly violated.
It seems that dropping the requirement of repeatability has allowed for the possibility of an accurate measurement, whereas before this was ruled out by the non-commutativity of $S_{x}$ with $J_{z}$. Furthermore, here Wigner has chosen $\phi$ to be an eigenstate of the conserved quantity with eigenvalue zero, whereas we saw in the previous subsection that he chose $\phi$ to have very many components in order to overcome the limitation imposed by the conservation law. Hence giving up repeatability also seems to take away the size constraint for the apparatus.
However, Wigner points out (and this has also been noted in [@yanase-opt]) that the issue of a measurement limitation due to the conservation law has been transferred from the system to the apparatus, since (as is made evident above) the final apparatus states must be eigenstates of the $x$-component of the apparatus’ angular momentum yielding a pointer observable that does not commute with $J_{z}$. It is natural to consider a pointer reading to be an instance of a repeatable measurement, since otherwise there would be no stable record of the measurement (see also [@ozawacon]). Here the WAY-type limitation reappears at the level of the pointer observable, which turns out not to commute with the apparatus’ conserved quantity. Hence the Yanase condition appears to be violated necessarily. Wigner, it seems, was moving toward a general no-go result: that if one wishes to have an accurate measurement, both repeatability and the Yanase condition must be abandoned. Indeed this is the case, as shall be proved in the next section.
The WAY Theorem {#sec:ay}
===============
The work of Araki and Yanase extended {#subsec:A+Y}
-------------------------------------
Araki and Yanase [@way] took up the work of Wigner and proved a general theorem which we state and prove in a somewhat extended and sharpened form. We show that for the same conclusion to be drawn the assumption of repeatability can be replaced by the Yanase condition.
Let $L=L_{1}\otimes \boldsymbol{1}+\boldsymbol{1}\otimes L_{2}$ denote the conserved quantity and $M$ the operator we wish to measure.
**Theorem** *Let ${\mathcal{M}}:=\langle \mathcal{K},U,\phi ,Z,f\rangle $ be a measurement of a discrete-spectrum self-adjoint operator $M$ on $\mathcal{H}$, and let $L_{1}$ and $L_{2}$ be bounded self-adjoint operators on $\mathcal{H}$ and $\mathcal{K}$, respectively, such that $[U,L_{1}\otimes \boldsymbol{1}+\boldsymbol{1}\otimes L_{2}]=0$. Assume that ${\mathcal{M}}$ is repeatable or satisfies the Yanase condition. Then $\left[L_{1},M\right] =0$.*
[*Proof.*]{} We choose orthonormal bases $\{\varphi _{\mu \rho }\}$ and $\left\{ \phi _{\mu \sigma }\right\} $ of eigenstates of $M$ and $Z$, respectively (with $\rho $,$\sigma$ as degeneracy parameters). The most general unitary coupling $U$ that constitutes a measurement of $M$ then takes the form $$\varphi _{\mu \rho }\otimes \phi \overset{U}\longrightarrow \sum\limits_{\sigma
}\varphi _{\mu \rho \sigma }^{\prime }\otimes \phi _{\mu \sigma },$$where $\{\varphi _{\mu \rho \sigma }^{\prime }\}$ in $\mathcal{H}$ is an arbitrary set of states such that $\sum_{\sigma}\left\Vert \varphi _{\mu \rho \sigma }^{\prime }\right\Vert^{2}=1$. Implementing the conservation law (given by $\left[ U,L\right] =0)$ we may now write the matrix elements of $L$ in the following way:$$\left\langle \varphi _{\mu ^{\prime }\rho ^{\prime }}\otimes \phi |L\varphi
_{\mu \rho }\otimes \phi \right\rangle =\sum\limits_{\sigma ,\sigma ^{\prime
}}\left\langle \varphi _{\mu ^{\prime }\rho ^{\prime }\sigma ^{\prime
}}^{\prime }\otimes \phi _{\mu ^{\prime }\sigma ^{\prime }}|L\varphi _{\mu
\rho \sigma }^{\prime }\otimes \phi _{_{\mu \sigma }}\right\rangle .
\label{WAY_matrix}$$The additivity of $L$ and the assumption that $\phi $ is normalized entails that (\[WAY\_matrix\]) can be written$$\begin{aligned}
\label{WAY_sum}
&\left\langle \varphi _{\mu ^{\prime }\rho ^{\prime }}|L_{1}\varphi _{\mu\rho }\right\rangle
+\left\langle \varphi _{\mu ^{\prime }\rho ^{\prime}}|\varphi _{\mu \rho }\right\rangle \left\langle \phi |L_{2}\phi
\right\rangle\nonumber \\
&\hspace{1cm}=\sum\limits_{\sigma ,\sigma ^{\prime }}\left[ \left\langle \varphi _{\mu
^{\prime }\rho ^{\prime }\sigma ^{\prime }}^{\prime }|L_{1}\varphi _{\mu
\rho \sigma }^{\prime }\right\rangle \left\langle \phi _{\mu ^{\prime
}\sigma ^{\prime }}^{\prime }|\phi _{\mu \sigma }^{\prime }\right\rangle
+\right.\\
&\hspace{2.5cm} \left.\left\langle \varphi _{\mu ^{\prime }\rho ^{\prime }\sigma ^{\prime
}}^{\prime }|\varphi _{\mu \rho \sigma }^{\prime }\right\rangle \left\langle
\phi _{\mu ^{\prime }\sigma ^{\prime }}|L_{2}\phi _{\mu \sigma}\right\rangle \right] . \nonumber \end{aligned}$$By the orthogonality of pointer eigenstates, $\left\langle \phi _{\mu
^{\prime }\sigma ^{\prime }}^{\prime }|\phi _{\mu \sigma }^{\prime
}\right\rangle =0$ for $\mu \neq \mu ^{\prime }$; examination of each of the remaining terms in the sum in the above expression tells us that these vanish if one of the following conditions holds:\
(a) $\left\langle \varphi _{\mu ^{\prime }\rho ^{\prime
}\sigma ^{\prime }}^{\prime }|\varphi _{\mu \rho \sigma }^{\prime
}\right\rangle =0$ for $\mu \neq \mu ^{\prime }$;\
(b) $\left\langle \phi _{\mu
^{\prime }\sigma ^{\prime }}|L_{2}\phi _{\mu \sigma }\right\rangle =0$ for $\mu \neq \mu ^{\prime }$.\
Condition (a) is satisfied whenever the measurement is repeatable. Condition (b) is satisfied exactly when the eigenspaces of the pointer observable are invariant under the action of $L_{2}$, i.e when $[L_{2},Z]=0$, so that the measurement satisfies the Yanase condition. If either of these are satisfied, then the right hand side of (\[WAY\_sum\]) is zero, and thus the left hand side must vanish also. Clearly the second term on the left hand side vanishes due to the orthogonality of the eigenstates of $M$, and the first vanishes if and only if $L_{1}$ leaves $M$–eigenspaces invariant, i.e. if and only if $\left[ L_{1},M\right] =0$. We interpret the theorem as follows: if ${\mathcal{M}}$ is a measurement of $M$ and $\left[
L_{1},M\right] \neq 0$, then the conservation of $L$ entails that ${\mathcal{M}}$ must violate both repeatability and the Yanase condition, in accordance with the expectation that emerged in the previous section.
As the proof shows, the commutativity of $M$ with $L_1$ follows from the condition (a), which is in fact a weakening of the repeatability requirement as it merely requires the distinguishability of the post-measurement states of the system. Repeatability is obtained by assuming that the $\varphi_{\mu\rho\sigma}'$ are eigenvectors of $M$. In [@miyadera] it has been shown that the distinguishability of the post measurement object states on one hand and of the post measurement apparatus states on the other are subject to a WAY-type trade-off relation. There the distinguishability is quantified by a measure of *fidelity*, and the measurement inaccuracy is manifested by final pointer states having non-maximal fidelity.
We note that a result of the form of the above theorem (i.e. using the weakened form of repeatability or the Yanase condition to derive the commutativity of the observable to be measured with the conserved quantity) has been proved by Beltrametti et al in 1990 [@BCL] for the special case of [*minimal unitary measurements*]{}, for which the spectra of both the measured observable and pointer are nondegenerate.
As noted above, the violation of the Yanase condition can be understood as disallowing accurate and repeatable measurements of the apparatus observable (since this observable is now subject to the same limitations as prescribed by the WAY theorem). We also observed that the repeatability of pointer measurements is required for ensuring stable and reproducible measurement records. Hence, even if repeatability is sacrificed at the object level, it would seem indispensable at the level of the pointer measurement, thereby enforcing fulfillment of the Yanase condition. Thus we argue that no “measurement” violating the Yanase condition may be called a measurement at all. One may talk only of information transfer between system and apparatus and must also consider how this information can be finally extracted. This conclusion applies to the class of pointer observables that are subject to the WAY theorem.
Technical developments {#subsec:tech}
----------------------
As demonstrated in a footnote in [@way], the case of $L_2$ being unbounded can be incorporated into the proof in a natural way. This is achieved by using the unitary operators $V(t)=\exp{(itL)}$ and $V_{i}(t)=\exp{(itL_i)}$ (with $i=1,2$, $t\in\mathbb{R}$) and noting that $V(t)=V_{1}(t)\otimes V_{2}(t)$. Then one can follow the previous line of proof, replacing the original operators with their exponentiated forms, and exploiting the boundedness of $L_1$.
Ghirardi et al [@Ghi-etal] have extended the WAY theorem to the case where $L_1$ may be unbounded, but all eigenvectors of $M$ are contained in the domain of $L_1$. The measurement is still stipulated to be repeatable. They note that their theorem constrains the feasibility of repeatable measurements of a component of the orbital angular momentum observable in the presence of the conservation of another angular momentum component for the system plus apparatus. Yet their extension still does not cover some physically important cases, namely, those involving observables with continuous spectra.
WAY-type Limitation for Approximate Measurements {#sec:approx}
================================================
Wigner’s paper [@wigner] not only demonstrated the strict impossibility of accurate and repeatable measurements given the conservation law, but also delineated a means by which approximate measurements with approximate repeatability could be recovered. It is also the case, as demonstrated by Araki and Yanase, that this positive part of Wigner’s example can be extended to a much more general class of observables and conserved quantities. Here we describe further developments in this area, examine WAY-type limitations for approximate measurements, and discuss how approximate repeatability also follows a trade-off relation with the size of the apparatus in certain circumstances. This helps to elucidate further the crucial role of the Yanase condition in discussions of WAY-type limitations to quantum measurements.
In the case where $\left[L_{1},M\right] \neq 0$, the limitation given by the WAY theorem can thus be re-expressed more quantitatively: There are approximate measurements of $M$, with some degree of approximate repeatability, which satisfy the Yanase condition, but where good approximations are achieved at the price of requiring a large apparatus, quantified by the magnitude of $\langle \phi
|(L_{2})^{2}\phi \rangle $.
Overview of results {#subsec:overview}
-------------------
Yanase [@yanase-opt] derives an “optimal” lower bound for the probability of the measurement failing to be accurate and repeatable; he considers measurements of a spin component $S_{x}$ where the conserved quantity is $S_{z}+J_{z}$, with $J_{z}$ the $z$-component of the apparatus’ (unbounded) angular momentum . The pointer observable is chosen so that it commutes with $J_{z}$. In this case, the lower bound for the probability of the apparatus malfunctioning is given by $[8\langle \phi | (J_{z})^{2}\phi \rangle]^{-1}$. This bound was also illustrated by Ghirardi [@Ghirardi] for rotationally invariant Hamiltonians. Yanase’s result, though claimed to be “optimal”, still only considered terms up to second moments in $(J_{z})$, and thus optimality was not proven rigorously. This was pointed out by Ozawa [@ozawacon] who obtained a sharper, tight bound without approximations.
Ghirardi et al [@Ghi-etal] have considered the case where measurement errors arise from the non-orthogonality of the final apparatus states. They consider both “distorting” and “non-distorting” (yet still repeatable) measurements. They derive lower bounds on the probability of the “malfunctioning” of the apparatus, and even consider the role that large apparatus size has in reducing these probabilities. However, they do not establish the [*necessity*]{} of a large apparatus for good measurements; they merely assume that the error probabilities can be made small by increasing the expectation of the square of the apparatus part of the conserved quantity.
Ozawa’s trade-off inequality {#subsec:Ozawa}
----------------------------
Ozawa [@ozawacon] develops an alternative formulation of the WAY theorem. He introduces a measure of noise to quantify measurement inaccuracy, and shows that this has a lower bound that can be decreased provided the variance of the apparatus’ conserved quantity is increased. This trade-off inequality follows as an application of the Cauchy–Schwarz inequality.
Given a measurement $\mathcal M$ that is to serve as an approximate determination of an observable $M$, the [*noise operator*]{} is defined as the difference $N:=Z(\tau )-M$, where $Z(\tau )$ represents the Heisenberg-evolved pointer observable after the interaction period $\tau $. A measure of *noise* is then given as $\epsilon (\varphi )^{2}:=\left\langle
\varphi \otimes \phi |N^{2}\varphi \otimes \phi \right\rangle \equiv \langle
N^{2}\rangle $. Clearly $\epsilon (\varphi )^{2}\geq (\Delta N)^{2}$. A global measure of *error* can be provided by taking the supremum over all (normalized) input states $\varphi$ of the quantity $\epsilon (\varphi )^{2}$, i.e. $\epsilon^{2}:=\sup_\varphi\epsilon(\varphi)^{2}$. This quantity should be finite for any measurement that would qualify as an approximate determination of $M$. Then the uncertainty relation entails $$\epsilon^{2}\geq
\epsilon (\varphi )^{2}\geq \frac{1}{4}\frac{\left\vert \left\langle \left[
Z(\tau )-M,L_{1}+L_{2}\right] \right\rangle \right\vert ^{2}}{(\Delta L)^{2}}, \label{Ozawa_error}$$ where it is found that $(\Delta L)^{2}= (\Delta _{\psi }L_{1})^{2}+(\Delta _{\phi }L_{2})^{2}$. The measurement is accurate if and only if $\epsilon=0$.
Thus, if the Yanase condition $(\left[ Z,L_{2}\right] =0)$ is satisfied and the interaction obeys the conservation law, then all that remains in the numerator is $\left\vert \left\langle \left[ M,L_{1}\right] \right\rangle \right\vert ^{2}$. If this is zero then there is no lower bound on the measurement accuracy, in accordance with the findings of WAY.
In the case that $\left\vert \left\langle \left[
M,L_{1}\right] \right\rangle \right\vert ^{2}$ is non-zero but finite, then clearly if $(\Delta L)^{2}$ becomes large the lower bound on the inaccuracy decreases. Furthermore, since the initial system state is arbitrary, only by fixing $\phi$ such that $(\Delta _{\phi }L_{2})^{2}$ is large may one increase the accuracy of the measurement, thus establishing the necessity of a large apparatus variance for good measurements.
It is also worthwhile investigating the case of a measurement scheme $\mathcal{M}$ that satisfies neither the Yanase condition nor the commutativity condition $[M,L_{1}]=0 $ but is such that the bound on the right hand side of (\[Ozawa\_error\]) vanishes; thus, $\left[ Z(\tau ),L_{1}+L_{2}\right] =\left[ M,L_{1}\right]=U^*\left[ Z,L_{2}\right] U $, by the conservation law. This is clearly satisfied if $\mathcal M$ happens to be accurate, $\epsilon=0$. Such a measurement scheme allows for perfectly accurate transfer of information from system to apparatus, and demonstrates the necessary failure of the Yanase condition for this to be achieved.
Trade-off relation for repeatability {#subsec:repeatability}
------------------------------------
Ozawa [@repeatable] has proved that observables with a continuous spectrum do not admit any repeatable measurements. This holds regardless of whether there are additive conserved quantities or not. In order to describe repeatability properties of measurements of such observables, it is therefore necessary to have notions of approximate repeatability, and methods for quantifying how repeatable a measurement is. One approach to weaken condition [@Davies; @OQP]. We will explain and use this in Sec. \[subsec:Stein/Shimony\] in the context of a measurement model.
Here we introduce a different intuitive quantification of repeatability that is somewhat similar to the construction of the noise operator. With this we can generically describe how repeatable a measurement is by utilizing a commutation relation with the conserved quantity. We define: $$\ \mu (\varphi )^{2}:=\langle \varphi \otimes \phi |(M(\tau )-Z(\tau
))^{2}\varphi \otimes \phi \rangle \text{;} \label{rep_meas}$$intuitively if this expectation is small, then the difference between the measured observable and the time-evolved system observable is small, and hence the measurement should display some level of repeatability. A state independent measure of repeatability may thus be defined as $\mu ^{2}:=\sup
\mu (\varphi )^{2}$, yielding $$\mu ^{2}\geq\sup_{\varphi } \frac{1}{4}\frac{\left\vert \left\langle \left[ M(\tau )-Z(\tau
),L_{1}+L_{2}\right] \right\rangle \right\vert ^{2}}{(\Delta_{\varphi }
L_{1})^{2}+(\Delta _{\phi }L_{2})^{2}} . \label{app_rep}$$If the Yanase condition is satisfied, then $\left[ Z(\tau),L_{1}+L_{2}\right] =0$ and so $$\mu ^{2}\geq\sup_{\varphi } \frac{1}{4}\frac{\left\vert \left\langle \left[ M(\tau ),L_{1}+L_{2}\right] \right\rangle \right\vert ^{2}}{(\Delta_{\varphi }
L_{1})^{2}+(\Delta _{\phi }L_{2})^{2}} ,$$which demonstrates that good repeatability may also be achieved when $(\Delta _{\phi }L_{2})^{2}$ is large. This condition becomes a necessity when $\left[ M,L_{1}\right] $ is non-zero.
“WAYs Out" {#sec:wayout}
==========
If an observable we wish to measure does not commute with an additive conserved quantity, we have seen that one may still obtain perfectly accurate information transfer between system and apparatus despite the WAY theorem. Here we note some realizations in which this is achieved, and show explicitly that these models violate both repeatability and the Yanase condition.
Ohira and Pearle {#Ohira and Pearle}
----------------
Ohira and Pearle [@ohira/pearle] provide a “WAY-out” of the limitation arising from the WAY theorem via a model in which both the object and the probe are given as spin-$\frac{1}{2}$ systems. The measurement coupling is generated by a rotationally invariant Hamiltonian of the form $H=(\mathbf{S}+\mathbf{J})\cdot (\mathbf{S}+\mathbf{J})$.
We proceed under the notation that $\psi _{\pm }$ represent both $S_{z}$ and $J_{z}$ eigenstates, and $\phi =\psi_+$. The evolution takes the form (with the interaction period $\tau = \pi /2$): $$\label{O+H_2}
\begin{aligned}
(\psi _{+}+\psi _{-})\otimes \phi &\longrightarrow &(-\psi _{+})\otimes
(\psi _{+}+\psi _{-}), \\
(\psi _{+}-\psi _{-})\otimes \phi &\longrightarrow &(-\psi _{+})\otimes
(\psi _{+}-\psi _{-}) .
\end{aligned}$$ Here the appropriate pointer observable is $Z=J_{x}$. This model is not repeatable, and also violates the Yanase condition.
Recalling equations and which appeared on Wigner’s final page, we see that these have precisely the same form as (\[O+H\_2\]), apart from an inconsequential difference of initial pointer states.
Our analysis of this model of Ohira and Pearle coincides with that of Wigner’s last page (Sec. \[subsec:alternative\]). They point out that this model has demonstrated that if repeatability is not insisted upon, one may achieve an accurate measurement despite the restrictions of the WAY theorem. However, as we have seen, the theorem does not stipulate *any* limitation to the accuracy (of information transfer) when both the repeatability and Yanase conditions are violated, as is the case here. This is precisely the setting in which perfect accuracy is achievable, and this model of Ohira and Pearle is therefore fully in accordance with the WAY theorem as we have given it.
Ohira and Pearle’s aim was to expose and correct a common misreading of the WAY theorem as prohibiting accurate measurements in the presence of an additive conserved quantity. This prohibition, they show, is removed at the expense of giving up the repeatability of the measurement. We know now that in addition the Yanase condition has to be violated as well.
Ozawa’s inequality shows how the zero-error measurement can be achieved; the condition for vanishing lower bound for the error takes the form $U^*\left[ Z,L_{2}\right] U=\left[ M,L_{1}\right]$. In this model, it is easily verified that $U^{\ast }\boldsymbol{1}\otimes S_{y}U=S_{y}\otimes \boldsymbol{1}$, which indeed entails that the expectation value in the numerator of Ozawa’s inequality vanishes.
The SWAP Map Example {#sebsec:SWAP}
--------------------
Following the work of Wigner and Ohira and Pearle, we note that these “WAYs out” are both examples of a remarkably simple structure. They violate both repeatability and the Yanase condition, and whenever the initial system state is an eigenstate of the observable to be measured, both take the form of an unentangled (product) state after the unitary interaction. It is known [@busch_unitary] that the only non-entangling unitary operators $U$ on $\mathcal{H}_1 \otimes \mathcal{H}_2 $ are either of the form: (i) $U(\varphi \otimes \phi) = (V \varphi) \otimes (W \phi)$ (with $V$ and $W$ unitary on $\mathcal{H}_1$ and $\mathcal{H}_2$ respectively), or (ii) $U(\varphi \otimes \phi) = (V_{21} \phi) \otimes (W_{12} \varphi)$ with $V_{21}: \mathcal{H}_2 \rightarrow \mathcal{H}_1$ and $W_{12}: \mathcal{H}_1 \rightarrow \mathcal{H}_2$ surjective isometries. This latter scenario is only possible if $\dim{\mathcal{H}_1}=\dim{\mathcal{H}_2}$ (with the dimension possibly infinite).
One of the simplest examples of a non-entangling unitary map (which is of type (ii), see above) is provided by the *SWAP* map $U_{S}$ on $\mathcal{H}\otimes\mathcal{H}$, defined by $U_{S}(\varphi\otimes\phi)=\phi\otimes\varphi$. If this unitary map is to be used in the context of a measurement, we see that takes the form $ \left\langle \varphi |{\mathsf{E}}(X)\varphi \right\rangle = \left\langle \varphi |{\mathsf{E}}^{Z}(f^{-1}(X))\varphi \right\rangle$ (for all $\varphi \in{\mathcal{H}}$), which can be satisfied if ${\mathsf{E}}={\mathsf{E}}^{M}={\mathsf{E}}^{Z}$, and hence $Z=M$. This also respects any conservation law that is additive and where each non-trivial operator in the sum takes the same form. The noise operator is given as $N=U^{\ast }(\boldsymbol{1}\otimes Z)U-M\otimes \boldsymbol{1=}Z\otimes
\boldsymbol{1}-M\otimes \boldsymbol{1}$. Thus, since we have chosen $Z=M$, the noise operator $N$ vanishes and we have a perfectly accurate information transfer between system and apparatus. However, as the SWAP map violates the Yanase condition, there remains the problem of recovering this information from the pointer observable.
Position Measurements Obeying Momentum Conservation {#sec:pos}
===================================================
Many of the observables that make up a coherent and complete view of (quantum) physical reality are not of the class that have been discussed thus far. Technical difficulties arise in the context of unbounded operators with continuous spectrum, position and (linear) momentum being two noteworthy examples. However if one wishes for a comprehensive understanding of WAY-type limitations to the measurability of physical quantities, it is critical to understand the fundamental case of position measurements that obey momentum conservation. In this section we discuss some results that have been obtained in this context. Any WAY-type theorem for these observables will have to take into account Ozawa’s result that as a continuous quantity, position cannot be measured repeatably.
In [@Loveridge/Busch] the present authors have provided strong evidence for the existence for such a theorem in the position–momentum case. They demonstrate that a model put forward by Ozawa claiming to demonstrate no WAY-type restriction is flawed. The model of Ozawa satisfies the Yanase condition, and one can show that only in the limit of the pointer preparation becoming a delta-function may the inaccuracy tend to zero, which comes at the expense of the apparatus’ momentum distribution having a large width (suitably defined). Furthermore [@Loveridge/Busch] provides a model that explicitly violates the Yanase condition, where arbitrarily accurate and repeatable measurements may still be achieved without resorting to a size constraint on the apparatus.
A General Argument {#gen}
------------------
It is again possible to implement the Ozawa inequality (\[Ozawa\_error\]) to obtain a general argument in favour of WAY-limitations in the continuous unbounded case when the Yanase condition is satisfied. The form of the position–momentum commutator allows the supremum on the right-hand side of (\[Ozawa\_error\]) to be taken in the following way: $$\epsilon ^{2}\geq \frac{1}{4}\frac{1}{\inf_{\varphi }(\Delta _{\varphi
}P)^{2}+(\Delta _{\phi }P_{\mathcal{A}})^{2}}=\frac{1}{4(\Delta _{\phi }P_{
\mathcal{A}})^{2}}.$$with $(\Delta _{\varphi}P)^{2}$ and $(\Delta _{\phi }P_{\mathcal{A}})^{2}$ the variance of the momentum in the system and apparatus respectively. This bound allows for an increase in accuracy only when $(\Delta _{\phi }P_{\mathcal{A}})^{2}$ is large, establishing the necessity of large apparatus size for good measurements.
Precisely the same bound arises when one considers the repeatability (defined in (\[app\_rep\]));$$\mu ^{2}\geq \frac{1}{4(\Delta _{\phi }P_{\mathcal{A}})^{2}}.$$This provides an indication that good repeatability can indeed be achieved if (and only if) there is a large momentum variance in the probe.
Notice that the non-zero lower bounds to both accuracy and repeatability arise after explicit implementation of the Yanase condition, $[Z,P_{\mathcal{A}}]=0$. If we relinquish this condition, there is nothing that would prevent $[
Z(\tau )-Q,P+P_{\mathcal{A}}] $ from vanishing. Indeed this would be the case in any model where one could choose the pointer observable as the apparatus’ position, $Q_\mathcal{A}$.
In the position–momentum case, the role of the Yanase condition must be considered very carefully. Previously (in the case where the WAY theorem certainly applied) we argued for the Yanase condition by applying the WAY theorem to the measurement of the pointer, of which we demanded accurate and repeatable measurements. However, since no such theorem has been proven in the continuous/unbounded case, one must be more tentative when stipulating this condition, and it may be considered as a precautionary manoeuvre. The models discussed in [@Loveridge/Busch], as well as the above model-independent relations point in the direction of a WAY-type limitation if the Yanase condition is satisfied and no such obstruction if it is not.
The last conclusion (of “no obstruction") contrasts, perhaps somewhat surprisingly, the WAY theorem for accurate measurements: Within the realm of that theorem, it is not sufficient to violate the Yanase condition in order to lift the obstruction against perfect accuracy and repeatability. The fact that no size constraint is required for good measurements of position if the pointer observable is a position itself can be understood by considering the lower bounds in equations (\[Ozawa\_error\]) and (\[app\_rep\]): If the object position does not change during the interaction, $M(t)=M=Q$, and the pointer is $Z=Q_{\mathcal A}$, the lower bounds become zero in both cases since the commutator of the noise operator $N=Q_{\mathcal A}(t)-Q$ with the conserved quantity $L_1+L_2=P+P_{\mathcal A}$ vanishes identically. This is a consequence of the fact that $[Q_{\mathcal A}(t),P+P_{\mathcal A}]=i \boldsymbol{1}=[Q,P]$. Such cancellation of commutators living on different Hilbert spaces can only arise for pairs of observables with constant commutators.
It is not known whether, under violation of the Yanase condition, there exist measurements of position that are fully accurate, and repeatable to a good approximation. It is also an open problem whether, again with giving up the Yanase condition, approximate spin measurements obeying angular momentum conservation are possible with good repeatability properties, without any constraint on the size of the apparatus.
The Problem of Stein and Shimony {#subsec:Stein/Shimony}
--------------------------------
In 1979 Stein and Shimony [@Stein-Shimony] posed a problem concerning the possibility of realizing a two-valued (and hence coarse-grained) position measurement that respects the conservation of momentum.
This problem takes the form of whether there exists a non-zero function $\phi \in{L^2(\mathbb{R})}$ and unitary operator $U:L^2(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}^2)$ that commutes with the shift operators (defined by $T_{t}(g)(x,y)=g(x+t,y+t)$ for $g \in{ L^2(\mathbb{R}^2)}$ and $t \in{\mathbb{R}}$)) and satisfy: $$\begin{aligned}
&{\operatorname{supp}}{[U(\varphi \otimes \phi)]} \subseteq\mathbb{R}^{+}\times \mathbb{R}^{+}\ {\rm if}\ {\operatorname{supp}}{\varphi} \subseteq\mathbb{R}^{+},\\
&{\operatorname{supp}}{[U(\varphi \otimes \phi)]}\subseteq\mathbb{R}^{-}\times \mathbb{R}^{-}\ {\rm if}\ {\operatorname{supp}}{\varphi} \subseteq\mathbb{R}^{-},\end{aligned}$$ where $\varphi\in L^2(\mathbb R)$. With the pointer being a two-valued, discretized position observable, this coupling necessarily violates the Yanase condition. The condition that the unitary $U$ commutes with $T_{t}$ is a mathematical expression of the conservation of the total momentum $P + P_{\mathcal{A}}$.
Here we provide a position measurement scheme [@OQP] that approximately satisfies the above requirements with the quality of the approximation becoming arbitrarily good as the value of the coupling parameter $\lambda$ becomes large. The momentum–conserving unitary operator $U$ which describes the interaction is given by $$U=\exp \left[ -i\frac{\lambda }{2}\bigl((Q-Q_{\mathcal{A}})P_{\mathcal{A}
}+P_{\mathcal{A}}(Q-Q_{\mathcal{A}})\bigr)\right] ,$$ where for example we have written $(Q-Q_{\mathcal{A}})P_{\mathcal{A}
}$ as shorthand for $(Q\otimes \mathbf{1}-\mathbf{1} \otimes Q_{\mathcal{A}})\mathbf{1} \otimes P_{\mathcal{A}
}$. The pointer observable is given as $Q_{\mathcal{A}}$, and the measured observable ${\mathsf{E}}$ \[Eq. (\[PRC\])\] is of the form ${\mathsf{E}}(X)=\chi _{X}\star e(Q)$, if the scaling function $f$ is chosen such that $f^{-1}(X)=(1-e^{-\lambda })X$. Here $\chi_X$ represents the characteristic set function. The probability density $e=e^{(\lambda )}$ depends on $\lambda $ in the following way: $$e^{(\lambda )}(q)=(e^{\lambda }-1)\left\vert \phi (-q(e^{\lambda}-1))\right\vert ^{2}.$$
In order to answer the question of Stein and Shimony, we first recast the conditions that need to be satisfied as follows. Firstly, the measurement must satisfy a stronger form of the probability reproducibility condition called the *calibration condition*, which requires that if the initial state is localized in the positive (or negative) half line, then this result is shown on the pointer with certainty. We shall denote the spectral measures of $Q$ and $Q_{\mathcal{A}}$ by $\mathsf{Q}$ and $\mathsf{Q}_{\mathcal{A}}$ respectively. Allowing for some error, this may be written (for $\alpha >0$)
$$\label{calib}
\left\langle \Psi _{\tau} |\mathbf{1}\otimes \mathsf{Q}_{\mathcal{A}}[-\alpha,\infty)
\Psi _{\tau}\right\rangle =1$$
if $ {\operatorname{supp}}{\varphi} \subseteq \lbrack 0,\infty )$, and we show that $[
-\alpha ,\infty )$ can become arbitrarily close to $[0,\infty )$ if $\lambda $ is made suitably large.
The second requirement is that of repeatability, which we give as a slightly modified version of whereby the immediate subsequent measurement is of the observable $Q$. This takes the form (with $\beta>0$) $$\begin{gathered}
\left\langle \Psi _{\tau }|\mathsf{Q}[-\beta,\infty) \otimes
\mathsf{Q}_{\mathcal{A}}\mathbb{(R}^{+}\mathbb{)}\Psi _{\tau }\right\rangle
=\\ \left\langle \Psi _{\tau }|\boldsymbol{1}\otimes \mathsf{Q}_{\mathcal{A}}\mathbb{(R}^{+}\mathbb{)}\Psi _{\tau }\right\rangle = \left\langle \varphi |\mathsf{E}(\mathbb{R}^{+})\varphi \right\rangle, \label{SSrep}\end{gathered}$$ where the last equality results from the probability reproducibility condition. We shall show that this may be satisfied for all $\varphi$ and that $\beta$ can be made arbitrarily small.
We shall make the immediate specification that the initial state wave function $\phi $ of the apparatus be supported on a fixed finite interval of width $2{\ell}$ around the origin; $ {{\operatorname{supp}}\phi} =[-{\ell},{\ell}]$. Therefore the distribution $
e^{(\lambda )}$ is supported on the $\lambda $-scaled interval $[-\delta
,\delta ]$, with $\delta =\ell / (e^{\lambda }-1)$.
After some manipulation the calibration requirement takes the form $$\int_{0}^{\infty} \left\vert \varphi (q)\right\vert ^{2}\chi _{\lbrack -\alpha ^{\prime
},\infty )}\ast e^{(\lambda )}(q)dq=1$$ with $\alpha ^{\prime }=f(\alpha)$. Thus we require $\chi _{\lbrack -\alpha ^{\prime },\infty
)}\ast e^{(\lambda )}(q)=1$ for all $q \geq{0}$ and so $$\int_{-\infty }^{\alpha ^{\prime }+q}e^{(\lambda )}(y)dy=1,$$ which is satisfied if $q\geq\delta -\alpha ^{\prime }$. The smallest $\alpha ^{\prime }$ consistent with this constraint occurs when $\alpha ^{\prime
}=\delta $, and so $\alpha ={\ell}e^{-\lambda}$. Therefore we see that indeed $\alpha
\rightarrow 0$ as $\lambda \rightarrow \infty $. It must also be shown that the same behaviour emerges in the case when ${\operatorname{supp}}{\varphi \subseteq{(-\infty,0]}}$ but we omit this essentially identical calculation, and this completes the proof.
We now address the repeatability requirement. Writing in integral form and rearranging, we see that $$\int \left\vert \varphi (q)\right\vert ^{2}(\chi _{\lbrack \mathbb{-}\beta
,\infty )}(q)-1)\chi _{[0, \infty)}\ast e^{(\lambda)}(q)dq=0,$$ and so $$\chi _{( \mathbb{-}\infty ,-\beta )}(q)\int_{-\infty
}^{q}e^{(\lambda )}(y)dy=0.$$ This expression certainly vanishes if $q\ge -\beta $. When $q< -\beta $, recalling that ${\operatorname{supp}}{e^{(\lambda )}}=[-\delta ,\delta ]$, we see that if $-\delta \geq -\beta $ (and thus $\beta \geq \delta $) then the integral vanishes. Since we are looking for the smallest $\beta $ for which this may be satisfied, we choose $\beta =\delta =\ell/(e^{\lambda} -1)$. Therefore in the large $\lambda $ limit, $\beta $ is arbitrarily small, showing that arbitrarily good repeatability may be achieved. Due to the symmetry of the support of $e^{(\lambda )}$, it follows that arbitrarily good repeatability holds also for the $\mathbb{R}^-$ outcome on the pointer.
Although this model provides only an approximate solution to the problem of Stein and Shimony, we note that from an operational perspective this does not differ from an exact solution. Since the accuracy and approximate repeatability can be made arbitrarily good by simply tuning the coupling parameter, in any experimental realization this could not be distinguished from a measurement in which perfect accuracy and repeatability can be achieved. This does not require a large momentum spread in the probe, and it has been shown that the present model indeed presents an approximate measurement scheme for the full position observable $Q$, with arbitrarily good accuracy and repeatability properties [@Loveridge/Busch].
Concluding Remarks {#sec:conclusion}
==================
The WAY theorem, with its generalizations, is applicable to a large class of physically important scenarios. In any situation in which, for example, spin or angular momentum is the relevant observable, the measurement accuracy is likely to be hampered by a WAY-type constraint. When considering the manipulation of individual quantum objects using other small objects as ‘apparatus’, it may not be possible to fulfill the requirement of large variance of the apparatus part of the conserved quantity. Such scenarios do occur in quantum information processing and quantum control. Ozawa and coworkers [@conquant] have in fact demonstrated a limitation to the realizability of quantum logic gates insofar as the observables involved are subject to the WAY theorem. This has led to an increased awareness that attention has to be paid to the presence of conserved quantities in the design of quantum gates.
In the case of position measurements that obey momentum conservation, no WAY-type obstruction exists if one asks only for a measurement of the relative distance between the object and a “reference system”. In this case, when the reference system is provided by part of the apparatus, the measured observable can be given as the relative position. As is alluded to in [@AR], it appears that there is a link to the theory of superselection rules and quantum reference frames (see, e.g. [@qfr]), which has been the subject of much interest and investigation recently. This possible link opens up an avenue that requires further systematic study.\
**Acknowledgments.** Thanks are due to Rebecca Ronke and Tom Potts for many helpful discussions and careful reading of drafts of this manuscript. This work was supported by EPSRC UK.
Appendix: Reconstructing Wigner’s last page {#sec:last-page .unnumbered}
===========================================
In this appendix we shall carefully reconstruct the argument that appears on the final page of Wigner’s 1952 paper [@wigner]. Although Wigner’s work is succinct and simple, the lack of detailed calculations makes reproducing his conclusions somewhat harder work than one might imagine. We also present some subtly different arguments from those found in the original work.
Wigner restricts his consideration to the case where the post-interaction states are of product form (unentangled) in the system–apparatus Hilbert space, and he makes the choice that the initial apparatus state $\phi$ be an eigenstate of $S_z$ with eigenvalue zero. He writes $$\begin{aligned}
(\psi _{0}+\psi _{1})\otimes \phi &\longrightarrow
&\sum\limits_{i=0}^{1}\psi _{i}^{\prime }\otimes \sum \phi _{j}^{\prime },
\label{wig1} \\
(\psi _{0}-\psi _{1})\otimes \phi &\longrightarrow
&\sum\limits_{i=0}^{1}\psi _{i}^{\prime \prime }\otimes \sum \phi
_{j}^{\prime \prime }, \label{wig2}\end{aligned}$$with $\psi _{i}^{\prime }$ and $\psi _{i}^{^{\prime \prime }}$ representing un-normalized $S_{z}$ eigenstates. In order that Wigner’s analysis be compelling, we must assume $\phi _{j}^{\prime }$ and $\phi _{j}^{^{\prime \prime }}$ to be eigenstates of the apparatus’ angular momentum, $J_{z}$. The reason for this choice will become clear shortly; this is the only way in which consistency with the conservation law can be maintained. We omit summation indices on the apparatus Hilbert space since it is assumed to run to infinity. However, the number of non-zero terms in this expansion is dramatically reduced due to the choice of initial apparatus state and the conservation law; the left hand side of (\[wig1\]) contains a superposition of $S_{z}$ eigenstates, and thus a superposition of states containing zero and one “unit” of the conserved quantity. The right hand side cannot, then, contain more than one such unit.
In order to correspond to Wigner’s analysis, we proceed under the restriction that [0]{} be the lowest eigenvalue for the apparatus’ conserved quantity, and from here it follows that ( \[wig1\]) and (\[wig2\]) take on a much simpler forms. With $\phi=\phi _{0}$ and dropping all terms with the apparatus containing two or more units of the conserved quantity, we have $$\begin{gathered}
(\psi _{0}+\psi _{1}) \otimes \phi _{0} \longrightarrow
\psi _{0}^{\prime}\otimes \phi _{0}^{\prime }+\psi _{0}^{\prime }\otimes
\phi _{1}^{\prime }+\psi _{1}^{\prime }\otimes \phi _{0}^{\prime }+\psi
_{1}^{\prime }\otimes \phi _{1}^{\prime } , \label{wig3} \end{gathered}$$ $$\begin{gathered}
(\psi _{0}-\psi _{1})\otimes \phi _{0}\longrightarrow
\psi _{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }+\psi
_{0}^{\prime \prime }\otimes \phi _{1}^{\prime \prime }+\psi _{1}^{\prime
\prime }\otimes \phi _{0}^{\prime \prime }+\psi _{1}^{\prime \prime }\otimes
\phi _{1}^{\prime \prime }, \label{wig3'}\end{gathered}$$ Indeed, the conservation law provides an even stronger restriction, and the last term on the right hand side of (\[wig3\]) must in fact be zero, and thus at least one of $\psi _{1}^{\prime }$ and $\phi
_{1}^{\prime }$ must always vanish. The same argument applies to and so (independently), at least one of $\psi
_{1}^{\prime \prime }$ and $\phi _{1}^{\prime \prime }$ must vanish too.
It follows from (\[wig3\]) and consistency with the conservation law that $\psi _{0}^{\prime }$ and $\phi _{0}^{\prime
} $ are necessarily non-zero. For if either did vanish, the right hand side would contain one unit of the conserved quantity with certainty, and the left hand side only with probability $\frac{1}{2}$. The same argument runs in clear analogy for the double-primed quantities. There are then four scenarios that require consideration:\
[*Case 1:*]{} $\psi _{1}^{\prime }\neq 0$, $\phi _{1}^{\prime }=0$, $\psi
_{1}^{\prime \prime }\neq 0$, $\phi _{1}^{\prime \prime }=0$; [*Case 2:*]{} $\psi _{1}^{\prime }\neq 0$, $\phi _{1}^{\prime }=0$, $\phi
_{1}^{\prime \prime }\neq 0$, $\psi _{1}^{\prime \prime }=0$; [*Case 3:*]{} $\phi _{1}^{\prime }\neq 0$, $\psi _{1}^{\prime }=0$, $\psi
_{1}^{\prime \prime }\neq 0$, $\phi _{1}^{\prime \prime }=0$; [*Case 4:*]{} $\phi _{1}^{\prime }\neq 0$, $\psi _{1}^{\prime }=0$, $\phi
_{1}^{\prime \prime }\neq 0$, $\psi _{1}^{\prime \prime }=0$.\
With this in mind, one can add (\[wig3\]) and (\[wig3’\]) to give the evolution of $\psi _{0}\otimes \phi _{0} $: $$\begin{aligned}
2\psi _{0}\otimes \phi _{0} &\longrightarrow &\psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }+\psi _{0}^{\prime }\otimes \phi _{1}^{\prime }+\psi
_{1}^{\prime }\otimes \phi _{0}^{\prime }+ \label{wig4} \\
&&\psi _{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }+\psi
_{0}^{\prime \prime }\otimes \phi _{1}^{\prime \prime }+\psi _{1}^{\prime
\prime }\otimes \phi _{0}^{\prime \prime }, \notag\end{aligned}$$and for the evolution of $\psi _{1}\otimes \phi $ we subtract:$$\begin{aligned}
2\psi _{1}\otimes \phi _{0} &\longrightarrow &\psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }+\psi _{0}^{\prime }\otimes \phi _{1}^{\prime }+\psi
_{1}^{\prime }\otimes \phi _{0}^{\prime }- \label{wig7} \\
&&\psi _{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }-\psi
_{0}^{\prime \prime }\otimes \phi _{1}^{\prime \prime }-\psi _{1}^{\prime
\prime }\otimes \phi _{0}^{\prime \prime }. \notag\end{aligned}$$We first consider Case 1 where (\[wig4\]) and (\[wig7\]) reduce to $$\begin{gathered}
2\psi _{0}\otimes \phi _{0} \longrightarrow
\psi _{0}^{\prime }\otimes \phi _{0}^{\prime }+\psi _{1}^{\prime }\otimes
\phi _{0}^{\prime }+\psi _{0}^{\prime \prime }\otimes \phi _{0}^{\prime
\prime }+\psi _{1}^{^{\prime \prime }}\otimes \phi _{0}^{\prime \prime } ,
\label{wig5'} \end{gathered}$$ and $$\begin{gathered}
2\psi _{1}\otimes \phi _{0} \longrightarrow
\psi _{0}^{\prime }\otimes \phi _{0}^{\prime }+\psi _{1}^{\prime }\otimes
\phi _{0}^{\prime }-\psi _{0}^{\prime \prime }\otimes \phi _{0}^{\prime
\prime }-\psi _{1}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }\label{wig34} . \end{gathered}$$ Since the left hand side of contains no units of the conserved quantity, so must the right, and therefore $\psi _{1}^{\prime }\otimes \phi _{0}^{\prime
}=-\psi _{1}^{^{\prime \prime }}\otimes \phi _{0}^{^{\prime \prime }}$. Similarly in (\[wig34\]) the left hand side contains one unit, and if the right hand side is to agree, we require that $\psi _{0}^{\prime
}\otimes \phi _{0}^{\prime }=\psi _{0}^{\prime \prime }\otimes \phi
_{0}^{\prime \prime }$.
With $\psi _{1}^{\prime }\otimes \phi _{0}^{\prime
}=-\psi _{1}^{^{\prime \prime }}\otimes \phi _{0}^{^{\prime \prime }}$ we get: $$2\psi _{0}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }+\psi _{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime },$$and thus, with $\psi _{0}^{\prime }\otimes \phi _{0}^{\prime }=\psi
_{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }$, $$\psi _{0}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }.$$Also,$$2\psi _{1}\otimes \phi _{0} \longrightarrow \psi _{1}^{\prime }\otimes \phi
_{0}^{\prime }-\psi _{1}^{\prime \prime }\otimes \phi _{0}^{\prime \prime },
\label{wig14}$$and finally, exploiting the condition $\psi _{1}^{\prime }\otimes \phi _{0}^{\prime
}=-\psi _{1}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }$, we arrive at $$\psi _{1}\otimes \phi _{0} \longrightarrow \psi _{1}^{\prime }\otimes \phi
_{0}^{\prime }. \label{wig17}$$
We now consider Case 2 which, with gives $$2\psi _{0}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }+\psi _{1}^{\prime }\otimes \phi _{0}^{\prime }+\psi
_{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }+\psi _{0}^{\prime
\prime }\otimes \phi _{1}^{\prime \prime },$$and thus one might wish to conclude that $\psi _{1}^{\prime }\otimes \phi
_{0}^{\prime }=-\psi _{0}^{\prime \prime }\otimes \phi _{1}^{\prime \prime }$. However, this can never be satisfied; these vectors must be distinct unless they are both zero (which is excluded, by assumption), since the unit of conserved quantity resides in different Hilbert spaces.
Case 3 gives $$2\psi _{0}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }+\psi _{0}^{\prime }\otimes \phi _{1}^{\prime }+\psi
_{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }+\psi _{1}^{\prime
\prime }\otimes \phi _{0}^{\prime \prime } \label{wig6}$$and we conclude that it must be the case that $\psi _{0}^{\prime }\otimes
\phi _{1}^{\prime }=-\psi _{1}^{\prime \prime }\otimes \phi _{0}^{\prime
\prime }$ which, again, cannot be fulfilled for both non-zero. We therefore must also reject Case 3.
Finally Case 4 gives $$2\psi _{0}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }+\psi _{0}^{\prime }\otimes \phi _{1}^{\prime }+\psi
_{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }+\psi _{0}^{\prime
\prime }\otimes \phi _{1}^{\prime \prime }$$ and $$2\psi _{1}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }+\psi _{0}^{\prime }\otimes \phi _{1}^{\prime }-\psi
_{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime }-\psi _{0}^{\prime
\prime }\otimes \phi _{1}^{\prime \prime },$$ and so $\psi _{0}^{\prime }\otimes \phi _{1}^{\prime }=-\psi _{0}^{\prime
\prime }\otimes \phi _{1}^{\prime \prime }$ and $\psi _{0}^{\prime
}\otimes \phi _{0}^{\prime }=\psi _{0}^{\prime \prime }\otimes \phi
_{0}^{\prime \prime }$.
It is now evident that each of the permissible cases gives the same state evolution for $\psi _{0}\otimes \phi $; Case 4 yields $$2\psi _{0}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }+\psi _{0}^{\prime \prime }\otimes \phi _{0}^{\prime \prime },
\label{wig8}$$and with $\psi _{0}^{\prime }\otimes \phi _{0}^{\prime }=\psi _{0}^{\prime
\prime }\otimes \phi _{0}^{\prime \prime }$, we arrive at
$$\psi _{0}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{0}^{\prime }. \label{wig18}$$
However, for the evolution of $\psi _{1}\otimes \phi $, using $\psi _{0}^{\prime }\otimes \phi _{1}^{\prime }=-\psi
_{0}^{\prime \prime }\otimes \phi _{1}^{\prime \prime }$, we see that a different form emerges than from Case 1: $$\psi _{1}\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes \phi
_{1}^{\prime }.$$
With these considerations, we now summarise the possible forms of the evolution of $(\psi _{0}+\psi
_{1})\otimes \phi _{0} $ and $(\psi _{0}-\psi _{1})\otimes \phi _{0} $. Remembering that the only cases which contain, *a priori*, no contradiction, are Cases 1 and 4, the first scenario is that Case 1 is satisfied, and we have: $$(\psi _{0}+\psi _{1})\otimes \phi _{0} \longrightarrow (\psi _{0}^{\prime }+\psi
_{1}^{\prime })\otimes \phi _{0}^{\prime } \label{wig15},$$and $$(\psi _{0}-\psi _{1})\otimes \phi _{0} \longrightarrow (\psi _{0}^{\prime }-\psi
_{1}^{\prime })\otimes \phi _{0}^{\prime }. \label{wig16}$$This cannot represent a measurement in any ordinary or physically meaningful sense, since the final states of the apparatus coincide on the right hand side of (\[wig15\]) and (\[wig16\]), leaving us in the position that there is no way of distinguishing which eigenstate of $S_{x}$ had been present on the left hand side. Furthermore, this product form does not correspond to a modification of equations (\[intro1\]) and (\[intro2\]) (as is claimed by Wigner).
The second scenario is that Case 4 is satisfied, and we see that summing (\[wig17\]) with (\[wig18\]) gives: $$(\psi _{0}+\psi _{1})\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes
(\phi _{0}^{\prime }+\phi _{1}^{\prime }) \label{wig12}$$and subtracting: $$(\psi _{0}-\psi _{1})\otimes \phi _{0} \longrightarrow \psi _{0}^{\prime }\otimes
(\phi _{0}^{\prime }-\phi _{1}^{\prime }) \label{wig13}$$ This coincides with and (Sec. \[subsec:alternative\]), and is the same result as Wigner obtained on his final page.
[99]{}
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"pile_set_name": "ArXiv"
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---
abstract: 'Grid (or comb) states are an interesting class of bosonic states introduced by Gottesman, Kitaev and Preskill [@Gottesman.etal.2001:GKPcode] to encode a qubit into an oscillator. A method to generate or ‘breed’ a grid state from Schrödinger cat states using beam splitters and homodyne measurements is known [@Vasconcelos.etal.2010:GKPprep], but this method requires post-selection. In this paper we show how post-processing of the measurement data can be used to entirely remove the need for post-selection, making the scheme much more viable. We bound the asymptotic behavior of the breeding procedure and demonstrate the efficacy of the method numerically.'
author:
- 'Daniel J. Weigand'
- 'Barbara M. Terhal'
bibliography:
- 'bibliography.bib'
title: 'Generating Grid States From Schrödinger Cat States without Post-Selection'
---
Introduction
============
Grid (or comb) states are a class of bosonic states with various interesting possible applications. Grid states were introduced in [@Gottesman.etal.2001:GKPcode] as simultaneous eigenstates of two commuting displacement operators. In this scheme grid states can be used to encode a qubit (or qudit) into an oscillator or bosonic mode so that small displacement errors can be corrected. As outlined in [@Gottesman.etal.2001:GKPcode], universal quantum computation can be achieved using grid states: Clifford gates can be implemented via linear optics while one may invoke magic-state-distillation techniques to get to universality. Grid states also play a crucial role in fault-tolerant continuous-variable computation using cluster states [@Menicucci.2014:ClusterStateComp].
It has also been shown that grid states can be used to generate maximal violations of CHSH-type inequalities [@WHG03:Grid_state-Bell; @Etesse.etal.2014:GridBellTest]. In recent work, we have shown that a grid state can be used to determine the two parameters of a small displacement accurately and simultaneously [@Duivenvoorden.etal.2017:Sensorstate], going beyond squeezed or coherent states.
First proposals to generate grid states use, [e.g. ]{}, the coupling between a micro-mirror and an optical mode [@Gottesman.etal.2001:GKPcode], the oscillatory motion of a trapped atom [@Pirandola.etal.2006:GKPprepNeutrAtoms; @Travaglione.Milburn.2002:GKPprepStandardPE] or a Kerr interaction between two bosonic modes [@Pirandola.etal.2004:GKPprepKerr]. Recent ideas on generated grid states in an atomic ensemble using squeezed light can be found in [@Motes.etal.2017:QuantumPhysics], while an optical breeding protocol for cat states was considered in [@Sychev.etal.2017:CatBreeding]. In earlier work, we have shown how grid states can be generated without post-selection using phase estimation and a qubit-bosonic mode coupling of the form $Z a^{\dagger}a$ [@Terhal.Weigand.2016:GKPprepPE], focusing on a circuit-QED setting. Very recent experiments [@fluehmann+home; @kienzler+:iontrap] show how a grid state can be constructed in the motional mode of an ion using post-selection.
In the linear optics setting, Vasconcelos [*et al. *]{}[@Vasconcelos.etal.2010:GKPprep] and Etesse [*et al. *]{}[@Etesse.etal.2014:GridBellTest] have independently developed a *breeding* protocol to generate grid states from Schrödinger cat states, using linear optics and homodyne post-selection [@Vasconcelos.etal.2010:GKPprep]. A similar breeding protocol, used to generate Schrödinger cat states from Fock states, has been demonstrated in an experiment by Etesse [*et al. *]{}[@Etesse.etal.2015:CatBreedingExp]. However, the protocol has an important drawback: The success probability of post-selection diminishes rapidly with the number of rounds.
In this paper, we show that classical post-processing can be used to correct the grid state generated by a breeding protocol. This allows the use of any state generated by breeding, independent of the measurement results, showing that no post-selection is necessary. Our understanding of the protocol is formed by showing that a breeding protocol has identical action as a phase estimation protocol of multiple rounds, with specific (known) feedback phases and measurement results. Through this identification the breeding protocol implements a particular phase estimation protocol which by definition gradually projects onto a grid state (since one is gradually learning bits of the phase). The feedback phases used and bits obtained in phase estimation inform us about the grid state that we have obtained, namely the information gives us an estimate of the eigenvalues of the commuting displacement operators thus fixing the eigenstate.
By describing a toy model, the so-called slow breeding protocol, we can show how breeding can be related to phase estimation. However, this slow breeding protocol is non-optimal in its requirement for very large cat states. We then examine an efficient breeding protocol, which is the protocol in [@Vasconcelos.etal.2010:GKPprep], and show how the measurement record can be used to correct any final state to a good grid state. Proving convergence of this breeding protocol towards a good grid state by invoking phase estimation is not simple. Instead, by using a new class of approximate grid states which is closed under the efficient breeding step, we can bound the asymptotic behavior of the protocol. Finally, we confirm the performance of the protocol with numerics.
We will first review some background concepts concerning grid states, squeezing parameters and phase estimation in \[sec:background\]. In \[sec:breeding\] we show how a breeding protocol can be mapped onto a phase estimation scheme, giving some intuition how a protocol works without post-selection. Then we focus on analyzing the efficient breeding protocol by Vasconcelos [*et al. *]{}[@Vasconcelos.etal.2010:GKPprep] without post-selection. In \[sec:bounds\] we introduce a very useful class of approximate grid states and present some bounds on the probability of improving the state in a breeding round using these approximate states. We close the paper with a numerical simulation of the breeding protocol in \[sec:simulation\] and a Discussion (\[sec:discussion\]).
Background
==========
In this section, we give a short review of previous results and the formalism needed in the rest of this paper. We start in \[sec:grid\] with a short introduction of grid states, following mostly the paper by Gottesman [*et al. *]{}[@Gottesman.etal.2001:GKPcode]. In \[sec:eff\_squeezing\], we review the effective squeezing parameters, a versatile metric for the quality of a grid states which we introduced in [@Duivenvoorden.etal.2017:Sensorstate]. In \[sec:ape\], we introduce a formalism which enables the construction of a map between breeding and phase estimation in an efficient manner.
\[sec:background\]
Grid states {#sec:grid}
-----------
Consider a bosonic mode with dimensionless quadrature operators $\hat{q}=\frac{1}{\sqrt{2}}(a+a^{\dagger})$ and $\hat{p}=\frac{i}{\sqrt{2}}(a^{\dagger}-a)$ obeying $[\hat{q},\hat{p}]=i$. A grid state in this mode is a simultaneous, approximate, $+1$ eigenstate of two commuting displacement operators $S_p=e^{i u \hat{p}}$ and $S_q=e^{i v \hat{q}}$ where $u\cdot v\mod 2\pi = 0$ ensures commutativity of $S_p$ and $S_q$. Note that it is not necessary that the displacements $S_p, S_q$ form a square lattice in phase space. In fact, grid states can be defined on any two dimensional lattice where the area of the unit cell is a multiple of $2\pi$ [@Gottesman.etal.2001:GKPcode].
In this paper, we will investigate grid states with a symmetric choice $u=v=\xi$. For example, for the choice $\xi=\sqrt{2\pi}$, the space fixed by $S_p=+1, S_q=+1$ is one-dimensional. This state will be referred to as the *sensor state*[@Duivenvoorden.etal.2017:Sensorstate].
Whenever a choice for $\xi$ is necessary ([e.g. ]{}for the numerical analysis or the Wigner function of a state), we investigate protocols generating this sensor state. In case of the choice $\xi=2\sqrt{\pi}$ the $+1$ eigenspace of $S_p$ and $S_q$ is two-dimensional and thus encodes a qubit [@Gottesman.etal.2001:GKPcode]. From here on, we will refer to $\xi$ as the [*spacing*]{} of a grid state. For both the wavefunction in quadrature space and the Wigner function of a grid state, the spacing corresponds to the distance between the sharp peaks in these functions. We use the notation for displacement $D(\alpha)=\exp(\alpha a^{\dagger}-\alpha^* a)$ so that $S_p=D(\sqrt{\pi})$ for the sensor state. Spacing $\xi$ thus corresponds to the action of a displacement with coherent amplitude $\xi/\sqrt{2}$.
a b c d e
Since a perfect eigenstate of these displacement operators, [i.e. ]{}an ideal grid state has infinite energy, it is only possible to generate approximate grid states. One possible approximation is a grid state of the form $$\begin{aligned}
\ket{\Psi} \propto \sum_{t=-\infty}^\infty e^{-\pi\kappa^2 t^2}S_p^t S(\Delta)\ket{\mathrm{vac}},
\label{eq:grid}\end{aligned}$$ where $S_p^t$ corresponds to the displacement $D(t\xi/2)$ and $S(\Delta)$ is the squeezing operator which has the action $\hat{q}\to\hat{q}\Delta, \hat{p}\to\hat{p}/\Delta$ (so that $\bra{{\rm vac}}S^{\dagger}(\Delta){\rm Var}(q) S(\Delta)\ket{{\rm vac}}=\Delta^2 \bra{{\rm vac}} {\rm Var}(q)\ket{{\rm vac}}=\frac{\Delta^2}{2}$). The squeezing parameter $\Delta < 1$ and the width of the Gaussian envelope can be chosen to be the same, [i.e. ]{}$\kappa = \Delta$ [@Gottesman.etal.2001:GKPcode].
In this form, the squeezed vacuum can be understood as an approximate $+1$ eigenstate of $S_q$, while the weighed sum over powers of $S_p$ is an approximation of the projector onto the $+1$ eigenspace of $S_p$. Essentially, the ideal grid state is invariant under the two translations $S_p$ and $S_q$ (and their inverses) in phase space, hence a $+1$ eigenstate of these operators. Any finite-photon number version of this state occupies a bounded volume in phase space and cannot be fully translationally-invariant, but a Gaussian envelope allows the non-translational invariance of the tails to play a relatively small role.
Effective squeezing parameters {#sec:eff_squeezing}
------------------------------
In order to characterize the quality of an approximate grid state we have introduced so-called effective squeezing parameters for both quadratures in [@Duivenvoorden.etal.2017:Sensorstate]. A ‘squeezing’ parameter can be generally used for capturing how well a state $\rho$ is an approximate eigenstate of a unitary operator $U$. The idea is based on the fact that a state $\rho$ is an eigenstate of the operator $U$ iff $|\operatorname*{Tr}\rho U| = 1$. For such a state the mean phase $\theta \in [-\pi,\pi)$ equals $\theta (\rho)= \arg(\operatorname*{Tr}U \rho)$. Because of the $2\pi$-periodicity of the phase, the variance should not be taken to be the standard variance, but can be chosen as a phase variance equal to $\operatorname*{Var}(\rho) = \ln(|\operatorname*{Tr}U \rho |^{-2})$ [@Duivenvoorden.etal.2017:Sensorstate]. This variance is identical to the more commonly used Holevo phase variance [@book:WM] for small $|\operatorname*{Tr}U \rho |$.
For a displacement $\mathcal{D}:=D(u e^{i\phi})$ with $\phi, u\in\mathds{R}$, the variance should be rescaled by $u$, [i.e. ]{}we define the mean phase $\theta_\mathcal{D}$ and the effective squeezing parameter $\Delta_\mathcal{D}$ as: $$\begin{aligned}
&\theta_\mathcal{D} := \arg (\operatorname*{Tr}\mathcal{D} \rho), &&\Delta_\mathcal{D} := \frac{1}{u}\sqrt{\ln(|\operatorname*{Tr}\mathcal{D} \rho|^{-2})}.
\label{eq:delta}\end{aligned}$$ As grid states are defined with respect to the displacement $S_p$ ($S_q$) along the real (imaginary) axis in phase space, it is convenient to use the short-hand $\Delta_p:=\Delta_{S_p}$ and $\Delta_q:=\Delta_{S_q}$ for the two *effective squeezing parameters*. The squeezing parameters of an approximate grid state as defined in \[eq:grid\] are $\Delta_q=\Delta$, $\Delta_p \approx \kappa$. For a squeezed vacuum state $S(\Delta) \ket{\rm vac}$, one has $\Delta_q=\Delta=1/\Delta_p$. The effective squeezing parameter and mean phase have a very natural relation to grid states:
Protocols to generate an approximate eigenstate of $S_p$ and $S_q$ will produce a state $\rho$ with certain values for $\theta_p:= \theta_{S_p}$, $\theta_q:=\theta_{S_q}$, $\Delta_p$ and $\Delta_q$. The effective squeezing parameters then give a direct measure of the quality of the state $\rho$. In case of the sensor state, they directly relate to the measurement precision that can be achieved using $\rho$ as a sensor [@Duivenvoorden.etal.2017:Sensorstate]. In case of the GKP code, the probability of a logical X (or Z) error in the encoding can be bounded as $P_{\text{error}} < \frac{2\Delta}{\pi}e^{-\pi/(4\Delta^2)}$ with $\Delta=\Delta_q=\Delta_p$ [@Gottesman.etal.2001:GKPcode].
The mean values $\theta_p$ and $\theta_q$ which are extracted from the protocol can be used to correct the resulting state by displacing this state by $D_{\rm correct}$, i.e. $\rho \rightarrow \rho'=D_{\rm correct} \rho D_{\rm correct}^{\dagger}$ such that $\theta_p(\rho')\approx \theta_q(\rho') \approx 0$. For example, to shift the mean phase $\theta_p$ back to 0 we choose $\alpha$ in $D_{\rm correct}=\exp(i \alpha \hat{q})$ such that $S_p D_{\rm correct}=\exp(-i \theta_p) D_{\rm correct} S_p$. A simple visual representation of this procedure is that the positive parts of the Wigner function form a grid in phase space for grid states and this grid is aligned with the $p=0,q=0$ axes for a $+1$ eigenstate of $S_p,S_q$ (see \[fig:setup,fig:grid\_state\]).
The final state is then an approximate $+1$ eigenstate of $S_p$ and $S_q$. However, it is not necessary to perform such a correcting displacement if one uses the concept of a phase or displacement frame [@Terhal.Weigand.2016:GKPprepPE] (in analogy with a Pauli frame for qubits).
Clearly, approximate grid states are not unique. For example, two grid states whose grid envelope is displaced or translated one unit cell over can have the same values for $\theta_p, \theta_q$ and $\Delta_p,\Delta_q$ but contain a different mean number of photons. Similarly, one can note that the corrective displacement is not unique: in practice one may opt for the smallest displacement shifting the grid envelope to the correct position, see \[fig:grid\_state\](e).
Adaptive phase estimation {#sec:ape}
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Phase estimation refers to a whole class of algorithms that measure the eigenvalue of a unitary operator $U$. A recent overview of some of these schemes can be found e.g. in [@Svore.etal.2014:PhaseEstimation]. All phase estimation procedures, including textbook phase estimation [@book:Nielsen.Chuang.2000:QCompAndQInfo], Kitaev’s phase estimation [@Kitaev.1995:RPE] and variants thereof, can be executed in an iterative form with a single-qubit applying controlled-$U^k$ gates. An in-depth analysis of some adaptive schemes can be found in works by Berry [*et al. *]{}[@Berry.etal.2001:ARPE]. We are interested in this case when the unitary operator to be measured is some displacement and we consider performing such measurement by repeating a circuit of the form \[fig:breeding\].
A convenient formalism to describe such adaptive phase estimation uses the following ‘measurement’ operator: $$\begin{aligned}
{\mathcal{M}_{}\left( \varphi, \alpha \right)} := 1 + e^{i\varphi}D\left(\alpha\right),
\label{eq:M}\end{aligned}$$ with $\varphi\in [0, 2\pi)$ and $\alpha$ is a coherent amplitude. In all what follows we focus on breeding an approximate eigenstate of $S_p=D(\xi/\sqrt{2})$ and assume that $\alpha$ is real, but the same method can be used for complex $\alpha$.
With this operator, a squeezed Schrödinger cat state has the form $D(-\alpha/2){\mathcal{M}_{}\left( 0,\alpha \right)}S(\Delta)\ket{\text{vac}}$, [i.e. ]{}a single application of the measurement operator onto a squeezed vacuum state, plus an additional displacement.
One can also see that the circuit shown in \[fig:PE\_round\] acts on an input state $\ket{\Phi_0}$ as ${\mathcal{M}_{}\left( \phi+\pi x, \alpha \right)}\ket{\Phi_0}$ where $x\in\{0,1\}$ is the measurement result, [i.e. ]{}it applies one additional measurement operator to the initial state. Thus *any* state generated by a sequence of $N$ of these circuits is of the form $$\begin{aligned}
\ket{\Psi} \propto \prod_{j=1}^N {\mathcal{M}_{}\left( \varphi_j, \alpha_j \right)}\ket{\Phi_0},
\label{eq:PE_form}\end{aligned}$$ where $\ket{\Phi_0}$ is the initial state and $\varphi_j=\phi_j+x_j \alpha_j$ with measurement outcome $x_j$, feedback phase $\phi_j$ of round $j$ and $\alpha_j$ possibly varying per round.
It can be observed that the class of states which is described by fixing the outcome to be $x=0$ and letting the feedback phase vary captures all states in \[eq:PE\_form\] since $\phi_j\in[0,2\pi)$ can be freely chosen. We will show that the state obtained by a breeding protocol is identical to a state obtained by such a phase estimation protocol with all outcomes $x_j=0$ and with varying $\phi_j=\varphi_j$. A (trivial) example is that a squeezed Schrödinger cat state is equivalent to a single round of phase estimation with $x=\phi=0$ applied to a squeezed vacuum state. This map gives some intuition why breeding gives rise to a grid state. Using the form of the state allows one to estimate the mean phase and the effective squeezing parameters of the state using \[eq:delta\].
As was mentioned before, even given $\theta_p$ and $\Delta_p$, a grid state is not unique since it can be shifted by any $S_p$ without affecting these parameters. Thus to place the grid state symmetrically around the vacuum state and minimize photon number, it is better to perform a pre-displacement by $D(-\alpha/2)$ in each phase estimation round in Fig. \[fig:PE\_round\] and similarly use the measurement operator $D(-\alpha/2)+ e^{i\varphi} D(\alpha/2)$. Since our analysis does not depend on these shifts, we have opted to not include them.
Breeding {#sec:breeding}
========
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Breeding protocols refer to a procedure where a grid state is gradually constructed from input (squeezed) Schrödinger cat states. One can view these input states as a very poor approximation (panel (a) in \[fig:grid\_state\]) to a grid state and the goal is to gradually improve these states. The circuit in \[fig:breeding\] shows a single round of breeding. We will denote the number of breeding rounds by $M$, while $N$, which is a function of $M$, refers to the number of measurement operators acting on some initial state as in \[eq:PE\_form\]. In a single breeding round partially bred grid states that will be of the same form as \[eq:PE\_form\] are fed into a beam-splitter. After the beam-splitter, the $p$-quadrature of one of the states is measured (for breeding of a $S_p$ eigenstate). For $N_2=1$ in \[fig:breeding\], the input of the bottom port (port 2) plays the role of squeezed cat state modulo the additional pre-displacement, i.e. $(D(-\beta/2) + D(\beta/2))\ket{\Phi_0}=D(-\beta/2) {\mathcal{M}_{}\left( 0, \beta \right)}\ket{\Phi_0}$. The aim of the Breed operation is to map the measurement operators in port $2$ to port $1$, [i.e. ]{}the state at the output port is still of the form of \[eq:PE\_form\], but with $N_1+N_2$ measurement operators.
Since we would like to produce a state which is both an approximate eigenstate of $S_p$ and $S_q$, we choose the input state $\ket{\Phi_0}$ as a squeezed vacuum state $\ket{\Phi_0}=S(\Delta) \ket{\rm vac.}$ providing an approximate eigenstate of $S_q$. It is important that the effective squeezing parameter $\Delta_q$ is approximately preserved under the breeding operation so that the outgoing state is both an approximate eigenstate of $S_p$ and $S_q$: we will verify this at the end of Section \[sec:eff\_breeding\].
The rounds of this breeding procedure could be repeated in at least two ways. In the first manner, which we call slow breeding, we always use a squeezed cat state at input port 2 and input port 1 contains the state that came out of port 1 in the previous breeding round. This protocol can be seen as a toy model in that it has several drawbacks, but we describe its functionality in order to understand how breeding works and how it maps onto phase estimation. In Section \[sec:eff\_breeding\] we describe a parallelized distillation protocol in which $2^M$ squeezed cat states are fed into beam-splitters, leading to $2^{M-1}$ output states, which are subsequently used to produce $2^{M-2}$ states etc., eventually extracting one grid state after $M$ breeding rounds, [i.e. ]{}the setting proposed in [@Etesse.etal.2014:GridBellTest; @Vasconcelos.etal.2010:GKPprep]. Then, we will show how a map to phase estimation can be constructed for this protocol, removing the need for post-selection.
Slow Breeding
-------------
Using that the action $\mathcal{B}$ of the beam splitter is given by $$\begin{aligned}
&\hat{q}_1\to (\hat{q}_1 - \hat{q}_2)/\sqrt{2}, && \hat{p}_1\to (\hat{p}_1 - \hat{p}_2)/\sqrt{2},\\
&\hat{q}_2 \to (\hat{q}_1 + \hat{q}_2)/\sqrt{2}, && \hat{p}_2\to (\hat{p}_1 + \hat{p}_2)/\sqrt{2},\end{aligned}$$ one can show that the output state of a breeding round in \[fig:breeding\] equals $$\begin{aligned}
\prod_{j=1}^{N_1} \prod_{k=1}^{N_2} \mathcal{\tilde{M}}_1(\varphi_j,\alpha_j)
\mathcal{\tilde{M}}_2(\psi_k, \beta_k)\mathcal{B} \ket{\Phi_0, \Phi_0}.
\label{eq:BSO_output}\end{aligned}$$ Here $\mathcal{\tilde{M}}_i(\varphi,\alpha)=\mathcal{B} \mathcal{M}_i(\varphi,\alpha) \mathcal{B}^{\dagger}$. For the input states $\ket{\Phi_0,\Phi_0}$ we use the invariance under beam-splitting, i.e. $\mathcal{B} S_{1}(\Delta)S_2(\Delta) \ket{\mathrm{vac, vac}}_{1,2} = S_{1}(\Delta)S_2(\Delta) \ket{\mathrm{vac, vac}}_{1,2}$. For real $\alpha$ we have $\mathcal{\tilde{M}}_1(\varphi,\alpha)=I+e^{i\varphi} D_1(\alpha/\sqrt{2}) D_2(-\alpha/\sqrt{2})$ and $\mathcal{\tilde{M}}_2(\psi,\beta)=I+e^{i\psi}D_1(\beta/\sqrt{2}) D_2(\beta/\sqrt{2})$. When mode 2 is then measured via homodyne measurement of $\hat{p}$ with outcome $p$, we can replace $D_2(\alpha)$ by $e^{-i \alpha \sqrt{2} p}$ (for real $\alpha$). This implies that the output state of the protocol is as claimed in Fig. \[fig:breeding\], [i.e. ]{}$$\begin{aligned}
&\prod_{j=1}^{N_1}\prod_{k=1}^{N_2} \mathcal{M}_1(\tilde{\varphi}_j, \tilde{\alpha}_j) \mathcal{M}_1(\tilde{\psi}_k, \tilde{\beta}_k) \ket{\Phi_0}, \label{eq:bso} \\
&\tilde{\varphi}_{j}=\varphi_j+ \alpha_j p,\ \tilde{\alpha}_{j}=\frac{\alpha_j}{\sqrt{2}},\ \tilde{\psi}_{k}=\psi_k-\beta_{k} p,\ \tilde{\beta}_{k}=\frac{\beta_k}{\sqrt{2}}. \notag\end{aligned}$$ The probability to find outcome $p$ for the homodyne measurement depends in detail on the state of the form \[eq:PE\_form\] that goes into the beam-splitter, but the variance of this probability distribution in $p$ scales as $\sim 1/\Delta^2$. Hence the more the input state $\ket{\Phi_0}$ is squeezed in $q$ (by $\Delta$), the large the spread of measured values for $p$ will be and hence the greater the need for not using post-selection on the outcome $p=0$.
Consider now the [*slow breeding*]{} case where the state at input $2$ is always a squeezed cat state [i.e. ]{}$N_2=1$ , and the output state is fed into port $1$ of the next round. In order to breed a grid state we take $\alpha_1=\beta_1=\alpha$ and $\varphi_1=\psi_1=0$ for the first breeding round, meaning that the inputs in both ports are squeezed cat states.
In the second breeding round one takes $\beta_2=\alpha/\sqrt{2}, \psi_2=0$ and in the $M$th round $\beta_M=\alpha/\sqrt{2^{M-1}}, \psi_M=0$ so that the final state has spacing $\xi=\alpha/\sqrt{2^{M-1}}$. The evolution of mode 1 under the slow breeding protocol without post-selection and $M=3$ rounds is shown in \[fig:grid\_state\](a-d).
By post-selecting the measurement result onto $p=0$, it is apparent from this choice for the $\beta_i$ and \[eq:bso\] that $M$ rounds of this procedure generate a binomial distribution of displacements, since all the phases are zero. Thus, clearly, when we post-select on outcome $p=0$, one obtains a grid state with a binomial envelope (similar to the protocols shown in [@Vasconcelos.etal.2010:GKPprep; @Etesse.etal.2014:GridBellTest]).
From \[eq:bso\] it follows immediately that $M$ rounds of breeding in this setup with a final spacing $\xi=\alpha/\sqrt{2^{M-1}}$ can be mapped to $N=M+1$ rounds of phase estimation with the choice $\varphi_m = \alpha (\sum_{k>m}^M 2^{-k/2} p_k - 2^{-m/2} p_m)$ for the feedback phase and measurement result $x_m=0$, where $p_m$ is the homodyne measurement result of $\hat{p}_2$ in round $m=1, \ldots, M$ and $p_0=0$ to fix the initial state ($m=0$) to the squeezed cat state $\propto (I + D(\alpha))\ket{\Phi_0}$. It is noteworthy that the feedback phase depends on the outcomes of many ‘later’ rounds: One can thus only construct the corresponding phase estimation protocol after the last homodyne measurement is done. This suggests that instead of post-selecting on $p=0$, one can simply process the measurement information to infer the values of $\theta_p$ in Eq. (\[eq:delta\]) of the final state. This correction is demonstrated in \[fig:grid\_state\](e), where a correcting displacement is applied to the final state of the protocol.
However, the slow breeding protocol suffers from a different problem. To get a grid state with final spacing $\xi=\sqrt{2\pi}$ after $M$ rounds, the number of photons in the squeezed cat state used in the first round $\bar{n}_{cat} \geq 2^M \pi$. This is exponentially larger than the mean photon number of the final grid state which scales as $\bar{n}_{grid} \sim M$ [@Terhal.Weigand.2016:GKPprepPE; @Gottesman.etal.2001:GKPcode], [i.e. ]{}the procedure is inefficient in its use of photons.
Efficient Breeding {#sec:eff_breeding}
------------------
A much better scheme is to use a partially-bred grid state in the ancilla mode as proposed in [@Vasconcelos.etal.2010:GKPprep; @Etesse.etal.2014:GridBellTest], effectively performing a grid state distillation scheme. In this scheme, one starts with two cat states ($N_1=N_2=1$), leading to a state with $N_{\rm out}=2$. Then one takes two such states ($N_1=N_2=2$) and feeds them into the beam-splitter to get a state with $N_{\rm out}=4$ etc. With \[eq:bso\], one can see that we have $N=2^M$ for $M$ repetitions of this scheme.
In this scheme one will always have $\beta_j=\alpha_j$ for the two input ports, but the phases can vary depending on measurement results and do not need to be the same for both inputs. This parallelization leads to a much faster built-up of the grid state. For a final grid state with $N=2^M$ applications of $\mathcal{M}$, one requires $M$ rounds of beam-splitters in sequence. For the final grid state to have spacing $\xi$ one starts the protocol with cat states with amplitude $\xi 2^{(M-3)/2}=\xi\frac{\sqrt{N}}{2\sqrt{2}}$, thus $\bar{n}_{cat} \sim \bar{n}_{grid}\sim N$. For example, generating a sensor state with $M=2$ rounds would require $\bar{n}_{cat} = \frac{\pi}{2} + \bar{n}_{sq}$ photons, where $\bar{n}_{sq}$ is the additional number of photons due to squeezing.\
![Left side: Efficient breeding protocol as proposed in [@Vasconcelos.etal.2010:GKPprep; @Etesse.etal.2014:GridBellTest], with $M=2$ rounds but without post-selection. The labeling of modes is according to the scheme introduced in \[sec:eff\_breeding\]: The initial $2^M=4$ Schrödinger cat states are labeled by the 2-bit strings $\{00,01,10,11\}$. Those are put pairwise into beam splitters, resulting in the states and measurement results labeled by the 1-bit strings $\{0,1\}$. The phases and final state of the corresponding phase estimation setup are determined using \[eq:effective\_correction\], the correcting displacement is then obtained with \[eq:delta\]. Right side: Shown are the Wigner functions of modes $\{00, 0, \text{Final}\}$ and the fianl state after applying the correction, zoomed in around the origin. The yellow crosses mark the ‘center’ of the state, for a $+1$ eigenstate of $S_p$ they lie at the origin. []{data-label="fig:setup"}](breeding_setup.pdf){width="\hsize"}
In order to estimate the effective squeezing after $M$ rounds as well as the phase $\theta_p$, one needs to describe the final state in terms of the measurement outcomes. A concise description of the output state of an $M$-round protocol is as follows. We label all $2^M$ ingoing modes of the protocol with a bit-string ${\bf x}[M]$ of length $M$. Two modes $x_1\ldots x_{M-1} x_M$ and $x_1 \ldots x_{M-1} \overline{x_M}$ which differ only on the last bit $x_M$ will enter into one beam-splitter and so the outgoing single mode can be labeled by the remaining $M-1$ bit string $x_1..\ldots x_{M-1}= {\bf x}[M-1]$. The outcomes of these $2^{M-1}$ measurements of the first round forms a vector ${\bf p}^1$ with $2^{M-1}$ entries $p^1_{{\bf x}[M-1]}$ which are labeled by the bitstrings ${\bf x}[M-1]$. The final measurement in round $M$ is then ${\bf p}^{M}$ with a single entry labeled by a bit string of length 0. An example of this labeling can be seen in \[fig:setup\]. With this notation the initial state is thus a product state proportional to $\prod_{\mathbf{x}[M]} {\cal M}_{\mathbf{x}[M]}(0, 2^{(M-1)/2}\xi)\ket{\Phi_0}_{\mathbf{x}[M]}$.
Similarly, the state of the system after the first round of breeding is the product state $$\begin{aligned}
\prod_{\mathbf{x}[M]} {\mathcal{M}_{{{\bf x}[M-1}\left( } \right)}]{\xi(-1)^{x_M}2^{\frac{M-2}{2}}p^1_{{\bf x}[M-1]}, 2^{\frac{M-2}{2}}\xi} \nonumber \\ \ket{\Phi_0}_{{{\bf x}[M-1]}}, \nonumber\end{aligned}$$ where each state now gets two measurement operators applied to it since we are taking the product over all bit-strings of length $M$. After all $2^M-1$ measurements, the final state is given by $2^M$ measurement operators acting on a single mode, i.e. $$\begin{aligned}
&\ket{\Psi_{\rm out}({\bf p}^{1},\ldots {\bf p}^{M})} \nonumber \\ &=\prod_{\mathbf{x}[M]}{\mathcal{M}_{{x[0}\left( } \right)}]{\xi\sum_{j=1}^{M} (-1)^{x_j} 2^{\frac{j-2}{2}}p^{M-j+1}_{{\bf x}[j-1]},\frac{\xi}{\sqrt{2}}} \ket{\Phi_0}_{\mathbf{x}[0]}.
\label{eq:effective_correction}\end{aligned}$$ with normalization $\mathds{P}({\bf p}^{1},\ldots, {\bf p}^{M})=\bra{\Psi_{\rm out}}\Psi_{\rm out}\rangle$.
In order to evaluate $\theta_p=\frac{\arg \bra{\Psi_{\rm out}} S_p \ket{\Psi_{\rm out}}}{\braket{\Psi_{\rm out}|\Psi_{\rm out}}}$ for a given series of outcomes ${\bf p}_{\rm all}\colon= {\bf p}^{1},\ldots, {\bf p}^{M}$, it is convenient to write down the initial state as a wavefunction in $p$ and use that $S_p \ket{p}=e^{i\xi p}\ket{p}$. The effect of this correction is shown in \[fig:setup\]. Similarly, one can evaluate the average $\langle \Delta_p\rangle =\sum_{{\bf p}_{\rm all}} \mathds{P}({\bf p}_{\rm all}) \Delta_p({\bf p}_{\rm all})$. Note that, if minimizing the run-time of this procedure is crucial ([e.g. ]{}for feedback in an experiment), the mean phases could be approximated using the mode of the probability distribution in $p$ corresponding to the final state.
While the map between breeding and phase estimation derived in the previous section suggests that $\langle \Delta_p\rangle$ will decrease rapidly with breeding rounds, it is in fact not simple to use this mapping to analytically prove this. The difficulty is that since the phases can vary per round (depending on the homodyne measurement outcomes), arguments which use laws of large numbers, which apply when identical experiments are repeated, are not directly applicable.
In order to understand the outgoing state in terms of the initial squeezing, we note that the final state of the breeding protocol after $M$ rounds consists of applying powers of $S_p=D(\xi/\sqrt{2})$ (with phases) to the initial state $\ket{\Phi_0}$ and $S_q$ commutes with $S_p$. However, the input state is not an [*exact*]{} eigenstate of $S_q$. Furthermore, the full description of the unitary evolution involves the beam-splitter and the measured ancilla modes and the full action does not commute with $S_q$.
This means that a few steps are required to show that the expectation value of $S_q$ of the output state is close to the expectation value of $S_q$ of the input state. The output state of any breeding protocol will be $\ket{\Psi_{\rm out}}=A \ket{\Phi_0}$ and $A=\sum_{j=-\infty}^{\infty} \alpha_j S_p^j$ (Where the number of non-zero coefficients $\alpha_j\in\mathds{C}$ is determined by the protocol). We can compute the normalization of $\ket{\Psi_{\rm out}}$ by writing the initial squeezed state as a wave function in $\ket{q}$ $$\begin{aligned}
&\bra {\Psi_{\rm out}} \Psi_{\rm out}\rangle \nonumber \\
& =\sum_{j,k=-\infty}^{\infty} \frac{\alpha_j \alpha_k^*}{\sqrt{\pi\Delta^2}} \iint \text{d}q \,\text{d}q'\ e^{-\frac{q^2}{2\Delta^2}} e^{-\frac{(q'-(j-k)\xi)^2}{2\Delta^2}} \braket{q|q'} \\
&= \sum_{j,k}\frac{\alpha_j \alpha_k^*}{\sqrt{\pi\Delta^2}} \int \text{d}q\ e^{-\frac{(q-(j-k)\xi/2)^2}{\Delta^2}} e^{-\frac{\xi^2(j-k)^2}{4\Delta^2}} \nonumber \\
& =\sum_{j,k}\alpha_j \alpha_k^*e^{-\frac{\xi^2(j-k)^2}{4\Delta^2}}.\end{aligned}$$ For small $\Delta \lesssim 0.5$, the last term vanishes for $j\neq k$ ($\xi$ is at least $\sqrt{2\pi}$), [i.e. ]{}$\sum_j |\alpha_j|^2 \approx \braket {\Psi_{\rm out} |\Psi_{\rm out}} = 1$. Using the same method one obtains $$\begin{aligned}
&\bra{\Psi_{\rm out}} S_q \ket{\Psi_{\rm out}}\\
&= \sum_{j,k}\frac{\alpha_j \alpha_k^*}{\sqrt{\pi\Delta^2}} \int \text{d}q\ e^{i\xi q}e^{-\frac{(q-(j-k)\xi/2)^2}{\Delta^2}} e^{-\frac{\xi^2(j-k)^2}{4\Delta^2}}, \\
&= \sum_{j,k}\frac{\alpha_j \alpha_k^*}{\sqrt{\pi\Delta^2}} e^{-\frac{\xi^2(j-k)^2}{4\Delta^2}}e^{-\frac{i(j-k)\xi^2}{2}}\int \text{d}q\ e^{i\xi q}e^{-\frac{q^2}{\Delta^2}}, \\
&= \sum_{j,k}\alpha_j \alpha_k^*e^{-\frac{\xi^2(j-k)^2}{4\Delta^2}}e^{-\frac{i(j-k)\xi^2}{2}}\bra {\Phi_0} S_q \ket{\Phi_0}\approx \bra {\Phi_0} S_q \ket{\Phi_0},\end{aligned}$$ where we used the normalization condition obtained before. This implies that $\Delta_q(\Psi_{\rm out}) \approx \Delta_q(\Phi_0)=\Delta$ for initial squeezing $\Delta \lesssim 0.5$ (which corresponds to large squeezing in $q$). The effective squeezing parameter of a squeezed Schrödinger cat state $\propto (D(-\sqrt{\pi}/2)+ D(\sqrt{\pi}/2))\ket{\Phi_0}$ is $\sqrt{\Delta^2 - \frac{2}{\pi}\ln(\tanh(\frac{\pi}{4\Delta^2}))}$, which differs from a squeezed vacuum state $\ket{\Phi_0}$ by $\mathcal{O}(10^{-17})$ for $\Delta=0.2$. This is also expected, as $\Delta_q = \Delta$ for a squeezed vacuum state and $\Delta_q \approx \Delta$ for an approximate grid state as defined in [@Gottesman.etal.2001:GKPcode].
Asymptotic Behavior {#sec:bounds}
===================
In this section we derive probability bounds for the breeding protocol showing how the effective squeezing parameter changes round-by-round. The known class of approximate grid states which are described by a perfect grid state to which a Gaussian distribution of shift errors is applied [@Gottesman.etal.2001:GKPcode] is not closed under a round of breeding, the same holds for squeezed Schrödinger cat states. Thus, analyzing the effect of the breeding map for many rounds is a nontrivial problem when using either class of states.
In order to solve this issue, we introduce a new class of approximate grid states which is closed under the breeding operation, enabling an analytical discussion. Since the breeding protocol changes the spacing of an approximate grid state round-by-round, the spacing of these states is round-dependent. To this end, we first define scale-dependent shifted grid states as $$\ket{u, v, m} = \frac{\sqrt{s_m \xi}}{2\pi}e^{i\frac{v}{s_m \xi}\hat{p}} e^{i\frac{s_m \xi u}{2\pi}\hat{q}} \ket{\Psi_m},
\label{eq:def_shifted}$$ where $u, v\in[-\pi,\pi)$. The parameter $s_m$ is some scale parameter that we will choose below, $\xi$ is the spacing of the final grid state and $\ket{\Psi_m}\propto \sum_{s=-\infty}^{\infty} \ket{p=s \xi s_m}$. With the choice $s_m=1,\xi=2\sqrt{\pi}$, one obtains the shifted code states introduced by Glancy and Knill in the context of the GKP code [@Glancy.Knill.2006:GKPfaultTolerance]: these states above can be viewed as an extension of this concept. For any choice of $m$ and $s_m \xi$, it can be verified that the class of states $\ket{u,v,m}$ forms an orthonormal basis of the whole Hilbert space of the oscillator, i.e. $\braket{u,v,m|u',v',m} =\delta(u-u') \delta(v-v')$ and $\int_{-\pi}^{\pi} du \int_{-\pi}^{\pi} dv \ket{u,v,m} \bra{u,v,m}=I$, see Appendix \[sec:ortho\].
For our application in the breeding protocol we will choose $s_m = \sqrt{2^{m-M}}$ and one can confirm that this choice yields a shifted grid state with spacing $\xi$ for $m=M$. Note that $\ket{\Psi_m}$ is a $+1$ eigenstate of the rescaled operators $S_q^{s_m}$ and $S_p^{2\pi/(\xi^2 s_m)}$, i.e. the spacing of the states is rescaled round by round since each beam-splitter will change the spacing by $\sqrt{2}$. We can see this by writing $\ket{\Psi_m}\propto \lim_{\Delta \rightarrow 0} \Pi_{S_{q}^{s_m}=1} S(1/\Delta) \ket{\rm vac}$ since $\lim_{\Delta\rightarrow 0} S(1/\Delta) \ket{\rm vac}=\ket{p=0}$ and $\Pi_{S_{q}^{s_m}=1} \propto \sum_{t=-\infty}^{\infty} S_{q}^{t s_m}$ is the projector onto the $+1$ eigenspace of $S_q^{s_m}$.
In general, a basis of shifted grid states can be used to write down an approximate code state as a Gaussian superposition of states with different shifts [@Gottesman.etal.2001:GKPcode; @Terhal.Weigand.2016:GKPprepPE]. Here, we will similarly use these states but the filter for the quadrature on which we apply the breeding will not be Gaussian but determined by a von Mises probability distribution. We thus define the class of approximate shifted grid states (for general $\xi$) as $$\begin{aligned}
\ket{V_{\kappa,\mu,m}} &:= \frac{1}{\mathcal{N}}\int_{-\pi}^\pi \text{d}u\ \int_{-\pi}^\pi \text{d}v\ V(u-\mu)_{\kappa}\notag \\
&\quad\quad\times \sum_{s=-\infty}^\infty e^{i u s}G(v+2\pi s)_{s_m \Delta} \ket{u, v, m}, \label{eq:mises}\\
V(u-\mu)_{\kappa} &:= \frac{1}{\sqrt{2\pi I_0(\kappa)}}\exp\left(\frac{\kappa}{2}\cos(u-\mu)\right), \notag\\
G(v)_{\sigma} &:= \frac{1}{\sqrt{\sigma \sqrt{\pi}}}\exp\left(-\frac{v^2}{2 \sigma^2}\right).
\label{eq:def_states}\end{aligned}$$ In the limit of large initial squeezing $\Delta \ll 1$, the normalization constant $\mathcal{N}$ goes to 1. Note that $\mathds{P}_{\sigma}(v)=G(v)_{\sigma}^2$ is a Gaussian distribution with mean 0 and standard deviation $\sigma/\sqrt{2}$ so that when $m=M$, the standard deviation of $\mathds{P}_{\Delta s_m}(v)$ is $\Delta/\sqrt{2}$. The choice of probability distribution on $u$ and $v$ is different because the breeding protocol acts differently on the $\hat{p}$ and $\hat{q}$ quadratures of the initial states. This choice ensures that the class of states $\ket{V_{\kappa,\mu,m}}$ is closed under breeding, see \[eq:breedstep\]. The probability distribution $\mathds{P}_{\kappa}(u-\mu) = V(u-\mu)^2_{\kappa}$ is the von Mises distribution and $I_\nu(\kappa)$ is the modified Bessel function of the first kind of order $\nu$. The von Mises distribution $\mathds{P}_{\kappa}(u-\mu)$ which models a Gaussian distribution for a circular phase variable $u$ has mean $\mu$. In the limit $\kappa \gg 1$ the probability distribution becomes Gaussian by approximating $\exp(\kappa \cos(u-\mu)) \approx \exp(-\kappa (u-\mu)^2)/2)\exp(\kappa)$ with standard deviation $1/\sqrt{\kappa}$.
The index $m=0,\dots,M$ will refer to the number of breeding rounds applied to the initial state, with $m=0$ the initial state and $m=M, s_M=1$ the final state. Note that the shift error distribution in $v$ gets rescaled each round: the standard deviation of $G(v)^2_{\Delta, m=0}$ is increasing in each round, but given a $v$, the shift induced in each round in \[eq:def\_shifted\] gets smaller, so that effectively the spread in $p$ stays the same. Thus $\Delta_q \approx \Delta$ where $\Delta$ is the initial squeezing.
For the approximate state with $m=M$, i.e. $\ket{V_{\kappa,\mu,M}}$, the mean phase $\theta_p$ is simply the mean $\mu$ of the distribution while the effective squeezing parameter $\Delta_p$ equals $$\Delta_p = \sqrt{\ln(I_0^2(\kappa)/I_1^2(\kappa))/\pi},
\label{eq:squeeze}$$ which for large $\kappa$ becomes $1/\sqrt{\pi \kappa}$, hence directly connecting to the standard deviation of the Gaussian distribution. Using the formula for linear combinations of trigonometric functions with a phase shift, one can show that the distribution over $u$ of the outgoing state after a round of breeding is again a von Mises distribution (see [e.g. ]{}[@book:Mardia.etal.2014:CircularStatistics; @book:Jammalamadaka.Sengupta.2001:CircularStatistics] in the context of the convolution of von Mises distributions). Using the convolution property of Gaussian distributions, one can show the same for the $v$ shifts. Combining these two properties, one can show that a round of breeding with measurement outcome $p_{\rm out}$ maps two input states of this form with label $m$ onto an output state of the same form with label $m+1$ $$\begin{aligned}
&\ket{V_{\kappa_1,\mu_1, m}}\ket{V_{\kappa_2,\mu_2, m}} \stackrel{\rm breeding}{\rightarrow} \ket{V_{\kappa_{out},\mu_{out}, m+1}},\nonumber \\
&\kappa_{out}^2 = \kappa_1^2 + \kappa_2^2 + 2\kappa_1 \kappa_2 \cos(\mu_1-\mu_2-2\tilde{p}),\nonumber \\
&\mu_{out} = -\operatorname*{atan2}[\kappa_1\cos(\mu_1-\tilde{p}) + \kappa_2\cos(\mu_2+\tilde{p}) \notag\\
& \hspace{2.35cm},\kappa_1\sin(\mu_1-\tilde{p}) + \kappa_2\sin(\mu_2+\tilde{p})],
\label{eq:breedstep}\end{aligned}$$ with $\tilde{p} = \frac{2\pi}{\xi s_{m+1}}p_{\rm out}$. The details of this derivation can be found in the Appendix \[sec:breed\].
Thus, if the two states fed into round $m$ have the error model of \[eq:mises\], the outgoing state is of the same type, with new parameters $\kappa_{out}, \mu_{out}$ which depend on measurement outcome $p_{\rm out}$ and the round $m$. Since the ingoing states are normalized, the probability of finding outcome $p_{\rm out}$ can be obtained by evaluating the norm of the outgoing state, see Appendix \[sec:breed\], and we obtain the oscillatory function $$\mathds{P}(p_{\rm out}) = \frac{\Delta I_0(\kappa_{out}) \mathcal{N}_{out}^2}{\sqrt{\pi} \xi I_0(\kappa_1)I_0(\kappa_2) \mathcal{N}_{1}^2 \mathcal{N}_{2}^2} e^{-\frac{p_{\rm out}^2 \Delta^2}{\xi^2}}.
\label{eq:prob}$$ Defining the variable $x = \mu_1-\mu_2-2\tilde{p}\mod 2\pi$ gives a concise description of the effect of one breeding round. The probability $\mathds{P}(x)$ can be simplified in the limit of large initial squeezing, $s_m \Delta \ll 1$ from Eq. (\[eq:prob\]). Since $x$ is $2\pi$-periodic, we can use that the limit of a wrapped normal distribution with large variance is simply a circular uniform density of $1/(2\pi)$. Together with the fact that the normalization constants $\mathcal{N}$ all go to $1$ for large initial squeezing, one obtains $$\begin{aligned}
\kappa_{out}(x) &= \sqrt{\kappa_1^2 +\kappa_2^2 + 2 \kappa_1 \kappa_2 \cos(x)} = (\kappa_1 + \kappa_2)\lambda,\\
\mathds{P}(x) &= \frac{I_0(\kappa_{out})}{2\pi I_0(\kappa_1) I_0(\kappa_2)},\end{aligned}$$ where we defined $\lambda:= \lambda(x, \kappa_1, \kappa_2)$ with $0\leq \lambda \leq 1$.
Not surprisingly the growth of $\kappa$ (or shrinking of $\Delta_p$) with the number of rounds is upperbounded as $\kappa_M \leq 2^M \kappa_0$ for any protocol with $M$ rounds and initial states all with equal $\kappa_0$.
To get insight into the probabilistic behavior we would like to bound the probability that $\lambda \leq 1- \epsilon$ for some $\epsilon$ assuming $\kappa_1 \geq 1 / (1-\epsilon)$ and $\kappa_2 \geq 1 / (1-\epsilon)$ in a given round $m$. Let $\mathds{A} = \{x |\lambda\leq 1-\epsilon\}$, i.e. the set of all events for which $\lambda \leq 1-\epsilon$. Then $$\begin{aligned}
\mathds{P}(\lambda\leq 1-\epsilon) &= \int_{\mathds{A}} \text{d}x\ \frac{I_0\left((\kappa_1+\kappa_2)\lambda\right)}{2\pi I_0(\kappa_1) I_0(\kappa_2)}, \\
&\leq \frac{I_0\left((\kappa_1+\kappa_2)(1-\epsilon)\right)}{I_0(\kappa_1) I_0(\kappa_2)},\end{aligned}$$ where we used that $I_0(x)<I_0(y)$ for $x<y$.
It has been shown by Pal’tsev that $\frac{1}{\sqrt{2\pi \kappa}} e^{\kappa-\frac{1}{2\kappa}} \leq I_0(\kappa) \leq \frac{1}{\sqrt{2\pi \kappa}} e^{\kappa+\frac{1}{2\kappa}}$, where the lower bound holds for $\kappa>0$ and the upper bound was only proved for $\kappa>(\sqrt{7}+2)/3$ [@Paltsev.1999:BesselBound]. The range for the upper bound is limited because Pal’tsev derived the bounds for $I_\nu(\kappa)$ with $\nu, \kappa\in \mathds{R}^+_0$. In the special case of $I_0(\kappa)$, it is simple to show that the bound holds for all $\kappa>0$: $\frac{1}{\sqrt{2\pi \kappa}} e^{\kappa+\frac{1}{2\kappa}}$ is minimal for $\kappa=1$ and $I_0(\kappa), 0\leq \kappa\leq (\sqrt{7}+2)/3$ is maximal for $\kappa=(\sqrt{7}+2)/3$. The bound holds because $\frac{1}{\sqrt{2\pi}} e^{\frac{3}{2}} > I_0((\sqrt{7}+2)/3)$. Using these bounds, we get $$\begin{aligned}
\mathds{P}(\kappa_{out} \geq (\kappa_1+\kappa_2)(1-\epsilon)) \geq \delta.
\label{eq:bound_kappa}\end{aligned}$$ with $$\delta\equiv 1 - \sqrt{\frac{2\pi \kappa_1 \kappa_2}{(\kappa_1 + \kappa_2)(1-\epsilon)}} \exp\left(-\epsilon(\kappa_1 + \kappa_2 + 1) +\frac{5}{4}\right).
\label{def:delta}$$
For any choice of $\epsilon >0$, this probability is exponentially close to 1 for large $\kappa_1$ or $\kappa_2$. As a simple example of this bound one can take $\kappa_1=\kappa_2=\kappa_{in}$ and $\epsilon=1/2$. Then we have $$\begin{aligned}
\mathds{P}(\kappa_{out} \geq \kappa_{in}) \geq 1 - \sqrt{2 \pi \kappa_{in}} &\exp\left(-\kappa_{in} +\frac{3}{4}\right).\end{aligned}$$ What we see in these bounds is that for sufficiently large $\kappa_{\rm in}$ the protocol produces states with larger $\kappa_{\rm out}$ with high probability. For example, the probability that $\kappa_{\rm out} \geq \kappa_{\rm in}$ is at least $0.92$ for $\kappa_{\rm in}=5$ (squeezing parameter roughly $\Delta\approx 0.25$). For $\kappa_{\rm in}=10$, the probability that $\kappa_{\rm out} \geq \frac{3}{2} \kappa_{\rm in}$ is at least $0.88$.
Alternatively, one can phrase Eq. (\[eq:bound\_kappa\]) for large $\kappa$, hence Gaussian-distributed states, in terms of the variance of the Gaussian distribution of shift errors: In this case, we have that with probability larger than $\delta$ in Eq. (\[def:delta\]), the variance of the outgoing state obeys $$\begin{aligned}
& {\rm Var}_{\rm out} \leq \frac{{\rm Var}_{\rm 1,\rm in}{\rm Var}_{2,\rm in}}{(1-\epsilon)({\rm Var}_{1,\rm in}+{\rm Var}_{2,\rm in})}.
\label{eq:vari}\end{aligned}$$ For a grid state with Gaussian distributed shift errors and spacing $\xi$, one has $\Delta_p \approx {\rm Var}/\xi$ so we can see how \[eq:vari\] expresses the stochastic improvement of the effective squeezing parameter per round.
These bounds are not tight, the probability $\delta$ scales more favorably in practice than these bounds would suggest. In the next section we examine how the mapping of the von Mises distributed states works out numerically as compared to an actual simulation of the protocol with squeezed cat states.
Simulation {#sec:simulation}
==========
![Simulated breeding of a sensor state ($S_p = D(\sqrt{\pi})$) with initial squeezing $\Delta=0.2$. Shown is the (dimensionless) effective squeezing parameter $\Delta_p$ (averaged over the homodyne measurement outcomes) versus the number of rounds $M$ of the protocol. ‘Post-select’ refers to the protocol by Vasconcelos [*et al. *]{}[@Vasconcelos.etal.2010:GKPprep], with squeezed Schrödinger cat states as input and post-selected onto the result $p_i=0$ for all measurements. ‘Breeding’ refers to the efficient breeding protocol without post-selection. ‘Mises’ is the same efficient breeding protocol, but with von Mises distributed initial states, see \[eq:mises\]. The error bounds in both Mises and Breeding are asymmetric, i.e. both the variance of all the data above the mean as well as the variance on all the data below the mean are plotted separately. ‘Lower’ is the lower bound for the effective squeezing parameter, namely at round $M$ $\Delta(\kappa_M)=\Delta(2^M \kappa_0)$ where $\Delta(\kappa)$ is given in \[eq:squeeze\]. []{data-label="fig:simulation"}](numerics.pdf){width="\hsize"}
To demonstrate the use of classical post-processing we simulate the breeding of a grid state numerically. All the simulated breeding protocols aim to generate an eigenstate of $S_p=D(\sqrt{\pi})$, using $M$ rounds with the efficient breeding protocol. The breeding is simulated by sampling each measurement result randomly from the state generated by the previous rounds. This is done for protocols with $M=0,\dots ,6$ rounds, each protocol leading to an approximate grid state with the required spacing $\sqrt{2\pi}$ ($M=0$ means just having a squeezed cat state).
In \[fig:simulation\] we show the mean and standard deviation of the effective squeezing parameter $\Delta_p$ over 1000 repetitions of this procedure. In this Figure, the line ‘Breeding’ shows the efficient breeding protocol using finitely-squeezed Schrödinger cat states, with $\Delta_q\approx\Delta=0.2$. This corresponds to states with $\bar{n} \approx 2^M \pi/2 + 25$ photons in all rounds (where 25 is the contribution from initial squeezing by $S(\Delta)$).
In addition, we simulate the same protocol using the von Mises states (with infinite squeezing, corresponding to $\lim_{\Delta\to 0}$ in \[eq:mises\]) as initial states, starting at a $\kappa$ and $\mu=0$ which gives the same $\Delta_p$ as the squeezed cat states in the real protocol. For comparison, we also show the effective squeezing achieved by post-selecting onto $p=0$ (‘Post-select’) and the lower bound (‘Lower’) on the decrease in the squeezing parameter for the von Mises states as follows from $\kappa_{\rm out} \leq \kappa_1+\kappa_2$. Since the lower bound has been derived only for von Mises distributed states and not for squeezed cat states as initial states it does not necessarily hold for the latter. However, it gives a good estimate for the asymptotic behavior, as grid states and the von Mises distributed states get arbitrarily close for small $\Delta_p$.
As can be seen the effective squeezing which is achieved on average is lower both for Breeding and Mises than for the post-selected protocol. Furthermore, the two lines are almost parallel after some rounds, showing that the von Mises error model is a good approximation after a small number of rounds. All lines show similar scaling with $M$ which we asymptotically expect to be $\sim 2^{-M}$ (this scaling is hard to verify for $M \leq 6$).
Discussion {#sec:discussion}
==========
In this paper we have shown that classical post-processing, combined with the breeding protocol by Vasconcelos [*et al. *]{}yields an efficient method to generate grid states. By providing a map between breeding and phase estimation, we have argued that any state generated by breeding results in an approximate eigenstate of the commuting displacement operators, i.e. a grid state with an additional known displacement. We have introduced a new class of approximate grid states which are mapped onto themselves by the application of breeding and allow one to bound the success of the stochastic process implemented by breeding. In numerical simulations, we could confirm that the protocol discussed in this paper generates grid states reliably, showing scaling close to the asymptotic behavior, even for a small number of rounds.
As we have observed, the action of each round of beam-splitting reduces the spacing of the grid, requiring one to use cat states with large spacing at the beginning of the protocol. An alternative solution is to squeeze the outgoing mode after each beam-splitter so one does not lose a $\sqrt{2}$ factor in each round, see e.g. the use of beam-splitting and $\sqrt{2}$-squeezing in [@Glancy.Knill.2006:GKPfaultTolerance]. However, this precisely counteracts the initial squeezing in the $q$-quadrature, hence requires more initial squeezing by $\Delta$. We thus expect that the average number of photons in the initial squeezed cat states scales the same in this alternative protocol, making it a slightly different but not necessarily better alternative.
In any real set-up the measurement of the $p$-quadrature will have some variance, determined for example by the duration of the measurement. Using the mapping onto phase estimation one can understand this as a spread or uncertainty in the circuit which has been applied to the state, leading to uncertainty of an estimate for the eigenvalue phase. In the efficient breeding protocol, the spread in $p$ also leads to the preparation of a noisy state which contains additional shift or displacement errors.
While generating optical squeezed Schrödinger cat states on demand is a hard task, squeezed cat states with sufficient amplitude to generate the sensor state have been experimentally demonstrated in [@Etesse.etal.2015:CatBreedingExp; @Huang.etal.2015:CatStatePrep]. The amplitudes of cat states demonstrated there are sufficiently large for 1-2 rounds of breeding with beam splitters only. Multiple rounds could be possibly achieved using additional squeezing in between rounds as suggested in the previous paragraph above. It might also be of interest to analyze the concrete implementation of this scheme for microwave cavities coupled to superconducting qubits where all components, i.e. the preparation of cat states [@Vlastakis.etal.2013:CatStatePrep; @Ofek.etal.2016:CatCode], beam-splitters and homodyne measurement read-out are readily available. The scheme would lend itself well to a set-up in which cat states are prepared in microwave cavities and are then released [@Pfaff.etal.2017:CatchRelease; @Yin.etal.2013:CatchRelease] onto transmission lines which couple via beam-splitters and allow for homodyne read-out. We would like to thank Kasper Duivenvoorden and Christophe Vuillot for useful discussions and acknowledge support through the EU via the ERC GRANT EQEC. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development $\&$ Innovation.
Orthonormality and Completeness of Shifted Grid States {#sec:ortho}
======================================================
Recall that the shifted grid states are defined as (see \[eq:def\_shifted\]) $$\begin{aligned}
\ket{u, v, m}
&=\frac{\sqrt{s_m \xi}}{2\pi} \sum_{s=-\infty}^{\infty} \exp(i v (s+\frac{u}{2\pi})) \ket{p=s_m \xi
( s+ \frac{u}{2\pi})},
\label{eq:uvm}\end{aligned}$$ with $u,v\in[-\pi,\pi)$, $s_m\in(0,1]$ and $m\in\mathds{N}_0$. In this section we show that this class of states forms an orthonormal basis. The proof will be split in two parts, showing orthonormality first and completeness afterwards, see the Lemma’s below.
If we extend the definition of these basis states so that $u \rightarrow x,v \rightarrow y$ with $x,y \in \mathds{R}$, then we can observe that $\ket{x+2\pi,y,m}=\ket{x,y,m}$ and $\ket{x,y\pm 2\pi,m}=e^{\pm i x} \ket{x,y,m}$. In \[sec:breed\] we will only consider states of the form $$\begin{aligned}
\ket{\Psi} = \int_{-\pi}^\pi \text{d} u \int_{-\pi}^\pi \text{d}v\; \Theta(u,v)\ket{u,v,m}, \end{aligned}$$ where the function $\Theta(x,y)$ is such that $\Theta(x+2\pi,y)=\Theta(x,y)$ and $\Theta(x,y\pm 2\pi)=e^{\mp i x}\Theta(x,y)$. For such choice we observe that $\Theta(x,y) \ket{x,y,m}$ is $2\pi$-periodic in both arguments, allowing us to write $$\begin{aligned}
\int_{-\pi}^\pi \text{d}x \int_{-\pi}^\pi \text{d}y \ \Theta(x,y)\ket{x,y,m} = \int_{-\pi+z_x}^{\pi+z_x} \text{d}x\ \int_{-\pi+z_y}^{\pi+z_y} \text{d}x\ \Theta(x,y)\ket{x,y,m}.
\label{eq:shift_int}
\end{aligned}$$ for any $z_x$ and $z_y$.
\[lem:uvm\_orthonormal\] The class of shifted grid states as defined in \[eq:uvm\] is orthonormal, i.e. it holds that $\braket{u', v', m|u, v, m} = \delta(u-u')\delta(v-v')$.
From the definition of shifted grid states (\[eq:uvm\]) and the orthonormality of the momentum eigenstates it follows $$\begin{aligned}
\braket{u', v', m|u, v, m}
&= \frac{s_m \xi}{(2\pi)^2} \sum_{s, t=-\infty}^\infty \exp\left(i v \left(s+\frac{u}{2\pi}\right) - i v'\left(t+\frac{u'}{2\pi}\right)\right) \delta\left(s_m \xi\left(s-t + \frac{u-u'}{2\pi}\right)\right).
\end{aligned}$$ The difference $u-u'$ needs to be an integer multiple of $2\pi$ for the Dirac delta-function to be non-zero. Since $u, u'\in [-\pi,\pi)$, [i.e. ]{}$u-u'\in (-2\pi,2\pi)$, the only solution is $u=u'$ and $s=t$. With $\delta(x) = |a| \delta(a x)$ and $\delta(x) = \frac{1}{2\pi}\sum_{s=-\infty}^\infty \exp(isx)$ the claim follows: $$\begin{aligned}
\braket{u', v', m|u, v, m}
&= \frac{1}{2\pi}\sum_{s} \exp\left(i (v - v')\left(s+\frac{u}{2\pi} \right)\right) \delta(u-u'),
\notag \\
&= \delta(v - v') \delta(u-u').
\end{aligned}$$
To complete the proof that the shifted grid states form an orthonormal basis, we also show their completeness. We do this by showing $\int\text{d}u\int\text{d}v {| u,v,m\rangle\! \langle u,v,m|}p\rangle = \ket{p}$ for any momentum eigenstate $\ket{p}$.
\[lem:uvm\_complete\] The class of shifted grid states as defined in \[eq:uvm\] is complete, i.e. it holds that $\int\text{d}u\int\text{d}v\ {| u,v,m\rangle\! \langle u,v,m|} = I$.
The wave function of a momentum state in the shifted grid state basis is $$\begin{aligned}
\braket{u,v,m|\hat{p}=p}
&=
\sqrt{\frac{s_m \xi}{(2\pi)^2}} \sum_{s=-\infty}^\infty \exp\left(-i v \left(s+\frac{u}{2\pi}\right)\right)
\delta\left(\xi s_m \left(s+\frac{u}{2\pi}\right)-p\right).
\end{aligned}$$ Since $u\in[-\pi,\pi)$, the Dirac delta distribution is only non-zero for a specific value $s=\tilde{s}$ with $\tilde{p} := p-\xi s_m \tilde{s}, \tilde{p} \in [-\pi,\pi)$. Using $\delta(x) = |a| \delta(a x)$, we can simplify the wave function of a momentum state in the basis of shifted grid states to $$\begin{aligned}
\braket{u,v,m|\hat{p}=p}
&=
\sqrt{\frac{1}{\xi s_m}} \exp\left(-i v \left(\tilde{s}+\frac{u}{2\pi}\right)\right)
\delta\left(u- \frac{2\pi}{\xi s_m}\tilde{p}\right).
\end{aligned}$$ Using the definition of a shifted grid state (see \[eq:uvm\]) and the wavefunction of a momentum state in the basis of shifted grid states, we obtain $$\begin{aligned}
\iint \text{d}u\ \text{d}v\ \ket{u,v,m}\braket{u,v,m|\hat{p}=p}
&=
\sqrt{\frac{1}{\xi s_m}} \iint \text{d}u\ \text{d}v\ e^{-i v (\tilde{s}+\frac{u}{2\pi}) }
\delta\left(u- \frac{2\pi}{\xi s_m}\tilde{p}\right)\ket{u,v,m}\\
&=
\frac{1}{2\pi} \int \text{d}v\ \sum_{s} e^{i v (s-\tilde{s})}
\ket{\hat{p}=s_m \xi s+ \tilde{p}} \\
&= \ket{\hat{p}=s_m \xi \tilde{s}+ \tilde{p}} = \ket{\hat{p}=p}
\end{aligned}$$ In the second step, we used the integral representation of the Kronecker delta, $\frac{1}{2\pi} \int_0^{2\pi} \text{d}x \exp\left(ix(n-m)\right) = \delta_{mn}$.
Analytic Discussion of Breeding {#sec:breed}
===============================
In this appendix, we discuss the breeding protocol analytically and show that the class of states used as initial states in \[sec:bounds\] is closed under the breeding operation. To this end, we first analyze the action of breeding on a superposition of shifted grid states $\ket{u,v,m}$ in \[sec:breed\_sum\], and simplify the state obtained after measurement. Then, we show in \[sec:breed\_conv\] that the action of breeding on the $v$ shifts is that of a convolution of the ingoing wavefunctions, and that a Gaussian error model for these shifts is preserved under breeding. There, we also see that the action on the $u$ shifts is that the ingoing wavefunctions of these shifts are multiplied. Finally in \[sec:wavechoice\], we show that for the $u$ shifts, an error model using the von Mises distribution is preserved under breeding, yielding the states used in \[sec:bounds\].
Breeding Shifted Grid States {#sec:breed_sum}
----------------------------
The action $\mathcal{B}$ of a beam splitter is given by $$\begin{aligned}
&\hat{q}_1\to (\hat{q}_1 - \hat{q}_2)/\sqrt{2}, && \hat{p}_1\to (\hat{p}_1 - \hat{p}_2)/\sqrt{2},\notag \\
&\hat{q}_2 \to (\hat{q}_1 + \hat{q}_2)/\sqrt{2}, && \hat{p}_2\to (\hat{p}_1 + \hat{p}_2)/\sqrt{2}, \label[pluralequation]{eq:bs}\end{aligned}$$ where mode $1$ is the target mode and mode $2$ is the control mode.
Using conjugation one can see that two shifted grid states are transformed as $$\begin{aligned}
\mathcal{B}\ket{x_1, y_1, m}_1 \ket{x_2, y_2, m}_2 &= \mathcal{B} \frac{s_m \xi}{(2\pi)^2} \sum_{s,t} e^{i (y_2 (s+\frac{x_2}{2\pi})+y_1(t+\frac{x_1}{2\pi}))} e^{i\hat{q}_2 \xi s_m( s+\frac{x_2}{2\pi})}
e^{i\hat{q}_1\xi s_m( t+\frac{x_1}{2\pi})}
\mathcal{B}^\dag \mathcal{B}
\ket{p=0}_1
\ket{p=0}_2
,\notag \\
&= \frac{s_m \xi}{(2\pi)^2} \sum_{s,t} e^{i (y_2 (s+\frac{x_2}{2\pi})+y_1(t+\frac{x_1}{2\pi}))} e^{i\frac{\hat{q}_1+\hat{q}_2}{\sqrt{2}}\xi s_m( s+\frac{x_2}{2\pi})}
e^{i\frac{\hat{q}_1-\hat{q}_2}{\sqrt{2}}\xi s_m( t+\frac{x_1}{2\pi})}
\ket{p=0}_1
\ket{p=0}_2,\notag \\
&=
\frac{s_m \xi}{(2\pi)^2} \sum_{s,t} e^{i(y_2 (s+\frac{x_2}{2\pi})+y_1(t+\frac{x_1}{2\pi}))}
\ket{p=\frac{\xi s_m}{\sqrt{2}} (t+s+\frac{x_2+x_1}{2\pi})}_1
\ket{p=\frac{\xi s_m}{\sqrt{2}} (s-t+\frac{x_2-x_1}{2\pi})}_2.
$$ The invariance of the formal state $\ket{p=0}_1\ket{p=0}_2$ under beam-splitting can be understood from writing $\lim_{\Delta\rightarrow 0} S(1/\Delta) \ket{\rm vac}=\ket{p=0}$ and conjugating the squeezing operators by beam-splitters.
Now, we can easily compute the action of a measurement of mode $2$ with result $p_{\rm out}$: $$\begin{aligned}
& \bra{\hat{p}_2=p_{\rm out}} \mathcal{B}\ket{x_1,y_1,m}_1\ket{x_2,y_2,m}_2=\notag\\
&\hspace{2cm}=
\frac{s_m \xi}{(2\pi)^2} \sum_{s,t} e^{i (y_2 (s+\frac{x_2}{2\pi})+y_1(t+\frac{x_1}{2\pi})) }
\delta(p_{\rm out}-\frac{\xi s_m}{\sqrt{2}} (s-t+\frac{x_2-x_1}{2\pi}))
\ket{p=\frac{\xi s_m}{\sqrt{2}} (t+s+\frac{x_2+x_1}{2\pi})}_1.
\label{eq:measure_uvm}\end{aligned}$$ As a warm-up, we consider the effect of the breeding step on two input modes both in a state of the form $$\begin{aligned}
\ket{\Psi_{\text{in}}} = \int_{-\pi}^\pi \text{d}u\ V(u) \ket{u, v, m},\end{aligned}$$ where $V(u)$ is a wave function with normalization $\int_{-\pi}^{\pi} \text{d}u \ |V(u)|^2=1$ that will be chosen in \[sec:wavechoice\]. Using \[eq:measure\_uvm\], switching to variables $x$ and $y$, and substituting $\tilde{x}_2 = x_2-\frac{2\pi \sqrt{2}p_{\rm out}}{\xi s_m}$, breeding then gives the output state $$\begin{aligned}
\ket{\Psi_{\rm out}} &=\frac{s_m \xi}{(2\pi)^2} \int_{-\pi+z}^{\pi+z} \text{d}\tilde{x}_2
\int_{-\pi}^{\pi} \text{d} x_1\
V_1(x_1) V_2(\tilde{x}_2+\frac{2\pi \sqrt{2}p_{\rm out}}{\xi s_m})
\sum_{s,t} e^{i (y_2 (s+\frac{\tilde{x}_2}{2\pi}+\frac{\sqrt{2}p_{\rm out}}{\xi s_m})+y_1(t+\frac{x_1}{2\pi})) } \notag \\
&\quad\quad\times
\delta\left(\frac{\xi s_m}{\sqrt{2}} (s-t+\frac{\tilde{x}_2-x_1}{2\pi})\right)
\ket{p=\frac{\xi s_m}{\sqrt{2}} (t+s+\frac{\tilde{x}_2+x_1}{2\pi})+p_{\rm out}}_1.\end{aligned}$$ where $z=\frac{2\pi \sqrt{2} p_{\rm out}}{\xi s_m}$. We can move the integration region for $\tilde{x}_2$ back as described in \[eq:shift\_int\]. Then, note that the Dirac delta distribution is only non-zero if $s=t$ and $\tilde{x}_2=x_1$: After moving the integration region back, $\tilde{x}_2 - x_1\in (-2\pi,2\pi)$. The solutions $\tilde{x}_2 - x_1 = \pm 2\pi$ are a nullset, after applying the Dirac delta distribution, the second integral vanishes for these two solutions. Using $\delta(x) = |a| \delta(a x)$ we obtain: $$\begin{aligned}
\ket{\Psi_{\rm out}}&= \frac{\sqrt{2}}{2\pi} \int_{-\pi}^\pi\text{d}\tilde{x}_2 \int_{-\pi}^\pi \text{d} x_1\
V_1(x_1) V_2(\tilde{x}_2+\frac{2\pi \sqrt{2}p_{\rm out}}{\xi s_m}) \notag \\
&\hspace{1cm}\times\sum_{s} e^{i (y_2 (s+\frac{\tilde{x}_2}{2\pi}+\frac{\sqrt{2}p_{\rm out}}{\xi s_m})+y_1(s+\frac{x_1}{2\pi})) }
\delta(\tilde{x}_2-x_1)
e^{ip_{\rm out}\hat{q}}
\ket{p=\frac{\xi s_m}{\sqrt{2}} (2s+\frac{\tilde{x}_2+x_1}{2\pi})}_1\\
&= \frac{\sqrt{2}}{2\pi} \int_{-\pi}^\pi \text{d} x_1\
V_1(x_1) V_2(x_1+\frac{2\pi \sqrt{2}p_{\rm out}}{\xi s_m})
\sum_{s} e^{i ((y_2+y_1) (s+\frac{x_1}{2\pi})+y_2\frac{\sqrt{2}p_{\rm out}}{\xi s_m})}
e^{ip_{\rm out}\hat{q}}
\ket{p=\frac{\xi s_m}{\sqrt{2}} (2s+\frac{2 x_1}{2\pi})}_1.\end{aligned}$$ With $s_{m+1} = \sqrt{2}s_m$, we finally have $$\begin{aligned}
\ket{\Psi_{\rm out}}&=
\frac{\sqrt{2}}{2\pi} \int_{-\pi}^\pi \text{d} x_1\
V_1(x_1) V_2(x_1+\frac{4\pi p_{\rm out}}{\xi s_{m+1}})
\sum_{s} e^{i ((y_2+y_1) (s+\frac{x_1}{2\pi})+y_2\frac{2p_{\rm out}}{\xi s_{m+1}} )}
e^{ip_{\rm out}\hat{q}}
\ket{p=\xi s_{m+1} (s+\frac{x_1}{2\pi})}_1.
\label{eq:measure}\end{aligned}$$
Choice of Wave Function $\Theta(u,v)$ {#sec:breed_conv}
-------------------------------------
We now take the input states in both modes with a wave function $\Theta(x,y)$ (obeying the conditions set forth previously), namely we choose $$\begin{aligned}
\Theta(u,v)=\frac{1}{\mathcal{N}} V(u) \sum_{s=-\infty}^{\infty}e^{i u s }G_{s_m \Delta}(v+2\pi s),
\label{eq:formtheta}\end{aligned}$$ where $V(u)$ is again the normalized wave function to be chosen in Section \[sec:wavechoice\], and $G_{s_m \Delta}$ is a Gaussian distribution $$\begin{aligned}
G_{\Delta}(v) &= \frac{1}{\sqrt{\Delta\sqrt{\pi }}}\exp(-\frac{v^2}{2\Delta^2}).\end{aligned}$$ The wave function’s dependence on $v$ is thus that of wrapped Gaussian distribution and the $e^{i us}$ factor in \[eq:formtheta\] is required for the $2\pi$-periodicity of the states as explained below \[eq:uvm\]. The normalization constant $\mathcal{N}$ is given by $$\begin{aligned}
\mathcal{N}^2 &= \int_{-\pi}^\pi \text{d} u \int_{-\pi}^\pi \text{d}v\ |V(u)|^2 \sum_{s,t=-\infty}^\infty e^{iu(s-t)} G_{s_m \Delta}(v+2\pi s)G_{s_m \Delta}(v+2\pi t) \notag \\
&= \int_{-\pi}^\pi \text{d} u \ |V(u)|^2 \sum_{s,t=-\infty}^\infty \frac{1}{2} e^{iu(s-t)} e^{-\frac{\pi^2(s-t)^2}{(s_m \Delta)^2}}\left( \operatorname*{erf}\left(\frac{\pi}{s_m \Delta}(s+t+1)\right) - \operatorname*{erf}\left(\frac{\pi}{s_m \Delta}(s+t-1)\right)\right) \xrightarrow{s_m\Delta \ll 1} 1.
\label{eq:norm}\end{aligned}$$ In the limit $s_m \Delta \ll 1$, the exponential $e^{-\frac{\pi^2(s-t)^2}{(s_m \Delta)^2}}$ enforces $s-t=0$, while for the difference of error functions to be non-zero, we need $s+t=0$, hence together one has $s=t=0$. Note that $s_m\in(0,1]$, [i.e. ]{}if $\Delta\ll 1$, then also $s_m\Delta\ll 1$.
Using this wave function we can write the input state in one of the modes as $$\begin{aligned}
\ket{\Psi_{\rm in}} = \frac{1}{\mathcal{N}} \int_{-\pi}^\pi \text{d} u \int_{-\pi}^\pi \text{d}v\ V(u) \sum_{s=-\infty}^\infty e^{ius} G_{s_m \Delta}(v+2\pi s) \ket{u, v, m} = \frac{1}{\mathcal{N}} \int_{-\pi}^\pi \text{d} x \int_{-\infty}^\infty \text{d}y\ V(x)G_{s_m \Delta}(y) \ket{x, y, m}.\end{aligned}$$
We will assume that the Gaussian wave function of both modes has the same variance, and mean equal to $0$. This choice is justified if the outgoing Gaussians only depend on the round $m$, which we will show below.
From the result for breeding states with arbitrary superpositions of shifts in $\hat{p}$, \[eq:measure\], it follows that $$\begin{aligned}
\ket{\Psi_{\rm out}}&=
\frac{\sqrt{2}}{2\pi \mathcal{N}^2} \int_{-\pi}^\pi \text{d} x_1 \int_{-\infty}^\infty\text{d}y_1
\int_{-\infty}^\infty\text{d}y_2\
V_1(x_1) V_2(x_1+\frac{4\pi p_{\rm out}}{\xi s_{m+1}})G_{s_m\Delta}(y_1)G_{s_m\Delta}(y_2) \notag \\
&\quad\quad\times \sum_{s} e^{i ((y_1+y_2) (s+\frac{x_1}{2\pi})+y_2\frac{2p_{\rm out}}{\xi s_{m+1}} )}e^{ip_{\rm out}\hat{q}}
\ket{p=\xi s_{m+1} (s+\frac{x_1}{2\pi})},\\
&=
\frac{\sqrt{2}}{2\pi \mathcal{N}^2} \int_{-\pi}^\pi \text{d} x_1 \int_{-\infty}^\infty\text{d}y_1
\int_{-\infty}^\infty\text{d}y_2\
V_1(x_1) V_2(x_1+\frac{4\pi p_{\rm out}}{\xi s_{m+1}}) G_{s_m\Delta}(\tilde{y}-y_2) G_{s_m\Delta}(y_2) \notag \\
&\quad\quad\times \sum_{s} e^{i (\tilde{y} (s+\frac{x_1}{2\pi})+y_2\frac{2p_{\rm out}}{\xi s_{m+1}} )}e^{ip_{\rm out}\hat{q}}
\ket{p=\xi s_{m+1} (s+\frac{x_1}{2\pi})}\\
&=
\frac{\sqrt{2s_{m+1}\Delta\sqrt{\pi}}}{2\pi \mathcal{N}^2} e^{-\frac{p_{\rm out}^2\Delta^2}{2\xi^2 }} \int_{-\pi}^\pi \text{d} x_1 \int_{-\infty}^\infty\text{d}\tilde{y}\
V_1(x_1) V_2(x_1+\frac{4\pi p_{\rm out}}{\xi s_{m+1}}) G_{s_{m+1}\Delta}(\tilde{y}) \notag \\
&\quad\quad\times \sum_{s} e^{i (\tilde{y} (s+\frac{x_1}{2\pi})+\tilde{y}\frac{p_{\rm out}}{\xi s_{m+1}} )}
e^{ip_{\rm out}\hat{q}}\ket{p=\xi s_{m+1} (s+\frac{x_1}{2\pi})}.\end{aligned}$$
Here, we used $s_{m+1}=\sqrt{2}s_m$ and the substitution $\tilde{y}=y_1+y_2$ to write the integral over $y_2$ as a [*convolution*]{} of Gaussian wave functions. Comparing this state with the definition of shifted grid states, \[eq:uvm\], we see that the outgoing state has a ‘simple’ expression in terms of the shifted grid states with extended parameters: $$\begin{aligned}
\ket{\Psi_{\rm out}}
&=
\frac{\sqrt{2\Delta\sqrt{\pi}}}{\sqrt{\xi} N^2} e^{-(\frac{p_{\rm out}\Delta}{\xi \sqrt{2}})^2} \int_{-\pi}^\pi \text{d} x_1 \int_{-\infty}^\infty\text{d}\tilde{y}\
V_1(x_1) V_2(x_1+\frac{4\pi p_{\rm out}}{\xi s_{m+1}})G_{s_{m+1}\Delta}(\tilde{y})
\ket{x_1+\frac{2 \pi p_{\rm out}}{\xi s_{m+1}}, \tilde{y},m+1}.\end{aligned}$$ With $\tilde{x} = x_1+\frac{ 2\pi p_{\rm out}}{\xi s_{m+1}}$ and \[eq:shift\_int\], we finally have $$\begin{aligned}
\ket{\Psi_{out}}
&=
\frac{\sqrt{2\Delta\sqrt{\pi}}}{\sqrt{\xi} \mathcal{N}^2} e^{-(\frac{p_{\rm out}\Delta}{\xi \sqrt{2}})^2} \int_{-\pi}^\pi \text{d} \tilde{x} \int_{-\infty}^\infty\text{d}\tilde{y}\
V_1(\tilde{x}-\frac{2\pi p_{\rm out}}{\xi s_{m+1}})V_2(\tilde{x}+\frac{2\pi p_{\rm out}}{\xi s_{m+1}})G_{s_{m+1}\Delta}(\tilde{y})
\ket{\tilde{x},\tilde{y},m+1} \label{eq:v_out_xy}\\
&=
\frac{\sqrt{2\Delta\sqrt{\pi}}}{\sqrt{\xi} \mathcal{N}^2} e^{-(\frac{p_{\rm out}\Delta}{\xi \sqrt{2}})^2} \int_{-\pi}^\pi \text{d} u \int_{-\pi}^\pi\text{d}v\
V_1(u-\frac{2\pi p_{\rm out}}{\xi s_{m+1}})V_2(u+\frac{2\pi p_{\rm out}}{\xi s_{m+1}})\sum_{s=-\infty}^\infty e^{ius} G_{s_{m+1}\Delta}(v+2\pi s)
\ket{u,v,m+1}
\label{eq:v_out_uv}\end{aligned}$$ Hence we conclude that the outgoing state has the same wave function dependence in $v$ as the ingoing states. The only change is $s_m \to s_{m+1}$. From this last equation we can also immediately see the action of breeding on the wave function $V(u)$, i.e. $V(u) \rightarrow V(u+\frac{2\pi p_{\rm out}}{\xi s_{m+1}})V(u-\frac{2\pi p_{\rm out}}{\xi s_{m+1}})$.
Choice for Wave Function $V(u)$ {#sec:wavechoice}
-------------------------------
As can be seen in \[eq:v\_out\_uv\], the output state depends on a product of the form $V_1(u)V_2(u)$. For some choices for the ingoing wavefunctions, one can simplify $V_1(u)V_2(u) = V_{\text{out}}(u)$, where all $V_i$ are in the same class of functions. One such class of functions is the set of von Mises distributions, which is closed under multiplication. Let $$\begin{aligned}
V(x-\mu)_{\kappa} = \frac{\exp\left(\frac{\kappa}{2}\cos(x-\mu)\right)}{\sqrt{2\pi I_0(\kappa)}}.
\label{eq:vonMises}\end{aligned}$$
Assuming a von Mises wave function in $u$ and a wrapped (signed) Gaussian wave function in $v$, the initial state of the system is thus chosen as $$\begin{aligned}
\ket{\Psi_{\rm in}} = \frac{1}{\mathcal{N}} \int_{-\pi}^\pi \text{d} u \int_{-\pi}^\pi \text{d}v\ V_\kappa(u-\mu) \sum_{s=-\infty}^\infty e^{ius} G_{s_m \Delta}(v+2\pi s) \ket{u, v, m} = \frac{1}{\mathcal{N}} \int_{-\pi}^\pi \text{d} x \int_{-\infty}^\infty \text{d}y\ V_\kappa(x-\mu) G_{s_m \Delta}(y) \ket{x, y, m},
\label{eq:u_in}\end{aligned}$$ where $V_{\kappa}(u)$ is the distribution defined in \[eq:vonMises\]. The normalization constant $\mathcal{N}$ has the same form as \[eq:norm\], with the von Mises wave function defined above taking the role of $V(x)$. This is also the initial state used in the main text, see \[eq:mises\]. Using the result for a Gaussian error model in $\hat{q}$ and an arbitrary wave function for $\hat{p}$, \[eq:v\_out\_xy\], the state after measurement is $$\begin{aligned}
\ket{\Psi_{\rm out}}
&=
\frac{\sqrt{2\Delta\sqrt{\pi}}}{\sqrt{\xi} \mathcal{N}_2 \mathcal{N}_1} e^{-(\frac{p_{\rm out}\Delta}{\xi \sqrt{2}})^2} \int_{-\pi}^\pi \text{d} \tilde{x} \int_{-\infty}^\infty\text{d}\tilde{y}\
V_{\kappa_2}(\tilde{x}+\frac{2\pi p_{\rm out}}{\xi s_{m+1}}-\mu_2)V_{\kappa_1}(\tilde{x}-\frac{2\pi p_{\rm out}}{\xi s_{m+1}}-\mu_1)G_{s_{m+1}\Delta}(\tilde{y})
\ket{\tilde{x},\tilde{y},m+1},\end{aligned}$$ where $\mathcal{N}_1, \mathcal{N}_2$ are the normalization constants of the initial state of modes $1$ and $2$, respectively. This expression can be simplified with the following lemma.
For a product of von Mises wave functions as defined in \[eq:vonMises\] it holds that $$\begin{aligned}
V_{\kappa_1}(x-\mu_1)V_{\kappa_2}(x-\mu_2) &= \sqrt{\frac{I_0(\kappa)}{2\pi I_0(\kappa_1) I_0(\kappa_2)}} V_{\kappa}(x-\mu),
\end{aligned}$$ with $$\begin{aligned}
&\mu = -\operatorname*{atan2}\left(\kappa_1\cos(\mu_1) + \kappa_2\cos(\mu_2), \kappa_1\sin(\mu_1) + \kappa_2\sin(\mu_2)\right),
&&\kappa^2 = \kappa_1^2+\kappa_2^2+2\kappa_1\kappa_2\cos(\mu_1-\mu_2).
\end{aligned}$$
We can use the properties of linear combinations of trigonometric functions to show that the set of von Mises distributions is closed under multiplication. We have $$\begin{aligned}
V(x-\mu_1)_{\kappa_1}V(x-\mu_2)_{\kappa_2} &= \frac{\exp\left(\frac{\kappa_1}{2}\cos(x-\mu_1) + \frac{\kappa_2}{2}\cos(x-\mu_2)\right)}{2\pi \sqrt{I_0(\kappa_1)I_0(\kappa_2)}}
\end{aligned}$$ For the exponent on the r.h.s. it holds that $$\begin{aligned}
\kappa_1\cos(x-\mu_1) + \kappa_2\cos(x-\mu_2) &=
\left(\kappa_1\cos(\mu_1) + \kappa_2 \cos(\mu_2)\right)\cos(x)
+ \left(\kappa_1\sin(\mu_1) + \kappa_2\sin(\mu_2)\right)\sin(x)\\
&=\sqrt{\kappa_1^2+\kappa_2^2+2\kappa_1\kappa_2\cos(\mu_1-\mu_2)} \cos(x-\mu) := \kappa \cos(x-\mu)
\end{aligned}$$ with $\mu, \kappa$ as in the claim. In the first step, we used $\cos(x-y) = \cos(x)\cos(y) + \sin(x)\sin(y)$. In the second step, we used $a\cos(x) + b\sin(x) = \sqrt{a^2+b^2}\cos(x+\operatorname*{atan2}(a, b))$.
Using this lemma, the outgoing state is given by $$\begin{aligned}
\ket{\Psi_{\rm out}}
&=
\sqrt{\frac{I_0(\kappa) \Delta}{\sqrt{\pi} I_0(\kappa_1) I_0(\kappa_2) \xi \mathcal{N}_2^2 \mathcal{N}_1^2}} e^{-\frac{p_{\rm out}^2\Delta^2}{2\xi^2}} \int_{-\pi}^\pi \text{d} u \int_{-\pi}^\pi\text{d}v\
V_{\kappa}(u-\mu)\sum_{s=-\infty}^\infty e^{ius} G_{s_{m+1}\Delta}(v+2\pi s),
\ket{u,v,m+1}\end{aligned}$$ with $$\begin{aligned}
\mu &= -\operatorname*{atan2}\left(\kappa_2\cos(\mu_2 - \frac{2\pi p_{\rm out}}{\xi s_{m+1}}) + \kappa_1\cos(\mu_1 + \frac{2\pi p_{\rm out}}{\xi s_{m+1}}), \kappa_2\sin(\mu_2 - \frac{2\pi p_{\rm out}}{\xi s_{m+1}}) + \kappa_1\sin(\mu_1 + \frac{2\pi p_{\rm out}}{\xi s_{m+1}})\right)\\
\kappa^2 &= \kappa_2^2+\kappa_1^2+2\kappa_2\kappa_1\cos(\mu_2-\mu_1-\frac{4\pi p_{\rm out}}{\xi s_{m+1}})\end{aligned}$$ This state is not yet normalized. However, we can use \[eq:norm\] to obtain $\mathcal{N}_{out}$ and both normalize this state and obtain the probability distribution of measurement results $p_{\rm out}$ as written in the main text.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We discuss some phenomenological aspects of $\gamma$-ray emitting jets. In particular, we present calculations of the $\gamma$-sphere and $\pi$-sphere for various target photon fields and employ them to demonstrate how $\gamma$-ray observations at very high energies can be used to constraint the Doppler factor of the emitting plasma and the production of VHE neutrinos. We also consider some implications of the rapid TeV variability observed in M87 and the TeV blazars, and propose a model for the very rapid TeV flares observed with HESS and MAGIC in some blazars, that accommodates the relatively small Doppler factors inferred from radio observations. Finally, we briefly discuss the prospects for detecting VHE neutrinos from relativistic jets.'
author:
- Amir Levinson
title: 'Phenomenology of Gamma-Ray Jets'
---
[ address=[School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel]{} ]{}
Introduction
============
Ejection of collimated relativistic outflows appears to be a common phenomena in astrophysics. The radio-to-$\gamma$-ray continuum emission observed in blazars, microquasars, and $\gamma$-ray bursts (GRBs) is believed to be produced in such outflows on various scales. The common view is that these flows are powered by a magnetized accretion disk and a spinning black hole, and collimated by magnetic fields and/or the medium surrounding the jet. However, there is as yet no universal agreement about the mechanisms responsible for the formation and acceleration the jet and the dissipation of its bulk energy. Even the composition of the jet is unknown in most sources.
Active states during which rapid, large amplitude variations of the high-energy emission are observed appear to be quite common in most classes of compact relativistic systems. This activity is presumably associated with violent ejection episodes, as directly inferred in a few cases. Some of the objects mentioned above may also provide sites for acceleration of the UHE cosmic rays detected by various experiments, if the latter are indeed produced in a bottom-up scenario, posing a great challenge to the theory of particle acceleration. Emission of VHE neutrinos should accompany the production of those UHECRs, and optimistic models predict fluxes in excess of detection limit of upcoming cubic-km scale neutrino telescopes. As discussed below, observations of VHE $\gamma$-rays can provide stringent constraints on the photopion opacity, which can be translated into upper limits on the neutrino flux.
Of particular interest is the class of TeV sources. There are at present over a dozen TeV blazars (for updated list see, e.g., Ref [@wag07]), all of which are exclusively associated with the class of high peak BL Lac objects, another (nonblazar) TeV AGN, M87 [@ah06], and several X-ray binaries (microquasars or gamma-ray binaries). The observed bolometric luminosity of TeV blazars during quiescent states is typically of the order of a few times $10^{44}$ ergs s$^{-1}$, with about 10 percents emitted as VHE $\gamma$-rays. The luminosity in the VHE band may be larger by a factor of 10 to 100 during flaring states. The intrinsic spectra (corrected for absorption on the extragalactic background light) appear to be hard, with a peak photon energy in excess of 10 TeV in the most extreme cases. The constraints on the dynamics of the system are most stringent in this class of sources, and are discussed in some greater detail below.
Structure and Dynamics of $\gamma$-ray jets
===========================================
The $\gamma$-sphere and the $\pi$-sphere
----------------------------------------
The pair production opacity is typically large within the inner jet region in essentially all classes of compact high-energy sources. This implies that $\gamma$-rays produced at small radii will not be able to escape the system before being converted to e$^\pm$ pairs. Both the synchrotron photons produced inside the jet and ambient radiation intercepted by the jet contribute an opacity to pair production. Results of detailed calculations of the pair production opacity are exhibited in figure 1, where the [*$\gamma$-spheric radius*]{}, defined as the radius $r_{\gamma}(\epsilon_{\gamma})$ beyond which the pair production optical depth to infinity is unity, viz., $\tau_{\gamma\gamma}(r_\gamma,\epsilon_\gamma)=1$, is plotted against $\gamma$-ray energy $\epsilon_{\gamma}$, for two target radiation fields: synchrotron radiation (dashed lines) and external radiation (solid lines). The spectra of the target radiation fields employed in those calculations are given in Ref [@lev06]. As seen the $\gamma$-spheric radius increases, quite generally, with increasing $\gamma$-ray energy, and for luminous sources can be much larger than the dissipation radius (indicated in the figure).
In the powerful blazars, like 3C279, and in microquasars the intensity of both external and synchrotron radiation is quite large, corresponding to the uppermost curves in fig. 1. This then implies that the $\gamma$-spheres at energies corresponding to the GLAST band should encompass a rather large range of radii. In these sources the $\gamma$-spheres can be mapped in principle by measuring temporal variations of the $\gamma$-ray flux in different energy bands during a flare. If the $\gamma$-ray emission is produced over many octaves of jet radius, where intense pair cascades at the observed energies are important [@bln95], then it is expected that a flare will propagate from low to high energies, or that the variations at higher $\gamma$-ray energy will be slower than at lower energies. With the limited sensitivity and energy band of the EGRET instrument it was practically impossible to resolve such effects. It is hoped that with the upcoming GLAST instrument this will be feasible. In contrast, in TeV blazars, which are much fainter, the target photon luminosity is small and the location of the $\gamma$-spheres is not constrained, except, perhaps, at the highest energies observed (a few TeV). This difference should be reflected in the variability pattern of the VHE emission. A particular model for the rapid TeV flares is discussed below.
![Dimensionless $\gamma$-spheric radius versus $\gamma$-ray energy, computed in Ref [@lev06]. The different curves correspond to a different normalization of the target radiation field intensity. The dissipation radius $r_d$ is indicated.](f1.eps){height=".5\textheight"}
The same target radiation field contributes also an opacity to photopion production. Because both the protons and the $\gamma$ rays interact locally with the same target radiation field the ratio of photomeson and pair production opacities depend solely on the ratio of cross sections, $\sigma_{p\gamma}/\sigma_{\gamma\gamma}\simeq 4\times 10^{-3}$, and the spectrum of the target radiation field. For a target photon spectrum $n_s(\epsilon_s)\propto \epsilon_s^{-\alpha}$ we have [@lev06] $${\tau_{p\gamma}(\epsilon_p,r)\over \tau_{\gamma\gamma}(\epsilon_{\gamma},r)}
\simeq4\times10^{-3}
\left(\frac{\epsilon_p}{3\times10^5\epsilon_{\gamma}}\right)^{\alpha}.
\label{tau_pg}$$ Detailed calculations of opacity ratios that employed more realistic target photon spectra are presented in Ref [@lev06]. It is found that at $\gamma$-ray energies above a few TeV the opacity ratio is smaller than unity even at the maximum proton energy (determined from the confinement limit). For the class of TeV blazars this implies neutrino yields well below detection limit. With GLAST it should be possible to constrain other sources and to use such constraints to identify the best candidates for the upcoming km$^3$ detectors.
Implications of variability
---------------------------
The observed variability limits the linear size of the emission region (as measured in the Lab frame) to $d{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}10^{14}(1-\beta\cos\theta_n)^{-1} t_{\rm var,h}/(1+z)$ cm, where $\beta$ is the bulk speed of the emitting fluid, $\theta_n$ is the viewing angle, $t_{\rm var,h}$ is the observed variability time in hours, and $z$ is the redshift of the source. The rapid variability observed in GRBs and blazars implies typically $d/r_g{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}\Gamma^2$, where $\Gamma$ is the bulk Lorentz factor of the emitting fluid. The rapid VHE flare recorded recently in the TeV blazar PKS 2155-304 requires $\Gamma\sim20$ in order that $d\sim r_g$, regardless of any other considerations.
The location of the emitting plasma is yet another issue. If the emission originates from radii $r_{\rm em}\sim d$, as often assumed, then compactness arguments yield a lower limit on the Doppler factor of the emission zone, as discussed further below. If, on the other hand, the emission is produced at radii $r_{\rm em}>>d$, as proposed recently for M87 and TeV blazars, then the fraction of jet energy that can be dissipated and converted to radiation in a conical jet of opening angle $\theta$ is at most $\eta{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}(d/\theta r_{em})^2$. As a consequence, either the opening angle of the jet must be very small, $\theta \sim d/r<<1$, or the radiative efficiency $L_{\rm rad}/L_j$ must be very small, implying unreasonably large jet power in the most extreme cases. In situations where the jet is underpressured relative to the confining medium the jet is expected to converge to the axis. Reflection of collimation shocks at the nozzle may then give rise to appreciable dissipation in a very small region. The pattern speed of the emission region can differ significantly from the speed of the fluid, and stationary features, as occasionally observed in radio jets of blazars [@Jorstad01] can be naturally produced. Such a model has been proposed to explain the rapid variability of the resolved X-ray emission and the unresolved TeV emission from the HST1 knot in M87 [@cheu07]. Alternative explanations have been offered for the rapidly varying TeV emission (e.g., Ref [@m87], and references therein). It should be noted though that in M87 the X-ray and TeV luminosities, $L_{\rm TeV}\sim L_{\rm x}{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}10^{41}$ erg s$^{-1}$ [@ah06; @cheu07], are much smaller than the TeV luminosity, $L_{\rm TeV}\sim 10^{44-45}$ erg s$^{-1}$, observed typically in the TeV blazars. Estimates of the jet power in M87 yield $L_j{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}10^{44}$ erg s$^{-1}$ [@bick96], implying a very small conversion fraction, $L_{\rm TeV}/L_j{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}10^{-3}$. Even with such a small conversion efficiency an opening angle $\theta<10^{-2}$ rad is required if the TeV emission were to originate from the HST1 knot, unless reconfinement can give rise to sufficient convergence of the jet at the location of HST1, as proposed in Ref [@cheu07]. This idea is compelling since even modest radiative cooling of the shocked jet layer in a proton dominated jet will lead to such a convergence, at least in the non-relativistic case [@eich82]. The effect of cooling on the collimation of relativistic jets needs to be explored. The stationary radio features observed in blazars seem to indicate that recollimation shocks may be an important dissipation channel in blazsrs, and this may apply also to other sources, e.g., GRBs [@brom07]. Whether the extreme TeV flares observed in VHE blazars can be accounted for by recollimation shocks at radii $r_{\rm em}>> d$ remains to be investigated. However, this would not resolve the ’Doppler factor crises’ if the IR emission would turn out to vary on timescales comparable to the duration of the TeV flare.
Constraints on Doppler factors
------------------------------
Constraints on the Doppler factor of the fluid emitting VHE $\gamma$-rays can be derived by measuring the low-energy flux (radio-to-IR) simultaneously with the variable $\gamma$-ray emission. The requirement that the pair production opacity should not exceed unity, viz., $\tau^{\rm syn}_{\gamma\gamma}(r_{\rm em},\epsilon_\gamma/\delta)<1$, constrains the density of target photons: $n_s(r_{\rm em}){\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}(\sigma_{\gamma\gamma} d)^{-1}$. If the emission is assumed to originate from the innermost jet radii, in which case $r_{\rm em}\sim d$, then we must have $r_{\gamma}(\epsilon_\gamma)< r_{\rm em}$. The latter condition on $r_{\gamma}(\epsilon_\gamma)$ can be solved for the Doppler factor to yield [@lev06] $$\delta^{5}>2\times10^{11}
t_{\rm var,h}^{-3/2}(\Gamma\theta)^{-2}z^2(\epsilon_\gamma/{\rm 1 TeV})^{1/2}S_{{\rm Jy}},
\label{Var-const}$$ where $S_{\rm Jy}$ is the measured synchrotron flux density in Janskys and $z$ is the redshift of the source. In deriving eq. (\[Var-const\]) a luminosity distance $d_L=10^{28} z$ has been adopted. In cases where $r_{\rm em}>> d$ the compactness of the TeV emission zone may be constrained by the variability of the IR flux observed simultaneously with the TeV flare. If the IR emission varies over time scales comparable to the duration of the VHE flare then eq. (\[Var-const\]) still applies. Lower values of $\delta$ are allowed if the variability time of the IR emission is much longer than the variability time of the VHE emission.
Adopting $\Gamma\theta\simeq1$, we estimate $\delta>35$ for the rapid flare observed in Mrk 421 and $\delta>190$ for the few minuets variability reported for PKS 2155-304, with $r_{\rm em}<10^{17}$ cm for the minimum condition in both sources.
A model for rapid flares in TeV blazars
---------------------------------------
The large values of the Doppler factor implied by opacity constraints and rapid variability in TeV blazars are consistent with those obtained from fits of the SED to a homogeneous SSC model, but are in clear disagreement with the much lower values inferred from unification models [@Urry91; @Hardcastle03] and superluminal motions on parsec scales [@mar99; @Jorstad01; @Giroletti04]. Various explanations, including a structure consisting of interacting spine and sheath [@Ghisellini05], opening angle effects [@Gopal-Krishna04] and jet deceleration [@Geo03; @Piner05] have been proposed in order to resolve this discrepancy.
It has been proposed recently [@lev07] that the rapid TeV flares observed in sources like Mrk 421, Mrk 501 and PKS 2155-304 are produced by radiative deceleration of fluid shells expelled during violent ejection episodes. These shells are envisaged to accelerate to a Lorentz factor $\Gamma_0>>1$ at some radius $r_d\sim10^2-10^3 r_g$, at which dissipation of their bulk energy occurs. The dissipation may be accomplished through formation of internal shocks in a hydrodynamic jet or dissipation of magnetic energy in a Poynting flux dominated jet [@rom92; @lev98], and it is assumed that a fraction $\xi_e$ of the total proper jet energy density, $u_j^\prime$, is tapped for acceleration of electrons to a maximum energy $\gamma_{max}m_ec^2$. The dynamics of the front is then governed by the equation [@lev07] $$\frac{d}{dr}(u^\prime_j\Gamma^2 \beta)= -\frac{4\sigma_T}{3m_ec^2}\chi\xi_e \Gamma^3\gamma_{\rm max}u_su^\prime_{j},
\label{eq-mot}$$ where $u_s$ is the energy density of the target radiation field, as measured in the Lab frame, and $\chi=<\gamma^2>/(<\gamma>\gamma_{\rm max})$ depends on the energy distribution of nonthermal electrons. For a power law distribution, $dn_e/d\gamma\propto \gamma^{-q}$ with $q\le2$, we have $1>\chi>\>0.1$. Under the assumptions that $u_s(r)\propto r^{-2}$ and that the proper density and average energy of the nonthermal electrons are independent of radius the solution of eq. (\[eq-mot\]) (in the limit $\beta=1$) reads: $$\Gamma_\infty=\Gamma_0 \frac{l}{l+r_d},$$ where $\Gamma_\infty$ is the asymptotic Lorentz factor downstream. The stopping length can be expressed in terms of the optical depth for $\gamma\gamma$ absorption of a $\gamma$-ray of energy $m_ec^2\epsilon_{\gamma}$ by a power law target photon field of the form $I_s(\epsilon_s)\propto \epsilon_s^{-\alpha}$; $\epsilon_{s,min}<\epsilon_s<\epsilon_{s,max}$, as $$\frac{l}{r_d}=\frac{1}{\chi\xi_e\tau_{\gamma\gamma}}\left(\frac{\sigma_{\gamma\gamma}}{\sigma_T}\right)
\left(\frac{\epsilon_\gamma}{\Gamma_0\gamma_{\rm max}}\right)g(\epsilon_\gamma),
\label{stopp2}$$ with $g(\epsilon_{\gamma})=(\epsilon_\gamma\epsilon_{s,min})^{\alpha-1}$ if $\alpha>1$ and $g(\epsilon_{\gamma})=(\epsilon_\gamma\epsilon_{s,max})^{\alpha-1}$ if $\alpha<1$, and $g(\epsilon_\gamma)\le1$ in both cases. We conclude that for a reasonably flat distribution of nonthermal electrons, $q\le2$, extension of the distribution to a maximum energy $\gamma_{\rm max}$ at which the pair production optical depth, $\tau(\Gamma_0\gamma_{\rm max})$, is a few is already sufficient to cause appreciable deceleration of the front.
From the above it can be shown [@lev07] that for the TeV blazars a background luminosity of $L_s\sim 10^{41}-10^{42}$ erg s$^{-1}$, roughly the luminosity of LLAGN, would lead to a substantial deceleration of the front and still be transparent enough to allow the TeV $\gamma$-rays produced by Compton scattering of the background photons to escape the system. The ambient radiation field is most likely associated with the nuclear continuum source. The bulk Lorentz factor of the jet during states of low activity may be appreciably smaller than that of fronts expelled during violent ejection episodes.
What is the prospect for detection of VHE neutrinos from relativistic jets?
===========================================================================
A new generation of experiments just started operating or will become operative soon, design to detect VHE neutrinos (IceCube, ANTARES, NESTOR, NEMO), and UHE cosmic rays (HiRes and the hybrid Auger detectors), are probing and will probe regions opaque to electromagnetic radiation, which are presently unaccessible, and will determine the composition of jets. Besides providing an important probe of the innermost regions of compact astrophysical systems, these experiments can also be exploited to test new physics.
As mentioned above, the detection of UHECRs motivated considerations of heavy jets that effectively accelerate protons to energies approaching the confinement limit. Effective neutrino production requires large photopion opacity which, as discussed above, can be constrained by VHE $\gamma$-ray observations. Present observations rules out TeV blazars as potential candidates for the upcoming km$^3$ neutrino telescopes (see Ref [@lev06] for further discussion). This leaves GRBs, microquasars, and perhaps powerful EGRET blazars as potential candidates. Powerful blazars at a redshift of $z=1$ may produce up to one event per year in a km$^3$ detector [@Atoyan01]. In the case of microquasars up to a few events can be detected during a strong outburst if the viewing angle is sufficiently small [@levw01; @dest02; @torr07]. The estimated neutrino flux from GRBs implies that only nearby sources can be individually detected by the upcoming experiments. However, the cumulative flux produced by the entire GRB population should be detectable assuming that cosmological GEBs are the sources of the observed UHECRs [@wax95].
This work was supported by an ISF grant for the Israeli Center for High Energy Astrophysics.
[9]{} R. Wagner, in *Proc. 30th ICRC*, in press (2007; arXiv:0706.4442) F. Aharonian, et al. *Science*, **314**, 1424-1427 (2006) A. Levinson, *Int. J. Mod. Phys. A*, **21**, 6015-6054 (2006) R. D. Blandford, R. D. & A. Levinson, A., *Astrophys. J.*, **441**, 79-95 (1995) S. Jorstad S., et al. *Astrophys. J. Supp.*, **134**, 181 (2001) C. C. Cheung, D. E. Hrris & L. Stawarz *Astrophys. J.*, **663**, L65-L68 (2007) A. Neronov & F. Aharonian, *Astrophys. J.*, submitted (2007; arXiv:0704.3282) G. V. Bicknell, & M. C. Begelman, *Astrophys. J.*, **467**, 597-621 (1996) W. Peter, & D. Eichler, 1995, *Astrophys. J.*, **438**, 244-249 (1995) O. Bromberg, & A. Levinson, 2007, *Astrophys. J.*, in press (2007; arXiv0705.2040) C. M. Urry & P. Padovani, *Astrophys. J.*, **371**, 60 (1991) M. J. Hardcastle, D. M. Worrel, M. Birkinshaw & C. M. Canosa C. M. *Mon. Not. Roy. Astron. Soc.*, **338**, 176 (2003) A. Marscher, *Astropart. Phys.*, **11**, 19-25 (1999) M. Giroletti et al. *Astrophys. J.*, **600**, 127 (2004) G. Ghisellini, F. Tavecchio & M. Chiaberg, *Astron. & Astrophys.*, **432**, 401-410 (2005) Gopal-Krishna, S. Dhurde & P. J. Wiita P. J. *Astrophys. J.*, **615**, L81-L84 (2004) M. Georganopoulos & D. Kazanaz, *Astrophys. J.*, **594**, L27-L30 (2003) B. G. Piner, & P. G. Edwards *Astrophys. J.*, **622**, 168 (2005) A. Levinson, *Astrophys. J.*, submitted (2007) M. M. Romanova, & R. V. E. Lovelace, *Astron. & Astrophys.*, **262**, 26 (1992) A. Levinson, *Astrophys. J.*, **507**, 145-155 (1998) A. Atoyan, and C. D. Dermer [*Phys. Rev. Lett.*]{}, [**87**]{}, 221102 (2001) A. Levinson, & E. Waxman, E., *Phys. Rev. Lett.*, **87**, 171101-171104 (2001) C. Distefano, D. Guetta, E. Waxman, and A. Levinson *Astrophys. J.*, **575**, 378-383 (2002) D. Torres, & F. Halzen, F., *Astropart. Phys.*, **27**, 500 (2007) E. Waxman, *Phys. Rev. Lett.*, **75**, 386 (1995)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The presence of superconducting and superfluid components in the core of mature neutron stars calls for the rethinking of a number of key magnetohydrodynamical notions like resistivity, the induction equation, magnetic energy and flux-freezing. Using a multi-fluid magnetohydrodynamics formalism, we investigate how the magnetic field evolution is modified when neutron star matter is composed of superfluid neutrons, type-II superconducting protons and relativistic electrons. As an application of this framework, we derive an induction equation where the resistive coupling originates from the mutual friction between the electrons and the vortex/fluxtube arrays of the neutron and proton condensates. The resulting induction equation allows the identification of two timescales that are significantly different from those of standard magnetohydrodynamics. The astrophysical implications of these results are briefly discussed.'
author:
- |
Vanessa Graber,$^1$[^1] Nils Andersson,$^1$ Kostas Glampedakis$^{2,3}$ and Samuel K. Lander$^1$\
$^1$Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom\
$^2$Departamento de Fisica, Universidad de Murcia, 30100 Murcia, Spain\
$^3$Theoretical Astrophysics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
bibliography:
- '../../Bibliography/library.bib'
title: |
Magnetic Field Evolution in\
Superconducting Neutron Stars
---
Magnetohydrodynamics (MHD) – stars: magnetic fields – stars: neutron.
Introduction {#sec-Intro}
============
Existing at the extremes of physics, neutron stars serve as excellent cosmic laboratories. Some of the most striking features are related to their magnetic properties. Measured magnetic field strengths, generally inferred from the star’s dipole spin-down, by far exceed the strengths of terrestrial magnets. Observations also suggest a link between the various classes of neutron stars. Old *millisecond pulsars* thought to be formed in low-mass X-ray binaries have fields between $10^{8}-10^{10} \, {{\rm G}}$, while the fields of ‘classical’ *rotation-powered pulsars* range between $10^{10}-10^{13} \, {{\rm G}}$. A third class of slow-rotating, highly magnetised neutron stars, so-called *magnetars*, reaches field strengths up to $10^{15} \, {{\rm G}}$. This class is believed to include both, soft gamma repeaters and anomalous X-ray pulsars. Understanding the long-term evolution of the stars’ magnetic fields might be key to establishing connections between the different classes and forming a unified picture of the neutron star ‘zoo’ [@Kaspi2010; @Vigano2013; @Harding2013].
Unsurprisingly, such enormous strengths suggest that magnetic fields are crucial for the neutron stars’ dynamics. As first pointed out by Thompson & Duncan ([-@Thompson1995; -@Duncan1996]), magnetic field decay on a timescale of $\sim 10^4 \, {{\rm yr}}$ could power the high activity of magnetars. The rotational energy is not sufficient to explain the observed emission of these objects. There are also observations indicating that the magnetic dipole fields of standard pulsars evolve on a timescale of the order $10^7
\, {{\rm yr}}$ [@Lyne1985; @Narayan1990]. The actual mechanisms, causing the magnetic field to change on these rather short timescales, are only poorly understood and there is no definitive answer to the question of which part of the neutron star dominates the magnetic field evolution. Most theoretical studies and numerical simulations focus on the crust as the source of the field decay and neglect the core contribution [@Pons2007; @Vigano2013; @Gourgouliatos2014]. However, one could argue that the core, which carries the majority of the star’s inertia and magnetic energy, should also play a role in the magnetic field evolution.
The problem of magnetic field evolution in isolated neutron stars has been discussed by a number of authors. @Goldreich1992 determined several mechanisms that are present in an ionised plasma consisting of neutrons, protons and electrons. *Ohmic diffusion* due to the interaction of relativistic electrons and lattice nuclei causes magnetic field dissipation in the crust. This mechanism is most effective on small scales and, thus, not expected to affect the large scale evolution of the crustal field. However, Ohmic decay could be enhanced by *Hall drift*, which is in itself conservative but may redistribute magnetic energy from large to gradually smaller lengthscales. The combined effect, sometimes referred to as the Hall cascade, could cause field evolution on a timescale of order $10^7$ years [@Goldreich1992]. However, recent neutron star crust simulations show no strong cascading behaviour but suggest the existence of a quasi-equilibrium established on timescales shorter than the Ohmic timescale [@Pons2010; @Gourgouliatos2014; @Wood2015]. In the core, standard Ohmic decay is negligible because the interior is expected to form a type-II superconductor (@Baym1969 [-@Baym1969], see also below). Electron-proton scattering, already acting on very long timescales in the star’s interior [@Baym1969b], is then restricted to the normal conducting cores of fluxtubes. These only contribute a tiny fraction to the star’s total cross section. Hence, the coupling timescale is increased further, making this dissipation mechanism irrelevant. Additionally, *ambipolar diffusion*, describing the motion of charged particles and magnetic field lines relative to the neutrons, could cause magnetic field decay and drive the flux from the core to the crust [@Goldreich1992]. This mechanism was originally considered by @Thompson1995 to explain the magnetar activity. However, more recent results seem to indicate that the timescale for ambipolar diffusion considerably increases when the superfluid nature of the neutrons or proton superconductivity is taken into account [@Glampedakis2011c].
Lacking a clear answer, the question of magnetic field evolution in a neutron star is revisited in this paper. We focus on the outer core and use the formalism developed by @Glampedakis2011a (see also @Mendell1991 [-@Mendell1991], @Mendell1991a [-@Mendell1991a]), which presents a general set of macroscopic hydrodynamic equations for a multi-fluid mixture. We address, in particular, the effect the superconducting component has on the evolution of the magnetic field.
The presence of superfluid and superconducting components in neutron stars is firmly supported by observations and microphysical calculations. Traditionally, glitches and post-glitch relaxation timescales on the order of months to years are seen as observational evidence of superfluidity. @Anderson1975 first proposed that the neutron star’s dynamical evolution during and after a glitch could be explained by the weak viscous properties of a superfluid component that is coupled to the crust. Moreover, recent spectral analyses of the neutron star in the supernova remnant Cassiopeia A indicate that the surface temperature of this young object decreases faster than one would expect from standard cooling models [@Page2011; @Shternin2011]. The rapid cooling could be explained by enhanced neutrino emission, resulting from the onset of neutron superfluidity and proton superconductivity in the core. In addition to observations, theoretical calculations provide strong reasons for macroscopic quantum condensates in neutron star cores. The idea of superfluid interiors was first put forward by @Migdal1959, several years before the first detection of a pulsar [@Hewish1968]. A few hundred years after birth, neutron stars are in thermal equilibrium and have temperatures of $10^6 - 10^8 \, {{\rm K}}$ [@Tsuruta1998; @Page2006; @Ho2012b]. While this is certainly hot in terms of terrestrial physics, the temperatures lie well below the Fermi temperature of nuclear matter, which is of the order $10^{12} \, {{\rm K}}$ [@Sauls1989]. Applying the microscopic theory of laboratory superconductors [@Bardeen1957] to the neutron star interior suggests that the neutrons and protons form Cooper pairs and, thus, condense into a superfluid and a superconductor, respectively.
The properties of the proton superconductor were further discussed in the seminal paper by @Baym1969. As the conductivity of normal conducting matter is very large [@Baym1969b], the authors argued that flux could not be expelled from the star’s interior. The phase transition into a superconducting state has to occur at constant magnetic flux. Two characteristic lengthscales determine the state of the superconducting protons. The *London penetration depth* describes the magnetic field’s exponential fall-off from the surface of a superconductor or the fluxtube cores due to the Meissner effect. The *coherence length*, which characterises the typical dimensions of a Cooper pair, is equivalent to the core radius of a fluxtube. The ratio of these two quantities, the Ginzburg-Landau parameter, $\kappa$, dictates the type of superconductivity. For $\kappa > 1/ \sqrt{2}$, it is energetically favourable for the material to increase the surface area between normal and superconducting regions, implying that magnetic flux can penetrate the medium in the form of quantised fluxlines [@Abrikosov1957]. @Baym1969 estimated $\kappa$ for the neutron star’s outer core and found that the protons would enter a metastable type-II state by forming a regular array of fluxtubes (see also Figure 1 of @Glampedakis2011a [-@Glampedakis2011a]).
The presence of fluxtubes significantly influences the magnetic properties of the star because the flux is no longer locked to the charged plasma but is mainly confined inside the fluxtubes. As standard coupling mechanisms, like Ohmic dissipation, are suppressed as a result of pairing, interactions of fluxlines with their surroundings determine the magnetic field evolution on macroscopic lengthscales. The most prominent of these effective coupling processes, known as mutual friction, is the scattering of electrons off the fluxtube magnetic field [@Alpar1984; @Mendell1991a; @Andersson2006b]. In the following, we derive an equation for the magnetic field evolution of a superfluid/superconducting mixture using a smooth-averaged formalism. We translate the mesoscopic phenomena, influencing individual fluxtubes, to the large scale picture by applying the framework known from standard resistive magnetohydrodynamics (MHD).
The structure of this paper is as follows. In Section \[sec-Background\], we introduce the hydrodynamical equations for a multi-fluid system and the corresponding Maxwell equations. In Section \[sec-StandardMHD\], the standard resistive MHD formalism is reviewed. We then discuss the case of superconducting matter in Section \[sec-SuperconMHD\], focussing on the standard resistive coupling introduced by @Alpar1984. After giving the most general form of the superconducting induction equation, we use a few assumptions to simplify and interpret our results. We finally conclude with a discussion in Section \[sec-Discussion\].
Note that we will work in an inertial frame and carry out the analysis in a coordinate basis. Vectors are, thus, denoted by their individual components. We also use Gaussian units in the remainder of this paper.
Theoretical Background {#sec-Background}
======================
Multi-fluid hydrodynamics {#subsec-MultifluidHydro}
-------------------------
The model presented in this section largely builds upon the recent work by @Glampedakis2011a. Making use of the Lagrangian formalism developed by Carter, Prix and collaborators [@Carter1995a; @Prix2004; @Andersson2006], a full set of MHD equations for the superfluid/superconducting bulk in the outer neutron star core is derived. The simplest representation of this Fermi liquid is a mixture of three components, namely relativistic electrons, superconducting protons and superfluid neutrons. In the following, the constituents are denoted by roman indices, ${{\rm x}}=\{{{\rm e}}, {{\rm p}},
{{\rm n}}\}$. Note that in order to keep the discussion clear, we neglect the presence of muons. However, generalising to the four-constituent case would be straightforward as electrons and muons are strongly coupled and move as one component on macroscopic lengthscales [@Mendell1991a].
The dynamics of the three-fluid system is governed by two Euler equations, one representing the superfluid neutrons and the other representing the electron-proton conglomerate. The charged fluids can be characterised as a single component if charge neutrality holds over macroscopic distances, i.e. $n_{{\rm e}}=n_{{\rm p}}$, where $n_{{\rm x}}$ denotes the particle number density. As the electrons are mobile enough to quickly equilibrate any local charge imbalances, this requirement is fulfilled in the neutron star core [@Jackson1999; @Glampedakis2011a]. Additionally, the electron mass is significantly smaller than the proton mass, $m_{{\rm e}}\ll m_{{\rm p}}$, which allows us to neglect any electron inertial terms. The resulting macroscopic Euler equations are $$\begin{aligned}
\left[\partial_t + v^j_{{\rm n}}\nabla_j \right] & ( v^i_{{\rm n}}+ \varepsilon_{{\rm n}}w^i_{\rm pn} ) -
{\varepsilon_{{\rm n}}}w_{\rm pn}^j \nabla^i v_j^{{\rm n}}\nonumber \\[1.3ex]
&= - \nabla^i \left( \tilde{\mu}_{{\rm n}}+ \Phi \right ) + f^i_{\rm mf} + f^i_{\rm mag,n},
\label{eulern_final} \\[2ex]
\left[ \partial_t + v^j_{{\rm p}}\nabla_j \right] & ( v^i_{{\rm p}}+ \varepsilon_{{\rm p}}w^i_{\rm np} )
+ {\varepsilon_{{\rm p}}}w_{\rm np}^j \nabla^i v_j^{{\rm p}}\nonumber \\[1.3ex]
&= - \nabla^i \left ( \tilde{\mu} + \Phi \right ) - \frac{\rho_{{\rm n}}}{\rho_{{\rm p}}} f^i_{\rm mf} + f^i_{\rm mag,p}.
\label{eulerp_final}\end{aligned}$$ The averaged fluid velocities are denoted by $v^i_{{\rm x}}$ and the mass densities by $\rho_{{\rm x}}= m n_{{\rm x}}$, where $m \equiv m_{{\rm n}}=m_{{\rm p}}$ is the baryon mass. $w_\mathrm{xy}^i \equiv v_{{\rm x}}^i-v_{{\rm y}}^i$ is the relative velocity, $\Phi$ the gravitational potential and $\varepsilon_{{\rm x}}$ the entrainment parameters. By definition, the latter satisfy the condition $n_{{\rm p}}\varepsilon_{{\rm p}}= n_{{\rm n}}\varepsilon_{{\rm n}}$. The specific chemical potentials are defined as $$\tilde{\mu}_{{\rm n}}\equiv \frac{\mu_{{\rm n}}}{m}, \qquad \tilde{\mu} \equiv \frac{\mu_{{\rm p}}+ \mu_{{\rm e}}}{m}.$$ The two Euler equations are supplemented by three continuity equations for the number densities, $$\partial_t n_{{\rm x}}+ \nabla_i \left( n_{{\rm x}}v_{{\rm x}}^i \right) = 0,
\label{eqn-Continuity}$$ which reflect the conservation of mass for each individual species, and the Poisson equation $$\nabla^2 \Phi = 4 \pi G \rho.$$ $\rho=\sum_{{\rm x}}\rho_{{\rm x}}$ is the total mass density of the system and $G$ the gravitational constant.
The variational approach used to derive the Euler and continuity equations explicitly distinguishes between the fluid momenta and velocities. This formalism provides the possibility to include any changes, caused by the superfluid and superconducting condensates, into the hydrodynamical model. In contrast to the momentum equations of standard plasma physics [@Jackson1999], Equations and incorporate new inertial terms due to entrainment, which arise from the strong coupling of Fermi liquids, and terms that go beyond the standard electromagnetic interaction given by the Lorentz force. For the fluid mixture in the outer neutron star core, the right-hand sides of the momentum equations contain the total magnetic and mutual friction forces per unit volume, $f^i_{\rm mag,x}$ and $f^i_{\rm mf}$, respectively. The former one is caused by interactions of the vortex/fluxtube magnetic field with the charged fluid. We point out that the neutron fluid experiences this magnetic force because protons are entrained around each neutron vortex and create an effective magnetic field. In the absence of entrainment, the magnetic force on the neutron component would vanish. Finally, the mutual friction forces arise from the dissipative coupling of the vortex and fluxtube array with the fluid components.
We keep in mind that our hydrodynamic model is solely based on averaged quantities and reflects the macroscopic behaviour of the fluid components. It is on these large scales that we have a method to deal with the presence of the quantum condensates in a consistent way. Taking advantage of the large numbers of vortices/fluxtubes, we over the respective arrays and obtain a smooth-averaged description of the magnetic and mutual friction forces. Using this formalism, it is possible to determine how the presence of vortices and fluxtubes influences the macroscopic dynamics of a neutron star, i.e. its rotational and magnetic evolution. If individual vortices/fluxtubes do not overlap and are distant enough so that interactions within one array can be neglected, the averaging procedure is obtained from the macroscopic quantisation conditions originally developed by @Onsager1949 and @Feynman1955 for the dynamics of rotating superfluid helium. Assuming that neutron vortices and proton fluxtubes are locally straight and directed along the unit vectors, $\hat{\kappa}^i_{{\rm n}}$ and $\hat{\kappa}^i_{{\rm p}}$, the arrays can be assigned vortex and fluxtube surface densities, ${{\cal N}}_{{\rm n}}$ and ${{\cal N}}_{{\rm p}}$, respectively. As the vorticities, $\mathcal{W}^i_{{\rm x}}$, are related to the circulation of the averaged canonical momenta, the macroscopic quantisation conditions are given by $$\begin{aligned}
\mathcal{W}_{{\rm n}}^i &= \epsilon^{ijk} \nabla_j \left( v_k^{{\rm n}}+ \varepsilon_{{\rm n}}w_k^{\rm pn} \right) = {{\cal N}}_{{\rm n}}\kappa^i_{{\rm n}},
\label{eqn-Quantisation1} \\[1.4ex]
\mathcal{W}_{{\rm p}}^i & = \epsilon^{ijk} \nabla_j \left( v_k^{{\rm p}}+ \varepsilon_{{\rm p}}w_k^{\rm np} \right) + a_{{\rm p}}B^i = {{\cal N}}_{{\rm p}}\kappa^i_{{\rm p}},
\label{eqn-Quantisation2}\end{aligned}$$ where $\kappa^i_{{\rm x}}= \kappa \hat{\kappa}^i_{{\rm x}}$ points along the local vortex direction with the quantum of circulation $$\kappa = \frac{h}{2 m} \approx 2.0 \times 10^{-3} \, {{\rm cm}}^2 \, {{\rm s}}^{-1},$$ and we define $$a_{{\rm p}}\equiv \frac{e}{m c} \approx 9.6 \times 10^3 \, {{\rm G}}^{-1} \, {{\rm s}}^{-1}.$$ The proton charge, the speed of light and the Planck constant are denoted by $e$, $c$ and $h$, respectively.
Macroscopic magnetic induction {#subsec-Induction}
------------------------------
In the averaged framework, the total magnetic induction, $B^i$, is the sum of three individual components, namely the averaged fluxtube and vortex field and the London field, $$B^i = B_{{\rm p}}^i + B_{{\rm n}}^i + b_{\rm L}^i .
\label{eqn-AveragedField}$$ The former two contributions are obtained by multiplying the surface densities, ${{\cal N}}_{{\rm x}}$, with the flux carried by a single line of the lattice, $\phi_{{\rm x}}$. The fluxes can be derived by considering the dynamics of a single vortex/fluxtube on mesoscopic lengthscales, denoted by bars on the respective quantities. Using the corresponding quantisation condition and the mesoscopic Ampère law (e.g. see Appendix A1 of @Glampedakis2011a for details), it is possible to derive generalised London equations for the mesoscopic magnetic fields, $\bar{B}_{{\rm x}}^i$, $$\lambda^2_* \nabla^2 \bar{B}_{{\rm x}}^i - \bar{B}_{{\rm x}}^i = - \phi_{{\rm x}}\hat{\kappa}_{{\rm x}}^i \delta(\vec{r}\, ),
\label{eqn-LondonModified}$$ where $\delta(\vec{r}\, )$ is the two-dimensional delta function located at the centre of each vortex/fluxtube, $\phi_{{\rm x}}$ is defined below and the effective London penetration depth is given by $$\lambda_* = \left(\frac{1}{4 \pi \rho_{{\rm p}}a_{{\rm p}}^2} \, \frac{1 -{\varepsilon_{{\rm n}}}-{\varepsilon_{{\rm p}}}}{1-{\varepsilon_{{\rm n}}}} \right)^{1/2}.
\label{eqn-LondonDepth}$$ In the absence of entrainment, ${\varepsilon_{{\rm n}}}= {\varepsilon_{{\rm p}}}=0$, this expression reduces to the standard result of superconductivity [@Tinkham2004]. Taking advantage of the symmetry and using cylindrical coordinates, the inhomogeneous Helmholtz equation can be solved using a Green’s function approach in two dimensions [@Fetter1969]. Integrating the resulting magnetic induction over a disc of radius $r \gg \lambda_*$ perpendicular to $\hat{\kappa}_{{\rm x}}^i$ gives for the magnetic flux $$\int \bar{B}_{{\rm x}}^i \, {{\rm d}}S = \phi_{{\rm x}}\hat{\kappa}_{{\rm x}}^i.$$ For a proton fluxtube, we obtain the expected unit of flux, $$\phi_{{\rm p}}= \phi_0 = \frac{\kappa}{a_{{\rm p}}} = \frac{hc}{2e} \approx 2.1 \times 10^{-7} \,{{\rm G}}\, {{\rm cm}}^2,$$ whereas the flux of a superfluid vortex is $$\phi_{{\rm n}}= - \frac{{\varepsilon_{{\rm p}}}}{1-{\varepsilon_{{\rm n}}}} \, \phi_0.$$ We note that the minus sign originates from $\hat{\kappa}_{{\rm n}}^i$ and $\bar{B}_{{\rm n}}^i$ pointing into opposite directions. In the absence of entrainment, the neutron flux would be zero. The averaged contributions from the two arrays to the macroscopic magnetic induction, $B^i$, are then obtained by $$B_{{\rm x}}^i = {{\cal N}}_{{\rm x}}\phi_{{\rm x}}\hat{\kappa}_{{\rm x}}^i.
\label{eqn-BAveraged}$$
The third contribution to the induction is the London field. It is a fundamental property of a superconductor and associated with its rotation [@Tilley1990]. While superfluids need to form vortices in order to support any circulation, the superconducting fluxtube array is not related to the macroscopic rotation. However, these dynamics induce an additional magnetic field inside the superconductor, whose axis is parallel to the rotation axis. In contrast to $B_{{\rm x}}^i$, the London field is not of microscopic origin but related to the macroscopic electromagnetic current (see Subsection \[subsec-Maxwell\]). Combining the quantisation conditions and with Equation , the London field can be related to the macroscopic fluid properties. Assuming that the hydrodynamical lengthscales are sufficiently small to ensure constant entrainment parameters, we have $$b_{{\rm L}}^i = - \frac{1}{a_{{\rm p}}} \, \frac{1 - {\varepsilon_{{\rm n}}}- {\varepsilon_{{\rm p}}}}{1-\varepsilon_{{\rm n}}} \,
\epsilon^{ijk} \nabla_j v^{\rm p}_k.
\label{eqn-LondonField2}$$ The proton entrainment parameter is related to the effective proton mass via $${\varepsilon_{{\rm p}}}= 1 - \frac{m_{{\rm p}}^*}{m} \approx 0.3,
\label{eqn-envalue}$$ where we have used the estimate $m_{{\rm p}}^* \approx (0.6-0.9)m$ given by @Chamel2006 for the outer neutron star core. For small proton fractions, $x_{{\rm p}}\equiv \rho_{{\rm p}}/\rho \ll 1$, an approximation which is valid in the interior of a neutron star, the neutron entrainment coefficient is negligible because $\varepsilon_{{\rm n}}\approx x_{{\rm p}}\varepsilon_{{\rm p}}\ll 1$. Taking the proton fluid to be tightly coupled to the neutron star’s crust through the magnetic field and, thus, rotating rigidly at the observable pulsar frequency, we can substitute a canonical rotation period to calculate an estimate for the magnitude of the London field. Using the normalisation $P_{10} \equiv P/(10 \, {{\rm ms}})$, we find $$b_{\rm L} \approx 9.2 \times 10^{-2} P^{-1}_{10} \, {{\rm G}},$$ which is many orders of magnitude smaller than the magnetic field strengths usually invoked for neutron star physics. Hence, it is generally justified to neglect the London field in Equation , an approach we will take in Subsection \[subsec-SimplifiedSet\].
We can simplify the magnetic induction, $B^i$, one step further by taking the properties of the vortex and fluxtube arrays into account. Although the individual fluxes, $\phi_{{\rm x}}$, are comparable, the contribution from the superconducting protons dominates: Consider smooth-averaged fluid velocities of the form $\epsilon^{ijk} \nabla_j v_k^{{\rm x}}= 2 \Omega_{{\rm x}}\hat{z}^i$. In this case, the vortices/fluxtubes are aligned with the $z$-axis, i.e. $\hat{z}^i = \hat{\kappa}^i_{{\rm x}}$, and Equation gives $${{\cal N}}_{{\rm n}}= \frac{2}{\kappa} \left[ \Omega_{{\rm n}}+ \epsilon_{{\rm n}}\Omega_{\rm pn} \right].$$ As we expect the lag, $\Omega_{\rm pn}=\Omega_{{\rm p}}- \Omega_{{\rm n}}$, to be small on macroscopic scales and the neutrons to be coupled to the crust, the vortex surface density is $${{\cal N}}_{{\rm n}}\approx \frac{4 \pi}{\kappa P} \approx 6.3 \times 10^5 P^{-1}_{10} \, {{\rm cm}}^{-2},$$ The neutron vortex density is, thus, fixed by the rotation of the neutron star. Using previous estimates for the entrainment parameters, the magnetic field strength of the neutron vortex array is $$B_{{\rm n}}\approx {{\cal N}}_{{\rm n}}{\varepsilon_{{\rm p}}}\phi_0 \approx 4.0 \times 10^{-2} P^{-1}_{10} \, {{\rm G}}.$$ This is again many orders of magnitude smaller than typical neutron star field strengths, which implies that the magnetic field of the outer neutron star core is mainly confined to the proton fluxtube cores. It is, therefore, important to investigate which mechanisms affect the motion of individual fluxtubes in order to link the small scale behaviour to the large scale evolution of the star’s magnetic field. The fluxtube density can be estimated to $${{\cal N}}_{{\rm p}}= \frac{B_{{\rm p}}}{\phi_0} \approx \frac{B}{\phi_0}
\approx 4.8 \times 10^{18} \, B_{12} \, {{\rm cm}}^{-2}
\label{eqn-FluxtubeDensity},$$ with the normalised magnetic field $B_{12} \equiv B/(10^{12} \, {{\rm G}})$.
Macroscopic Maxwell equations {#subsec-Maxwell}
-----------------------------
In order to capture the electromagnetic response of the fluid mixture correctly, the Euler equations and have to be supplemented by Maxwell’s equations. Taking these to be valid in our multi-fluid mixture, we have to redefine, or rather reinterpret, the various fields accordingly in order to make Maxwell’s equations suitable for a type-II superconductor.
As mentioned before, the London field, despite being of small magnitude, plays an important role for the electrodynamics. As discussed by @Carter2000 and @Glampedakis2011a, the London field is closely connected to the macroscopic electromagnetic current, $$J^i = e n_{{\rm e}}w_{\rm pe}^i,
\label{eqn-Current}$$ which enters the macroscopic Ampère law. In contrast to standard MHD, where the equality $H^i=B^i$ is satisfied, the averaged magnetic induction, $B^i$, and the macroscopic magnetic field, $H^i$, are no longer equivalent in a type-II superconducting sample. Instead, Ampère’s law reads $$\epsilon^{ijk} \nabla_j H_k = \epsilon^{ijk} \nabla_j b^{{\rm L}}_k = \frac{4 \pi}{c} J^i,
\label{eqn-AmpereLaw}$$ where the displacement current has been neglected because the fluid motion is slow compared to the speed of light. This deviation from standard MHD can be also understood in terms of the classification generally applied to terrestrial superconductors (see for example @Tinkham2004 [-@Tinkham2004]). In laboratory experiments, one distinguishes between macroscopic electromagnetic currents that generate a macroscopic field, $H^i$, and magnetisation currents *only* affecting the mesoscopic induction, which is $\bar{B}^i$ in our notation. A supercurrent, circulating around each vortex/fluxtube and generating $\bar{B}^i_{{\rm x}}$, is attributed to the second class. It does not contribute to the field $H^i=b^i_{{\rm L}}$, which is created by the current $J^i$. Hence, the macroscopic magnetic induction, $B^i$, given in Equation differs from the magnetic field, $H^i$. For comparison, in vacuum or normal conductors, no magnetisation currents are present and the identification $H^i=B^i=\bar{B}^i$ can be made. In the present case, $$\epsilon^{ijk} \nabla_j H_k = \epsilon^{ijk} \nabla_j B_k = \frac{4 \pi}{c} J^i.
\label{eqn-AmpereLawStandard}$$ In addition to Ampère’s law, we use $$\nabla_i B^i =0
\label{eqn-MaxwellB},$$ which has to hold everywhere in the superconducting fluid, and the macroscopic Faraday law, $$\partial_t B^i = - c \, \epsilon^{ijk} \nabla_j E_k.
\label{eqn-Faraday}$$ Instead of defining the macroscopic electric field as the average over the microscopic equivalent, we take advantage of the remaining fluid degree of freedom, namely the electron Euler equation, to obtain an expression for $E^i$. Neglecting again the electron inertial terms, we have $$E^i = -\frac{1}{c} \epsilon^{ijk} v_j^{{\rm e}}B_k
- \frac{m_{{\rm e}}}{e} \nabla^i \left ( \tilde{\mu}_{{\rm e}}+ \Phi \right ) - \frac{F^i_{{\rm e}}}{c a_{{\rm p}}\rho_{{\rm p}}},
\label{eqn-ElectricField}$$ where $F^i_{{\rm e}}$ represents the total force exerted on the electrons due to interactions with the surrounding fluid components.
Combining Equations and leads to an evolution equation for the magnetic induction that only depends on macroscopic fluid variables. However, the procedure on the forces, $F^i_{{\rm e}}$, and we address this in the following sections.
Magnetic field evolution in standard MHD {#sec-StandardMHD}
========================================
Before discussing the more complicated problem of magnetic field evolution in a superfluid/superconducting mixture, we briefly review the approach taken in normal resistive matter, which allows us to compare our new results with a well studied model.
The MHD induction equation {#subsec-MHDInduction}
--------------------------
In a charged plasma containing electrons and protons, the only relative flow present is the motion between the two components. Assuming that a frictional mechanism would damp these dynamics and try to bring the two species into co-motion is straightforward. Hence, resistive coupling acting on a timescale $\tau_{{\rm e}}$ leads to a dissipative force on the electron fluid, $$F_{{\rm e}}^i = \frac{n_{{\rm e}}m_{{\rm e}}}{\tau_{{\rm e}}} w_{\rm pe}^i = - \frac{m_{{\rm e}}}{e \tau_{{\rm e}}} J^i.
\label{eqn-ForceMHD}$$ Substituting this force into Equation gives a generalised Ohm’s law $$E^i = - \frac{1}{c} \, \epsilon^{ijk} \left( v_j^{{\rm p}}- \frac{J_j}{e n_{{\rm e}}} \right) B_k
- \frac{m_{{\rm e}}}{e} \nabla^i \left ( \tilde{\mu}_{{\rm e}}+ \Phi \right ) + \frac{J^i}{\sigma_{{\rm e}}}
\label{eqn-OhmsLaw}$$ with the standard electrical conductivity defined by $$\sigma_{{\rm e}}\equiv \frac{c \rho_{{\rm p}}a_{{\rm p}}e \tau_{{\rm e}}}{m_{{\rm e}}} = \frac{n_{{\rm e}}e^2 \tau_{{\rm e}}}{m_{{\rm e}}}.$$ Equation combined with Faraday’s law leads to an evolution equation for the magnetic induction, $$\begin{aligned}
\partial_t B^i &= \epsilon^{ijk} \nabla_j \epsilon_{klm} \left( v_{{\rm p}}^l B^m \right)
- \epsilon^{ijk} \nabla_j \epsilon_{klm} \left( \frac{c^2}{4 \pi \sigma_{{\rm e}}} \nabla^l B^m \right) \nonumber \\[1.6ex]
&- \epsilon^{ijk} \nabla_j \epsilon_{klm} \left[ \frac{m c}{4 \pi e \rho_{{\rm p}}} \, \epsilon^{lsp} \left(
\nabla_s B_p \right) B^m \right].
\label{eqn-NormalMHDInduction}\end{aligned}$$ We have used Ampère’s law for normal matter to eliminate the macroscopic current in the last expression. The second and the third term on the right-hand side represent the Ohmic decay and the Hall evolution. We can extract the following well-known timescales, $$\tau_{\rm Ohm} = \frac{4 \pi \sigma_{{\rm e}}L^2}{c^2}$$ and $$\tau_{\rm Hall} = \frac{4 \pi e \rho_{{\rm p}}L^2}{m c B},$$ where $L$ is the characteristic lengthscale over which the magnetic field changes.
Flux-freezing in MHD {#subsec-MHDFreezing}
--------------------
We can estimate the two characteristic timescales for a neutron star core. According to @Baym1969b, the electrical conductivity associated with the interaction of highly relativistic electrons and normal, degenerate protons is given by $$\sigma_{{\rm e}}\approx 5.5 \times 10^{28} \, T_8^{-2} \, \rho_{14}^{3/2} \left(\frac{x_{{\rm p}}}{0.05} \right)^{3/2} \, {{\rm s}}^{-1},$$ where $T_8 \equiv T/(10^8 \, {{\rm K}})$ is the star’s normalised temperature, $\rho_{14} \equiv \rho/( 10^{14} \, {{\rm g}}\, {{\rm cm}}^{-3})$ the normalised total density and $x_{{\rm p}}$ the proton fraction. Approximating the characteristic lengthscale by the radius of the neutron star, the Ohmic diffusion timescale is $$\tau_{\rm Ohm} \approx 2.4 \times 10^{13} \, T_8^{-2} \, L_6^2 \, \rho_{14}^{3/2} \left(\frac{x_{{\rm p}}}{0.05} \right)^{3/2} \, {{\rm yr}},
\label{eqn-Ohmicestimate}$$ with the normalised lengthscale $L_6 \equiv L/(10^6 \, {{\rm cm}})$. For the Hall timescale, we obtain $$\tau_{\rm Hall} \approx 1.9 \times 10^{10} \, B_{12}^{-1} \, L_6^2 \, \rho_{14} \left(\frac{x_{{\rm p}}}{0.05} \right) \, {{\rm yr}}.
\label{eqn-Hallestimate}$$ Both estimates are many orders of magnitude larger than the typical spin-down ages of radio pulsars. We would, therefore, expect Ohmic decay and Hall term to be negligible for the evolution of the plasma’s magnetic field. In this idealised case, which is commonly used to approximate astrophysical or laboratory plasmas, the induction equation reduces to $$\begin{aligned}
\partial_t B^i &= \epsilon^{ijk} \nabla_j \epsilon_{klm} \left( v_{{\rm p}}^l B^m \right).
\label{eqn-NormalMHDInductionIdeal}\end{aligned}$$ Using Equation , we can rewrite the last expression and simplify the result using the standard Lie derivative, $$\begin{aligned}
\partial_t B^i + \mathcal{L}_{v_{{\rm p}}} B^i = - B^i \nabla_j v_{{\rm p}}^j.
\label{eqn-Lie2}\end{aligned}$$ The left-hand side describes how the magnetic field vector, $B^i$, is transported with the fluid flow, $v_{{\rm p}}^i$. Taking into account that the mass of the proton plasma is conserved, we use Equation to further simplify, $$\begin{aligned}
\frac{\partial}{\partial t} \left( \frac{B^i}{\rho_{{\rm p}}} \right)
+ \mathcal{L}_{v_{{\rm p}}} \left( \frac{B^i}{\rho_{{\rm p}}} \right) = 0.
\label{eqn-Lie4}\end{aligned}$$ This implies that the magnetic field is moving with the fluid, i.e. the fluxlines are frozen into the proton plasma.
As soon as Ohmic and Hall terms play a role for the dynamics, this frozen-in condition is destroyed and field lines are no longer forced to follow the protons. In particular, if Ohmic decay characterised by the conductivity, $\sigma_{{\rm e}}$, is included, the induction equation resembles a diffusion equation. It encodes how the magnetic field lines diffuse through the fluid and reconnect, leading to the decay of magnetic energy as discussed in Section \[subsec-MHDEnergy\]. If the Hall term is present but Ohmic decay is negligible, the induction equation reduces to $$\begin{aligned}
\partial_t B^i &= \epsilon^{ijk} \nabla_j \epsilon_{klm} \left( v_{{\rm e}}^l B^m \right).
\label{eqn-NormalMHDInductionHall}\end{aligned}$$ In contrast to Equation , the electron velocity enters the magnetic evolution law. This implies that the relative motion between electrons and protons becomes important and the magnetic field is frozen into the electron fluid. The Hall term in Equation itself is not dissipative but may act to redistribute magnetic energy from large scales to smaller ones, where it can decay ohmically. Many studies of the induction equation’s non-linear behaviour are based on results from hydrodynamic turbulence [@Kolmogorov1941], as it has several similarities with the vorticity equation of a viscous fluid [@Goldreich1992]. However, recent numerical simulations in the context of neutron stars have shown no evidence of strong cascading behaviour. Instead the Hall cascade appears to be saturated at long lengthscales [@Pons2010; @Gourgouliatos2014; @Wood2015].
Magnetic energy in standard MHD {#subsec-MHDEnergy}
-------------------------------
The conservative and dissipative nature of the different pieces in Equation is illustrated by considering the evolution of the magnetic energy. In order to compare the standard MHD plasma with the superconducting mixture later on, we calculate the magnetic energy associated with the work done by the Lorentz force. In its standard form, the Lorentz force density, given by $$F_{{\rm L}}^i = \frac{1}{4 \pi} \left[ B_j \nabla^j B^i - \frac{1}{2} \nabla^i \left( B_k B^k \right) \right],
\label{eqn-LorentzForce}$$ is composed of a tension and a pressure term. The work is obtained by calculating the product with the position vector, $r^i$, and integrating over the volume, $V$. Using the product rule and Equation , we arrive at $$\begin{aligned}
W_{{\rm L}}&= \int r_i F_{{\rm L}}^i \, {{\rm d}}V = \frac{1}{4 \pi} \int \nabla^i \left( r_j B^j B_i -
\frac{1}{2} B^2 r_i \right) {{\rm d}}V \nonumber \\[1.2ex]
&- \frac{1}{4 \pi} \int \left( B^i B_j \nabla^j r_i - \frac{1}{2} B^2 \nabla^i r_i \right) {{\rm d}}V.
\label{eqn-Energy}\end{aligned}$$ The total gradient term can be rewritten as a surface integral using Gauss’ theorem. As no discontinuities are present at the boundary of the plasma region, we can push the integration radius to infinity. Provided that the magnetic induction vanishes at infinity, the surface contribution is zero. The integrand of the second term in Equation simplifies to the well-known magnetic energy density $$\begin{aligned}
W_{{\rm L}}&= \int \frac{B^2} {8 \pi} \, {{\rm d}}V = \int {{\cal E}}_{\rm mag} \, {{\rm d}}V.\end{aligned}$$ Changes in the magnetic energy are, thus, determined by $$\frac{\partial {{\cal E}}_{\rm mag}}{\partial t} = \frac{\partial}{\partial t} \left( \frac{B^2}{8\pi} \right)
= \frac{B_i}{4\pi} \frac{\partial B^i}{\partial t}.
\label{eqn-Energychange}$$ Calculating the product of the induction equation with $B_i$ and using the product rule to rewrite the result, we find $$\begin{aligned}
\frac{\partial {{\cal E}}_{\rm mag}}{\partial t} &= \frac{1}{4 \pi} \, \epsilon^{isp} (\nabla_s B_p)
\bigg[ \epsilon_{ijk} v_{{\rm p}}^j B^k - \frac{c^2}{4 \pi \sigma_{{\rm e}}} \,
\epsilon_{ijk} (\nabla^j B^k) \nonumber \\[1.2ex]
&- \frac{m c}{4 \pi e \rho_{{\rm p}}} \, \epsilon_{ijk} \epsilon^{jlm} ( \nabla_l B_m ) B^k \bigg]
- \nabla^i \Sigma_i.
\label{eqn-MagneticEnergy}\end{aligned}$$ The last term contains all contributions that can be written as a divergence. After integrating over the volume, it is possible to convert this part into a surface integral using Gauss’ theorem. Additionally, the third term has to be zero due to the properties of the Levi-Civita tensor. As expected, the Hall term is conservative and does not contribute to the change in the magnetic energy density. Using Ampère’s law and the generalised Ohm’s law , the remaining terms are simplified to $$\begin{aligned}
\frac{\partial {{\cal E}}_{\rm mag}}{\partial t} &= \frac{1}{c} \, J^i \epsilon_{ijk} \, v^j_{{\rm p}}B^k - \frac{J^2}{\sigma_{{\rm e}}}
\nonumber \\[1.4ex]
&- \nabla^i \left[\frac{c}{4 \pi} S_i - \frac{m_{{\rm e}}}{e} \left( \tilde{\mu}_{{\rm e}}+ \Phi \right) J_i \right],
\label{eqn-MagneticEnergy2}\end{aligned}$$ where $S^i = \epsilon^{ijk} E_j B_k$ is the Poynting vector. Equation clearly shows that any resistive plasma is subject to the decay of magnetic energy due to Ohmic diffusion and the energy loss is proportional to $J^2$. The inertial term vanishes if the protons are not able to move, which is, for example, the case in a standard metal, or the macroscopic current and the proton velocity are aligned.
Magnetic field evolution in superconducting neutron stars {#sec-SuperconMHD}
=========================================================
The coupling force: ‘standard’ resistivity {#subsec-CouplingForce}
------------------------------------------
We now return to the question of magnetic field evolution in the superconducting outer neutron star core. In order to apply a formalism similar to the resistive MHD discussion, we need to determine the forces, $F^i_{{\rm e}}$, exerted on the electron component by the various fluid constituents and the vortices/fluxtubes. However, this is where things get complicated. Due to the multi-fluid nature of the superfluid/superconducting mixture, there are not simply two components coupled by a single resistive force. We could imagine a variety of ways for the components to interact with each other ranging from electron scattering [@Sauls1982; @Alpar1984; @Andersson2006b] and vortex-fluxtube interactions [@Ruderman1998; @Jahan-Miri2000; @Link2003] to shear or bulk viscosity [@Andersson2005; @Shternin2008; @Manuel2013]. Choosing a more pedagogical approach to our problem, we pick one specific mechanism, determine how it affects the electrons on mesoscopic scales and translate this into a macroscopic picture. While we won’t provide a complete picture of the magnetic field evolution in the core, this method provides more insight to how different mechanisms could play a role.
We keep in mind that the magnetic field is locked to the superconducting fluxtubes and their motion determines the evolution of the magnetic field. We, therefore, consider the scattering of electrons off the vortex/fluxtube magnetic fields as a source of mutual friction. This ‘standard’ resistive coupling in a superfluid/superconducting mixture, first discussed by @Alpar1984, results in two forces acting on the electrons, $$F^i_{{{\rm e}}} = F^i_{\rm pe} + F^i_{\rm ne} \approx F^i_{\rm pe}.$$ We neglect the contribution from electrons scattering off the neutron vortices because ${{\cal N}}_{{\rm p}}\gg {{\cal N}}_{{\rm n}}$. This implies that electron-fluxtube interactions are markedly more common and, thus, dominate the electron coupling. The macroscopic force, $F^i_{{{\rm e}}}$, is obtained by multiplying the electron drag force, $f^i_{\rm d}$, exerted on a single fluxtube, with the fluxtube density, ${{\cal N}}_{{\rm p}}$, $$F^i_{\rm pe} = {{\cal N}}_{{\rm p}}f^i_{\rm d} = {{\cal N}}_{{\rm p}}\rho_{{\rm p}}\kappa {{\cal R}}\left(v^i_{{\rm e}}-u^i_{{\rm p}}\right).
\label{eqn-DragElectron}$$ The dimensionless drag coefficient, ${{\cal R}}$, contains all the information about the coupling on mesoscopic scales and $u^i_{{\rm p}}$ is the velocity of a single fluxtube.
In order to determine an evolution equation for the macroscopic magnetic field in superconducting matter, we have to eliminate any quantities from Equation that are defined on mesoscopic lengthscales. Hence, the next step is to rewrite the fluxtube velocity, $u^i_{{\rm p}}$, in terms of the macroscopic fluid variables. This can be achieved by using the force balance for an individual fluxtube, an approach introduced by @Hall1956 for the description of superfluid helium. We have $$\sum f^i = f^i_{\rm d} + f^i_{\rm M} + f^i_{\rm t} + f^i_{\rm em} =0,
\label{eqn-ForceBalanceFluxtube}$$ where the fluxline inertia is neglected. Our force balance equation includes the electron drag force given above, the Magnus force, $f^i_{\rm M}$, the tension force, $f^i_{\rm t}$, and the electromagnetic Lorentz force, $f^i_{\rm em}$. The different forces have been calculated by @Glampedakis2011a and are given by $$\begin{aligned}
f^i_{\rm M} &= - \rho_{{\rm p}}\kappa \, \epsilon^{ijk} \hat{\kappa}_j^{{\rm p}}\left(v_k^{{\rm p}}-u_k^{{\rm p}}\right),
\label{eqn-MagnusForce} \\[1.8ex]
f^i_{\rm t} &= \frac{H_{{{\rm c}}1} \kappa}{4 \pi a_{{\rm p}}} \, \hat{\kappa}^j_{{\rm p}}\nabla_j \hat{\kappa}^i_{{\rm p}},
\label{eqn-TensionForce}\end{aligned}$$ where $H_{{{\rm c}}1}$ is the lower critical field for superconductivity [@Tilley1990], and $$f^i_{\rm em} = \rho_{{\rm p}}\kappa \, \epsilon^{ijk} \hat{\kappa}_j^{{\rm p}}w_k^{{{\rm p}}{{\rm e}}}.
\label{eqn-EmForce}$$ Note at this point that we are interested in the linear analysis of one specific resistive mechanism. For this reason, our force balance does not include a ‘pinning’ force, resulting from the magnetic short-range interaction between the two arrays [@Ruderman1998; @Jahan-Miri2000; @Link2003; @Glampedakis2011].
Calculating repeated cross products of the force balance equation with $\hat{\kappa}^i_{{\rm p}}$, pointing along the local orientation of a fluxtube, it is possible to express the mesoscopic fluxtube velocity in terms of the averaged fluid velocities, $$u _{{\rm p}}^i = v_{{\rm e}}^i + \frac{1}{1+{{\cal R}}^2} \left( {{\cal R}}f_\star^i + \epsilon^{ijk} \hat{\kappa}_j^{{\rm p}}f^{\star}_k \right),
\label{eqn-VelocityProtons1}$$ where $$f_{\star}^i = \epsilon^{ijk} \hat{\kappa}^{{\rm p}}_j w^{\rm ep}_k + \frac{1}{\rho_{{\rm p}}\kappa}
\left(f^i_{\rm t} + f^i_{\rm em} \right).
\label{eqn-ForceStar1}$$ Combining the previous relations, we observe that the first term in Equation and the electromagnetic force, $f^i_{\rm em}$, cancel each other. Then, the effective force, $f_{\star}^i$, is equivalent to the fluxtube tension, $$f_{\star}^i = \frac{1}{\rho_{{\rm p}}\kappa} \, f^i_{\rm t}
= \frac{H_{{{\rm c}}1}}{4 \pi a_{{\rm p}}\rho_{{\rm p}}} \hat{\kappa}^j_{{\rm p}}\nabla_j \hat{\kappa}^i_{{\rm p}}.
\label{eqn-ForceStar2}$$ Substituting Equations and back into Equation finally gives for the macroscopic drag force, $$F_{{{\rm e}}}^i = - \frac{H_{\rm c1} \phi_0 {{\cal N}}_{{\rm p}}{{\cal R}}}{4\pi (1+{{\cal R}}^2)} \left({{\cal R}}\, \hat{\kappa}_{{\rm p}}^j
\nabla_j \hat{\kappa}^i_{{\rm p}}+ \epsilon^{ijk} \hat{\kappa}_j^{{\rm p}}\hat{\kappa}^l_{{\rm p}}\nabla_l \hat{\kappa}_k^{{\rm p}}\right).
\label{eqn-ForceElectronsSC}$$ For a straight fluxtube array, the tension force and, thus, the electron coupling would vanish.
The superconducting induction equation {#subsec-SuperconInduction}
--------------------------------------
Having determined the force, $F^i_{{\rm e}}$, exerted on the electron component due to scattering off the fluxtube magnetic fields, we use Equation for the macroscopic electric field to derive a generalised Ohm’s law that is valid in the superfluid/superconducting mixture, $$\begin{aligned}
E^i &= -\frac{1}{c} \epsilon^{ijk} v_j^{{\rm e}}B_k -\frac{m_{{\rm e}}}{e} \nabla^i \left ( \tilde{\mu}_{{\rm e}}+ \Phi \right )
+ \frac{H_{\rm c1} \phi_0 {{\cal N}}_{{\rm p}}}{4\pi c a_{{\rm p}}\rho_{{\rm p}}} \nonumber \\[1.2ex]
&\times \frac{{{\cal R}}}{1+{{\cal R}}^2} \left( {{\cal R}}\, \hat{\kappa}_{{\rm p}}^j
\nabla_j \hat{\kappa}^i_{{\rm p}}+ \epsilon^{ijk} \hat{\kappa}_j^{{\rm p}}\hat{\kappa}^l_{{\rm p}}\nabla_l \hat{\kappa}_k^{{\rm p}}\right).
\label{eqn-ElectricCurrentSC2}\end{aligned}$$ With Faraday’s law , we obtain a superconducting induction equation describing the evolution of the macroscopic magnetic field in the outer neutron star core, $$\begin{aligned}
\partial_t B^i &= \epsilon^{ijk} \nabla_j \bigg[ \epsilon_{klm} \left( v^l_{{\rm e}}B^m \right)- \frac{H_{\rm c1} \phi_0 {{\cal N}}_{{\rm p}}}
{4\pi a_{{\rm p}}\rho_{{\rm p}}} \nonumber \\[1.2ex] &\times \frac{{{\cal R}}}{1+{{\cal R}}^2} \left( {{\cal R}}\, \hat{\kappa}_{{\rm p}}^l \nabla_l
\hat{\kappa}^{{\rm p}}_k + \epsilon_{klm} \hat{\kappa}^l_{{\rm p}}\hat{\kappa}^s_{{\rm p}}\nabla_s \hat{\kappa}_{{\rm p}}^m
\right) \bigg].
\label{eqn-InductionFull}\end{aligned}$$ Let us specify the lower critical field, $H_{\rm c1}$, to simplify this expression further [@Tilley1990]. The field is related to the energy per unit length of the fluxtube, $\mathcal{E}_{{\rm p}}$, via $$H_{\rm c1} = \frac{4 \pi \mathcal{E}_{{\rm p}}}{\phi_0}.
\label{eqn-LowerCriticalField}$$ The fluxtube energy, on the other hand, is determined by the characteristic lengthscales of the superconducting phase and given by $$\mathcal{E}_{{\rm p}}= \left( \frac{\phi_0} {4 \pi \lambda_\star} \right)^2 \text{ln} \left( \frac{\lambda_*}{\xi_{{\rm p}}}\right).
\label{eqn-SelfEnergy1}$$ Here, $\xi_{{\rm p}}$ denotes the proton coherence length and the effective London penetration depth was defined in Equation . According to @Tinkham2004, one can estimate $\text{ln} \left( \lambda_*/ \xi_{{\rm p}}\right)
\approx 2$, which leads to $$\mathcal{E}_{{\rm p}}\approx \frac{\rho_{{\rm p}}\kappa^2} {2 \pi} \frac{1-{\varepsilon_{{\rm n}}}}{1 -{\varepsilon_{{\rm n}}}-{\varepsilon_{{\rm p}}}}
\approx \frac{\rho_{{\rm p}}\kappa^2} {2 \pi} \frac{m}{m_{{\rm p}}^*} .
\label{eqn-SelfEnergy2}$$ Combining the previous equations, we find $$\begin{aligned}
\partial_t B^i &= \epsilon^{ijk} \nabla_j \bigg[ \epsilon_{klm} \left( v^l_{{\rm e}}B^m \right) - \frac{\kappa \phi_0 {{\cal N}}_{{\rm p}}}
{2 \pi} \frac{m}{m_{{\rm p}}^*} \nonumber \\[1.2ex] &\times \frac{{{\cal R}}}{1+{{\cal R}}^2} \left( {{\cal R}}\, \hat{\kappa}_{{\rm p}}^l \nabla_l
\hat{\kappa}^{{\rm p}}_k + \epsilon_{klm} \hat{\kappa}^l_{{\rm p}}\hat{\kappa}^s_{{\rm p}}\nabla_s \hat{\kappa}_{{\rm p}}^m
\right) \bigg],
\label{eqn-InductionFullFinal}\end{aligned}$$ which is the main result of our paper.
A simplified set of equations and the field evolution timescales {#subsec-SimplifiedSet}
----------------------------------------------------------------
At this point, it seems natural to make several assumptions about the actual physics of the multi-fluid mixture inside a neutron star in order to find a simplified version of Equation . As discussed previously, the main contribution to the macroscopic magnetic induction is given by the fluxtubes. In this case, we can neglect the weak London field and the superconducting Ampère law dictates that the protons and electrons are co-moving on large scales, i.e. $v^i_{{\rm p}}\approx v^i_{{\rm e}}$, and the macroscopic current vanishes. This also implies that the local direction of the fluxtube array is aligned with the direction of the magnetic field because $$B^i = B \hat{B}^i \approx {{\cal N}}_{{\rm p}}\phi_0 \hat{\kappa}^i_{{\rm p}}\qquad \text{gives} \qquad \hat{B}^i \approx \hat{\kappa}^i_{{\rm p}}.$$ Using these simplifications to rewrite the force on the electron fluid, we obtain $$F_{{{\rm e}}}^i \approx - \frac{H_{\rm c1} B {{\cal R}}}{4\pi (1+{{\cal R}}^2)} \left({{\cal R}}\hat{B}^j
\nabla_j \hat{B}^i + \epsilon^{ijk} \hat{B}_j \hat{B}^l \nabla_l \hat{B}_k
\right).
\label{eqn-ForceElectronsSCSimplifieda}$$ The induction equation, on the other hand, reduces to $$\begin{aligned}
\partial_t B^i &\approx \epsilon^{ijk} \nabla_j \bigg[ \epsilon_{klm} \left( v^l_{{\rm p}}B^m \right) - \frac{\kappa B}
{2 \pi} \frac{m}{m_{{\rm p}}^*} \nonumber \\[1.2ex] &\times \frac{{{\cal R}}}{1+{{\cal R}}^2} \left( {{\cal R}}\hat{B}^l \nabla_l
\hat{B}_k + \epsilon_{klm} \hat{B}^l \hat{B}^s \nabla_s \hat{B}^m
\right) \bigg].
\label{eqn-InductionSimplifieda}\end{aligned}$$ We will compare this form of the superconducting induction equation with the standard MHD result in Equation . As in the resistive MHD case, we can extract two timescales, $$\tau_{\rm diss} = \frac{2 \pi L^2 (1+{{\cal R}}^2)} {\kappa {{\cal R}}}\, \frac{m_{{\rm p}}^*} {m}
\label{eqn-tau1eqn}$$ and $$\tau_{\rm cons} = \frac{\tau_{1}}{{{\cal R}}} = \frac{2 \pi L^2(1+{{\cal R}}^2)} {\kappa {{\cal R}}^2} \, \frac{m_{{\rm p}}^*} {m},
\label{eqn-tau2eqn}$$ where $L$ is again the characteristic lengthscale over which the magnetic field changes. The naming convention of the two timescales might seem arbitrary at this point, but our choice will become clear later on.
We can estimate these two timescales provided the strength of the mutual friction is known. A method to calculate the dimensionless drag parameter, ${{\cal R}}$, for the coupling of relativistic electrons and a single fluxtubes is discussed in @Alpar1984 (see also @Sauls1982 [-@Sauls1982]; @Andersson2006b [-@Andersson2006b])[^2]. The authors give the relaxation timescale, $\tau_{\rm pe}$, for the ‘resistive’ interaction and include effects caused by the finite fluxtube size and the increase in moment of inertia due to the coupling of electrons and protons on much shorter timescales. The relaxation timescale is related to the drag coefficient via $${{\cal R}}= \left(\kappa \, {{\cal N}}_{{\rm p}}\tau_{\rm pe} \right)^{-1},$$ which leads to the following numerical estimate $$\hspace{-0.01cm}
{{\cal R}}\approx 1.9 \times 10^{-4} \left( \frac{m}{m_{{\rm p}}^*} \right)^{1/2} \rho_{14}^{1/6} \left(\frac{x_{{\rm p}}}{0.05} \right)^{1/6}.$$ This gives ${{\cal R}}\ll 1$ and implies that the standard friction mechanism is rather weak. Adopting this limit, we can approximate for the neutron star core, $$\tau_{\rm diss} \approx 3.1 \times 10^{11} L_6^2 \, \rho_{14}^{-1/6} \left( \frac{x_{{\rm p}}}{0.05}\right)^{-1/6} \, {{\rm yr}}\label{eqn-tau1estimate}$$ and $$\tau_{\rm cons} \approx 1.3 \times 10^{15} L_6^2 \, \rho_{14}^{-2/6} \left(\frac{x_{{\rm p}}}{0.05}\right)^{-2/6} \, {{\rm yr}}.
\label{eqn-tau2estimate}$$ We have used Equation to estimate the effective proton mass, i.e. the entrainment parameter.
Flux-freezing and magnetic energy {#subsec-SuperconEnergy}
---------------------------------
For conventional electron-fluxtube coupling, the timescales for the magnetic field evolution are rather long and the dynamics of the macroscopic induction are dominated by the inertial term in the induction equation. In the weak mutual friction limit, we are, thus, left with an equation that is equivalent to the ones discussed in Subsection \[subsec-MHDFreezing\]. The magnetic field in the superconducting sample is frozen to proton fluid, which implies that the superconducting fluxtubes are locked to the proton plasma, i.e. $v_{{\rm p}}^i \approx u_{{\rm p}}^i$. Hence, electrons, protons and fluxtubes are comoving on large scales, which is different to the weakly resistive case of standard MHD, where the relative motion between the charged particles was important.
In order to determine whether the additional terms in Equation are conservative or dissipative and which timescale dominates, we discuss the evolution of the superconducting magnetic energy. As before, we evaluate the energy associated with the work done by the magnetic force. However, in a superfluid/superconducting mixture the standard Lorentz force has to be changed accordingly. For a non-rotating star in the absence of entrainment the total magnetic force is given by [@Easson1977; @Akgun2008; @Glampedakis2011a; @Lander2013] $$F_{\rm mag}^i = \frac{1}{4 \pi} \left[ B_j \nabla^j H_{\rm c1}^i -
\nabla^i \left( \rho_{{\rm p}}B \frac{\partial H_{\rm c1}}{\partial \rho_{{\rm p}}} \right) \right]
\label{eqn-SuperconForce},$$ where $H_{\rm c1}^i = H_{\rm c1} \hat{B}^i$. This expression also contains a tension and a pressure term but both scale with $H_{\rm c1}^i $ and $B^i$ instead of $B^2$. The work associated with this force is given by $$\begin{aligned}
W_{\rm mag} &= \frac{1}{4 \pi} \int \nabla^i \left( r_j H_{\rm c1}^j B_i -
\rho_{{\rm p}}B \frac{\partial H_{\rm c1}}{\partial \rho_{{\rm p}}} \, r_i \right) {{\rm d}}V \nonumber \\[1.2ex]
&- \frac{1}{4 \pi} \int \left( H_{\rm c1}^i B_j \nabla^j r_i - \rho_{{\rm p}}B \frac{\partial H_{\rm c1}}{\partial \rho_{{\rm p}}}
\nabla^i r_i \right) {{\rm d}}V,
\label{eqn-EnergySuperconA}\end{aligned}$$ where we have used the product rule and Equation . Similar to the standard MHD case, the first term can be rewritten using Gauss’ theorem. However, in the superconducting outer core, this contribution does not simply vanish because discontinuities are likely to be present at the fluids’ boundaries. The dynamics that might arise due the presence of a current sheet at the crust-core interface or the type-II to type-I transition region in the neutrons star’s inner core are only poorly understood and significantly complicate the problem. Incorporating these interfaces would require a much more detailed understanding of the microphysics involved. In the following, we, therefore, omit a discussion of the surface terms and focus on the much simpler problem of magnetic field evolution in the bulk fluid.
Taking into account that $H_{\rm c1}$ is a function linear in $\rho_{{\rm p}}$, the second integral in Equation reduces to $$\begin{aligned}
W_{\rm mag, bulk} & = \int \frac{H_{\rm c1} B} {2 \pi} \, {{\rm d}}V = \int {{\cal E}}_{\rm mag,sc} \, {{\rm d}}V.\end{aligned}$$ Taking the time derivative of the magnetic energy density gives two contributions $$\frac{\partial {{\cal E}}_{\rm mag,sc}}{\partial t} = \frac{B}{2 \pi} \frac{\partial H_{\rm c1}} {\partial t}
+ \frac{H_{\rm c1}}{2 \pi} \hat{B}_i \frac{\partial B^i}{\partial t}.
\label{eqn-EnergychangeSC}$$ Comparison with the corresponding expression of standard MHD given in Equation shows that the superconducting nature of the mixture gives rise to a new contribution for the change in energy density. In contrast to resistive MHD, the evolution of matter and the magnetic induction are no longer decoupled in the condensate. Equation demonstrates that modifying the properties of the superconductor, such as the lower critical field, $H_{\rm c1}$, alters the magnetic energy. This implies that an evolving matter configuration can be closely linked to a changing magnetic field.
The second term in Equation is similar to the result for normal conducting matter. Calculating the product of the induction equation with $\hat{B}_i$ and using the product rule to simplify, we find $$\begin{aligned}
\hat{B}_i \frac{\partial B^i}{\partial t} &= \epsilon^{isp} (\nabla_s \hat{B}_p) \bigg[ \epsilon_{ijk} v_{{\rm p}}^j B^k -
\frac{\kappa B {{\cal R}}^2} {2 \pi (1+{{\cal R}}^2)} \frac{m}{m_{{\rm p}}^*} \, \hat{B}^l \nabla_l \hat{B}_i \nonumber \\[1.2ex]
&- \frac{\kappa B {{\cal R}}} {2 \pi (1+{{\cal R}}^2)} \frac{m}{m_{{\rm p}}^*} \, \epsilon_{ijk} \hat{B}^j \hat{B}^l \nabla_l \hat{B}^k \bigg]
- \nabla^i \Sigma_i.
\label{eqn-SuperconMagneticEnergy}\end{aligned}$$ As before, $\Sigma^i$ denotes the divergence terms. This equation bears some resemblance with the result found in standard MHD and we would equivalently expect to obtain a conservative and a dissipative contribution. In order to determine which of the terms are nonzero or zero, we rewrite the tension using the following identity $$\begin{aligned}
\hat{B}^l \nabla_l \hat{B}_i = \epsilon_{ijk} \epsilon^{jlm} ( \nabla_l \hat{B}_m ) \hat{B}^k.
\label{eqn-TensionIdentity}\end{aligned}$$ The second term in Equation is, thus, proportional to $$\begin{aligned}
\epsilon^{isp} (\nabla_s \hat{B}_p) \hat{B}^l \nabla_l \hat{B}_i =
\epsilon^{isp} (\nabla_s \hat{B}_p) \, \epsilon_{ijk} \epsilon^{jlm} (\nabla_l \hat{B}_m) \hat{B}^k.
\label{eqn-SuperconHall}\end{aligned}$$ Analogous to the Hall term of standard resistive MHD, this vanishes due to the properties of the antisymmetric Levi-Civita tensor. Thus, the second term in the superconducting induction equation is conservative and does not modify the total magnetic energy of the superconducting mixture. The last term in Equation , on the other hand, is proportional to $$\begin{aligned}
\epsilon^{isp} (\nabla_s \hat{B}_p) \, \epsilon_{ijk} \hat{B}^j \hat{B}^l \nabla_l \hat{B}^k
= \mathcal{J}^i \epsilon_{ijk} \hat{B}^j \epsilon^{klm} \mathcal{J}_l \hat{B}_m,\end{aligned}$$ where we defined the vector $$\mathcal{J}^i \equiv \epsilon^{ijk} \nabla_j \hat{B}_k.$$ Rewriting the remaining two Levi-Civita tensors in terms of Kronecker deltas, $\delta^i_j$, gives the projection $$\begin{aligned}
\mathcal{J}^i \, \epsilon_{ijk} \hat{B}^j \epsilon^{klm} \mathcal{J}_l \hat{B}_m
= \mathcal{J}_i \mathcal{J}^i - \left( \mathcal{J}^i \hat{B}_i \right) \left( \mathcal{J}^j \hat{B}_j\right).
\label{eqn-IdentityOhm}\end{aligned}$$ Decomposing the vector $\mathcal{J}^i$ into a component parallel to $\hat{B}_i$ and one perpendicular to the magnetic field direction, i.e. $\mathcal{J}^i = \mathcal{J}_{\parallel} \hat{B}_i + \mathcal{J}_{\perp}^i$, we see that Equation only depends on the component of $\mathcal{J}^i$ that is perpendicular to $\hat{B}_i$, $$\begin{aligned}
\mathcal{J}_i \mathcal{J}^i - \left( \mathcal{J}^i \hat{B}_i \right) \left( \mathcal{J}^j \hat{B}_j\right)
= \mathcal{J}_{\perp}^2.
\label{eqn-IdentityOhm2}\end{aligned}$$ Similar to the Ohmic term in standard MHD, we also retain a dissipative contribution to the total magnetic energy in the case of superconducting MHD. It is given by $$\begin{aligned}
\frac{\partial {{\cal E}}_{\rm mag,sc}}{\partial t} &= \frac{B}{2 \pi} \frac{\partial H_{\rm c1}} {\partial t}
+ \frac{H_{\rm c1}}{2 \pi} \mathcal{J}^i_\perp \epsilon_{ijk} v_{{\rm p}}^j B^k \nonumber \\[1.4ex]
&- \frac{H_{\rm c1} \kappa B {{\cal R}}} {4 \pi^2 (1+{{\cal R}}^2)} \frac{m}{m_{{\rm p}}^*} \,\mathcal{J}_{\perp}^2
- \frac{H_{\rm c1}}{2 \pi} \nabla^i \Sigma_i.
\label{eqn-SuperconMagneticEnergyFinal}\end{aligned}$$
Having calculated the change in the magnetic energy density, we can associate the timescale $\tau_{\rm diss}$, given in Equation , with a dissipative mechanism. Comparing the numerical estimate to the Ohmic decay timescale , we observe that the resistive coupling in a superfluid/superconducting mixture acts on a timescale, which is two orders of magnitude smaller than the standard MHD diffusion, $$\frac{\tau_{\rm diss}}{\tau_{\rm Ohm}} \approx 1.3 \times 10^{-2} \,
T_8^2 \, \rho_{14}^{-5/3} \left( \frac{x_{{\rm p}}}{0.05}\right)^{-5/3}.$$ On the other hand, the timescale for the Hall evolution in a normal conducting plasma can be compared to $\tau_{\rm cons}$ in Equation . In contrast to the standard MHD case, the conservative timescale emerging from the superconducting induction equation is several orders of magnitude larger than the Hall timescale, $$\frac{\tau_{\rm cons}}{\tau_{\rm Hall}} \approx 6.8 \times 10^{4} \,
B_{12} \, \rho_{14}^{-4/3} \left( \frac{x_{{\rm p}}}{0.05}\right)^{-4/3}.$$ We also note that due to the dependence on the dimensionless drag coefficient, ${{\cal R}}$, the dissipative term in the superconducting induction equation governs the magnetic field evolution, whereas in standard MHD the conservative Hall term acts on shorter timescales. In the latter case, the order of the two timescales is necessary for any cascading behaviour to take place; the Hall term drives the magnetic field to shorter lengthscales, where it can decay ohmically. However, if diffusion is dominating the evolution, the redistribution of magnetic energy will happen on much longer timescale not causing a cascade. Hence, despite the close similarities between the conservative terms in Equation and standard MHD, we conjecture that the analysis of the Hall cascade by @Goldreich1992 is not transferable to the superconducting case.
Discussion {#sec-Discussion}
==========
Strong magnetic fields are a key ingredient for many phenomena observed in neutron stars. Understanding the fields’ long-term evolution might give insight into the ‘metamorphosis’ between the different neutron star classes, the changing fields of standard radio pulsars or the high activity of magnetars. As the mechanisms causing field changes are only poorly understood, we revisited the question of magnetic field evolution and, in particular, discussed the influence of a superconducting component. Our aim was to rethink key notions of magnetohydrodynamics and develop a better intuition for the magnetic field evolution in a superconductor.
In this work, we used a multi-fluid formalism to describe the mixture in the outer neutron star core. The model introduced in Section \[sec-Background\] translates the presence of mesoscopic vortices/fluxtubes into the large scale dynamics of the fluid. As an application of this framework, we analysed the conventional dissipative mechanism, i.e. the scattering of electrons off the fluxtube magnetic field. Based on the approach of standard resistive MHD, we combined a generalised Ohm’s law with Faraday’s law and the respective force on the electron fluid to derive a superconducting induction equation. Considering the London field as a negligible contribution led to a simplified equation, which should be applicable to most astrophysical scenarios. Caution is in order when discussing highly magnetised objects. For field strengths above the upper critical field, $B > H_{\rm c2} \sim 10^{16} \, {{\rm G}}$ [@Tilley1990], the superconducting state breaks down and our averaged formalism no longer applies. According to @Goldreich1992, ambipolar diffusion could potentially become important in this regime and drive field decay on the order of typical magnetar ages.
To compare our new results for magnetic field evolution with the standard MHD case, the magnetic energies associated with the total magnetic forces were calculated. In our analysis, we significantly simplified the problem by omitting a detailed discussion of the surface terms, a key problem which needs to be addressed in future studies. This implies that effects originating at the crust-core interface or the type-II to type-I transition in the inner core, which could potentially drive the magnetic field evolution, are not taken into account. Instead, we focused on the evolution of the averaged magnetic field in the bulk. We found that in the limit of weak mutual friction, the inertial term dominates the field evolution. The fluxtubes move with the proton fluid and the flux is, as in the standard MHD case, frozen to the charged particles. We additionally showed that the new induction equation contains a dissipative and a conservative contribution, similar to the Ohmic and the Hall term in normal conducting matter. However, the evolution timescales extracted from the superconducting induction equation for weak mutual friction, $10^{11} \, {{\rm yr}}$ and $10^{15} \, {{\rm yr}}$ respectively, are notably longer than the typical spin-down ages of neutron stars. We conclude that the conventional mutual friction mechanism cannot serve as an explanation for the field changes in pulsars or the activity of magnetars, which would require timescales of order $10^7\, {{\rm yr}}$ and $10^4 \, {{\rm yr}}$, respectively. Simply increasing the strength of the mutual friction cannot provide a solution to this problem either. Due to the ${{\cal R}}$-dependence of $\tau_{\rm diss}$, the minimum dissipation timescale one could obtain with a frictional mechanism of the form is $$\tau_{\min} = \frac{4\pi L^2}{\kappa} \, \frac{m_{{\rm p}}^*}{m} \approx 1.4 \times 10^{8} L_6^2 \, \rho_{14}^{-1/6}
\left( \frac{x_{{\rm p}}}{0.05}\right)^{-1/6} \, {{\rm yr}}$$ for ${{\cal R}}=1$. We note at this point that all numerical estimates crucially depend on the lengthscale $L$. It can be identified with the curvature radius of the magnetic field and we chose the neutron star radius, $R$, to normalise the previous results. Recent work on field equilibria in superconducting neutron star cores by @Lander2013 suggests that the field configuration actually supports structures on a shorter lengthscale of $L \approx 10^5 \, {{\rm cm}}$. Adopting such an estimate would reduce the characteristic timescales by two orders of magnitude. In particular, the minimum dissipation timescale, $\tau_{\min}$, would be shortened to a million years, which is closer to the timescales of astrophysical interest.
Our results additionally suggest that the highly conducting neutron star core might affect the crustal field and slow down its evolution. Making a precise statement at this point is, however, not possible due to the poorly known physics at the crust-core boundary. This transition is crucial in understanding how changes of the core magnetic field are translated to the crust. The analysis presented in this paper does, therefore, not reconcile the discrepancy between short crustal decay timescales [@Pons2009] and the much longer core evolution. In order to significantly reduce the latter different dissipative mechanisms have to be invoked. The typical candidate for strong coupling is vortex-fluxtube ‘pinning’ due to the short-range magnetic interaction between the two arrays. While we did not address pinning specifically, discussing the vortex-fluxtube interaction would be the natural continuation of this paper as our prescription can deal with any coupling mechanism. Based on a mesoscopic description of the pinning process, one would have to determine how the coupling affects the electron fluid and substitute the respective force, $F_{{\rm e}}^i$, into the generalised Ohm’s law. Determining the superconducting induction equation that would result from the pinning interaction will be left for future work.
Acknowledgements {#acknowledgements .unnumbered}
================
VG is supported by Ev. Studienwerk Villigst and NA acknowledges support from STFC in the UK. KG is supported by the Ramón y Cajal Programme of the Spanish Ministerio de Ciencia e Innovación and by the German Science Foundation (DFG) via SFB/TR7.
[^1]: E-mail: [email protected]
[^2]: Note that according to @Jones2006, magnetic scattering off individual fluxtubes is suppressed for very large fluxtube densities. Instead electron scattering by a cluster of fluxtubes dominates the coupling, leading to a much smaller drag coefficient, ${{\cal R}}$.
|
{
"pile_set_name": "ArXiv"
}
|
In one of their recent works, Byland and Scialom studied the evolution of the Bianchi I, the Bianchi III and the Kantowski–Sachs universe in a model with a real scalar field and a convex positive potential [@BS]. A considerable part of the investigation was devoted to the analysis of the asymptotic behavior and the stability properties of the solutions of the Einstein–Klein–Gordon equations $$\begin{aligned}
&& \dot{\theta} = -\frac{1}{3} \, \theta^2
-2 \sigma^2 + V(\varphi) - \psi^2, \\
&& \dot{\sigma} = -\frac{1}{3 \sqrt{3}} \, \theta^2
- \theta \sigma
+ \frac{1}{\sqrt{3}} \left(\sigma^2
+ V(\varphi) + \frac{1}{2} \, \psi^2 \right),
\\
&& \dot{\varphi}= \psi , \\
&& \dot{\psi} = -\theta \psi - \frac{dV}{d \varphi},\end{aligned}$$ where $\theta$ is the function of the expansion rate, $\sigma_{\mu\nu}$ is the shear tensor of the hypersurface of constant time, $\sigma=\frac{1}{2} \, \sigma_{\mu\nu} \sigma^{\mu\nu}$, $\varphi$ is the scalar field, and $V$ is a convex positive potential; an overdot stands for derivatives with respect to $t$ (see Eqs. (8)–(11) in [@BS]).
The analysis was based on determining the stability properties of the critical points of the dynamical system $$\begin{aligned}
\label{MainDS}
&& S' = - \frac{1}{3 \sqrt{3}} - \frac{2}{3} S
+ \frac{1}{2 \sqrt{3}} (2 S^2 + 2 U^2 + P^2)
+ F S, \nonumber \\
&& U' = \left(\frac{1}{3} - \frac{\lambda}{2} P + F \right) U, \\
&& P' = -\frac{2}{3} P + \lambda U^2 + F P, \nonumber\end{aligned}$$ where $S = \sigma/\theta$, $U = \sqrt{V}/\theta$, $P = \psi/\theta$, $F = 2 S^2 - U^2 + P^2$, $\lambda$ is given by $V = V_0 \, e^{-\lambda \phi}$, and a prime stands for derivatives with respect to $\tau$ defined by $d\tau = \theta \, dt$ (see Eqs. (24)–(26) in [@BS]). Here we omit an equation for $\theta$, since Eqs. (\[MainDS\]) do not contain this function.
It was found in [@BS] that the dynamical system (\[MainDS\]) has the following critical points: $$\begin{aligned}
&P_1:& \quad S = -\frac{1}{2 \sqrt{3}}, \; U = 0, \; P = 0 \\
&P_2:& \quad S = 0, \; U = \sqrt{\frac{6 - \lambda^2}{18}}, \;
P = \frac{\lambda}{3}, \quad \lambda \le \sqrt{6} \\
&P_3:& \quad S = \frac{1}{2 \sqrt{3}}
\frac{2 - \lambda^2}{1 + \lambda^2}, \;
U = \frac{\sqrt{2 + \lambda^2}}{\sqrt{2}\,(1 + \lambda^2)},
\;
P = \frac{\lambda}{1 + \lambda^2} \\
&\Sigma:& \quad U = 0, \; 3 S^2 + \frac{3}{2} P^2 = 1, \quad
S \in [-1/\sqrt{3}, 1/\sqrt{3}].\end{aligned}$$ In particular, it was shown that the point $P_2$ has the eigenvalues $\varepsilon_1 = -1 + \lambda^2/6$ (twice) and $\varepsilon_2 = -2/3 + \lambda^2/3$. Thus, it is stable for $\lambda < \sqrt{2}$ and unstable for $\sqrt{2} < \lambda < \sqrt{6}$. It was also found that $P_2$ is also unstable for $\lambda = \sqrt{2}$. The problem is to determine the stability properties of $P_2$ for the case $\lambda = \sqrt{6}$, in which $P_2$ is degenerate with the eigenvalues $\varepsilon_1 = 0$ (twice) and $\varepsilon_2 = 4/3$. Here we shall demonstrate that the point $P_2$ is unstable (both to the future and to the past).
Notice that for the case $\lambda = \sqrt{6}$, $P_2$ belongs to the ellipsis $\Sigma$. Thus, one of the zero eigenvalues corresponds to the fact that $\Sigma$ is a one-dimensional critical set. In order to prove that the critical point $P_2$ is unstable both to the past and to the future, it is sufficient to show that there is a projection of this point, which is unstable. To do this, we rewrite the dynamical system (\[MainDS\]) as $$\begin{aligned}
\label{ShDS4P2}
&& S' = 2 \alpha
\left( \frac{1}{2 \sqrt{3}} + S \right) \bar{P}
+ \frac{1}{2 \sqrt{3}} (2 S^2 + 2 U^2 + \bar{P}^2)
+ \bar{F} S, \nonumber \\
&& U' = \left( \frac{\alpha}{2} \, \bar{P} + \bar{F} \right) U, \\
&& \bar{P}' = \frac{4}{3} \bar{P}
+ \alpha \, (3 U^2 + 2 \bar{P}^2)
+ (\alpha + \bar{P}) \, \bar{F}, \nonumber\end{aligned}$$ where $\bar{P} = P - \alpha$, $\alpha = \sqrt{2/3}$, and $\bar{F} = 2 S^2 - U^2 + \bar{P}^2$. The point $P_2$ now corresponds to the origin, $(S,U,\bar{P}) = (0,0,0)$.
Consider the projection of (\[ShDS4P2\]) in the plane $S = 0$. The equations for $U$ and $\bar{P}$ may now be written as
\[eqs4S=0\] $$\begin{aligned}
&& U' = \left( \frac{\alpha}{2} \, \bar{P}
- U^2 + \bar{P}^2 \right) U, \label{e4U} \\
&& \bar{P}' = \frac{4}{3} \bar{P}
+ (2 \alpha - \bar{P}) \, U^2
+ (3 \alpha + \bar{P}) \, \bar{P}^2. \label{e4P}\end{aligned}$$
Denote the right hand sides of Eqs. (\[e4U\]) and (\[e4P\]) as ${\cal U}(U,\bar{P})$ and ${\cal P}(U,\bar{P})$ respectively. The idea is to find a solution $\bar{P} = f(U)$ of the equation ${\cal P}(U,\bar{P}) = 0$ in a neighborhood of $U=0$, to substitute it into ${\cal U}(U,\bar{P})$: $${\cal U}(U,f(U)) = a_m U^m + \dots ,$$ and then to examine whether the power $m$ of the leading term is even or odd and, in the latter case, to check the sign of $a_m$ (see, e.g., [@Perko]).
It is easy to see that the corresponding solution in our case assumes the form $$\bar{P} = f(U) = - \sqrt{\frac{3}{2}} \, U^2 + O(U^4).$$ Therefore, $${\cal U}(U,f(U)) = - \frac{3}{2} \, U^3 + O(U^7).$$ Thus, $m = 3$ is odd, and $a_m$ is negative. It follows immediately from the standard results for the two-dimensional dynamical systems that the point $(U,\bar{P}) = (0,0)$ is a topological saddle. Hence, it is unstable both to the future and to the past.
Let us mention that the same conclusion may be obtained basing on the results of the center manifold theory (see, e.g., [@Wiggins]).
We remark that $P_2$ is not the only degenerate point of $\Sigma$. Recall that the eigenvalues of $\Sigma$ are [@BS] $$\varepsilon_1 = 0, \quad
\varepsilon_2 = 1 - \frac{\lambda}{2} \, P, \quad
\varepsilon_3 = \frac{2}{3} (2 + \sqrt{3} \, S).$$ Thus, for any point of $\Sigma$, except for the points $(P,S) = (0, \pm 1/\sqrt{3})$, there exists $\lambda = 2/P$, $\lambda \in [-\sqrt{6}, \sqrt{6}]$ such that $\varepsilon_2$ turns to zero, thus making this point degenerate. The stability properties of these degenerate points may be studied in a way similar to the above.
Finally, let us make some comments on the behavior of the trajectories of (\[MainDS\]) in the vicinity of the point $P_1$. It was found in [@BS] that this point has the eigenvalues $\varepsilon_{S,P} = -1/2$ and $\varepsilon_U = 1/2$. It was also claimed that starting around $P_1$, the critical point can never be reached by any solutions of the dynamical system. This is not quite correct. It follows from the standard results of the dynamical systems theory (see, e.g., [@Hartman], Theorem 6.1) that in a neighborhood of $P_1$ there exists a two-dimensional stable manifold $W^s$ and a one-dimensional unstable manifold $W^u$. The trajectories lying on these manifolds tend to $P_1$ as $\tau \to \infty$ and $\tau \to -\infty$ respectively. One can easily find the asymptotic behavior of these solutions.
Namely, let us introduce $\bar{S} = S + 1/(2 \sqrt{3})$. Then the dynamical system (\[MainDS\]) reads as $$\begin{aligned}
\label{ShDS}
&& \bar{S}' = - \frac{1}{2} \bar{S}
- \frac{1}{2 \sqrt{3}} \left(4 \bar{S}^2 - 3 U^2 \right)
+ \bar{F} \bar{S}, \nonumber \\
&& U' = \frac{1}{2}
\left(1 - \lambda P - \frac{4}{\sqrt{3}} \, \bar{S}
+ 2 \bar{F} \right) U, \\
&& P' = -\frac{1}{2} P - \frac{2}{\sqrt{3}} \, \bar{S} P
+ \lambda U^2 + \bar{F} P, \nonumber\end{aligned}$$ where $\bar{F} = 2 \bar{S}^2 - U^2 + P^2$. The point $P_1$ now corresponds to the origin, $(\bar{S},U,P) = (0,0,0)$. The eigenvectors of this point are $\zeta_{\bar{S}} = (1,0,0)$, $\zeta_P = (0,0,1)$, and $\zeta_U = (0,1,0)$. Now one can see that the trajectories on $W^s$ take the form $$U \equiv 0, \quad
S \approx -\frac{1}{2 \sqrt{3}} + C_S \, e^{-\tau/2}, \quad
P \approx C_P \, e^{-\tau/2},$$ as $\tau \to \infty$, where $C_S$ and $C_P$ are arbitrary constants, $C_S^2 + C_P^2 \ne 0$. (One can also obtain these solutions in a parametric and partially in an explicit form (for $\bar{S} \equiv 0$).) Notice that $S$ here is just a linear function of $P$. We also mention that one of the trajectories on $W^s$, namely, $$U \equiv 0, \quad
S = -\frac{1}{2 \sqrt{3}} + \frac{1}{2 \sqrt{2}} \, P \quad
\text{for } P \in \; ]0, \sqrt{2/3}[,$$ joins $P_1$ to $P_2$ if $\lambda = \sqrt{6}$.
In order to obtain the asymptotic behavior of the trajectories on $W^u$, we notice that for these trajectories $\bar{S} = o(U)$ and $P = o(U)$ as $U \to 0$. This allows us to consider only the leading terms in (\[ShDS\]): $$\bar{S}' = - \frac{1}{2}\, \bar{S} + \frac{3}{2}\, U^2, \quad
U' = \frac{1}{2} \, U, \quad
P' = -\frac{1}{2}\, P + \lambda U^2.$$ It follows immediately that the outgoing trajectories take the form $$U \approx C_U \, e^{\tau/2}, \quad
S \approx -\frac{1}{2 \sqrt{3}} + \frac{1}{\sqrt{3}} \, U^2, \quad
P \approx \frac{2}{3} \, \lambda U^2,$$ as $\tau \to -\infty$, where $C_U$ is a nonzero constant.
I would like to thank Professor D. V. Gal’tsov for attracting my attention to the theory of dynamical systems.
S. Byland and D. Scialom, [*Phys. Rev.*]{} [**D57**]{}, 6065 (1998) \[gr-qc/9802043\].
L. Perko, [*Differential Equations and Dynamical Systems*]{} (Springer-Verlag, New York, 1991).
S. Wiggins, [*Introduction to Applied Nonlinear Dynamical Systems and Chaos*]{} (Springer-Verlag, New York, 1990).
P. Hartman, [*Ordinary Differential Equations*]{} (John Wiley & Sons, New York, 1964).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.'
address: 'Naval Postgraduate School, Monterey, CA 93943'
author:
- 'I. M. Ross'
- 'R. J. Proulx'
- 'M. Karpenko'
title: An Optimal Control Theory for the Traveling Salesman Problem and Its Variants
---
traveling salesman problems, functionals, optimal control theory, neighborhoods, forbidden neighborhoods, profits, time window, routing
Introduction
============
Recently, we showed that a continuous-variable, nonlinear static optimization problem can be framed as a dynamic optimization problem [@ross-jcam-1]. In this theory, a generic point-to-set algorithmic map is defined in terms of a controllable continuous-time[^1] trajectory, where, the decision variable is a continuous-time search vector. Starting with this simple idea, many well-known algorithms, such as the gradient method and Newton’s method, can be derived as optimal controllers over certain metric spaces [@ross-jcam-1]. If the control is set to the acceleration of a double-integrator model, then a similar theory[@ross-accel] generates accelerated optimization techniques such as Polyak’s heavy ball method[@polyak64] and Nesterov’s accelerated gradient method[@nesterov83]. In this paper, we further the theory of framing optimization problems in terms of an optimal control problem. More specifically, we use a menu of modifications to the traveling salesman problem (TSP) and its variants [@cook-2012; @TSP-variants] to address a class of combinatorial optimization problems. To motivate the new mathematical paradigm, consider the following questions:
How do we define the distance between two sets if the vertices in a TSP graph are cities equipped with the power of the continuum? Assume the cities to be disjoint sets; see Fig. \[fig:2-City-Blobs\].
A version of this question was posed in [@TSPN] more than two decades ago. It continues to be an active research topic; see for example, [@TSPN-robotics] and [@wang2019]. To appreciate this question, consider a distance function defined by, $$\label{eq:d(AB)}
d(A,B) := \displaystyle\mathop\text{min}_{{\boldsymbol x}\in A, {{\boldsymbol y}}\in B}d({\boldsymbol x}, {{\boldsymbol y}}) = \displaystyle\mathop\text{min}_{{\boldsymbol x}\in A, {{\boldsymbol y}}\in B} {\left\Vert{\boldsymbol x}-{{\boldsymbol y}}\right\Vert}_2
$$ Besides the fact that $d(A,B)$ is not a metric, if is used to construct the edge weights in the TSP graph, the resulting solution may comprise disconnected segments because the entry and exit points for a city may not necessarily be connected. Adding a “continuity segment” *a posteriori* does not generate an optimal solution as indicated by the three-city tour illustrated in Fig. \[fig:3-City-Blobs\].
This solution is clearly not optimal because the triangle tour shown in Fig. \[fig:3-City-Blobs\] is shorter. In other words, an optimal tour is obtained by not using the shortest distance between two cities.
\[rem:que-not-soln\] It is critically important to note that Question 1 is not centered on solving the problem of the type illustrated in Fig. \[fig:3-City-Blobs\]; rather, this question and others to follow, are focused more fundamentally on simply framing the problem mathematically.
\[rem:newTSP-objFun\] It is apparent by a cursory examination of Fig. \[fig:3-City-Blobs\] that the values of the arc weights are not independent of the path. That is, the objective function in the TSP must somehow account for the functional dependence of the sequence of cities in the computation of the distance between any two cites.
How do we define distances between two cities if the entry and exit points are constrained by some angle requirement?
This question was first addressed in [@TSP-angle] for the case when the cities are points; the new question posed here pertains to the the additional issues when the cities are not points as in Fig. \[fig:3-City-Blobs\].
How do we define distances between two cities in the presence of a no-drive, no-fly zone? See Fig. \[fig:no-fly-zone\].
This question was addressed in [@TSPFN] for the case of point-cities. It is apparent that any difficulty encountered in answering Questions 2 and 3 is further amplified by the issues resulting from the discussions related to Question 1; see also Remarks \[rem:que-not-soln\] and \[rem:newTSP-objFun\].
\[que:moving-cities\] How do we define distances between neighborhoods that are in deterministic motion?
This question is related to the one posed in [@DTSP] with respect to the dynamic TSP. It remains a problem of ongoing interest; see, for example [@DTSP-2019].
Assuming we can answer the preceding questions, is distance a correct measure for a minimum-time TSP (with neighborhoods)? If not, what is the proper mathematical problem formulation for a minimum-time TSP?
While limited versions of the aforementioned questions have been addressed in the literature as noted in the preceding paragraphs, the totality of Questions 1–5 appear in many practical and emerging mathematical problems that lie at the intersection of physics, operations research and engineering science. For example, the problem of touring the 79 moons of Jupiter [@witze] by a remote sensing spacecraft has all the elements of Questions 1–5. Relative orbits around the moons are the “cities” and the spacecraft is the “traveling salesman;” see Fig. \[fig:Grand-tour\].
The moons are in various non-circular orbits. The measure of “distance” (i.e., weights) is the amount of propellant it takes to transfer the spacecraft between two (moving) relative orbits. Not all visits to the moons are valued equally by the science team. The objective of a grand tour mission is to maximize the science return by orbiting around as many moons as possible under various constraints arising from the physics of gravitational motion, electromagnetic instrumentation, thermodynamics, electrical power, dollar cost and lifetime of the spacecraft. It is clear that modeling this optimization problem using the available constructs of a TSP [@cook-2012] is neither apparent nor easy. In fact, the computation of the weights associated with the arcs of the graph (that represent the transfer trajectories) involve solving a constrained, nonlinear optimal control problem with variable endpoints [@ross-book; @vinter]. Consequently, even generating the data[^2] to define this problem as a standard TSP is a nontrivial task.
The main contribution of this paper is a new mathematical problem formulation for addressing a class of information-rich, operations-research-type mathematical problems such as the modified TSPs discussed in the preceding paragraphs.
A New Mathematical Paradigm
===========================
Throughout this paper we use the word functional in the sense of mathematical analysis: a mapping from a space of measurable functions to the field of real numbers.
An $\mathcal{F}$-graph is a finite collection of functionals that constitute the arcs (edges) and vertices of a graph.
Let $V^i: \text{dom}(V^i) \to {\mathbb R}, i \in \mathbb{N}_+ $ and $E^k: \text{dom}(E^k) \to {\mathbb R}, k \in \mathbb{N}_+$ be a finite collection of functionals, where $\text{dom}(\cdot)$ is the domain of $(\cdot)$. Let $\mathcal{F}$ be an $\mathcal{F}$-graph whose vertices and arcs/edges are given by $V^i$ and $E^k$ respectively. From standard graph theory, a walk in $\mathcal{F}$ may be defined in terms of an alternating sequence of $V$- and $E$-functionals. In order to perform evaluations in $\mathcal{F}$, it is necessary to define some new constructs.
\[def:F-control\] An $\mathcal{F}$-control is a sequence of functions $$\langle \psi_0, \psi_1, \ldots, \psi_n \rangle \quad n \in \mathbb{N}$$ where each $\psi_j, \ j = 0, \ldots, n$ is selected from the domain of $V^i$ or the domain of $E^k$.
\[rem:F-control\] An $\mathcal{F}$-control involves two simultaneous actions: selecting functions from the domain of the functionals that comprise $\mathcal{F}$, and ordering the selections in some sequence. Thus, different selections ordered the same way is a different $\mathcal{F}$-control. Conversely, the same selection ordered differently is a also a different $\mathcal{F}$-control (provided the reordering is consistent with the selection process).
A control walk (in $\mathcal{F}$) is an $\mathcal{F}$-control with the following property: $$\psi_j \in \text{dom}(E^k) \Rightarrow \psi_{j-1} \in \text{dom}(V^l) \ \text{and } \psi_{j+1} \in \text{dom}(V^m)$$ and $E^k$ is the arc/edge that joins $V^l$ to $V^m$.
The preceding definition of a control walk requires a sequence of at least three functions. In order to complete this definition and accommodate certain special situations we define a trivial control walk as follows:
A control walk $\omega_c$ is said to be trivial if:
1. $\omega_c = \langle \psi_0 \rangle$ and $\psi_0 \in \text{dom}(V^i)$ for some $i \in \mathbb{N}$; or,
2. $\omega_c = \langle \psi_0, \psi_1 \rangle$ and $$\big(\psi_0 \in \text{dom}(V^i), \ \psi_1 \in \text{dom}(E^k) \big) \vee \big(\psi_0 \in \text{dom}(E^k), \ \psi_1 \in \text{dom}(V^i)\big)$$ for some $i, k$ in $\mathbb{N}_+$ and $E^k$ joins $V^i$ with itself or with some other vertex; or
3. $\omega_c = \langle \psi_0, \psi_1, \psi_2 \rangle$ and $$\psi_0 \in \text{dom}(E^k), \psi_1 \in \text{dom}(V^i), \psi_2 \in \text{dom}(E^m)$$ for some $i, k, m$ in $\mathbb{N}_+$, where $E^k$ and $E^m$ join $V^i$ with itself or with some other vertex.
\[rem:isomorp\] Let $G$ be a graph that is isomorphic to an $\mathcal{F}$-graph. Furthermore, let $G$ be such that its vertices and arcs are a selection of functions from the domains of the functionals the constitute the $\mathcal{F}$-graph. Then, by construction, there is an uncountably infinite set of isomorphic graphs $G$. A control walk may be interpreted as a walk in one of these isomorphic graphs.
It follows from Remarks \[rem:F-control\] and \[rem:isomorp\] that a control walk involves the simultaneous action of choosing an isomorphism and a walk in the chosen isomorphic graph, $G$.
Let $\omega_c$ be a control walk defined over an $\mathcal{F}$-graph. An objective functional is a functional, $\omega_c \mapsto {\mathbb R}$, defined over all possible control walks.
An optimal control walk is a control walk that optimizes an objective functional.
From Remark \[rem:F-control\], it follows that an optimal control walk jointly optimizes the selection of functions from the domains of the functionals that constitute an $\mathcal{F}$-graph as well as the walk itself. This feature of the optimal control walk directly addresses the comments in Remark \[rem:newTSP-objFun\] in the context of Fig. \[fig:3-City-Blobs\].
For the remainder of this paper, we will limit the discussions to a special type of an $\mathcal{F}$-graph defined over some measure space ${\mathbb{L}}$.
A measure space ${\mathbb{L}}$ is called a label space if ${\mathbb{L}}^i, i = 1, \ldots, |N_v| \in \mathbb{N}_+$ are a given finite collection of disjoint measurable subsets of ${\mathbb{L}}$.
Suppose we are given a label space. Let,
\[eq:dom4Tg\] $$\begin{aligned}
{\mathbb{L}}^a &:={\mathbb{L}}\setminus\displaystyle\bigcup_{i=1}^{|N_v|}{\mathbb{L}}^i \\
\text{dom}(V^i) &:= {\left\{{\mathbb R}\to {\mathbb{L}}^i:\ {\mathbb R}\to {\mathbb{L}}^i \text{ is measurable}\right\}} \label{eq:domK4Tg}\\
\text{dom}(E^a) &:= {\left\{{\mathbb R}\to {\mathbb{L}}^a:\ {\mathbb R}\to {\mathbb{L}}^a \text{ is measurable}\right\}}\label{eq:domJ4Tg}\end{aligned}$$
Let $E^{l,m}$ denote the arcs of an $\mathcal{F}$-graph with the property that $E^{l,m}$ joins $V^l$ to $V^m$ for all $l$ and $m$ in $N_v$. Set $E^{l,m} = E^a$, where the domain of $E^a$ is given by . Set the domains of $V^i$ according to . The resulting $\mathcal{F}$-graph is called a $\mathcal{T}$-graph.
A label space trajectory is a measurable function ${{\boldsymbol l}}(\cdot): {\mathbb R}\supseteq [t_0, t_f] \ni t \mapsto {\mathbb{L}}$ where $t_f - t_0 > 0$ is some non-zero time interval in ${\mathbb R}$.
\[prop:1\] A label space trajectory is an $\mathcal{F}$-control for the $\mathcal{T}$-graph.
Let ${{\boldsymbol l}}(\cdot): [t_0, t_f] \to {\mathbb{L}}$ be a label space trajectory. Because ${\mathbb{L}}^a$ and ${\mathbb{L}}^i$ are all disjoint sets, we have, $$\label{eq:l_is_iore}
{{\boldsymbol l}}(t) \in {\mathbb{L}}^1 \vee {\mathbb{L}}^2 \vee \ldots \vee{\mathbb{L}}^{|N_v|}\vee {\mathbb{L}}^a \text{ for a.a. } t \in [t_0, t_f]$$ As a result, we can perform the following construction: Let, $$t_0 < t_1, < \ldots, < t_{n} < t_{n+1}=t_f$$ be an increasing sequence of clock times such that for each $ j = 0, 1, \ldots, n,\ n \in \mathbb{N}$, $(t_{j+1} - t_j)$ is the maximum time duration for which we have, $$\label{eq:lseg_is_iore}
{{\boldsymbol l}}(t) \in \left\{
\begin{array}{ll}
{\mathbb{L}}^i \text{ for some } i \in N_v & \hbox{and for a.a. } t \in (t_j, t_{j+1}) \\
& \text{or}\\
{\mathbb{L}}^a & \hbox{for a.a. } t \in (t_j, t_{j+1})
\end{array}
\right.$$ The feasibility of constructing follows from . Let, $$\label{eq:psi-def-restrict}
\psi_j:= {{\boldsymbol l}}(\cdot)\big|_{(t_{j} , t_{j+1})}, \ j = 0, 1, \ldots, n$$ be the restrictions of ${{\boldsymbol l}}(\cdot)$ to $(t_{j} , t_{j+1})$ for $j = 0, 1, \ldots, n$. Then, by and , we have, $$\label{eq:Impsi}
\text{Im}(\psi_j) \subseteq {\mathbb{L}}^i \vee {\mathbb{L}}^a \text{ for some } i \in N_v \text{ and } \forall\ j = 0, 1, \ldots, n$$ Hence from , it follows that, $$\psi_j \in \text{dom}(V^i) \vee \text{dom}(E^a), \text{ for some } i \in N_v$$ Consequently the sequence $\langle \psi_0, \ldots, \psi_n \rangle$ is an $\mathcal{F}$-control for the $\mathcal{T}$-graph.
A label space trajectory is said to be trivial if $\textrm{Im}({{\boldsymbol l}}(\cdot)) \subseteq {\mathbb{L}}^a$.
A nontrivial, continuous label space trajectory generates a control walk in a $\mathcal{T}$-graph.
Let $\psi_j, j = 0, \ldots, n \in \mathbb{N}$ be the restrictions of ${{\boldsymbol l}}(\cdot)$ as defined by . By Proposition \[prop:1\], $\langle \psi_0, \psi_1, \ldots, \psi_n \rangle$ is an $\mathcal{F}$-control; hence, any given $\psi_j$ is either in the domain of $E^a$, or of $V^i$, for some $i \in N_v$. The rest of the proof is broken down to three cases:
**Case(a)**: $n \ge 2$ and $\psi_j \in \text{dom}(E^a)$ for some $0 < j < n$.\
From and we get the following conditions:
1. $\psi_{j-1} \in \text{dom}(V^l)$ for some $l \in N_v$
2. $\psi_{j+1} \in \text{dom}(V^m)$ for some $m \in N_v$
Hence, $\langle \psi_{j-1}, \psi_j, \psi_{j+1} \rangle$ is a control subwalk.
**Case(b)**: $n \ge 2$ and $\psi_j \in \text{dom}(V^i)$ for some $0 < j < n $ and some $i \in N_v$.\
By continuity of ${{\boldsymbol l}}(\cdot)$, we have, $$\label{eq:l-psi-continuity}
\lim_{t \uparrow t_{k+1} }\psi_k(t) = \lim_{t \downarrow t_{k+1} } \psi_{k+1}(t), \quad k = 0, \ldots, n$$ Setting $k = j-1$ and $k = j$ in we get the two continuity conditions,
$$\begin{aligned}
\lim_{t \uparrow t_{j} }\psi_{j-1}(t) &= \lim_{t \downarrow t_{j} } \psi_{j}(t)\\
\lim_{t \uparrow t_{j+1} }\psi_j(t) &= \lim_{t \downarrow t_{j+1} } \psi_{j+1}(t)\end{aligned}$$
Hence, from , , and , we have, $$\psi_{j-1} \in \text{dom}(E^a) \text{ and } \psi_{j+1} \in \text{dom}(E^a)$$ If $n=2$ ($\Rightarrow j=1$), we get a trivial control walk. If $ j >1$, by using the same arguments as in Case(a), we get $\psi_{j-2} \in \text{dom}(V^k) \text{ for some } l \in N_v \text{ and } \psi_{j+2} \in \text{dom}(V^m) \text{ for some } m \in N_v$; hence, $\langle \psi_{j-2}, \psi_{j-1}, \psi_j, \psi_{j+1}, \psi_{j+2} \rangle$ is a control subwalk.
**Case(c)**: $n<2$.\
By similar arguments as in Case (a) and (b), it is straightforward to show that $\langle \psi_0, \psi_1 \rangle$ is trivial control walk. From Cases (a), (b) and (c), it follows that $\langle \psi_0, \psi_1, \ldots, \psi_n \rangle, n \in \mathbb{N}$ is a control walk.
Two $\mathcal{T}$-Graph-Based Formulations of a TSP
===================================================
Let ${\mathbb{L}}$ be a finite dimensional normed space. If we set ${\mathbb{L}}^i$ as the “cities” in ${\mathbb{L}}$, then a TSP and its many variants may be framed in terms of finding nontrivial optimal label-space trajectories. To illustrate the theoretical simplicity of our approach, we allow ${\mathbb{L}}^i$ to be time dependent (i.e., in deterministic motion) so that some answers to the questions posed in Section 1 are readily apparent.
For a fixed time $t$, an atomic return function $R_a : ({{\boldsymbol l}}, {\mathbb{L}}^i(t)) \mapsto {\mathbb R}$ is defined by, $$\label{eq:atomic-ret-fun}
R_a({{\boldsymbol l}}, {\mathbb{L}}^i(t)) \left\{
\begin{array}{lll}
\neq 0 & \hbox{if} \ {{\boldsymbol l}}\in {\mathbb{L}}^i(t) \\
= 0 & \hbox{otherwise}
\end{array}
\right.$$
\[def:atomic-ret-func\] An atomic return functional $R^i$ is defined by, $$\label{eq:return-functional-def}
R^i[{{\boldsymbol l}}(\cdot), t_0, t_f] := \int_{t_0}^{t_f} R_a\big({{\boldsymbol l}}(t), {\mathbb{L}}^i(t)\big) dt$$ where $t_f - t_0 > 0$ is the time horizon of interest.
Atomic return functionals may be used as vertex functionals whenever ${{\boldsymbol l}}(t) \in {\mathbb{L}}^i(t)$. Whether or not a vertex functional is defined for a given problem, we define the Kronecker indicator function, $$\label{eq:kron-indicator}
\mathcal{I} ({{\boldsymbol l}}, {\mathbb{L}}^i(t)) := \left\{
\begin{array}{ll}
1 & \hbox{if } {{\boldsymbol l}}\in {\mathbb{L}}^i(t) \\
0 & \hbox{otherwise}
\end{array}
\right.$$ as a fundamental return function. Equation thus generates by way of a fundamental return functional:
A time-on-task functional is defined by, $$\label{eq:Ti-def}
T^i[{{\boldsymbol l}}(\cdot), t_0, t_f]:= \displaystyle \int_{t_0}^{t_f} \mathcal{I} ({{\boldsymbol l}}(t), {\mathbb{L}}^i(t))\, dt$$
Because generates a dwell time over vertex $i$, we define a visit in the context of a walk in the following manner:
The vertex $i$ is said to have been visited in $[t_0, t_f]$ if $$\label{eq:visit-def}
T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \ne 0$$ Correspondingly, we say the vertex $i$ has not been visited if $T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] = 0$.
In the context of a $\mathcal{T}$-graph formulation of a basic TSP,[^3] a vertex functional $V^i$ assigns a value of one for a visit, zero otherwise. Furthermore, an arc functional $E^a$ is any functional that assigns a numerical value (of “distance”) for segments of $t \mapsto {{\boldsymbol l}}$ that are not associated with a vertex. If a visit is preceded and followed by an arc functional with no other visits of a vertex in between (including itself), we say vertex $i$ has been visited once. In this context, we say ${{\boldsymbol l}}(\cdot)$ is Hamiltonian if all vertices are visited exactly once.
A Derivative-Based Formulation of a TSP
---------------------------------------
Let $D^i[{{\boldsymbol l}}(\cdot), t_0, t_f]$ be the functional defined by, $$D^i[{{\boldsymbol l}}(\cdot), t_0, t_f] := \int_{t_0}^{t_f}{\left\vertd_t\mathcal{I} \left({{\boldsymbol l}}(t), {\mathbb{L}}^i(t)\right)\right\vert}dt$$ where, $d_t$ denotes the distributional derivative with respect to $t$. Let ${{\boldsymbol l}}(\cdot): [t_0, t_f] \to {\mathbb{L}}$ be a continuous label space trajectory such that $T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \ne 0$. Furthermore, let ${{\boldsymbol l}}(t_0) \not\in {\mathbb{L}}^i(t_0)$ and ${{\boldsymbol l}}(t_f) \not\in {\mathbb{L}}^i(t_f)$. Then, $$\label{eq:Di=even}
D^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \in 2\mathbb{N}_+$$
Consider the time intervals $\Delta t_{out}$ and $\Delta t_{in}$ defined by, $$\Delta t_{out}:= {\left\{t \in [t_0, t_f]:\ \mathcal{I}({{\boldsymbol l}}(t), {\mathbb{L}}^i(t) = 0 \right\}} \quad \Delta t_{in}:= {\left\{t \in [t_0, t_f]:\ \mathcal{I}({{\boldsymbol l}}(t), {\mathbb{L}}^i(t) = 1 \right\}}$$ By assumption $T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \ne 0$; hence, we have $\mu\left(\Delta t_{in}\right) > 0 $. Likewise, we have $\mu\left(\Delta t_{out}\right) > 0 $. Hence, we can write, $$\frac{d\mathcal{I} \left({{\boldsymbol l}}(t), {\mathbb{L}}^i(t)\right)}{dt} = 0 \quad\forall\ t \in {\left\{\text{int}\left(\Delta t_{out}\right)\right\}} \cup {\left\{\text{int}\left(\Delta t_{in}\right)\right\}}$$ where, int$(\cdot)$ denotes the interior of the set $(\cdot)$. Let $ t_x \in \partial\Delta t_{int} $ where $\partial(\cdot)$ denotes the boundary of the set $(\cdot)$. Then, the distributional derivative of the function $t \mapsto \mathcal{I} \left({{\boldsymbol l}}(t), {\mathbb{L}}^i(t)\right)$ may be written as, $$\label{eq:diff-I}
d_t\mathcal{I} \left({{\boldsymbol l}}(t), {\mathbb{L}}^i(t)\right)= -\delta(t-t_x) \vee \delta(t-t_x)$$ where, $\delta(t-t_x)$ is the Dirac delta function centered at $t=t_x$. Hence, $$\label{eq:diff-I-abs=delta}
{\left\vertd_t\mathcal{I} \left({{\boldsymbol l}}(t), {\mathbb{L}}^i(t)\right)\right\vert} = \delta(t-t_x)$$ Because ${{\boldsymbol l}}(t_0)$ and ${{\boldsymbol l}}(t_f)$ are not in ${\mathbb{L}}^i(t_0)$ and ${\mathbb{L}}^i(t_f)$ respectively, integrating we get, $$\int_{t_0}^{t_f}{\left\vertd_t\mathcal{I} \left({{\boldsymbol l}}(t), {\mathbb{L}}^i(t)\right)\right\vert}dt \in 2\mathbb{N}_+$$ where, we have used the fact that the integral of a delta function is unity.
If ${{\boldsymbol l}}(t_0) \in {\mathbb{L}}^i(t_0)$ and ${{\boldsymbol l}}(t_f) \in {\mathbb{L}}^i(t_f)$, then generalizes to $D^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \in 2\mathbb{N}$. If ${{\boldsymbol l}}(t_0) \in {\mathbb{L}}^i(t_0)$ or ${{\boldsymbol l}}(t_f) \in {\mathbb{L}}^i(t_f)$, then the statement of Lemma 1 may be further generalized to $D^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \in \mathbb{N}$.
Let ${{\boldsymbol l}}(\cdot): [t_0, t_f] \to {\mathbb{L}}$ be a continuous label space trajectory. Then, ${{\boldsymbol l}}(\cdot)$ is Hamiltonian if and only if $$\label{eq:Di=2}
D^i[{{\boldsymbol l}}(\cdot), t_0, t_f] = 2 \quad \forall\ i \in N_v$$
If ${{\boldsymbol l}}(\cdot)$ is Hamiltonian, then every vertex has been visited just once; hence, follows from Lemma 1. If holds, then every vertex has been visited just once in $[t_0, t_f]$ (with the understanding of a visit given by ). Although continuity of a label space trajectory is sufficiently smooth to generate a Hamiltonian cycle (Cf. Theorem 1), we now consider the space of absolutely continuous functions for the convenience of using the derivative of $t \mapsto {{\boldsymbol l}}$ to define arc lengths. To this end, let $\dot{{\boldsymbol l}}(t)$ denote the time derivative of ${{\boldsymbol l}}(t)$; then, a distance functional may be defined according to, $$\label{eq:J=dist}
J_{dist}[{{\boldsymbol l}}(\cdot), t_0, t_f] := \displaystyle \int_{t_0}^{t_f} {\left\Vert\dot{{\boldsymbol l}}(t)\right\Vert} dt$$ The integrand in is any finite-dimensional norm. If the two-norm is used, then the numerical value of $J_{dist}$ is consistent with the notion of Euclidean distance illustrated in Fig. \[fig:3-City-Blobs\]. Combining with Theorem 2, a shortest distance TSP can be formulated as, $$\begin{aligned}
&\big(\text{$D$-TSP}\big) \left\{
\begin{array}{lll}
\text{Minimize} & J_{dist}[{{\boldsymbol l}}(\cdot), t_0, t_f] := \displaystyle \int_{t_0}^{t_f} {\left\Vert\dot{{\boldsymbol l}}(t)\right\Vert} dt \\[1.1em]
\text{Subject to }
& \displaystyle D^i[{{\boldsymbol l}}(\cdot), t_0, t_f] = 2 \quad \forall\ i \in N_v\\[0.7em]
& {{\boldsymbol l}}(t_f) = {{\boldsymbol l}}(t_0)
\end{array} \right.& \label{eq:prob-TSP-classic}\end{aligned}$$ The last constraint equation in simply ensures that the resulting label-space trajectory is a closed control-walk.
In comparing it with the various discrete variable optimization formulations of a TSP [@Dantzig-54; @Flood; @MTZ], it is apparent that contains no explicit subtour-type elimination constraints. This is because the (absolute) continuity of the label space trajectory ensures (by Theorem 1) that the closed control-walk generated by is a single Hamilton cycle.
An Integral-Based Formulation of a TSP
--------------------------------------
A TSP may also be formulated (in the sense of a $\mathcal{T}$-graph) using the time-on-task functional to construct a new indicator-type functional.
A control-walk indicator functional is defined by, $$\label{eq:Wi-def}
W^i[{{\boldsymbol l}}(\cdot), t_0, t_f] := \left\{
\begin{array}{lll}
1 & \hbox{if } \ T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \ne 0 \\[1.1em]
0 & \hbox{if } \ T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] = 0
\end{array}
\right.$$
Using $W^i$ as an indicator of a vertex visit, we arrive at an alternative formulation of a TSP: $$\begin{aligned}
&\big(\text{$I$-TSP}\big) \left\{
\begin{array}{lll}
\text{Minimize} & J_{dist}[{{\boldsymbol l}}(\cdot), t_0, t_f] := \displaystyle \int_{t_0}^{t_f} {\left\Vert\dot{{\boldsymbol l}}(t)\right\Vert} dt \\[1.1em]
\text{Subject to }
& \displaystyle W^i[{{\boldsymbol l}}(\cdot), t_0, t_f] = 1 \quad \forall\ i \in N_v\\[0.7em]
& {{\boldsymbol l}}(t_f) = {{\boldsymbol l}}(t_0)
\end{array} \right.& \label{eq:prob-TSP-classic-I}\end{aligned}$$ Equations and are identical except for the imposition of degree constraints. Formulation ($I$-TSP) requires that the vertex be visited at least once.
No Entity Discrete Optimization $\mathcal{T}$-Graph Formulations
---- ---------------------------------------------- ----------------------- ----------------------------------
1 optimization variable discrete continuous function
2 cost matrix; i.e., explicit arc/edge weights required not required
3 explicit degree constraints required required in ; not in
4 explicit subtour elimination constraints required not required
: A discriminating comparison between a discrete-variable and $\mathcal{T}$-graph formulations of a TSP
\
A comparison between the $\mathcal{T}$-graph and discrete-variable problem formulations is summarized in Table 1. As noted in the second row of Table 1, the $\mathcal{T}$-graph formulations do not require the explicit construction of the arc/edge weights; i.e., the cost matrix for the computation of the objective function in the discrete-variable formulation. This data is “generated” simultaneously via as part of the process of solving the problem. That is, in terms of the concepts introduced in Section 2, the $\mathcal{T}$-graph formulation incorporates the simultaneous selection of the function sequence and the functions themselves from the domains of the vertex and arc functionals.
The $\mathcal{T}$-graph formulations presented in this section are not transformations of the well-established discrete-optimization models of TSPs; rather, they are a realization of a fundamentally new domain of analysis.
Sample $\mathcal{T}$-Graph Formulations of Several Variants of a TSP
====================================================================
Because of the large number of variants of a TSP, we limit the scope of this section to a small sample of selective problems to illustrate the new modeling framework.
An Orienteering Problem
-----------------------
Let $\sigma^i > 0$ be the score associated with each ${\mathbb{L}}^i(t)$. Define a score functional according to: $$S^i[{{\boldsymbol l}}(\cdot), t_0, t_f] := \sigma^i W^i[{{\boldsymbol l}}(\cdot), t_0, t_f]$$ where, $W^i$ is given by . An orienteering problem (OP) may now be defined as, $$\begin{aligned}
&\big(\text{OP}\big) \left\{
\begin{array}{lll}
\text{Maximize} & \displaystyle \sum_{i \in N_v} S^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \\[1.1em]
\text{Subject to}
& \displaystyle \int_{t_0}^{t_f} 1\, dt \le t_{max} \\[0.7em]
& {{\boldsymbol l}}(t_0) \in {\mathbb{L}}^0 \\
& {{\boldsymbol l}}(t_f) \in {\mathbb{L}}^{|N_v|}
\end{array} \right.& \label{eq:prob-OP-classic}\end{aligned}$$ The payoff functional is the sum of the score functionals. The maximum allowable time is $t_{max} > 0$. The time constraint in is written in terms of the integral of one to merely illustrate the fact that the resource constraint is a functional. The domain of ${{\boldsymbol l}}(\cdot)$ in is the space of continuous functions in accordance with Theorem 1.
An Orienteering Problem With Neighborhoods
------------------------------------------
Problem (OP) given by was motivated by the usual orienteering problem discussed in the literature [@OP-survey]. Using the atomic return function concept introduced in , we may now score a traversal through a neighborhood based on the values of the atomic return functional (Cf. ). Although there are many ways to score a traversal, we illustrate a problem formulation based on the following construction, $$\label{eq:S=nbhd-step}
S^i_{nbd}[{{\boldsymbol l}}(\cdot), t_0, t_f] := \left\{
\begin{array}{lll}
\sigma^i & \hbox{if} \ R^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \ge r^i\\
0 & \hbox{if} \ R^i[{{\boldsymbol l}}(\cdot), t_0, t_f] < r^i
\end{array}
\right.$$ In , $r^i > 0$ is a required “revenue” from a visit to ${\mathbb{L}}^i(t)$. If a label-space traversal $t \mapsto {{\boldsymbol l}}$ across the city does not generate a revenue of at least $r^i$ as measured by the return functional $R^i$, then the salesman gets no commission (i.e., zero score). On the other hand, if the salesman performs a judicious travel through the city to generate a revenue of at least $r^i$, then, he is rewarded by $\sigma^i > 0$. The salesman makes no extra commission for generating a revenue greater than $r^i$ at ${\mathbb{L}}^i$; i.e., he is encouraged to expand the market by visiting a different city. This situation arises in astronomy [@cook-2012] where a telescope is required to scan a portion of the sky to collect a specific frequency of electromagnetic (EM) radiation [@TSP-telescope]. Scientific value is generated based on the integration of EM collects. A critical amount of EM collects ($r^i$) is necessary to perform useful science. No extra science is generated once the targeted amount $r^i$ is reached; hence, the telescope is awarded $\sigma^i$ for performing a task successfully with no “extra” credit. Thus, the orienteering problem with neighborhoods may be defined according to, $$\begin{aligned}
&\big(\text{OP-nbd}\big) \left\{
\begin{array}{lll}
\text{Maximize} & \displaystyle \sum_{i \in N_v} S^i_{nbd}[{{\boldsymbol l}}(\cdot), t_0, t_f] \\[1.1em]
\text{Subject to }
& \displaystyle C^k[{{\boldsymbol l}}(\cdot), t_0,t_f] \le C^k_{max} \quad \forall\ k \in N_C \\[0.7em]
& {{\boldsymbol l}}(t_0) \in {\mathbb{L}}^0 \\
& {{\boldsymbol l}}(t_f) \in {\mathbb{L}}^{|N_v|}
\end{array} \right.& \label{eq:prob-OP-nbhd}\end{aligned}$$ In , the constraints $C^k[{{\boldsymbol l}}(\cdot), t_0, t_f] \le C^k_{max},\ \forall\ k \in N_C$ are $N_C \subset \mathbb{N}$ generic resource constraint, where $C^k_{max} > 0$ is the maximum “capacity” associated with the $k$-th resource.
A Fast TSP
----------
Minimizing distance traveled does not necessarily equate to minimum time; this fact has been known since Bernoulli posed his famous Brachistochrone problem as a mathematical challenge in the year 1696 [@ross-book; @vinter]. In framing a more interesting minimum-time TSP, we let $\tau^i > 0$ be an additional constraint of a minimum required dwell-time over ${\mathbb{L}}^i(t)$. In this case, a fast (i.e., minimum-time) TSP may be framed as follows: $$\begin{aligned}
&\big(\text{fastTSP}\big) \left\{
\begin{array}{lll}
\text{Minimize} &J_{time}[{{\boldsymbol l}}(\cdot), t_0, t_f]:= \displaystyle \int_{t_0}^{t_f} 1\, dt \\[1.1em]
\text{Subject to}
& T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \ge \tau^i \qquad \forall\ i \in N_v\\[0.7em]
& {{\boldsymbol l}}(t_f) = {{\boldsymbol l}}(t_0)
\end{array} \right.& \label{eq:prob-TSP-minTime}\end{aligned}$$ For the same reason as the formulation of the constraint in , the cost function in is written as an integral to emphasize the fact that the travel-time is a functional. The functional $T^i$ in is given by . Furthermore, while the dwell-time constraint in may, in principle, be written as an equality, $ T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] = \tau^i $, the advantage of an inequality is that it allows the solution to satisfy the stipulated requirement without incurring a penalty in performance for “navigating through the city” to find an optimal exit point at an optimal exit time.
Dynamic TSP with Time Windows
-----------------------------
Because ${\mathbb{L}}^i(t), i \in N_v$ are “moving cities,” dynamic TSPs are are explicitly incorporated in all of the preceding problem formulations. These formulations have also implicitly incorporated time windows because of the following argument: Let $t_a^i \in [t_0, t_f]$ and $t_b^i \in [t_0, t_f]$ be two given clock-times associated with each ${\mathbb{L}}^i(t)$ with $t_a^i < t_b^i$. Consider the augmented set defined by, $$\label{eq:Lspace-i-window}
{\mathbb{L}}^i_{aug}(t):= {\mathbb{L}}^i(t) \times [t_a^i, t_b^i]$$ Then, it is clear that the space defined by, $${\mathbb{L}}_{aug}(t):= {\mathbb{L}}(t) \times [t_0, t_f]$$ is a label space, whose disjoint sets are defined by . Hence, by defining a new label-space variable ${{\boldsymbol l}}_{aug} := ({{\boldsymbol l}}, t) $, it is apparent that no additional theoretical developments are necessary to incorporate time-window constraints in the proposed $\mathcal{T}$-graph formulations. Furthermore, note that the time window in may itself also vary with respect to time.
In view of economic considerations that go beyond distance and time, a significantly greater degree of flexibility in modeling can be obtained by transforming the preceding $\mathcal{T}$-graph “variational” formulations to their optimal-control versions.
A $\mathcal{T}$-Graph Optimal Control Framework
===============================================
By limiting the space of allowable label space trajectories to the space of absolutely continuous functions, it is possible to frame a significantly richer class of traveling-salesman-type problems. To develop this transformed framework, we first set the derivative of ${{\boldsymbol l}}(t)$ as a candidate optimization variable, $$\label{eq:ldot=w}
\dot{{\boldsymbol l}}(t) = {{\boldsymbol w}}(t)$$ where, ${{\boldsymbol w}}\in {{\mathbb R}^{N_l}}, N_l \in \mathbb{N}$ is the label-space tangent control variable. Thus, for example, transforms according to, $$\begin{aligned}
&\big(\text{fastTSP: OC-ver}\big) \left\{
\begin{array}{lll}
\displaystyle\mathop\text{Minimize } & \displaystyle J_{time}[{{\boldsymbol l}}(\cdot), {{\boldsymbol w}}(\cdot), t_0, t_f] := t_f - t_0 \\[1.1em]
\text{Subject to }
& \dot{{\boldsymbol l}}(t) = {{\boldsymbol w}}(t)\\[0.7em]
& T^i[{{\boldsymbol l}}(\cdot), t_0, t_f] \ge \tau^i \qquad \forall\ i \in N_v\\[0.7em]
& {{\boldsymbol l}}(t_f) = {{\boldsymbol l}}(t_0)
\end{array} \right.& \label{eq:prob-TSP-minTime-ocp-basic}\end{aligned}$$ In comparing it with , note that contains an additional decision variable ${{\boldsymbol w}}(\cdot)$ and an additional constraint given by . This aspect of the transformation may be used advantageously; for example, if the objective functional in is replaced by, $$J_{dist}[{{\boldsymbol l}}(\cdot), {{\boldsymbol w}}(\cdot), t_0, t_f] := \displaystyle \int_{t_0}^{t_f} {\left\Vert{{\boldsymbol w}}(t)\right\Vert}dt$$ then, the resulting problem transforms to yet another formulation of a minimum-distance TSP (Cf. Section 3). In fact, the most important aspect of is that it provides a clear avenue for generalization that is conducive to modeling traveling-salesman-type problems driven by constrained, nonlinear dynamical systems such as those illustrated in Fig. \[fig:Grand-tour\].
Generalization Based on Nonlinear Dynamics
------------------------------------------
In many applications such as robotics [@TSP-Dubins], the natural home space for the “salesman” is some state space ${\mathbb{X}}$ that is not necessarily the label space. Furthermore, the dynamics of the salesman is given by some nonlinear controllable ordinary differential equation of the type, $$\label{eq:xdot=f}
\dot{\boldsymbol x}= {{\boldsymbol f}}({\boldsymbol x}, {{\boldsymbol u}}, t)$$ where, ${\boldsymbol x}\in {{\mathbb R}^{N_x}}$ is a state variable, ${{\boldsymbol u}}\in {{\mathbb R}^{N_u}}$ is the control variable and ${{\boldsymbol f}}: {{\mathbb R}^{N_x}} \times {{\mathbb R}^{N_u}} \times {\mathbb R}\to {{\mathbb R}^{N_x}}$ is the nonlinear dynamics function. Because all aspects of the problem definition so far are described in terms of a label space, it is necessary to connect it to the state space. Let this connection be given by some algebraic equation, $$\label{eq:g=0-nou}
{{\boldsymbol g}}({{\boldsymbol l}}, {\boldsymbol x}, t) = {{\bf 0}}$$ where, ${{\boldsymbol g}}: {{\mathbb R}^{N_l}} \times {{\mathbb R}^{N_x}} \times {\mathbb R}\to {{\mathbb R}^{N_g}} $ is called a connexion function. An example of a physical description of state space, label space and the connexion function is illustrated in Fig. \[fig:UAV\].
In this example, an uninhabited aerial vehicle (UAV) is tasked to collect over a geographic region [@TSP-UAV-shima; @TSP-UAV-ross]. The areas of interest vary from a point area to a broad area of scan. Technical properties associated with the areas of interest are defined in label space. The state variables of the UAV is given in terms of of its position, velocity and the orientation of the maneuverable camera. The connexion function is the mathematical model that connects the label space variables to the state space variables based on the precise position and orientation of the camera at a given instant of time.
From and , it follows that may be written implicitly as a differential algebraic equation, $$\label{eq:ldot-via-xdot}
{\left.
\begin{array}{l}
\dot{\boldsymbol x}= {{\boldsymbol f}}({\boldsymbol x}, {{\boldsymbol u}}, t) \\
{{\bf 0}}= {{\boldsymbol g}}({{\boldsymbol l}}, {\boldsymbol x}, t)
\end{array}
\right\} } \quad \Longrightarrow \quad \dot{{\boldsymbol l}}= {{\boldsymbol w}}$$ The significance of replacing by the state-space representation is that it facilitates a direct means to incorporate the full nonlinear dynamics of a salesman in the optimization problem formulation.
Generalization Based on Economics-Driven Cost/Payoff/Constraint Functionals
---------------------------------------------------------------------------
As a result of , the decision variables expand to the tuple, $({{\boldsymbol l}}(\cdot), {\boldsymbol x}(\cdot), {{\boldsymbol u}}(\cdot), t_0, t_f)$. This can be taken to great advantage by formulating cost, payoff and/or constraint functionals in their more natural “home spaces.” For example, in the electric vehicle-routing problem [@TSP-Electric], the capacity constraint of the battery is more naturally expressed as a functional constraint in terms of the state ${\boldsymbol x}$ of the vehicle [@EV-battery], $$\label{eq:batt-state}
\int_{t_0}^{t_f}S({\boldsymbol x}(t))\,dt \ge C_{safe}$$ In , $S$ is the state-of-charge and $C_{safe}$ is a battery charge required for safe operations. In the absence of extending the decision variables to incorporate the state vector, would need to be transformed to a label space constraint using . This difficult task is completely circumvented by incorporating the state vector as part of the optimization variable. The alternative is to generate proxy models [@TSP-Electric] as a means to extend the scope of vehicle routing problems [@vidal].
In , the travel time is an implicit functional of the label-space trajectory ${{\boldsymbol l}}(\cdot)$. Replacing by , the travel time now becomes an implicit functional of ${{\boldsymbol l}}(\cdot), {\boldsymbol x}(\cdot)$ and ${{\boldsymbol u}}(\cdot)$. That is, the functional, $$\label{eq:J=travel-time}
J_{time}[{{\boldsymbol l}}(\cdot), {\boldsymbol x}(\cdot), {{\boldsymbol u}}(\cdot), t_0, t_f]:= t_f - t_0$$ with constraints given by offers a more accurate model of travel time with obvious business implications in practical routing problems. Note that also allows the clock time to be optimized. Equation may be further modified to take into account gas/power consumption depending upon the vehicle type (standard/hybrid/electric). Analogous to , gas/power consumption may be written in terms of an integral, $$\label{eq:burn-rate-integ}
\displaystyle \int_{t_0}^{t_f} f_0\big({\boldsymbol x}(t), {{\boldsymbol u}}(t), t\big)\, dt$$ where, $f_0$ is a function that models the time-varying gas/power consumption rate. This function may be well-modeled using the physics of the automobile power train [@EV-battery]. A convex combination of with generates the functional, $$\begin{gathered}
\label{eq:J=hybrid}
J_{hybrid}[{{\boldsymbol l}}(\cdot), {\boldsymbol x}(\cdot), {{\boldsymbol u}}(\cdot), t_0, t_f]:= \alpha (t_f - t_0) \\
+ (1-\alpha) \displaystyle \int_{t_0}^{t_f} f_0\big({\boldsymbol x}(t), {{\boldsymbol u}}(t), t\big)\, dt \qquad \alpha \in [0, 1]\end{gathered}$$ The parameter $\alpha$ in offers a sliding scale over the “trade-space” of time and energy consumption.
An Optimal Control Framework for a TSP and its Variants
--------------------------------------------------------
Substituting in while simultaneously adding additional levels of abstraction for the functionals associated with the vertices and arcs of the underlying $\mathcal{T}$-graph, we arrive at: $$\begin{aligned}
&\big(\mathcal{T_X}P\big) \left\{
\begin{array}{lll}
\displaystyle\mathop\text{Minimize} & \displaystyle J[{{\boldsymbol l}}(\cdot), {\boldsymbol x}(\cdot), {{\boldsymbol u}}(\cdot), t_0, t_f] \\
\text{Subject to }
& \dot{\boldsymbol x}(t) = {{\boldsymbol f}}({\boldsymbol x}(t), {{\boldsymbol u}}(t), t)\\
& {{\boldsymbol g}}({{\boldsymbol l}}(t), {\boldsymbol x}(t), t) = {{\bf 0}}\\
& K^m\left[{{\boldsymbol l}}(\cdot), {\boldsymbol x}(\cdot), {{\boldsymbol u}}(\cdot), t_0, t_f\right] \left\{
\begin{array}{ll}
\le 0, & \forall\ m \in N_{\ge 0} \\
= 0, & \forall\ m \in N_{=0}
\end{array}
\right. \\[0.1em]
& \big({{\boldsymbol l}}(t_0), {{\boldsymbol l}}(t_f) \big) \in {\mathbb{L}}^b \subseteq {\mathbb{L}}\\
& {\boldsymbol x}(t) \in \mathbb{X}(t)\\
& {{\boldsymbol u}}(t) \in \mathbb{U}(t, {\boldsymbol x}(t))
\end{array} \right.& \label{eq:prob-TXP}\end{aligned}$$ In , the objective function is simply stated in terms of an abstract functional $J$ in order to facilitate the formulation of a generic cost functional that go beyond those discussed in Sec. 5.2. Likewise, the functionals $K^m$ in may be time-on-task functionals, walk-indicator functionals, degree functionals, capacity-constraint functionals or some other functionals. Furthermore, because each vertex may have several constraints, the total number of these constraints may be greater than ${\left\vertN_v\right\vert}$. The index sets $N_{=0}$ and $N_{\ge 0}$ simply organize the constraint functionals into equalities and inequalities. The set ${\mathbb{L}}^b \subseteq {\mathbb{L}}$ stipulates the boundary conditions for the label space trajectory. Also included in are state variable constraints ${\boldsymbol x}(t) \in {\mathbb{X}}(t)$ and state-dependent control constraints given by ${{\boldsymbol u}}(t) \in \mathbb{U}(t, {\boldsymbol x}(t))$. Such constraints are included in Problem $(\mathcal{T_X}P) $ because they are critically important in practical applications[@ross-book; @vinter] as well as coordinate transformations of optimal control problems[@ross-book].
Because is a generalization of the problems discussed in Sections 3–5, it is evident that a fairly large class of traveling-salesman-like problems can be modeled as instances of Problem $(\mathcal{T_X}P)$.
Illustrative Numerical Examples
===============================
A motorized TSP was first introduced in [@TSP-motorized]. In essence, a motor is a differential equation; hence, a motorized TSP is one with dynamical constraints. Here, we combine this problem with several other variants of the TSP [@CETSP; @TSPFN] to generate a minimum-time close-enough motorized traveling salesman problem with forbidden neighborhoods given by: $$\begin{aligned}
& {{\boldsymbol l}}\in {{\mathbb R}^{4}}, \quad {\boldsymbol x}\in {{\mathbb R}^{4}}, \quad {{\boldsymbol u}}\in {{\mathbb R}^{2}} \nonumber\\
&\big(\textit{fastCEMTSPFN-1}\big) \left\{
\begin{array}{lll}
\displaystyle\mathop\text{Minimize} & \displaystyle J[{{\boldsymbol l}}(\cdot), {\boldsymbol x}(\cdot), {{\boldsymbol u}}(\cdot), t_0, t_f]:= t_f-t_0 \\ \text{Subject to }
& \dot x_1(t) = x_3(t), \quad {\left\vertx_3(t)\right\vert} \le 1 \quad \forall\ t \in [t_0, t_f] \\
& \dot x_2(t) = x_4(t), \quad {\left\vertx_4(t)\right\vert} \le 1 \quad \forall\ t \in [t_0, t_f] \\
& \dot x_3(t) = u_1(t), \quad {\left\vertu_1(t)\right\vert} \le 1 \quad \forall\ t \in [t_0, t_f] \\
& \dot x_4(t) = u_2(t), \quad {\left\vertu_2(t)\right\vert} \le 1 \quad \forall\ t \in [t_0, t_f] \\
& l_i(t)-x_i(t) = 0, \quad \forall\ t \in [t_0, t_f]\\
& \qquad i = 1, 2, 3, 4 \\
& T^i\left[{{\boldsymbol l}}(\cdot), t_0, t_f\right] \ge \tau^i_{min}, \quad \forall\ i = 1, \ldots, N_v \\[0.5em]
& \displaystyle \left( \frac{x_1(t) - x_1^i}{a^i} \right)^2 + \left( \frac{x_2(t) - x_2^i}{b^i} \right)^2 \ge 1,\\
& \qquad \forall\ i = 1, \ldots N_{obs} \\
& \big({{\boldsymbol l}}(t_0), {{\boldsymbol l}}(t_f) \big) = ({{\bf 0}}, {{\bf 0}})
\end{array} \right.& \label{eq:prob-EX-1}\end{aligned}$$ It is apparent that is a special case of . In fact, it is a generalization of the fast TSP posed in . Equation is motorized by the four differential equations resulting from an elementary application of Newtonian dynamics: The position and velocity of the salesman are $(x_1, x_2) \in {{\mathbb R}^{2}}$ and $(x_3, x_4) \in {{\mathbb R}^{2}}$ respectively. Both the velocity and acceleration $(u_1, u_2) \in {{\mathbb R}^{2}}$ vectors are constrained in the $\ell_\infty$ norm (i.e., components are constrained in terms of their absolute values). The $N_{obs}$ ellipsoidal constraints in are the forbidden neighborhoods whose centroids are given by $(x_1^i, x_2^i), \ i = 1, \ldots, N_{obs}$. The parameters $a^i > 0$ and $b^i > 0$ determine the shape of the ellipse. The last constraint in requires the salesman to start and end at the origin.
A sample data set for with $N_v = 11$ and $N_{obs} = 2$ is shown in Fig. \[fig:TSPResult11BoxFN\].
Also shown in Fig. \[fig:TSPResult11BoxFN\] is the solution obtained by solving using DIDO’s implementation[@ross-dido] of pseudospectral optimal control theory [@PSOC]. Some key points to note regarding the solution presented in Fig. \[fig:TSPResult11BoxFN\] are:
1. Nearly all the “arcs” between city-pairs are curvilinear. This is because the salesman’s trajectory is required to satisfy the Newtonian dynamical constraints as well as the instantaneous $\ell_\infty$ bounds on velocity and acceleration as dictated in .
2. The city-pair arcs as well as the entry and exit points to the city are a natural outcome of solving the posed problem (Cf. ). That is, the city-pair arcs were not determined either *a priori* or *a posteriori* to the determination of the city sequence. See also Remark \[rem:newTSP-objFun\].
3. The ellipsoidal forbidden neighborhood marked FN2 overlaps neighborhood No. 9. Thus, the allowable region for neighborhood No. 9 is nonconvex.
Next, consider a “variant” of obtained by replacing the $\ell_\infty$ constraints on velocity and acceleration by its $\ell_2$ version, $$\label{eq:prob-EX-2}
(\textit{fastCEMTSPFN-2}): \quad \sqrt{x_3^2(t) + x_4^2(t)} \le 1, \ \sqrt{u_1^2(t) + u_2^2(t)} \le 1 \qquad \forall\ t \in [t_0, t_f]$$ That is, Problem *(fastCEMTSPFN-2)* is identical to Problem *(fastCEMTSPFN-1)* except for the additional nonlinear constraint given by . A solution to this problem is shown in Fig. \[fig:TSPResult11DiskFN\].
It is apparent that Fig. \[fig:TSPResult11DiskFN\] differs from Fig. \[fig:TSPResult11BoxFN\] in several ways:
1. The turn at neighborhood No. 6 in Fig. \[fig:TSPResult11BoxFN\] is significantly sharper than the corresponding one in Fig. \[fig:TSPResult11DiskFN\]. The curvilinear turn in Fig. \[fig:TSPResult11DiskFN\] is a clear and direct result of the $\ell_2$ constraint given by .
2. The sequence of visits in Fig. \[fig:TSPResult11DiskFN\] is different from that of Fig. \[fig:TSPResult11BoxFN\]; for example, compare the visit to neighborhood No. 8.
3. The entry and exit points to the same numbered cities in Figs. \[fig:TSPResult11BoxFN\] and \[fig:TSPResult11DiskFN\] are different; for example, compare neighborhoods Nos. 8, 11 and 4.
For the third and final case, we change the problem “data set” with a random distribution of a large number of forbidden zones as shown in Fig. \[fig:TSPFNgaloreNo1\]. Only the neighborhood marked “FNB” was not randomly selected; rather, its size and location were purposefully set as indicated in Fig. \[fig:TSPFNgaloreNo1\] to generate a more interesting case study.
Also shown in Fig. \[fig:TSPFNgaloreNo1\] is a solution to the problem whose vehicle is constrained by . Beyond the apparent efficacy of the process in obtaining a solution, noteworthy points pertaining to Fig. \[fig:TSPFNgaloreNo1\] are:
1. Some of the forbidden neighborhoods are nonconvex. The nonconvex sets are generated randomly in the sense that the circular neighborhoods are allowed to overlap.
2. Some of the allowable neighborhoods are nonconvex because the forbidden circular disks overlap the allowable regions.
3. Comparing Figs. \[fig:TSPFNgaloreNo1\] and \[fig:TSPResult11DiskFN\], it is clear that the addition of forbidden neighborhoods has drastically altered the sequence of visits.
Conclusions
===========
Many types of objective functions and constraints in emerging variations of the traveling salesman problem (TSP) are more naturally defined in their continuous-time home space. Modeling these variants of the TSP using a classic discrete optimization framework is neither straightforward nor easy. By inverting the traditional modeling process, that is, by formulating the naturally discrete quantities in terms of continuous-time nonsmooth functions, it is possible to generate a new framework for the TSP and some of its variants. In this context, the discrete-optimization formalism of a TSP may be viewed as a problem formulation in the co-domain of the functionals that constitute its $\mathcal{T}$-graph. In sharp contrast, the problem formulation presented in this paper is in the domain of the functionals of the TSP $\mathcal{T}$-graph. This perspective suggests that the discrete-optimization- and the variational formulations are effectively two sides of the same $\mathcal{T}$-graph constructs introduced in this paper.
There is no doubt that there are a vast number of open theoretical issues in the proposed framework. Nonetheless, we have demonstrated, by way of a fast, close-enough motorized TSP with forbidden neighborhoods, that it is indeed possible to solve some challenging problems using the new formalisms. One of the interesting revelations indicated by the numerical studies is the big impact of seemingly small changes in the motion-constraints of the traveling salesman. This suggests that an exploitation of nonlinear models may provide a discriminating edge to a business/engineering operation by way of a non-intuitive economical utilization of its end-to-end systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
Funding from the Defense Advanced Research Projects Agency, the Center for Multi-INT Studies, and the Department of Defense (DoD) are gratefully acknowledged. The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the DoD or the U.S. Government.
Strong support from the Secretary of the Navy and United States Navy’s Judge Advocate General’s Office is gratefully acknowledged. Portions of this work are protected by the following patents issued by the United States Patent and Trademark Office: Patent Nos. US 9,983,585 B1; US 10,476,584 B1; and 62/971,068 (patent pending).
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[^1]: Time is just a proxy for an independent variable.
[^2]: To produce the TSP data, it is necessary to solve $\mathcal{O}(N^2)$ optimal control problems, where $N$ is the number of moons.
[^3]: By a basic TSP, we mean a problem that does not come with any additional qualifiers.
|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'Let ${\mathscr{Q}}(m,q)$ and ${\mathscr{S}}(m,q)$ be the sets of quadratic forms and symmetric bilinear forms on an $m$-dimensional vector space over ${\mathbb{F}}_q$, respectively. The orbits of ${\mathscr{Q}}(m,q)$ and ${\mathscr{S}}(m,q)$ under a natural group action induce two translation association schemes, which are known to be dual to each other. We give explicit expressions for the eigenvalues of these association schemes in terms of linear combinations of generalised Krawtchouk polynomials, generalising earlier results for odd $q$ to the more difficult case when $q$ is even. We then study $d$-codes in these schemes, namely subsets $X$ of ${\mathscr{Q}}(m,q)$ or ${\mathscr{S}}(m,q)$ with the property that, for all distinct $A,B\in X$, the rank of $A-B$ is at least $d$. We prove tight bounds on the size of $d$-codes and show that, when these bounds hold with equality, the inner distributions of the subsets are often uniquely determined by their parameters. We also discuss connections to classical error-correcting codes and show how the Hamming distance distribution of large classes of codes over ${\mathbb{F}}_q$ can be determined from the results of this paper.'
address: 'Department of Mathematics, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany'
author:
- 'Kai-Uwe Schmidt'
date: 11 March 2018
title: Quadratic and symmetric bilinear forms over finite fields and their association schemes
---
Introduction
============
Let $q$ be a prime power and let $V=V(m,q)$ be an $m$-dimensional ${\mathbb{F}}_q$-vector space. Let ${\mathscr{Q}}={\mathscr{Q}}(m,q)$ be the space of quadratic forms on $V$ and let ${\mathscr{S}}={\mathscr{S}}(m,q)$ be the space of symmetric bilinear forms on $V$. These spaces are naturally equipped with a metric induced by the rank function. The main motivation for this paper is to study $d$-codes in ${\mathscr{Q}}$ and ${\mathscr{S}}$, namely subsets $X$ of ${\mathscr{Q}}$ or ${\mathscr{S}}$ such that, for all distinct $A,B\in X$, the rank of $A-B$ is at least $d$. We are in particular interested in the largest cardinality of $d$-codes in ${\mathscr{Q}}$ and ${\mathscr{S}}$ and in the structure of such sets when this maximum is attained. One of the applications is that $d$-codes in ${\mathscr{Q}}$ can be used to construct optimal subcodes of the second-order generalised Reed-Muller code and our theory can be used to determine the Hamming distance distributions of such codes.
For odd $q$, most of the results in this paper have been obtained by the author in [@Sch2015]. The new results of this paper concern the more difficult case that $q$ is even, although whenever possible we aim for a unified treatment of the two cases. For even $q$, some partial results were obtained previously by the author in [@Sch2010].
The main tool for studying subsets of ${\mathscr{Q}}$ and ${\mathscr{S}}$ is the beautiful theory of association schemes. It is known that ${\mathscr{Q}}(m,q)$ and ${\mathscr{S}}(m,q)$ carry the structure of a translation association scheme with ${\lfloor3m/2\rfloor}$ classes. These have been studied by Wang, Wang, Ma, and Ma [@WanWanMaMa2003]. In particular the two schemes are dual to each other and, for odd $q$, they are isomorphic. Hence the association scheme on ${\mathscr{Q}}$ is self-dual for odd $q$. These association schemes differ considerably from the classical association schemes typically studied by coding theorists, in the sense that the association schemes on ${\mathscr{Q}}$ and ${\mathscr{S}}$ are neither $P$-polynomial, nor $Q$-polynomial. This means that their most important parameters, namely the $P$- and $Q$-numbers (also known as the first and second eigenvalues), do not just arise from evaluations of sets of orthogonal polynomials. For odd $q$, the $Q$-numbers (and the $P$-numbers by self-duality) of ${\mathscr{S}}$ have been determined by the author in [@Sch2015]. For even $q$, the computation of these numbers appears to be more difficult and, so far, only very limited partial results are known. Recursive formulae were given by Feng, Wang, Ma, and Ma [@FenWanMaMa2008] and some special cases were computed by Bachoc, Serra, and Zémor [@BacSerZem2017]. The $P$-numbers of ${\mathscr{Q}}(m,2)$ have been determined by Hou [@Hou2001] using an interesting coding-theoretic approach, which implicitly identifies ${\mathscr{Q}}(m,2)$ with $R(2,m)$ and ${\mathscr{S}}(m,2)$ with $R(m,m)/R(m-3,m)$, where $R(r,m)$ is the Reed-Muller code of order $r$ and length $2^m$. The main result of the present paper is the determination of the $P$- and $Q$-numbers of ${\mathscr{Q}}$ and, by duality, the $Q$- and $P$-numbers of ${\mathscr{S}}$. Although these numbers do not directly arise from evaluations of orthogonal polynomials, they can be expressed in terms of linear combinations of generalised Krawtchouk polynomials. This also simplifies the expressions given by Hou [@Hou2001].
Using the $Q$-numbers of ${\mathscr{Q}}$ and ${\mathscr{S}}$, we then obtain tight bounds on the size of $d$-codes in ${\mathscr{Q}}$ and ${\mathscr{S}}$, except in ${\mathscr{S}}$ when $d$ is even and (by self-duality) in ${\mathscr{Q}}$ when $d$ is even and $q$ is odd, and give explicit expressions for the inner distributions of $d$-codes when these bounds are attained. In the remaining cases, we obtain tight bounds for $d$-codes that are subgroups $({\mathscr{Q}},+)$ or $({\mathscr{S}},+)$.
In the final section of this paper we briefly discuss the connection between $d$-codes in ${\mathscr{Q}}$ and classical error-correcting codes. It turns out that error-correcting codes obtained from maximal $d$-codes in ${\mathscr{Q}}$ often compare favourably to the best known codes. We show that the Hamming distance enumerators of these error-correcting codes are uniquely determined by their parameters. This at once gives the distance enumerators of large classes of error-correcting codes for which many special cases have been obtained previously using different methods, for example results for (extended) binary cyclic codes obtained by Berlekamp [@Ber1970] and Kasami [@Kas1971], recent results for $q$-ary cyclic codes obtained by Li [@Li2017], and many results for $q$-ary cyclic codes and odd $q$, as explained in [@Sch2015]. In particular, Li [@Li2017] recently determined the true minimum distance of some narrow-sense primitive BCH codes and obtained their distance enumerators in the case that $q$ is odd. These results are recovered in this paper and the distance enumerators are obtained for all $q$ as corollaries of our results.
Quadratic forms and symmetric bilinear forms
============================================
In this section we recall the definitions and basic properties of the association schemes of symmetric bilinear forms and quadratic forms from [@WanWanMaMa2003]. We refer to [@Del1973], [@DelLev1998], and [@BanIto1984] for more background on association schemes and to [@MacSlo1977 Chapter 21] and [@vLiWil2001 Chapter 30] for gentle introductions.
A (symmetric) *association scheme* with $n$ classes is a pair $({\mathscr{X}},(R_i))$, where ${\mathscr{X}}$ is a finite set and $R_0,R_1,\dots,R_n$ are nonempty relations on ${\mathscr{X}}$ satisfying:
1. $\{R_0,R_1,\dots,R_n\}$ is a partition of ${\mathscr{X}}\times{\mathscr{X}}$ and $R_0=\{(x,x):x\in {\mathscr{X}}\}$;
2. Each of the relations $R_i$ is symmetric;
3. If $(x,y)\in R_k$, then the number of $z\in {\mathscr{X}}$ such that $(x,z)\in R_i$ and $(z,y)\in R_j$ is a constant $p^k_{ij}$ depending only on $i$, $j$, and $k$, but not on the particular choice of $x$ and $y$.
Let $({\mathscr{X}},(R_i))$ be a symmetric association scheme with $n$ classes and let $D_i$ be the adjacency matrix of the graph $(X,R_i)$. The vector space generated by $D_0,D_1,\dots,D_n$ over the real numbers has dimension $n+1$ and is in fact an algebra, called the *Bose-Mesner algebra* of the association scheme. There exists another uniquely defined basis for this vector space, consisting of minimal idempotent matrices $E_0,E_1,\dots,E_n$. We may write $$D_i=\sum_{k=0}^nP_i(k)E_k\quad\text{and}\quad E_k=\frac{1}{{\lvertX\rvert}}\sum_{i=0}^nQ_k(i)D_i$$ for some uniquely determined numbers $P_i(k)$ and $Q_k(i)$, called the *$P$-numbers* and the *$Q$-numbers* of $({\mathscr{X}},(R_i))$, respectively.
Now let $V=V(m,q)$ be an $m$-dimensional ${\mathbb{F}}_q$-vector space. We denote by ${\mathscr{Q}}={\mathscr{Q}}(m,q)$ the set of quadratic forms on $V$ and by ${\mathscr{S}}={\mathscr{S}}(m,q)$ the set of symmetric bilinear forms on $V$. Notice that ${\mathscr{Q}}$ and ${\mathscr{S}}$ are themselves ${\mathbb{F}}_q$-vectors spaces of dimension $m(m+1)/2$.
Let $G=G(m,q)$ be the direct product ${\mathbb{F}}_q^*\times\operatorname{GL}_m({\mathbb{F}}_q)$. Then $G$ acts on ${\mathscr{Q}}$ by $(g,Q)\mapsto Q^g$, where $Q^g$ is given by $Q^g(x)=aQ(Lx)$ and $g=(a,L)$. The semidirect product ${\mathscr{Q}}\rtimes G$ acts transitively on ${\mathscr{Q}}$ by $$((A,g),Q)\mapsto Q^g+A.$$ The action of ${\mathscr{Q}}\rtimes G$ extends to ${\mathscr{Q}}\times{\mathscr{Q}}$ componentwise and partitions ${\mathscr{Q}}\times {\mathscr{Q}}$ into orbits, which define the relations of a symmetric association scheme. Two pairs of quadratic forms $(Q,Q')$ and $(R,R')$ are in the same relation if and only if there is a $g\in G$ such that $(Q-Q')^g=R-R'$. This shows that the relation containing $(Q,Q')$ depends only on $Q-Q'$, which is the defining property of a translation scheme [@DelLev1998 Chapter V].
The group $G$ also acts on ${\mathscr{S}}$ by $(g,S)\mapsto S^g$, where $S^g$ is given by $S^g(x,y)=aS(Lx,Ly)$ and $g=(a,L)$. The semidirect product ${\mathscr{S}}\rtimes G$ acts transitively on ${\mathscr{S}}$ by $$((A,g),S)\mapsto S^g+A.$$ Again, the action of ${\mathscr{S}}\rtimes G$ extends to ${\mathscr{S}}\times{\mathscr{S}}$ componentwise and so partitions ${\mathscr{S}}\times {\mathscr{S}}$ into orbits, which define the relations of a symmetric association scheme. Two pairs of symmetric bilinear forms $(S,S')$ and $(T,T')$ are in the same relation if and only if there is a $g\in G$ such that $(S-S')^g=T-T'$, which again makes the association scheme a translation scheme.
When $q$ is odd, every quadratic form $Q\in{\mathscr{Q}}$ gives rise to a symmetric bilinear form $S\in{\mathscr{S}}$ via $$S(x,y)=Q(x+y)-Q(x)-Q(y), \label{eqn:S_from_Q}$$ from which we can recover $Q$ by $Q(x)=\tfrac{1}{2}S(x,x)$. This shows that the association schemes on ${\mathscr{Q}}$ and ${\mathscr{S}}$ are isomorphic when $q$ is odd. We shall see that this is not the case when $q$ is even.
Now let $\{\alpha_1,\alpha_2,\dots,\alpha_m\}$ be a basis for $V(m,q)$. For every quadratic form $Q\in{\mathscr{Q}}$, there exist $A_{ij}\in{\mathbb{F}}_q$ such that $$Q\bigg(\sum_{i=1}^mx_i\alpha_i\bigg)=\sum_{i,j=1}^mA_{ij}x_ix_j \label{eqn:coordinate_rep_Q}$$ for all $(x_1,x_2,\dots,x_m)\in{\mathbb{F}}_q^m$. We say that the right hand side is the *coordinate representation* of $Q$ (with respect to the basis chosen). The matrix $A=(A_{ij})$ is only unique modulo the subgroup of $m\times m$ alternating matrices over ${\mathbb{F}}_q$. Accordingly we associate with $Q$ the coset $[A]$ of alternating matrices containing $A$.
Let $\{\beta_1,\beta_2,\dots,\xi_m\}$ be another basis for $V(m,q)$. For every symmetric bilinear form $S\in{\mathscr{S}}$ we then have $$S\bigg(\sum_{i=1}^mx_i\beta_i,\sum_{j=1}^my_j\beta_j\bigg)=\sum_{i,j=1}^mB_{ij}x_iy_j,$$ for all $(x_1,x_2,\dots,x_m)\in{\mathbb{F}}_q^m$, where $B_{ij}=S(\beta_i,\beta_j)$. Again, we refer to the right hand side as the *coordinate representation* of $S$. We associate with every symmetric bilinear form the corresponding $m\times m$ symmetric matrix $B=(B_{ij})$.
Let $\chi:{\mathbb{F}}_q\to{\mathbb{C}}^*$ be a fixed nontrivial character of $({\mathbb{F}}_q,+)$. Hence, if $q=p^k$ for a prime $p$ and an integer $k$, then $\chi(y)=\omega^{\operatorname{Tr}(\theta y)}$ for some fixed $\theta\in{\mathbb{F}}_q^*$ and some fixed primitive complex $p$-th root of unity $\omega$. Here, $\operatorname{Tr}:{\mathbb{F}}_q\to{\mathbb{F}}_p$ is the *absolute trace function* on ${\mathbb{F}}_q$ defined by $$\operatorname{Tr}(y)=\sum_{i=1}^ky^{p^i}.$$ For $Q\in{\mathscr{Q}}$ and $S\in{\mathscr{S}}$, write $${\langleQ,S\rangle}=\chi(\operatorname{tr}(AB)), \label{eqn:inner_product}$$ where $[A]$ is the coset of alternating matrices associated with $Q$ and $B$ is the symmetric matrix associated with $S$ and $\operatorname{tr}$ is the matrix trace. Note that ${\langleQ,S\rangle}$ is well defined since $\operatorname{tr}(CD)=0$ if $C$ is alternating and $D$ is symmetric. It is readily verified that ${\langle\,\cdot\,,S\rangle}$ ranges through all characters of $({\mathscr{Q}},+)$ when $S$ ranges over ${\mathscr{S}}$ and that ${\langleQ,\,\cdot\,\rangle}$ ranges through all characters of $({\mathscr{S}},+)$ when $Q$ ranges over ${\mathscr{Q}}$. Notice that this correspondence depends on the choice of the bases.
The following duality result was observed in [@WanWanMaMa2003].
[[@WanWanMaMa2003 Proposition 3.2]]{} \[pro:duality\] For every $g\in G$ with $g=(a,L)$, we have $${\langleQ^g,S\rangle}={\langleQ,S^h\rangle},$$ where $h=(a,L^T)$.
Proposition \[pro:duality\] shows that the association schemes on ${\mathscr{Q}}$ and ${\mathscr{S}}$ are dual to each other in the strong sense of [@DelLev1998 Definition 11].
In what follows, we shall describe the relations of ${\mathscr{Q}}$ and ${\mathscr{S}}$ explicitly. For a symmetric bilinear form $S\in{\mathscr{S}}$, the *radical* is defined to be $$\operatorname{rad}(S)=\{x\in V:\text{$S(x,y)=0$ for every $y\in V$}\}.$$ The *rank* of $S$ is the codimension of the radical and coincides with the rank of the symmetric matrix associated with $S$. For a quadratic form $Q\in{\mathscr{Q}}$, let $S_Q(x,y)=Q(x+y)-Q(x)-Q(y)$ be the associated symmetric bilinear form, and define the *radical* of $Q$ to be $$\operatorname{rad}(Q)=\{x\in \operatorname{rad}(S_Q):Q(x)=0\}.$$ The *rank* of $Q$ is defined to be the codimension of its radical.
The following result describes the orbits of the action of $G$ on ${\mathscr{Q}}$ and was essentially obtained by Dickson [@Dic1958].
\[pro:canonical\_quad\_forms\] The action of $G$ on ${\mathscr{Q}}(m,q)$ partitions ${\mathscr{Q}}(m,q)$ into ${\lfloor3m/2\rfloor}+1$ orbits, one of them contains just the zero form. There is one orbit for each odd rank $r$ and one representative is, in coordinate representation, $$\sum_{i=1}^{(r-1)/2}x_{2i-1}x_{2i}+x_r^2.$$ There are two orbits for each nonzero even rank $r$ and representatives from the two orbits are, in coordinate representation, $$\begin{gathered}
\sum_{i=1}^{r/2}x_{2i-1}x_{2i}, \label{eqn:Q_hyperbolic}\\
\sum_{i=1}^{r/2-1}x_{2i-1}x_{2i}+Q_0, \label{eqn:Q_elliptic}\end{gathered}$$ where $$Q_0=\begin{cases}
x_{r-1}^2+x_{r-1}x_r+\alpha x_r^2 & \text{for $q$ even}\\
x_{r-1}^2-\beta x_r^2 & \text{for $q$ odd}
\end{cases}$$ and $\alpha\in{\mathbb{F}}_q^*$ is a fixed element satisfying $\operatorname{Tr}(\alpha)=1$ (for even $q$) and $\beta\in{\mathbb{F}}_q^*$ is a fixed nonsquare in ${\mathbb{F}}_q^*$ (for odd $q$).
If $Q$ belongs to an orbit corresponding to , then $Q$ is called *hyperbolic* or of *type* $1$ and if $Q$ belongs to an orbit corresponding to , then $Q$ is called *elliptic* or of *type* $-1$. By convention, the zero form is a hyperbolic quadratic form of rank $0$. Let ${\mathscr{Q}}_{2s+1}$ be the set of quadratic forms on $V$ of rank ${2s+1}$ and let ${\mathscr{Q}}_{2s,1}$ and ${\mathscr{Q}}_{2s,-1}$ be the sets of hyperbolic and elliptic quadratic forms on $V$ of rank $2s$, respectively. Write $$I=\left\{2s+1:s\in{\mathbb{Z}}\right\}\cup \left\{(2s,{\tau}):s\in{\mathbb{Z}},{\tau}=\pm 1\right\}.$$ and, for every $i\in I$, define the relations $$R_i=\{(Q,Q')\in{\mathscr{Q}}\times {\mathscr{Q}}:Q-Q'\in {\mathscr{Q}}_i\}. \label{eqn:relations_Q}$$ The nonempty relations are then precisely the relations of the association scheme of quadratic forms.
The following result describes the orbits of the action of $G$ on ${\mathscr{S}}$ and was essentially obtained by Albert [@Alb1938] (and, for odd $q$, also follows from Proposition \[pro:canonical\_quad\_forms\] via ).
\[pro:canonical\_sym\_bil\_forms\] The action of $G$ on ${\mathscr{S}}(m,q)$ partitions ${\mathscr{S}}(m,q)$ into ${\lfloor3m/2\rfloor}+1$ orbits, one of them contains just the zero form. There is one orbit for each odd rank $r$ and one representative is, in coordinate representation, $$\sum_{i=1}^{(r-1)/2}(x_{2i-1}y_{2i}+x_{2i}y_{2i-1})+x_ry_r.$$ There are two orbits for each nonzero even rank $r$ and representatives from the two orbits are, in coordinate representation, $$\begin{gathered}
\sum_{i=1}^{r/2}(x_{2i-1}y_{2i}+x_{2i}y_{2i-1}), \label{eqn:S_hyperbolic}\\
\sum_{i=1}^{r/2-1}(x_{2i-1}y_{2i}+x_{2i}y_{2i-1})+S_0, \label{eqn:S_elliptic}\end{gathered}$$ where $$S_0=\begin{cases}
x_{r-1}y_r+x_ry_{r-1}+x_ry_r & \text{for even $q$}\\
x_{r-1}y_{r-1}-\beta x_ry_r & \text{for odd $q$},
\end{cases}$$ and $\beta$ is a fixed nonsquare of ${\mathbb{F}}_q^*$ (for odd $q$).
Let ${\mathscr{S}}_{2s+1}$ be the set of symmetric bilinear forms on $V$ of rank $2s+1$ and let ${\mathscr{S}}_{2s,1}$ and ${\mathscr{S}}_{2s,-1}$ be the sets of symmetric bilinear forms on $V$ of rank $2s$ corresponding to the orbits and , respectively. Symmetric bilinear forms in ${\mathscr{S}}_{2s,{\tau}}$ are said to be of *type* ${\tau}$. For even $q$, it can be shown [@Alb1938] that ${\mathscr{S}}_{2s,1}$ contains precisely the alternating bilinear forms of rank $2s$. For every $i\in I$, define the relations $$R'_i=\{(S,S')\in{\mathscr{S}}\times {\mathscr{S}}:S-S'\in {\mathscr{S}}_i\}. \label{eqn:relations_S}$$ The nonempty relations are then precisely the relations of the association scheme of symmetric bilinear forms.
Now write $v_i={\lvert{\mathscr{Q}}_i\rvert}$ and $\mu_i={\lvert{\mathscr{S}}_i\rvert}$, whose nonzero values are called the *valencies* of the association schemes on ${\mathscr{Q}}$ and ${\mathscr{S}}$, respectively. The numbers $v_i$ have been determined by McEliece [@McE1969], following the work of Dickson [@Dic1958]. Since the association schemes on ${\mathscr{Q}}$ and ${\mathscr{S}}$ are isomorphic for odd $q$, we have $\mu_i=v_i$ for odd $q$. For even $q$, the numbers $\mu_i$ were determined by MacWilliams [@Mac1969]. We summarise the results in the following form.
\[pro:valencies\_multiplicities\] We have $$\begin{aligned}
v_{2s+1}&=\mu_{2s+1}=\frac{1}{q^s}\;\frac{\prod\limits_{i=0}^{2s}(q^m-q^i)}{\prod\limits_{i=0}^{s-1}(q^{2s}-q^{2i})},\\[1ex]
v_{2s,{\tau}}&=\frac{q^s+{\tau}}{2}\;\frac{\prod\limits_{i=0}^{2s-1}(q^m-q^i)}{\prod\limits_{i=0}^{s-1}(q^{2s}-q^{2i})},\\[1ex]
\mu_{2s,{\tau}}&=\big(\alpha_{\tau}q^s+{\tau}\beta_sq^{-s}\big)\;\frac{\prod\limits_{i=0}^{2s-1}(q^m-q^i)}{\prod\limits_{i=0}^{s-1}(q^{2s}-q^{2i})},\end{aligned}$$ where $$\alpha_{\tau}=\begin{cases}
\tfrac{1}{2}(1-{\tau}) & \text{for even $q$}\\[1ex]
\tfrac{1}{2} & \text{for odd $q$}
\end{cases}
\quad\text{and}\quad
\beta_s=\begin{cases}
1 & \text{for even $q$}\\[1ex]
\tfrac{1}{2}\, q^s & \text{for odd $q$}.
\end{cases}$$
We conclude this section by noting that our association scheme on ${\mathscr{S}}$ is slightly different from the association schemes on ${\mathscr{S}}$ in [@WanWanMaMa2003] and [@Sch2015]. The difference is that in [@WanWanMaMa2003] and [@Sch2015] the group $G$ is just $\operatorname{GL}_m({\mathbb{F}}_q)$, which increases the number of orbits from ${\lfloor3m/2\rfloor}+1$ to $2m+1$ in the case that $q$ is odd. Another difference to [@Sch2015] is that the sets ${\mathscr{S}}_{2s,1}$ and ${\mathscr{S}}_{2s,-1}$, and so also the relations $R'_{2s,1}$ and $R'_{2s,-1}$ on ${\mathscr{S}}$, are interchanged when $s$ is odd and $q\equiv 3\pmod 4$.
Computation of the $Q$- and $P$-numbers
=======================================
Throughout this section we identify quadratic forms with the corresponding cosets of alternating matrices and symmetric bilinear forms with the corresponding symmetric matrices. For $A,B\in{\mathbb{F}}_q^{m\times m}$, we write $${\langleA,B\rangle}=\chi(\operatorname{tr}(AB)),$$ where $\chi$ is the same nontrivial character as in . The $Q$-numbers and the $P$-numbers of the association scheme on ${\mathscr{Q}}$ are given by the character sums (see [@DelLev1998 Section V], for example) $$\begin{aligned}
Q_k(i)&=\sum_{B\in{\mathscr{S}}_k}{\langleA,B\rangle}\quad \text{for $[A]\in{\mathscr{Q}}_i$}, \label{eqn:Q_char_sum}\\
P_i(k)&=\sum_{[A]\in{\mathscr{Q}}_i}{\langleA,B\rangle}\quad \text{for $B\in{\mathscr{S}}_k$}, \label{eqn:P_char_sum}\end{aligned}$$ respectively, where $k,i\in I$. The $Q$-numbers $Q'_i(k)$ and the $P$-numbers $P'_k(i)$ of the association scheme on ${\mathscr{S}}$ satisfy $$Q'_i(k)=P_i(k)\quad\text{and}\quad P'_k(i)=Q_k(i),$$ respectively. For convenience, we define $Q_k(i)=0$ if ${\mathscr{Q}}_i=\emptyset$ and $P_i(k)=0$ if ${\mathscr{S}}_k=\emptyset$.
In order to give explicit expressions for these numbers, it is convenient to use *$q^2$-analogs of binomial coefficients*, which are defined by $${n\brack k}=\prod_{i=0}^{k-1}\frac{q^{2n}-q^{2i}}{q^{2k}-q^{2i}}$$ for integral $n$ and $k$. These numbers satisfy the following identities $${n\brack k}=q^{2k}{n-1\brack k}+{n-1\brack k-1}={n-1\brack k}+q^{2(n-k)}{n-1\brack k-1}. \label{eqn:Pascal_triangle}$$ We also need the following numbers, which can be derived from generalised Krawtchouk polynomials [@DelGoe1975], [@Del1976]. We define $$F^{(m)}_r(s)=\sum_{j=0}^r(-1)^{r-j}q^{(r-j)(r-j-1)}{n-j\brack n-r}{n-s\brack j}\, c^j,$$ where $$n=\left\lfloor m/2\right\rfloor,\quad\text{and}\quad c=q^{m(m-1)/(2n)},$$ whenever this expression is defined and let $F^{(m)}_r(s)=0$ otherwise. Equivalently, these numbers can be defined via the $n+1$ equations $$\sum_{r=0}^j{n-r\brack n-j} F^{(m)}_r(s)={n-s\brack j}c^j\quad\text{for $j\in\{0,1,\dots,n\}$} \label{eqn:ev_transform}$$ (see [@DelGoe1975 (29)]).
The following theorem contains explicit expressions for the $Q$-numbers of ${\mathscr{Q}}(m,q)$. For odd $q$, this is follows from [@Sch2015 Theorem 2.2]. For even $q$, the result is new.
\[thm:Q\_numbers\] The $Q$-numbers of the association scheme of quadratic forms ${\mathscr{Q}}(m,q)$ are as follows. We have $Q_{0,1}(i)=1$ and $Q_k(0,1)=\mu_k$ for all $i,k\in I$ and the other $Q$-numbers are given by $$\begin{aligned}
Q_{2r+1}(2s+1)&=-q^{2r}F^{(m-1)}_r(s),\\
Q_{2r+1}(2s,{\tau})&=-q^{2r}F^{(m-1)}_r(s-1)+\tau\, q^{m-s+2r}F^{(m-2)}_r(s-1),\\
Q_{2r,{\epsilon}}(2s+1)&=\alpha_{\epsilon}\, q^{2r}F^{(m-1)}_r(s)+{\epsilon}\,\beta_r\,F^{(m)}_r(s),\\
Q_{2r,{\epsilon}}(2s,{\tau})&=\alpha_{\epsilon}\,[q^{2r}F^{(m-1)}_r(s-1)-\tau\, q^{m-s+2r-2}F^{(m-2)}_{r-1}(s-1)]+{\epsilon}\,\beta_r\,F^{(m)}_r(s),\end{aligned}$$ where $\alpha_{\epsilon}$ and $\beta_r$ are given in Proposition \[pro:valencies\_multiplicities\].
It is well known (and can be easily verified) that the $P$-numbers of ${\mathscr{Q}}$ can be computed from the $Q$-numbers of ${\mathscr{Q}}$ via $$P_i(k)=\frac{v_i}{\mu_k}Q_k(i).$$ Proposition \[pro:valencies\_multiplicities\] then shows that $P_i(k)=Q_k(i)$ for odd $q$ (as it should since the association scheme on ${\mathscr{Q}}$ is isomorphic to its dual in this case). For even $q$, the $P$-numbers of ${\mathscr{Q}}$ are given in the following theorem.
\[thm:P\_numbers\] For even $q$, the $P$-numbers of the association scheme of quadratic forms ${\mathscr{Q}}(m,q)$ are as follows. We have $P_{0,1}(k)=1$ and $P_i(0,1)=v_i$ for all $i,k\in I$ and the other $P$-numbers are given by $$\begin{aligned}
P_{2s+1}(2r+1)&=-q^{2s}F^{(m-1)}_s(r),\\
2P_{2s,{\tau}}(2r+1)&=q^{2s}F^{(m-1)}_s(r)+{\tau}\, q^s F^{(m)}_s(r),\\
2P_{2s,{\tau}}(2r,1)&=q^s(q^s+{\tau})F^{(m)}_s(r),\\
2P_{2s,{\tau}}(2r,-1)&=q^{2s}F^{(m-1)}_s(r-1)+{\tau}\,q^sF^{(m)}_s(r),\\
P_{2s+1}(2r,1)&=(q^m-q^{2s})F^{(m)}_s(r),\\
P_{2s+1}(2r,-1)&=-q^{2s}F^{(m-1)}_s(r-1).\end{aligned}$$
In the remainder of this section, we shall prove Theorems \[thm:Q\_numbers\] and \[thm:P\_numbers\]. We begin with the following result, which is essentially known.
\[pro:Q\_alt\] Let $\alpha_{\epsilon}$ and $\beta_r$ be as in Proposition \[pro:valencies\_multiplicities\]. The $Q$-numbers of the association scheme ${\mathscr{Q}}(m,q)$ satisfy $$\begin{aligned}
\beta_rF^{(m)}_r(s)&=\alpha_{-1}Q_{2r,1}(2s,{\tau})-\alpha_1Q_{2r,-1}(2s,{\tau})\\
&=\alpha_{-1}Q_{2r,1}(2s+1)-\alpha_1Q_{2r,-1}(2s+1).\end{aligned}$$
For odd $q$, the statement in the lemma can be deduced from [@Sch2015 Lemma 6.3], so assume that $q$ is even. Then ${\mathscr{S}}_{2r,1}$ is the set of alternating bilinear forms of rank $2r$ on $V$. By Proposition \[pro:canonical\_quad\_forms\], every quadratic form in ${\mathscr{Q}}_{2s+1}$ can be represented by an $m\times m$ block diagonal matrix with the block $(1)$ in the top left corner, followed by $s$ copies of $$\begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix}. \label{eqn:antidiagonal_matrix}$$ It can be shown using Proposition \[pro:canonical\_quad\_forms\] that the quadratic form $x^2+xy+\lambda y^2$ in ${\mathscr{Q}}(2,q)$ is of type $(-1)^{\operatorname{Tr}(\lambda)}$. Hence a quadratic form in ${\mathscr{Q}}_{2s,{\tau}}$ can be represented by the zero matrix or by an $m\times m$ block diagonal matrix with the block $$\begin{bmatrix}
\lambda & 1\\
0 & 1
\end{bmatrix}\label{eqn:top_left_block}$$ in the top left corner, followed by $s-1$ copies of , where $(-1)^{\operatorname{Tr}(\lambda)}={\tau}$. It follows from these observations that, for $k=(2r,1)$ the character sums have been evaluated by Delsarte and Goethals [@DelGoe1975 Appendix], which gives $$Q_{2r,1}(2s,{\tau})=Q_{2r,1}(2s+1)=F^{(m)}_r(s),$$ as required.
In what follows we write, for every $i\in I$, $$Q_{2r}(i)=Q_{2r,1}(i)+Q_{2r,-1}(i).$$ We shall also write $Q^{(m)}_k(i)$ for $Q_k(i)$ and ${\mathscr{S}}_k^{(m)}$ for ${\mathscr{S}}_k$ whenever we need to indicate dependence on $m$.
We have the following recurrences for the $Q$-numbers.
\[lem:recurrence\_Q\] For $k\ge 1$ and $s\ge 0$, we have $$Q^{(m)}_k(2s+1)=Q^{(m)}_k(2s,1)-q^{m-s}\,Q^{(m-1)}_{k-1}(2s,1) \label{eqn:rec_Q_1}$$ and for $k\ge 1$ and $s\ge 1$, we have $$Q^{(m)}_k(2s,{\tau})=Q^{(m)}_k(2s-1)+\tau\,q^{m-s}\,Q^{(m-1)}_{k-1}(2s-1) \label{eqn:rec_Q_2}.$$
For odd $q$, the lemma can be deduced from [@Sch2015 Lemma 6.1], so henceforth we assume that $q$ is even.
To prove the identity , fix an integer $s$ with $0\le s\le (m-1)/2$ and let $A$ be the $m\times m$ block diagonal matrix with the block $(1)$ in the top left corner, followed by $s$ copies of . Then $A$ is a matrix of a quadratic form of rank $2s+1$. Let $A'$ be the $(m-1)\times (m-1)$ matrix obtained from $A$ by deleting the first row and the first column. Then we have $$\begin{aligned}
Q^{(m)}_k(2s,1)-Q^{(m)}_k(2s+1)&=\sum_{B\in{\mathscr{S}}^{(m)}_k}({\langleA',B'\rangle}-{\langleA,B\rangle}) \nonumber\\
&=\sum_{B\in{\mathscr{S}}^{(m)}_k}{\langleA',B'\rangle}(1-\chi(a)), \label{eqn:diff_Q_1}\end{aligned}$$ where $\chi$ is the nontrivial character of $({\mathbb{F}}_q,+)$ used to define the pairing in and we write $B$ as $$B=\begin{bmatrix}
a & u^T\\
u & B'
\end{bmatrix} \label{eqn:matrix_B}$$ for some $a\in{\mathbb{F}}_q$, some $u\in{\mathbb{F}}_q^{m-1}$ and some $(m-1)\times (m-1)$ matrix $B'$ over ${\mathbb{F}}_q$. The summand in is zero for $a=0$, so assume that $a\ne 0$. Writing $$L=\begin{bmatrix}
1 & -a^{-1}u^T\\
0 & I
\end{bmatrix},$$ we have $$L^TBL=\begin{bmatrix}
a & 0\\
0 & C
\end{bmatrix},
\quad\text{where}\quad
C=B'-a^{-1}uu^T.$$ As $a$ ranges over ${\mathbb{F}}_q^*$ and $u$ ranges over ${\mathbb{F}}_q^{m-1}$ and $C$ ranges over ${\mathscr{S}}^{(m-1)}_{k-1}$, the matrix $B$ in ranges over ${\mathscr{S}}^{(m)}_k$ with the constraint $a\ne 0$. Therefore the sum is $$\sum_{a\in{\mathbb{F}}_q^*}\sum_{u\in{\mathbb{F}}_q^{m-1}}\sum_{C\in{\mathscr{S}}^{(m-1)}_{k-1}}{\langleA',C\rangle}{\langleA',a^{-1}uu^T\rangle}(1-\chi(a)).$$ We have $$\sum_{C\in{\mathscr{S}}^{(m-1)}_{k-1}}{\langleA',C\rangle}=Q^{(m-1)}_{k-1}(2s),$$ and $$\begin{aligned}
\sum_{u\in{\mathbb{F}}_q^{m-1}}{\langleA',a^{-1}uu^T\rangle}&=q^{m-2s-1}\sum_{u_1,\dots,u_{2s}\in{\mathbb{F}}_q}\chi\bigg(a^{-1}\bigg(\sum_{i=1}^su_{2i-1}u_{2i}\bigg)\bigg)\\
&=q^{m-2s-1}\bigg(\sum_{u,v\in{\mathbb{F}}_q}\chi(uv)\bigg)^s\\
&=q^{m-s-1}\end{aligned}$$ for every $a\in{\mathbb{F}}_q^*$, and $$\sum_{a\in{\mathbb{F}}_q^*}(1-\chi(a))=q.$$ Substitute everything into to obtain the first identity in the lemma.
To prove the identity , fix an integer $s$ with $1\le s\le m/2$ and $\tau\in\{-1,1\}$. Let $\lambda\in{\mathbb{F}}_q$ be such that $(-1)^{\operatorname{Tr}(\lambda)}={\tau}$ and let $A$ be the $m\times m$ block diagonal matrix with the block in the top left corner, followed by $s-1$ copies of . Then $A$ is a matrix of a quadratic form of rank $2s$ and type $\tau$. Let $A'$ be the $(m-1)\times (m-1)$ matrix obtained from $A$ by deleting the first row and the first column. Then we have $$Q^{(m)}_k(2s-1)-Q^{(m)}_k(2s,\tau)=\sum_{B\in{\mathscr{S}}^{(m)}_k}{\langleA',B'\rangle}(1-\chi(a\lambda+u_1)), \label{eqn:diff_Q_2}$$ where we write $B$ as and where $u=(u_1,\dots,u_{m-1})^T$. We split the summation in into two parts: the sum $S_1$ is over all $B$ with $a\ne 0$ and the sum $S_2$ is over all $B$ with $a=0$. Similarly as in the proof of the first identity , we have $$\begin{aligned}
S_1&=\sum_{a\in{\mathbb{F}}_q^*}\sum_{u\in{\mathbb{F}}_q^{m-1}}\sum_{C\in{\mathscr{S}}^{(m-1)}_{k-1}}{\langleA',C\rangle}{\langleA',a^{-1}uu^T\rangle}(1-\chi(a\lambda+u_1)) \nonumber\\
&=Q^{(m-1)}_{k-1}(2s-1)\sum_{a\in{\mathbb{F}}_q^*}\sum_{u\in{\mathbb{F}}_q^{m-1}}{\langleA',a^{-1}uu^T\rangle}(1-\chi(a\lambda)\chi(u_1)). \label{eqn:sum_S1}\end{aligned}$$ For every $a\in{\mathbb{F}}_q^*$, we have $$\begin{aligned}
\sum_{u\in{\mathbb{F}}_q^{m-1}}{\langleA',a^{-1}uu^T\rangle}&=q^{m-2s}\sum_{u_1,\dots,u_{2s-1}\in{\mathbb{F}}_q}\chi\bigg(a^{-1}\bigg(u_1^2+\sum_{i=1}^{s-1}u_{2i}u_{2i+1}\bigg)\bigg)\\
&=q^{m-2s}\bigg(\sum_{u,v\in{\mathbb{F}}_q}\chi(uv)\bigg)^{s-1}\sum_{w\in{\mathbb{F}}_q}\chi(w)\\
&=0\end{aligned}$$ since the inner sum is zero, and similarly, $$\begin{aligned}
\sum_{u\in{\mathbb{F}}_q^{m-1}}{\langleA',a^{-1}uu^T\rangle}\chi(u_1)&=q^{m-2s}\bigg(\sum_{u,v\in{\mathbb{F}}_q}\chi(uv)\bigg)^{s-1}\sum_{w\in{\mathbb{F}}_q}\chi(a^{-1}w^2+w)\\
&=q^{m-s-1}\sum_{y\in{\mathbb{F}}_q}\chi(a(y^2+y)),\end{aligned}$$ by applying the substitution $w=ay$. The mapping is $y\mapsto y^2+y$ is $2$-to-$1$ and its image is the set of elements in ${\mathbb{F}}_q$ whose absolute trace ist zero. Since $\chi$ is nontrivial, there exists $\theta\in{\mathbb{F}}_q^*$ such that $$\sum_{y\in{\mathbb{F}}_q}\chi(a(y^2+y))=\sum_{y\in{\mathbb{F}}_q}(-1)^{\operatorname{Tr}(\theta a(y^2+y))},$$ which equals $q$ if $a=1/\theta$ and equals zero otherwise. Substitute everything into to obtain $$S_1=-\tau\, q^{m-s}\,Q^{(m-1)}_{k-1}(2s-1),$$ since $\chi(\lambda/\theta)=(-1)^{\operatorname{Tr}(\lambda)}=\tau$.
We complete the proof by showing that the sum $S_2$, namely the summation in over all $B$ with $a=0$, equals zero. Let $A''$ be the matrix obtained from $A$ by deleting the first two rows and the first two columns. Then we have $$S_2={\sum_{\substack{B\in{\mathscr{S}}^{(m)}_k\\a=0}}}{\langleA'',B''\rangle}\chi(c)(1-\chi(b)), \label{eqn:sum_S2}$$ where we now write $$B=\begin{bmatrix}
E & U^T\\
U & B''
\end{bmatrix}
\quad\text{and}\quad
E=\begin{bmatrix}
a & b\\
b & c
\end{bmatrix}$$ for some $b,c\in{\mathbb{F}}_q$, some $(m-2)\times 2$ matrix $U$ and some $(m-2)\times(m-2)$ matrix $B''$. Henceforth we put $a=0$. For $b=0$, the summand in equals zero, so we assume that $b$ is nonzero and so $E$ is invertible. Writing $$M=\begin{bmatrix}
I & -E^{-1}U^T\\
0 & I
\end{bmatrix},$$ we have $$M^TBM=\begin{bmatrix}
E & 0\\
0 & D
\end{bmatrix},
\quad\text{where}\quad
D=B''-UE^{-1}U^T.$$ Then, arguing similarly as before, we obtain $$\begin{aligned}
S_2&=\sum_{b\in{\mathbb{F}}_q^*}\sum_{c\in{\mathbb{F}}_q}\sum_{U\in{\mathbb{F}}_q^{(m-2)\times 2}}\sum_{D\in{\mathscr{S}}^{(m-2)}_{k-2}}{\langleA'',D\rangle}{\langleA'',UE^{-1}U^T\rangle}\chi(c)(1-\chi(b))\\
&=Q^{(m-2)}_{k-2}(2s-2,1)\sum_{b\in{\mathbb{F}}_q^*}\sum_{c\in{\mathbb{F}}_q}\sum_{U\in{\mathbb{F}}_q^{(m-2)\times 2}}{\langleA'',UE^{-1}U^T\rangle}\chi(c)(1-\chi(b)).\end{aligned}$$ There exists an invertible matrix $2\times 2$ matrix $N$ over ${\mathbb{F}}_q$ such that $NE^{-1}N^T$ is either the $2\times 2$ identity matrix or $$F=\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix},$$ depending on whether $c\ne 0$ or $c=0$, respectively. It is readily verified that $$\begin{aligned}
\sum_{U\in{\mathbb{F}}_q^{(m-2)\times 2}}{\langleA'',UU^T\rangle}&=\sum_{U\in{\mathbb{F}}_q^{(m-2)\times 2}}{\langleA'',UFU^T\rangle}\\
&=q^{2(m-2s)}\bigg(\sum_{u,v\in{\mathbb{F}}_q}\chi(uv)\bigg)^{2s-2}\\
&=q^{2(m-s-1)}.\end{aligned}$$ Therefore we have $$S_2=q^{2(m-s-1)}\,Q^{(m-2)}_{k-2}(2s-2,1)\sum_{b\in{\mathbb{F}}_q^*}(1-\chi(b))\sum_{c\in{\mathbb{F}}_q}\chi(c)=0,$$ since the inner sum is zero. This completes the proof of the identity .
We shall now solve the recurrence relations in Lemma \[lem:recurrence\_Q\] using the initial values $$\begin{aligned}
Q^{(m)}_{0,1}(i)&=1 \label{eqn:initial_1}
\intertext{for each $i\in I$ with ${\mathscr{Q}}_i\ne\emptyset$,}
Q^{(m)}_{2r}(0,1)&=\mu^{(m)}_{2r,1}+\mu^{(m)}_{2r,-1}, \label{eqn:initial_2}\\
Q^{(m)}_{2r+1}(0,1)&=\mu^{(m)}_{2r+1} \label{eqn:initial_3}\end{aligned}$$ for each $r\ge 0$, where we write $\mu^{(m)}_k$ for $\mu_k$. These initial values follow immediately from .
\[pro:Q\_sum\] For $k\ge 1$, the numbers $Q_k(2s+1)$ for $s\ge 0$ and the numbers $Q_k(2s,\tau)$ and $s\ge 1$ satisfy $$\begin{aligned}
Q_{2r+1}(2s+1) &=-q^{2r}F^{(m-1)}_r(s), \label{eqn:QS1}\\
Q_{2r+1}(2s,{\tau})&=-q^{2r}F^{(m-1)}_r(s-1)+{\tau}\, q^{m-s+2r}F^{(m-2)}_r(s-1), \label{eqn:QS2}\\
Q_{2r}(2s+1) &=\phantom{-}q^{2r}F^{(m-1)}_r(s), \label{eqn:QS3}\\
Q_{2r}(2s,{\tau}) &=\phantom{-}q^{2r}F^{(m-1)}_r(s-1)-{\tau}\, q^{m-s+2r-2}F^{(m-2)}_{r-1}(s-1). \label{eqn:QS4}\end{aligned}$$
For odd $q$, the statements in the lemma are given by [@Sch2015 Lemma 6.2]. However we prove the lemma for odd and even $q$ simultaneously. Write $$n={\lfloor(m-1)/2\rfloor}\quad\text{and}\quad c=q^{(m-1)(m-2)/(2n)}.$$ From with $s=0$ and the initial values and we have $$\begin{aligned}
Q^{(m)}_{2r}(1) &=\mu^{(m)}_{2r,1}+\mu^{(m)}_{2r,-1}-q^m\mu^{(m-1)}_{2r-1},\\
Q^{(m)}_{2r+1}(1)&=\mu^{(m)}_{2r+1}-q^m\big[\mu^{(m-1)}_{2r,1}+\mu^{(m-1)}_{2r,-1}\big].\end{aligned}$$ From Proposition \[pro:valencies\_multiplicities\] we then find that $$Q^{(m)}_{2r}(1)=-Q^{(m)}_{2r+1}(1)=\frac{1}{q^r}\;\frac{\prod\limits_{i=1}^{2r}(q^m-q^i)}{\prod\limits_{i=0}^{r-1}(q^{2r}-q^{2i})},$$ which we can write as $$q^{2r}{n\brack r}\prod_{j=0}^{r-1}(c-q^{2j}).$$ This latter expression equals $q^{2r}F^{(m-1)}_r(0)$ (see [@DelGoe1975] or [@Sch2015], for example) and therefore and hold for $s=0$. Using the initial value , we see that also holds for $r=0$. Now substitute into to obtain $$Q^{(m)}_k(2s+1)=Q^{(m)}_k(2s-1)-cq^{2(n-s+1)}\,Q^{(m-2)}_{k-2}(2s-1).$$ Using , we verify by induction that and hold for all $r,s\ge 0$. The identities and then follow and and the recurrence .
Theorem \[thm:Q\_numbers\] now follows directly from Propositions \[pro:Q\_sum\] and \[pro:Q\_alt\].
We shall now determine the $P$-numbers of ${\mathscr{Q}}(m,q)$ for even $q$ from the $Q$-numbers and thereby prove Theorem \[thm:P\_numbers\]. We begin with stating the $Q$-numbers of ${\mathscr{Q}}(m,q)$ in the following alternative form.
\[pro:Q\_numbers\_sums\] The $Q$-numbers $Q_k(i)$ of the association scheme of quadratic forms ${\mathscr{Q}}(m,q)$ satisfy $$\begin{aligned}
F^{(m+1)}_r(s)&=Q_{2r,1}(2s-1)+Q_{2r,-1}(2s-1)+Q_{2r-1}(2s-1) \label{eqn:Q_sum_1}\\
&=Q_{2r,1}(2s,{\tau})+Q_{2r,-1}(2s,{\tau})+Q_{2r-1}(2s,{\tau}), \nonumber\\[1ex]
{\tau}\,q^{m-s}F^{(m)}_r(s)&=Q_{2r,1}(2s,{\tau})+Q_{2r,-1}(2s,{\tau})+Q_{2r+1}(2s,{\tau}), \label{eqn:Q_sum_2}\\
0&=Q_{2r,1}(2s+1)+Q_{2r,-1}(2s+1)+Q_{2r+1}(2s+1), \label{eqn:Q_sum_2b}\\[1ex]
\beta_rF^{(m)}_r(s)&=\alpha_{-1}Q_{2r,1}(2s,{\tau})-\alpha_1 Q_{2r,-1}(2s,{\tau}) \label{eqn:Q_sum_3}\\
&=\alpha_{-1}Q_{2r,1}(2s+1)-\alpha_1Q_{2r,-1}(2s+1) \nonumber\end{aligned}$$ where $\alpha_{\epsilon}$ and $\beta_r$ are as in Proposition \[pro:valencies\_multiplicities\].
This follows from Propositions \[pro:Q\_alt\] and \[pro:Q\_sum\] using the identity $$F^{(m+1)}_{r}(s)=q^{2r}F^{(m-1)}_{r}(s-1)-q^{2r-2}F^{(m-1)}_{r-1}(s-1), \label{eqn:F_identity}$$ which can be proved using .
We now use Proposition \[pro:Q\_numbers\_sums\] to prove the following counterpart of Proposition \[pro:Q\_numbers\_sums\] for the $P$-numbers.
\[pro:P\_numbers\_sums\] The $P$-numbers of the association scheme of quadratic forms ${\mathscr{Q}}(m,q)$ satisfy $$\begin{aligned}
F^{(m+1)}_s(r)&=P_{2s,1}(2r-1)+P_{2s,-1}(2r-1)+P_{2s-1}(2r-1) \label{eqn:P_sum_1}\\
&=P_{2s,1}(2r,{\epsilon})+P_{2s,-1}(2r,{\epsilon})+P_{2s-1}(2r,{\epsilon}), \nonumber\\[1ex]
q^s\,F^{(m)}_s(r)&=P_{2s,1}(2r,{\epsilon})-P_{2s,-1}(2r,{\epsilon}) \label{eqn:P_sum_2}\\
&=P_{2s,1}(2r+1)-P_{2s,-1}(2r+1), \nonumber\\[0.3ex]
\frac{\alpha_{-{\epsilon}}}{\beta_r}\,{\epsilon}\,q^m\,F^{(m)}_s(r)&=P_{2s,1}(2r,{\epsilon})+P_{2s,-1}(2r,{\epsilon})+P_{2s+1}(2r,{\epsilon}), \label{eqn:P_sum_3}\\[-1ex]
0&=P_{2s,1}(2r+1)+P_{2s,-1}(2r+1)+P_{2s+1}(2r+1), \label{eqn:P_sum_4}\end{aligned}$$ where $\alpha_{\epsilon}$ and $\beta_r$ are as in Proposition \[pro:valencies\_multiplicities\].
We use the orthogonality relation $$\sum_{s=0}^{{\lfloorm/2\rfloor}}F^{(m)}_p(s)F^{(m)}_s(r)=q^{m(m-1)/2}\delta_{r,p} \label{eqn:orthogonality_F}$$ (see [@DelGoe1975 (17)], for example). This shows that the matrix $$F=\Big(F^{(m)}_r(s)\Big)_{0\le r,s\le{\lfloorm/2\rfloor}}$$ is invertible and its inverse is $q^{-m(m-1)/2}F$.
Let ${\mathscr{Q}}^-_s$ be the set of all quadratic forms in ${\mathscr{Q}}(m,q)$ of rank $2s$ or $2s-1$. Similarly, let ${\mathscr{S}}^-_r$ be the set of all symmetric matrices in ${\mathscr{S}}(m,q)$ of rank $2r$ or $2r-1$. From and we find that, for every $[A]\in{\mathscr{Q}}^-_s$, we have $$F_p^{(m+1)}(s)=\sum_{B\in{\mathscr{S}}^-_p}{\langleA,B\rangle}.$$ Therefore, letting $B'\in{\mathscr{S}}^-_r$, we have $$\begin{aligned}
\sum_{s=0}^{{\lfloor(m+1)/2\rfloor}}F^{(m+1)}_p(s)\sum_{[A]\in{\mathscr{Q}}^-_s}{\langleA,B'\rangle}&=
\sum_{B\in{\mathscr{S}}^-_p}\sum_{[A]\in{\mathscr{Q}}}{\langleA,B+B'\rangle}\\[1ex]
&=q^{m(m+1)/2}\,\delta_{r,p},\end{aligned}$$ by the orthogonality of characters. From the orthogonality relation we then conclude that $$F^{(m+1)}_s(r)=\sum_{[A]\in{\mathscr{Q}}^-_s}{\langleA,B'\rangle},$$ which, in view of the character sum representation of the $P$-numbers, proves .
The other identities can be proved similarly. Let ${\mathscr{Q}}^+_s$ be the set of all quadratic forms in ${\mathscr{Q}}(m,q)$ of rank $2s$ or $2s+1$ and let ${\mathscr{S}}^+_r$ be the set of all symmetric matrices in ${\mathscr{S}}(m,q)$ of rank $2r$ or $2r+1$. From , , and we see that $$\sum_{B\in{\mathscr{S}}^+_p}{\langleA,B\rangle}=\begin{cases}
{\tau}\,q^{m-s}F^{(m)}_p(s) & \text{for $[A]\in{\mathscr{Q}}_{2s,{\tau}}$}\\
0 & \text{for $[A]\in{\mathscr{Q}}_{2s+1}$}.
\end{cases}$$ Let $B'\in{\mathscr{S}}_r^+$. We then find that $$\begin{aligned}
\sum_{s=0}^{{\lfloorm/2\rfloor}}F^{(m)}_p(s)\sum_{{\tau}\in\{-1,1\}}{\tau}\,q^{m-s}\sum_{[A]\in{\mathscr{Q}}_{2s,{\tau}}}{\langleA,B'\rangle}&=\sum_{s=0}^{{\lfloorm/2\rfloor}}\sum_{[A]\in{\mathscr{Q}}_s^+}{\langleA,B'\rangle}\sum_{B\in{\mathscr{S}}_p^+}{\langleA,B\rangle}\\[1ex]
&=\sum_{B\in{\mathscr{S}}_p^+}\sum_{[A]\in{\mathscr{Q}}}{\langleA,B+B'\rangle}\\[1ex]
&=q^{m(m+1)/2}\,\delta_{r,p}.\end{aligned}$$ From we conclude that $$\sum_{[A]\in{\mathscr{Q}}_{2s,1}}{\langleA,B'\rangle}-\sum_{[A]\in{\mathscr{Q}}_{2s,-1}}{\langleA,B'\rangle}=q^s\,F^{(m)}_s(r),$$ which together with proves .
To prove , we invoke and to obtain, for $B'\in{\mathscr{S}}_{2r,{\epsilon}}$, $$\begin{aligned}
\sum_{s=0}^{{\lfloorm/2\rfloor}}\beta_pF^{(m)}_p(s)\sum_{[A]\in{\mathscr{Q}}_s^+}{\langleA,B'\rangle}&=\sum_{s=0}^{{\lfloorm/2\rfloor}}\sum_{[A]\in{\mathscr{Q}}_s^+}{\langleA,B'\rangle}\sum_{\kappa\in\{-1,1\}}\kappa\,\alpha_{-\kappa}\!\sum_{B\in{\mathscr{S}}_{2p,\kappa}}{\langleA,B\rangle}\\[1ex]
&=\sum_{\kappa\in\{-1,1\}}\kappa\,\alpha_{-\kappa}\sum_{B\in{\mathscr{S}}_{2p,\kappa}}\sum_{[A]\in{\mathscr{Q}}}{\langleA,B+B'\rangle}\\[1ex]
&={\epsilon}\,\alpha_{-{\epsilon}}\,q^{m(m+1)/2}\,\delta_{r,p},\end{aligned}$$ which, using , gives $$\sum_{[A]\in{\mathscr{Q}}_s^+}{\langleA,B'\rangle}=\frac{\alpha_{-{\epsilon}}}{\beta_r}\,{\epsilon}\,q^m\,F^{(m)}_s(r).$$ Now follows from . To prove , we take $B'\in{\mathscr{S}}_{2r+1}$ and obtain similarly as above, $$\sum_{s=0}^{{\lfloorm/2\rfloor}}\beta_pF^{(m)}_p(s)\sum_{[A]\in{\mathscr{Q}}_s^+}{\langleA,B'\rangle}=0.$$ This implies that the inner sum is zero for every $s$ and this gives .
We now complete the proof of Theorem \[thm:P\_numbers\], which gives explicit expressions for the $P$-numbers.
The $P$-numbers of ${\mathscr{Q}}(m,q)$ are uniquely determined by Proposition \[pro:P\_numbers\_sums\]. We therefore just need to verify that the $P$-numbers claimed in the theorem satisfy the equations in Proposition \[pro:P\_numbers\_sums\]. The identities , , and are trivially satisfied. The identity is verified using and $$F^{(m+1)}_s(r)=q^{2s}F^{(m)}_s(r)+(q^m-q^{2s-2})F^{(m)}_{s-1}(r).$$ For even $m$, this last identity can be proved directly using . For odd $m$, first apply and then .
Subsets of quadratic and symmetric bilinear forms
=================================================
Inner distributions, codes, and designs
---------------------------------------
In what follows, let ${\mathscr{X}}={\mathscr{X}}(m,q)$ be either ${\mathscr{Q}}(m,q)$ or ${\mathscr{S}}(m,q)$. Accordingly, for $i\in I$, let ${\mathscr{X}}_i$ be either ${\mathscr{Q}}_i$ or ${\mathscr{S}}_i$ and let $(R_i)$ be the corresponding relations on ${\mathscr{X}}$ defined in and . Let $X$ be a subset of ${\mathscr{X}}$ and associate with $X$ the rational numbers $$a_i=\frac{{\lvert(X\times X)\cap R_i\rvert}}{{\lvertX\rvert}},$$ so that $a_i$ is the average number of pairs in $X\times X$ whose difference is contained in ${\mathscr{X}}_i$. The sequence of numbers $(a_i)_{i\in I}$ is called the *inner distribution* of $X$. Let $Q_k(i)$ be the $Q$-numbers of $({\mathscr{X}},(R_i))$. The *dual inner distribution* of $X$ is the sequence of numbers $(a'_k)_{k\in I}$, where $$a'_k=\sum_{i\in I}\,Q_k(i)\,a_i. \label{eqn:def_dual_distribution}$$ It is a well known fact of the general theory of association schemes that the numbers $a'_k$ are nonnegative (see [@DelLev1998 Theorem 3], for example).
It is readily verified that the mapping $\rho:X\times X\to{\mathbb{Z}}$, given by $$\rho(A,B)=\operatorname{rank}(A-B),$$ is a distance function on ${\mathscr{X}}$. Accordingly, given an integer $d$ satisfying $1\le d\le m$, we say that $X$ is a *$d$-code* in ${\mathscr{X}}$ if $\operatorname{rank}(A-B)\ge d$ for all distinct $A,B\in X$. Alternatively, writing $$I_\ell=\{2s-1:s\in{\mathbb{Z}},\,1\le 2s-1\le \ell\}\cup\{(2s,\pm1):s\in{\mathbb{Z}},\,2\le 2s\le \ell\},$$ we can define $X$ to be a $d$-code if $$a_i=0\quad\text{for each $i\in I_{d-1}$}.$$ We say that $X$ is a *$t$-design* if $$a'_k=0\quad\text{for each $k\in I_t$}.$$ A subset $X$ of ${\mathscr{X}}$ is *additive* if $X$ is a subgroup of $({\mathscr{X}},+)$. Note that the inner distribution $(a_i)_{i\in I}$ of an additive subset $X$ of ${\mathscr{X}}$ satisfies $$a_i={\lvertX\cap {\mathscr{X}}_i\rvert},$$ for every $i\in I$. The *annihilator* of an additive subset $Y$ of ${\mathscr{Q}}$ is defined to be $$Y^\circ=\{S\in {\mathscr{S}}:\langle Q,S\rangle=1\;\text{for each $Q\in Y$}\}$$ and the *annihilator* of an additive subset $Z$ of ${\mathscr{S}}$ is defined to be $$Z^\circ=\{Q\in {\mathscr{Q}}:\langle Q,S\rangle=1\;\text{for each $S\in Z$}\}.$$ Note that $(Y^\circ)^\circ=Y$ and $(Z^\circ)^\circ=Z$ and $${\lvert{\mathscr{Q}}\rvert}={\lvertY\rvert}\,{\lvertY^\circ\rvert}={\lvertZ\rvert}\,{\lvertZ^\circ\rvert}={\lvert{\mathscr{S}}\rvert}.$$ The following MacWilliams-type identity is a special case of a general property of association schemes (see [@DelLev1998 Theorem 27], for example).
\[thm:dual\] Let $X$ be an additive subset of ${\mathscr{X}}$ with inner distribution $(a_i)_{i\in I}$ and dual inner distribution $(a'_k)_{k\in I}$ and let $X^\circ$ be its annihilator with inner distribution $(a^\circ_k)_{k\in I}$. Then we have ${\lvertX\rvert}a^\circ_k=a'_k$.
Subsets of symmetric bilinear forms
-----------------------------------
In this section, we prove bounds on the size of $d$-codes in ${\mathscr{S}}$. We begin with the following proposition.
\[pro:inner\_dist\_S\] Let $Z$ be a subset of ${\mathscr{S}}(m,q)$ with inner distribution $(a_i)_{i\in I}$ and dual inner distribution $(a'_k)_{k\in I}$. Write $$\begin{aligned}
A_r&=a_{2r,1}+a_{2r,-1}+a_{2r-1}, &
A'_s&=a'_{2s,1}+a'_{2s,-1}+a'_{2s-1},\\
B_r&=a_{2r,1}+a_{2r,-1}+a_{2r+1},&
B'_s&=a'_{2s,1}+a'_{2s,-1}+a'_{2s+1},\\
C_r&=\frac{\alpha_{-1}}{\beta_r}a_{2r,1}-\frac{\alpha_{1}}{\beta_r}a_{2r,-1},&
C'_s&=q^{-s}(a'_{2s,1}-a'_{2s,-1}),\end{aligned}$$ where $\alpha_{\epsilon}$ and $\beta_r$ are given in Proposition \[pro:valencies\_multiplicities\]. Then we have $$\begin{aligned}
A'_s&=\sum_rF^{(m+1)}_s(r)A_r,\\
C'_s&=\sum_rF^{(m)}_s(r)B_r,\\
B'_s&=q^m\sum_rF^{(m)}_s(r)C_r.\end{aligned}$$
Since the $Q$-numbers $Q_k(i)$ of ${\mathscr{S}}(m,q)$ are the $P$-numbers $P_i(k)$ of ${\mathscr{Q}}(m,q)$, the result follows directly from and Proposition \[pro:P\_numbers\_sums\].
The following theorem was obtained in [@Sch2015] in the case that $q$ is odd and in [@Sch2010] in the case that $q$ is even and $d$ is odd. The case that $q$ and $d$ are even is new.
\[thm:bounds\_S\] Let $Z$ be a $d$-code in ${\mathscr{S}}(m,q)$, where $Z$ is required to be additive if $d$ is even. Then $${\lvertZ\rvert}\le\begin{cases}
q^{m(m-d+2)/2} & \text{for $m-d$ even}\\
q^{(m+1)(m-d+1)/2} & \text{for $m-d$ odd}.
\end{cases}$$ Moreover, in the case of odd $d$, equality occurs if and only if $Z$ is a $t$-design for $$t=2\left(\left\lfloor\frac{m+1}{2}\right\rfloor-\frac{d-1}{2}\right).$$
As remarked above, the only new case arises when $q$ is even. When $q$ is odd, the theorem was proved in [@Sch2015 Lemmas 3.5 and 3.6] using the identities for $A'_s$ and $C'_s$ in Proposition \[pro:inner\_dist\_S\]. Since these do not involve $C_r$ (which is the only quantity in the conclusion of Proposition \[pro:inner\_dist\_S\] that crucially depends on the parity of $q$), the proofs of [@Sch2015 Lemmas 3.5 and 3.6] carry over verbatim to the case that $q$ is even.
We call a $d$-code $Y$ in ${\mathscr{S}}(m,q)$ *maximal* if $d$ is odd and equality holds in Theorem \[thm:bounds\_S\]. We shall see in Section \[sec:Constructions\] that maximal $d$-codes in ${\mathscr{S}}$ exist for all possible parameters.
The situation for even $d$ is somewhat mysterious. Theorem \[thm:bounds\_S\] gives bounds for the largest additive $d$-codes in ${\mathscr{S}}(m,q)$ in this case and there certainly exist $d$-codes that are larger than the largest possible additive $d$-code [@Sch2016]. For example, the largest additive $2$-code in ${\mathscr{S}}(3,2)$ has $16$ elements by Theorem \[thm:bounds\_S\], whereas the largest $2$-code in ${\mathscr{S}}(3,2)$ has $22$ elements [@Sch2016]. In fact, this $2$-code is essentially unique and can be constructed by taking the zero matrix together with all $21$ nonalternating $3\times 3$ symmetric matrices of rank $2$. Moreover, [@Sch2016] contains (not necessarily optimal) $d$-codes in ${\mathscr{S}}(m,q)$ for many small values of $q$, $m$, and even $d$, which are larger than the largest additive $d$-codes in ${\mathscr{S}}(m,q)$.
Subsets of quadratic forms
--------------------------
In this section, we prove bounds on the size of $d$-codes in ${\mathscr{Q}}$. We begin with the following counterpart of Proposition \[pro:inner\_dist\_S\].
\[pro:inner\_dist\_Q\] Let $Y$ be a subset of ${\mathscr{Q}}(m,q)$ with inner distribution $(a_i)_{i\in I}$ and dual inner distribution $(a'_i)_{i\in I}$. Write $$\begin{aligned}
A_s&=a_{2s,1}+a_{2s,-1}+a_{2s-1}, & A'_r&=a'_{2r,1}+a'_{2r,-1}+a'_{2r-1},\\
B_s&=a_{2s,1}+a_{2s,-1}+a_{2s+1}, & B'_r&=a'_{2r,1}+a'_{2r,-1}+a'_{2r+1},\\
C_s&=q^{-s}(a_{2s,1}-a_{2s,-1}), & C'_r&=\frac{\alpha_{-1}}{\beta_r}a'_{2r,1}-\frac{\alpha_{1}}{\beta_r}a'_{2r,-1},\end{aligned}$$ where $\alpha_{\epsilon}$ and $\beta_r$ are given in Proposition \[pro:valencies\_multiplicities\]. Then we have $$\begin{aligned}
A'_r&=\sum_sF^{(m+1)}_r(s)A_s,\\
C'_r&=\sum_sF^{(m)}_r(s)B_s,\\
B'_r&=q^m\sum_sF^{(m)}_r(s)C_s.\end{aligned}$$
This follows directly from and Proposition \[pro:Q\_numbers\_sums\].
In the next theorem, we give bounds for $d$-codes in ${\mathscr{Q}}$. Since the association schemes on ${\mathscr{Q}}(m,q)$ and ${\mathscr{S}}(m,q)$ are isomorphic for odd $q$, the statement of Theorem \[thm:bounds\_S\] still holds when $Z$ is a $d$-code in ${\mathscr{Q}}(m,q)$ and $q$ is odd. We therefore give bounds for $d$-codes in ${\mathscr{Q}}(m,q)$ only for even $q$.
\[thm:bound\_Q\] Let $q$ be even and let $Y$ be a $d$-code in ${\mathscr{Q}}(m,q)$. Then $${\lvertY\rvert}\le\begin{cases}
q^{m(m-d+2)/2} & \text{for odd $m$ and odd $d$},\\
q^{(m+1)(m-d+1)/2} & \text{for even $m$ and odd $d$},\\
q^{(m-1)(m-d+2)/2} & \text{for even $m$ and even $d$},\\
q^{m(m-d+1)/2} & \text{for odd $m$ and even $d$}.
\end{cases}$$ Moreover, in the case of odd $d$, equality occurs if and only if $Y$ is a $t$-design for $$t=2\left(\left\lfloor\frac{m+1}{2}\right\rfloor-\frac{d-1}{2}\right).$$
Let $(a_i)_{i\in I}$ be the inner distribution of $Y$. First assume that $d$ is odd, say $d=2{\delta}-1$. Let $A_s$ and $A'_r$ be as defined in Proposition \[pro:inner\_dist\_Q\] and put $$n=\lfloor (m+1)/2\rfloor\quad\text{and}\quad c=q^{m(m+1)/(2n)}.$$ From Proposition \[pro:inner\_dist\_Q\] and we obtain $$\sum_{r=0}^{n-\delta+1}{n-r\brack \delta-1}A'_r=c^{n-\delta+1}\sum_{s=0}^n{n-s\brack n-\delta+1}A_s.$$ Since $A'_0={\lvertY\rvert}$ and $A_0=1$ and $A_s=0$ for $0<s<\delta$, we obtain $$\sum_{r=1}^{n-\delta+1}{n-r\brack \delta-1}A'_r={n\brack \delta-1}(c^{n-\delta+1}-{\lvertY\rvert}).$$ Since the numbers $A'_r$ are nonnegative, the left-hand side is nonnegative, and therefore ${\lvertY\rvert}\le c^{n-\delta+1}$, as required. Moreover, this inequality is an equality if and only if $A'_1=\cdots=A'_{n-{\delta}+1}=0$, which is equivalent to $Y$ being a $t$-design for $t=2(n-{\delta}+1)$.
Now assume that $d$ is even, say $d=2\delta$. Let $B_s$ and $C'_r$ be as defined in Proposition \[pro:inner\_dist\_Q\] and put $$n=\lfloor m/2\rfloor\quad\text{and}\quad c=q^{m(m-1)/(2n)}.$$ From Proposition \[pro:inner\_dist\_Q\] and we obtain $$\sum_{r=0}^{n-\delta+1}{n-r\brack \delta-1}C'_r=c^{n-\delta+1}\sum_{s=0}^n{n-s\brack n-\delta+1}B_s$$ and find, similarly as above, $$\sum_{r=1}^{n-\delta+1}{n-r\brack \delta-1}C'_r={n\brack \delta-1}(c^{n-\delta+1}-{\lvertY\rvert}).$$ Again, we have ${\lvertY\rvert}\le c^{n-\delta+1}$, which completes the proof.
We call a $d$-code $Y$ in ${\mathscr{Q}}(m,q)$ *maximal* if equality holds in Theorem \[thm:bound\_Q\] or in Theorem \[thm:bounds\_S\], unless $q$ is odd and $d$ is even. We shall see in Section \[sec:Constructions\] that maximal $d$-codes in ${\mathscr{Q}}$ exist for all possible parameters.
An interesting situation, in particular from the coding-theoretic viewpoint of Section \[sec:applications\], occurs for $d$-codes in ${\mathscr{Q}}$, when $d$ and $m$ are even and no difference between distinct elements is hyperbolic of rank $d$. We call such a set an *elliptic* $d$-code.
\[thm:bound\_elliptic\_Q\] Let $m$ and $d$ be even and let $Y$ be an elliptic $d$-code in ${\mathscr{Q}}(m,q)$. Then $${\lvertY\rvert}\le q^{m(m-d+1)/2}.$$ Moreover, equality occurs if and only if $Y$ is a $t$-design for $t=m-d+1$.
Write $\delta=d/2$ and $n=m/2$. Let $(a_i)_{i\in I}$ be the inner distribution of $Y$ and let $A_s$, $A'_r$, $C_s$, and $B'_r$ be as defined in Proposition \[pro:inner\_dist\_Q\]. From Proposition \[pro:inner\_dist\_Q\] and we obtain $$\sum_{r=0}^{n-\delta}{n-r\brack \delta}(q^\delta A'_r+B'_r)=q^{(m+1)(n-\delta)}q^\delta\sum_{s=0}^n{n-s\brack n-\delta}(A_s+q^\delta C_s)$$ and therefore, since $A_s=C_s=0$ for $0<s<\delta$, $$\sum_{r=0}^{n-\delta}{n-r\brack \delta}(q^\delta A'_r+B'_r)=q^{(m+1)(n-\delta)}q^\delta\left({n\brack \delta}(A_0+q^\delta C_0)+A_\delta+q^\delta C_\delta\right).$$ We have $$\begin{aligned}
A_\delta+q^\delta C_\delta&=a_{2\delta,1}+a_{2\delta,-1}+a_{2\delta-1}+(a_{2\delta,1}-a_{2\delta,-1})\\
&=2a_{2\delta,1}+a_{2\delta-1}\\
&=0\end{aligned}$$ since $Y$ is an elliptic $(2\delta)$-code. Since $A_0=C_0=1$ and $A'_0={\lvertY\rvert}$ and $B'_0={\lvertY\rvert}+a'_1$, we then obtain $${n\brack \delta}a'_1+\sum_{r=1}^{n-\delta}{n-r\brack \delta}(q^\delta A'_r+B'_r)={n\brack \delta}(1+q^\delta)(q^{(m+1)(n-\delta)}q^\delta-{\lvertY\rvert}).$$ Since the left-hand side is nonnegative, we find that $${\lvertY\rvert}\le q^{(m+1)(n-\delta)}q^\delta.$$ Moreover, equality occurs if and only if $q^\delta A'_r+B'_r=0$ for all $r$ satisfying $1\le r\le n-\delta$, or equivalently if and only if $Y$ is a $t$-design for $t=m-d+1$.
We call an elliptic $(2{\delta})$-code $Y$ in ${\mathscr{Q}}(2n,q)$ *maximal* if equality holds in Theorem \[thm:bound\_elliptic\_Q\]. We shall see in Section \[sec:Constructions\] that maximal elliptic $d$-codes in ${\mathscr{Q}}(m,q)$ exist for all possible parameters.
Inner distributions of maximal codes
------------------------------------
If $Z$ is a subset of ${\mathscr{S}}(m,q)$ such that the bound in Theorem \[thm:bounds\_S\] holds with equality, then in many cases [@Sch2010] and [@Sch2015] give explicit expressions for the inner distribution of $Z$. These results carry over to subsets of ${\mathscr{Q}}(m,q)$ in the case that $q$ is odd.
In this section we provide explicit expressions for the inner distributions of maximal $d$-codes in ${\mathscr{Q}}(m,q)$. We note that, once we know Proposition \[pro:inner\_dist\_Q\] for even $q$, the results in this section can be proved with methods that are very similar to those of [@Sch2015 Section 3.3]. Hence the proofs in this section are sketched only.
Our first result holds for $d$-codes in ${\mathscr{Q}}(m,q)$, where $d$ is odd.
\[thm:inner\_dist\_d\_odd\] If $Y$ is a maximal $(2{\delta}+1)$-code in ${\mathscr{Q}}(2n+1,q)$, then its inner distribution $(a_i)_{i\in I}$ satisfies $$\begin{gathered}
a_{2s-1}={n\brack s-1}\sum_{j=0}^{s-{\delta}-1}(-1)^jq^{j(j-1)}{s\brack j}\big(q^{(2n+1)(s-{\delta}-j)}-1\big),\\[1ex]
a_{2s,{\tau}}=\frac{1}{2}q^s\big(q^s+{\tau}\big)\,{n\brack s}\sum_{j=0}^{s-{\delta}-1}(-1)^jq^{j(j-1)}{s\brack j}\big(q^{(2n+1)(s-{\delta}-j)}-1\big)\end{gathered}$$ for $s>0$. If $Y$ is a maximal $(2{\delta}+1)$-code in ${\mathscr{Q}}(2n,q)$, then its inner distribution $(a_i)_{i\in I}$ satisfies $$\begin{aligned}
a_{2s-1,{\tau}}&=\frac{1}{2}(q^{2s}-1){n\brack s}\sum_{j=0}^{s-{\delta}-1}(-1)^jq^{j(j-1)}{s-1\brack j}q^{(2n+1)(s-{\delta}-j-1)+2j},\\[1ex]
a_{2s,{\tau}}&=\frac{1}{2}{n\brack s}\sum_{j=0}^{s-{\delta}}(-1)^jq^{j(j-1)}{s\brack j}\big(q^{(2n+1)(s-{\delta}-j)+2j}-1\big)\\
&+\frac{{\tau}}{2}\,q^s{n\brack s}\sum_{j=0}^{s-{\delta}-1}(-1)^jq^{j(j-1)}{s\brack j}\big(q^{(2n+1)(s-{\delta}-j)+2(j-s)}-1\big)\end{aligned}$$ for $s>0$.
If $Y$ is a maximal $d$-code in ${\mathscr{Q}}(m,q)$, where $d$ is odd, then Theorems \[thm:bounds\_S\] and \[thm:bound\_Q\] imply that $Y$ is a $t$-design for $$t=2\left(\left\lfloor\frac{m+1}{2}\right\rfloor-\frac{d-1}{2}\right).$$ For odd $q$, the theorem is then [@Sch2015 Theorem 3.9] and its proof relies just on Proposition \[pro:inner\_dist\_S\]. For even $q$, the proof is almost identical if we use Proposition \[pro:inner\_dist\_Q\] instead of Proposition \[pro:inner\_dist\_S\].
The next result holds for maximal $d$-codes in ${\mathscr{Q}}(m,q)$ when $q$ is even and $d$ is even. In this case, the inner distribution is only partially determined. It is not clear whether there exist such $d$-codes with different inner distribution.
If $q$ is even and $Y$ is a maximal $(2{\delta})$-code in ${\mathscr{Q}}(m,q)$, then its inner distribution $(a_i)_{i\in I}$ satisfies $$a_{2s,1}+a_{2s,-1}+a_{2s+1}={n\brack s}\sum_{j=0}^{s-{\delta}}(-1)^jq^{j(j-1)}{s\brack j}\big(c^{s-{\delta}-j+1}-1\big)$$ for $s>0$, where $n={\lfloorm/2\rfloor}$ and $c=q^{m(m-1)/(2n)}$.
Let $B_s$ and $C'_r$ be defined as in Proposition \[pro:inner\_dist\_Q\], so that $$C'_r=\sum_sF^{(m)}_r(s)B_s.$$ In particular $B_s=a_{2s,1}+a_{2s,-1}+a_{2s+1}$. If $Y$ is a maximal $(2{\delta})$-code in ${\mathscr{Q}}(m,q)$, then we conclude from the proof of Theorem \[thm:bound\_Q\] that $C'_r=0$ for all $r$ satisfying $1\le r\le n-{\delta}+1$. This gives enough equations to solve for the numbers $B_s$. The solution is given by [@Sch2015 Lemma 3.8].
The final result of this section concerns maximal elliptic $(2{\delta})$-codes in ${\mathscr{Q}}(2n,q)$.
\[thm:inner\_dist\_elliptic\] If $Y$ is a maximal elliptic $(2{\delta})$-code in ${\mathscr{Q}}(2n,q)$, then its inner distribution $(a_i)_{i\in I}$ satisfies $$\begin{aligned}
a_{2s-1}&=\frac{1}{2}(q^{2s}-1){n\brack s}\sum_{j=0}^{s-{\delta}-1}(-1)^jq^{j(j-1)}{s-1\brack j}\big(q^{2n(s-{\delta}-j-1)}q^{s+j-1}-1\big),\\[1ex]
a_{2s,{\tau}}&=\frac{1}{2}(q^s+{\tau}){n\brack s}\sum_{j=0}^{s-{\delta}}(-1)^jq^{j(j-1)}{s\brack j}\big(q^{2n(s-{\delta}-j)}q^j-{\tau}\big)\end{aligned}$$ for $s>0$.
If $Y$ is a maximal elliptic $(2{\delta})$-code in ${\mathscr{Q}}(2n,q)$, then by Theorem \[thm:bound\_elliptic\_Q\] we have ${\lvertY\rvert}=q^{2n(n-{\delta}+1/2)}$ and $Y$ is a $(2n-2{\delta}+1)$-design. The proof is then identical to that of the first part of [@Sch2015 Proposition 3.10].
Constructions {#sec:Constructions}
=============
In this section we provide constructions of maximal $d$-codes in ${\mathscr{S}}(m,q)$ and ${\mathscr{Q}}(m,q)$ using field extensions of ${\mathbb{F}}_q$. Throughout this section we take $V={\mathbb{F}}_{q^m}$ and use the *relative trace function* $\operatorname{Tr}_m:{\mathbb{F}}_{q^m}\to{\mathbb{F}}_q$, which is given by $$\operatorname{Tr}_m(y)=\sum_{i=1}^my^{q^i}.$$
Canonical representations
-------------------------
In what follows we give canonical representations of quadratic forms $Q$ and symmetric bilinear forms $S$ on ${\mathbb{F}}_{q^m}$ and describe the pairing ${\langleQ,S\rangle}$ in terms of these representations.
\[thm:sym\_quad\_unique\_representation\] Let $Q\in{\mathscr{Q}}(m,q)$ be a quadratic form and let $S\in{\mathscr{S}}(m,q)$ be a symmetric bilinear form.
(1) If $m$ is odd, say $m=2n-1$, then there exist unique $f_0,\dots,f_{n-1}\in{\mathbb{F}}_{q^m}$ and $g_0,\dots,g_{n-1}\in{\mathbb{F}}_{q^m}$ such that $Q$ is given by $$Q(x)=\sum_{i=0}^{n-1}\operatorname{Tr}_m(f_ix^{q^i+1})$$ and $S$ is given by $$S(x,y)=\operatorname{Tr}_m(g_0xy)+\sum_{i=1}^{n-1}\operatorname{Tr}_m(g_i(xy^{q^i}+x^{q^i}y)).$$ Moreover, there are ${\mathbb{F}}_q$-bases for ${\mathbb{F}}_{q^m}$ such that with respect to these bases we have $${\langleQ,S\rangle}=\chi\left(\sum\limits_{i=0}^{n-1}\operatorname{Tr}_m(f_ig_i)\right).$$
(2) If $m$ is even, say $m=2n$, then there exist unique $f_0,\dots,f_{n-1}\in{\mathbb{F}}_{q^m}$ and $g_0,\dots,g_{n-1}\in{\mathbb{F}}_{q^m}$ and $f_n,g_n\in{\mathbb{F}}_{q^n}$ such that $Q$ is given by $$Q(x)=\sum_{i=0}^{n-1}\operatorname{Tr}_m(f_ix^{q^i+1})+\operatorname{Tr}_n(f_nx^{q^n+1})$$ and $S$ is given by $$S(x,y)=\operatorname{Tr}_m(g_0xy)+\sum_{i=1}^{n-1}\operatorname{Tr}_m(g_i(xy^{q^i}+x^{q^i}y))+\operatorname{Tr}_m(g_nxy^{q^n}).$$ Moreover, there are ${\mathbb{F}}_q$-bases for ${\mathbb{F}}_{q^m}$ such that with respect to these bases we have $${\langleQ,S\rangle}=\chi\left(\sum\limits_{i=0}^{n-1}\operatorname{Tr}_m(f_ig_i)+\operatorname{Tr}_n(f_ng_n)\right).$$
To prove Theorem \[thm:sym\_quad\_unique\_representation\], we require some notation and a lemma. Given a linearised polynomial $L\in{\mathbb{F}}_{q^m}[X]$ of the form $$L=\sum_{k=0}^{m-1}c_kX^{q^k}, \label{eqn:linearised_poly}$$ we associate with $L$ its *Dickson matrix* $D_L$, given by $(D_L)_{1\le i,j\le m}=c_{j-i}^{q^i}$, where the index of $c_k$ is taken modulo $m$. Henceforth the entries of an $m\times m$ matrix $M$ are denoted by $M_{ij}$, where $1\le i,j\le m$.
\[lem:lin\_poly\_to\_mat\] Let $L\in{\mathbb{F}}_{q^m}[X]$ be the linearised polynomial . Let $\{\xi_1,\xi_2,\dots,\xi_m\}$ be an ${\mathbb{F}}_q$-basis for ${\mathbb{F}}_{q^m}$ and let $M\in{\mathbb{F}}_q^{m\times m}$ be given by $M_{ij}=\operatorname{Tr}_m(\xi_iL(\xi_j))$. Then we have $$M=PD_LP^T,$$ where $P\in{\mathbb{F}}_q^{m\times m}$ is given by $P_{ij}=\xi_i^{q^j}$.
We can write $M=PR$, where $R_{ij}=L(\xi_j)^{q^i}$. For every $x\in{\mathbb{F}}_{q^m}$, we have $$L(x)^{q^i}=\sum_{k=1}^mc_{k-i}^{q^i}\,x^{q^k},$$ where the index is taken modulo $m$. We conclude that $R=D_LP^T$, as required.
We now prove Theorem \[thm:sym\_quad\_unique\_representation\].
It is easy to see that the possible choices for the $f_i$’s and the $g_i$’s yield $q^{m(m+1)/2}$ quadratic forms and $q^{m(m+1)/2}$ symmetric bilinear forms. In order to prove that these are distinct, it is sufficient to show that $Q$ or $S$ is the zero form if and only if the $f_i$’s are all zero or the $g_i$’s are all zero, respectively. For ${\mathscr{S}}(m,q)$ and odd $m$, this is accomplished by the proof of Theorem \[thm:constr\_sym\]. The other cases can be proved similarly, which we leave to the reader. This proves the existence and uniqueness of the $f_i$’s and the $g_i$’s.
It remains to prove the expressions for the pairing ${\langleQ,S\rangle}$. We present the proof only in the case that $m$ is odd. Slight modifications also give a proof for even $m$. Let $\{\alpha_1,\alpha_2,\dots,\alpha_m\}$ and $\{\beta_1,\beta_2,\dots,\beta_m\}$ be a pair of dual ${\mathbb{F}}_q$-bases for ${\mathbb{F}}_{q^m}$, that is $$\operatorname{Tr}_m(\alpha_i\beta_j)=\delta_{ij}\quad\text{for all $i,j$}.$$ We use the former basis to associate cosets of alternating matrices with quadratic forms and the latter to associate symmetric matrices with symmetric bilinear forms. It will be convenient to define the $m\times m$ matrices $U$ and $V$ by $U_{ij}=\alpha_i^{q^j}$ and $V_{ij}=\beta_i^{q^j}$. Notice that the duality of the two involved bases implies $UV^T=I$, and so $U^TV=I$.
Define the linearised polynomials $$\begin{aligned}
{3}
F_0&=f_0X, \quad & F_1&=\sum_{i=1}^{n-1}(f_iX^{q^i}+f_i^{q^{m-i}}X^{q^{m-i}}),\quad F_2&\;=\sum_{i=1}^{n-1}f_iX^{q^i},\\
G_0&=g_0X, \quad & G_1&=\sum_{i=1}^{n-1}(g_iX^{q^i}+g_i^{q^{m-i}}X^{q^{m-i}}).\end{aligned}$$ Then we have $$S(x,y)=\operatorname{Tr}_m(x(G_0(y)+G_1(y)))$$ and so the matrix $B$ of $S$ is given by $B=B_0+B_1$, where $$\begin{aligned}
(B_0)_{ij}&=\operatorname{Tr}_m(\beta_iG_0(\beta_j)),\\
(B_1)_{ij}&=\operatorname{Tr}_m(\beta_iG_1(\beta_j)).\end{aligned}$$ To associate cosets of alternating matrices with quadratic forms, we distinguish the cases that $q$ is odd or even.
For odd $q$, let $A$ be the unique symmetric matrix associated with the quadratic form $Q$. From we find that this matrix is given by $$\begin{aligned}
A_{ij}&=\tfrac{1}{2}(Q(\alpha_i+\alpha_j)-Q(\alpha_i)-Q(\alpha_j))\\[1ex]
&=\operatorname{Tr}_m(\alpha_i(F_0(\alpha_j)+\tfrac{1}{2}F_1(\alpha_j))).\end{aligned}$$ Write $F=F_0+\tfrac{1}{2}F_1$ and $G=G_0+G_1$ and use Lemma \[lem:lin\_poly\_to\_mat\] to obtain $$\operatorname{tr}(AB)=\operatorname{tr}(UD_FU^TVD_GV^T)=\operatorname{tr}(D_FD_G),$$ and therefore $$\operatorname{tr}(AB)=\sum\limits_{i=0}^{n-1}\operatorname{Tr}_m(f_ig_i),$$ as required.
For even $q$, let $A'$ be the unique upper triangular matrix associated with $Q$. From we find that this matrix is given by $A'_{ii}=Q(\alpha_i)$ and $A'_{ij}=Q(\alpha_i+\alpha_j)-Q(\alpha_i)-Q(\alpha_j)$ for $i<j$. In fact, it is more convenient to work with a slightly different matrix of $Q$, namely $A=A_0+A_1$, where $A_0$ and $A_1$ are given by $$\begin{aligned}
(A_0)_{ij}&=\operatorname{Tr}_m(\alpha_iF_0(\alpha_j)) \nonumber\\[1ex]
(A_1)_{ij}&=\begin{cases}
\operatorname{Tr}_m(\alpha_iF_1(\alpha_j)) & \text{for $i<j$}\\
\operatorname{Tr}_m(\alpha_iF_2(\alpha_j)) & \text{for $i=j$}\\
0 & \text{otherwise}.
\end{cases} \label{eqn:def_A1}\end{aligned}$$ Notice that $A-A'$ is alternating, which is in fact the off-diagonal part of $A_0$. Therefore $A$ and $A'$ represent the same quadratic form. We have $$\operatorname{tr}(AB)=\operatorname{tr}(A_0B_0)+\operatorname{tr}(A_1B_1)+\operatorname{tr}(A_0B_1)+\operatorname{tr}(A_1B_0).$$ Using Lemma \[lem:lin\_poly\_to\_mat\] we have $$\operatorname{tr}(A_0B_0)=\operatorname{tr}(UD_{F_0}U^TVD_{G_0}V^T)=\operatorname{tr}(D_{F_0}D_{G_0})=\operatorname{Tr}_m(f_0g_0).$$ Now define an inner product on alternating matrices in ${\mathbb{F}}_q^{m\times m}$ by $$(X,Y)=\sum_{i<j}X_{ij}Y_{ij}.$$ This inner product satisfies $$(WXW^T,Y)=(X,W^TYW)$$ for every $W\in{\mathbb{F}}_q^{m\times m}$. Using this property and Lemma \[lem:lin\_poly\_to\_mat\], we obtain $$\operatorname{tr}(A_1B_1)=(A_1+A_1^T,B_1)=(UD_{F_1}U^T,VD_{G_1}V^T)=(D_{F_1},D_{G_1}),$$ and therefore $$\operatorname{tr}(A_1B_1)=\sum_{i=1}^{n-1}\operatorname{Tr}_m(f_ig_i).$$ Now, since $A_0$ is symmetric and $B_1$ is alternating, we have $\operatorname{tr}(A_0B_1)=0$. From Lemma \[lem:lin\_poly\_to\_mat\], we find that $B_0=VD_{G_0}V^T$, where $D_{G_0}$ is a diagonal matrix. We claim that $A_1=UEU^T$, where $E$ has only zeros on the main diagonal. This implies that $$\operatorname{tr}(A_1B_0)=\operatorname{tr}(UEU^TVD_{G_0}V^T)=\operatorname{tr}(ED_{G_0})=0,$$ and so completes the proof.
It remains to prove the claim. The required matrix $E$ is given by $E=V^TA_1V$, and so for every $i$, we have using $$\begin{aligned}
E_{ii}&=\sum_{k,\ell}\beta_k^{q^i}(A_1)_{k\ell}\,\beta_\ell^{q^i}\\
&=\sum_k\beta_k^{q^i}\operatorname{Tr}_m(\alpha_kF_2(\alpha_k))\beta_k^{q^i}+\sum_{k<\ell}\beta_k^{q^i}\operatorname{Tr}_m(\alpha_kF_1(\alpha_\ell))\beta_\ell^{q^i}\\
&=\sum_k\beta_k^{q^i}\operatorname{Tr}_m(\alpha_kF_2(\alpha_k))\beta_k^{q^i}+\sum_{k<\ell}\beta_k^{q^i}\operatorname{Tr}_m(\alpha_kF_2(\alpha_\ell)+\alpha_\ell F_2(\alpha_k))\beta_\ell^{q^i}\\
&=\sum_{k,\ell}\beta_k^{q^i}\operatorname{Tr}_m(\alpha_kF_2(\alpha_\ell))\beta_\ell^{q^i}\\
&=v_i^TUD_{F_2}U^Tv_i=(v_i^TU)D_{F_2}(v_i^TU)^T,\end{aligned}$$ where $v_i$ is the $i$-th column of $V$. Since $V^TU=I$, we find that the main diagonal of $E$ equals the main diagonal of $D_{F_2}$, which is zero. This proves the claim.
The constructions
-----------------
We now give constructions of maximal $d$-codes in ${\mathscr{S}}(m,q)$ and ${\mathscr{Q}}(m,q)$. We begin with recalling constructions from [@Sch2010] and [@Sch2015] of additive $d$-codes in ${\mathscr{S}}(m,q)$.
\[thm:constr\_sym\] Let $d$ be an integer with the same parity as $m$ satisfying $1\le d\le m$ and let $Z$ be the subset of ${\mathscr{S}}(m,q)$ formed by the symmetric bilinear forms $$\begin{gathered}
S:{\mathbb{F}}_{q^m}\times {\mathbb{F}}_{q^m}\to{\mathbb{F}}_q\\
S(x,y)=\operatorname{Tr}_m(g_0xy)+\sum_{i=1}^{(m-d)/2}\operatorname{Tr}_m(g_i(xy^{q^i}+x^{q^i}y)),\quad g_i\in{\mathbb{F}}_{q^m}.\end{gathered}$$ Then $Z$ is an additive $d$-code in ${\mathscr{S}}(m,q)$ of size $q^{m(m-d+2)/2}$. In particular, $Z$ is a maximal $d$-code in ${\mathscr{S}}(m,q)$ for odd $m$ and is maximal among additive $d$-codes in ${\mathscr{S}}(m,q)$ for even $m$.
Whenever $m-d$ is odd, Theorem \[thm:constr\_sym\] gives $(d+2)$-codes $Z$ in ${\mathscr{S}}(m+1,q)$ for which equality holds in Theorem \[thm:bounds\_S\]. Let $W$ be an $m$-dimensional subspace of $V(m+1,q)$ and define the *punctured* set (with respect to $W$) of $Z$ to be $$Z^*=\big\{S|_W:S\in Z\big\},$$ where $S|_W$ is the restriction of $S$ onto $W$. Then $Z^*$ is a $d$-code in ${\mathscr{S}}(m,q)$ for which again equality holds in Theorem \[thm:bounds\_S\]. This shows that $Z^*$ is a maximal $d$-code in ${\mathscr{S}}(m,q)$ for odd $d$ and is maximal among additive $d$-codes in ${\mathscr{S}}(m,q)$ for even $d$.
For odd $q$, Theorem \[thm:sym\_quad\_unique\_representation\] of course also gives corresponding sets of quadratic forms by associating a quadratic form $Q$ with $S$ via $Q(x)=\tfrac{1}{2}S(x,x)$. It therefore remains to give constructions of maximal $d$-codes in ${\mathscr{Q}}(m,q)$ for even $q$. The following consequence of Theorems \[thm:sym\_quad\_unique\_representation\] and \[thm:constr\_sym\] gives a construction for $d$-codes in ${\mathscr{Q}}(m,q)$ when both $m$ and $d$ are odd (and where $q$ can have either parity).
\[thm:constr\_m\_odd\_d\_odd\] Let $m$ and $d$ be odd integers satisfying $1\le d\le m$ and let $Y$ be the subset of ${\mathscr{Q}}(m,q)$ formed by the quadratic forms $$\begin{gathered}
Q:{\mathbb{F}}_{q^m}\to{\mathbb{F}}_q\\
Q(x)=\sum_{i=(d-1)/2}^{(m-1)/2}\operatorname{Tr}_m(f_ix^{q^i+1}),\quad f_i\in{\mathbb{F}}_{q^m}.\end{gathered}$$ Then $Y$ is additive and a maximal $d$-code in ${\mathscr{Q}}(m,q)$ of size $q^{m(m-d+2)/2}$.
It is plain that $Y$ is additive and has size $q^{m(m-d+2)/2}$. From Theorems \[thm:sym\_quad\_unique\_representation\] and \[thm:constr\_sym\] we find that the annihilator of $Y^\circ$ of $Y$ is a maximal $(m-d+3)$-code in ${\mathscr{S}}(m,q)$. Theorem \[thm:bounds\_S\] implies that $Y^\circ$ is a $(d-1)$-design and Theorem \[thm:dual\] then implies that $Y$ is a $d$-code. From Theorem \[thm:bound\_Q\] we find hat $Y$ is maximal.
Whenever $d$ is odd and $m$ is even, Theorem \[thm:constr\_m\_odd\_d\_odd\] gives maximal $(d+2)$-codes $Y$ in ${\mathscr{Q}}(m+1,q)$. In fact, Theorem \[thm:bound\_Q\] implies that $Y$ is also a maximal $(d+1)$-code in ${\mathscr{Q}}(m+1,q)$. Let $W$ be an $m$-dimensional subspace of $V(m+1,q)$ and define the *punctured* set (with respect to $W$) of $Y$ to be $$Y^*=\big\{Q|_W:Q\in Y\big\},$$ where $Q|_W$ is the restriction of $Q$ onto $W$. Then $Y^*$ is a maximal $d$-code in ${\mathscr{Q}}(m,q)$. This leaves the case that $m$ and $d$ are both even. In this case we have the following construction, which identifies $V$ with ${\mathbb{F}}_{q^{m-1}}\times{\mathbb{F}}_q$ and is essentially contained in [@DelGoe1975].
\[thm:constr\_elliptic\_codes\] Let $q$ be even, let $m$ and $d$ be even integers satisfying $1\le d\le m$, and let $Y$ be the subset of ${\mathscr{Q}}(m,q)$ formed by the quadratic forms $$\begin{gathered}
Q:{\mathbb{F}}_{q^{m-1}}\times{\mathbb{F}}_q\to{\mathbb{F}}_q\\
Q(x,u)=\sum_{i=1}^{m/2-1}\operatorname{Tr}_{m-1}\big((f_0x)^{q^i+1}\big)+u\operatorname{Tr}_{m-1}(f_0x)+\sum_{i=1}^{(m-d)/2}\operatorname{Tr}_{m-1}\big(f_ix^{q^i+1}\big),\end{gathered}$$ where $f_i\in{\mathbb{F}}_{q^{m-1}}$. Then $Y$ is a maximal $d$-code in ${\mathscr{Q}}(m,q)$ of size $q^{(m-1)(m-d+2)/2}$.
The quadratic form $Q$ polarises to the bilinear form $$\operatorname{Tr}_{m-1}(f_0^2xy+f_0y\operatorname{Tr}_{m-1}(f_0x)+f_0(uy+vx))+\sum_{i=1}^{(m-d)/2}\operatorname{Tr}_{m-1}(f_i(xy^{q^i}+x^{q^i}y)).$$ It is known [@DelGoe1975 Theorem 9] that the difference between two such forms for distinct $(f_0,f_1,\dots,f_{(m-d)/2})$ has rank at least $d$. Therefore $Y$ is a $d$-code in ${\mathscr{Q}}(m,q)$ of size $q^{(m-1)(m-d+2)/2}$, hence a maximal $d$-code in ${\mathscr{Q}}(m,q)$ by Theorem \[thm:bound\_Q\].
We close this section by giving a construction for maximal elliptic $d$-codes in ${\mathscr{Q}}(m,q)$.
\[thm:constr\_elliptic\] Let $m$ be even and write $m=2n$. Let $\delta$ be an integer satisfying $1\le\delta\le n$ and let $Y$ be the subset of ${\mathscr{Q}}(m,q)$ formed by the quadratic forms $$\begin{gathered}
Q:{\mathbb{F}}_{q^m}\to{\mathbb{F}}_q\\
Q(x)=\sum_{i=\delta}^{n-1}\operatorname{Tr}_m(f_ix^{q^i+1})+\operatorname{Tr}_n(f_nxy^{q^n}),\quad f_i\in{\mathbb{F}}_{q^m},f_n\in{\mathbb{F}}_{q^n}.\end{gathered}$$ Then $Y$ is a maximal elliptic $(2\delta)$-code in ${\mathscr{Q}}(m,q)$ of size $q^{m(n-{\delta}+1/2)}$.
It is plain that $Y$ is additive. A straightforward computation gives $$Q(x+y)-Q(x)-Q(y)=\operatorname{Tr}_m(yL(x)),$$ where $$L(x)=f_nx^{q^n}+\sum_{i=\delta}^{n-1}\left(f_ix^{q^i}+f_i^{q^{2n-i}}x^{q^{2n-i}}\right).$$ Since $L(x^{q^{2n-\delta}})$ is induced by a polynomial of degree at most $2n-2\delta$, we find that $Q$ has rank at least $2\delta$, unless $f_\delta=\dots=f_n=0$. Hence $Y$ is $(2\delta)$-code of size $q^{m(n-\delta+1/2)}$.
Let $(a_i)_{i\in I}$ be the inner distribution of $Y$ and let $A_s$, $A'_r$, $C_s$, and $B'_r$ be as defined in Proposition \[pro:inner\_dist\_Q\]. By Theorems \[thm:sym\_quad\_unique\_representation\] and \[thm:constr\_sym\], the annihilator $Y^\circ$ of $Y$ is a $(2n-2\delta+2)$-code in ${\mathscr{S}}(m,q)$. Thus Theorem \[thm:dual\] implies that $A'_r=B'_r=0$ for all $r$ satisfying $1\le r\le n-\delta$. As in the proof of Theorem \[thm:bound\_elliptic\_Q\], we find that $${n\brack \delta}(q^\delta A'_0+B'_0)=q^{(m+1)(n-\delta)}q^\delta\left({n\brack \delta}(A_0+q^\delta C_0)+A_\delta+q^\delta C_\delta\right).$$ Since $A'_0=B'_0={\lvertY\rvert}=q^{m(n-\delta+1/2)}$ and $A_0=C_0=1$, we conclude that $$A_\delta+q^\delta C_\delta=0.$$ We have $A_\delta+q^\delta C_\delta=2a_{2\delta,1}+a_{2\delta-1}$ by definition and $a_{2\delta-1}=0$ since $Y$ is a $(2\delta)$-code. Therefore $a_{2\delta,1}=0$, and so $Y$ is an elliptic $(2\delta)$-code, hence a maximal elliptic $(2\delta)$-code in ${\mathscr{Q}}(m,q)$ by Theorem \[thm:bound\_elliptic\_Q\].
Applications to classical coding theory {#sec:applications}
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In this section we construct classical error-correcting codes over finite fields from subsets of ${\mathscr{Q}}(m,q)$, extending results from [@Sch2015] for odd $q$.
A *code* over ${\mathbb{F}}_q$ of *length* $n$ is a subset of ${\mathbb{F}}_q^n$; such a code is *additive* if it is a subgroup of $({\mathbb{F}}_q^n,+)$. The (Hamming) *weight* of $c\in{\mathbb{F}}_q^n$, denoted by $\operatorname{wt}(c)$, is the number of nonzero entries in $c$. This weight induces a distance on ${\mathbb{F}}_q^n$ and the smallest distance between two distinct elements of a code ${\mathcal{C}}$ is called the *minimum distance* of ${\mathcal{C}}$. We associate with a code ${\mathcal{C}}$ the polynomials $$\alpha(z)=\sum_{c\in{\mathcal{C}}}z^{\operatorname{wt}(c)}$$ and $$\beta(z)=\frac{1}{{\lvert{\mathcal{C}}\rvert}}\sum_{b,c\in{\mathcal{C}}}z^{\operatorname{wt}(c-b)},$$ which are called the *weight enumerator* and the *distance enumerator* of ${\mathcal{C}}$, respectively. Note that, if ${\mathcal{C}}$ is additive, then its weight enumerator coincides with its distance enumerator.
As usual, we let $V=V(m,q)$ be an $m$-dimensional ${\mathbb{F}}_q$-vector space and ${\mathscr{Q}}(m,q)$ the set of quadratic forms on $V$. Since for every quadratic form $Q:V\to{\mathbb{F}}_q$ we have $Q(0)=0$, we shall identify functions from $V$ to ${\mathbb{F}}_q$ with vectors ${\mathbb{F}}_q^{V^*}$, where $V^*=V-\{0\}$.
Let $R_q(1,m)^*$ be the set of all $q^{m+1}$ affine functions from $V$ to ${\mathbb{F}}_q$. This code has length $q^m-1$ and is the punctured version of the generalised first-order Reed-Muller code $R_q(1,m)$ of length $q^m$. If we identify $V$ with ${\mathbb{F}}_{q^m}$, then $R_q(1,m)^*$ consists of the functions $$\begin{gathered}
{\mathbb{F}}_{q^m}\to{\mathbb{F}}_q\\
x\mapsto\operatorname{Tr}_m(ax)+c,\quad a\in{\mathbb{F}}_{q^m},\,c\in{\mathbb{F}}_q.\end{gathered}$$ We shall associate codes with subsets $Y$ of ${\mathscr{Q}}(m,q)$ by taking cosets of $R_q(1,m)^*$ with coset representatives from $Y$. Care must be taken in the case that $q=2$ since $x^2=x$ for all $x\in{\mathbb{F}}_2$, which implies that every quadratic form in ${\mathscr{Q}}(m,2)$ of rank $1$ is in fact also a linear function. Accordingly, we define a subset $Y$ of ${\mathscr{Q}}(m,q)$ to be *nondegenerate* if $q>2$ or if $q=2$ and $Y$ contains no forms of rank $1$. For every nondegenerate subset $Y$ of ${\mathscr{Q}}(m,q)$, we define the code ${\mathcal{C}}(Y)$ of size $q^{m+1}\,{\lvertY\rvert}$ by $${\mathcal{C}}(Y)=\bigcup_{Q\in Y}Q+R_q(1,m)^*.$$ If $Y$ equals ${\mathscr{Q}}(m,q)$, then ${\mathcal{C}}(Y)$ is the punctured version $R_q(2,m)^*$ of the generalised second-order Reed-Muller code $R_q(2,m)$ of length $q^m$.
For $i\in I$, define the polynomial $$\omega_i(z)=n_1z^{w_1}+n_2z^{w_2}+n_3z^{w_3}+n_4z^{w_4}+n_5z^{w_5}+n_6z^{w_6},$$ where $$\begin{aligned}
w_1&=q^{m-1}(q-1)-q^{m-s-1}-1 & n_1&=\tfrac{1}{2}(q^{2s}(q-1)-q^s)(q-1)\\
w_2&=q^{m-1}(q-1)-q^{m-s-1} & n_2&=\tfrac{1}{2}(q^{2s}+q^s)(q-1)\\
w_3&=q ^{m-1}(q-1)-1 & n_3&=(q^m-q^{2s}(q-1))(q-1)\\
w_4&=q ^{m-1}(q-1) & n_4&=q^m-q^{2s}(q-1)\\
w_5&=q^{m-1}(q-1)+q^{m-s-1}-1 & n_5&=\tfrac{1}{2}(q^{2s}(q-1)+q^s)(q-1)\\
w_6&=q^{m-1}(q-1)+q^{m-s-1} & n_6&=\tfrac{1}{2}(q^{2s}-q^s)(q-1)
\intertext{for $i=2s+1$ and}
w_1&=(q^{m-1}-\tau q^{m-s-1})(q-1)-1 & n_1&=(q^{2s-1}-\tau q^{s-1})(q-1)\\
w_2&=(q^{m-1}-\tau q^{m-s-1})(q-1) & n_2&=q^{2s-1}+\tau q^{s-1}(q-1)\\
w_3&=q^{m-1}(q-1)-1 & n_3&=(q^m-q^{2s})(q-1)\\
w_4&=q^{m-1}(q-1) & n_4&=q^m-q^{2s}\\
w_5&=q^{m-1}(q-1)+\tau q^{m-s-1}-1 & n_5&=(q^{2s-1}(q-1)+\tau q^{s-1})(q-1)\\
w_6&=q^{m-1}(q-1)+\tau q^{m-s-1} & n_6&=(q^{2s-1}-\tau q^{s-1})(q-1)\end{aligned}$$ for $i=(2s,{\tau})$. The following result relates the polynomial $\omega_i(z)$ with the weight enumerator of cosets of $R_q(1,m)^*$. This result can be proved using the standard theory of quadratic forms. (Recall that ${\mathscr{Q}}_{2s+1}$ contains all quadratic forms of rank $2s+1$ and ${\mathscr{Q}}_{2s,{\tau}}$ contains all quadratic forms of rank $2s$ and type ${\tau}$.)
\[lem:coset\_weight\_distribution\] Let $Q\in{\mathscr{Q}}(m,q)$ be a quadratic form with $Q\in{\mathscr{Q}}_i$. Then $\omega_i(z)$ is the weight enumerator of the coset $Q+R_q(1,m)^*$.
Now, since $R_q(1,m)^*$ is additive, the distance enumerator of ${\mathcal{C}}(Y)$ equals $$\frac{1}{{\lvertY\rvert}}\sum_{b,c\in Y}\;\sum_{a\in R_q(1,m)^*}z^{\operatorname{wt}(a+c-b)}.$$ The inner sum is the weight enumerator of the coset $c-b+R_q(1,m)^*$ and so Lemma \[lem:coset\_weight\_distribution\] gives the distance enumerator of ${\mathcal{C}}(Y)$ in terms of the inner distribution of $Y$.
\[thm:dist\_C2\] Let $Y$ be a nondegenerate subset of ${\mathscr{Q}}(m,q)$ with inner distribution $(a_i)_{i\in I}$. Then the distance enumerator of ${\mathcal{C}}(Y)$ is $\sum_{i\in I}a_i\omega_i(z)$.
If $Y$ equals ${\mathscr{Q}}(m,q)$, then Theorem \[thm:inner\_dist\_d\_odd\] with ${\delta}=0$ and Theorem \[thm:dist\_C2\] give the distance enumerator of $R_q(2,m)^*$. This complements results of McEliece [@McE1969], who determined the distance enumerator of the second-order generalised Reed-Muller code $R_q(2,m)$ itself. This latter result can also be recovered from Theorem \[thm:inner\_dist\_d\_odd\] and a slightly modified version of Theorem \[thm:dist\_C2\].
Now let $m$ and $d$ be two integers of equal parity satisfying $1\le d\le m$. If $d$ is odd, let $Y$ be a nondegenerate maximal $d$-code in ${\mathscr{Q}}(m,q)$ and if $d$ is even, let $Y$ be a maximal elliptic $d$-code in ${\mathscr{Q}}(m,q)$. Writing $\delta={\lfloord/2\rfloor}$, we have by Theorems \[thm:bounds\_S\], \[thm:bound\_Q\], and \[thm:bound\_elliptic\_Q\] $${\lvertY\rvert}=q^{m(m-2\delta+1)/2}.$$ The code ${\mathcal{C}}(Y)$ has length $q^m-1$, cardinality $q^{m(m-2\delta+3)/2+1}$, and minimum distance $$q^{m-1}(q-1)-q^{m-\delta-1}-1. \label{eqn:designed_distance}$$ The distance enumerator of ${\mathcal{C}}(Y)$ is determined by Theorems \[thm:dist\_C2\] and \[thm:inner\_dist\_d\_odd\] for odd $d$ and by Theorems \[thm:dist\_C2\] and \[thm:inner\_dist\_elliptic\] for even $d$.
Now assume that $Y$ is obtained from the specific constructions in Theorems \[thm:constr\_m\_odd\_d\_odd\] and \[thm:constr\_elliptic\], according to whether $d$ is odd or even, respectively. Then ${\mathcal{C}}(Y)$ is a linear code and, in many cases, ${\mathcal{C}}(Y)$ is an optimal linear code or has the same parameters as the best known linear code [@Gra2007]. Generalising work of Berlekamp [@Ber1970], it was shown by Li [@Li2017 Proposition 2.5] that if $$\frac{m}{3}\le \delta\le \frac{m}{2},$$ then ${\mathcal{C}}(Y)$ is a narrow-sense primitive BCH code of designed minimum distance . Hence, in this case, the true minimum distance of ${\mathcal{C}}(Y)$ equals its designed minimum distance. This recovers principal results of [@Ber1970] for $q=2$ and of [@Li2017] for odd $q$. Using the results of [@Sch2015] and additional arguments, the distance enumerator of ${\mathcal{C}}(Y)$ was obtained in [@Li2017] for odd $q$. Using entirely different methods, the distance enumerator of the extended version of ${\mathcal{C}}(Y)$ was also obtained for $q=2$ in [@Ber1970]. Our results give, in a uniform way, the distance enumerator of ${\mathcal{C}}(Y)$ for every prime power $q$.
Berlekamp [@Ber1970] and Kasami [@Kas1971] studied cyclic codes of the form ${\mathcal{C}}(Y)$ and related codes for other specific subsets $Y$ of ${\mathscr{Q}}(m,2)$. They determined the distance enumerators of such codes using methods that are completely different from our methods. Many of these results can be recovered and generalised to $q>2$ using Theorems \[thm:inner\_dist\_d\_odd\] and \[thm:inner\_dist\_elliptic\] together with Theorem \[thm:dist\_C2\] or some suitable modification.
We close this section by noting that, if $Y$ is a maximal $(2\delta)$-code in ${\mathscr{Q}}(m,q)$ and $m$ and $q$ are even, then ${\mathcal{C}}(Y)$ has length $q^m-1$, cardinality $q^{m(m-2{\delta}+4)/2-(m-2{\delta})/2}$, and minimum distance $$(q^{m-1}-q^{m-\delta-1})(q-1)-1.$$ For $q=2$, the extended version of ${\mathcal{C}}(Y)$ is known as the Delsarte-Goethals code and for $2{\delta}=m$ it is known as the Kerdock code [@MacSlo1977 Ch. 15].
Acknowledgment {#acknowledgment .unnumbered}
==============
I would like to thank Shuxing Li for helpful discussions on applications to error-correcting codes.
[10]{}
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E. Bannai and T. Ito, *Algebraic combinatorics [I]{}: Association schemes*, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984.
E. R. Berlekamp, *The weight enumerators for certain subcodes of the second order binary [R]{}eed-[M]{}uller codes*, Information and Control **17** (1970), 485–500.
Ph. Delsarte, *An algebraic approach to the association schemes of coding theory*, Philips Res. Rep. Suppl. **10** (1973).
[to3em]{}, *Properties and applications of the recurrence [$F(i+1,k+1,n+1)=q^{k+1}F(i,k+1,n)-q^{k}F(i,k,n)$]{}*, SIAM J. Appl. Math. **31** (1976), no. 2, 262–270.
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GF}(q)$]{}*, J. Combin. Theory Ser. A **19** (1975), no. 1, 26–50.
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X.-D. Hou, *The eigenmatrix of the linear association scheme on [$R(2,m)$]{}*, Discrete Math. **237** (2001), no. 1-3, 163–184.
T. Kasami, *The weight enumerators for several classes of subcodes of the [$2$]{}nd order binary [R]{}eed-[M]{}uller codes*, Information and Control **18** (1971), 369–394.
Sh. Li, *The minimum distance of some narrow-sense primitive [BCH]{} codes*, SIAM J. Discrete Math. **31** (2017), no. 4, 2530–2569.
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[to3em]{}, *Symmetric bilinear forms over finite fields with applications to coding theory*, J. Algebraic Combin. **42** (2015), no. 2, 635–670.
M. Schmidt, *Rank metric codes*, Master’s thesis, University of Bayreuth, Germany, 2016.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Central European University\
Amadou Keita (keita\[email protected])
bibliography:
- 'references.bib'
date: '[December 2018]{}'
nocite: '[@*]'
title: 'The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases'
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The radiation shift of the quark mass in a constant chromomagnetic field at finite temperature and density was calculated. The limiting cases of a weak and a strong chromomagnetic field were considered. It was shown that in a strong field there is no imaginary part in the contribution of the finite density effects to the quark mass shift, and its real part can considerably exceed the corresponding part of the purely field contribution.'
---
Moscow University\
Physics Bulletin\
Vol. 51, No. 1. (1996) 1-6
**Radiative Shift of the Quark Mass in a Constant\
Chromomagnetic Field at Finite\
Temperature and Density**
V.Ch. Zhukovskii, K. G. Levtchenko, T. L. Shoniya\
[ *Faculty of Physics,\
Department of Theoretical Physics, Moscow State University,\
119899, Moscow, Russia*]{}\
P. A. Eminov\
[ *Department of Physics\
Moscow State Institute of Electronics and Mathematics\
(Technical University)\
109028, Moscow, Russia*]{}
In spite of considerable efforts that have been undertaken in recent years, there is no consistent theory of the QCD vacuum state at present. All recently proposed QCD vacuum models are based upon investigation of the interaction of quantized quark and gluon fields against the background of a gluon condensate, i.e., a classical gauge field. Along with instanton models of the QCD vacuum \[1\], efforts have been undertaken to construct models that are based on replacement of stochastic vacuum gluon fields by various regular configurations of external non–Abelian fields, the simplest of which is a constant chromomagnetic field of the Abelian type proposed by Matinyan and Savvidy \[2\].
In this connection various radiation effects in non–Abelian external fields became the subject of intensive studies \[3–5\]. On the other hand, many problems of elementary particle physics have statistical aspects, such as problems in phase transitions in gauge theories at finite temperature and non–zero chemical potential \[6\], the question on the possibility of stabilizing the classical Yang–Mills field configurations through temperature effects \[7\], the problem of the particle energy radiative shift and modifications of the particle electromagnetic properties at the expense of finite temperature and density effects \[8, 9\], etc.
In \[10\], spectra of gluon and quark–like excitations in the hot quark–gluon plasma were calculated in the one–loop approximation, and it was pointed out that the results obtained could be used in the analysis of experimental data on collisions of relativistic nuclei (which are expected to generate the quark–gluon plasma \[11\]) and also in the analysis of the processes that take place at the early stages of evolution of the Universe and in quark stars.
In the present paper we consider the contribution of the finite density effects to the radiation shift of the quark energy with consideration for the vacuum fluctuations of gauge bosons. The vacuum state here is approximated by the Matinyan and Savvidy constant chromomagnetic field.
In the real time representation used in the present paper, the quark energy radiative shift is written in the form of the sum of two terms, one of which corresponds to the energy radiative shift at $T=\mu=0$, and the other term corresponds to the contributions of the finite density and temperature effects \[12\].
Following \[3\], we represent the total gluon field $A^a_{\mu}$ as a sum of a classical (non–Abelian) external field $\bar A^a_{\mu}$ and small quantum fluctuations around it: $$ A^a_{\mu}=\bar A^a_{\mu}+Q^a_{\mu}.$$ The external field is chosen in the Matinyan–Savvidy form specified by the potential $$\bar A^a_{\mu}=\delta^{a8}A_{\mu},\quad
A_{\mu}=B\delta^2_{\mu}x_1,
\label{aa}$$ and the interaction of the quark with the external field $\bar
A^a_{\mu}$ will be taken into account exactly, while its interaction with the quantized gluon field $Q^a_{\mu}$ will be treated by the perturbation theory.
The one–loop energy shift of the color $k$ quark in the external chromomagnetic field has the form \[3–5\] $$\triangle E=\frac1T \int d^4xd^4x'\bar\psi_k(x)M_{(k)}(x,x')\psi_k(x'),
\label{ve}$$ where the mass operator, diagonal with respect to color in view of the specific choice of the external field (1), is defined by the formulas $$M_{kl}(x,x')=\delta_{kl}M_{(k)}(x,x')$$ $$M_{(k)}(x,x')=-ig^2 \gamma^{\mu}{\left({\hat T}_a
\right)}_{kn}S_{(n)}(x,x')
\gamma^{\nu}{\left({\hat T}^{+}_a
\right)}_{nk}D^{(a)}_{\mu\nu}(x,x').
\label{mas}$$
In the following, for definiteness, we will consider, as in \[5\], the case when the quark color is $k=1$, i.e., its charge is $e_1=\bar g/3$. Then >from Eqs. (2), (3) and the explicit form of matrices $T_a=\lambda_a/2$ it follows that both neutral $(a=1, 2, 3, 8)$ and charged gluons contribute to the quark energy shift. In the most interesting case of a charged gluons contribute to the quark energy shift. In the most interesting case of a charged gluon $(a=4,\quad g_a=\bar g)$, due to the law of conservation of the color charge, the intermediate quark has the charge $e_n=-2\bar g/3\quad(n=3)$. This is just the case we consider here.
In the real time representation, in order to compute the one–loop contributions of the finite density and temperature to the quark energy shift one has to replace in formula (3) the causal Green functions $S_{(n)}(x,x')$ and $D^{(a)}_{\mu\nu}(x,x')$ by the time Green functions of the ideal quark–antiquark and gluon gases \[13\].
The following representation for the time Green function \[8\] will be used: $$S(x,x')=S_{(n)}(x,x')
-\sum_{\{s\},\varepsilon=\pm1}\frac{
\varepsilon\Psi^{(\varepsilon)}_s(\vec x)
{\bar\Psi}^{(\varepsilon)}_s({\vec x}^{\,\prime})}
{\exp{\left\{\frac{E_s-\varepsilon\mu}{T}\right\}}+1}
\exp{\{-i\varepsilon E_n(x_0-x^\prime_0)\}}.
\label{qpn}$$ Here $\Psi^{(\pm)}_s$ and $E^{(\pm)}_s$ are, respectively, the wave functions and energy levels in the constant chromomagnetic field (1) of the quark and antiquark, summation is performed over all quantum states ${s}$ of quarks $(\varepsilon=1)$ and antiquarks $(\varepsilon=-
1)$, and the first term is the causal Green function of a quark at $T=\mu=0$ \[3–5\].
Thus, the calculation of the quark energy shift in the case of a degenerate quark gas is reduced to the replacement in (3) of the causal Green function by the time Green function (4), and the explicit form of the causal Green function of the charged gluon in (3) is presented in \[5\].
As a result, the ground state energy shift of a quark of the color $k=1$ is represented by the sum of two terms: $$\triangle E=\triangle E(H,T=\mu=0)+\triangle E(H,\mu\ne0,T\ne0).
\label{tel}$$ The first term in (5) corresponds to the radiative energy shift of the quark in the field (1) at $T=\mu=0$ and it has been considered in detail in \[3–5\]; the second term describes the contribution of the exchange interaction to the quark energy shift and is of interest to us.
Calculating the quantity $\triangle E(H,T\ne0,\mu\ne0)$ leads to the expression $$\triangle E(H,T,\mu)=-g^2m\frac{H_0}H\frac
i{(2\pi)^3}\sum_{\varepsilon}
\int\limits^{\infty}_0r^2\,dr
\int\limits^{\infty}_0\,dt\int\limits^{+\infty}_{-
\infty}ds\varepsilon\,A(r,s,t).
\label{det}$$ Here $$A(r,s,t)=\frac{\sqrt{\pi}}2\frac{\exp{\{i(s+t)\rho^2\}}}{\rho\sqrt{i(s+t)}}
\Phi(\rho\sqrt{i(s+t)})
$$ 1[+1]{} $$$$ 1[23|gH23|gHs+|gH|gHt+ 3]{} $$$$ , $$ where $\Phi(z)$ is the error integral and $\rho=m r$.
The case of fermions with the isotopic spin 1 in the theory with the $SU(2)$ gauge group (the adjoint representation) is considered in a similar way. It is well known \[3, 4\] that in the representation chosen, fermions and bosons have the following charges with respect to the external field: $-g,0,g$. We will restrict ourselves to the case of fermions with the charge $g$ in the ground state which, as well as the case considered above in the framework of QCD, does not have any electrodynamical analog. Then the intermediate boson in the one–loop diagram for the radiation correction to the energy has the charge $+g$, and the intermediate fermion is neutral.
The contribution of the exchange interaction to the fermion energy shift is calculated according to the above scheme: $$\triangle E(H,T,\mu)=-
g^2m\frac{H_0}H\frac{2i}{(2\pi)^3}\sum_{\varepsilon}
\int\limits^{\infty}_0r^2\,dr\int\limits^{\infty}_0d\tau\varepsilon
A(r,\tau),
\label{etw}$$ where $$A(r,\tau)=\frac{\sqrt{\pi}}2\frac{\Phi(Z)}Z\exp{\{Z^2\}}
\exp{\left\{-2i\tau\frac{H_0}H[\varepsilon\sqrt{1+r^2}-1]\right\}}\times$$ $$\times\frac1{\exp{\left\{\frac{m(\sqrt{r^2+1}-
\frac{\mu}m)}T\right\}}+1}
\left[\frac{\exp{\{-i\tau\}}}{\sqrt{r^2+1}}-\left(1-
\frac1{\sqrt{r^2+1}}\right)
\exp{\{i\tau\}}\right]$$ and the following notation is adopted $$H_0=\frac{m^2}{\bar g},\qquad
\tau=\bar g Ht,\qquad
Z=-ir^2\frac{H_0}H(\sin\tau\exp{\{-i\tau\}}-\tau).$$
In the limiting case of a highly degenerate quark gas, when $$T\ll E_F=\mu\quad(T=0),$$ the first term in the expansion in formula (5) with respect to the parameter $T/E_F\ll1$ corresponds to the replacement of the Fermi distribution by the $\theta$–function \[14\]: $$\frac1{\exp{\left\{\frac{m(\sqrt{r^2+1}-\frac{\mu}m)}T\right\}}+1}
\longrightarrow\theta\left(\frac{\mu}m-\sqrt{r^2+1}\right)
=\left\{
\begin{array}{l}
0,\;\mu<m\sqrt{r^2+1}\\
1,\;\mu>m\sqrt{r^2+1}
\end{array}\right.$$
In the limiting case of zero temperature which is of interest to us, when there are no antiqurks in the gas, only the contribution of the positive– frequency states should be left in formulas (6) and (7).
We consider first the energy shift of a quark in the QCD in a comparatively strong chromomagnetic field, when the following condition $$2|e_3|H>\mu^2-m^2
\label{str}$$ is fulfilled. For the contribution of the finite density effects to the energy shift of the quark ground state we obtain the following exact results: $$\triangle E=m\frac{g^2}{6\pi^2}\left(\frac{\bar gH}{m^2}\right)
\left\{\ln\left(y+\sqrt{y^2-1}\right)+
x\frac1{\displaystyle\sqrt{x(2+x)}}\times\right.
\label{ebh}$$ $$\left.\times\ln\left|\frac{1-y(1+x)+\sqrt{(y^2-1)x(2+x)}}{y-1-x}
\right|_{y=\frac{\mu}m}\right\},
\quad x=\frac{\displaystyle\bar gH}{\displaystyle 2m^2}.$$
We note that the chemical potential, or, to be exact, the Fermi energy in formula (9) are related to the quark concentration by the relationship $$n_e=\frac{|e_3|H}{(2\pi)^2}
\int\limits^{+\sqrt{\mu^2-m^2}}_{-\sqrt{\mu^2-m^2}}dp_3=
\frac{|e_3|H}{2\pi^2}\sqrt{\mu^2-m^2}.
\label{conqa}$$ In the limiting case of a strong field when $$\sqrt{\left(\frac{\mu}m\right)^2-1}\ll\frac{H}{H_0},$$ it follows from Eq. (9) with regard for (10) that $$\triangle m=-m\frac{g^2}{2\pi^2}\sqrt{\frac{H_0}H}
\left(\frac{n_e}{m^3}\right)^{1/2}.$$ Now we pass to the case of a comparatively weak chromomagnetic field: $$H\ll H_0,\quad 2|e_3|H\ll\mu^2-m^2.
\label{rwf}$$ In the free case when $H=0$ we find from (6) $$\triangle m(H=0,T=0,\mu\ne0)=\frac{\alpha_s}{4\pi}m
\left\{\sqrt{\left(\frac{\mu}m\right)^2-1}\left(\frac{\mu}m-2\right)-\right.
\label{den}$$ $$\left.-3\ln\left(\frac{\mu}m+\sqrt{\left(\frac{\mu}m\right)^2-
1}\;\right)\right\},$$ where $\alpha_s=g^2/(4\pi)$ and the chemical potential of the free quark gas is related to the density $n$ by the equation \[14\] $$\frac{\mu}m=\left[1+\left(\frac{3\pi^2n}{m^3}\right)^{2/3}
\right]^{1/2}.$$
After the replacement $\alpha_s/2\to\alpha_s$, the result (12) coincides with the corresponding result for the theory with the $SU(2)$ gauge group, and the latter, as was to be expected after the replacement $\alpha_s\to\alpha$ ($\alpha$ is the fine structure constant), coincides with the well–known expression for the contribution of finite density effects to the electron mass shift in the QED \[8\].
To calculate the field contribution to the quark mass shift in the case of the $SU(2)$ gauge theory, when conditions (11) are fulfilled, one has to make use of the known representation of the error integral in the form of the series \[15\] $$\Phi(z)=\frac2{\sqrt{\pi}}\exp{\{-
z^2\}}\sum^{\infty}_{n=0}\frac{2^nz^{2n+1}}{(2n+1)!!}.
\label{int}$$
When conditions (11) are fulfilled, the first term in series (13) corresponds to the main term in the asymptotics (6). In particular, in the limiting case of low densities, when $$\frac{H}{H_0}\ll\left(\frac{\mu}m\right)^2-1\ll 1,
\label{wn}$$ we find that $$\triangle m(H\ne0,\mu\ne0,T=0)=-2m\frac{\alpha_s}{\pi}
\sqrt{\left(\frac{\mu}m\right)^2-1}\left[1-
\sqrt{\frac{gH}{\mu^2-m^2}}\right].
\label{wecon}$$
As seen from (15), the contribution of the finite density effects to the quark mass shift is negative in the case of low densities, whereas the purely field contribution is positive. In the other limiting case, when $\mu/m\gg1$, we have $$\triangle m=\frac{\alpha_s}{4\pi}m\left(\frac{\mu}m\right)^2
\left[1+\frac{2gH}{m\mu}\right].
\label{ln}$$
Thus, as the density of the quark gas grows, the radiative shift of the quark mass, which is negative at low densities, passes the zero value and increases proportionally to $\left(\frac{\mu}m\right)^2$ at high densities, whereas the leading field contribution is positive both at low and high densities.
We also note that up to the replacement $\alpha_s\longrightarrow\alpha_s/2$ the asymptotics (15) is also valid in the case of the $SU(3)$ gauge group. Similar to the calculation of the radiative shift of the quark mass at $T=\mu=0$ \[3–5\], this is related to the fact that a weak external field $H\ll H_0=m^2/g$ turns out to be weak for the quark but superstrong for the massless charged gluon.
Comparison of the results obtained with the radiative correction to the quark mass shift in a constant chromomagnetic field at zero temperature and nonzero chemical potential \[3–5\] shows that under the condition (14) the field correction, as in the case of $T=\mu=0$ has, according to (15), the order of magnitude $(gH)^{\frac12}$ though the sign is opposite.
As was first reported in \[3\], the ground state of a quark in the cromomagnetic field (1) at $T=\mu=0$ has a positive imaginary part, i.e., it is unstable. This instability is caused by the presence of a tachionic mode in the gluon energy spectrum in such a field.
In \[3\], the imaginary part of the quantity $\triangle E_0$ is interpreted in the following way: as a result of gluon absorption from the gluon pair created by the unstable external field, the quark passes from the ground to an excited state. It seems interesting that in the case of a comparatively strong magnetic field (when condition (8) is fulfilled) no imaginary part in the energy shift of the ground state of a quark arises on account of the finite density effects. This is immediately seen from the exact result (9) and it can be explained by the fact that all quarks under condition (8) may stay only in the ground state, so that the above–mentioned transitions to excited states on account of gluon absorption cannot take place. At the same time in a comparatively weak magnetic field, when conditions (11) are fulfilled, allowance for the terms of series (13) that follow the main term leads to the appearance of an imaginary part in the contribution of the exchange interaction to the quark energy shift, too. This can be qualitatively explained by the fact that transitions of the quark to excited states are no longer forbidden under condition (11).
The authors are grateful to A.V. Borisov for the discussion of the results of this work.
[15]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Savaş Arapoğlu,'
- Cemsinan Deliduman
- 'and K. Yavuz Ekşi'
title: |
Constraints on perturbative\
$f(R)$ gravity via neutron stars
---
Introduction
============
The current accelerated expansion of the universe has been confirmed by many independent observations. The supporting evidence comes from the supernovae Ia data [@Perlmutter; @Riess1; @Riess2], cosmic microwave background radiation [@Spergel1; @Spergel2; @Komatsu], and the large scale structure of the universe [@Tegmark1; @Tegmark2]. Although the cosmological constant is arguably the simplest explanation and the best fit to all observational data, its theoretical value predicted by quantum field theory is many orders of magnitude greater than the value to explain the current acceleration of the universe. This problematic nature of cosmological constant has motivated an intense research for alternative explanations and the reasonable approaches in this direction can be divided into two main categories, both of them introducing new degrees of freedom [@Uzan]: The first approach is to add some unknown energy-momentum component to the right hand side of Einstein’s equations with an equation of state $p/\rho
\approx -1$, dubbed *dark energy*. In the more radical second approach, the idea is to modify the left hand side of Einstein’s equations, so-called *modified gravity*. Trying to explain such perplexing observations by modifying gravity rather than postulating an unknown dark energy has been an active research area in the last few years and in this paper we adopt this path.
A modified theory of gravity has to explain the late time cosmology, and also be compatible with the constraints obtained from solar system and laboratory tests. However, it is not easy to construct theories of gravity with these requirements. Nevertheless, a class of theories, called $f(R)$ models [@Odintsov-rev; @Sotiriou-rev; @deFelice-rev], has attracted serious attention possibly because of its (deceptive) simplicity. Today there exist viable $f(R)$ models which are constructed carefully to be free of instabilities, and to pass the current solar system and laboratory tests [@Nojiri1; @Nojiri2; @Nojiri3; @Cognola; @Hu-Sawicki; @Appleby-Battye; @Starobinsky; @Miranda].
The strong gravity regime [@Dimitri-rev] of these theories is another way of checking their viability. In this regime, divergences stemming from the functional form of f(R) may prevent the existence of relativistic stars in such theories [@Briscese; @Abdalla; @Nojiri4; @Bamba; @Kobayashi-Maeda; @Frolov; @Nojiri5], but thanks to the chameleon mechanism the possible problems jeopardizing the existence of these objects may be avoided [@Tsujikawa; @Upadhye-Hu]. Furthermore, there are also numerical solutions corresponding to static star configurations with a strong gravitational field [@Babichev1; @Babichev2] where the choice of the equation of state for the star is crucial for the existence of solutions, and therefore a polytropic equation of state is used in these works to overcome the possible problems related to the equation of state.
Another approach to probe the viability of f(R) theories in the strong gravity regime is to use a method called perturbative constraints, or order reduction [@Eliezer; @Jaen]. The motivation behind using this technique in the present context is the thought that the reason of all the problems encountered in modifying gravity may be the outcome of considering these modifications as exact ones. The main issue with the exact modifications are the problems arising in curvature scales which are not originally aimed by these modifications. In the perturbative constraints approach, the modifications are viewed as next to leading order terms to the terms coming from Einstein’s General Relativity. Treating f(R) gravity via perturbative constraints at cosmological scales is considered in [@DeDeo; @Cooney1] and for compact objects in [@Cooney2]. In this manuscript we further examine the existence and properties of relativistic stars in the context of f(R) models via perturbative constraints.
Specifically, we study a $f(R)$ model of the form $f(R)=R+\alpha
R^2$ and constrain the value of $\alpha$ with the recent constraints on the mass-radius relation [@OBG10]. Such a gravity model in the weak field limit is known to reduce to Yukawa-like potentials and has been recently constrained by binary pulsar data [@naf] as $\alpha \lesssim 5 \times 10^{15}$ cm$^2$. Here we find that in the strong gravity regime the constraint on perturbative parameter is $\alpha \lesssim 10^{10}$ cm$^2$. This value does not contrast with the value obtained in [@naf] as they argue that the value of $\alpha$ could be different at different length scales.
The plan of the paper is as follows: In section II, we assume a perturbative form of $f(R)$ modified gravity model and obtain the field equations. Assuming also perturbative forms of metric functions and the hydrodynamical quantities we obtain the modified Tolman–Oppenheimer–Volkoff (TOV) equations. In section III, the modified TOV equations are solved numerically for various forms of the equation of state, the functional form of $f(R)$, and various values of the perturbation parameter $\alpha$. Finally, in the discussion section, we comment on the results of numerical study and on the significance of the scale of the perturbation parameter $\alpha$.
Modified TOV equations of $f(R)$ gravity
========================================
The action of $f(R)$ gravity models is the simplest generalization of the Einstein–Hilbert action. Here, instead of having a linear function of Ricci scalar as the Lagrangian density we have a function of it: $$\label{action}
S=\frac{1}{16\pi}\int d^4x \sqrt{-g}f(R) + S_{{\rm matter}}\quad ,$$ where $g$ denotes the determinant of the metric $g_{\mu\nu}$, and $R$ is the Ricci scalar. We set $G=1$ and $c=1$ in the action and in the rest of this section. Here we are considering the metric formalism of gravity and therefore matter only couples to the metric, and the Levi–Civita connection is derived from the metric.
A straightforward variation of the action (\[action\]) with respect to the metric gives fourth order differential equations of $g_{\mu\nu}$. However, treating higher than second order differential equations in 4 dimensions is problematic. For this reason, we adopt a perturbative approach as suggested in [@Cooney2] and choose $f(R)$ such that all terms higher than second order will be multiplied by a small parameter $\alpha$. The meaning of smallness of the parameter $\alpha$ is explained in the next section.
Since this is a perturbative approach, the action and the field equations must have the form of those of general relativity for $\alpha=0$. So we choose the function $f(R)$ to have the form $$\label{fR}
f(R)=R+\alpha h(R)+\mathcal{O}(\alpha ^{2})$$ without a constant piece, i.e. without a cosmological constant. Here $h(R)$ is, for now, an arbitrary function of $R$ and $\mathcal{O}(\alpha ^{2})$ denotes the possible higher order corrections in $\alpha$. Variation of the action (\[action\]) with respect to the metric, with the form of $f(R)$ as given in (\[fR\]), results in field equations which are $$\label{field}
(1+\alpha h_{R})G_{\mu \nu }-\frac{1}{2}\alpha(h-h_{R}R)g_{\mu \nu
}-\alpha (\nabla _{\mu }\nabla _{\nu }-g_{\mu \nu }\Box )h_{R}=8\pi
T_{\mu \nu }$$ where $G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu}$ is the Einstein tensor and $h_R=\frac{dh}{dR}$ is the derivative of $h(R)$ with respect to the Ricci scalar.
We are interested in spherically symmetric solutions of these field equations inside a neutron star, so we choose a spherically symmetric metric with two unknown, independent functions of $r$, $$\label{metric}
ds^2= -e^{2\phi_\alpha}dt^2 +e^{2\lambda_\alpha}dr^2 +r^2 (d\theta^2
+\sin^2\theta d\phi^2).$$
Perturbative solution of the field equations means that $g_{\mu\nu}$ can be expanded perturbatively in $\alpha$ and therefore the metric functions have the expansions $\phi_\alpha =
\phi +\alpha\phi_1 + \ldots$ and $\lambda_\alpha = \lambda
+\alpha\lambda_1 + \ldots$. The energy–momentum tensor, which is on the right hand side of the field equations, is the energy-momentum tensor of the perfect fluid. For a perturbative solution, then, hydrodynamics quantities are also defined perturbatively: $\rho_\alpha = \rho +\alpha\rho_1 + \ldots$ and $P_\alpha = P +\alpha P_1 + \ldots$. Note that we denote zeroth order quantities, which can be obtained solving Einstein equations, without a subscript.
Then the “tt” and “rr” components of field equations become $$\begin{aligned}
-8\pi \rho_\alpha &=& -r^{-2} +e^{-2\lambda_\alpha}(1-2r\lambda_\alpha')r^{-2}
+\alpha h_R(-r^{-2} +e^{-2\lambda}(1-2r\lambda')r^{-2}) \nonumber \\
&& -\frac12\alpha(h-h_{R}R) +e^{-2\lambda}\alpha[h_R'r^{-1}(2-r\lambda')+h_R''] \label{f-tt},\\
8\pi P_\alpha &=& -r^{-2} +e^{-2\lambda_\alpha}(1+2r\phi_\alpha')r^{-2}
+\alpha h_R(-r^{-2} +e^{-2\lambda}(1+2r\phi')r^{-2}) \nonumber \\
&& -\frac12\alpha(h-h_{R}R) +e^{-2\lambda}\alpha h_R'r^{-1}(2+r\phi'), \label{f-rr}\end{aligned}$$ where prime denotes derivative with respect to radial distance, $r$.
We would like to solve $\rho_\alpha$ and $P_\alpha$ up to order $\alpha$. On the right hand side, terms containing $h_R$ and its derivatives are already first order in $\alpha$. Therefore, for the quantities that are multiplied by them, we use the zeroth order quantities which can be obtained from the Einstein equations written for this metric. In the equations above, hence, in terms that are multiplied by $\alpha$ we have already written zeroth order quantities. Using the “tt" and “rr" components of Einstein equations, $$\begin{aligned}
-8\pi \rho &=& -r^{-2} +e^{-2\lambda}(1-2r\lambda')r^{-2} \label{e-tt},\\
8\pi P &=& -r^{-2} +e^{-2\lambda}(1+2r\phi')r^{-2}, \label{e-rr}\end{aligned}$$ we replace terms multiplied by $h_R$ in (\[f-tt\]) and (\[f-rr\]) with $-8\pi \rho$ and $8\pi P$, respectively. Additionally, combining $\alpha$ order terms on the left hand sides of (\[f-tt\]) and (\[f-rr\]) we obtain, $$\begin{aligned}
8\pi r^{2} \rho_\alpha &=& 1-e^{-2\lambda_\alpha}(1-2r\lambda_\alpha') \nonumber \\
&& +\alpha h_R r^{2}\left[ 8\pi \rho +\frac12\left(\frac{h}{h_R}-R\right)
-e^{-2\lambda}\left(r^{-1}(2-r\lambda')\frac{h_R'}{h_R}+\frac{h_R''}{h_R}\right)\right] \label{f-tt2},\\
8\pi r^{2} P_\alpha &=& -1+e^{-2\lambda_\alpha}(1+2r\phi_\alpha') \nonumber \\
&& +\alpha h_R r^{2}\left[ 8\pi P -\frac12\left(\frac{h}{h_R}-R\right)
+e^{-2\lambda}r^{-1}(2+r\phi')\frac{h_R'}{h_R}\right]. \label{f-rr2}\end{aligned}$$
To define a mass parameter, we assume a solution that has the same form of the exterior solution for the metric function $\lambda_\alpha$. This form of solution has been previously suggested by the authors of [@Cooney2]. Therefore we define $$\label{mass}
e^{-2\lambda_\alpha}=1-\frac{M_\alpha}{r}.$$ Here, similar to $\rho_\alpha$, $M_\alpha$ is expanded in $\alpha$ as $M_\alpha = M +\alpha M_1 + \ldots$, where $M$ is the zeroth order solution, which, in general relativity, is given in terms of $\rho$ as $$\label{mass_GR}
M=8\pi \int \rho(r) r^2 dr.$$ Taking the derivative of $M_\alpha$ with respect to $r$ one obtains $$\label{dMa/dr}
\frac{dM_\alpha}{dr}= 1-e^{-2\lambda_\alpha}(1-2r\lambda_\alpha').$$ Substituting this into (\[f-tt2\]) and arranging terms, one gets the first modified TOV equation, $$\begin{aligned}
\frac{dM_\alpha}{dr} &=& 8\pi r^{2} \rho_\alpha -\alpha h_R \left[
\begin{array}{l}
8\pi r^{2}\rho +\frac{r^2}{2} (\frac{h}{h_R}-R) \\
+(4\pi\rho r^3-2r+\frac32 M)\frac{h_R'}{h_R}-r(r-M)\frac{h_R''}{h_R}
\end{array}
\right] \label{1stTOV}.\end{aligned}$$ To obtain this equation, we also substitute general relativistic form of (\[mass\]), $e^{-2\lambda}=1-\frac{M}{r}$.
The conservation equation of energy-momentum tensor of a perfect fluid, $\nabla^\mu T_{\mu\nu}=0$, is equivalent to the hydrostatic equilibrium equation, $$\label{hydro}
\frac{dP_\alpha}{dr}=-(\rho_\alpha
+P_\alpha)\frac{d\phi_\alpha}{dr}.$$ Therefore in order to get the second modified TOV equation we pull $\frac{d\phi_\alpha}{dr}$ from the “rr” field equation, eq.(\[f-rr2\]). After some straightforward manipulations one gets, $$\begin{aligned}
2(r-M_\alpha)\frac{d\phi_\alpha}{dr} &=& 8\pi r^{2}P_\alpha + \frac{M_\alpha}{r}
-\alpha h_R \left[
\begin{array}{l}
8\pi r^{2}P +\frac{r^2}{2} (\frac{h}{h_R}-R) \\
+(2r-\frac32M+4\pi Pr^3)\frac{h_R'}{h_R}
\end{array}
\right] \label{2ndTOV}.\end{aligned}$$
Note that when one sets $\alpha$ to zero, one gets the original TOV equations for general relativistic quantities. Similar to the case in general relativity, the modified TOV equations, (\[1stTOV\]), (\[hydro\]) and (\[2ndTOV\]), are solved numerically for some special functional form of $h(R)$. Obviously perturbation expansion parameter $\alpha$ introduces a new scale into the theory. By choosing a realistic equation of state we compute mass–radius relation for various values of $\alpha$ and this way we put a bound on $\alpha$ for perturbative $f(R)$ gravity models with various forms of $h(R)$.[^1] This numerical analysis is explained in the next section.
Numerical model and astrophysical constraints on the value of $\alpha$
======================================================================
Eqs. (\[1stTOV\]) and (\[2ndTOV\]) describe any spherical mass distribution in a general $f(R)$ theory in perturbative approach. Interesting results would be obtained in the case of neutron stars which have the highest compactness ratio $\eta=2GM_{\ast}/c^2R_{\ast}$ and curvatures $\xi=GM_{\ast}/c^2R_{\ast}^3$ [@Dimitri-rev]. In order to specialize these equations for describing neutron stars they must be supplemented by an appropriate equation of state (EoS). However, the EoS of nuclear matter at the densities prevailing in neutron stars is not very well constrained by nuclear scattering data and there is a number of EoS leading to different mass-radius (M-R) relations for neutron stars.
For integrating the TOV equations in GR, it is possible to interpolate the tabulated relation between density and pressure. In $f(R)$ theories, where one needs high order derivatives of pressure with respect to density, interpolation leads to numerical problems. In order to circumvent this problem, we employ analytical expressions obtained by fitting the tabulated data. Such analytical representations are already provided by [@HP04] for two EoS’, FPS and SLy. For the rest of the EoS’, namely AP4, GS1, MPA1, and MS1, we used analytical representations provided by [@GE11] obtained by fitting the tabulated data with a function which is an extension of the function provided in [@HP04]. The 6 EoS’ we have chosen constitute a representative sample (see figure 1 in [@OBG10]) for multi-nucleonic and condensate inner composition possibilities (see [@LP01] for the description of all these EoS’). We have not employed any strange quark matter EoS. As such stars are not gravitationally bound, alternative gravity models does not produce different mass-radius relations for such objects.
We numerically integrate Eqs. (\[1stTOV\]), (\[hydro\]) and (\[2ndTOV\]) supplemented by the analytical expression for the EoS, employing a Runge-Kutta scheme with fixed step size of $\Delta r=0.001$ km. We obtain a sequence of equilibrium configurations by varying the central density $\rho_c$ from $2\times 10^{14}$ g cm$^{-3}$ to $1\times
10^{16}$ g cm$^{-3}$ (to $2\times
10^{16}$ g cm$^{-3}$ in some cases) in 200 logarithmically equal steps. This traces a mass-radius relation for a certain equation of state. We then repeat this procedure for a range of $\alpha$ to see the effect of the higher order terms in perturbative $f(R)$ gravity.
Recently, the authors of [@OP09] showed that the measurement of masses and radii of three neutron stars are sufficient for constraining the pressure of nuclear matter at densities a few times the density of nuclear saturation. These data are provided by the measurements on the neutron stars EXO 1745-248 [@OGP09], 4U 1608-52 [@GOCW09] and 4U 1820-30 [@GWCO10] by the methods described in the cited papers. We use the constraints on the M-R relation of neutron stars given in [@OBG10], which is a union of these three constraints.[^2] The constraint of [@OBG10] is shown in all M-R plots as the region bounded by the thin black line. Note that the M-R constraint in the first version of [@OBG10] is larger in the published version. In the earlier versions of our work we employed the earlier tighter constraint and reached somewhat different conclusions for FPS and SLy EoS.
Apart from the above constraint, the recent measurement [@dem10] of the mass of the neutron star PSR J1614-2230 with $1.97 \pm 0.04\, M_{\odot}$ provides a stringent constraint on any M-R relation that can be obtained with a combination of $\alpha$ and EoS. This constraint is shown as the horizontal black line with its error shown in grey. Any viable combination of $\alpha$ and EoS must yield a M-R relation with a maximum mass exceeding this measured mass.
The constraints on the M-R relation obtained by [@OBG10] and the 2 solar mass neutron star PSR J1614-2230 exclude many of the possible EoS’ if one assumes GR as the ultimate classical theory of gravity. In the gravity model employed here, the value of $\alpha$ provides a new degree of freedom such that some of the EoS’, which are excluded within the framework of GR, can now be reconciled with the observations for certain values of $\alpha$. In the following we discuss this for all EoS’ individually. To save space in the figures, we define the parameter $\alpha_9 \equiv \alpha/10^9$ cm$^2$. We show the stable configurations ($dM/d\rho_c > 0$) with solid lines and the unstable configurations with dashed lines of the same color.
In figures, we show our results (mass versus radius, M-R, relations) for 6 representative EoS’ for the $f(R)=R+\alpha R^2$ gravity model. Results for each EoS are summarized as follows:
- [**FPS (Figure \[fig\_FPS\]):**]{} For FPS [@ref_FPS], the maximum mass that a neutron star can have, within GR, is about $1.8\,M_{\odot}$ and is less than the measured mass of PSR J1614-2230. This means FPS can not represent the EoS of neutron stars in GR ($\alpha =0$). The maximum mass increases with decreasing value of $\alpha$ and we find that, for $\alpha_9<-2$, the maximum mass becomes $M_{\max}=2.04 M_{\odot}$. We thus find that FPS is consistent with the measurement of the maximum mass for $\alpha<-2 \times 10^9$ cm$^2$. Nevertheless, this does not mean $\alpha= -10 \times 10^{9}$ cm$^2$ will be satisfying both constraints. In fact for such large values of $|\alpha|$ we can not obtain M-R relations resembling known properties of neutron stars. Furthermore for $|\alpha|>10^{11}$ cm$^2$ the validity of the perturbative approach is dubious, as we mention in the discussion section.
- [**SLy (Figure \[fig\_SLY\]):**]{} SLy [@ref_SLY] is consistent with both constraints within the framework of GR. For $\alpha>2\times 10^9$ cm$^2$, however, we see that $M_{\max}$ is less than the measured mass of PSR J1614-2230. We conclude for the gravity model employed here, $f(R)=R+\alpha R^2$, that SLy is consistent with the observations only if $\alpha<2\times 10^9$ cm$^2$.
- [**AP4 (Figure \[fig\_AP4\]):**]{} AP4 [@ref_AP4] is consistent with the constraints as long as $\alpha<4\times 10^9$ cm$^2$. Interestingly, we find that a new stable solution branch, for which $dM/d\rho_c>0$ is satisfied, for values of $\alpha$ different than zero. This stable branch is obtained, for $\alpha_9=-2$ starting from central densities $8.6 \times 10^{15}$ g cm$^{-3}$.
- [**GS1 (Figure \[fig\_GS1\]):**]{} For GS1 [@ref_GS1], the maximum mass in GR remains well below the measured mass of PSR J1614-2230. The maximum mass of neutron stars for this EoS can reach up to $\sim 2\,M_{\odot}$ for $\alpha_9=-4$. Starting from $\alpha_9=-2$ the stability condition ($dM/d\rho_c > 0$) is satisfied for the whole range of central densities considered.
- [**MPA1 (Figure \[fig\_MPA1\]):**]{} MPA1 [@ref_MPA1] provides a maximum mass above the observed mass of PSR J1614-2230 in GR, though it does not pass through the M-R constraint of [@OBG10]. For $\alpha_9 > 6$ it can not satisfy the maximum mass constraint as well.
- [**MS1 (Figure \[fig\_MS1\]):**]{} The maximum mass for MS1 [@ref_MS1] satisfies the observed mass of PSR J1614-2230 only for $\alpha_9 <2$ though it moves away from the M-R constraint of [@OBG10] for such low values of $\alpha$.
For all EoS’ we observe that the maximum stable mass of a neutron star, $M_{\max}$, and its radius at this mass, $R_{\min}$, increases for decreasing values of $\alpha$, for the ranges we consider in the figures. There is no change in the behavior of $M_{\max}$ and $R_{\min}$ values while $\alpha$ changes sign. Thus the structure of neutron stars in GR ($\alpha=0$) does not constitute an extremal configuration in terms of $M_{\max}$ and $R_{\min}$. In figure \[fig\_alpha\] we show the dependence of these quantities on the value of $\alpha$ for the polytropic EoS $$\rho =\left(\frac{P}{K}\right)^{1/\Gamma} +\frac{P}{\Gamma -1}
\label{polytropic}$$ used in [@Cooney2] where $\Gamma=9/5$ is the polytropic index and $K$ is a constant. We fit the numerical results with cubic polynomials. For the maximum mass fitted with $$M_{\max} = A \alpha_9^3 + B \alpha_9^2 + C \alpha_9 + M_0$$ where $M_0$ is the maximum mass obtained for general relativity, we find that $A= -1.30796\times 10^{-6} \pm 5.547 \times 10^{-8} \, M_{\odot}$, $B = 1.44851 \times 10^{-4} \pm 1.625 \times 10^{-6}\, M_{\odot}$ and $C = -3.82907 \times 10^{-3} \pm 1.234 \times 10^{-5} \, M_{\odot}$. Interestingly, we find that for $\alpha \cong 15 \times 10^9$ cm$^2$ the maximum mass and minimum radius of the neutron star attains their minimum values starting to increase (again) beyond this value. The analysis is complicated by increased numerical oscillations with increasing values of $|\alpha|$. Similarly, we fit the minimum value of NS radius which is attained at the maximum mass with a cubic function $$R_{\min} = a \alpha_9^3 + b \alpha_9^2 + c \alpha_9 + M_0.$$ We find that $a = 6.18915 \times 10^{-6} \pm 1.625\times 10^{-6}$ km, $b = 0.00144416 \pm 4.76\times 10^{-5}$ km, $c = -0.0469634 \pm 0.0003614 $ km and $R_0 = 11.2939 \pm 0.003055$ km.
The upper and lower bounds on the value of $\alpha$ presented for each EoS are in the range of $|\alpha| \sim 10^{9}$ cm$^2$. Values of $|\alpha|$, that are an order of magnitude smaller than this value, produce results that can not be distinguished from the results obtained within GR. This corresponds to a curvature scale of $R_0 \sim \alpha^{-1} \sim 10^{-10}$ cm$^{-2}$ and a corresponding length scale of $L\sim \alpha^{1/2} \sim 10^5$ cm, which is an order of magnitude smaller than the radius of the neutron star.
![\[fig\_FPS\] M-R relation obtained with $f(R)=R+\alpha R^2$ using the FPS. The observational constraints of [@OBG10] is shown with the thin black contour; the measured mass $M=1.97 \pm 0.04\, M_{\odot}$ of PSR J1614-2230 [@dem10] is shown as the horizontal black line with grey errorbar. Each solid line corresponds to a stable configuration for a specific value of $\alpha$. Dashed lines show the solutions for unstable configurations ($dM/d\rho_c < 0$). The grey shaded region shows where the total mass would be enclosed within its Schwarzschild radius. The red line ($\alpha=0$) shows the result for GR. $M_{\max}$ and $R_{\min}$ increase for decreasing values of $\alpha$. Variations in the M-R relation comparable to employing different EoS’ can be obtained for $|\alpha| \sim 10^{9}$ cm$^{2}$. Using $\alpha \lesssim 10^{8}$ cm$^{2}$ gives M-R relations that can not be distinguished from the GR result on this plot. ](FPS_MR)
![\[fig\_SLY\] M-R relation for the SLy. The notation in the figure is the same as that of Figure \[fig\_FPS\] and the results are discussed in the text.](SLY_MR)
![\[fig\_AP4\] M-R relation for the AP4. The notation in the figure is the same as that of Figure \[fig\_FPS\] and the results are discussed in the text.](AP4_MR)
![\[fig\_GS1\] M-R relation for the GS1. The notation in the figure is the same as that of Figure \[fig\_FPS\] and the results are discussed in the text.](GS1_MR)
![\[fig\_MPA1\] M-R relation for the MPA1. The notation in the figure is the same as that of Figure \[fig\_FPS\] and the results are discussed in the text.](MPA1_MR)
![\[fig\_MS1\] M-R relation for the MS1. The notation in the figure is the same as that of Figure \[fig\_FPS\] and the results are discussed in the text.](MS1_MR)
![$M_{\max}$ (left panel) and $R_{\min}$ (right panel) changing with $\alpha$ for the polytropic EoS given in Equation (\[polytropic\]).[]{data-label="fig_alpha"}](Malpha)
![$M_{\max}$ (left panel) and $R_{\min}$ (right panel) changing with $\alpha$ for the polytropic EoS given in Equation (\[polytropic\]).[]{data-label="fig_alpha"}](Ralpha)
Discussion
==========
In this work we analyze the neutron star solutions with realistic EoS’ in perturbative $f(R)$ gravity. Among the modified gravity theories the $f(R)$ theories are relatively simple to handle. However, even for these theories, the field equations are complicated and obtaining modified TOV equations in a standard fashion is difficult. This difficulty is mainly due to field equations being fourth order unlike in the case of general relativity, which has second order field equations. In order to resolve this situation, we adapt a perturbative approach [@Cooney2] in which the extra terms in the gravity action are multiplied by a ‘dimensionful’ parameter $\alpha$. The extra terms with some appropriate value of $\alpha$ are supposed to act perturbatively and modify the results obtained in the case of general relativity. We present how the perturbative $f(R)$ modifications affect the TOV equations.
A drawback in using neutron stars for testing alternative theories of gravity has been the weakly constrained M-R relation. After the tight constraints obtained in [@OBG10], it seems this is no longer quite true. The result of [@OBG10] and the measured mass of PSR J1614-2230 [@dem10] excludes many EoS’ in the framework of GR. In the $f(R)=R+\alpha R^2$ gravity model, the value of $\alpha$ provides a new degree of freedom and we show in this paper that some of the EoS’, which are excluded within the framework of GR, can now be reconciled with the observations for certain values of $\alpha$. This then brings the question of degeneracy between using different EoS’ and modifying gravity. In the gravity model studied here, variations in M-R relation comparable to that of using different EoS are induced for $\alpha$ being in the order of $10^9$ cm$^2$, which we specify for each EoS. An order of magnitude larger values of $\alpha$, which is still 5 orders of magnitude smaller than the constraint obtained via Gravity Probe B [@naf], does not produce neutron stars with observed properties. Thus, we argue that $|\alpha| \lesssim 10^{10}$ cm$^2$ is a reasonable constraint independent of the EoS. We conclude that the presence of uncertainties in the EoS does not cloak the effect of the free parameter $\alpha$ on the results.
For some EoS’ (AP4 and GS1) a new stable solution branch ($dM/d\rho_c>0$), which does not exist in general relativity, is found. This solution branch, for larger values of $|\alpha|$ has a larger domain and joins the conventional stable branch beyond some $\alpha$. In this case there is no critical maximum mass to the neutron stars.
One might be curious whether the perturbative approach followed in this paper holds for the range of $\alpha$ considered. For a neutron star the typical value of the Ricci curvature is calculated to be roughly on the order of $\sim 10^{-12}$ cm$^{-2}$. Therefore in the case of $f(R)=R+\alpha R^2$ model one easily sees that the perturbative term is $10^{-2}$ orders of magnitude smaller than the Einstein–Hilbert term $R$, which justifies our approach.
Although we consider 6 representative EoS’ here, the above analysis repeated with other realistic EoS’ will not alter the order of magnitude of the constraint on $\alpha$ for the following reason: the constraints we obtained implies a length scale of $R_0^{-1/2}\sim 10^5$ cm which is only an order of magnitude smaller than the typical radius of a neutron star, the probe used in this test. This is actually the length scale below which the gravity models used here *should* induce modifications on the M-R relation of an object of size 10 km. This implies that real deviations from general relativity should be hidden at even much smaller values of $|\alpha|$.
We thank F. Özel and D. Psaltis for useful discussions and valuable comments, and J. Lattimer for providing the EoS data. This work is supported by the Turkish Council of Research and Technology (TÜBİTAK) through grant number 108T686.
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[^1]: A realistic equation of state employs different physics for different regions of the neutron star, as opposed to a single polytropic relation prevailing throughout the star.
[^2]: For a critic of these constraints see [@ste10].
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bibliography:
- 'kerrcftspinors.bib'
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[**Boundary Terms, Spinors and Kerr/CFT** ]{}\
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(1,0)[447]{}
.8cm [**Melanie Becker, Waldemar Schulgin**]{}
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-0.1cm [*George and Cynthia Mitchell Institute*]{} -0.15cm [*-.05cm for Fundamental Physics and Astronomy*]{} -0.15cm [*-.05cm Texas A &M University, College Station*]{} -0.15cm [*-.05cm TX 77843–4242, USA*]{} -0.15cm 0.5cm [-.5cm mbecker AT physics.tamu.edu, schulgin AT physics.tamu.edu]{}
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[Abstract:]{} Similarly as in AdS/CFT, the requirement that the action for spinors be stationary for solutions to the Dirac equation with fixed boundary conditions determines the form of the boundary term that needs to be added to the standard Dirac action in Kerr/CFT. We determine this boundary term and make use of it to calculate the two-point function for spinor fields in Kerr/CFT. This two-point function agrees with the correlator of a two dimensional relativistic conformal field theory.
Introduction
============
AdS/CFT and its generalizations play a major role in recent developments of theoretical physics. Examples which are related to realistic physical objects are, however, still rare. The ultimate goal of the Kerr/CFT correspondence is to describe the black holes of our universe in terms of a dual two dimensional conformal field theory. The concrete proposal of [@Guica:2008mu] is that the near horizon region of a near-extremal Kerr black hole (the so called near-NHEK geometry) is dual to a two dimensional conformal field theory. Even though we are still far from describing a real black hole, many tests supporting this conjecture have appeared in the literature so far (see [@Bredberg:2011hp] for a review). In particular, the scattering amplitudes for spinor fields computed in [@Hartman:2009nz] (see also [@Chen:2010ni]) were found to be in agreement with the conformal field theory result. Spinor fields in AdS/CFT and its non-relativistic generalizations are particularly interesting to study, as their correlation functions are many times related to semi-realistic physical observables such as the spectral function.
In this note we would like to revisit spinor fields in Kerr/CFT. We would like to consider spinor fields in the near-NHEK geometry $$\label{nearNHEK}
ds^2=2J\Gamma\Big(-r(r+4\pi T_R)dt^2+\frac{dr^2}{r(r+4\pi T_R)}+d\theta^2+\Lambda^2\left(d\phi+(r+2\pi T_R)dt\right)^2\Big) \ ,$$ where $$\Gamma(\theta)=\frac{1+\cos^2\theta}{2} \ , \qquad \Lambda(\theta)=\frac{2\sin\theta}{1+\cos^2\theta} \ , \qquad\phi\sim\phi+2\pi,\ 0\le\theta\le\pi\ .$$ More concretely, we would like to calculate two-point correlation functions for spinor fields in this geometry.
Recall that in AdS/CFT spinor field correlation functions are slightly more involved than correlation functions for scalars. Let’s recapitulate some highlights for spinor fields in AdS/CFT, which will become handy later on (see [@Henningson:1998cd], [@Mueck:1998iz], [@Henneaux:1998ch], [@Iqbal:2009fd] for more details).
The key assumption of the correspondence is the equivalence between the partition functions of a CFT in $d$-dimensions and a bulk gravitational theory in $(d+1)$-dimensions $$\label{eins}
\langle \exp\left( \int d^d x\, \left(\bar\chi_0{\cal O}+\bar{\cal O}\chi_0\right)\right)\rangle_{{\rm QFT}}=e^{-S_{\rm{grav}}(\chi_0,\bar \chi_0)}.$$ In this formula $\chi_0$ is the asymptotic value of the $(d+1)$-bulk spinor $\psi$ $$\label{zwei}
\lim_{r\rightarrow\infty}\, \psi\sim\chi_0 \, ,$$ that couples to the conformal field theory operator ${\cal{O}}$. The above formula tells us that to calculate correlation functions of the CFT operator ${\cal O}$, one needs to evaluate the gravitational action $S_{\rm{grav}}$ for solutions to the Dirac equation with proper boundary conditions. The gravitational action functional contains a bulk term described by the standard Dirac action that vanishes for solutions to the equations of motion. In addition there is a boundary term [@Henningson:1998cd], [@Mueck:1998iz],[@Iqbal:2009fd] which is non-vanishing for solutions to the equations of motion. Correlation functions of the CFT operator $\cal{O}$ are determined by this boundary term. For example, the two point function of two conformal field theory operators ${\cal O},\bar {\cal {O}}$ is given by the functional derivative of the boundary term $S_{\rm {bdry}}$ $$\label{2ptfct}
\langle {\cal O}\bar{{\cal} {O}}\rangle = \frac{\delta^2S_{\rm {bdry} }}{ \delta \bar \chi_0\delta {\chi}_0}.$$ As nicely shown in [@Henneaux:1998ch], the form of the gravitational boundary term is dictated by the variational principle. More recently it was shown that different boundary terms all satisfying the variational principle can be added to the bulk action [@Laia:2011zn]. Different boundary terms lead to different conformal field theories.
Having the explicit form of the boundary term, the main challenge is to find solutions to the Dirac equation with proper boundary conditions. The situation here is a bit more involved than for a simple scalar, because bulk and boundary spinors live in different dimensions and thus have different number of components in the minimal representation. A formula like needs to be interpreted with more care. Since $\psi$ is a $(d+1)$-dimensional spinor it contains twice as many degrees of freedom as $\chi_0$ that lives in $d$-dimensions. Only half of the components of $\psi$ can be fixed by $\chi_0$. The other half is determined in terms of the first by the Dirac equation. Therefore, $\psi$ is decomposed into two eigenstates of a projection operator $$\psi=\psi_++\psi_-\ , \qquad \psi_\pm=\Gamma_\pm \psi, \quad \rm{with}\quad \Gamma_{\pm}=\frac{1}{2}(1\pm\Gamma^r).$$ The explicit form of the projection operator depends on the dimension of the boundary. Details can be found in e.g [@Iqbal:2009fd]. The upshot is that for generic values of the spinor mass $\mu$, the $\psi_+$ spinor is the leading component in the large $r$ expansion. This spinor corresponds to the source that is fixed by the boundary condition and which couples to the conformal field theory operator[^1] $$\lim_{r\rightarrow \infty} r^{d/2-\mu}\psi_+=\chi_0\, .$$ The spinor $\psi_-$ is determined in terms of $\psi_+$ by the Dirac equation and vanishes as it approaches the boundary. Once the spinors solving the equations of motion with proper boundary conditions are known, the evaluation of the gravitational action functional (more precisely the boundary term) will lead to the CFT correlation functions, as previously mentioned.
Similar in spirit, in this paper we show that a boundary term needs to be added to the Dirac action for spinor fields in the near-NHEK geometry for the variational principle to be satisfied. The boundary term is the key ingredient for the calculation of the fermionic correlation functions. Using the proposed boundary term, it is shown that the bulk fermionic two-point function agrees with the two-point function of a two dimensional conformal field theory. Some additional care, however, is required because we shall perform our calculation in Lorentzian signature, rather than analytically continuing to Euclidean signature. The reason is that we are not aware of an Euclidean version of the near-NHEK metric. A Lorentzian version of AdS/CFT (where the action carries an ‘$i$’)
$$\label{minpart}
\langle \exp\left( i\int d^d x\, \left(\bar\chi_0{\cal O}+\bar{\cal O}\chi_0\right)\right)\rangle_{{\rm QFT}}=e^{-iS_{\rm{grav}}(\chi_0,\bar \chi_0)},$$
leads to some additional subtleties that are well known in the context of AdS/CFT (see [@Marolf:2004fy] for a discussion). As first explained in [@Son:2002sd] (and later reformulated by [@Iqbal:2009fd]), having complex solutions to the equations of motion requires us to amend the Lorentzian version of AdS/CFT with some further constraints:
\(1) To evaluate the action functional appearing on the right hand side of we should consider the solutions to the equations of motion with incoming boundary conditions. (2) To evaluate boundary terms of the action, we should not consider any contributions coming from the horizon. (3) Applying the Euclidean AdS/CFT prescription to the Lorentzian theory means that the desired correlator plus its complex conjugate appear once the functional derivative of the gravitational action functional is taken. The correct result for the correlator is given by one of these contributions, while the other should be discarded. We shall see that these three constraints plus the equivalence of partition functions provides the correct fermionic correlation function in Kerr/CFT. This paper is organized as follow. In Section 2 we consider the variational principle for spinor fields in Kerr/CFT and determine the boundary term. In Section 3 we perform the calculation of the fermionic two-point function using the proposed boundary term. In Section 4 we show how the result of Section 3 can be matched with a two dimensional relativistic conformal field theory. Our conclusions appear in Section 5. In Appendix \[appA\] we present the features of the near-NHEK geometry we need for the calculation of correlation functions, while Appendix \[appB\] is left for notations and conventions.
Variational Principle
=====================
The bulk action for fermions with mass $\mu$ in the near-NHEK geometry is the standard Dirac action $$\label{bulk}
S_{\rm {bulk}}=i\int d^4 x \sqrt{-g}\bar \psi \left( {\slashed D} -\mu\right)\psi,$$ where we have dropped an overall normalization factor. We use the representation of the four dimensional bulk gamma matrices as given in appendix \[appB\]. To determine the boundary conditions on the spinor we calculate the variation of the action which is given by [^2]
$$\label{nine}
\delta S_{\rm{bulk}}=i\int_{r=r_B}\, d^3x \sqrt{-g_B} \bar \psi \Gamma^r \delta \psi+\ldots,$$
where the dots denote terms that vanish by the equations of motion. Here $g_B=g g^{rr}$ describes the induced boundary metric and $r_B$ is the cutoff describing the boundary of the near-NHEK geometry. The gamma matrix $\Gamma^r$ in the near-NHEK geometry takes the form
$$\Gamma^r=-\frac{r(r+4\pi T_R)}{8J\Gamma}(\Gamma^0+\Gamma^3)+\frac{1}{2}(\Gamma^0-\Gamma^3).$$
Boundary conditions need to be imposed so that the variation of the gravitational action vanishes. To do so, we take a closer look at the spinor solving the Dirac equation $$({\slashed D} -\mu)\psi=0\, .$$ The solution to this equation in the Kerr geometry was worked out by Chandrasekhar in the late seventies [@Chandrasekhar:1976ap]. Using the Newman-Penrose formalism he showed that the Dirac equation can be separated into a radial and an angular equation. Finding an analytical expression for the solution proved, nevertheless, to be very difficult. For a long time only numerical solutions were available. More than thirty years later, an analytic expression for the solution to the Dirac equation in the near-NHEK limit was obtained in [@Hartman:2009nz]. In this limit (described to the necessary details in Appendix \[appA\]) the spinor computed in [@Hartman:2009nz] takes the form $$\label{spinorsolution}
\psi=e^{-i n_R t+i n_L \phi}
\left(\begin{matrix}
-R_{1/2}S_{1/2}\\ \\
\frac{R_{-1/2}S_{-1/2}}{\sqrt{2}M(1-i\cos(\theta))}\\\\
-\frac{R_{-1/2}S_{1/2}}{\sqrt{2}M(1+i\cos(\theta))}\\\\
R_{1/2}S_{-1/2}\\
\end{matrix}\right)\, ,$$ where $R_{\pm 1/2}= R_{\pm 1/2}(r)$ describes the radial dependence and $S_{\pm 1/2}=S_{\pm 1/2}(\theta)$. Even though the radial part $R_{\pm 1/2}$ of the solution is in general a hypergeometric function, we only need its asymptotic expression (for large but finite $\lambda r$. The solution with infalling boundary conditions is $$\begin{aligned}
\label{R}
R_{1/2}(r)&=&N_{1/2}T_R^{-in_R/2-1/2}\left(A_{1/2}\left(\frac{ r}{T_R}\right)^{-1+\beta}+B_{1/2}\left(\frac{ r}{T_R}\right)^{-1-\beta}\right)+\ldots\\
R_{-1/2}(r)&=&N_{-1/2}T_R^{-in_R/2+1/2}\left(A_{-1/2}\left(\frac{ r}{T_R}\right)^{\beta}+B_{-1/2}\left(\frac{r}{T_R}\right)^{-\beta}\right)+\ldots\end{aligned}$$ The coefficients appearing in these expressions are defined in terms of gamma functions $$\begin{aligned}
\label{AB}
A_s&=&\frac{\Gamma(1-i(n_R+n_L)-s)\Gamma(2\beta)}{\Gamma(\frac{1}{2}+\beta-in_R)\Gamma(\frac{1}{2}+\beta-in_L-s)}\ , \nonumber\\
B_s&=&\frac{\Gamma(1-i(n_R+n_L)-s)\Gamma(-2\beta)}{\Gamma(\frac{1}{2}-\beta-in_R)\Gamma(\frac{1}{2}-\beta-in_L-s)} \ .\end{aligned}$$ The $N$’s describe normalization factors $$\label{beta}
\frac{N_{1/2}}{N_{-1/2}}=\frac{1/2-i(n_R+n_L)}{M(\Lambda_\ell+i\mu M)}\, ,\qquad \beta^2+n_L^2=\Lambda_\ell^2+\mu^2 M^2.$$ In this paper we restrict to real values of $\beta$ for simplicity. Similarly as in AdS/CFT, only half of the components of $\psi$ can be fixed at the boundary (the other half is related to the first half by the Dirac equation and will vanish at the boundary). To decide which components of $\psi$ we would like to fix, it is convenient to introduce projection operators $$P_\pm=\frac{1}{2}\left(1\pm\Gamma^0\Gamma^3\right)\, ,$$ which satisfy $P_+^2=P_+$, $P_-^2=P_-$ and $$\Gamma^0\pm \Gamma^3=\Gamma^0 P_\pm=P_\mp \Gamma^0.$$ These operators allow us to write the bulk spinor in terms of projector eigenstates as $$\psi=\psi_++\psi_- \, .$$ Here $\psi_+$ satisfies $$P_+\psi=\psi_+=e^{-in_R t+i n_L\phi}R_{1/2}
\left(\begin{matrix}
-S_{1/2} \\
0\\
0\\
S_{-1/2}\\
\end{matrix}\right)\, ,$$ while $\psi_-$ obeys $$P_-\psi=\psi_-=e^{-in_R t+i n_L\phi}\frac{R_{-1/2}}{\sqrt{2}M}
\left(\begin{matrix}
0\\ \frac{S_{-1/2}}{1-i\cos \theta} \\
-\frac{S_{1/2}}{1+i\cos \theta}\\
0
\end{matrix}\right),$$ and conjugate spinors satisfy $$\bar\psi P_\pm=\bar\psi_\mp.$$ To decide whether $\psi_+$ or $\psi_-$ is the source (which gets fixed at the boundary), we notice that there is a relation between both boundary spinors[^3] $$\psi^{B}_+\sim { R_{1/2}^B \over R_{-1/2}^B}\psi^{B}_-,$$ where the index $B$ denotes boundary quantities. Taking into account , this relation tells us that for real $\beta$ we should treat $\psi^B_-$ as the source, while $\psi^B_+$ vanishes at the boundary.
We can now proceed to evaluate the boundary term. To do so it is convenient to write the $\Gamma^r$ matrix in terms of the projection operators $$\label{ten}
\Gamma^r=\frac{1}{8J\Gamma}r(r+4\pi T_R)P_-\Gamma^0 P_+-\frac{1}{2}P_+\Gamma^0 P_-.$$ It is easy to see that the boundary term becomes
$$\label{eleven}
\delta S_{\rm {bulk}}=i\int_{r=r_B}\, d^3x\, \sqrt{-g_B}\left( \frac{1}{8J\Gamma}\, r_B(r_B+4\pi T_R)\, \psi^\dagger_+\delta\psi_+-\frac{1}{2}\psi_-^\dagger\delta\psi_-\right).$$
We had seen that $\psi_-$ is the source, so this spinor and its conjugate are fixed at the boundary $$\label{cond}
\delta\psi_-\Big|_{r_B}=0 \, ,\quad \delta\bar \psi_-\Big|_{r_B}=0 \, .$$ To cancel the contribution proportional to $\delta \psi_+$ we need to add a boundary term $$\label{bndry_term}
S_{\rm{bdry}}
=-\frac{r_B(r_B+4\pi T_R)}{8J\Gamma}\, i\int_{r=r_B} d^3 x \, \sqrt{-g_B}\, \psi_+^\dagger\psi_+.$$ This guarantees that the variation of the total action vanishes[^4] $$\label{twelve}
\delta S_{\rm {total}}=\delta S_{\rm {bulk}} + \delta S_{\rm {bdry}}=0.$$ It is interesting to observe that the boundary term looks similar to the non-relativistic boundary terms recently considered in [@Laia:2011zn]. There it was argued that non-relativistic conformal field theories can be generated through Lorentz violating boundary terms, even though the underlying bulk theory is Lorentz invariant. One may wonder if the conformal field theory dual to the near-NHEK geometry could be non-relativistic. Some recent discussion on the possible connection between Kerr/CFT and non-relativistic conformal field theory has appeared recently in the literature [@ElShowk:2011cm]. A more extensive analysis is needed to answer this question.
It is interesting to notice that the boundary term can be written as $$\label{bound}
S_{\rm {bdry}}=i\int_{r=r_B}\, d^3 x \sqrt{-g_B} \, \bar \psi \Gamma^r\psi,$$ up to contact terms. This expression is familiar from the fermionic flux derived in [@Martellini:1977qf], [@Iyer:1978du]. There it was shown that superradiance does not occur for a fermionic field in a Kerr geometry as the particle flux into the black hole is always positive. Precisely the same expression for the fermionic flux entered the scattering calculation done in [@Hartman:2009nz], so the above boundary term does not come as a surprise. It is nice to see this expression emerge from the variational principle.
Fermionic Two-Point Function: the Bulk
======================================
To calculate correlation functions for spinors in the near-NHEK geometry we need to evaluate the boundary term for spinors satisfying the equations of motion. We would like to express the bulk spinor in terms of its value on the boundary. To do so it is convenient to off the $\theta$ dependence by introducing spinors $a^{\pm}$
$$\psi=e^{-i n_R t+i n_L \phi}\left( R_{1/2}
\underbrace{\left(\begin{matrix}-S_{1/2}\\0 \\ 0 \\S_{-1/2}
\end{matrix}
\right)}_{a_+}
+R_{-1/2}
\underbrace{\left(
\begin{matrix}
A & 0&0&0\\
0&\frac{1}{\sqrt{2}M(1-i\cos \theta)}&0&0\\
0&0&\frac{1}{\sqrt{2}M(1+i\cos \theta)}&0\\
0 & 0&0&B
\end{matrix}
\right)}_{Z}
\underbrace{\left(\begin{matrix}0\\S_{-1/2}\\-S_{1/2}\\0
\end{matrix}
\right)}_{a_-}
\right)$$
$A$ and $B$ are arbitrary non-zero entries, so that $Z$ is invertible. The eigenstates of the projection operator $\psi_\pm$ can be conveniently written as $$\begin{aligned}
\label{fourfour}
\psi_+&=&e^{-in_R t+in_L \phi}R_{1/2} a_+,\nonumber\\
\psi_-&=&e^{-in_R t+in_L \phi}R_{-1/2}Za_-.\end{aligned}$$
Since there is a relation between $a_+$ and $a_-$, $\Gamma^0 a^+=a^-$, we can write the bulk spinor in terms of $a_-$ only $$\psi=e^{-in_R t+i n_L \phi}\left( R_{1/2}\Gamma^0+R_{-1/2}Z \right) a_-.$$ Using equation () we can express this spinor and its conjugate in terms of boundary data $$\begin{aligned}
\psi&=&\left(R_{1/2}\Gamma^0 Z^{-1}+R_{-1/2}\right)\frac{\psi^B_-}{R_{-1/2}^B},\nonumber\\
\bar \psi& =& \frac{\bar \psi^{B}_-}{\bar R_{-1/2}^B}\left(\bar R_{1/2} \Gamma^0Z^{*-1}+\bar R_{-1/2}\right),\end{aligned}$$ where the bar on $\bar R_{\pm 1/2}$ means complex conjugation. To apply the prescription for computing the boundary two-point function, we would like to express the boundary term as a double integral over momenta. This will allow us to take the functional derivative. Fourier transforming along the $t$ are $\phi$ directions we introduce new spinors $$\begin{aligned}
\psi_F (r,\theta,n_L,n_R)&=&\delta(n_L-n_L')\delta(n_R-n_R')\Big(R_{1/2} (r, n_L',n_R')\Gamma^0 Z^{-1}+R_{-1/2}(n_L',n_R')\Big)\frac{ \psi^B_-(\theta,n_L',n_R')}{R_{-1/2}^B(n_L',n_R')}\nonumber\\
{\bar \psi}_F (r,\theta, n_L,n_R)&=&\delta(n_L-n_L')\delta(n_R-n_R')\,\frac{{\bar \psi}^{B}_-(\theta,n_L',n_R')}{\bar R_{-1/2}^B(n_L',n_R')} \Big(\bar R_{1/2}(r,n_L',n_R')\Gamma^0 Z^{*-1}+\bar R_{-1/2}(r,n_L',n_R')\Big)\nonumber\end{aligned}$$ where $n_L$ and $n_R$ are the momenta dual to the coordinates $t$ and $\phi$. We insert $\psi$ and $\bar\psi$ into the boundary term $$\begin{aligned}
\int d\theta\, \int dt\, d\phi\, \sqrt{-g} \,\bar \psi \Gamma^r\psi\Big|_{r=r_B}&=&\int d\theta\, \sqrt{-g_B}\, \int dn_L' dn_R' \, \int dn_L'' dn_R''\, \delta(n_L'-n_L'')\, \delta(n_R'-n_R'')\times \nonumber\\
&&\times {\bar \psi}_F (r_B,\theta, n_L',n_R')\, \Gamma^r\, { \psi}_F (r_B, \theta,n_L'',n_R''),\end{aligned}$$ where the determinant of the metric only depends on $\theta$ and the cutoff $r_B$ $$\sqrt{-g_B}=(2J\Gamma(\theta))^{3/2}\Lambda(\theta)\, r_B.$$ Using the explicit form of $\Gamma^r$ and the properties of the projection operator listed in appendix \[appB\], we can evaluate the integrand of the boundary term $$\begin{aligned}
\label{bdr}
{\bar \psi}_F (r_B,\theta, n_L',n_R')&\Gamma^r&{ \psi}_F (r_B,\theta, n_L'',n_R'')=-\frac{r_B^2}{8J\Gamma}{\bar \psi}_+\Gamma^0\psi_+
+\frac{1}{2}{\bar \psi}_-\Gamma^0\psi_-\nonumber\\
&=&-\frac{r_B^2}{8J\Gamma}\frac{R^B_{1/2}\bar R^B_{1/2}}{R_{-1/2}^B
\bar R_{-1/2}^B}{\bar \psi}_-^B\Gamma^0 |Z^{-1}|^2\psi_-^B+\frac{1}{2}{\bar\psi}^B_-\Gamma^0 \psi_-^B,\end{aligned}$$ where we have dropped the coordinate dependency on the rhs for simplicity of the notation. We would like to factor out the $\theta$-dependency of the boundary term. To do so notice that $\psi_-^{B}$ can be split into a two dimensional chiral spinor $\chi_0$ and a boundary spinor describing the theta dependence $$\psi_{-}^B=\chi_{0}\otimes(S^+\oplus S^-).$$ In the four dimensional representation space we know each spinor explicitly $$\psi^{B}_-(\theta, n_R,n_L)=\underbrace{\delta(n_L-n_L')\delta(n_R-n_R')R_{-1/2}^B(n_L',n_R')}_{\chi_0(n_L',n_R')}
\left (
\underbrace{Z\left(
\begin{matrix}
0\\S_{-1/2}\\0\\0
\end{matrix}
\right)}_{S^+}+
\underbrace{Z\left(
\begin{matrix}
0\\0\\-S_{1/2}\\0
\end{matrix}
\right)}_{S^-}
\right ).$$ $$\bar\psi^{B}_-(\theta, n_R,n_L)=\underbrace{\delta(n_L-n_L')\delta(n_R-n_R')\bar R_{-1/2}^B(n_L',n_R')}_{\bar \chi_0(n_L',n_R')}
\left (
\underbrace{Z\left(
\begin{matrix}
0\\S_{-1/2}\\0\\0
\end{matrix}
\right)}_{S^+}+
\underbrace{Z\left(
\begin{matrix}
0\\0\\-S_{1/2}\\0
\end{matrix}
\right)}_{S^-}
\right )^\dagger\, \Gamma^0.$$
Inserting this into the boundary term we notice that the theta dependence of the relevant contribution (the first term of the expression below) can be factored out $$\begin{aligned}
&&\int d\theta \sqrt{-g_B}(|S_{1/2}|^2+|S_{-{1/2}}|^2) \int dn_L' dn_R'\, \int \, dn_L'' dn_R''\, \bar \chi_0(n_L',n_R')\chi_0(n_L'',n_R''))\times\nonumber\\
&&\times\delta(n_L'-n_L'')\delta(n_R'-n_R'')\times \left(-\frac{r_B^2}{8J\Gamma}\frac{R^B_{1/2}(r_B, n_L'',n_R'')\bar R^B_{1/2}(r_B, n_L',n_R')}{R_{-1/2}^B(r_B, n_L'',n_R'')\bar R_{-1/2}^B(r_B, n_L',n_R')}\ +\frac{1}{4M^2(1+\cos ^2\theta)}\right)\, .\nonumber\\\end{aligned}$$ The second term in this expression describes a contact term that can be ignored, so we evaluate the first term. To do so we expand $R_{1/2}$ and $R_{-1/2}$ around $r_B$ using equations - to evaluate individual contributions. We are left with
$$\begin{aligned}
&&\frac{{\delta}^2 S}{\delta\chi_0(n_L,n_R)\bar\delta\chi_0(n_L,n_R)}\sim r_B^3\,\frac{R^B_{1/2}( n_L,n_R)\bar R^B_{1/2}( n_L,n_R)}{R_{-1/2}^B(n_L,n_R)\bar R_{-1/2}^B( n_L,n_R)}\nonumber\\
&=&\frac{N_{1/2}}{N_{-1/2}}\, \frac{\bar N_{1/2}}{\bar N_{-1/2}}\,\left(\frac{A_{1/2}}{A_{-1/2}}\,\frac{\bar A_{1/2}}{\bar A_{-1/2}}r_B
+\frac{B_{1/2}}{A_{-1/2}}\,\frac{\bar A_{1/2}}{\bar A_{-1/2}}T_R^{2\beta}r_B^{-2\beta+1}+\frac{A_{1/2}}{A_{-1/2}}\,\frac{\bar B_{1/2}}{\bar A_{-1/2}}T_R^{2\beta}r_B^{-2\beta+1}+{\cal O}(r^{-4\beta+1})\right)
\nonumber\\
&=&\frac{1}{M^2}r_B+\left(\mu+\frac{i\Lambda_\ell}{M}\right) \frac{\bar N_{1/2}\bar A_{1/2}} {\bar N_{-1/2}\bar A_{-1/2} }\, G_R(n_L,n_R)r_B^{-2\beta+1}+\left(\mu-i\frac{\Lambda_\ell}{M}\right) \frac{ N_{1/2} A_{1/2}} { N_{-1/2} A_{-1/2} }G^*_R(n_L,n_R)r_B^{-2\beta+1}\nonumber\\ &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+{\cal O}\left(r^{-4\beta+1}\right)\end{aligned}$$
with $$\label{GR}
G_R(n_L,n_R)=-\frac{i}{\beta+in_L}\frac{\Gamma(-2\beta)}{\Gamma(2\beta)}\frac{\Gamma(\beta-in_L)}{\Gamma(-\beta-in_L)}\frac{\Gamma(\frac{1}{2}+\beta-in_R)}{\Gamma(\frac{1}{2}-\beta-in_R)}T_R^{2\beta}$$ The first term above is obviously the contact term. The second and third terms are complex conjugate to each other. Similarly as for the scalar two point function in Lorentzian AdS/CFT considered in [@Son:2002sd], this means that the two point function is real, which is not what we want. The proposal of [@Son:2002sd] is to drop the complex conjugate solution. The $r_B$-factor here can be absorbed into $\chi_0$ by rescaling $$\chi_0\rightarrow r_B^{\beta-1/2}\chi_0.$$ Last, we factored out the ratio $\frac{N_{1/2}A_{1/2}}{N_{-1/2}A_{-1/2}}$ which is momentum dependent but not part of the two point function. A similar factor emerges in AdS/CFT calculations [@Iqbal:2009fd].
The expression agrees with the proposal of [@Chen:2010ni]. Recall that the relation between $G_R$ and the absorption probability $\sigma$ is ${\rm{Im}}\, G_R\sim \sigma$. The Greens function we calculated precisely gives the absorption probability of [@Bredberg:2009pv].[^5]
Fermionic Two-Point Function: CFT Result {#CFTcomp}
========================================
This section serves as a reminder for some basics on finite temperature conformal field theory. We would like to write the finite temperature two-point function of a two dimensional CFT in momentum space and compare with the result of the previous section. We start with the more familiar zero temperature correlation function in coordinate space. The zero temperature two-point function of a conformal field theory operator with conformal weights $$h_L=\frac{1}{2}\left(\Delta-\frac{1}{2}\right)\ , \qquad h_R=\frac{1}{2}\left(\Delta+\frac{1}{2}\right)\ ,$$ takes (up to a constant) the following form in coordinate space $$\label{retgreens}
\langle {\cal O}(\vec x)\bar {\cal O}(\vec y)\rangle
\sim\frac{\gamma^i(x^i-y^i)}{|\vec x-\vec y|^{(2\Delta+1)}} \ .$$ More explicitly we can use the following representation of the gamma matrices $$\label{gammas}
\gamma^0=\left(
\begin{array}{cc}
0&-1\\1&0
\end{array}
\right)\ ,
\qquad
\gamma^1=\left(
\begin{array}{cc}
0&1\\1&0
\end{array}
\right) \ ,$$ and the coordinates $$\begin{aligned}
t^+_1&=&x^0+x^1 \ , \qquad t^-_1=x^0-x^1\ ,\nonumber\\
t^+_2&=&y^0+ y^1 \ , \qquad t^-_2=y^0-y^1 \ ,\nonumber\end{aligned}$$ to rewrite (\[retgreens\]) as $$\langle {\bar \cal O}(t_1^+,t_1^-) {\cal O}(t_2^+,t_2^-)\rangle \sim \gamma^0
\left(
\begin{array}{cc}
\frac{1} {
(t^+_{12})^{2h_R-1} (t^-_{12})^{2h_L+1}
}&0\\
0&\frac{1}{{(t^+_{12})}^{2h_R}{(t^-_{12})}^{2h_L}}
\end{array}
\right) \ ,$$ where we have introduced $t_{12}^+=t_1^+-t_2^+$ and similarly for $t_{12}^-$. The finite temperature correlation function is obtained by mapping the above result to a torus with circumferences $1/T_L$ and $1/ T_R$ $$\label{resexpr}
\langle {\cal O}(t_1^+,t_1^-)\bar {\cal O}(t_2^+,t_2^-)\rangle
\sim\gamma^0\left(
\begin{array}{cc}
\left(\frac{\pi T_R}{\sinh(\pi T_R t^+_{12})}\right)^{2h_R-1} \left(\frac{\pi T_L}{\sinh(\pi T_L t^-_{12})}\right)^{2h_L+1} &0\\
0&\left(\frac{\pi T_R}{\sinh(\pi T_R t^+_{12})}\right)^{2h_R} \left(\frac{\pi T_L}{\sinh(\pi T_L t^-_{12})}\right)^{2h_L}
\end{array}
\right).$$
The formula (\[resexpr\]) is the two-point function $\langle {\cal O}\bar {\cal O}\rangle$ for a non-chiral spinor operator ${\cal O}$. The AdS/CFT correspondence gives a correlator only between chiral/antichiral parts of the operator ${\cal O}$. $$\begin{aligned}
\label{chiral}
{\cal O}^{\pm}=\frac{1}{2}\left(1\pm\gamma^0\gamma^1\right){\cal O} \ ,\qquad \bar{\cal O}^{\pm}=\bar {\cal O}\frac{1}{2}\left(1\mp\gamma^0\gamma^1\right) \ .\end{aligned}$$ Inserting (\[chiral\]) into $\langle {\cal O}\bar {\cal O}\rangle$ with ${\cal O}=\left(\begin{array}{c}{\cal O}_1\\{\cal O}_2\end{array}\right)$ we see that the non-zero elements of (\[resexpr\]) can be identified with $$\langle {\cal O}^+\bar {\cal O}^+\rangle=\gamma^0\left(
\begin{array}{cc}
\langle{\cal O}_2 {\cal O}_2\rangle & 0\\
0&0
\end{array}
\right) \ ,\qquad
\langle {\cal O}^-\bar {\cal O}^-\rangle=\gamma^0\left(
\begin{array}{cc}
0&0\\ 0&
\langle{\cal O}_1 {\cal O}_1\rangle
\end{array}
\right)\, .$$
After analytic continuation $t^\pm\rightarrow it^\pm$, we Fourier transform the two-point function assuming only integer frequencies $\omega_E=2\pi k T$ by using $$\int_{0}^{1/T}dt e^{i\omega_E t}\left(\frac{\pi T}{\sin(\pi Tt)}\right)^{2h}= \frac{(\pi T)^{2h-1}2^{2h}e^{i\omega_E/2T}\Gamma(1-2h)}{\Gamma\left(1-h+\frac{\omega_E}{2\pi T}\right)\Gamma\left(1-h-\frac{\omega_E}{2\pi T}\right)}\ , .$$
Once we identify $k_L=-in_L , k_R=-in_R$ and $h_L=\beta , h_R=\beta+\frac{1}{2}, T_L=\frac{1}{2\pi}, T_R=T_R$ the two-point function $\langle {\cal O}^-{\bar{\cal O}}^-\rangle$ on the CFT side becomes[^6] $$\langle {{\cal O}}^-{\bar{\cal O}}^-\rangle\sim T_R^{2\beta}\frac{1}{\beta+in_L}\frac{\Gamma(-2\beta)\Gamma(\beta-in_L)\Gamma(\frac{1}{2}+\beta-in_R)}{\Gamma(2\beta)\Gamma(-\beta-in_L)\Gamma(\frac{1}{2}-\beta-in_R)}\, .$$ This matches the expression computed on the bulk side.
Conclusions
===========
In this note we have calculated finite temperature two point correlations functions for fermionic fields in Kerr/CFT using the variational principle. Fermionic fields are particularly interesting because their correlation functions describe semi-realistic physical observables, such as the spectral function.
To perform this calculation we have followed an approach well known for AdS/CFT. After analyzing the variational principle we have seen that a boundary term needs to be added to the Dirac action for the variational principle to be satisfied. This boundary term is responsible for generating non-trivial fermion correlation functions. Kerr/CFT is a duality in which a four-dimensional bulk geometry is dual to a two-dimensional conformal field theory. The fact that the conformal field theory lives in two dimensions less than the original bulk theory may sound at first surprising because from AdS/CFT we are used to the fact that the conformal field theory lives in one dimension less than the bulk rather than two. Fermions allow us to very nicely understand this aspect of Kerr/CFT because fermions, as opposed to scalars, are very sensitive to the number of space-time dimensions they live in. The boundary of the near-NHEK geometry is a three-dimensional theory described by the coordinates $t$,$\phi$ and $\theta$, while the radial coordinate approaches a large but finite cutoff $r_B$. Performing the calculation of the two point-function for two spinors living on the 3D boundary, we have seen that the theta dependence of the correlation function factors out. Therefore, the fermion correlation function effectively becomes that of a two dimensional relativistic conformal field theory.
Our calculation was performed in Lorentzian signature rather than with an analytic continuation to Euclidean signature. We are not aware of a sensible Euclidean analytic continuation of the near-NHEK metric. For this reason we needed to impose some additional constraints on the two-point function that are well known from Lorentzian approaches to AdS/CFT [@Son:2002sd].
An interesting observation is that the gravitational action functional needed the inclusion of a boundary term that breaks Lorentz invariance and one may wonder if the boundary conformal field theory could be a non-relativistic theory once corrections to the leading terms are included. This would be similar in spirit to the recent discussion appearing in [@Laia:2011zn] in the context of AdS/CFT. Here a bulk theory in $AdS_4$ space-time is supplied with boundary conditions on the spinor field that break Lorentz invariance and it is argued that the dual conformal field theory is non-relativistic. Some recent discussion on the connection between Kerr/CFT and non-relativistic conformal field theories has recently appeared in [@ElShowk:2011cm]. It would be interesting to explore this connection in more detail.
Finally, it would be interesting to extend our calculation to the Kerr-Newman geometry, as well as to other correlation functions involving e.g fermions and gauge fields. We hope to report on this in the future.
Acknowledgments {#acknowledgments .unnumbered}
===============
We benefited from discussions with Katrin Becker, David Chow, Sera Cremonini, Umut Gursoy, Chris Pope, Daniel Robbins, Jan Troost as well as the correspondence with Aaron Amsel, Geoffrey Compere, Monica Guica, Tom Hartman, Gary Horowitz, Wei Song and Andrew Strominger. We would like to thank Tom Hartman and Andrew Strominger for comments on the manuscript. This work was supported by NSF under PHY-0505757, DMS-0854930 and the University of Texas A&M.
Near-NHEK Geometry {#appA}
==================
The Kerr/CFT correspondence relates the near horizon geometry of a near extreme Kerr black hole (near NHEK) to a two-dimensional conformal field theory with central charges $c_L=c_R=12J/\hbar$. The near-NHEK metric is constructed by taking a special limit of the Kerr metric. Let us summarize the main steps of this construction, as they are needed for the calculation of the fermionic two-point function. The geometry of the Kerr black hole is described by the metric $$\label{kerrmetric}
ds^2=-\frac{\Delta}{\hat \rho^2}\left(d\hat t-a\sin^2\theta d\hat\phi\right)^2+\frac{\sin^2\theta}{\hat \rho^2}\left(\left(\hat r^2+a^2\right)d\hat\phi-a d\hat t\right)^2+\frac{\hat \rho^2}{\Delta}d\hat r^2+\hat\rho^2d\theta^2,$$ with $\Delta=\hat r^2-2M\hat r +a^2$, $\hat\rho^2=\hat r^2+a^2\cos^2\theta$. In general, there are two horizons at $$r_\pm=M\pm\sqrt{M^2-a^2},$$ where $a$ is the proportionality factor between the angular momentum and the mass $J=aM$. The Hawking temperature and the angular velocity of the horizon are $$T_H=\frac{r_+-r_-}{8\pi Mr_+}=\frac{\tau_H}{8\pi M}\ , \qquad\Omega_H=\frac{a}{2Mr_+} \ .$$ The near horizon limit of the near extremal Kerr black hole can be defined by taking the limit $T_H\rightarrow 0, \ \hat r\rightarrow r_+$ with the dimensionless near-horizon temperature $T_R=\frac{2MT_H}{\lambda}$ fixed when $\lambda\rightarrow 0$. Following [@Hartman:2009nz] the metric of the near-NHEK space-time is obtained by performing the expansions $$r_+=M+\lambda M 2\pi T_R+{\cal O} (\lambda^2) \ , \qquad a=M-2M (\lambda\pi T_R)^2 +{\cal O}(\lambda^3) \ ,$$ coordinate redefinitions $$\label{coordredef}
t=\lambda \frac{\hat t}{2M} \ , \qquad r=\frac{\hat r-r_+}{\lambda r_+} \ , \qquad \phi=\hat\phi-\frac{\hat t}{2M},$$ and taking limit $\lambda\ll 1$ while keeping $T_R$ fixed. The near-NHEK metric is then $$\label{nearNHEK}
ds^2=2J\Gamma\Big(-r(r+4\pi T_R)dt^2+\frac{dr^2}{r(r+4\pi T_R)}+d\theta^2+\Lambda^2\left(d\phi+(r+2\pi T_R)dt\right)^2\Big) \ ,$$ where $$\Gamma(\theta)=\frac{1+\cos^2\theta}{2} \ , \qquad \Lambda(\theta)=\frac{2\sin\theta}{1+\cos^2\theta} \ , \qquad\phi\sim\phi+2\pi,\ 0\le\theta\le\pi\ .$$
The appearance of $\lambda$ in (\[coordredef\]) may look confusing. The range of the near-NHEK space is parametrized by the radial coordinate $r$, where it takes values $0< \lambda r\ll1$. Notice that since $\lambda\ll1$, the position of the near-NHEK boundary is at some large but still finite value of $r$.
Notations and Conventions {#appB}
=========================
- We define $\slashed D=\Gamma^MD_M$, $D_M=\partial_M+\frac{1}{4}\omega_{ab\,M}\Gamma^{ab}$ with $\omega$ being the bulk spin connection, $\Gamma^{ab}={1\over 2}[\Gamma^a, \Gamma^b]$, while the conjugate spinor is defined as $\bar \psi={\psi}^{\dagger}{\Gamma}^0$. Capital indices denote bulk space-time indices and $a,b$ denote bulk tangent indices.
- The flat gamma matrices are $$\begin{aligned}
\Gamma^0&=&\left(
\begin{matrix}
0&0&1&0\\
0&0&0&1\\
1&0&0&0\\
0&1&0&0
\end{matrix}
\right) , \ \
\Gamma^1=\left(
\begin{matrix}
0&0&0&-1\\
0&0&-1&0\\
0&1&0&0\\
1&0&0&0
\end{matrix}\right), \ \
\nonumber\\
\Gamma^2&=&\left(
\begin{matrix}
0&0&0&i\\
0&0&-i&0\\
0&-i&0&0\\
i&0&0&0
\end{matrix}
\right) , \ \
\Gamma^3=\left(
\begin{matrix}
0&0&-1&0\\
0&0&0&1\\
1&0&0&0\\
0&1&0&0
\end{matrix}\right)\nonumber\\\end{aligned}$$
- The curved gamma for the near NHEK geometry are given in [@Hartman:2009nz] $$\Gamma^\mu=\sqrt{2}\left(
\begin{matrix}
0&\sigma^\mu_{AB'}\\ \bar \sigma^{\mu A' B} &0
\end{matrix}
\right)\ , \qquad
\sigma_{AB'}^\mu=\left(
\begin{matrix}
l^\mu &m^\mu\\
\bar m^\mu & n^\nu
\end{matrix}
\right)\, ,$$ where the Newman-Penrose tetrad for the near-NHEK was worked out in [@Chen:2010ni] $$\begin{aligned}
l^\mu&=&\frac{1}{r(r+4\pi T_R)}(1,r(r+4\pi T_R),0,-(r+2\pi T_R))\ ,\nonumber\\
n^\mu &=&\frac{1}{4 J\Gamma (\theta)}(1,-r(r+4\pi T_R),0,-(r+2\pi T_R))\ ,\nonumber\\
m^\mu&=&\frac{1}{2\sqrt{J\Gamma(\theta)}}(0,0,1,i\Lambda^{-1}(\theta)) \ .\end{aligned}$$
For the near-NHEK geometry $\Gamma^r$ is given by $$\begin{aligned}
\Gamma^r&=&\left(
\begin{matrix}
0&0&1&0\\
0&0&0&-\frac{r(r+4\pi T_R)}{4J\Gamma}\\
-\frac{r(r+4\pi T_R)}{4J\Gamma}&0&0&0\\
0&1&0&0
\end{matrix}
\right)\\
&=&-\frac{r(r+4\pi T_R)}{8J\Gamma(\theta)}(\Gamma^0+\Gamma^3)+\frac{1}{2}(\Gamma^0-\Gamma^3)\end{aligned}$$
[^1]: There is a small range of values for the mass $\mu$ in which $\psi_-$ rather than $\psi_+$ is fixed by boundary conditions.
[^2]: The boundary of the near-NHEK geometry is described by large but finite $r$, $r_B\gg 1$, such that $r_+-r_-\ll \lambda r_B\ll 1$, where $\lambda$ goes to zero. $r_+, r_-$ are the positions of the outer and inner horizons of the Kerr black hole.
[^3]: The precise relation is given in the next section.
[^4]: Here we used $\delta\psi^\dagger_+\Big|_{r_B}=0$, since $\psi_+^\dagger\sim \bar \psi_-$.
[^5]: We thank Tom Hartman for pointing this out.
[^6]: Here we have absorbed the $m\Omega_R$ appearing in eq. (5.13) of [@Bredberg:2009pv] into our definition of $n_R$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a unified Boltzmann-transport theory for the drag resistivity $\rho_{\rm D}$ in two-component systems close to a second-order phase transition. We find general expressions for $\rho_{\rm D}$ in two and three spatial dimensions, for arbitrary population and mass imbalance, for particle- and hole-like bands, and show how to incorporate, at the Gaussian level, the effect of fluctuations close to a phase transition. We find that the proximity to the phase transition enhances the drag resistivity upon approaching the critical temperature from above, and we qualitatively derive the temperature dependence of this enhancement for various cases. In addition, we present numerical results for two concrete experimental systems: i) three-dimensional cold atomic Fermi gases close to a Stoner transition and ii) two-dimensional spatially-separated electron and hole systems in semiconductor double quantum wells.'
author:
- 'M. P. Mink'
- 'H. T. C. Stoof'
- 'R. A. Duine'
- Marco Polini
- 'G. Vignale'
title: 'Unified Boltzmann-transport theory for the drag resistivity close to a second-order phase transition'
---
Introduction {#sec:int}
============
The behavior of linear-response coefficients close to a phase transition has a long history as an interesting field of study. Two prime examples are the vanishing of the resistivity at the superconducting phase transition and the divergence of the magnetic susceptibility at a ferromagnetic transition. Indeed, the behavior of these coefficients is often the most important experimental signature in assessing whether or not the system has reached an ordered state. For example, deviations from the standard low temperature $T^2$ dependence of the resistivity of a metal may indicate non-Fermi-liquid behavior that could result from a phase transition to a symmetry-broken state.
However, in electronic condensed-matter systems, the theoretical calculation of these coefficients is often very difficult, due to many competing phenomena: electron-phonon coupling, presence of impurities, localization, and Coulomb interactions between the carriers. The prime advantage of a drag experiment is that it singles out the effect of Coulomb interactions on a transport coefficient: consider a system consisting of two layers, separated by a barrier so that tunneling between the layers is absent. In a drag experiment in this bilayer system, a current is driven through one of the layers, denoted as the drive layer. Due to momentum transfer, the carriers in the other (the passive) layer are dragged along and a voltage drop over the passive layer is observed. The drag resistance is defined as the ratio between the current in the active or drive layer and the voltage drop over the passive layer. Due to the spatial separation of the layers, the Coulomb interaction between carriers in both layers can be singled out as solely responsible for this effect, which for this reason is called “Coulomb drag". In cold two-component Fermi gases a similar phenomenon can be observed, where now the two (hyperfine) spin species play the role of the carriers in the two layers. When a cloud of atoms with one spin state moves relative to another, interactions lead to momentum transfer and the second spin species is also set into motion. This phenomenon is called spin drag.
Coulomb drag was first observed in 1990 [@gramilla] for electron-electron bilayers and later also for electron-hole bilayers, [@sivan] for a review see Ref. \[\]. In electron-hole bilayers the electrons in one layer and holes in the other can form excitons, which are expected to condense for low enough temperatures. The behavior of the drag resistivity in this condensed state was first studied in Ref. \[\], and its enhancement above the critical temperature was calculated in Ref. \[\]. An enhancement of the drag resistivity has been measured experimentally, [@ehexp] although exciton condensation has not been confirmed. Also for a topological insulator thin film, an enhancement of the drag resistivity upon approaching the critical temperature for (in this case topological) exciton condensation has recently been predicted. [@mink] Spin drag was first considered in semiconductors,[@scd_giovanni] where it gives rise to a temperature-dependent difference between spin and charge diffusion constants. This latter difference was indeed observed experimentally. [@weber_nature_2005] For ultracold fermions with repulsive interactions, the one-dimensional situation was discussed in Ref. \[\] and an enhancement of the spin-drag resistivity was predicted close to the ferromagnetic (Stoner) transition [@duinemag] and Bardeen-Cooper-Schrieffer (BCS) transition. [@mink2012] Enhancement of the collision rate was observed in the BCS regime. [@riedl] Experimentally, the spin-drag resistivity was measured in the strongly interacting (unitary) regime of fermionic cold atoms but an enhancement was not clearly observed.[@sommer] For ultracold bosons close to the Bose-Einstein condensation transition, Bose-enhanced scattering between atoms was predicted to lead to an enhanced spin-drag resistivity in three dimensions, [@hedwig] which was indeed observed experimentally. [@koller2012]
In this work we present a unified theory of drag phenomena near a second-order phase transition – a theory that encompasses the effects described in the previous paragraph and provides a framework for the study of similar effects yet to be discovered. This theory is based on quantum kinetic theory and Fermi’s golden rule for the scattering amplitudes. Within these approximations, the theory is valid in both two and three dimensions, for particle- and hole-like bands, and for arbitrary imbalances in density and mass. We show how to incorporate the effects of Gaussian critical fluctuations close to a phase transition due to an instability in a specific (Hartree, Fock, or Cooper) channel. More precisely, adopting the Gaussian model of critical fluctuations means that the dominant energy and wave vector dependence of the quasiparticle scattering amplitude near the phase transition is calculated in terms of the non-interacting, i.e., Gaussian, propagator of the fluctuations of the order parameter. The benefit of this approach is that we are therefore able to incorporate these fluctuations within the transparent Boltzmann formalism which allows a straightforward calculation of the transport coefficients. Truly critical fluctuations, resulting from interactions between order-parameter fluctuations, are not straightforwardly taken into account within this formalism as it relies on a quasi-particle description, which usually breaks down near the phase transition. When critical fluctuations beyond the Gaussian level are important the diagrammatic approach is a more natural starting point.[@rakpong] Such fluctuations are important only in the Ginzburg region,[@amitbook] where our Boltzmann approach ceases to be a good approximation.
The purpose of this article is twofold: on the one hand, we present a unified view of the special cases considered in our previous publications.[@duinemag; @mink; @mink2012] On the other hand, we present new details and improved results from our general formalism for three-dimensional Fermi gases and new results for electron-hole bilayers.
The remainder of the paper is organized as follows. In Sec. \[sec:ss\] we introduce our formalism in the simple case of the Boltzmann equation for a single species. In Sec. \[sec:BM\] we solve the coupled Boltzmann equations of the two species and find an expression for the drag resistivity in terms of the collision integral. We derive a general expression for this collision integral in Sec. \[sec:CI\]. Our results for quadratic-dispersion systems, e.g. electrons in semiconductors and cold atoms, are given in Sec. \[sec:quad\] and for linear-dispersion systems, e.g. massless Dirac fermions in graphene, in Sec. \[sec:lin\]. In Sec. \[sec:allresults\] we present analytical results for the behavior of the drag resistivity close to the critical temperature, as well as numerical results for specific systems. Our conclusions are in Sec. \[sec:con\]. An appendix is included that details some calculational steps that were skipped in the main text, but may yet benefit the reader who is interested in applying the theory to other systems.
Single species results {#sec:ss}
======================
We start by specifying the single-particle dispersion relations that we will use: $$\label{eq:dispS}
\xi({\bm k}) = s (\hbar^2 k^2/2 m - \mu) \quad \text{or} \quad
\xi({\bm k}) = s (\hbar v k - \mu)~,$$ for a quadratic and linear dispersion, respectively. Here, $\hbar k$ is the momentum of the carrier and $m$ their mass (in case of a parabolic band). Alternatively, $v$ is the carrier velocity for linear dispersion. When the band is “particle-like" $s=1$ and when the band is “hole-like" $s=-1$. Alternatively, we will say that for $s=1$ the band has a positive sign and for $s=-1$ a negative sign. Note that when $s = -1$ the $\mu$ we introduce is actually the negative of the chemical potential measured from the top of the hole band (see Fig. \[fig:dispersion\]).
Before writing two coupled Boltzmann equations to calculate the drag resistivity, it is instructive to consider the ordinary resistivity of a single species. Along the way we define some quantities we need later on. To determine this resistivity, we need to determine the non-equilibrium carrier distribution $f$ in the presence of a uniform force field ${\bm F}$. Note that, since we consider both mass transport (in the case of cold-atom systems) and charge transport (in the case of solid-state systems), we prefer to keep the discussion general and use force rather than electric field ${\bm E}$. In the case of electrons the force is of course equal to ${\bm F} = -|e| {\bm E}$, with $-|e|$ the charge of a single electron. The distribution function $f({\bm k}(t))$ is independent of position and, in the absence of relaxation, obeys the equation of motion f([k]{}(t)) = \_[k]{} f([k]{}) . In the relaxation-time approximation, one adds a phenomenological term which relaxes $f$ back to the equilibrium Fermi-Dirac distribution on a time scale $\tau$. As we are interested in drag effects due to interactions we take the simplest version of the relaxation-time approximation and ignore momentum dependence of the relaxation time. Under the influence of a force ${\bm F}$, we have $\hbar \dot{{\bm k}} = {\bm F}$. Then, for a steady-state solution $$\label{eq:slBM}
\frac{1}{\hbar} {\bm F} \cdot \partial_{\bm k} f({\bm k}) = -\frac{1}{\tau} [f({\bm k}) - n_{\rm F}(\xi({\bm k}))]~,$$ where $\xi({\bm k})$ is the bare dispersion introduced above and $n_{\rm F}(\epsilon) = 1/(1+\exp( \beta \epsilon))$ is the Fermi-Dirac distribution with $\beta = (k_{\rm B} T)^{-1}$ the inverse thermal energy.
We are interested in linear-response transport coefficients, so we take as an [*ansatz*]{} for the solution of Eq. (\[eq:slBM\]) the first-order expansion of a Fermi-Dirac distribution shifted by a drift momentum $\hbar {\bm k}^\text{drift}$: \[eq:f1\] f([k]{}) &= &n\_[F]{}\[([k]{})\] - s \[[k]{}\^ \_[k]{} ([k]{})\] n\_[F]{}’\[([k]{})\]\
& & n\_[F]{}(([k]{})) + f\^[(1)]{}([k]{}) , where $n_{\rm F}'(\ep) = \partial_\ep n_{\rm F}(\ep)$. The inclusion of the extra factor $s$ ensures that the average momentum of the ensemble of particles is proportional to the carrier density $n$, i.e., the number of electrons when $s=1$ and the number of holes when $s = -1$, so that \_[[k]{}]{} f\^[(1)]{}([k]{}) = \^ n .
We also need to evaluate the current density ${\bm j}$: = \_[[k]{}]{} \[\_[k]{} ([k]{})\] f\^[(1)]{}([k]{}) . Note that, for the same reasons as mentioned before we consider mass, rather than charge current, and omit a prefactor $-|e|$. As a consequence, for the case of electrons the resistivities found below should be multiplied with a factor $e^2$ to convert them to electrical resistivities. Introducing the current-to-momentum conversion factor $C$ C = - \_[[k]{}]{} \[\_[k]{} ([k]{})\]\^2 n\_[F]{}’(\_[s]{}([k]{})) , where $d$ is the dimensionality, we obtain ${\bm j} = s C {\bm k}^\text{drift}$. The conversion factor is easily determined to be C = C = , for a quadratic and linear dispersion, respectively.
Using these definitions, we solve the Boltzmann equation Eq. (\[eq:slBM\]) which, in the linear-response approximation, reads \[eq:slBMfo\] \_[k]{} n\_[F]{}(([k]{})) = - f\^[(1)]{}([k]{}) . Performing the differentiation with respect to ${\bm k}$ on the left-hand side of the above leads to ${\bm k}^\text{drift} =s (\tau/\hbar) {\bm F}$ and ${\bm F} = (\hbar/C \tau) {\bm j}$, from which we identify the resistivity ${\bm F} = \rho {\bm j}$ as = = , for a quadratic and linear dispersion, respectively. For the quadratic dispersion we recognize the familiar Drude result.
Drag resistivity and coupled Boltzmann equations {#sec:BM}
================================================
In this section we first introduce and then solve two coupled Boltzmann equations and determine the drag resistivity which is the focus of this article. We denote the two species by the pseudospin label $\sigma = \uparrow,\downarrow$ which can either be hyperfine spin for the case of cold Fermi gases or layer index in the case of double-layer systems. The dispersions in Eq. (\[eq:dispS\]) acquire the species label $\sigma$ and are $$\label{eq:disp}
\xi_\sigma ({\bm k}) = s_\sigma (\hbar^2 k^2/2 m_\sigma - \mu_\sigma) \quad \text{or} \quad
\xi_\sigma ({\bm k}) = s_\sigma (\hbar v k - \mu_\sigma)~,$$ for the quadratic and linear dispersion, respectively. Note that we allow for a mass “imbalance" (i.e. $m_\uparrow \neq m_\downarrow$) and population imbalance (i.e. $\mu_\uparrow \neq \mu_\downarrow$).
We will denote the number of degenerate fermion types in a species by $N_f$, which will always be equal for both species. For example, in an electron-hole double-layer, $N_f = 2$, because both layers have spin degeneracy. In double-layer graphene (two graphene sheets separated by a tunnel barrier), $N_f = 4$, due to the presence of spin degeneracy and two Dirac cones in each layer. The density $n$ is always the density of a single fermion type, the total density of species $\sigma$ is $N_f n_\sigma$ and the total carrier density in the system $N_f(n_\uparrow + n_\downarrow)$. We apply a species-dependent force ${\bm F}_\sigma$ so that the equivalent of the linearized Boltzmann equation in Eq. (\[eq:slBMfo\]) is the following system of two coupled equations: $$\begin{aligned}
\frac{1}{\hbar} {\bm F}_\uparrow \cdot \partial_{\bm k} n_{\rm F}(\xi_\uparrow({\bm k})) &=- \frac{1}{\tau_\uparrow} f^{(1)}_\uparrow({\bm k}) + \Gamma_\uparrow({\bm k})~;\\
\frac{1}{\hbar} {\bm F}_\downarrow \cdot \partial_{\bm k} n_{\rm F}(\xi_\downarrow({\bm k})) &=- \frac{1}{\tau_\downarrow} f^{(1)}_\downarrow({\bm k}) + \Gamma_\downarrow({\bm k})~,\end{aligned}$$ where we introduced species-dependent intra-species relaxation times $\tau_\sigma$. The $\Gamma_\sigma({\bm k})$ are the collision integrals which give the net flux of particles into the state ${\bm k}$ of species $\sigma$. We will specify them in the next section. We substitute the expressions for $f^{(1)}_\sigma$ to introduce the drift momenta ${\bm k}^\text{drift}_\sigma$ $$\begin{aligned}
\left[\frac{1}{\hbar} {\bm F}_\uparrow - \frac{s_\uparrow}{\tau_\uparrow} {\bm k}^\text{drift}_\uparrow \right]\cdot [\partial_{\bm k} \xi_\uparrow({\bm k})] n_{\rm F}'(\xi_\uparrow({\bm k})) &= \Gamma_\uparrow({\bm k})~;\\
\left[\frac{1}{\hbar} {\bm F}_\downarrow - \frac{s_\downarrow}{\tau_\downarrow} {\bm k}^\text{drift}_\downarrow \right]\cdot [\partial_{\bm k} \xi_\downarrow({\bm k})] n_{\rm F}'(\xi_\downarrow({\bm k})) &= \Gamma_\downarrow({\bm k})~.\end{aligned}$$ To make connection to the current density, we multiply with the group velocity and sum over ${\bm k}$ with the result $$\begin{aligned}
- C_{\uparrow} \left[\frac{1}{\hbar} {\bm F}_\uparrow - \frac{s_\uparrow}{\tau_\uparrow} {\bm k}^\text{drift}_\uparrow \right] &
= {\bm \Gamma}_\uparrow~;\\
- C_{\downarrow}\left[\frac{1}{\hbar} {\bm F}_\downarrow - \frac{s_\downarrow}{\tau_\downarrow} {\bm k}^\text{drift}_\downarrow \right] &
= {\bm \Gamma}_\downarrow~,\end{aligned}$$ where we defined ${\bm \Gamma}_\sigma = (N_f^2 /V\hbar)\sum_{\bm k} (\partial_{\bm k} \xi_\sigma({\bm k})) \Gamma_\sigma ({\bm k})$.
Below, we find that to first order in the drift momenta the above yields ${\bm \Gamma}_\uparrow = N_f^2(s_\uparrow \Gamma^{\rm S}_\uparrow {\bm k}^\text{drift}_\uparrow + s_\downarrow \Gamma^{\rm D} {\bm k}^{\rm drift}
_\downarrow)$ and ${\bm \Gamma}_\downarrow = N_f^2(s_\downarrow \Gamma^{\rm S}_\downarrow {\bm k}^{\rm drift}
_\downarrow + s_\uparrow \Gamma^{\rm D} {\bm k}^{\rm drift}_\uparrow)$, where “S" labels the contribution of collisions between particle of the same species, and “D" labels the contribution of collisions from particles of different species. Note that the coefficients for the cross dependence are equal in both relations (i.e., $\Gamma^{\rm D}$ does not depend on pseudospin). After substituting these expansions, we obtain the resistivity matrix relating ${\bm F}$ to ${\bm j}$
[F]{}\_\
[F]{}\_
=
\_ & \_\
\_ & \_
[j]{}\_\
[j]{}\_
, with the drag resistivity $\rho_{\rm D} \equiv \rho_{\downarrow\uparrow} = \rho_{\uparrow\downarrow} = - N_f \hbar \Gamma^{\rm D}/ C_{\downarrow} C_{\uparrow}$. The intra-species resistivities are $\rho_{\sigma \sigma} = \hbar/N_f C_\sigma \tau_\sigma - N_f \hbar \Gamma^{\rm S}_\sigma/C^2_\sigma$. When the interspecies collision integrals are zero, $\rho_{\sigma \sigma}$ reduces to the result obtained in the previous section for the single-species problem and $N_f=1$.
Collision Integral {#sec:CI}
==================
In this section we determine the collision integral ${ \bm \Gamma}_\sigma$. We start with the expression for the scattering rate from Fermi’s golden rule and expand it to first order in the drift momenta. Let $\uparrow,{\bm k}_1$ and $\downarrow,{\bm k}_2$ be the incoming states, which are scattered onto the final states $\downarrow,{\bm k}_3$ and $\uparrow,{\bm k}_4$. The rate for this process can be calculated from Fermi’s golden rule: \[eq:rate\] R &=& |W([k]{}\_1,[k]{}\_2,[k]{}\_3,[k]{}\_4)|\^2 \_[[k]{}\_1 + [k]{}\_2,[k]{}\_3 + [k]{}\_4]{}\
&&(\_([k]{}\_1) + \_([k]{}\_2) - \_([k]{}\_3) - \_([k]{}\_4))\
&& f\_([k]{}\_1) f\_([k]{}\_2) (1-f\_([k]{}\_3 )) (1- f\_([k]{}\_4)) . We note the presence of the standard Fermi’s golden rule factors in Eq. (\[eq:rate\]): the delta functions ensuring energy and momentum conservation and the interaction matrix element $W$ squared. Additionally, we see that each contribution is weighted with the distribution functions of the two incoming particles, and with one minus the distribution functions for the two outgoing particles. For two-dimensional chiral electron systems described by a massless Dirac equation, such as electrons in double-layer graphene or topological insulator thin films, we should multiply the right-hand side of the last equation with the following form factors \[eq:ff\] , with $\phi_i$ the angle between ${\bm k}_i$ and the $x$-axis. These form factors encode suppressed backscattering in chiral electron systems. For simplicity, we will drop these factors in the manipulations carried out in this section, and reinstate them when we consider systems with linear dispersions in section \[sec:lin\].
To make our manipulations more tractable, we introduce some shorthand notations: for $i=1,4$ we have $\xi_i = \xi_\uparrow({\bm k}_i)$ and $f_i = f_\uparrow({\bm k}_i)$, and for $i=2,3$ we have $\xi_i = \xi_\downarrow({\bm k}_i)$ and $f_i = f_\downarrow({\bm k}_i)$. Furthermore, the delta-function expressing the conservation of momentum ($\delta_{{\bm k}_1 + {\bm k}_2,{\bm k}_3 + {\bm k}_4}$) will be indicated by $\delta_{{\bm k}_i}$, while the one expressing the conservation of energy \[$\delta(\xi_\uparrow({\bm k}_1) + \xi_\downarrow({\bm k}_2) - \xi_\downarrow({\bm k}_3) - \xi_\downarrow({\bm k}_4))$\] by $\delta(\xi_i)$. Using this new notation and dropping the arguments of the many-particle scattering amplitude $W$, the rate $R$ becomes R= \_[[k]{}\_i]{}(\_i) |W|\^2 f\_1 f\_2 (1-f\_3) (1- f\_4). For concreteness, we will now determine the collision integral for the $\uparrow$-species. The expression for the $\downarrow$-species can be easily obtained by appropriately changing the labels. The net flux of particle into the state $\uparrow,{\bm k}$ which is $\Gamma_\uparrow({\bm k})$ is then \_([k]{}) = \_[[k]{}\_1,[k]{}\_2,[k]{}\_3,[k]{}\_4]{} R (\_[[k]{}\_4,[k]{}]{} - \_[[k]{}\_1,[k]{}]{}). The expression for ${\bm \Gamma}_\uparrow$ is then \[eq:gav\] \_&=& \_[k]{} (\_[k]{} \_([k]{})) \_([k]{}) = \_[[k]{}\_i]{} \_[[k]{}\_i]{} (\_i)\
&& |W|\^2 f\_1 f\_2 (1-f\_3) (1- f\_4) (\_[k]{} \_[4]{} -\_[k]{} \_[1]{}), where $\sum_{{\bm k}_i}$ is a shorthand notation for $\sum_{{\bm k}_1,{\bm k}_2,{\bm k}_3,{\bm k}_4} $ and where $\partial_{\bm k} \xi_{i} \equiv [\partial_{\bm k} \xi_\uparrow({\bm k})]_{{\bm k}={\bm k}_i}$ and $\partial_{\bm k} \xi_{i} \equiv [\partial_{\bm k} \xi_\downarrow({\bm k})]_{{\bm k}={\bm k}_i}$ for $i=1,4$ and $i=2,3$, respectively.
We perform the first-order expansion in the drift momenta ${\bm k}^\text{drift}_\sigma$ of the distribution functions $f_\sigma$ and obtain to first order in the $f^{(1)}_\sigma$’s \[eq:linearization\] && f\_1 f\_2 (1-f\_3) (1- f\_4)\
&& n\_1 n\_2 (1-n\_3 )(1- n\_4)\
&&( + - - ), where we dropped the zeroth-order term since it evaluates to zero in the momentum summation. In the previous equation we introduced the shorthand notation $n_i = n_{\rm F}(\xi_i)$. Substituting Eq. (\[eq:linearization\]) and the expression for $f^{(1)}_\sigma$ from Eq. (\[eq:f1\]) in Eq. (\[eq:gav\]) and using that $n_{\rm F}'(\epsilon) = - \beta n_{\rm F}(\epsilon)[1-n_{\rm F}(\epsilon)]$ yields $$\begin{gathered}
{\bm \Gamma}_\uparrow = \frac{2 \pi \beta N_f^2}{\hbar^2 V^3} \sum_{{\bm k}_i} \delta_{{\bm k}_i} \delta(\xi_i)
|W|^2 n_1 n_2 (1-n_3) (1- n_4) \\ (\partial_{\bm k} \xi_{4} -\partial_{\bm k} \xi_{1})
\left\{
s_\uparrow {\bm k}^\text{drift}_\uparrow \cdot \left[(1-n_1)\partial_{\bm k} \xi_{1} - n_4\partial_{\bm k} \xi_{4}\right] +\right.\\
\left. s_\downarrow {\bm k}^\text{drift}_\downarrow \cdot \left[(1-n_2) \partial_{\bm k} \xi_{2} - n_3\partial_{\bm k} \xi_{3}\right]
\right\}~.\end{gathered}$$ When we interchange the labels of the incoming and outgoing states (specifically, when we transform ${\bm k}_1 \leftrightarrow {\bm k}_4$ and ${\bm k}_2 \leftrightarrow {\bm k}_3$), ${\bm \Gamma}_\uparrow$ remains invariant. Taking the average of the original expression for ${\bm \Gamma}_\uparrow $ and the one transformed in this way, and using that due to energy conservation $n_1 n_2 (1-n_3) (1- n_4) = n_4 n_3 (1-n_2) (1- n_1)$, and also that the interaction $W$ remains invariant under this transformation, we find that \[eq:gav2\] \_&=& - \_[[k]{}\_i]{} \_[[k]{}\_i]{} (\_i) |W|\^2 n\_1 n\_2 (1-n\_3) (1- n\_4)\
&& (\_[k]{} \_[4]{} -\_[k]{} \_[1]{}) . In the Appendix we show that we may take the drift momenta out of the summation provided that we introduce a factor $1/d$, so that we obtain $$\begin{aligned}
{\bm \Gamma}_\uparrow &= N_f^2 (s_\uparrow \Gamma^{\rm S}_\uparrow {\bm k}^{\rm drift}_\uparrow
+ s_\downarrow \Gamma^{\rm D} {\bm k}^\text{drift}_\downarrow)~; \\
{\bm \Gamma}_\downarrow &= N_f^2 (s_\downarrow \Gamma^{\rm S}_\downarrow {\bm k}^{\rm drift}_\downarrow + s_\uparrow \Gamma^{\rm D} {\bm k}^\text{drift}_\uparrow)~,\end{aligned}$$ where we again note that the cross contributions have the same coefficient $\Gamma^{\rm D}$. The explicit expressions for these coefficients are $$\begin{gathered}
\label{eq:gasu}
\Gamma^{\rm S}_\uparrow = -\frac{ \pi \beta}{ d \hbar^2 V^3} \sum_{{\bm k}_i} \delta_{{\bm k}_i} \delta(\xi_i)
|W|^2 n_1 n_2 (1-n_3) (1- n_4) \\
(\partial_{\bm k} \xi_{4} -\partial_{\bm k} \xi_{1})^2~,\end{gathered}$$ $$\begin{gathered}
\label{eq:gasd}
\Gamma^{\rm S}_\downarrow = - \frac{ \pi \beta}{ d \hbar^2 V^3} \sum_{{\bm k}_i} \delta_{{\bm k}_i} \delta(\xi_i)
|W|^2 n_1 n_2 (1-n_3) (1- n_4) \\(\partial_{\bm k} \xi_{3} -\partial_{\bm k} \xi_{2})^2~,\end{gathered}$$ and $$\begin{gathered}
\label{eq:gad}
\Gamma^{\rm D} = - \frac{ \pi \beta}{ d \hbar^2 V^3} \sum_{{\bm k}_i} \delta_{{\bm k}_i} \delta(\xi_i)
|W|^2 n_1 n_2 (1-n_3) (1- n_4) \\ (\partial_{\bm k} \xi_{4} -\partial_{\bm k} \xi_{1}) \cdot (\partial_{\bm k} \xi_{3} - \partial_{\bm k} \xi_{2})~.\end{gathered}$$ We note that these results and the expression for the drag resistivity $\rho_{\rm D} = - \hbar \Gamma^{\rm D}/ C_{\downarrow} C_{\uparrow}$ are valid for two and three dimensions, for a linear (after reinstating the form factors Eq. (\[eq:ff\]) in needed) and quadratic dispersion, for particle-like and hole-like bands, and arbitrary density and mass imbalance.
We now comment on the signs of $\Gamma^{\rm D}$ and $\rho_{\rm D}$. We stress that $\rho_{\rm D}$ is the transport coefficient relating the mass (and not the charge) current to the force. The sign of $\Gamma^{\rm D}$ is $s_\uparrow s_\downarrow$, as can be seen by considering a quadratic dispersion, so that the sign of $\rho_{\rm D}$ is $- s_\uparrow s_\downarrow$. Consider the situation in which there is a non-zero current in the “active" $\uparrow$ species ${\bm j}_\uparrow$, and that the current in the passive $\downarrow$ species is held to zero by the force ${\bm F}_\downarrow$, so that ${\bm F}_\downarrow = \rho_{\rm D} {\bm j}_\uparrow$. Recall that the intra-species resistivities are positive for both particle-like and hole-like bands. For bands of equal character, interaction tend to equalize the currents in both layers (or spin projections). Thus, for bands of equal character, $\rho_{\rm D}$ is negative. When one of the bands is particle-like and the other hole-like, $\rho_{\rm D}$ is positive.
It is interesting to consider the relations between $\Gamma^{\rm S}_{\uparrow,\downarrow}$ and $\Gamma^{\rm D}$. For a quadratic dispersion we substitute Eq. (\[eq:disp\]) into Eqs. (\[eq:gasu\],\[eq:gasd\],\[eq:gad\]) and obtain $$\begin{aligned}
{\bm \Gamma}_\uparrow &= -s_\downarrow m_\downarrow N_f^2|\Gamma^{\rm D}| \left( \frac{{\bm k}^\text{drift}_\uparrow}{m_\uparrow} - \frac{{\bm k}^\text{drift}_\downarrow}{m_\downarrow} \right) \\
{\bm \Gamma}_\downarrow &= -s_\uparrow m_\uparrow N_f^2|\Gamma^{\rm D}| \left(\frac{{\bm k}^\text{drift}_\downarrow}{m_\downarrow} - \frac{{\bm k}^\text{drift}_\uparrow}{m_\uparrow}\right)~.\end{aligned}$$ When, ${\bm k}^\text{drift}_\uparrow/m_\uparrow = {\bm k}^\text{drift}_\downarrow/m_\downarrow$ the integrated collision integral vanishes and the resistivities are just given by the decoupled single-species result. This condition is satisfied when the drift velocities have the same magnitude, and the drift momenta have the same direction. This is what one expects for a Galilean invariant system. For the case of a linear dispersion the Hamiltonian is not Galilean invariant and no simple relations between the $\Gamma^{\rm S}_\sigma$ and $\Gamma^{\rm D}$ has been obtained.
Instability channels
--------------------
At the methodological level, the main purpose of this article is to show how to determine the collision integral $\Gamma^{\rm D}$ in Eq. (\[eq:gad\]) incorporating the effect of Gaussian fluctuations close to a phase transition. In the case of a ferromagnetic transition these are magnetic fluctuations, while in the case of superconductivity or exciton condensation, these are pairing fluctuations. These fluctuations increase in strength when the system approaches the transition and ultimately lead to a divergence of the scattering amplitude at the critical temperature $T_{\rm c}$. This divergence occurs at energy $\hbar \omega=0$ and momentum ${\bm k}_{\rm p}=0$, that are related to the energies and momenta of the incoming particles in a way that depends on the channel in which the instability occurs. To take into account these fluctuations and account for the dependence of the scattering amplitude on $\hbar \omega$ and ${\bm k}_{\rm p}$ that is dominant close to the phase transition, we should “decouple" Eq. (\[eq:gad\]) in the correct channel. This decoupling is carried out by introducing the appropriate auxiliary energy variable $\hbar\omega$ through an integral over it. For example, in the case of superconductivity between two bands with the same character, the pole in the scattering amplitude (in this case it is usually called the “many-body $T$ matrix") lies at zero energy and zero center-of-mass momentum of the two incoming particles $\hbar\omega = \xi_1 + \xi_2 = 0$ and ${\bm k}_{\rm p} ={\bm k}_1 + {\bm k}_2= {\bm 0}$. Thus, in this specific case, we would introduce $\omega$ as follows (\_1 + \_2 - \_3 - \_4) &=& d() (\_1 + \_2 - )\
&&(\_3 + \_4 - ) . The three possibilities of combining the incoming $\uparrow$ particle with energy $\xi_1$ with i) the incoming $\downarrow$ particle with energy $\xi_2$, ii) the outgoing $\downarrow$ particle with energy $\xi_3$, and iii) the outgoing $\uparrow$ particle with energy $\xi_4$, are denoted as decoupling in the Cooper, Fock, and Hartree channels, respectively. We summarize the properties of these instability channels in Table \[tab:1\] and give their Feynman diagrams in Fig. \[fig:Channels\].
Instability channel Combination $\hbar \omega$ ${\bm k}_p$
--------------------- ------------------------------------- ----------------- ----------------------------------------
Hartree In $\uparrow$ with Out $\uparrow$ $\xi_1 - \xi_4$ ${\bm k}_1 - {\bm k}_4 \equiv {\bm q}$
Fock In $\uparrow$ with Out $\downarrow$ $\xi_1 - \xi_3$ ${\bm k}_1 - {\bm k}_3 \equiv {\bm Q}$
Cooper In $\uparrow$ with In $\downarrow$ $\xi_1 + \xi_2$ ${\bm k}_1 + {\bm k}_2 \equiv {\bm K}$
: \[tab:1\] Overview of the three instability channels. In the third and fourth columns we specify the energy and momentum variables appropriate for this instability channel.
In the next section, we discuss how to perform the decoupling of Eq. (\[eq:gad\]) in the three channels in full detail for a quadratic dispersion. These results are valid for arbitrary spatial dimensionality, band character, mass, and density imbalance. In section \[sec:lin\] we consider a linear dispersion and restrict ourselves to the evaluation of Eq. (\[eq:gad\]) in a system with exciton-condensation, i.e., the decoupling of Eq. (\[eq:gad\]) in the Fock channel for bands of opposite character. Then, in section \[sec:res\] we give some qualitative results and present quantitative results for the resistivity of three-dimensional cold Fermi gases close to a Stoner transition and two-dimensional spatially-separated electron and hole systems in semiconductor double quantum wells. A list of phase transitions in various systems and their instability channels can be found in Table \[tab:2\].
System Transition $s_\uparrow,s_\downarrow$ Channel
------------------------ ------------------------ --------------------------- ---------
Cold gases Magnetism longitudinal 1,1 Hartree
Cold gases Magnetism transverse 1,1 Fock
Cold gases BCS 1,1 Cooper
Electron-hole bilayers Exciton condensation 1,-1 Fock
TI thin films Exciton condensation 1,-1 Fock
: \[tab:2\] Overview of phase transitions in various systems and their instability channels.
Quadratic Dispersion {#sec:quad}
====================
For a quadratic dispersion, Eq. (\[eq:gad\]) simplifies to $$\begin{gathered}
\label{eq:gadq}
\Gamma^{\rm D} = s_\uparrow s_\downarrow \frac{\pi \beta}{d V^3} \frac{\hbar^2}{m_\uparrow m_\downarrow} \sum_{{\bm k}_i} \delta_{{\bm k}_i} \delta(\xi_i)\\
|W|^2 n_1 n_2 (1-n_3) (1- n_4) ({\bm k}_1 - {\bm k}_4)^2~,\end{gathered}$$ where we note that ${\bm k}_1 - {\bm k}_4$ is just the momentum transferred in the scattering event. Before we discuss the various instabilities in different channels we note that when we replace the scattering amplitude $W$ by a constant, all three instability channels should give the same result for $\Gamma^{\rm D}$. This serves as an important check for the, sometimes lengthy, analytical calculations presented below.
Hartree instability {#sec:qh}
-------------------
When the dominant energy and momentum dependence of the effective interaction $W$ is on the transferred energy $\hbar \omega = \xi_1 - \xi_4$ and momentum ${\bm k}_{\rm p} = {\bm k}_1 - {\bm k}_4 \equiv {\bm q}$ in the scattering event, the Hartree channel is the appropriate instability channel. We introduce an additional integral over $\hbar \omega$ using the identity (\_1 + \_2 - \_3 - \_4) &=& d() (\_1 - \_4 - )\
&&(\_2 - \_3 + ) , and solve the delta function for momentum conservation by writing ${\bm k}_1 = {\bm k}$, ${\bm k}_2 = {\bm k}'$, ${\bm k}_3 = {\bm k}'+{\bm q}$, and ${\bm k}_4={\bm k}-{\bm q}$, so that the ${\bm k}$ and ${\bm k}'$ summations in Eq. (\[eq:gadq\]) decouple: \^[D]{} &=& s\_s\_ \_[[q]{}]{} d() q\^2 |W([q]{},)|\^2\
&& \_[[k]{}]{} (\_([k]{}) - \_([k]{}-[q]{}) - ) n\_[F]{}(\_([k]{}))\
&& \[1- n\_[F]{}(\_([k]{}-[q]{}))\]\
&& \_[[k]{}’]{} (\_([k]{}’) - \_([k]{}’+[q]{}) + ) n\_[F]{}(\_([k]{}’))\
&&\[1-n\_[F]{}(\_([k]{}’+[q]{}))\] . Note that in the above result we have now explicitly indicated the dominant dependence of the scattering amplitude on ${\bm q}$ and $\hbar \omega$ and ignored any other dependence. We will do this also for the other channels discussed below. Making use of the identities $$n_{\rm F} (x)[1-n_{\rm F} (y)]=[n_{\rm F} (y)-n_{\rm F}(x)]n_{\rm B}(x-y)~,$$ $$n_{\rm B}(x)n_{\rm B} (-x)=\frac{1}{4\sinh^2(\beta x/2)}~,$$ where $n_{\rm B} (x)=[e^{\beta x}-1]^{-1}$ is the Bose distribution, we rewrite this expression as $$\begin{gathered}
\label{eq:gadh}
\Gamma^{\rm D} = s_\uparrow s_\downarrow \frac{\beta \hbar^2}{4 d \pi V m_\uparrow m_\downarrow} \sum_{{\bm q}} \int d \hbar \omega \frac{q^2 |W({\bm q},\hbar \omega)|^2}{\sinh^2(\beta \hbar \omega/2)} \\ \Im m\Pi_\uparrow({\bm q},\hbar \omega) \Im m\Pi_\downarrow({\bm q},\hbar \omega)~,\end{gathered}$$ where we have introduced the well-known polarizability (Lindhard function) \[eq:PI\] \_([q]{},) = \_[[k]{}]{} , and we used that $\Pi_\sigma({\bm q},\hbar \omega) = \Pi_\sigma( - {\bm q},\hbar \omega)$ and $\Pi_\sigma({\bm q},-\hbar \omega) = \Pi^*_\sigma({\bm q},\hbar \omega)$.
In the three-dimensional case we have [@giovannibook] $$\begin{gathered}
\Im m \Pi_\sigma({\bm q},\hbar \omega) = -\frac{m_\sigma^2}{4 \pi \hbar^4 \beta q}
\left\{ \beta \hbar \omega \right. + \\
\log\left[1 + \exp\left(\beta(\epsilon_\sigma(q/2) + m_\sigma \omega^2/2 q^2- \hbar \omega /2- \mu_\sigma)\right) \right] \\-
\left.
\log\left[1 + \exp\left(\beta(\epsilon_\sigma(q/2) + m_\sigma \omega^2/2 q^2 + \hbar \omega /2- \mu_\sigma)\right) \right]
\right\}.\end{gathered}$$ In the two-dimensional case, we must resort to numerical methods to determine $\Im m \Pi_\sigma$ at arbitrary temperatures. Substitution of the result for $\Im m \Pi_\sigma$ into Eq. (\[eq:gadh\]) leads to our final expression for $\Gamma^{\rm D}$ after decoupling in the Hartree channel. The real part of $\Pi_\sigma$, which is typically present in the interaction $W$, can be obtained by a Kramers-Kronig transform of $\Im m \Pi_\sigma$. We note that since $\Pi$ is an intra-species quantity, the presence of imbalance in the chemical potential or mass does not make the determination of $\Pi_\sigma$ more difficult as compared to the balanced case.
Cooper instability
------------------
When the effective interaction $W$ depends strongly on the center-of-mass momentum ${\bm K} = {\bm k}_1 + {\bm k}_2$ and total energy $\hbar \omega = \xi_1 + \xi_2$ of the incoming particles, the Cooper channel is the most appropriate channel to decouple the collision integral in Eq. (\[eq:gadq\]). In this case we thus introduce the energy variable $\hbar \omega$ using the identity (\_1 + \_2 - \_3 - \_4) &=& d() (\_1 + \_2 - )\
&&(\_3 + \_4 - ) , and solve the delta function of momentum conservation by setting ${\bm k}_1 = {\bm K}/2 + {\bm k}$, ${\bm k}_2 = {\bm K}/2 - {\bm k}$, ${\bm k}_3 = {\bm K}/2 - {\bm k}'$, and ${\bm k}_4 = {\bm K}/2 + {\bm k}'$. We find: $$\begin{gathered}
\label{eq:gadc}
\Gamma^{\rm D} = s_\uparrow s_\downarrow
\frac{\pi \beta \hbar^2}{d V^3 m_\uparrow m_\downarrow} \sum_{{\bm k}, {\bm k}', {\bm K}} |W({\bm K},\hbar \omega)|^2
({\bm k}'-{\bm k})^2 \\
[\delta(\xi_\uparrow({\bm K}/2 + {\bm k})+\xi_\downarrow({\bm K}/2 - {\bm k}) - \hbar \omega) \\ n_{\rm F}(\xi_\uparrow({\bm K}/2 + {\bm k})) n_{\rm F}(\xi_\downarrow({\bm K}/2 - {\bm k})) ] \\
\{\delta(\xi_\uparrow({\bm K}/2 + {\bm k}')+\xi_\downarrow({\bm K}/2 - {\bm k}') - \hbar \omega) \\
[1-n_{\rm F}(\xi_\uparrow({\bm K}/2 + {\bm k}'))]
[1- n_{\rm F}(\xi_\downarrow({\bm K}/2 - {\bm k}'))]\}~.\end{gathered}$$ Expanding the factor $({\bm k}'-{\bm k})^2$ we can decouple the ${\bm k}$ and ${\bm k}'$ summations and obtain $$\begin{gathered}
\label{eq:gadc2}
\Gamma^{\rm D} = s_\uparrow s_\downarrow
\frac{\beta \hbar^2}{2 \pi d V m_\uparrow m_\downarrow} \int d(\hbar \omega) \sum_{{\bm K}} \frac{|W({\bm K},\hbar \omega)|^2}{\sinh^2(\beta \hbar \omega/2)}\\ [\Im m~\Xi_0({\bm K}, \hbar \omega) \Im m~\Xi_2({\bm K},\hbar \omega) - \Im m~\Xi_1^2({\bm K},\hbar \omega)],\end{gathered}$$ where we have introduced the following three “generalized pairing susceptibilities" ($n=0,1$, and $2$) $$\begin{gathered}
\label{eq:imxn}
\Im m~\Xi_n({\bm K},\hbar\omega) = \frac{1}{V} \Im m \sum_{{\bm k}} C_n({\bm k}) \\ \frac{1-n_{\rm F}(\xi_\uparrow({\bm K}/2+{\bm k}))- n_{\rm F}(\xi_\downarrow({\bm K}/2-{\bm k}))}{\hbar \omega + i 0^+ - (\xi_\uparrow({\bm K}/2+{\bm k}) + \xi_\downarrow({\bm K}/2-{\bm k}))}~.\end{gathered}$$ Here $C_0({\bm k}) = 1$, $C_1({\bm k})= {\bm k}$, and $C_2({\bm k}) = |{\bm k}|^2$. Note that we introduced a minor abuse of notation for the sake of accessibility of our formalism: although $\Im m\Xi_1$ and $C_1$ are vector valued functions, they are not denoted by a bold-faced symbol. In determining the susceptibility $\Xi$, we restrict ourselves to the case of bands of equal character $s_\uparrow = s_\downarrow = 1$. It will turn out that the case $s_\uparrow = - s_\downarrow$ can be obtained by considering the susceptibility $\Delta$ defined in Eq. (\[eq:de\]), obtained by performing the Fock decoupling in the case that the bands have equal character $s_\uparrow = s_\downarrow$. We will show this point explicitly in the next section in which we perform the Fock decoupling.
To determine the generalized susceptibility it is convenient to make a shift of the summation variable ${\bm k}$ such that the denominator in Eq. (\[eq:imxn\]) becomes independent of the angle $\theta$ between ${\bm k}$ and ${\bm K}$.[^1] We do not need to shift the ${\bm k}$ as argument of the $C_n$’s, as can be seen by making the shift for both ${\bm k}$ and ${\bm k}'$ in Eq. (\[eq:gadc\]) before expanding the factor $({\bm k}'-{\bm k})^2$. The result is $$\begin{gathered}
\Im m\Xi_n({\bm K},\hbar\omega) = \frac{1}{V} \Im m \sum_{{\bm k}} C_n({\bm k}) \\ \frac{1-n_{\rm F}(\xi_\uparrow(R_\uparrow {\bm K}+{\bm k}))- n_{\rm F}(\xi_\downarrow(R_\downarrow {\bm K}-{\bm k}))}{\hbar \omega + i 0^+ - (\xi_\uparrow(R_\uparrow {\bm K}+{\bm k}) + \xi_\downarrow(R_\downarrow {\bm K}-{\bm k}))},\end{gathered}$$ where $R_\sigma = m_\sigma / (m_\uparrow + m_\downarrow)$ and $R_\uparrow + R_\downarrow = 1$. We convert the summation to an integration and introduce the delta function $$\begin{gathered}
\Im m\Xi_n({\bm K},\hbar\omega) = - \frac{\pi}{(2 \pi)^d} \Im m \int d {\bm k} C_n({\bm k})
\\ \left[1-n_{\rm F}(\xi_\uparrow(R_\uparrow {\bm K}+{\bm k}))- n_{\rm F}(\xi_\downarrow(R_\downarrow {\bm K}-{\bm k}))\right]\\
\delta(\mu_\uparrow + \mu_\downarrow + \hbar \omega - (\hbar^2 k^2/2 m_\uparrow + \hbar^2 k^2/2 m_\downarrow + \hbar^2 K^2/2 (m_\uparrow + m_\downarrow))),\end{gathered}$$ where we choose ${\bm K}$ along the $x$-axis in the two-dimensional case and ${\bm K}$ along the $z$-axis in the three-dimensional case, and where we noted that $\Im m\Xi_1({\bm K},\hbar\omega)$ can only have a component in the direction of ${\bm K}$. The solution of the delta function is k\_0 = . This solution only exists when K\^2 K\^2\_, which gives an upper bound for the length of ${\bm K}$ in Eq. (\[eq:gadc2\]). For $K> K_{\rm max}$ the imaginary part of $\Xi$ is zero. We solve the delta function and take into account the Jacobian and the transformation rule for the delta function, which leads for the three-dimensional case to
m\_n([K]{},) = \_0\^d C\_n([k]{}) \[1 - n\_[F]{}(E\_()) - n\_[F]{}(E\_())\]. The expression for the two-dimensional case differs in the Jacobian and the upper-integration boundary for $\theta$: m\_n([K]{},) = \_0\^[2]{} d C\_n([k]{}) \[1 - n\_[F]{}(E\_()) - n\_[F]{}(E\_())\]. In the previous equation, we defined the energies E\_() = \^2 k\_0\^2/2 m\_+ m\_\^2 K\^2/2 (m\_+ m\_)\^2 + () \^2 k\_0 K /2 (m\_+ m\_) - \_and E\_() = \^2 k\_0\^2/2 m\_+ m\_\^2 K\^2/2 (m\_+ m\_)\^2 - () \^2 k\_0 K /2 (m\_+ m\_) - \_.
Using the integrals \_0\^d = 2 + , and $$\begin{gathered}
\label{eq:polylocation}
\int_0^\pi d \theta \frac{\cos(\theta) \sin(\theta)}{1 + \exp(a + b\cos(\theta))} \\=
- \frac{1}{b} \log\left[(1+\exp(a-b))(1+\exp(a+b))\right]+ \\
\frac{1}{b^2}\left(\text{Li}_2(-\exp(a-b)) -\text{Li}_2(-\exp(a+b))\right),\end{gathered}$$ where $\text{Li}_s(z) = \sum_{k=1}^\infty z^k/k^s$ is the polylogarithm, we can determine the susceptibility $\Im m\Xi_n({\bm K},\hbar\omega)$ for the three-dimensional case in closed form. However, the resulting expressions are not very enlightening and will not be given explicitly. Substitution of the results for $\Im m \Xi_n$ into Eq. (\[eq:gadc2\]) leads to our final expression for $\Gamma^{\rm D}$ after decoupling in the Cooper channel.
Fock instability {#sec:qf}
----------------
When the interaction depends strongly on the energy and momentum difference of the incoming $\uparrow$ particle and outgoing $\downarrow$ particle, the Fock channel is the appropriate channel to decouple the collision integral Eq. (\[eq:gadq\]). We introduce a new energy variable $\hbar \omega = \xi_1 - \xi_3$ using the identity (\_1 + \_2 - \_3 - \_4) = d (\_1 - \_3 - ) (\_2 - \_4 + ), and the “conjugate" momentum variable ${\bm Q}= {\bm k}_1 - {\bm k}_3$ by solving the momentum-conserving delta function by ${\bm k}_1 = {\bm k}+{\bm Q}$, ${\bm k}_2 = {\bm k}'$, ${\bm k}_3 = {\bm k}$, and ${\bm k}_4= {\bm k}'+{\bm Q}$. Then, we have for $\Gamma^{\rm D}$ $$\begin{gathered}
\Gamma^{\rm D} = s_\uparrow s_\downarrow \frac{\pi \beta \hbar^2}{d V^3 m_\uparrow m_\downarrow} \sum_{{\bm k},{\bm k}',{\bm Q}} \int d \hbar \omega |W({\bm Q},\hbar \omega)|^2 ({\bm k}' -{\bm k})^2 \\
\delta(\xi_\uparrow({\bm k}+{\bm Q}) - \xi_\downarrow({\bm k}) - \hbar \omega) n_{\rm F}(\xi_\uparrow({\bm k}+{\bm Q})) (1-n_{\rm F}(\xi_\downarrow({\bm k}))) \\
\delta(\xi_\downarrow({\bm k}') - \xi_\uparrow({\bm k}'+{\bm Q}) + \hbar \omega) n_{\rm F}(\xi_\downarrow({\bm k}')) (1-n_{\rm F}(\xi_\uparrow({\bm k}'+{\bm Q}))).\end{gathered}$$ We again expand the factor $({\bm k}' -{\bm k})^2$ and obtain $$\begin{gathered}
\label{eq:gadfoq}
\Gamma^{\rm D} = s_\uparrow s_\downarrow \frac{\beta \hbar^2}{2 \pi d V m_\uparrow m_\downarrow} \sum_{{\bm Q}} \int d \hbar \omega
\frac{|W({\bm Q},\hbar \omega)|^2 }{\sinh^2(\beta \hbar \omega/2)} \\ [ \Im m\Delta_0({\bm Q},\hbar \omega) \Im m\Delta_2({\bm Q},\hbar \omega) - \Im m\Delta^2_1({\bm Q},\hbar \omega)],\end{gathered}$$ where we defined: $$\begin{gathered}
\label{eq:de}
\Im m\Delta_n({\bm Q},\hbar\omega) = \frac{1}{V} \Im m \sum_{{\bm k}} F_n({\bm k}) \\ \frac{n_{\rm F}(\xi_\uparrow({\bm k}+{\bm Q}))- n_{\rm F}(\xi_\downarrow({\bm k}))}{\xi_\uparrow({\bm k}+{\bm Q}) - \xi_\downarrow({\bm k})- \hbar \omega - i 0^+}.\end{gathered}$$ Here $F_0({\bm k}) = 1$, $F_1({\bm k})= {\bm k}$, and $F_2({\bm k}) = |{\bm k}|^2$. Now, we compare the susceptibilities for the Cooper and the Fock channel, $\Xi$ and $\Delta$, respectively. Defining $\tilde{\xi}_\downarrow(k) = -\xi_\downarrow(k)$ we rewrite the fraction in Eq. (\[eq:de\]) as , which is exactly the expression for $\Xi_n({\bm Q},\hbar\omega)$ in Eq. (\[eq:imxn\]) with a switched character for the $\downarrow$-dispersion. Thus, when determining $\Xi_n$ and $\Delta_n$, we can restrict ourselves to the case $s_\uparrow = s_\downarrow =1$. When we need $\Xi_n$ for dispersions of opposite character ($s_\uparrow=-s_\downarrow$), this is just obtained by considering $\Delta_n$ for bands of the same character ($s_\uparrow = s_\downarrow =1$).
Following up on the remarks made above, we now determine $\Delta_n ({\bf Q}, \hbar \omega)$ for bands of the same character $s_\uparrow = s_\downarrow =1$. The way the analysis proceeds depends on whether or not there is mass imbalance. The reason is that when $m_\uparrow = m_\downarrow$ the terms with $k^2$ cancel from the denominator of the integrand of Eq. (\[eq:de\]), whereas they do not cancel in the mass-imbalanced case. For sake of simplicity, we only consider the fully balanced case with $m_\uparrow = m_\downarrow \equiv m$ and $\mu_\uparrow = \mu_\downarrow \equiv \mu$. Shifting ${\bm k}$ with ${\bm Q}/2$, we arrive at $$\begin{gathered}
\Im m\Delta_n({\bm Q},\hbar\omega) = \frac{\pi}{(2 \pi)^d} \Im m \int d {\bm k} F_n({\bm k}) \\ [n_{\rm F}(\xi_\uparrow({\bm k}+{\bm Q}/2))- n_{\rm F}(\xi_\downarrow({\bm k}-{\bm Q}/2))] \delta((\hbar^2 k Q/m) \cos(\theta) - \hbar \omega),\end{gathered}$$ where $\theta$ is the angle between ${\bm k}$ and ${\bm Q}$. In three dimensions, we choose ${\bm Q}$ along the $z$-axis, so that $\theta$ is the polar angle of ${\bm k}$. In two dimensions, we choose ${\bm Q}$ along the $x$-axis so that $\theta$ is the angle between ${\bm k}$ and the $x$-axis. After some calculations, we arrive at the result $$\begin{gathered}
\Im m \Delta_n({\bm Q},\hbar \omega) = \frac{\pi}{(2 \pi)^d} \int_{k_{\text{min}}}^\infty d k F_{n,d}(k)
\\
\left[
n_{\rm F}\left(\epsilon(k) + \epsilon(Q/2) - \mu + \hbar \omega/2 \right)\right. \\ - \left.
n_{\rm F}\left(\epsilon(k) + \epsilon(Q/2) - \mu - \hbar \omega/2 \right)
\right],\end{gathered}$$ where $\epsilon(k) = \hbar^2 k^2/2m$ and where $k_{\text{min}} = m |\hbar \omega|/\hbar^2 Q$. The functions $F_{0,d}$ depend on dimensionality: F\_[0,2]{}(k) = F\_[0,3]{}(k) = . Finally, $F_{1,d}(k)=(\omega m/\hbar Q) F_{0,d}(k) $, and $F_{2,d}(k) = k^2 F_{0,d}(k)$. In the three-dimensional case, we can find closed form expressions for $\Delta_n$ using the following integrals d k = - (1+(a(b+k\^2))), and $$\begin{gathered}
\int d k \frac{k^3}{1+\exp(a(b+k^2))} = \frac{k^4}{4} - \frac{k^2}{2a} \log(1+\exp(a(b+k^2))) \\ - \frac{1}{2a^2} \text{Li}_2(-\exp(a(b+k^2))).\end{gathered}$$ where $\text{Li}_s(z)$ the polylogarithm defined after Eq. (\[eq:polylocation\]). These expressions are, however, not particularly enlightening and will not be given explicitly here.
Linear Dispersion {#sec:lin}
=================
For a linear dispersion, Eq. (\[eq:gad\]) simplifies to $$\begin{gathered}
\label{eq:gadl}
\Gamma^{\rm D} = - s_\uparrow s_\downarrow \frac{1}{4} \frac{\pi \beta v^2}{d V^3} \sum_{{\bm k}_i} \delta_{{\bm k}_i} \delta(\xi_i)
|W|^2 n_1 n_2 (1-n_3) (1- n_4) \\ (\hat{{\bm k}}_4 -\hat{{\bm k}}_1) \cdot (\hat{{\bm k}}_3 - \hat{{\bm k}}_2)
\left(1+\hat{{\bm k}}_1 \cdot \hat{{\bm k}}_4 \right)
\left(1+\hat{{\bm k}}_2 \cdot \hat{{\bm k}}_3 \right)~,\end{gathered}$$ where we reinstated the chirality factors. The main purpose of this section is to derive the expressions that were the starting point of Ref. \[\], where the drag resistivity was determined in a topological insulator thin film close to exciton condensation. Then, $s_\uparrow =1$ and $s_\uparrow =-1$ and the appropriate decoupling channel for this transition is the Fock channel. For comparison, we also perform the decoupling of Eq. (\[eq:gadl\]) in the Hartree channel.
In Ref. \[\], we used an approximation for $\Gamma^{\rm D}$ in which the chirality factors $\left(1+\hat{{\bm k}}_1 \cdot \hat{{\bm k}}_4 \right)\left(1+\hat{{\bm k}}_2 \cdot \hat{{\bm k}}_3 \right)$ were set to unity. The effect of the chirality factors is the suppression of backscattering: $1+\hat{{\bm k}}_1 \cdot \hat{{\bm k}}_4 $ vanishes when ${\bm k}_1$ and ${\bm k}_4$ are directed oppositely. The pole in the scattering amplitude $W$ for the case of exciton condensation occurs at ${\bm Q} = {\bm k}_1 - {\bm k}_3 =0 $. The motivation for this approximation is that the condition for backscattering $\hat{{\bm k}}_1 \cdot \hat{{\bm k}}_4=-1$ is largely independent of the value of ${\bm Q}$. In other words, the suppression of backscattering does not favor or suppress a particular value for the exciton momentum ${\bm Q}$. In particular, discarding the chirality factors will not have a qualitative influence on the behavior of $\Gamma^{\rm D}$ close to exciton condensation, which is determined by the ${\bm Q}$ dependence of the integrand of Eq. (\[eq:gadl\]).
Hartree Instability {#hartree-instability}
-------------------
When the interaction depends only on the transferred energy $\hbar \omega = \xi_1 - \xi_4$ and momentum ${\bm k}_p = {\bm k}_1 - {\bm k}_4 \equiv {\bm q}$ in the scattering event, the Hartree channel is the appropriate instability channel. We introduce an additional integral over $\hbar \omega$ using the identity (\_1 + \_2 - \_3 - \_4) = d (\_1 - \_4 - ) (\_2 - \_3 + ), and solve the momentum-conserving delta function by ${\bm k}_1 = {\bm k}$, ${\bm k}_2 = {\bm k}'$, ${\bm k}_3 = {\bm k}'+{\bm q}$, and ${\bm k}_4={\bm k}-{\bm q}$, so that the ${\bm k}$ and ${\bm k}'$ summations in Eq. (\[eq:gadl\]) decouple $$\begin{gathered}
\Gamma^{\rm D} = - s_\uparrow s_\downarrow \frac{\pi \beta v^2}{d V^3} \sum_{{\bm k},{\bm k}',{\bm q}} \int d \hbar \omega |W({\bm q}, \hbar \omega)|^2 \\
\delta(\xi_\uparrow({\bm k}) - \xi_\uparrow({\bm k}-{\bm q}) - \hbar \omega) n_{\rm F}(\xi_\uparrow({\bm k})) (1- n_{\rm F}(\xi_\uparrow({\bm k}-{\bm q})))
\\ \left(\frac{{\bm k} - {\bm q}}{|{\bm k} - {\bm q}|} -\hat{{\bm k}}\right) \cdot \left(\frac{{\bm k}' + {\bm q}}{|{\bm k}' + {\bm q}|} -\hat{{\bm k}'}\right) \\ \delta(\xi_\downarrow({\bm k}') - \xi_\downarrow({\bm k}'+{\bm q}) + \hbar \omega) n_{\rm F}(\xi_\downarrow({\bm k}')) (1-n_{\rm F}(\xi_\downarrow({\bm k}'+{\bm q}))).\end{gathered}$$ This expression can be rewritten as $$\begin{gathered}
\Gamma^{\rm D} = s_\uparrow s_\downarrow \frac{\beta v^2}{4 \pi d V} \sum_{{\bm k},{\bm k}',{\bm q}} \frac{ |W({\bm q}, \hbar \omega)|^2}{\sinh^2(\beta\hbar \omega/2)} \\
\Im m \Pi_\uparrow({\bm q},\hbar \omega) \cdot \Im m \Pi_\downarrow({\bm q},\hbar \omega),\end{gathered}$$ where we defined the polarizability $$\begin{gathered}
\Pi_\sigma({\bm q},\hbar \omega) = \frac{1}{V} \sum_{{\bm k}} \frac{n_{\rm F}(\xi_\sigma({\bm k}+{\bm q})) - n_{\rm F}(\xi_\sigma({\bm k}))}{\xi_\sigma({\bm k}+{\bm q}) - \xi_\sigma({\bm k}) - \hbar \omega-i0} \\ \left(\frac{{\bm k} + {\bm q}}{|{\bm k} + {\bm q}|} -\hat{{\bm k}}\right),\end{gathered}$$ and used that $\Pi_\sigma({\bm q},\omega) = -\Pi_\sigma(-{\bm q},\omega)$ and $\Pi_\sigma({\bm q},-\omega) = -\Pi^*_\sigma({\bm q},\omega)$. Using that $\Pi_\sigma({\bm q},\omega)$ is independent of $s_\sigma$ we find $$\begin{gathered}
\Pi_\sigma({\bm q},\hbar \omega) = \frac{1}{2 \pi} \int_0^\pi d \theta \int_0^\infty k dk \\ \left(\frac{q + k \cos(\theta)}{\sqrt{q^2 + k^2 + 2 k q \cos(\theta)}} - \cos(\theta)\right) \\ (n_{\rm F}(\hbar v \sqrt{k^2 + q^2 + 2 k q \cos(\theta)}-\mu_\sigma) - n_{\rm F}(\hbar v k - \mu_\sigma)) \\ \delta(\hbar v \sqrt{k^2 + q^2 + 2 k q \cos(\theta)} - \hbar v k - \hbar \omega),\end{gathered}$$ where we restricted the angular integration between $0$ and $\pi$. The delta function has the solution $k = k_0$ with k\_0 = when $|\omega|/vq<1$ and $\cos^{-1}(\omega/v)<\theta\equiv \theta_{\rm min}$. After solving the delta function, $\Pi_\sigma$ becomes $$\begin{gathered}
\Pi_\sigma({\bm q},\hbar \omega) = \frac{\hbar}{2 \pi} \frac{(v q)^2 - (\omega)^2}{2 \epsilon_{\rm F}} \int_{\theta_{min}}^\pi d \theta \frac{q v - \omega \cos(\theta)}{(\omega - q v \cos(\theta))^2 } \\
\left\{
n_{\rm F}\left[\left(\frac{\hbar(\omega^2-q^2 v^2)}{2 (q v \cos(\theta)-\omega)}-\mu_\sigma+\hbar\omega\right)/T\right] \right.\\
\left.-n_{\rm F}\left[\left(\frac{\hbar(\omega^2-q^2 v^2)}{2 (q v \cos(\theta)-\omega)}-\mu_\sigma\right)/T\right] \right\}.\end{gathered}$$ Substitution of the result for $\Im m \Pi_\sigma$ into Eq. (\[eq:gadl\]) leads to our final expression for $\Gamma^{\rm D}$ after decoupling in the Hartree channel.
Fock Instability {#fock-instability}
----------------
When the interaction depends mainly on the energy and momentum difference of the incoming $\uparrow$ particle and outgoing $\downarrow$ particle, the Fock channel is the appropriate channel to decouple the collision integral Eq. (\[eq:gadq\]). We introduce the energy variable $\hbar \omega = \xi_1 - \xi_3$ using the identity (\_1 + \_2 - \_3 - \_4) = d (\_1 - \_3 - ) (\_2 - \_4 + ), and the momentum variable ${\bm Q}= {\bm k}_1 - {\bm k}_3$ by solving the momentum-conserving delta function by ${\bm k}_1 = {\bm k}+{\bm Q}$, ${\bm k}_2 = {\bm k}'$, ${\bm k}_3 = {\bm k}$, and ${\bm k}_4= {\bm k}'+{\bm Q}$. Then, we have for $\Gamma^{\rm D}$ $$\begin{gathered}
\label{eq:gadfoli}
\Gamma^{\rm D} = s_\uparrow s_\downarrow \frac{\pi \beta v^2}{ d V^3} \sum_{{\bm k},{\bm k}',{\bm Q}} \int d \hbar \omega |W({\bm Q},\hbar \omega)|^2\\
\left(\frac{{\bm k}'+{\bm Q}}{|{\bm k}'+{\bm Q}|}-\frac{{\bm k}+{\bm Q}}{|{\bm k}+{\bm Q}|}\right) \cdot (\hat{{\bm k}}' - \hat{{\bm k}}) \\
\delta(\xi_\uparrow({\bm k}+{\bm Q}) - \xi_\downarrow({\bm k}) - \hbar \omega) n_{\rm F}(\xi_\uparrow({\bm k}+{\bm Q})) (1-n_{\rm F}(\xi_\downarrow({\bm k}))) \\
\delta(\xi_\downarrow({\bm k}') - \xi_\uparrow({\bm k}'+{\bm Q}) + \hbar \omega) n_{\rm F}(\xi_\downarrow({\bm k}')) (1-n_{\rm F}(\xi_\uparrow({\bm k}'+{\bm Q}))).\end{gathered}$$ Expanding the inner product between the unit vectors, we obtain after some rewriting $$\begin{gathered}
\label{eq:gadfol}
\Gamma^{\rm D} = s_\uparrow s_\downarrow \frac{\beta v^2}{2 \pi d V} \sum_{{\bm k},{\bm k}',{\bm Q}} \int d \hbar \omega |W({\bm Q},\hbar \omega)|^2 \\
[\Im m\Delta_0({\bm Q},\hbar \omega) \Im m\Delta_2({\bm Q},\hbar \omega) \\ - \Im m\Delta_{1a}({\bm Q},\hbar \omega) \cdot \Im m \Delta'_{1b}({\bm Q},\hbar \omega)].\end{gathered}$$ where we have defined the generalized susceptibility $$\begin{gathered}
\label{eq:den1}
\Im m\Delta_n({\bm Q},\hbar \omega) = \frac{1}{V} \Im m \sum_{{\bm k}} F_n({\bm k})\\
\frac{n_{\rm F}(\xi_\uparrow({\bm k}+{\bm Q})) -n_{\rm F}(\xi_\downarrow({\bm k}))}{\xi_\uparrow({\bm k}+{\bm Q})-\xi_\downarrow({\bm k}) - \hbar \omega - i 0},\end{gathered}$$ where $F_0({\bm k}) = 1$, $F_{1a}({\bm k}) = \hat{{\bm k}}$, $F_{1b}({\bm k}) = ({\bm k}+{\bm Q})|{\bm k}+{\bm Q}|$, $F_{2}({\bm k}) = \hat{{\bm k}} \cdot ({\bm k}+{\bm Q})|{\bm k}+{\bm Q}|$. The reason that these $F$’s are more complicated than for the quadratic case is that the exciton momentum ${\bm Q}$ does not cancel from the inner product in Eq. (\[eq:gadfoli\]). We note that from Eq. (\[eq:gadfol\]) we obtain Eq. (3) of Ref. \[\]. We evaluate $\Im m\Delta_n$ for $s_\uparrow = 1$ and $s_\downarrow = - 1$, as is appropriate for the case of exciton condensation. Evaluating the imaginary part of Eq. (\[eq:den1\]), we find $$\begin{gathered}
\label{eq:den2}
\Im m\Delta_n({\bm Q},\hbar \omega) = -\frac{1}{4 \pi} \int k d k \int_0^{2\pi} d \theta F_n(k,\theta)\\
(1 - n_{\rm F}(\hbar v \sqrt{k^2 + Q^2 + 2 k Q \cos(\theta)} - \mu_\uparrow) - n_{\rm F}(\hbar v k - \mu_\downarrow))\\
\delta(\hbar \omega + \mu_\uparrow + \mu_\downarrow - \hbar v k - \hbar v \sqrt{k^2 + Q^2 + 2 k Q \cos(\theta)}).\end{gathered}$$ The solution of the delta function is k\_0 = , when $\hbar \omega + \mu_\uparrow + \mu_\downarrow > \hbar v Q$. Solving the delta function in Eq. (\[eq:den2\]) leads to $$\begin{gathered}
\label{eq:den3}
\Im m\Delta_n({\bm Q},\hbar \omega) = \frac{(\hbar \omega + \mu_\uparrow + \mu_\downarrow)^2 - (\hbar v Q)^2}{8 \pi (\hbar v)^2} \int_0^{\pi} d \theta F_n(\theta)\\
\frac{(\hbar \omega + \mu_\uparrow + \mu_\downarrow)^2 + (\hbar v Q)^2+2 \hbar v Q (\hbar \omega + \mu_\uparrow + \mu_\downarrow)\cos(\theta)}{[(\hbar \omega + \mu_\uparrow + \mu_\downarrow) + \hbar v Q \cos(\theta)]^3}\\
\left[1 - n_{\rm F}\left(-\frac{(\hbar \omega + \mu_\uparrow + \mu_\downarrow)^2 - (\hbar v Q)^2}{2[(\hbar \omega + \mu_\uparrow + \mu_\downarrow) + \hbar v Q \cos(\theta)]} + \mu_\downarrow + \hbar \omega \right) - \right. \\ \left.
n_{\rm F}\left(\frac{(\hbar \omega + \mu_\uparrow + \mu_\downarrow)^2 - (\hbar v Q)^2}{2[(\hbar \omega + \mu_\uparrow + \mu_\downarrow) + \hbar v Q \cos(\theta)]} - \mu_\downarrow\right) \right].\end{gathered}$$ The functions $F_n(\theta)$ are given by $F_0(\theta) = 1$, $F_{1a}(\theta) = \cos(\theta)$, $$\begin{gathered}
F_{1b}(\theta) = \\\frac{[(\hbar \omega + \mu_\uparrow + \mu_\downarrow)^2 + (\hbar v Q)^2]\cos(\theta)+2 \hbar v Q (\hbar \omega + \mu_\uparrow + \mu_\downarrow)}{(\hbar \omega + \mu_\uparrow + \mu_\downarrow)^2 + (\hbar v Q)^2+2 \hbar v Q (\hbar \omega + \mu_\uparrow + \mu_\downarrow)\cos(\theta)},\end{gathered}$$ and
$$F_{2}(\theta) = \frac{(\hbar \omega + \mu_\uparrow + \mu_\downarrow)^2 + 2 \hbar v Q (\hbar \omega + \mu_\uparrow + \mu_\downarrow)\cos(\theta)+ (\hbar v Q)^2 \cos(2 \theta)}{(\hbar \omega + \mu_\uparrow + \mu_\downarrow)^2 + (\hbar v Q)^2+2 \hbar v Q (\hbar \omega + \mu_\uparrow + \mu_\downarrow)\cos(\theta)}.$$
Substitution of the result for $\Im m \Pi_\sigma$ into Eq. (\[eq:gadl\]) leads to our final expression for $\Gamma^{\rm D}$ after decoupling in the Fock channel.[@mink]
Basic applications {#sec:allresults}
==================
Before we quantitatively (and therefore numerically) consider two basic applications of our theory, we present some qualitative results on the dependence of the drag resistivity close to a phase transition. These results depend only on (i) the assumed spectrum of Gaussian fluctuations close to the transition and (ii) power counting. Therefore they are very general.
Qualitative results {#subsec:qualitative}
-------------------
Consider first the Hartree channel in three dimensions. At low temperatures we have in the balanced case that $\Im m \Pi_\sigma({\bm q},\hbar \omega) \propto \omega/q$, so that the leading order low-temperature behavior of the drag resistivity is determined by the integral $$\label{eq:lowtrho}
\rho_{\rm D} \propto \int d \omega \int d q q^4 \left[\frac{\omega }{q{\rm sinh} \left(\frac{ \beta \hbar \omega}{2}\right)}\right]^2 w (q,\omega)~,$$ where the relevant momentum and energy dependence that results from the scattering amplitude (derived in detail below), is given by $$w(q,\omega) \propto \frac{1}{\alpha(T)+\left(\frac{q}{k_{\rm F}}\right)^2+\left(\frac{c_1 \omega}{qv_{\rm F}}\right)^2}~,$$ where $\hbar k_{\rm F}$ and $v_{\rm F}$ are, respectively, the Fermi momentum and the Fermi velocity, and $c_1$ is a constant. The important temperature dependence is determined by $\alpha (T) \propto T-T_{\rm c}$, which approaches zero as $T$ approaches the critical temperature $T_{\rm c}$ from above. At low temperatures only small energies are relevant: we therefore expand ${\rm sinh} \left( \beta \hbar \omega/2 \right) \simeq \beta \hbar \omega/2$. The frequency integral is then carried out and we find that the low-temperature behavior of the drag resistivity is determined by the integral $$\label{eq:lowtrhoonlymomentum}
\rho_{\rm D} \propto \int d q \frac{q^k}{\alpha (T) + c_2 q^2},$$ with $c_2>0$ a constant independent of temperature, and $k=3$ for the Hartree channel in three dimensions. Carrying out the remaining momentum integral for $k=3$ leads to the conclusion that the drag resistivity remains finite within the Boltzmann theory presented in this paper.
For the Fock channel in three dimensions, a similar calculation leads to an expression of the form in Eq. (\[eq:lowtrhoonlymomentum\]) with $k=1$, so that the drag resistivity in that case diverges logarithmically. The difference between Hartree and Fock channels is understood from the fact that the momentum ${\bf q}$ that controls the critical fluctuations in the Hartree channel is precisely the momentum transferred in the collision, which leads to an additional factor $q^2$ in the integral in Eq. (\[eq:lowtrhoonlymomentum\]). This is absent in the Fock channel, where the critical momentum [**Q**]{} is different from the momentum transferred in the collisions, ${\bf k'}-{\bf k}$ (see Fig. \[fig:Channels\]).
For the Cooper channel in three dimensions the critical momentum ${\bf K}$ is again different from the momentum transfer $\kv'-\kv$ (see Fig. 2). However, in this case the generalized susceptibilities are independent of momentum in the low-frequency limit: $\Im m \Xi({\bm q},\hbar \omega) \propto \omega$ – a fact that compensates the “loss" of the $q^2$ factor. Furthermore, the critical scattering amplitude is given by $$w(q,\omega) \propto \frac{1}{\alpha(T)+\left(\frac{q}{k_{\rm F}}\right)^2+\left(\frac{c_3 \omega}{\epsilon_{\rm F}}\right)^2}~,$$ where $\epsilon_{\rm F}$ is the Fermi energy and $c_3$ is a constant. The presence of $\omega/\epsilon_{\rm F}$ rather than $\omega/qv_{\rm F}$ in the denominator of this expression causes one less power of $q$ to appear after the integral over frequency is done. The final result is of the form of Eq. (\[eq:lowtrhoonlymomentum\]) with $k=2$ so that the drag resistivity, in that case, remains finite at $T_{\rm c}$.
The above discussion of the Cooper-channel instability was for three dimensions. In two dimensions, we have one power of $q$ less so that $k=1$ and we again find a logarithmic divergence (irrespective of band dispersion) for the drag resistivity if the instability occurs in the Cooper channel. In the next two sections we find that our numerical results agree with these power-counting arguments.
Ferromagnetism in an ultracold Fermi gas {#sec:res}
----------------------------------------
In this section we extend the results of Ref. \[\], which considers the behavior of the spin-drag resistivity in an ultracold Fermi gas with repulsive interactions close to the ferromagnetic transition. This transition is accompanied by magnetic fluctuations of two different characters, longitudinal and transverse fluctuations. By summing all bubble diagrams that contribute to the interaction (see Fig. \[fig:bubbles\]) we obtain $$\begin{gathered}
|W({\bm k}_1,{\bm k}_2,{\bm k}_3,{\bm k}_4) |^2 \\ = |W_{\rm L}({\bm k}_1 - {\bm k}_4,\xi_1 - \xi_4)|^2 + |W_{\rm T}({\bm k}_1 - {\bm k}_3,\xi_1 - \xi_3)|^2,\end{gathered}$$ where $W_{\rm L}$ is the contribution from longitudinal fluctuations and $W_{\rm T}$ is the contribution from transverse fluctuations, which will be specified below. To determine the spin-drag resistivity we need to evaluate the collision integral in Eq. (\[eq:gad\]) for this scattering amplitude. We do this by splitting the collision integral into two integrals, one with scattering amplitude $W_{\rm L}$ and one with $W_{\rm T}$. The former integral can be evaluated by decoupling it in the Hartree channel as described in section \[sec:qh\], and the latter by decoupling it in the Fock channel as described in section \[sec:qf\]. The final result for the drag resistivity is then the sum of these two contributions. In Ref. \[\], the transverse fluctuations were included as if they were part of the Hartree channel. Instead, using the formalism described in this article, we can easily take into account the effect of both types of fluctuations.
The specific form of the scattering amplitude incorporating the longitudinal fluctuations is $$\begin{gathered}
W_{\rm L}({\bm q},\hbar \omega) = U +
\frac{U^2}{4} \frac{\Pi({\bm q},\hbar \omega)}{1-U \Pi({\bm q},\hbar \omega)/2} \\
- \frac{U^2}{4} \frac{\Pi({\bm q},\hbar \omega)}{1+U \Pi({\bm q},\hbar \omega)/2},\end{gathered}$$ where $\Pi({\bm q},\hbar \omega) = \Pi_\uparrow({\bm q},\hbar \omega) + \Pi_\downarrow({\bm q},\hbar \omega)$ with $\Pi_\sigma({\bm q},\hbar \omega)$ given in Eq. (\[eq:PI\]), and where $U$ is the contact interaction strength given by $U = 4 \pi a \hbar^2/m$ with $a$ the $s$-wave scattering length. The specific form of the scattering amplitude incorporating the transverse fluctuations is $$W_{\rm T}({\bm Q},\hbar \omega) = - \frac{U^2}{2} \frac{\Delta({\bm Q},\hbar \omega)}{1+U \Delta({\bm Q},\hbar \omega)/2},$$ where $\Delta({\bm Q},\hbar \omega) = 2\Delta_0({\bm Q},\hbar \omega)$ with $\Delta_0$ given in Eq. (\[eq:de\]).
The result of the complete calculation is shown in Fig. \[fig:mag\], where we plot the dimensionless spin-drag relaxation rate versus temperature. The spin-drag relaxation rate is obtained from the drag resistivity using the Drude formula $\rho_{\rm D} = m/2 n \tau$, where $2n$ is the total density, and $n=k_{\rm F}^3/6\pi^2$ is the density of one spin state. In Fig. \[fig:mag\] we see that the spin-drag relaxation rate is enhanced close to the critical temperature and diverges when approaching $T_{\rm c}$ from above. Closer inspection shows that this divergence is logarithmic, as concluded in the previous section.
Exciton condensation in an electron-hole bilayer
------------------------------------------------
As a second application of our formalism we consider an electron-hole bilayer, which can e.g. be obtained in a GaAs-AlGaAs double quantum well. [@ehexp] This system was considered previously close to exciton condensation using a diagrammatic approach [@hu]. The transition is caused by the formation of pairs (excitons) between electrons in one layer and holes in the other. We consider a system in which the top layer is electron doped with dispersion $\xi_\uparrow({\bm k}) = \hbar^2 k^2/2 m_e - \mu_e$ and the bottom layer is hole doped in which the electrons have the dispersion $\xi_\downarrow({\bm k}) = -\hbar^2 k^2/2 m_h + \mu_h$. In terms of the bare electron mass $m_{\text{bare}}$, the band masses of electrons and holes are $m_e = 0.067 m_{\text{bare}}$ and $m_h = 0.51 m_{\text{bare}}$, respectively.
Close to the transition, the interlayer interaction is determined by the momentum and the energy of an exciton. In the notation of Sec. \[sec:qf\] these are ${\bm Q} = {\bm k}_1 - {\bm k}_3$ and $\hb \om = \xi_\uparrow({\bm k}_1) - \xi_\downarrow({\bm k}_3)$, respectively. If we approximate the bare interlayer interaction by a contact interaction, the effective interlayer interaction can be determined to be \[eq:Weh\] W([k]{}\_1,[k]{}\_2,[k]{}\_3,[k]{}\_4) = W\_([Q]{},) = , where the contact interaction strength $V_0 = 4 \pi \hb^2 / m \log(2 \hb^2/ m \mu a^2)$, where $a$ is the two-dimensional scattering length, $\mu$ is the mean chemical potential $\mu = (\mu_e + \mu_h)/2$ and $m$ is the harmonic mean of the electron and hole masses $1/m = 1/2(1/m_e+1/m_h)$. The susceptibility $\Delta({\bm Q},\hbar\omega)$ is $$\begin{gathered}
\Delta({\bm Q},\hbar\omega) = \frac{1}{V} \sum_{{\bm k}} \left[\frac{n_{\rm F}(\xi_\uparrow({\bm k}+{\bm Q}))- n_{\rm F}(\xi_\downarrow({\bm k}))}{\xi_\uparrow({\bm k}+{\bm Q}) - \xi_\downarrow({\bm k})- \hbar \omega - i 0}
\right. \\ \left. -\frac{1}{\ep_\uparrow(\kv) + \ep_\downarrow(\kv) +\mu_\uparrow+\mu_\downarrow}\right],\end{gathered}$$ where $\ep_\uparrow(\kv) = \hbar^2 k^2/2 m_e$ and $\ep_\downarrow(\kv) = \hbar^2 k^2/2 m_h$.
To determine the drag resistivity we need to evaluate the collision integral in Eq. (\[eq:gad\]) for the scattering amplitude in Eq. (\[eq:Weh\]) by decoupling it in the Fock channel, as described in Sec. \[sec:qf\]. The result of this calculation is shown in Fig. \[fig:EH\], where we show the dimensionless drag resistivity $\rho_{\rm D}/\hb$ versus the scaled temperature $T/T_{\rm F}$ for the density-balanced case. The solid line corresponds to $k_{\rm F} a = 0.34$ with $T_{\rm c}/T_{\rm F} = 0.05$, the dashed line corresponds to $k_{\rm F} a = 0.68$ with $T_{\rm c}/T_{\rm F} = 0.10$, and the dotted line corresponds to $k_{\rm F} a = 1.0$ with $T_{\rm c}/T_{\rm F} = 0.15$. As the transition temperature is approached from above, we find that the drag resistivity diverges as $\log(T-T_{\rm c})$, which is in agreement with the theoretical prediction of Ref. \[\] and our qualitative arguments in Sec. \[subsec:qualitative\].
Conclusions {#sec:con}
===========
In summary, we have presented a general formalism to determine the drag resistivity in systems close to a phase transition. We have shown that by decoupling the collision integral in the appropriate channel, we can take into account the effect of Gaussian fluctuations close the phase transition. Our theory is valid for both two and three dimensions, linear and quadratic dispersions, arbitrary mass and population imbalances, and bands of both “positive" and “negative" character. The approach presented in this article is valid outside the critical region, by which we mean the region in which the Gaussian approximation breaks down and non-trivial critical exponents become relevant. In this region critical fluctuations will alter the temperature dependence qualitatively. The width, in temperature, of this region depends on the system at hand. For the case of a ferromagnetic Fermi gas we have checked that the upturn predicted by our Boltzmann theory takes place well outside the critical region. [@duinemag] As an application, we determined the spin-drag relaxation rate in an ultracold Fermi gas close to the ferromagnetic transition and the drag resistivity in an electron-hole bilayer close to exciton condensation.
This work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), the Netherlands Organization for Scientic Research (NWO), and by the European Research Council (ERC). G.V. was supported by NSF grant DMR-1104788.
APPENDIX {#sec:app .unnumbered}
========
We show how to take the drift momenta ${\bm k}^\text{drift}$ out of the summation in Eq. (\[eq:gav2\]) $$\begin{gathered}
{\bm \Gamma}_\uparrow = - \frac{2 \pi \beta}{2 \hbar^2 V^3} \sum_{{\bm k}_i} \delta_{{\bm k}_i} \delta(\xi_i)
|W|^2 n_1 n_2 (1-n_3) (1- n_4) \\ (\partial_{\bm k} \xi_{4} -\partial_{\bm k} \xi_{1})
\left[
s_\uparrow {\bm k}^\text{drift}_\uparrow \cdot (\partial_{\bm k} \xi_{4} - \partial_{\bm k} \xi_{1}) +
s_\downarrow {\bm k}^\text{drift}_\downarrow \cdot (\partial_{\bm k} \xi_{3} - \partial_{\bm k} \xi_{2})
\right].\end{gathered}$$ For a quadratic dispersion this is easy, since $\partial_{\bm k} \xi_{i} \propto {\bm k}_i$. Using momentum conservation ${\bm k}_1 + {\bm k}_2 = {\bm k}_3 + {\bm k}_4$ and that d [q]{} F(q) [q]{} ([q]{} \^) = d [q]{} F(q) [q]{}\^2, where $F(q)$ is a function that only depends on the length of $q$, immediately leads to the results Eqs. (\[eq:gasu\],\[eq:gasd\],\[eq:gad\]). When we have a two-dimensional system with linear dispersion, then $\partial_{\bm k} \xi \propto \hat{{\bm k}}$. The collision integral ${\bm \Gamma}_\uparrow$ in Eq. (\[eq:gav2\]) is invariant under a global rotation of all integration vectors, in particular, operating with d where the angle $\phi$ is a global rotation angle around an $z$-axis of all vectors ${\bm k}_i$, leaves the integral invariant. The $\delta$’s, the interaction and the Fermi functions do not depend on $\phi$. We write the integral for a general $\phi$-dependent term $$\begin{gathered}
\frac{1}{2 \pi} \int d \phi {\bm k}_a ({\bm k}^\text{drift} \cdot {\bm k}_b) = \frac{k_a k_b}{2 \pi} \int d \phi
\begin{pmatrix}
\cos(\phi + \phi_a) \\ \sin(\phi + \phi_a)
\end{pmatrix} \\
\left[
\begin{pmatrix}
k^\text{drift}_x \\ k^\text{drift}_y
\end{pmatrix}
\cdot
\begin{pmatrix}
\cos(\phi + \phi_b) \\ \sin(\phi + \phi_b)
\end{pmatrix}
\right],\end{gathered}$$ where ${\bm k}_a$ and ${\bm k}_b$ are momenta being summed in Eq. (\[eq:gav2\]), and we choose the $x$-axis along the external drift momentum ${\bm k}^\text{drift}$, where ${\bm k}_a$ and ${\bm k}_b$ can either be the same or different. Now, reflecting the whole integrand in the $x$-axis and then doing the integral gives the same answer. This sends all angles $\phi_i$ to $-\phi_i$. Again, the $\delta$’s, interaction, and Fermi functions do not depend on $\phi$ and the sign of the angles, so we get the term $$\begin{gathered}
\frac{1}{2 \pi} \int d \phi {\bm k}_a ({\bm k}^\text{drift} \cdot {\bm k}_b) = \frac{k_a k_b}{2 \pi} \int d \phi
\begin{pmatrix}
\cos(\phi - \phi_a) \\ \sin(\phi - \phi_a)
\end{pmatrix} \\
\left[
\begin{pmatrix}
k^\text{drift}_x \\ k^\text{drift}_y
\end{pmatrix}
\cdot
\begin{pmatrix}
\cos(\phi - \phi_b) \\ \sin(\phi - \phi_b)
\end{pmatrix}
\right]\,.\end{gathered}$$ Now, we can take the integrals over $\phi$ and perform the average of the two terms, leading to the replacement rule \_a ([k]{}\^ \_b) \^ ([k]{}\_a \_b), which can be used to obtain Eqs. (\[eq:gasu\],\[eq:gasd\],\[eq:gad\]). Note that for the special case ${\bm k}_a = {\bm k}_b$, we recover the result for used for the quadratic dispersion.
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C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein, J. Stephens and D. D. Awschalom, Nature [**437**]{}, 1330 (2005).
Marco Polini and G. Vignale, Phys. Rev. Lett. [**98**]{}, 266403 (2007). R.A. Duine, Marco Polini, H.T.C. Stoof, G. Vignale, , 220403 (2010); R. A. Duine, M. Polini, A. Raoux, H. T. C. Stoof, and G. Vignale, New J. Phys. [**13**]{} 045010 (2011). M. P. Mink, V. P. J. Jacobs, H. T. C. Stoof, R. A. Duine, Marco Polini, and G. Vignale, Phys. Rev. A [**86**]{}, 063631 (2012). S. Riedl, E. R. Sánchez Guajardo, C. Kohstall, A. Altmeyer, M. J. Wright, J. Hecker Denschlag, R. Grimm, G. M. Bruun, and H. Smith, , 053609 (2008). A. Sommer, M. Ku, G. Roati, and M. W. Zwierlein, Nature [**472**]{}, 201–204(2011). R.A. Duine and H.T.C. Stoof, , 170401 (2009); H. J. van Driel, R. A. Duine, and H. T. C. Stoof, , 155301 (2010). S.B. Koller, A. Groot, P.C. Bons, R.A. Duine, H.T.C. Stoof, P. van der Straten, arXiv:1204.6143v2 \[cond-mat.quant-gas\].
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[^1]: The shifted ${\bm k}$ is the relative momentum, defined as the relative velocity times the effective mass $\frac{m_\uparrow m_\downarrow}{m_\uparrow+m_\downarrow}$ of the two species. Then the kinetic energy of the pair is simply the sum of the energy of the center of mass and the relative energy.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The “Island of Inversion” for neutron-rich nuclei in the vicinity of N=20 has become the testing ground [*par excellence*]{} for our understanding and modelling of shell evolution with isospin. In this context, the structure of the transitional nucleus $^{29}$Mg is critical. The first quantitative measurements of the single particle structure of $^{29}$Mg are reported, using data from the d($^{28}$Mg,p $\gamma$)$^{29}$Mg reaction. Two key states carrying significant $\ell =3$ ($f$-wave) strength were identified at $2.40 \pm 0.10$ ($J^\pi = 5/2^-$) and $4.28 \pm 0.04$ MeV ($7/2^-$). New state-of-the-art shell model calculations have been performed and the predictions are compared in detail with the experimental results. Whilst the two lowest $7/2^-$ levels are well described, the sharing of single-particle strength disagrees with experiment for both the $3/2^-$ and $5/2^-$ levels and there appear to be general problems with configurations involving the $p_{3/2}$ neutron orbital and core-excited components. These conclusions are supported by an analysis of the neutron occupancies in the shell model calculations.'
author:
- 'A. Matta'
- 'W.N. Catford'
- 'N.A. Orr'
- 'J. Henderson'
- 'P. Ruotsalainen'
- 'G. Hackman'
- 'A. B. Garnsworthy'
- 'F. Delaunay'
- 'R. Wilkinson'
- 'G. Lotay'
- Naofumi Tsunoda
- Takaharu Otsuka
- 'A.J. Knapton'
- 'G. C. Ball'
- 'N. Bernier'
- 'C. Burbadge'
- 'A. Chester'
- 'D. S. Cross'
- 'S. Cruz'
- 'C. Aa. Diget'
- 'T. Domingo'
- 'T.E. Drake'
- 'L.J. Evitts'
- 'F.H. Garcia'
- 'S. Hallam'
- 'E. MacConnachie'
- 'M. Moukaddam'
- 'D. Muecher'
- 'E. Padilla-Rodal'
- 'O. Paetkau'
- 'J. Park'
- 'J.L. Pore'
- 'U. Rizwan'
- 'J. Smallcombe'
- 'J.K. Smith'
- 'K. Starosta'
- 'C.E. Svensson'
- 'J. Williams'
- 'M. Williams'
title: 'Shell evolution approaching the $N=20$ island of inversion: Structure of $^{29}$Mg'
---
\[intro\]Introduction
=====================
Changes in the relative energies of shell model orbits, depending on the neutron/proton balance in the nucleus [@Brown], cause the energy spacings of orbitals to evolve as one goes away from stability and this can therefore change the shell gaps and hence the corresponding magic numbers [@Sorlin-Review]. This evolution can be studied most effectively by means of single nucleon transfer reactions. In particular, the ($d$,$p$) reaction selectively populates states with a significant single particle character and, importantly, allows the spectroscopic strength to be mapped.
The “island of inversion” in which the neutron-rich (N$\approx$20) isotopes of Ne, Na and Mg exhibit ground states dominated by cross-shell intruder configurations has garnered much attention since the first measurements of their masses at ISOLDE [@Thibault; @Campi]. The intruder configurations become energetically favoured owing, in part, to a significant reduction in the energy gap at N=20 between the $1s0d$ and $0f1p$ shells. Importantly, over recent years, this region has become a prime testing ground for our understanding of many of the concepts of shell evolution away from $\beta$-stability, including the development of sophisticated shell-model interactions.
One of the keys to understanding the island of inversion lies in the evolution of the energies of the neutron orbitals as we move from near stable nuclei into this region. In the case of the Mg isotopes, the single-particle structure of $^{29}$Mg is of key importance to probing the transition into the island of inversion (Fig. \[isotope-chains\]). The object of the present work is, therefore, to investigate the $^{28}$Mg($d$,$p$)$^{29}$Mg reaction which permits the transfer of a neutron into the $0d_{3/2}$, $0f_{7/2}$, $1p_{3/2}$ and higher lying orbitals. As such, the energies of the observed strongly populated (or “single-particle”) states may be related to the spacing between the neutron $sd$ and $fp$ orbitals.
Evolution of intruder state energies for neutron-rich Mg isotopes approaching the island of inversion. The $3/2^+$ level is chosen as the energy reference (adapted from ref. [@Baumann89]). The transitional character of $^{29}$Mg is apparent. ](mg-isotopes.eps){width="\columnwidth"}
Very recently, new effective shell model interactions have been developed from first principles (using the Extended Kuo-Krenciglowa (EKK) method [@EKK]) and including specifically three-body forces [@EEdf1-Tsunoda]. The effective interaction designated “EEdf1”, developed for the $sd-pf$ shells [@EEdf1-Tsunoda], has proven capable of reproducing many of the properties of the neutron-rich Ne, Mg and Si isotopes and has provided new insights into the mechanisms underlying the related shell evolution and therefore the formation of the island of inversion [@EEdf1-Tsunoda]. These shell model calculations using the EEdf1 interaction have been key to understanding the structure of $^{30}$Mg as studied via intermediate-energy single-neutron removal from $^{31}$Mg [@Bea]. In particular, this work indicated that the transition into the island of inversion is far more gradual and complex than previously thought [^1], and suggested a much more nuanced picture whereby intruder particle-hole configurations ($2p$-$2h$,$4p$-$4h$, $\ldots$) represent major components of the wavefunctions of the ground and low lying levels.
As indicated above, the most direct means to understand the changes in shell structure in this region – and indeed to test the new interaction – is to establish the neutron single particle structure of $^{29}$Mg.
\[structure\]Levels and structure of $^{29}{\rm Mg}$
====================================================
The structure of $^{29}$Mg has previously been studied by $\beta \gamma$ coincidences in the $\beta$-decay of $^{29}$Na [@Baumann87; @Shimoda2014], by $\beta n\gamma$ coincidences in the $\beta$-decay of $^{30}$Na [@Baumann89] by three-neutron transfer using the reactions ($^{11}$B,$^{8}$B) [@DavidScott] and ($^{18}$O,$^{15}$O) [@Fifield85] with a $^{26}$Mg target, by a multinucleon transfer reaction that adds a single neutron $^{30}$Si($^{13}$C,$^{14}$O) [@Woods88] and by high-energy single-neutron removal from $^{30}$Mg [@RussTerry]. The presently known levels of $^{29}$Mg are summarized in the final columns of Table \[exp-sm-levels\]. The selectivity observed in the $^{30}$Si($^{13}$C,$^{14}$O)$^{29}$Mg reaction led to the suggestion [@Fifield85; @Woods88] that the states observed at 1.095 and 1.431 MeV were intruder levels with spin-parity $3/2^-$ and $7/2^-$ respectively. These assignments were consistent with the $\beta$-decay results [@Baumann87; @Baumann89] and received further support from the $^{30}$Mg neutron-removal experiment where the angular momenta were suggested to be $\ell = 1$ and 3 respectively [@RussTerry], for the removed neutron. The evolution of the energies of the $fp$ intruder states along the Mg isotopic chain is shown in Figure \[isotope-chains\]. The significance of $^{29}$Mg on the edge of the island of inversion is clear.
The shell model predictions included in the first columns of Table \[exp-sm-levels\] are from a new calculation using the EEdf1 interaction of ref. [@EEdf1-Tsunoda]. This interaction is calculated from a nucleon-nucleon interaction with various computed corrections, and is not fitted to data. The basis for the calculation allowed for cross-shell excitations up to $6\hbar \omega$ for positive parity states and $7\hbar \omega$ for negative parities, which was found to be sufficient for good convergence. The results labelled as [*wbc*]{} were obtained using the code [*nushellx*]{} [@nushellx; @nushellx-RAE; @nushellx-MSU] together with a modification of the WBP interaction [@WBP] wherein the relative energy of the $pf$-shell was lowered by 0.7 MeV as described in an earlier study [@Brown] of the $^{29}$Mg isotone $^{27}$Ne where this modification was labelled WBP-M. The [*wbc*]{}, in addition, replaces the USD interaction for the $sd$-shell [@USD] with the USD-a interaction [@USDA] which is a more appropriate choice in the neutron-rich region. The calculations were restricted to $0\hbar \omega$ for positive parity states as required by the effective interactions, and for negative parity states they included $1\hbar \omega$ excitations from either the $0p$ shell to $1s0d$ or from $1s0d$ to $0f1p$ as described in the original WBP paper [@WBP]. The shell model predicts another six states over the next 2 MeV of excitation (with spins of $3/2^-$ and $5/2^-$) that have values of $(2J+1)C^2 S$ between 0.10 and 0.33. These together add to just one unit in $(2J+1)C^2 S$ which means effectively that all observable states up to 6.5 MeV (according to the predictions) are included in the table.
Before reviewing all of the experimental levels, some general comments can be made. A key feature is the pair of $3/2^-$ and $7/2^-$ states just above 1 MeV which represent intruder configurations from the $fp$ shell in which a neutron in the $0f_{7/2}$ or $1p_{3/2}$ orbital is coupled to a $^{28}$Mg core. In this picture, the core can be in its $0^+$ ground state or excited to a higher energy configuration such as $2^+$ but the neutron-transfer reaction can populate these states only via the component with the $0^+$ core, leaving aside any two-step contributions to the reaction mechanism. A second pair of $3/2^-$ and $7/2^-$ states is predicted to lie near 4 MeV in $^{29}$Mg. Of these, the $3/2^-$ is predicted to carry 10-20% of the single-particle strength that it shares with the 1 MeV partner. The higher lying $7/2^-$ is predicted to carry 30-40% of the shared single-particle strength. According to the theory, there is evidently a significant mixing between the $7/2^-$ states of $0^+ \otimes 0f_{7/2}$ character and excited-core nature, such as $2^+ \otimes 1p_{3/2}$. There is mixing predicted also between the $3/2^-$ states with $0^+$ and $2^+$ cores. Furthermore, the excited core configurations can include coupling to a neutron in the $0f_{7/2}$ orbital. Another $3/2^-$ state predicted below 3.5 MeV appears not to contain significant single-particle strength relative to the $0^+$ core. Two additional states, each arising from a single excited-core configuration, are the $11/2^-$ state near 3.5 MeV, which arises from $2^+ \otimes 0f_{7/2}$, and the $1/2^-$ state near 2 MeV which arises from $2^+ \otimes 1p_{3/2}$. Of these, just the $1/2^-$ can have a component of single-particle nature with a $0^+$ core, and according to the theory, there is significant mixing and hence an appreciable spectroscopic factor. Finally, the lowest $5/2^-$ state must result from a coupling with an excited core and can mix with the much higher-lying $0^+ \otimes 0f_{5/2}$ configuration, but the mixing is small, at least according to the theory. To summarise, the states built upon excited cores can mix with the $3/2^-$ and $7/2^-$ single particle states and this would result in a significant population of states near 4 MeV that have yet to be identified.
-------------- -------------------------------- -------------- ------------------------------ -------------- -------------------- ------------------------------------------------------------
$J^\pi $ E$_{\rm{x}}^{\rm{SM}}$ (EEdf1) $(2J+1)C^2S$ E$_{\rm{x}}^{\rm{SM}}$ (wbc) $(2J+1)C^2S$ E$_{\rm{x}}$ (exp) Ref. for
(MeV) (MeV) (MeV) Assignment
$3/2^+ _1 $ 0.000 1.41 0.090 1.61 0.000 [@Dominique-paper]
$1/2^+ _1 $ 0.026 0.70 0.000 0.79 0.055 [@Fifield85; @Baumann89]
$3/2^- _1 $ 0.872 1.66 1.350 2.50 1.095 [@Fifield85; @Woods88; @Baumann87; @Baumann89; @RussTerry]
$7/2^- _1 $ 1.456 3.45 1.867 3.40 1.431 [@Fifield85; @Woods88; @Baumann87; @Baumann89; @RussTerry]
$5/2^+ _1 $ 1.713 0.05 1.611 0.01 1.638 [@Baumann87]
$1/2^- _1 $ 1.915 0.38 2.421 0.61 2.266 [@Baumann89]
$5/2^+ _2 $ 2.106 0.26 3.147 0.33 3.228 [@Baumann87; @Shimoda2014]
$3/2^+ _2 $ 2.129 0.77 2.269 1.00 2.500 [@Baumann87; @Shimoda2014]
$7/2^+ _1 $ 2.195 $-$ 2.249 $-$
$1/2^+ _1 $ 2.509 0.00 2.905 0.00 2.615 [@Baumann87; @Shimoda2014]
$5/2^- _1 $ 2.914 0.10 3.073 0.15
$3/2^+ _3 $ 2.924 0.01 3.619 0.02 3.223 [@Baumann87; @Shimoda2014]
$5/2^+ _3 $ 3.120 0.08 3.628 0.00 3.673 [@Baumann87; @Shimoda2014]
$3/2^- _2 $ 3.261 0.03 3.480 0.05 3.090 [@DavidScott; @Fifield85; @Woods88]
$5/2^+ _4 $ 3.262 0.00 4.253 0.00 3.985 [@Baumann87; @Shimoda2014]
$7/2^+ _2 $ 3.301 $-$ 3.992 $-$
$11/2^- _1 $ 3.491 $-$ 3.629 $-$
$5/2^+ $ 3.516 0.01 5.160 0.00
$7/2^+ _3 $ 3.642 $-$ 4.718 $-$
$1/2^- _2 $ 3.767 0.66 3.646 0.83
$3/2^- _3 $ 3.832 0.42 3.973 0.34
$9/2^+ _1 $ 4.104 $-$ 4.077 $-$
$7/2^- _2 $ 4.050 1.89 4.157 2.71 4.280 [@DavidScott; @Fifield85; @Woods88]
$5/2^- _2 $ 4.254 0.00 4.363 0.11
-------------- -------------------------------- -------------- ------------------------------ -------------- -------------------- ------------------------------------------------------------
The ground state of $^{29}$Mg was deduced to have spin-parity $3/2^+$ [@Dominique-paper] on the basis of its decay scheme to known states in $^{29}$Al. This assignment and others for experimentally observed excited states are included in Table \[exp-sm-levels\]. The higher energy state in the ground state doublet, at 0.054 MeV, was first proposed to have spin-parity $1/2^+$ by Fifield [*et al.*]{} [@Fifield85] in a re-interpretation of the early $\beta$-decay data [@Dominique; @Dominique-paper] and this was later confirmed in further $\beta$-decay studies [@Baumann89]. The states at 1.095 and 1.431 MeV were postulated [@Fifield85] to have spin-parity $3/2^-$ and $7/2^-$ respectively according to the selectivity observed in the $^{30}$Si($^{13}$C,$^{14}$O)$^{29}$Mg reaction. These assignments were consistent with the $\beta$-decay results [@Baumann87; @Baumann89; @Tajiri; @Shimoda2014] and the intermediate-energy reaction study mentioned above [@RussTerry]. The next higher state at 1.638 MeV was not populated at all in the multinucleon transfer [@DavidScott; @Fifield85; @Woods88] but was observed in the $\beta$-decay of $^{29}$Na (ground state $3/2^+$) and deduced to be $5/2^+$ [@Baumann87]. The $\beta$-decay study did not observe the 2.266 MeV state, but did measure and deduce spins and parities for the 2.500 MeV ($3/2^+$) and 2.615 MeV ($1/2^+$) states. These positive parity assignments are supported, where the work overlaps, by a recent study of $\beta$-decay using polarized $^{29}$Na [@Shimoda2014]. The 2.266 MeV state was subsequently observed in $\beta$-delayed neutron decay of $^{30}$Na [@Baumann89] and was interpreted to have negative parity on the basis of its non-population in the $\beta$-decay of the $3/2^+$ $^{29}$Na ground state; noting also the observed $\gamma$-ray decays (which populate both states in the ground state doublet) and evidence from neutron penetrability arguments, a spin-parity of (1/2,3/2)$^-$ was assigned. The next two states given in the most recent compilation [@NDS29] are those at 3.224 MeV and 3.228 MeV that were first observed in $\beta$-decay [@Baumann87]. The more recent polarized $\beta$-decay work [@Shimoda2014] assigns these as $3/2^+$ and ($5/2$)$^+$ with energies of 3.223 and 3.227 MeV. A level reported in ($^{13}$C,$^{14}$O) at $3.20 \pm 0.04$ MeV [@Woods88], also measured at $3.09 \pm 0.04$ MeV and $3.07 \pm 0.09$ MeV in three-neutron transfer [@DavidScott; @Fifield85], was suggested [@Woods88] to be a negative parity intruder state. In the compilation [@NDS29] this level is associated with the 3.223 MeV $3/2^+$ level, but the interpretation based on the multi-nucleon population [@Woods88] suggests that this should be retained as an additional observed state (which is denoted here as 3.090 MeV). Next highest in energy are states at 3.673 MeV and 3.985 MeV that were first observed in $\beta$-decay [@Baumann87] and have recently both been assigned as having spin-parity ($5/2^+$) in polarized $\beta$-decay [@Shimoda2014]. These two states are above the neutron separation energy of $^{29}$Mg (3.66 MeV). The highest state reported in the compilation is at 4.280 MeV and has previously been seen only in the three multinucleon transfer reactions [@DavidScott; @Fifield85; @Woods88].
Table \[exp-sm-levels\] suggests that there are about ten states in $^{29}$Mg predicted by the shell model below 4.3 MeV that are yet to be discovered experimentally. On the other hand, the known experimental states all have reasonable counterparts in the theory.
\[experimental\]Experimental details
====================================
A secondary beam of $^{28}$Mg was obtained from the ISAC2 facility at TRIUMF using a primary beam of 100 $\mu$A of 520 MeV protons bombarding a SiC production target. The extraction of the $^{28}$Mg$^{1+}$ ions was compromised by the failure to hold a sufficiently high voltage on the source and it was necessary to employ a charge state breeder (CSB [@Ames2014]) to produce $^{28}$Mg$^{5+}$ ions for injection into the radiofrequency quadrupole at the start of the ISAC acceleration system [@Laxdal2014]. The efficiency of the CSB was $10^{-3}$ and it inevitably introduced contaminants. These included radioactive nuclei which had mass to charge ratios close to that of $^{28}$Mg$^{5+}$ and moreover there were stable contaminants derived from the CSB itself. The beam transmitted to the secondary target station comprised $\sim$99% of the stable isobar $^{28}$Si at a rate of 300,000 pps. Approximately 1% of the beam, or 3000 pps was found to be the intended isotope $^{28}$Mg ($t_{1/2}=20.9$ h), as discussed below. A smaller amount, estimated as up to $\sim$300 pps, was deduced to be $^{28}$Al ($t_{1/2}=2.2$ m). The energy of the $A=28$ beam was 8.0 MeV/u. The beam spot size on target was $\sim 2$ mm in diameter. The secondary reaction target comprised deuterated polythene (CD$_2$)$_n$ with a thickness of 0.5 mg/cm$^2$.
Elimination of $^{28}$Si-induced reactions from the analysis was achieved using a thin scintillator detector (the <span style="font-variant:small-caps;">trifoil</span>, described below) mounted downstream of the target and preceded by a passive stopper foil. This setup was employed previously [@wilson-rutherford] in a similar experiment [@wilson-plb] with a radioactive $^{25}$Na beam. In the present work the intensity of the $^{28}$Mg beam was lower than the earlier $^{25}$Na beam by a factor of 10,000 and the mode of operation was different: the passive stopper was used to filter out the higher-Z contaminants, so that only the $^{28}$Mg and $^{29}$Mg reaction products could reach the <span style="font-variant:small-caps;">trifoil</span> and be recorded. The stopper was a 90 $\mu$m thick Al foil. This thickness was sufficient to stop the $^{28}$Si projectiles (and $^{29}$Si reaction products) and to allow all $^{28,29}$Mg ions to reach the <span style="font-variant:small-caps;">trifoil</span> with sufficient energy to be recorded. The Al foil also, as in the earlier experiment [@wilson-plb], stopped any fusion-evaporation reaction products (arising from reactions on the carbon in the target) from reaching the <span style="font-variant:small-caps;">trifoil</span>. The small component of $^{28}$Al in the beam was not anticipated and it was found (see below) that the $^{29}$Al products were able to reach the <span style="font-variant:small-caps;">trifoil</span> in some cases, but only for a particular range of Q-values and only for events with a proton recorded in the backward-most particle detectors.
 Schematic layout of the experiment, with the beam incident on a deuterated polythene target at the centre of the SHARC silicon strip detector array [@SHARC] which is surrounded by 12 TIGRESS clover Ge detectors [@TIGRESS] arranged at angles of 90$^\circ$ and 135$^\circ$. Downstream of the target, a passive Al stopper foil prevented fusion-evaporation residues and other contaminant particles from reaching a plastic scintillator detector (<span style="font-variant:small-caps;">trifoil</span>).](experimental-setup.eps){width="\linewidth"}
The experimental setup is shown schematically in Figure \[experiment\]. The CD$_2$ target was surrounded by the SHARC array [@SHARC] which comprises double-sided silicon strip detectors (DSSDs). The downstream box (covering laboratory scattering angles of less than $90^\circ$) was used primarily to detect elastically scattered deuterons for cross section normalisation. The upstream box (laboratory angles from $95^\circ$ to $143^\circ$) and the backward-angle “CD”annular array (angles $147^\circ$ to $172^\circ$) were employed to record protons from (d,p) reactions.
The <span style="font-variant:small-caps;">trifoil</span> detector was located 400 mm downstream from the target and for the present experiment comprised a square 25 $\mu$m thick BC400 plastic scintillator foil of area $40\times 40$ mm$^2$ aligned axially with the beam. The scintillator was viewed by three photomultipliers and a NIM logic signal was generated if any two photomultipliers responded in coincidence. The reaction angle spanned by the largest circle inscribed within the square scintillator foil was $2.8^\circ$, fully encompassing the $^{29}$Mg products from (d,p) reactions ($<2^\circ$ for protons recorded in the upstream detectors) and elastically scattered $^{28}$Mg particles (for centre-of-mass scattering angles up to $40^\circ$).
Gamma-rays were recorded in the TIGRESS array of HPGe clover detectors [@TIGRESS; @Hackman2014], mounted at a distance of 110 mm from the target and operated without any active escape suppression. A total of 12 clovers were deployed, of which 8 were centred at $90^\circ$ and 4 at $135^\circ$ with respect to the beam, spanning all polar angles. An add-back algorithm was implemented to recover the energies for gamma-rays scattered between different crystals within individual clovers. For all gamma-ray events, the segment signal corresponding to the largest energy was assumed to indicate the location of the initial gamma-ray interaction. This allowed the appropriate correction to be applied to the measured energy to account for the Doppler shift arising from the velocity ($\sim 0.10c$) of the emitting beam-like particle.
The TIGRESS data acquisition system [@TIGRESS-electronics] required a validation (trigger) signal to initiate the data readout. This trigger was derived from the SHARC silicon detectors such that a signal from any strip in SHARC led to the readout of any signals from the <span style="font-variant:small-caps;">trifoil</span> and any coincident silicon and gamma-ray detectors. For the <span style="font-variant:small-caps;">trifoil</span>, the NIM coincidence signal was digitized with a 10 ns sample period over an interval centred on the time of true coincidence pulses. The digital trace was processed to identify signals occurring at the true coincidence time. The adjacent beam pulses, which occurred with a spacing of 86 ns and could easily be distinguished, would also occasionally show signals in the trace if they randomly contained a non-reacting $^{28}$Mg projectile (probability $\sim 3000/(10^9/86)=0.00026=0.026$%). The times of all logic pulses in the time window were extracted and an example of the relative timing spectrum between the SHARC silicon array and the <span style="font-variant:small-caps;">trifoil</span> is shown in Figure \[trifoil\]. The subsidiary peaks are also randomly populated when events induced in the SHARC array by the 100-times more-intense $^{28}$Si beam are accompanied by unreacted $^{28}$Mg projectiles in nearby beam pulses (probability $\sim 100 \times 0.00026 = 2.6$%).
![\[trifoil\] (Color online) Spectrum of the relative time between SHARC and <span style="font-variant:small-caps;">trifoil</span> signals for events in which a particle was recorded in SHARC. The main peak corresponds to $^{28}$Mg-induced direct reactions and the small peaks correspond to $^{28}$Mg projectiles by chance being found in nearby beam pulses (see text). By selecting events in a region away from the main peak, a quantitative estimate of the background underlying the peak was obtained. ](Trifoil-WNCb.eps){width="\linewidth"}
\[analysis\]Analysis
====================
\[IVa\]Overview of analysis
---------------------------
As described above, it was possible to use the <span style="font-variant:small-caps;">trifoil</span> detector to select the events arising from direct reactions induced by the small $^{28}$Mg component in the beam. In particular these reactions included (d,p), (d,d) and (p,p). Without the <span style="font-variant:small-caps;">trifoil</span> selection, the kinematic loci for the (d,p) reaction induced by $^{28}$Si projectiles could clearly be observed, along with an underlying background from evaporation protons and alpha-particles. With the <span style="font-variant:small-caps;">trifoil</span> condition imposed, it was clear that the events induced by $^{28}$Si were successfully removed and the kinematic loci corresponding to reactions induced by $^{28}$Mg were observed (Fig. \[Mg-kinematic\]).
The energies recorded in SHARC were assumed to correspond to protons for laboratory angles greater than 90$^\circ$ and deuterons for angles forward of 90$^\circ$ and corrections were applied for the energy losses occurring in the target (assuming reactions at the midpoint) and the dead layers of the silicon detectors. As usual for double-sided silicon strip detectors, the energies recorded on the front and back strips were required to be equal. The position of the beam spot on the target was determined using the observed kinematic line for $^{28}$Si+d elastic scattering as recorded in the various downstream barrel detectors. The d($^{28}$Si,p)$^{29}$Si kinematic lines allowed the positions of the upstream detectors to be fine-tuned. Combined with the known geometry of SHARC, the laboratory scattering angle of the particles recorded in the silicon array could then be determined.
In order to extract absolute cross sections, the integrated luminosity (product of beam exposure and target thickness) was determined using measurements of the deuteron elastic scattering. The differential cross section in counts/msr was first extracted. Since the deuteron energy varies rapidly with the laboratory angle and is measured with good resolution, the energy is the best way to define the scattering angle. Thin cuts in energy were therefore used to define corresponding bins in centre of mass angle. The number of counts in each bin, with suitable background subtraction, was combined with the corresponding solid angle as determined by a Monte-Carlo calculation using [*Geant4*]{} implemented via [*NPTool*]{} [@NPTool]. In this manner the differential cross section over a range of angles corresponding to $22^\circ$ to $32^\circ$ in the centre-of-mass frame was obtained. A comparison of the measured elastic scattering cross section in counts/msr with an optical model calculation expressed in mb/sr allowed the luminosity to be deduced. Three optical potentials suitable for this beam-target combination were employed [@Daehnick; @PereyPerey; @Lohr] and these showed a variation between them of 10% in absolute magnitude over the angular range of interest. The number of counts in each angle bin was determined to an accuracy of 5%. The value adopted [@Daehnick] for the integrated luminosity was thus ascribed an uncertainty conservatively estimated as 15%. The analysis of the elastic scattering was validated using the much more intense $^{28}$Si component of the beam (and in fact it was this procedure that gave the best measure of the beam composition, viz. 99% $^{28}$Si and 1% $^{28}$Mg).
For (d,p) transfer events the energy and angle of the particle observed in SHARC were used, together with the beam energy and assumed reaction kinematics, to calculate the excitation energy of the final nucleus. This procedure was validated using the data for the $^{28}$Si beam which showed peaks in the excitation energy spectrum at the correct energies in $^{29}$Si, including the $1/2^+$ ground state and the strongly populated $3/2^-$ state at 4.93 MeV [@Mermaz; @Peterson]. In order to derive differential cross sections expressed as mb/msr, the integrated luminosity was taken from the elastic scattering and the solid angle was taken from the calculation using [*Geant4*]{} and [*NPTool*]{} [@NPTool]. The differential cross sections were extracted in terms of laboratory angles rather than centre-of-mass angle because it was then possible to identify most clearly the angles that needed to be eliminated due to detector edges, or gaps in the detector coverage, or due to energy detection thresholds.
\[IVb\]Results for $^{28}$Mg projectiles
----------------------------------------
--------- -------------------- -------------------- ----------------- ----------------- ----------------- ----------------- ------------------
Peak ID E$_{\rm{x}}$ (min) E$_{\rm{x}}$ (max) Expected
(MeV) (MeV) (MeV) $\ell = 0$ $\ell = 1$ $\ell = 2$ $\ell = 3$ states ($J^\pi$)
0.0 $-0.5$ 0.5 $0.68 \pm 0.06$ $1.20 \pm 0.12$ $3/2^+ , 1/2^+ $
1.2 0.6 1.8 $0.44 \pm 0.04$ $3.04 \pm 0.16$ $3/2^- , 7/2^- $
2.4 1.9 2.9 $0.32 \pm 0.12$ $1.80 \pm 0.18$ $3/2^+ , 5/2^- $
4.2 3.7 4.7 $2.40\pm 0.40$ $7/2^-$
--------- -------------------- -------------------- ----------------- ----------------- ----------------- ----------------- ------------------
The kinematical plot for the data from $^{28}$Mg projectiles is shown in Figure \[Mg-kinematic\] for angles backward of $90^\circ$. In order to eliminate low energy signals arising from noise and $\beta$-radiation not eliminated by use of the <span style="font-variant:small-caps;">trifoil</span>, a lower limit was imposed on the detected proton energy (before correction). The kinematic plot shows a small background of counts above the line corresponding to the ground state of $^{29}$Mg and this is noticeably more intense at angles larger than 145$^\circ$. Whereas the low level of background forward of 145$^\circ$ is explained by the small fraction of $^{28}$Si-induced reactions that escape rejection by the <span style="font-variant:small-caps;">trifoil</span> requirement (owing to random coincidences) the increase at more backward angles has a different origin. This additional background is attributed to a small and unanticipated component (about ten times smaller than the $^{28}$Mg) of $^{28}$Al in the beam. The more positive Q-value for the (d,p) reaction involving $^{28}$Al gives the protons extra energy and they extend to negative excitation energies if the kinematics is assumed to be d($^{28}$Mg,p)$^{29}$Mg.
Assuming that the events in Figure \[Mg-kinematic\] correspond to the d($^{28}$Mg,p)$^{29}$Mg reaction, the excitation energy in $^{29}$Mg was computed and is shown in Figure \[Mg-Ex\]. Fortunately, the [*Geant4*]{} simulation of the d($^{28}$Al,p)$^{29}$Al reaction shows that the <span style="font-variant:small-caps;">trifoil</span> requirement eliminates any background from this source in the region of positive excitation energies in $^{29}$Mg (cf. Figure \[Mg-Ex\]). That is, there is an abrupt change in the background at the ground state and the spectrum of $^{29}$Mg states should therefore have no significant underlying background. In more detail, the simulation also shows that the $^{29}$Al reaction products are only able to reach the <span style="font-variant:small-caps;">trifoil</span> and be recorded if the proton is detected in the CD detector that covers the most backward angles of the SHARC array. This is because the backward-going proton imparts a small extra kick to the forward-going $^{29}$Al ion and also the smaller deflection angle of these $^{29}$Al ions gives them the shortest paths through the passive stopping foil. The clear drop in the background intensity for angles below $145^\circ$ (Figure \[Mg-kinematic\]) is in excellent agreement with the simulations.
As may be seen in Figure \[Mg-Ex\], there are strong peaks observed in the spectrum for $^{29}$Mg at excitation energies of 0.0, 1.2, 2.4 and 4.2 MeV. The possible origins of these peaks are now discussed, keeping in mind that the expected resolution is $\sim$700 keV FWHM (limited mostly by the differential energy loss of protons escaping the target). The peak near 0.0 MeV is likely to contain contributions from both levels comprising the ground state doublet at 0.000 MeV ($3/2^+$) and 0.054 MeV ($1/2^+$). The peak near 1.2 MeV must correspond to the negative parity intruder doublet of 1.095 ($3/2^-$) and 1.431 MeV ($7/2^-$). The peak near 2.4 MeV is open to some speculation, but it does occur close to the known states at 2.266 MeV ($1/2^-$) and 2.500 ($3/2^+$) which can reasonably be expected to be populated (Table \[exp-sm-levels\]). Additional information from the differential cross sections as discussed below indicates that a previously unobserved negative parity state also contributes. The peak near 4.2 MeV is close to the level reported at 4.28 MeV in multinucleon transfer reactions [@DavidScott; @Fifield85; @Woods88] which was speculated [@Woods88] to have negative parity. The asymmetry on the left hand side may point to the population of a weaker state at a slightly lower energy. Interestingly, there is a marked absence of strength at 3.09 MeV where another prominent peak was observed in the multinucleon transfer.
The energy spectrum for all gamma-rays recorded in coincidence with SHARC and giving a <span style="font-variant:small-caps;">trifoil</span> signal is shown in Figure \[complete-gamma\](a). Clear peaks are observed in the <span style="font-variant:small-caps;">trifoil</span>-gated $^{29}$Mg spectrum, corresponding to the known transitions at 1.095 and 0.336 MeV. It is possible that other peaks occur at several different energies (discussed below) but the limited counting statistics are not conclusive. The spectrum with no <span style="font-variant:small-caps;">trifoil</span> requirement, shown in Figure \[complete-gamma\](d), serves to illustrate that any contribution to the <span style="font-variant:small-caps;">trifoil</span>-gated spectrum from the $^{28}$Si projectiles (from both direct and compound reactions) is essentially eliminated. The gamma-ray energy resolution (after Doppler correction) is 42 keV (FWHM) at 1.095 MeV which is of course far better than the 700 keV (FWHM) resolution for the excitation energy deduced using the protons.
 Kinematic plot showing proton energy as a function of laboratory angle, after correction for energy losses in the target and in the dead-layer of the silicon detector. The calculated kinematic line for protons populating the ground state of $^{29}$Mg is shown. The origin of the background above this line is discussed in the text.](Kinematics.eps){width="\linewidth"}
{width="\linewidth"}
 Doppler-corrected gamma-ray energy spectra for $\theta _{{\rm lab}}$(p)$>90^\circ$ : (a) events in which the <span style="font-variant:small-caps;">trifoil</span> is triggered (i.e. mostly corresponding to the $^{28}$Mg-induced (d,p) reaction), (b) with an additional gate of E$_{{\rm x}}$($^{29}$Mg)= $0.8-1.5$ MeV, (c) as for (b) but $2.0-2.8$ MeV, (d) no <span style="font-variant:small-caps;">trifoil</span> gating (i.e. mostly arising from $^{28}$Si-induced reactions). The well-known gamma-rays at 336 keV and 1040 keV from the decay of the 1.431 MeV state are clearly seen in the upper two spectra. Several other tentative peaks from $^{29}$Mg are discussed in the text. ](gamma-montage.eps){width="\linewidth"}
The two states contributing to the peak at 0.0 MeV cannot be distinguished using gamma-rays since the 54 keV transition was not detectable in this experiment (due primarily to the detection threshold and exacerbated by the 1.27 ns lifetime [@NDS29] of the state). The gamma-ray energy spectrum for the excitation energy peak near 1.3 MeV is shown in Figure \[complete-gamma\](b). The yield of the 1.041 MeV transition exceeds that of the 0.336 MeV by a factor of 3, after correction for efficiency. Given that the 1.431 MeV state decays via a cascade through the 1.095 MeV level, resulting in these two gamma-ray lines, it is clear that both of these states were directly populated in the (d,p) reaction [^2].
Unfortunately the gamma-ray statistics for other states are extremely limited and also the experimental spectrum enhances the Compton edge because the add-back is only within each individual clover (this gives an enhancement at $\sim 230$ keV below the full energy peak). There is very tentative evidence in Figure \[complete-gamma\](a) for peaks near 1.6, 1.8, 2.4 and 3.2 MeV. The tentative 2.4 MeV is the highest energy seen in the spectrum in Figure \[complete-gamma\](c) gated on E$_{\rm x}=2.0-2.6$ MeV, along with weak indications of a 1.0 MeV peak. It may be that the 3.2 MeV peak is associated with decays to either or both of the ground state doublet by a state near 3.2 MeV that could not be clearly discerned in the proton spectrum of Figure \[Mg-Ex\]. Similarly the 1.6 and 1.8 MeV gamma-rays could arise in part from a gamma-ray decay branch of the unbound states making up the 4.2 MeV peak.
As it was impossible to select individual states by gating on gamma-ray energy, the differential cross sections d$\sigma$/d$\Omega$ of the four prominent peaks in Figure \[Mg-Ex\] have been extracted. There was no reliable way to fit a smooth underlying background in the excitation energy spectrum but, on the other hand, the background evident at negative excitation energies should not extend into the positive energy region (as discussed above) and the only other background that should be present would arise from weakly populated states that lie near to the strongly populated states. To the extent that the strongly selected states very much dominate the yield (which is discussed again, at the end of the analysis), it was possible to use the simple integrated number of the counts in each peak over the relevant range of energies. In view of the resolution (FWHM) of 700 keV, a range of 1.0 MeV was generally adopted as shown in Table \[2j+1c2s\]. As discussed above, the peak near 1.2 MeV is known to comprise the two states at 1.095 MeV and 1.431 MeV, separated by 0.336 MeV and hence this gate was widened to 1.2 MeV so as to include as much as possible of both contributions without extending into other adjacent peaks. The region near 3 MeV appears to contain contributions from several less-strongly populated states, but the limitations of the statistics preclude any quantitative analysis.
The angle bins were chosen to be $4^\circ$ in width (in the laboratory frame) and spanned the angles from $96^\circ$ to $172^\circ$, excluding those from $136^\circ$ to $148^\circ$. This avoided the angles at which the solid angle acceptance was varying rapidly and might be incorrectly calculated if there were small residual misalignments in the setup. Expressed in terms of centre-of-mass angles, the range spanned was approximately $2^\circ$ to $40^\circ$ depending on the excitation energy. The peak near 4.3 MeV required a modified procedure, because it is clear in Figure \[Mg-kinematic\] that the protons fall below the energy threshold for the largest laboratory angles and hence it was possible only to use the angle bins from $96^\circ$ to $116^\circ$.
In order to determine the angular momentum of the transferred neutron, the differential cross sections were compared with theoretical distributions calculated using the Adiabatic Distorted Wave Approximation (ADWA) of Johnson and Soper [@Johnson-Soper]. The code TWOFNR [@TWOFNR] was used with standard input parameters [@Lee] and the Chapel-Hill (CH89) nucleon–nucleus optical potential [@CH89]. As may be seen in Figure \[angdis-assembly\] the angular momenta were well-determined by the data and multiple $\ell$ contributions were employed where necessary. Spectroscopic factors were deduced by normalising the theoretical curves to the data. The results are collated in Table \[2j+1c2s\] and discussed below.
![\[angdis-assembly\] (Color online) Differential cross sections for the four main peaks identified in the excitation energy spectrum and listed in Table \[2j+1c2s\], solid lines are the sum of the different contributions: (a) 0.0 MeV dashed ($\ell=0$, $S=0.34$) and dot-dashed ($\ell=2$, $S=0.30$), (b) 1.2 MeV dotted ($\ell=1$, $S=0.11$) and dot-dashed ($\ell=3$, $S=0.38$), (c) 2.4 MeV dot-dashed ($\ell=2$, $S=0.08$) and dash-three-dots ($\ell=3$, $S=0.30$), (d) 4.2 MeV dash-three-dots ($\ell=3$, $S=0.30$). ](differential-cross-sections2.eps){width="\linewidth"}
The peak at 0.0 MeV displays $\ell =2$ and $\ell =0$ contributions to the angular distribution (see Fig. \[angdis-assembly\](a)) which is consistent with the known assignments for the $3/2^+$ ground state and the $1/2^+$ first excited 0.054 MeV state, respectively. The distribution (b) for the 1.2 MeV peak is well described by a sum of $\ell = 1$ and $\ell = 3$ contributions, in agreement with the gamma-ray data that indicate the population of both the 1.095 ($3/2^-$) and 1.431 MeV ($7/2^-$) states. The only other known state of similar energy is the 1.638 MeV ($5/2^+$) level [@Baumann87; @Shimoda2014] and this cannot be populated in single step transfer. The peak at 2.4 MeV in the excitation energy spectrum is less prominent than the other three and hence is the most problematic in the analysis. The differential cross section (c) has a maximum near $105^\circ$ as seen for the peak at 1.2 MeV, and this requires a contribution from $\ell =3$. The behaviour near $180^\circ$ ($0^\circ$ in the centre of mass frame) is slightly different to that in (b), and the fit in this case also demands a contribution from $\ell =2$. The $\ell =2$ component must arise from the level at 2.500 MeV $3/2^+$ if it is from a state that is already known. Regarding the $\ell =3$ component, the only known negative parity state in the region is the 2.266 MeV level that was assigned negative parity in a $\beta$-delayed neutron study [@Baumann89]. In the subsequent study of intermediate-energy neutron removal [@RussTerry] it was then possible to deduce a spin-parity $(1/2,3/2)^-$. Therefore, the $\ell =3$ strength identified here must correspond to a newly-observed level. From the shell model calculations in Table \[exp-sm-levels\] the best candidate on the basis of excitation energy is the lowest $5/2^-$ level, predicted according to the EEdf1 calculation at 2.914 MeV. As is clear from Figure \[angdis-assembly\](c) the yield in this peak is dominated by the $\ell =3$ state. Hence the peak energy in Figure \[Mg-Ex\] can be interpreted as the excitation energy of the state, which gives $2.40 \pm 0.10$ MeV. The peak at 4.3 MeV is, unfortunately, observable only for a small range of angles as discussed above. Nevertheless, the distributions shown for $\ell =1$ and $\ell =2$ in (b) and (c) show that the corresponding shapes would give poor descriptions of the data if a single $\ell$-value were dominant. The $\ell =3$ and, less plausibly, $\ell =0$ distributions could account for the data. The two states in the shell model that are consistent with this (cf. Fig. \[compare-strengths\]) are both populated via $\ell =3$: the $7/2^-$ at 4.050 MeV and the $5/2^-$ at 4.254 MeV. Of these, as shown in the figure, it is only the $7/2^-$ that is predicted to have a strong population in (d,p). While the shell model is under test here, it is reasonable to associate this strong peak near 4.3 MeV with the second $7/2^-$ level. The excitation energy for this level is determined from the spectrum of Figure \[Mg-Ex\] to be $4.30 \pm 0.10$ MeV and it is natural to associate it also with the 4.28 MeV level reported in multinucleon transfer [@DavidScott; @Fifield85; @Woods88] and listed in the compilation [@NDS29]. This level lies above the neutron separation energy, but the experimental resolution is such that it is not possible to set any useful limits on the natural width. In the ADWA calculation for this state, the form factor was derived by assuming a small positive binding energy (and the inferred spectroscopic factor was not sensitive to the precise value).
The doublet at 1.2 MeV can be examined in more detail. Although the two contributions are unresolved, they are separated by half of the FWHM for an isolated peak, so the distribution of counts within the energy window can be explored for angle-dependent effects. Three angular ranges were chosen, each of width $10^\circ$ to contain reasonable statistics: $100^\circ$-$110^\circ$, $125^\circ$-$135^\circ$ and $160^\circ$-$170^\circ$. According to the best fit displayed in Figure \[angdis-assembly\](b), the state populated with $\ell =3$ should clearly dominate in the first angular range. It should be less dominant in the second angular range, and the $\ell =1$ state should dominate for the third angle (the solid filled spectrum in Figure \[three-angles\]). It is clear, therefore, that the higher-energy state has $\ell =3$ character and the lower-energy state has $\ell =1$. This then gives the first direct measurement of the orbital angular momenta for these two states and confirms the previous tentative assignments of Refs. [@Baumann89; @RussTerry].
![\[compare-strengths\] Comparison of the experimental values of $(2J+1)C^2 S$ and excitation energies from Table \[2j+1c2s\] with shell model values from Table \[exp-sm-levels\] (level associations given in Table \[SF-table\]). Key: (Color online) red $\ell=0$, green $\ell=1$, orange $\ell=2$, blue $\ell=3$.](compare-strengths4.eps){width="0.95\linewidth"}
![\[three-angles\] (Color online) Excitation energy spectra for $^{29}$Mg corresponding to three restricted angular ranges for protons. [*Green cross hatched:*]{} $100^\circ$-$110^\circ$, [*Red cross hatched:*]{} $125^\circ$-$135^\circ$ and [*Blue solid fill:*]{} $160^\circ$-$170^\circ$. The number of counts is not corrected for solid angle, which varies sinusoidally with angle and is weighted approximately as 7:6:2 for the three spectra. ](three-angles.eps){width="\linewidth"}
\[discussion\]Discussion
========================
With the spin-parity assignments proposed in section \[analysis\], the spectroscopic factors can be deduced from the values of $(2J+1)C^2 S$ presented in Table \[2j+1c2s\]. These experimental values of $S$ are compared with those predicted by the shell model in Table \[SF-table\] and the distributions of the strengths $(2J+1)C^2 S$ are compared in Figure \[compare-strengths\]. There is fairly reasonable agreement, which is discussed in more detail below, but also one notable disagreement. The large value for the spectroscopic factor for the $5/2^-$ level at 2.3 MeV is surprising and is hard to reconcile with the shell model expectations. However, as discussed in detail in section \[IVb\], the angular distribution including this level (Fig. \[angdis-assembly\](c)) clearly requires a contribution from $\ell =3$. It may be noted that, of the four peaks discussed here, this is the least strongly populated and potentially there could be unidentified background contributions.
-------------------- ------------ ------------------ --------- -------
E$_{\rm{x}}$ (exp) $J^\pi _n$ $S$ $S$ $S$
(MeV) (exp) (EEdf1) (wbc)
0.000 $3/2^+_1$ $0.30 \pm 0.03 $ 0.35 0.40
0.055 $1/2^+_1$ $0.34 \pm 0.03 $ 0.35 0.40
1.095 $3/2^-_1$ $0.11 \pm 0.01 $ 0.42 0.63
1.431 $7/2^-_1$ $0.38 \pm 0.02 $ 0.43 0.42
2.40\* $5/2^-_1$ $0.30 \pm 0.03 $ 0.02 0.03
2.500 $3/2^+_2$ $0.08 \pm 0.03 $ 0.19 0.25
4.280$\ddag$ $7/2^-_2$ $0.30 \pm 0.05 $ 0.24 0.34
-------------------- ------------ ------------------ --------- -------
: \[SF-table\]Values of the experimentally deduced spectroscopic factors $S$, using the level identifications discussed in section \[analysis\], compared with shell model predictions. The quoted errors in $S$ are statistical. The systematic uncertainties are detailed in Table \[2j+1c2s\]. Excitation energies are from the literature [@NDS29], cf. Table \[exp-sm-levels\], and have experimental uncertainties of $\leq 1$ keV except where indicated (\* = present work $\pm 10$ keV; $\ddag$ = $\pm 40$ keV).
One of the other striking features of the excitation energy spectrum in Figure \[Mg-Ex\] is the absence of any strong population of the 3.090 MeV state that dominated the spectra seen in three-neutron transfer [@DavidScott; @Fifield85]. This state was also populated in the single-neutron transfer (and two-proton pickup) reaction ($^{13}$C,$^{14}$O) [@Woods88]. Its most natural association with a shell model state, as shown in Table \[exp-sm-levels\], is with the second $3/2^-$ state which has a predicted spectroscopic factor of $S \leq 0.01$. On the other hand, the spectroscopic factors for the overlap of this state with excited core configurations are larger. For the $^{28}$Mg($2_1^+$) core (in the [*wbc*]{} calculation) these are 0.09 for $2^+ \otimes \nu(0f_{7/2})$ and 0.57 for $2^+ \otimes \nu(1p_{3/2})$. A structure like this would be consistent with the observed strong population of the state in ($^{18}$O,$^{15}$O) and ($^{13}$C,$^{14}$O) – where the single-neutron transfer could be accompanied by a di-neutron or di-proton transfer with $\ell =2$ – and also with weak or insignificant population via the (d,p) reaction. The experiment appears to support the predicted lack of mixing between the different $3/2^-$ configurations, but the spectroscopic factor deduced here is significantly smaller than the prediction. In contrast to the situation seen with the first two $3/2^-$ states, there appears to be much more mixing between the $0f_{7/2}$ single particle and $2^+ \otimes \nu(1p_{3/2})$ configurations so that the first and second $7/2^-$ states each have significant spectroscopic factors for the (d,p) reaction. The spectroscopic factors for the overlap of these states with the $^{28}$Mg($2_1^+$) excited core (in the [*wbc*]{} calculation) are 0.34 and 0.36 respectively, for $2^+ \otimes \nu(1p_{3/2})$, and rather smaller for $2^+ \otimes \nu(0f_{7/2})$. As such, in both theory and experiment, there is significant single-particle strength in each of these first two $7/2^-$ states.
-------------------- ------------------ ------- ---------------------- ---------------------- ---------------------- ---------------------- ---------
E$_{\rm{x}}$ (exp) $J^\pi _n$ SM $\langle n \rangle $ $\langle n \rangle $ $\langle n \rangle $ $\langle n \rangle $ SM
(MeV) int $\nu {f_{7/2}}$ $\nu {p_{3/2}}$ $\nu {p_{1/2}}$ $\nu {f_{5/2}}$ $S$
0.000 $^{28}$Mg($0^+$) EEdf1 0.36 0.10 0.05 0.10 $^{a)}$
0.000 $^{28}$Mg($0^+$) wbc 0.00 0.00 0.00 0.00 $^{b)}$
1.095 $3/2^-_1$ EEdf1 0.17 0.01 $-0.03~$ 0.42
wbc 0.23 0.03 0.01 0.63
1.431 $7/2^-_1$ EEdf1 0.53 0.22 $-0.02~$ $-0.02~$ 0.43
wbc 0.61 0.36 0.01 0.01 0.42
2.266 $1/2^-_1$ EEdf1 $-0.04~$ 0.56 0.20 $-0.02~$ 0.19
wbc 0.05 0.58 0.32 0.02 0.30
2.40\* $5/2^-_1$ EEdf1 0.43 0.01 $-0.01~$ 0.02
wbc 0.37 0.04 0.04 0.03
$1/2^-_2$ EEdf1 0.25 0.11 0.34 0.02 0.33
wbc 0.22 0.24 0.43 0.04 0.42
3.090 $3/2^-_2$ EEdf1 0.33 0.11 $-0.02~$ 0.01
wbc 0.24 0.08 0.02 0.01
4.280 $7/2^-_2$ EEdf1 0.31 0.38 0.03 $-0.02~$ 0.24
wbc 0.44 0.43 0.04 0.02 0.34
$5/2^-_2$ EEdf1 0.26 $-0.02~$ 0.00
wbc 0.25 0.08 0.02
-------------------- ------------------ ------- ---------------------- ---------------------- ---------------------- ---------------------- ---------
: \[occupancies\]Neutron occupancies of the $fp$-shell orbitals according to shell model predictions. For the EEdf1 calculations, the numbers shown are [*in addition*]{} to the average numbers for the $^{28}$Mg ground state which are shown at the top of the table. The occupancies for [*wbc*]{} add to slightly less than unity because of excitations from the proton $0p$-shell. The underlined values indicate where the two models differ by more than their overall [*rms*]{} variation (see text). The spectroscopic factors to the ground state, $S$, are also included. (\* = present work).
$^{a)}$The EEdf1 calculation includes excitations up to $4\hbar \omega $ for the $^{28}$Mg g.s.\
$^{b)}$The wbc calculation requires $0\hbar \omega $ for the $^{28}$Mg g.s.
The $fp$-shell neutron occupancies predicted in the two shell model calculations are shown in Table \[occupancies\]. In the case of the EEdf1 results the table gives the [*excess occupancy*]{} relative to the $^{28}$Mg ground state since the $^{28}$Mg already includes occupation of the $fp$-shell (excitations up to $6\hbar \omega$ or $7\hbar \omega$ are included for positive and negative parity states, respectively). Thus, there is a “base level” of excitation into the $fp$-shell (in the EEdf1) that is present in the $^{28}$Mg ground state and which is outside of the WBC basis (and therefore subsumed into the effective interaction). For this reason, we look beyond the inevitable differences between the $^{29}$Mg wave functions as calculated in the two models, and instead focus on comparing the orbitals occupied by the additional neutron in $^{29}$Mg (as given in Table \[occupancies\]). This highlights those aspects of the wave functions that are most relevant to (d,p) spectroscopic factors. The two calculations are generally in good agreement, with the average difference between the adjusted EEdf1 results and the [*wbc*]{} being just 0.06 (and the $rms$ difference equal to 0.14). In just five instances the discrepancy exceeds 0.15 and these are underlined in the Table. Intriguingly, all but one of these involve the $\nu$($p_{3/2}$) orbital. Three of the discrepancies concern the two $5/2^-$ wave functions and they reveal differences in the coupling with the excited core, since they occur in the orbitals having a spin different to that of the state. The other two substantial discrepancies concern the two $3/2^-$ states, where the component without any excited core is the source of the disagreement. Interestingly, it is the spectroscopic factors for the $5/2^-$ and $3/2^-$ states that show the largest discrepancy between theory and experiment (cf. Table \[SF-table\]) as well as between the different theoretical predictions. This indicates that further data for the (d,p) reaction, and in particular a clarification of the $J^\pi$ assignment for the 2.40 MeV state (identified here as the lowest $5/2^-$), would be valuable in distinguishing between the quality of different theoretical predictions and thus refining the models.
Finally, we note that the aforegoing discussion makes no attempt to address the reduction, or “quenching’’ of shell model spectroscopic factors that may be expected to arise from effects such as short- and long-range correlations that lie outside the shell model basis [@electron; @LeeTostevin2006]. The method of analysis employed in the present work has been demonstrated [@Tsang2005; @Lee] to reproduce (within an accuracy of 20%) the spectroscopic factors as calculated in conventional large-basis shell model calculations. Thus, this analysis affords a direct comparison of the experimental results with the theory. A modification to incorporate more realistic bound state wave functions [@Kramer2001] – using, for example, a potential geometry for the bound state wavefunction based on the Hartree-Fock matter density [@LeeTostevin2006] – leads to a reduction of around 30% in the spectroscopic factors deduced from the data. These reduced values show no significant dependence on the nucleon binding energy for isotopes of oxygen [@Flavigny2013; @Flavigny2018] and argon [@Lee2010] and are consistent with the values typically deduced from $(e,e^\prime p)$ scattering [@electron]. Recent results from higher-energy quasi-free knockout, viz. two different studies of $(p,2p)$ reactions induced by oxygen isotopes [@Atar2018; @Kawase2018], show similar results. Previous studies of nucleon removal reactions at intermediate energies, in contrast, showed a marked dependence of the quenching factor upon the nucleon binding energy [@JeffAlex2014] that is not apparent for any other reaction. None of these effects change in any significant fashion the conclusions of the present work.
{width="90.00000%"}
\[summary\]Summary and conclusions
==================================
The first quantitative measurements of the single-particle structure of $^{29}$Mg have been obtained using the d($^{28}$Mg,p $\gamma$)$^{29}$Mg reaction. In particular, substantial evidence was found for a previously unknown $5/2^-$ state at $2.40 \pm 0.10$ MeV excitation. Furthermore, considerable $\ell =3$ strength was observed just above the neutron decay threshold in a state at 4.28 MeV that is identified as the second $7/2^-$ level. The present data have also allowed the spins and parities of the two lowest lying intruder states to be confirmed, viz. the $3/2^- _1$ at 1.095 and the $7/2^- _1$ at 1.431 MeV. These results offer new insights into the development of nuclear structure approaching the Island of Inversion surrounding $^{32}$Mg.
As summarised in Figure \[levels\], and also highlighted in Figure \[compare-strengths\], the measurements reveal a marked difference in the spectroscopic strengths associated with the two low-lying negative parity intruder states below 1.5 MeV. This is in contrast to shell model predictions, even though the excitation energies are quite well reproduced. The measurements have also removed the ambiguities that existed in the interpretation of the three-nucleon transfer data [@DavidScott; @Woods88; @Fifield85] and as noted above have located the main part of the remaining intruder strength. As such, the distribution of single-particle strength between the negative parity states appears to be poorly described by the shell model. This is true for both large-basis shell model calculations presented here, despite their very different characteristics. Otherwise, the predicted excitation energies of states and spectroscopic factors for positive parity states are in general in good agreement with experiment.
Whilst the present work has clarified the dominant features of the single-particle structure of $^{29}$Mg, the measurements were compromised by the poor quality and intensity of the $^{28}$Mg radioactive beam. Without the unfortunate factor of 1000 reduction in intensity, the coincident gamma-ray data would have been exploited in the style of Ref. [@wilson-plb]. In particular, with an effective resolution in excitation energy of some 50 keV, the less strongly populated levels in the region 2.4 to 4.0 MeV could be identified and characterised. Moreover, the gamma-ray decay patterns would provide complementary information concerning the spins of the states. As such, further measurements using a higher beam intensity would be very worthwhile.
The authors are grateful to the beam operations team at TRIUMF and in particular F. Ames for exceptional efforts in finding some $^{28}$Mg beam, to B. Alex Brown for assistance with the [*wbc*]{} interaction and the calculations using [*NuShellX*]{} and to Ahmed Blenz for ensuring that shifts ran smoothly. This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), The Canada Foundation for Innovation and the British Columbia Knowledge Development Fund. TRIUMF receives federal funding via a contribution agreement through the National Research Council of Canada. The UK authors acknowledge support from STFC. This work was partially supported by STFC Grants ST/L005743/1, ST/P005314/1, ST/L005727/1 and EP/D060575/1. Partial support from the IN2P3-CNRS Projet Internationale de Coopération Scientifique PACIFIC is acknowledged. The shell model work was supported in part by HPCI Strategic Program (hp150224), in part by MEXT and JICFuS and a priority issue (Elucidation of the fundamental laws and evolution of the universe) to be tackled by using Post “K” Computer (hp160211, hp170230), in part by the HPCI system research project (hp170182), and by CNS- RIKEN joint project for large-scale nuclear structure calculations.
[^1]: Specifically, the transition into the island of inversion was considered to be very clear between $^{30}$Mg and $^{31}$Mg.
[^2]: Given the limited gamma-ray statistics and the significant lifetime of the 1.431 MeV state ($t_{1/2}=1.4 \pm 0.5$ ns [@RussTerry]) a more detailed quantitative analysis was not justified.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In 1887 W. Voigt published a paper that marked the birth of physical symmetries in field theories. He demanded form invariance of the wave equation in inertial frames and obtained a set of spacetime transformations that broke two pillars of Newtonian physics when introducing the non-absolute time: $t''=t-vx/c^2$ and implying the constancy of the speed of light: $c''=c$. In 1905 H. Poincaré showed that the wave equation was also invariant under the Lorentz transformations. Voigt and Lorentz transformations are then closely related, but this relation is not widely known in the standard literature. In this paper we derive the Lorentz transformations from the invariance of the D’Alambert operator $[\Box^{2}=\Box''^{2}]$ and the Voigt transformations from its conformal invariance $[\Box^{2}=(1/\gamma^2)\Box''^{2},$ where $\gamma=(1-v^2/c^2)^{-1/2}$\]. The homogeneous scalar wave equation is then invariant under the Lorentz transformations and conformally invariant under the Voigt transformations. We suggest a pedagogical presentation in which the Voigt transformations are introduced after discussing the Galilean transformations but before presenting the Lorentz transformations.'
author:
- Ricardo Heras
---
Woldemar Voigt published in 1887 the article [@1]: “On Doppler’s Principle" which has unfortunately received little recognition by physicists and historians of physics [@2; @3; @4; @5; @6]. Apparently, he was the first —or at least one of the firsts— who demanded form invariance of a physical law to obtain a set of transformation equations. This remarkable idea began the search for physical symmetries in field theories. More precisely, Voigt demanded form invariance of the homogeneous wave equation in inertial frames and obtained a set of spacetime transformations now known as the Voigt transformations [@7]: $$\begin{aligned}
x'=x -vt,\; t'=t-\frac{vx}{c^2},\; y'=y/\gamma,\; z'=z/\gamma,\end{aligned}$$ where $\gamma=1/\sqrt{1-v^2/c^2}$ is the Lorentz factor. The reader will have noticed that the Voigt transformations are similar to the well-known Lorentz transformations of special relativity: $$x'=\gamma[x -vt],\; t'=\gamma\bigg[t-\!\frac{vx}{c^2}\bigg], \; y'=y, \; z'=z.$$ If the right-hand of the Voigt transformations in Eq. (1) is multiplied by the factor $\gamma$ then the Lorentz transformations in Eq. (2) are obtained. In 1905 Poincaré [@8] demonstrated that the homogeneous wave equation was invariant under the transformations in Eqs. (2). Therefore the Voigt and Lorentz transformations are closely related and it would be worthwhile to discuss this relation from a pedagogical point of view [@9].
Three brief comments enlighten the conceptual and historical importance of Voigt’s 1887 paper: (i) Voigt transformations were obtained by the requirement of keeping the same form of the wave equation under inertial frames, and this is one application of which would be known later as the first postulate of special relativity. (ii) Form invariance of the wave equation carried the invariance of the speed of light, which constitutes the second postulate of special relativity. Remarkably, Voigt assumed the constancy of the speed of light about 18 years before Einstein’s celebrated paper on special relativity [@10]. (iii) As a consequence of the invariance of the speed of light: $c'=c$, the well-established Newtonian notion of absolute time: $t'=t$ should be replaced by the non-absolute time: $t'=t-vx/c^2$. According to Ives [@11] this was the first suggestion that: “...a ‘natural’ clock would alter its rate on motion.” In this same sense Simonyi [@12] claims that when demanding form invariance of the wave equation, Voigt was “...opening the possibility for the first time in the history of physics to call into question the concept of the absolute time.” Voigt’s non-absolute time was independently introduced in 1895 by H. A. Lorentz [@13] who called it “the local time.” Interestingly, in a paper devoted to the Doppler effect, Voigt was inadvertently laying the fundamentals of special relativity nearly two decades before Einstein. As pointed out by Ernst and Hsu [@2]: “He was very close to suggesting a conceptual framework for special relativity.”
The reader might be surprised by the fact that Voigt’s transformations are not mentioned in standard textbooks [@14]. His surprise would be even greater when he could have observed that these transformations imply the same velocity transformation law of the special relativity: $${\rm v}'_{x}=\frac{{\rm v}_{x}-v}{1-v{\rm v}_{x}/c^2},$$ which is fully consistent with Voigt’s assumption about the invariance of the speed of light, that is, if ${\rm v}_{x}=c$ is inserted in Eq. (3) then it yields ${\rm v}'_{x}=c$. In addition, a further application shows that the Doppler effect predicted by the Voigt transformations turns out to be identical to that predicted by special relativity [@4; @5]. The reader might be even more puzzled by the fact that Voigt obtained a set of transformations different from those of Lorentz, despite the fact that he essentially applied the two postulates of special relativity to the homogeneous wave equation.
Unfortunately, the Voigt transformations do not form a group [@3; @4] and this seems to be the main reason for which these transformations have been overlooked in textbooks. Another reason seems to be the absence of a simple and pedagogical approach to derive these transformations. The original derivation presented by Voigt [@1] is certainly not pedagogical nor easy-to-follow. In contrast, there are many pedagogical and easy-to-follow derivations of the Lorentz transformations, some of which are given in Ref. [@15].
In this paper we hope to call attention to the Voigt transformations and add clarity to the close relation between Lorentz and Voigt transformations (a) by deriving the Lorentz transformations from the invariance of the D’Alambert operator $[\Box^{2}=\!\Box'^{2}]$ and the Voigt transformations from its conformal invariance $[\Box^{2}=(1/\gamma^2)\Box'^{2}]$, (b) by pointing out that the homogeneous scalar wave equation is invariant under the Lorentz transformations and conformally invariant under the Voigt transformations, (c) by writing the Voigt transformations in the four-dimensional spacetime and showing that these transformations do not form a group, and (d) by suggesting a pedagogical presentation in which the Voigt transformations are introduced after discussing the Galilean transformations and before presenting the Lorentz transformations. We do not present here a full discussion on the physical or unphysical predictions of Voigt’s transformations. We only point out some of these predictions briefly to emphasize the pedagogical usefulness of introducing these transformations as well as their historical and conceptual importance.
In Sec. II we derive the Lorentz transformations from the invariance $\Box^{2}=\!\Box'^{2}$. In Sec. III we obtain the Voigt transformations from the conformal invariance $\Box^{2}=(1/\gamma^2)\Box'^{2}$. In Sec. IV we discuss the ideas of invariance and conformal invariance in the context of the Lorentz and Voigt transformations. In Sec. V we point out the relation between Lorentz and Voigt transformation in the four-dimensional spacetime and show that the latter do not form a group. In Sec. VI we suggest the pedagogical presentation: Galilean Trans. $\to$ Voigt Trans. $\to$ Lorentz Trans. In Sec. VIII we emphasize the conceptual importance of the Voigt’s 1887 paper. In the Appendix A we derive the Voigt transformations for the space and time derivative operators.
It is well-known that the D’Alambert operator is invariant under the Lorentz transformations. This result naturally suggests that the Lorentz transformations may be obtained from the invariance of the D’Alambert operator. Although this suggestion is correct, its explicit proof does not seem to be widely known in the standard literature. In this section we present such a proof which is inspired in one given by Parker and Schmieg [@16].
In order to introduce notation and for future reference, we will first show that the D’Alambert operator is invariant under the Lorentz transformations. For simplicity, we consider the standard configuration in which two inertial frames $S$ and $S'$ are in relative motion with speed $v$ along their common $xx'$ direction. The origins of the two frames coincide at the instant $t\!=\!t'\!=\!0$. The coordinates transverse to the relative motion of the frames $S$ and $S'$ are assumed to be invariant: $y'=y$ and $z'=z$. The corresponding derivative operators are also assumed to be invariant: $\partial /\partial y=\partial/\partial y'$ and $\partial /\partial z=\partial /\partial z'$. The D’Alambert operator is defined by $\Box^{2}\equiv\nabla^2\!-\!(1/c^2)\partial^2/\partial t^2$ in the frame $S$ and $\Box'^{2}\equiv\nabla'^2-(1/c^2)\partial^2/\partial t'^2$ in the frame $S'$. From the Lorentz transformations $x'=\gamma[x -vt], $ and $t'=\gamma [t-vx/c^2]$, we derive the transformation laws for the derivative operators [@17]: $\partial/\partial x\!= \gamma[\partial/\partial x'-(v/c^2)\partial/\partial t']$ and $ \partial/\partial t= \gamma[\partial/\partial t'-\!v\partial/\partial x']$. It follows that $$\begin{aligned}
\frac{\partial^2}{\partial x^2}&=\gamma^2\bigg[\frac{\partial^2}{\partial x'^2}-\frac{2v}{c^2}\frac{\partial}{\partial x'}\frac{\partial}{\partial t'} +\frac{v^2}{c^4}\frac{\partial^2}{\partial t'^2}\bigg],\nonumber\\
\frac{\partial^2}{\partial t^2}&=\gamma^2\bigg[\frac{\partial^2}{\partial t'^2}-2v\frac{\partial}{\partial t'}\frac{\partial}{\partial x'} +v^2\frac{\partial^2}{\partial x'^2}\bigg].\end{aligned}$$ Using these transformations and $\partial^2 /\partial y^2=\partial^2/\partial y'^2$ and $\partial^2 /\partial z^2=\partial^2 /\partial z'^2$ we obtain $$\begin{aligned}
&\frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} -\frac{1}{c^2}\frac{\partial^2}{\partial t^2}=\nonumber\\ &\gamma^2\bigg[1-\frac{v^2}{c^2}\bigg]\frac{\partial^2}{\partial x'^2}+ \frac{\partial^2}{\partial y'^2} +\frac{\partial^2}{\partial z'^2}
-\frac{1}{c^2}\gamma^2\bigg[1-\frac{v^2}{c^2}\bigg]\frac{\partial^2}{\partial t'^2}.\end{aligned}$$ From the definition of the factor $\gamma$ we have $\gamma^2[1-v^2/c^2]\equiv 1$ and therefore Eq. (5) clearly shows the invariance of the D’Alambert operator: $\Box^{2}=\Box'^{2}$.
We will now travel the inverse route and demand the invariance of the D’Alambertian to obtain the Lorentz transformations. We consider again the standard configuration and assume $\partial /\partial y=\partial/\partial y'$ and $\partial /\partial z=\partial /\partial z'$. The invariance of the D’Alambertian can be expressed as $$\frac{\partial^2}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}+\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}=\frac{\partial^2}{\partial x'^2}-\frac{1}{c^2}\frac{\partial^2}{\partial t'^2}+ \frac{\partial^2}{\partial y'^2} +\! \frac{\partial^2}{\partial z'^2}.$$ A simple mathematical manipulation shows that Eq. (6) can be expressed as $$\begin{aligned}
& \bigg[\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\bigg]\bigg[\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\bigg]+\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}=\nonumber\\&\bigg[\frac{\partial}{\partial x'}-\frac{1}{c}\frac{\partial}{\partial t'}\bigg]\bigg[\!\frac{\partial}{\partial x'}+\frac{1}{c}\frac{\partial}{\partial t'}\bigg]+ \frac{\partial^2}{\partial y'^2} + \frac{\partial^2}{\partial z'^2}.\end{aligned}$$ By assuming linearity for the involved transformations of operators, we can write the quantities $$\begin{aligned}
\bigg[\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\bigg]&=A \bigg[\frac{\partial}{\partial x'}-\frac{1}{c}\frac{\partial}{\partial t'}\bigg],\nonumber\\
\bigg[\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\bigg]&=A^{-1} \bigg[\frac{\partial}{\partial x'}+\frac{1}{c}\frac{\partial}{\partial t'}\bigg],\end{aligned}$$ and $\partial /\partial y=\partial/\partial y'$ and $\partial /\partial z=\partial /\partial z'$. Insertion of these quantities in the left-hand side of Eq. (7) leads to an identity. The factor $A$ is independent of the derivative operators but can depend on the velocity $v$ and $A^{-1}=1/A$. In order to determine $A$, we demand that the expected linear transformation relating primed and unprimed time-derivative operators should appropriately reduce to the corresponding Galilean transformation [@18]: $\partial/\partial t= \partial/\partial t'-v\partial/\partial x'.$ Our demand is consistent with a linear transformation of the general form: $\partial/\partial t= F(v)[\partial/\partial t' -v\partial/\partial x'],$ where $F(v)$ depends on the velocity $v$ so that $F(v)\!\to\! 1$ when $v<<c$. From this general transformation it follows that if $\partial/\partial t=0$ then $\partial/\partial t'=v\partial/\partial x'$ because $F(v)\neq 0$. Using this result in Eq. (8) we obtain $$\begin{aligned}
\frac{\partial}{\partial x}=A \bigg[\frac{\partial}{\partial x'}-\frac{v}{c}\frac{\partial}{\partial x'}\bigg],\;
\frac{\partial}{\partial x}=A^{-1}\bigg[\frac{\partial}{\partial x'}+\frac{v}{c}\frac{\partial}{\partial x'}\bigg].\end{aligned}$$ By combining these equations we can derive the expressions $$A=\frac {[1+v/c]^{1/2}}{[1-v/c]^{1/2}},\;
A^{-1}=\frac {[1-v/c]^{1/2}}{[1+v/c]^{1/2}},$$ which can conveniently be written as $$A=\gamma\bigg[1+\frac{v}{c}\bigg],\; A^{-1}=\gamma\bigg[1-\frac{v}{c}\bigg].$$ Insertion of these quantities into Eq. (8) gives $$\begin{aligned}
&\bigg[\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\bigg]=\gamma\bigg[1+\frac{v}{c}\bigg] \bigg[\frac{\partial}{\partial x'}-\frac{1}{c}\frac{\partial}{\partial t'}\bigg],\nonumber\\
&\bigg[\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\bigg]=\gamma\bigg[1-\frac{v}{c}\bigg] \bigg[\frac{\partial}{\partial x'}+\frac{1}{c}\frac{\partial}{\partial t'}\bigg].\end{aligned}$$ By adding and subtracting these equations, we obtain the transformation laws $$\begin{aligned}
\frac{\partial}{\partial x}= \gamma\bigg[\frac{\partial}{\partial x'}-\frac{v}{c^2}\frac{\partial}{\partial t'}\bigg],\;
\frac{\partial}{\partial t}= \gamma\bigg[\frac{\partial}{\partial t'}-v\frac{\partial}{\partial x'}\bigg],\end{aligned}$$ which must be completed with $\partial /\partial y=\partial/\partial y'$ and $\partial /\partial z=\partial /\partial z'$. These laws are the Lorentz transformations for derivative operators of the standard configuration. We note that the relations in Eq. (13) imply coordinate transformations of the form: $x'=x'(x,t),$ $y'=y,$ $z'=z$ and $t'=t'(x,t).$ To find the explicit form of these transformations we can use Eq. (13) to obtain $$\frac{\partial x'}{\partial x}= \gamma,\;
\frac{\partial x'}{\partial t}= -\gamma v,\; \frac{\partial t'}{\partial t}= \gamma,\;
\frac{\partial t'}{\partial x}=-\frac{\gamma v}{c^2}.$$ The first relation in Eq. (14) implies (I): $x'=\gamma x + f_1(t),$ where $f_1(t)$ can be determined (up to a constant) by deriving (I) with respect to the time $t$ and using the second relation in Eq. (14): $\partial x'/\partial t=\partial f_1(t)/\partial t=-\gamma v.$ This last equality implies (II): $f_1(t)=-\gamma vt+ x_0,$ where $x_0$ is a constant. From (I) and (II) we obtain $$x'=\gamma[ x -vt] + x_0.$$ The third relation in Eq. (14) implies (III): $t'=\gamma t + f_2(x),$ where $f_2(x)$ can be obtained (up to a constant) from deriving (III) with respect to $x$ and using the last relation in Eq. (14): $\partial t'/\partial x=\partial f_2(x)/\partial x=-\gamma v/c^2$. This last equality implies (IV): $f_2(x)= -\gamma v x/c^2 +t_0,$ where $t_0$ is a constant. From (III) and (IV) we obtain $$t'=\gamma \bigg[t-\frac{vx}{c^2}\bigg] + t_0.$$ The origins of the frames $S$ and $S'$ coincide at the time $t\!=\!t'\!=\!0$ and therefore we have $x_0\!=\!0$ and $t_0\!=\!0$. In this way we obtain the Lorentz transformations of the standard configuration: $$\begin{aligned}
x'=\gamma[x -vt],\; t'=\gamma \bigg[t-\frac{vx}{c^2}\bigg],\end{aligned}$$ which must be completed with the remaining transformations: $y'=y$ and $z'=z.$
The Voigt conformal invariance of the D’Alambert operator is defined by the set of transformations that satisfy the relation [@19] $$\Box^{2}=\frac{1}{\gamma^{2}}\Box'^{2}.$$ We adopt again the standard configuration of the inertial reference frames $S$ and $S'$ but now we do not assume that the coordinates transverse to the relative motion of these frames are invariant. As a first step, we will verify that Voigt transformations in Eq. (1) satisfy Eq. (18). In the Appendix A we explicitly show that the transformations in Eq. (1) imply the following transformation laws [@4]: $\partial/\partial x=[\partial/\partial x'- (v/c^2)\partial/\partial t'], \partial/\partial t= [\partial/\partial t'-v\partial/\partial x'], \partial/\partial y=(1/\gamma)\partial/\partial x'$ and $ \partial/\partial z=(1/\gamma)\partial/\partial z'.$ From these relations it follows that $$\begin{aligned}
&\frac{\partial^2}{\partial x^2}=\frac{\partial^2}{\partial x'^2}-\frac{2v}{c^2}\frac{\partial}{\partial x'}\frac{\partial}{\partial t'} +\frac{v^2}{c^4}\frac{\partial^2}{\partial t'^2},\nonumber\\
&\frac{\partial^2}{\partial t^2}=\frac{\partial^2}{\partial t'^2}-2v\frac{\partial}{\partial t'}\frac{\partial}{\partial x'} +v^2\frac{\partial^2}{\partial x'^2},\\
&\frac{\partial^2}{\partial y^2}=\frac{1}{\gamma^2}\frac{\partial^2}{\partial y'^2},\;
\frac{\partial^2}{\partial z^2} =\frac{1}{\gamma^2}\frac{\partial^2}{\partial z'^2}.\end{aligned}$$ Making use of Eqs. (19) and (20) we obtain $$\begin{aligned}
&\frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} -\frac{1}{c^2}\frac{\partial^2}{\partial t^2}=\nonumber\\&\frac{1}{\gamma^2}\!\bigg[\frac{\partial^2}{\partial x'^2}+\frac{\partial^2}{\partial y'^2}\! +\! \frac{\partial^2}{\partial z'^2}
-\frac{1}{c^2}\frac{\partial^2}{\partial t'^2}\bigg].\end{aligned}$$ This equation is the explicit form of Eq. (18).
Now we will demand the covariance of the D’Alambert operator to derive the Voigt transformations. This derivation is similar to that proposed in the previous section to obtain the Lorentz transformations. We consider again the standard configuration and assume Eq. (18), or equivalently, Eq. (21). A simple manipulation shows that Eq. (21) can be expressed as $$\begin{aligned}
&\bigg[\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\!\bigg]\bigg[\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\bigg]+\!\frac{\partial^2}{\partial y^2}+ \frac{\partial^2}{\partial z^2}=\nonumber\\&\frac{1}{\gamma^2}\Bigg(\bigg[\!\frac{\partial}{\partial x'}-\frac{1}{c}\frac{\partial}{\partial t'}\bigg]\bigg[\!\frac{\partial}{\partial x'}+\frac{1}{c}\frac{\partial}{\partial t'}\!\bigg]+ \frac{\partial^2}{\partial y'^2} + \frac{\partial^2}{\partial z'^2}\Bigg).\end{aligned}$$ By assuming linearity for the corresponding transformations of derivative operators, we can write $$\begin{aligned}
&\bigg[\frac{\partial}{\partial x}\!-\!\frac{1}{c}\frac{\partial}{\partial t}\bigg]=\frac{A}{\gamma}\bigg[\frac{\partial}{\partial x'}-\frac{1}{c}\frac{\partial}{\partial t'}\bigg],\nonumber\\&
\bigg[\frac{\partial}{\partial x}\!+\!\frac{1}{c}\frac{\partial}{\partial t}\bigg]=\frac{A^{-1}}{\gamma} \bigg[\frac{\partial}{\partial x'}+\frac{1}{c}\frac{\partial}{\partial t'}\bigg],\nonumber\\&
\frac{\partial}{\partial y}=\frac{1}{\gamma}\frac{\partial}{\partial y'},\;
\frac{\partial}{\partial z}=\frac{1}{\gamma}\frac{\partial}{\partial z'},\end{aligned}$$ where the quantity $A$ is independent of the derivative operators but can depend on the velocity $v$. Following essentially the same argument leading to Eq. (9) we can arrive at the expressions $$\begin{aligned}
\frac{\partial}{\partial x}=\frac{A}{\gamma}\bigg[\frac{\partial}{\partial x'}\!-\!\frac{v}{c}\frac{\partial}{\partial x'}\bigg],\quad
\frac{\partial}{\partial x}=\frac{A^{-1}}{\gamma}\bigg[\frac{\partial}{\partial x'}\!+\!\frac{v}{c}\frac{\partial}{\partial x'}\bigg].\end{aligned}$$ Dividing these relations, we obtain the same equations for $A$ and $A^{-1}$ given in Eq. (11). Therefore, using Eq. (11) in the first two relations displayed in Eq. (23) we obtain $$\begin{aligned}
&\bigg[\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\bigg]=\bigg[1+\frac{v}{c}\bigg] \bigg[\frac{\partial}{\partial x'}-\frac{1}{c}\frac{\partial}{\partial t'}\bigg],\nonumber\\
&\bigg[\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\bigg]=\bigg[1-\frac{v}{c}\bigg] \bigg[\frac{\partial}{\partial x'}+\frac{1}{c}\frac{\partial}{\partial t'}\bigg].\end{aligned}$$ By adding and subtracting these equations, we obtain the corresponding transformation laws connecting unprimed and primed derivative operators, which are added to the transformation laws for $\partial/\partial y$ and $\partial/\partial z$ to obtain [@4] $$\begin{aligned}
\frac{\partial}{\partial x}&=\frac{\partial}{\partial x'}-\!\frac{v}{c^2}\frac{\partial}{\partial t'},\;
\frac{\partial}{\partial t}= \frac{\partial}{\partial t'}-\!v\frac{\partial}{\partial x'},\nonumber\\ \frac{\partial}{\partial y}&=\frac{1}{\gamma}\frac{\partial}{\partial y'},\;
\frac{\partial}{\partial z}=\frac{1}{\gamma}\frac{\partial}{\partial z'}.\end{aligned}$$ These are the Voigt transformations for derivative operators of the standard configuration. A direct manipulation of Eq. (26) yields the corresponding inverse transformations [@4]: $$\begin{aligned}
\frac{\partial}{\partial x'}&=\gamma^2\bigg[\frac{\partial}{\partial x}+\frac{v}{c^2}\frac{\partial}{\partial t}\bigg],\;
\frac{\partial}{\partial t'}= \gamma^2\bigg[\frac{\partial}{\partial t}+v\frac{\partial}{\partial x}\bigg],\nonumber\\ \frac{\partial}{\partial y'}&=\gamma\frac{\partial}{\partial y},\;
\frac{\partial}{\partial z'}=\gamma\frac{\partial}{\partial z}.\end{aligned}$$
The first two relations in Eq. (26) imply coordinate transformations of the form: $x'=x'(x,t)$ and $t'=t'(x,t).$ To find the explicit form of these transformations we can use Eqs. (26) to obtain $$\frac{\partial x'}{\partial x}= 1,\;
\frac{\partial x'}{\partial t}= -v,\; \frac{\partial t'}{\partial t}= 1,\;
\frac{\partial t'}{\partial x}=-\frac{v}{c^2}.$$ From the first relation in Eq. (28) it follows the equation (A): $x'= x + g_1(t),$ where $g_1(t)$ can be determined (up to a constant) by deriving (A) with respect to the time $t$ and using the second relation in Eq. (28): $\partial x'/\partial t=\partial g_1(t)/\partial t=-v.$ This last equality implies (B): $g_1(t)=-vt+ x_0,$ where $x_0$ is a constant. From (A) and (B) we obtain $$x'=x -vt + x_0.$$ The third relation in Eq. (28) implies (C): $t'\!=\!t\! +\! g_2(x),$ where $g_2(x)$ can be obtained (up to a constant) from deriving (C) with respect to $x$ and using the last relation in Eq. (28): $\partial t'/\partial x\!=\!\partial g_2(x)/\partial x\!=\!-v/c^2$. This last equality implies (D): $g_2(x)\!=\! -v x/c^2\!+\! t_0,$ where $t_0$ is a constant. From (C) and (D) we conclude $$t'=t\!-\!\frac{vx}{c^2} + t_0.$$ The origins of the frames $S$ and $S'$ coincide at $t\!=\!t'\!=\!0$. It follows that $x_0\!=\!0$ and $t_0\!=\!0$. In this way we obtain the Voigt transformations for the $x$ and $t$ coordinates of the standard configuration: $$x'=x -vt,\; t'\!=t\!-\!\frac{vx}{c^2}.$$ The transformations for the $y$ and $z$ coordinates are easily derived. From the last two relations in Eq. (26) we obtain $\partial y'/\partial y=1/\gamma$ and $\partial z'/\partial z=1/\gamma$ which in turn imply $y'=y/\gamma + y_0$ and $z'=y/\gamma + z_0$, where $y_0$ and $z_0$ are constants, which are vanished because the origins of the frames $S$ and $S'$ coincide at the time $t=t'=0$. Thus $$y'=y/\gamma, \; z'=z/\gamma.$$ Equations (31) and (32) are the Voigt transformations of the standard configuration. A direct manipulation of Eqs. (31) and (32) yields to the corresponding inverse transformations [@5]: $$\begin{aligned}
x&=\gamma^2[x'+vt'],\; t=\gamma^2\bigg[t'+\frac{vx'}{c^2}\bigg],\nonumber\\
y&=\gamma y',\; z=\gamma z'.\end{aligned}$$ We note that the inverse transformations in Eqs. (33) are obtained from the transformations in Eqs. (31) and (32) by changing the roles of the primed and unprimed variables, reversing the sign of the velocity $v$ and multiplying the right-hand side of Eq. (31) and (32) by the factor $\gamma^2$.
Lorentz and Voigt transformations are useful to illustrate the subtle difference between invariance and conformal invariance. The next examples emphasize this difference.
Using Eq. (2) we can directly show the result $$x'^2+y'^2+z'^2-c^2t'^2= x^2+y^2+z^2-c^2t^2.$$ Evidently, the quantity: $x^2+y^2+z^2-c^2t^2$ preserves its form under the Lorentz transformations. Consider now the wavefront equation: $x^2+y^2+z^2-c^2t^2=0$, which describes a spherical light pulse sent out from the origin of the frame $S$ at $t=0$. From Eq. (34) it follows that $x'^2+y'^2+z'^2-c^2t'^2=0$, which is the wavefront equation associated with a spherical light pulse sent out from the origin of the frame $S'$ at $t'=0$. We conclude that the quantity $x^2+y^2+z^2-c^2t^2$ is invariant under the Lorentz transformations and therefore the light wavefront equation $x^2+y^2+z^2-c^2t^2=0$ is also invariant under the same transformations.
Using Eq. (1) we can directly show the result $$x'^2+y'^2+z'^2-c^2t'^2=\frac{1}{\gamma^2} [x^2+y^2+z^2-c^2t^2].$$ The quantity $x^2+y^2+z^2-c^2t^2$ does not preserve its form under the Voigt transformations because of the presence of the conformal factor $1/\gamma^2$. However, Eq. (35) states that the quantity $x'^2+y'^2+z'^2-c^2t'^2$ in the frame $S'$ is linear and homogeneously connected with the quantity $x^2+y^2+z^2-c^2t^2$ in the frame $S$. Quantities satisfying this kind of connections are generally called “covariant” and equations involving covariant quantities are called “covariant equations.” Voigt’s conformal invariance is then a kind of covariance. Therefore we can say that the quantity $x^2+y^2+z^2-c^2t^2$ is “conformally invariant” or simply “covariant” under the Voigt transformations. If we consider again the light wavefront equation $x^2+y^2+z^2-c^2t^2=0$ in the frame $S$, then Eq. (35) implies that $x'^2+y'^2+z'^2-c^2t'^2=0$ in the frame $S'$ because of $1/\gamma^2\not=0$. Thus the light wavefront equation is conformally invariant or covariant under the Voigt transformations.
We apply now the same order of ideas to the case of the D’Alambert operator. As already noted, this operator is invariant under the Lorentz transformations: $\Box^{2}=\!\Box'^{2}$. Consider now the invariant scalar field $F(\v x,t)$, which satisfies $F(\v x,t)=F'(\v x',t')$. It follows that the quantity $\Box^{2}F$ is Lorentz invariant: $\Box^{2}F=\Box'^{2}F'$. Therefore the homogeneous scalar wave equation is also Lorentz invariant. In short, $$\begin{aligned}
&{\rm If}\; \Box^{2}F=0\; {\rm then}\; \Box'^{2}F'=0\; {\rm because\; of}\nonumber\\
&\Box^{2}= \Box'^{2}\; {\rm and}\; F=F'.\end{aligned}$$
We have pointed out that the D’Alambert operator is conformally invariant or covariant under the Voigt transformations: $\Box^{2}=(1/\gamma^2)\Box'^{2}$. If we consider again the field $F(\v x,t)$ then the quantity $\Box^{2}F$ is Voigt conformally invariant \[or Voigt covariant\]: $\Box^{2}F=(1/\gamma^2)\Box'^{2}F'$. The homogeneous scalar wave equation is then Voigt conformally invariant \[or Voigt covariant\]. In few words: $$\begin{aligned}
&{\rm If}\; \Box^{2}F=0\; {\rm then}\;\Box'^{2}F'=0\; {\rm because\; of}\;\nonumber\\ & \Box^{2}=(1/\gamma^2) \Box'^{2}, F=F'\; {\rm and}\; 1/\gamma^2\not=0.\end{aligned}$$ Notice that the following statement: “if an equation holds in an inertial frame then it must hold in every inertial frame” is supported by the notions of invariance and covariance (or more precisely conformal invariance in our case). This explains why one can obtain two different sets of transformations by demanding the validity of the wave equation in two inertial frames. Let us elaborate on this point. In order to satisfy the demand: $\Box^{2}F=0$ in $S$ and $\Box'^{2}F'=0$ in $S'$, we can simply assume $F=F'$ and $\Box^{2}= \Box'^{2}$ and the latter leads to the Lorentz transformations. However, we can equally assume $F=F'$ and $\Box^{2}=(1/\gamma^2) \Box'^{2}$ and the latter leads to the Voigt transformations. Invariance and covariance (or conformal invariance) are connected with the same idea: “unchange in form” but in different forms. Roughly speaking, covariance implies invariance but the latter does not imply the former.
The close relation between Lorentz and Voigt transformations is easily established using four-dimensional spacetime coordinates. Greek indices $\alpha, \beta, \ldots$ run from 0 to 3; Latin indices $i,j,\ldots$ run from 1 to 3. Points are labeled as $x^{\alpha}=(x^0, x^1,x^2, x^3)=(ct,x,y,z)$ in the frame $S$ and $x'^{\alpha}=(x'^0, x'^1,x'^2, x'^3 )=(ct',x',y',z')$ in the frame $S'$. Summation convention on repeated indices is adopted. As is well-known, the Lorentz transformations in Eq. (2) can be written as $$x'^\alpha=\Lambda_\beta^\alpha x^\beta,$$ where $$\Lambda^\alpha_\beta=
\begin{bmatrix}
\gamma & -v\gamma/c & 0 &\quad 0\;\\
-v\gamma/c & \gamma & 0 &\quad 0\;\\
0 & 0& 1&\quad 0\;\\
0& 0 & 0 & \quad 1\;\\
\end{bmatrix},$$ is the Lorentz matrix. Analogously, the Voigt transformations in Eq. (1) can be written as $$x'^\alpha={\rm V}_\beta^\alpha x^\beta,$$ where $${\rm V}^\alpha_\beta=
\begin{bmatrix}
1 & -v/c & 0 & 0\\
-v/c & 1 & 0 & 0\\
0 & 0& 1/\gamma & 0\\
0& 0 & 0 & 1/\gamma\\
\end{bmatrix},$$ is the Voigt matrix. The relation between the Lorentz and Voigt matrices is given by $$\Lambda^\alpha_\beta=\gamma{\rm V}_\beta^\alpha,$$ that is, the Lorentz matrix is proportional to the Voigt matrix. Using Eq. (42) we can find directly properties of the Voigt transformations from those of the Lorentz transformations.
The Lorentz matrices satisfy the relation $\Lambda^\alpha_\theta\Lambda^\theta_\beta=\delta^\alpha_\beta,$ where $\delta^\alpha_\beta$ is the kronecker delta. This means that $\Lambda^\theta_\beta$ is the inverse of $\Lambda^\alpha_\theta$, which can be denoted as $(\Lambda^{-1})^\theta_\beta$. From $\Lambda^\alpha_\theta\Lambda^\theta_\beta=\delta^\alpha_\beta$ and Eq. (42) we find $${\rm V}^\alpha_\theta [\gamma^2{\rm V}^\theta_\beta]=\delta^\alpha_\beta.$$ Therefore $\gamma^2{\rm V}^\theta_\beta$ can be interpreted as the inverse of ${\rm V}^\alpha_\theta$, which can be denoted as ${\rm (V^{-1}})^\theta_\beta$. In its explicit form, this inverse reads $$({\rm V^{-1}})^\theta_\beta=
\begin{bmatrix}
\gamma^2 & -v\gamma^2/c & 0 &\quad 0\;\\
-v\gamma^2/c & \gamma^2 & 0 &\quad 0\;\\
0 & 0& \gamma&\quad 0\;\\
0& 0 & 0 & \quad \gamma\;\\
\end{bmatrix}.$$ The Lorentz matrices are defined to be those matrices satisfying $\Lambda^\mu_\alpha\eta_{\mu\nu}\Lambda^\nu_\beta=\eta_{\alpha\beta}$, where $$\eta_{\alpha\beta}=
\begin{bmatrix}
1 & 0 & 0 & 0\\
0& -1 & 0 & 0\\
0 & 0& -1 & 0\\
0& 0 & 0 & -1\\
\end{bmatrix}.$$ It follows that the Voigt matrices are defined to be those matrices satisfying $${\rm V}^\mu_\alpha\gamma^2\eta_{\mu\nu}{\rm V}^\nu_\beta=\eta_{\alpha\beta}.$$ Despite the close relation between Lorentz and Voigt matrices, the latter do not form a group [@3; @4]. To see this we can write Eq. (46) in the more compact form: ${\rm V^T}\gamma^2\eta{\rm V}=\eta$, where ${\rm V^T}$ is the transposed matrix of ${\rm V}$ \[notice that ${\rm V^{-1}}=\gamma^2{\rm V}^T]$. We consider the Voigt matrices ${\rm V_1}$ and ${\rm V_2}$ to investigate if their product ${\rm V_1}{\rm V_2}$ is also another Voigt matrix (closure property). We have ${\rm V_1^T}\gamma_1^2\eta{\rm V_1}=\eta$ and ${\rm V_2^T}\gamma_2^2\eta{\rm V_2}=\eta$. Let ${\rm V_3}={\rm V_1}{\rm V_2}.$ Therefore ${\rm V_3^T}\gamma_3^2\eta{\rm V_3}=[{\rm V_1^T}{\rm V_2^T}]\gamma_3^2\eta[{\rm V_1}{\rm V_2}].$ If $\gamma_3=\gamma_1\gamma_2$ then $${\rm V_3^T}\gamma_3^2\eta{\rm V_3}={\rm V_1^T}\gamma_1^2[{\rm V_2^T}\gamma_2^2\eta{\rm V_2}]{\rm V_1}={\rm V_1^T}\gamma_1^2\eta{\rm V_1}=\eta.$$ At first glance, it appears to be that the Voigt matrices satisfy the closure property. But this is not so because the assumption $\gamma_3=\gamma_1\gamma_2$ is incorrect. It can be shown that [@20]: $\gamma_3=\gamma_1\gamma_2[1+v_1v_2/c^2].$ Therefore the Voigt matrices do not satisfy the closure property. Consequently, the Voigt transformations do not form a group since two successive Voigt transformations do not yield another Voigt transformations. This makes them unattractive from a physical point of view because they break the equivalence of the inertial frames. As pointed out by Levi-Leblond [@21]: “The physical equivalence of the inertial frames implies a group structure for the set of all inertial transformations.” However, we should argue in favour of Voigt’s transformations that some of their predictions, like the transformation law for velocities and the formula for the relativistic Doppler effect, are also predictions of the Lorentz transformations [@22].
As above pointed out, the Voigt transformations are not mentioned in textbooks despite the fact that these transformations can be derived from considerations of invariance in the wave equation, being this equation one of the most discussed in textbooks. However, Voigt’s transformations preserving the Galilean transformation $x'=x -vt$ and rejecting the absolute time $t'\not=t$, could be pedagogically used as a previous step before introducing the Lorentz transformations. This means that the conventional presentation of the Lorentz transformations starting with the Galilean transformations could be pedagogical improved by including an intermediate step in which the Voigt transformations are discussed. In a symbolic form, we suggest the following order of a pedagogical presentation $${\rm Galilean \;Trans.\; \rightarrow Voigt\; Trans. \rightarrow Lorentz \;Trans.}$$ After all, the usual justification to introduce the Lorentz transformations is the fail of the Galilean transformations in preserving the form of the wave equation [@23], failure that do not exhibit the Voigt transformations. Furthermore, the implications displayed in Eq. (48) follow the historical development. The instructor can adopt the point of view that the non-invariance of the wave equation in inertial frames was not the historical cause of a conflict between Galilean and Lorentz transformations but between Galilean and Voigt transformations. In a second step he could provide physical arguments why the Voigt transformations must be abandoned in favour of the Lorentz transformations [@24]. More importantly, he could emphasize the transcendental idea, first pointed by Voigt, that the origin of physical symmetries lies in the invariance properties of physical equations.
Some brief reflections could explain the little impact caused by Voigt’s 1887 paper among their contemporaries: (I) The main purpose of Voigt’s paper was not to propose the ambitious idea of a new theory of space and time but simply to study the propagation of oscillating disturbances through an elastic incompressible medium and deduce the formula for the observed Doppler effect. (II) The process of finding a set of transformations that leave invariant the wave equation was not stressed in Voigt’s paper as a new and fundamental idea. The only four words used by Voigt to justify this invariance were [@1]: “as it must be” (or “da ja sein muss” in the original German version). (III) Voigt did not provide any physical interpretation of his non-absolute time: $t'=t-vx/c^2$ nor commented anything about his proposal of the invariant character of the speed of light in inertial frames. Apparently, Voigt did not realized the great conceptual importance of his results. In connection with Voigt’s ideas presented in his 1887 paper, Hsu has pointed [@25]: “If the physicists of the time had been imaginative enough, they might have recognized the potential of these ideas to open up a whole new view of physics.”
In the creation of special relativity, we traditionally find the names of Lorentz, Larmor, Poicaré and Einstein. They appear to be the main actors. Voigt is relegated to being a minor player, in the best of cases. But this tradition is not faithful to the history of physics, since Voigt was the first in applying the two postulates of special relativity. He deserves a place in textbooks. The idea of demanding that the wave equation should not change its form when observed by different inertial frames, was the great conceptual contribution of Voigt, since it opened the gate to the world of physical symmetries. This is the legacy of Voigt’s 1887 paper.
I wish to express my gratitude to J. A. Heras for his pedagogical suggestions. I am grateful to M. Foley and T. Zaldivar for educational financial support.
Consider a function $F(x',y',z',t')$ in the frame $S'$. The total differential of $F$ reads $$dF=\frac{\partial F}{\partial x'}dx'+\frac{\partial F}{\partial y'}dy'+\frac{\partial F}{\partial z'}dz'+\frac{\partial F}{\partial t'}dt'.$$ The coordinates $x',y',z'$ and $t'$ are all functions of the coordinates $x,y,z$ and $t$ of the frame $S$. The total differentials of $x'$ and $t'$ read $$\begin{aligned}
dx'&=\frac{\partial x'}{\partial x}dx+\frac{\partial x'}{\partial y}dy+\frac{\partial x'}{\partial z}dz+\frac{\partial x'}{\partial t}dt,\\
dt'&=\frac{\partial t'}{\partial x}dx+\frac{\partial t'}{\partial y}dy+\frac{\partial t'}{\partial z}dz+\frac{\partial t'}{\partial t}dt.\end{aligned}$$ From the Voigt transformations in Eq. (1) we have $$\begin{aligned}
&\frac{\partial x'}{\partial x}= 1,\;\frac{\partial x'}{\partial y}= 0,\; \frac{\partial x'}{\partial z}= 0,\;\frac{\partial x'}{\partial t}=-v,\;
\frac{\partial t'}{\partial x}= -\frac{v}{c^2},\nonumber\\
&\frac{\partial t'}{\partial y}= 0,\; \frac{\partial t'}{\partial z}= 0,\;
\frac{\partial t'}{\partial t}=1.\end{aligned}$$ From Eqs. (A2)-(A4) we obtain $$\begin{aligned}
dx'=dx-vdt,\; dt'=-\frac{v}{c^2}dx+dt.\end{aligned}$$ By a similar procedure we can derive $$\begin{aligned}
dy'=dy/\gamma,\; dz'=dz/\gamma.\end{aligned}$$ Using Eqs. (A5) and (A6) into Eq. (A1) and rearranging terms we have $$\begin{aligned}
dF=&\bigg[\frac{\partial F}{\partial x'}-\!\frac{v}{c^2}\frac{\partial F}{\partial t'}\bigg]dx+\frac{1}{\gamma}\frac{\partial F}{\partial y'}dy\nonumber\\&+\frac{1}{\gamma}\frac{\partial F}{\partial z'}dz+\bigg[ \frac{\partial F}{\partial t'}-\!v\frac{\partial F}{\partial x'}\bigg]dt.\end{aligned}$$ If now we consider $F$ as a function of the coordinates $x,y ,z$ and $t$ of the frame $S$ then the total differential can be written as $$dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}dz+\frac{\partial F}{\partial t}dt.$$ Comparing the coefficients of $dx,dy,dz$ and $dt$ in Eqs. (A7) and (A8) we conclude that $$\begin{aligned}
&\frac{\partial F}{\partial x}=\frac{\partial F}{\partial x'}-\!\frac{v}{c^2}\frac{\partial F}{\partial t'}, \; \frac{\partial F}{\partial t}= \frac{\partial F}{\partial t'}-\!v\frac{\partial F}{\partial x'},\nonumber\\& \frac{\partial F}{\partial y}=\frac{1}{\gamma}\frac{\partial F}{\partial y'},\;
\frac{\partial F}{\partial z}=\frac{1}{\gamma}\frac{\partial F}{\partial z'}.\end{aligned}$$ If we drop $F$ in this equation then we obtain the Voigt transformations for the derivative operators. Notice that we have derived these transformations two times. In the first derivation we demanded conformal invariance of the D’Alambertian and obtained Eq. (26). In the second one, we directly use Eq. (1) to derive Eq. (A9).
[99]{}
W. Voigt, “Über das Doppler’sche Princip," Nachr. Ges. Wiss. Göttingen, **8**, 41-51 (1887). An English version of this paper can be found in Ref. 2. There is also another translation in the website: [](http://en.wikisource.org/wiki/Translation:On_the_Principle_of_Doppler) [](http://en.wikisource.org/wiki/Translation:On_the_Principle_of_Doppler).
A. Ernst, J.-P Hsu, “First Proposal of the Universal Speed of Light by Voigt in 1887," Chinese J. Phys. **39**, 211-230 (2001).
A. G. Gluckman, “Coordinate Transformations of W. Voigt and the Principle of Special Relativity," Am. J. Phys. **36**, 226-231 (1968).
A. G. Gluckman, “Voigt Kinematics and Electrodynamic Consequences," Found. Phys. **6**, 305-316 (1976).
C. J. Masreliez, “Special relativity and inertia in curved spacetime," Adv. Studies Theor. Phys. **2**, 795-815 (2008).
J. P. Wesley, “Michelson-Morley result, a Voigt-Doppler effect in absolute space-time," Found. Phys. **16**, 817-824 (1986).
In stating the relations in Eq. (1) we are using the standard configuration defined by two inertial frames $S$ and $S'$ in relative motion with speed $v$ along their common $xx'$ direction and whose origins coincide at the instant $t\!=\!t'\!=\!0$.
H. Poincaré, “Sur la Dynamique de l’Électron," Compt. Rend. Acad. Sci. Paris **140**, 1504-1508 (1905) and Rend. Circ. Mat. Palermo **21**, pp. 129-175 (1906). Excerpts of this paper in English appear in [*The Genesis of General Relativity Boston Studies in the Philosophy of Science Vol. 3: Gravitation in the Twilight of Classical Physics: The Promise of Mathematics*]{}. Eds. J. Renn and M. Schemmel (Springer-Verlag, Netherlands, 2007), pp. 253-251. See the website: [](https://www.univ-nancy2.fr/poincare/bhp/pdf/hp2007gg.pdf).
Most discussions on Voigt’s transformations appearing in the literature have been essentially motived by historical considerations. See, e.g., the recent Arxiv contribution: W. Engelhardt, “On the Origin of the Lorentz Transformation,” in arXiv:1303.5309. Unfortunately, disussions on pedagogical aspects of Voigt’s transformations have not been still reported as far as the author is aware.
A. Einstein, “Zur Elektrodynamik bewegter Körper," Ann. Phys. **322**, 891-921 (1905). The English translation: “On the Electrodynamics of Moving Bodies" can be found in [*The Principle of Relativity*]{} (Methuen, 1923, reprinted by Dover Publications, New York, 1952), pp. 35-65. See also the website: [](http://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)) [](http://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)).
H. E. Ives, “Historical Note on the Rate of a Moving Clock", J. Opt. Soc. Am. **37**, 810-813 (1947).
K. Simonyi, [*A Cultural History of Physics*]{} (CRC, Taylor and Francis, Boca Raton, Fl, 2012), p 405.
H. A. Lorentz, *Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern,* E. J. Brill, Leiden 1895. An English translation can be found in the following website: [](http://en.wikisource.org/wiki/Translation:Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies) [](http://en.wikisource.org/wiki/Translation:Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies).
After an extensive bibliographyc reaserch, the author has not been enable to find a current textbook that discusses the Voigt transformations. However, he would be interested to learn of any textbook presenting these transformations.
See, e.g., L. Parker and G. M. Schmieg, “Special Relativity and Diagonal Transformations,” Am. J. Phys. [**38**]{}, 218-222 (1970); J. M. Levy-Leblond, “One more derivation of the Lorentz Transformation,” Am. J. Phys. [**44**]{}, 271-277 (1975); A. Macdonald, “Derivation of the Lorentz transformation,” Am. J. Phys. [**49**]{}, 493 (1981); and J. M. Levy-Leblond, “A simple derivation of the Lorentz transformation and of the accompanying velocity and acceleration changes,” Am. J. Phys. [**75**]{}, 615-618 (2007).
See Parker and Schmieg in Ref. 15.
See, e.g., W.G.V. Rosser, [*Interpretation of Classical Electromagnetism*]{} (Kluwer, Dordrecht, 1997), p. 378.
See, e.g., R. K. Wagness, [*Electromagnetic fields*]{} 2nd ed. (Wiley, New York, 1986), p. 496.
The conformal group is generally defined as the group of transformations that leave the metric invariant upto a scale: $g'_{\mu\nu}(x')= \lambda(x)g_{\mu\nu}(x)$,where we have used the conventional four-dimensional notation. For the case of the Voigt transformations we have: $\eta'_{\mu\nu}= (1/\gamma^2)\eta_{\mu\nu}(x)$.
See, e.g., A. Zangwill, [*Modern Electrodynamics*]{} (Cambridge University Press, NY, 2013), p. 829.
See J. M. Levi-Leblond 1975 paper in Ref. 15. The authors of Ref. 2 have exhibited (but not discussed) a set of transformations which are, according to these authors, implied by the Voigt transformations and form a four-dimensional conformal group with two parameters.
See, e.g., R. K. Wagness in Ref. 18, pp. 496 and 500; and A. Zangwill in Ref. 22, pp. 824 and 829. The instructor could argue that the lack of a structure group for the Voigt transformations is a physical reason enough to abandon these transformations in favour of the Lorentz transformations, which form a group.
J. P. Hsu, [*Einstein’s Relativity and Beyond: New Symmetry Approaches.*]{} Adavanced Series on Theoretical Physical Science, Vol. 7 (World Scientific, Signapore, 2000), p. 31.
|
{
"pile_set_name": "ArXiv"
}
|
---
address:
- |
Courant Institute of Mathematical Sciences\
251 Mercer Street\
New York, NY 10012
- |
Department of Mathematics\
Johns Hopkins University\
3400 N. Charles St.\
Baltimore, MD 21218
author:
- 'Tobias H. Colding'
- 'William P. Minicozzi II'
title: Sharp estimates for mean curvature flow of graphs
---
[^1]
Introduction
============
A one-parameter family of smooth hypersurfaces $\{ M_t \} \subset
\RR^{n+1}$ [*flows by mean curvature*]{} if $$z_t = {\bf{H}} (z) = \Delta_{M_t} z \, ,$$ where $z$ are coordinates on $\RR^{n+1}$ and ${\bf{H}} = - H {{\bf{n}}}$ is the mean curvature vector.
In this note, we prove sharp gradient and area estimates for graphs flowing by mean curvature. Thus, each $M_t$ is assumed to be the graph of a function $u(\cdot , t)$. So, if $z = (x,y)$ with $x \in \RR^n$, then $M_t$ is given by $y = u (x,t)$. Below, $du$ is the $\RR^n$ gradient of a function $u$, $\|u\|_{\infty}$ is the sup norm, and $B_{s}$ is the ball in $\RR^n$ with radius $s$ centered at the origin.
Our gradient estimate is the following (see Section \[s:area\] for the sharp area estimate):
\[t:gbp\] There exists $C= C(n)$ so if the graph of $u: B_{\sqrt{2n+1} r}
\times [0,r^2] \to \RR$ flows by mean curvature, then $$\label{e:gbp}
\log |du|(0,r^2/[4n]) \leq C \, ( 1 + r^{-1} \,
\|u(\cdot , 0) \|_{\infty} )^2 \, .$$
The quadratic dependence on $\|u(\cdot , 0) \|_{\infty}$ in [(\[e:gbp\])]{} should be compared with the linear dependence which holds when the graph of $u$ is minimal (i.e., $u_t = 0$). In the minimal case, Bombieri, De Giorgi, and Miranda proved in [@BDM] that $$\log |du|(0) \leq C \, ( 1 + r^{-1} \,
\|u \|_{\infty} )$$ (the case of surfaces was done by Finn in [@F1]). By an earlier example of Finn, this exponential dependence cannot be improved even in the minimal case (see [@F2] and cf. [@GiTr]).
In [@K], Korevaar gave a maximum principle proof of a weaker form of [@BDM]; this weaker form had $\|u\|^2_{\infty}$ in place of $\|u\|_{\infty}$. Ecker and Huisken adapted Korevaar’s argument to mean curvature flow in theorem 2.3 of [@EH2] to get $$\label{e:eh2}
\log |du|(0,r^2/[4n]) \leq 1/2 \, \log
\left( 1 + \|du (\cdot,0)\|^2_{\infty} \right)
+ C \, ( 1 + r^{-1} \, \|u(\cdot , 0) \|_{\infty} )^2 \, .$$ Note that, unlike [(\[e:gbp\])]{}, the gradient bound [(\[e:eh2\])]{} depends also on the initial bound for the gradient.
Using the so-called grim reaper, one can see that the quadratic dependence on $\| u (\cdot , 0) \|_{\infty}$ in Theorem \[t:gbp\] is sharp (see Proposition \[p:grr\] below). The [*grim reaper*]{} is the translating solution to the mean curvature flow given by that for each $t$ it is a graph of the function $$\label{e:gr}
u(x,t) = t - \log \sin x \, ,$$ where $x \in (0 , \pi )$ and $t \in [0,\infty)$. Note that $-u(x,t)$ is a downward translating solution. More generally, a parabolic rescaling by $\lambda>0$ gives that the graph of $$u^{\lambda}(x,t)=\frac{1}{\lambda}u(\lambda x ,\lambda^2 t) =
\lambda \, t - \frac{\log \sin (\lambda x )}{\lambda} \, ,$$ where $x \in (0 , \pi / \lambda )$, is a translating solution flowing with speed $\lambda$; see figure \[f:f1\]. Since $\lim_{x\to 0} \frac{\sin x}{x} = 1$, an easy calculation shows that for $\lambda > 0$ sufficiently large $$\label{e:ulam}
u^{\lambda} ( {{\text {e}}}^{-\lambda^2} , 1) = \lambda
- \frac{\log \sin (\lambda {{\text {e}}}^{-\lambda^2} )}{\lambda} \leq 2
\lambda \, .$$
\[p:grr\] Given $\lambda > 1$ sufficiently large, there is a solution $w(x,t)$ on $\RR \times [0,\infty)$ of the mean curvature flow with $$\begin{aligned}
\label{e:grr1}
3 \lambda &< \|w (\cdot , 0 ) \|_{\infty} \leq 4 \lambda \, , \\
\lambda {{\text {e}}}^{\lambda^2} & \leq \max_{ |x| \leq {{\text {e}}}^{-\lambda^2} } |d w (x,1) | \, .
\label{e:grr2}\end{aligned}$$
Define solutions $u^{+} (x,t)= u^{\lambda}(x + \pi/\lambda ,t) - 3
\lambda$ for $-\pi/\lambda < x < 0$ and $u^{-}(x,t) = -u^{\lambda}(x,t) + 3\lambda$ for $0<x<\pi/\lambda$ of the mean curvature flow to be used as barriers. Since $u^{+} \geq -3\lambda $ and $u^{-} \leq
3\lambda$, it is easy to choose (see figure \[f:f2\]) a smooth compactly supported function $w(\cdot , 0): \RR \to \RR$ satisfying [(\[e:grr1\])]{} and so $$\begin{aligned}
w(x,0) &< u^{+} (x,0) {\text{ for }} -\pi/\lambda < x < 0
\, , \label{e:up} \\
u^{-}(x,0) &< w (x,0) {\text{ for }} 0 < x < \pi/\lambda
\, . \label{e:down}\end{aligned}$$ (We can choose $w(\cdot , 0)$ so that $w(x,0) = 0$ for $|x|
> \pi / \lambda$.) The existence results of [@EH1] or [@EH2] (see, e.g., theorem 1.7 in [@E]) extend $w(x,0)$ to a solution $w(x,t)$ of the mean curvature flow defined for $x
\in \RR$ and $t \in [0, \infty)$; see figure \[f:f3\]. Moreover, the maximum principle extends [(\[e:up\])]{} and [(\[e:down\])]{} to all $t \geq 0$. In particular, using this at $x = \pm {{\text {e}}}^{-\lambda^2}$, $t=1$ and substituting [(\[e:ulam\])]{}, we get that $$\begin{aligned}
\label{e:punchline1}
w(-{{\text {e}}}^{-\lambda^2} , 1) &< u^{+} (-{{\text {e}}}^{-\lambda^2} , 1) \leq - \lambda \, , \\
\lambda
&\leq u^{-} ({{\text {e}}}^{-\lambda^2} , 1) < w({{\text {e}}}^{-\lambda^2} , 1) \,
. \label{e:punchline2}\end{aligned}$$ Finally, combining [(\[e:punchline1\])]{}, [(\[e:punchline2\])]{}, and the mean value theorem gives [(\[e:grr2\])]{}; see figure \[f:f3\].
Throughout, $\nabla$, $\Delta$, and ${{\bf{n}}}$ are the induced covariant derivative, laplacian, and unit normal on the submanifold $M_t$ of $\RR^{n+1}$. The graph of a function $u$ flows by mean curvature if $$\label{e:evq}
u_t = (1 + |du|^2)^{1/2} \,
{{\text {div}}}\left( \frac{du}{(1 + |du|^2)^{1/2} } \right)
\, ,$$ where ${{\text {div}}}$ is divergence in $\RR^n$.
Interior gradient estimate for graphs
=====================================
Theorem \[t:gbp\] will follow immediately from the next proposition and a standard maximum principle bounding $u$ at future times in terms of the initial bound, see Lemma \[l:sphere\].
\[p:gbp\] If the graph of $u: B_r \times [0,r^2] \to \RR$ flows by mean curvature, then $$\label{e:gbp2}
\log \left( 1 + |du|^2 (0,r^2/[4n]) \right)
\leq 2 \log 10
+ 16 \, n \, (1 + 2 \, r^{-1} \, \|u \|_{\infty} )^2 \, .$$
The strategy of the proof of Proposition \[p:gbp\] is as follows. By the maximum principle (Lemma \[l:mp2\]) at the maximum of $\phi\,v$ (where $\phi$ is a cutoff function and $v=
\left( 1 + |du|^2 \right)^{1/2}$) the heat operator of the cutoff function is nonnegative. By choosing an appropriate cutoff function in terms of $u$ (Lemma \[l:cop\]), we can bound the heat operator of the cutoff from above in terms of the gradient of $u$. Playing off this lower and upper bound at the max against each other gives the proposition.
\[l:mp2\] Suppose $( \partial_t - \Delta ) v \leq - 2 |\nabla v|^2 / v$ for a function $v \geq 0$ on $\{ M_t \}_{t \in [0,1]}$. If the function $\phi$ is $\leq 0$ on $M_0 \cup_t (\partial M_t)$ but $\max \phi v > 0$, then $(\partial_t - \Delta ) \phi \geq 0$ at the maximum of $\phi v$.
At the maximum of $\phi v$, we get $$\begin{aligned}
\nabla (\phi v) &= v \nabla \phi + \phi \nabla v = 0 \, , \label{e:mp1} \\
(\partial_t - \Delta) (\phi v) &= v \partial_t \phi +
\phi \partial_t v -
v \Delta \phi - 2 \langle \nabla \phi ,
\nabla v \rangle - \phi \Delta v \geq 0 \, . \label{e:mp2}\end{aligned}$$ Substituting [(\[e:mp1\])]{} into [(\[e:mp2\])]{} and using $( \partial_t
- \Delta ) v \leq - 2 |\nabla v|^2 / v$ gives $(\partial_t -
\Delta ) \phi \geq 0$.
We will apply Lemma \[l:mp2\] to the volume element $v = (1 +
|du|^2)^{1/2}$. We will first need some elementary formulas. If the graph of $u$ flows by mean curvature and $a \in \RR$, then $$( \partial_t - \Delta ) \,
{{\text {e}}}^{ay^2/t} = - \frac{{{\text {e}}}^{ay^2/t}}{t^2} \,
\left[ a y^2 + 4 a^2 y^2 |\nabla y|^2 + 2a t
|\nabla y|^2 \right] \, , \label{e:y2}$$ and (see, e.g., lemma 1.1 in [@EH2] or 2.11 in [@E]) $$\begin{aligned}
( \partial_t - \Delta ) v = -|A|^2 \, v -
2 \frac{|\nabla v|^2}{v} &\leq - 2 \frac{|\nabla v|^2}{v} \,
, \label{e:jac} \\
( \partial_t - \Delta ) (1 -|x|^2-2nt) &\leq 0 \, . \label{e:dx2}\end{aligned}$$
The next lemma introduces the cutoff $\phi$ which will be used in Lemma \[l:mp2\].
\[l:cop\] Set $\phi = \eta \, {{\text {e}}}^{ay^2/t}$ for $a \in \RR$ and $\eta =
(1 -|x|^2-2nt)$. If the graph of $u:B_1 \times [0,1] \to \RR$ flows by mean curvature, then $$\label{e:co1p}
(\partial_t - \Delta ) \phi \leq -
\frac{ {{\text {e}}}^{ay^2/t} }{t^2} \, \left[ a y^2 \eta + (4 a^2 y^2 + 2at)
\eta \frac{|du|^2}{1 + |du|^2}
- 8 |ay| \, t \frac{|du|}{1 + |du|^2} \right] \,
\, .$$
Using [(\[e:y2\])]{} and [(\[e:dx2\])]{} gives $$\begin{aligned}
\label{e:co2p}
(\partial_t - \Delta) \phi &= \eta (\partial_t - \Delta)
\, {{\text {e}}}^{ay^2/t} + {{\text {e}}}^{ay^2/t}
(\partial_t - \Delta) \eta - 4 {{\text {e}}}^{ay^2/t} \frac{a y}{t}
\langle \nabla \eta ,
\nabla y \rangle \notag \\
& \leq - \frac{{{\text {e}}}^{ay^2/t} }{t^2} \left[ (a y^2 + 4 a^2 y^2
|\nabla y|^2 + 2at |\nabla y|^2) \eta - 4 ay \, t \langle \nabla
y , \nabla |x|^2 \rangle \right] \, .\end{aligned}$$ The lemma follows since $|x| \leq 1$ and the $y$ component of the normal is $(1 + |du|^2)^{-1/2}$.
(of Proposition \[p:gbp\].) By scaling, it suffices to prove the proposition when $r=1$. Set $\eta = (1-|x|^2 - 2nt)$ and $\phi =
\eta {{\text {e}}}^{a y^2/t}$ for $a \leq -2$ to be chosen. After replacing $u$ by $u + \|u\|_{\infty} + 1$ (i.e., translating), we can assume that $u \geq 1$; in particular, $\phi$ vanishes when $t=0$. If the maximum of $\phi v$ for $t\in [0,1]$ is at $(x_0 , y_0 ,t_0)
\in B_1 \times \RR \times (0,1]$, then [(\[e:jac\])]{} together with Lemmas \[l:mp2\] and \[l:cop\] give $$\label{e:q}
a y_0^2 \eta + (4 a^2 y_0^2 +
2at_0) \eta \frac{|du|^2(x_0,t_0) }{1 + |du|^2 (x_0,t_0)}
- 8 |ay_0| \, t_0 \frac{|du|(x_0,t_0)}{1 + |du|^2(x_0,t_0) } \leq 0 \, .$$ There are now two cases. Namely, either $$\label{e:case1}
|ay_0| \eta
|du| (x_0 , t_0) < 8 \, ,$$ or $|ay_0| \eta |du| (x_0 , t_0) \geq 8$; in the second case, [(\[e:q\])]{} (and $4 a^2 y_0^2 + 2at_0 > 2 a^2 y_0^2$) yields $$\label{e:case2}
a y_0^2 \eta + a^2 y_0^2 \eta \frac{|du|^2(x_0,t_0) }{1 + |du|^2 (x_0,t_0)}
\leq 0 \, .$$ Since $\eta \leq 1$, we get in either case that $$\label{e:cases}
\eta |du| (x_0 , t_0) \leq 4 \, ,$$ Since $\max_{[0,1]} \, (\phi v) = \phi v(x_0, y_0 , t_0)$ and $a < 0$, we get $$\label{e:gb3p}
\phi v = \eta \, {{\text {e}}}^{a \, y^2/t} \, \left( 1 + |du|^2 \right)^{1/2}
\leq 5
\, .$$
A standard barrier argument using shrinking spheres bounds the future height by the initial height:
\[l:sphere\] If $\rho \geq (2n+1)^{1/2}$ and the graph of $u: B_{ \rho r }
\times [0,r^2] \to \RR$ flows by mean curvature, then $$\begin{aligned}
\label{e:sphere}
\max_{ B_r \times [0,r^2]} |u(x,t)|
&\leq r \, \left[ \rho - \left( \rho^2 - (2n+1)
\right)^{1/2} \right] + \max_{B_{ \rho r } } |u (x,0)| \notag \\
&\leq \frac{(2n+1) \, r}{ \rho} + \max_{B_{ \rho r } } |u (x,0)| \, .\end{aligned}$$
By scaling, it suffices to prove the lemma when $r=1$. Recall that the one-parameter family $M_t$ of concentric spheres in $\RR^{n+1}$ of radius $( \rho^2 - 2nt )^{1/2}$ centered at $x=0$, $y= \rho + \max_{B_{ \rho } } u (x,0) + \epsilon$ is a solution to mean curvature flow. For $\epsilon > 0$, $M_0$ does not intersect the graph of $u(\cdot , 0)$. Applying the maximum principle and letting $\epsilon \to 0$, we get that $$\max_{ B_1 \times [0,1]} u(x,t) \leq
\max_{B_{ \rho } } u (x,0) + \rho - \left( \rho^2 - (2n+1)
\right)^{1/2} \, .$$ This, and a similar argument for the minimum of $u$, gives [(\[e:sphere\])]{}.
Area estimates for graphs flowing by mean curvature {#s:area}
===================================================
In this section, we prove an area bound for graphs flowing by mean curvature which depends quadratically on the $L^{\infty}$ norm of the initial height (integrating our gradient estimate gives an exponential bound). We also give an example showing that this is sharp.
\[t:areabound\] There exists $C= C(n)$ so if the graph of $u: B_{\sqrt{2n+1} r}
\times [0,r^2] \to \RR$ flows by mean curvature, then $$\label{e:abp}
{{\text {Area}}}( u (B_{r/2} , r^2) )=\int_{ B_{r/2} }(1+|du|^2)^{1/2} (x,r^2) \, dx
\leq C \,r^{n}\, \left( 1 + r^{-1} \, \|u(\cdot,0)\|_{\infty}
\right)^2 \, .$$
Before proving Theorem \[t:areabound\], we first argue as in Proposition \[p:grr\] to see that the quadratic dependence on $\|u(\cdot,0)\|_{\infty}$ is sharp:
\[p:grr2\] Given an integer $k > 1$, there is a solution $w(x,t)$ on $\RR
\times [0,\infty)$ of the mean curvature flow with $$\begin{aligned}
\label{e:grr4}
2 k &< \|w (\cdot , 0 ) \|_{\infty} \leq 3 k \, , \\
4 k^2 - 2k & \leq \int_{-\pi}^{\pi} \left( 1 + |d w (x,1) |^2 \right)^{1/2} \, dx \, .
\label{e:grr5}\end{aligned}$$
For $-k \leq j \leq k$, define translating solutions $u_j$ on $j \pi/k < x < (j+1) \pi / k$ by $$\label{e:ujxt}
u_j (x,t)= (-1)^j \, \left[ u^{k}(x - j\pi / k ,t) - 2 \, k \right] \,
,$$ where $u^k$ is the scaled grim reaper. The solutions given by [(\[e:ujxt\])]{}, which alternate between translating up and down, will be used as barriers; see figure \[f:f4\]. As in the proof of Proposition \[p:grr\], we can choose a compactly supported function $w(\cdot,0) : \RR \to \RR$ satisfying [(\[e:grr4\])]{} which is below the upward translating solutions and above the downward translating solutions. Combining the existence results of [@EH1] or [@EH2] with the maximum principle as before gives a solution $w(x,t)$ with $$\begin{aligned}
w( (j+1/2) \pi / k , 1) < u_j ( (j+1/2) \pi / k , 1) = - k &{\text{ for }} j {\text{ even}}, \\
k = u_j ( (j+1/2) \pi / k , 1) < w( (j+1/2) \pi / k , 1)
&{\text{ for }} j {\text{ odd}}.\end{aligned}$$ The lower bound on length in [(\[e:grr5\])]{} follows immediately.
We will prove Theorem \[t:areabound\] by showing that the (weighted) area of the graph satisfies a differential inequality which will imply the desired bound (see Lemma \[l:di\]).
We begin with an elementary area bound for the graph of a general function $w$:
\[l:graph\] If $w$, $\phi : \RR^n \to \RR$ are functions and $\phi$ has compact support, then $$\label{e:a2}
\int \phi^2 (1 + |dw|^2)^{1/2} \, dx \leq
\int \phi^2 \, dx + \|w\|_{\infty} \int |d\phi^2| \, dx
+ \|w\|_{\infty} \, \int \phi^2 \, |H| \, dx \, ,$$ where $H = - {{\text {div}}}\left( \frac{dw}{(1 + |dw|^2)^{1/2} } \right)$ is the mean curvature of the graph of $w$.
Applying Stokes theorem to ${{\text {div}}}\left( \frac{ \phi^2 w dw}{(1 + |dw|^2)^{1/2}}
\right)$ gives $$\label{e:a1}
\int \phi^2 \frac{ |dw|^2 }{(1 + |dw|^2)^{1/2}} \, dx
\leq \int |d\phi^2| \, \frac{ |w| \, |dw| }
{(1 + |dw|^2)^{1/2}} \, dx
+ \int \phi^2 |w| \, |H| \, dx \, .$$ Adding $\int \phi^2 \, dx$ to each side gives [(\[e:a2\])]{}.
When the graph of $w$ is minimal (i.e., $H=0$), Lemma \[l:graph\] gives the well-known area bound $C \, r^n \, ( 1 + r^{-1} \, \| w \|_{\infty})$. This linear dependence on $\| w \|_{\infty}$ is easily seen to be sharp.
\[l:di\] If $f (t) \geq 0$ and $f^2 \leq - a \, f' + b$ with $a , b
>0$, then $f(T) \leq \sqrt{2b} + 2a / T$.
If $f^2 (T) < 2b$, then we are done. If $ f^2 (t) \geq 2b$, then $f^2(t) \leq - 2 a \, f'(t) $ so $$\left( 1/f \right)' (t) = - f'(t) / f^2(t) \geq 1/ (2 a) \, .$$ In particular, if $ f^2 (t) \geq 2b$ on $[t_0, T]$, then $$\label{e:t0}
1/ [f(T) ] \geq 1/ [f(t_0)] + (T-t_0)/2a \, .$$ We consider two cases. First, if $ f^2 (t) > 2b$ on $[0, T]$, then [(\[e:t0\])]{} yields $f(T) \leq 2a / T$. Otherwise, if $f(t_0)
= \sqrt{2b}$ for some $t_0 < T$, then [(\[e:t0\])]{} gives $f(T) \leq
f(t_0) = \sqrt{2b}$.
(of Theorem \[t:areabound\]). By scaling, we can assume that $r=1$. Within this proof, we write $\| u \|_{\infty}$ for the $L^{\infty}$ norm of $u$ on $B_1 \times [0,1]$. Set $\eta (x) =
\max \{(1-|x|), 0 \}$ and define $$f(t) = \int \eta^4 \, (1 + |du|^2 )^{1/2} (x,t) \, dx \, .$$ (We will omit the $(x,t)$ below.) Differentiating $f(t)$ and using Stokes theorem gives $$\label{e:diffA}
f'(t) = \int \eta^4 \, \frac{\langle du , du_t \rangle }{(1 + |du|^2)^{1/2}} \, dx
= - \int \eta^4 H^2 \, (1 + |du|^2)^{1/2} \, dx
+ 4 \int H \eta^3 \langle d \eta , du \rangle \, dx \, .$$ The absorbing inequality $4 |H| \, \eta^3 \leq H^2 \eta^4 / 2
+ 8 \eta^2 $ then gives $$\label{e:diffA2}
\int \eta^4 H^2 \, (1 + |du|^2)^{1/2} \, dx \leq - 2 f'(t)
+ 16 \int \eta^2 (1 + |du|^2)^{1/2} \, dx \, .$$ Applying Lemma \[l:graph\] with $\phi = \eta$ and using an absorbing inequality gives $$\label{e:diffA3}
16 \, \int \eta^2 (1 + |du|^2)^{1/2} \, dx
\leq C_1 \, (1 + \|u\|_{\infty}) + C_2 \,
\|u\|_{\infty}^2 + 1/2
\, \int \eta^4 \, H^2 \, dx
\, .$$ Combining [(\[e:diffA2\])]{} and [(\[e:diffA3\])]{} gives $$\label{e:diffA4}
\int \eta^4 H^2 \, (1 + |du|^2)^{1/2} \, dx \leq - 4 f'(t)
+ C_3 ( 1 + \|u\|_{\infty}^2) \, .$$ After applying Lemma \[l:graph\] with $\phi = \eta^2$, the Cauchy-Schwarz inequality and [(\[e:diffA4\])]{} give $$\begin{aligned}
\label{e:A}
f^2(t) &
\leq C_4 \left(1 + \|u\|_{\infty}^2 + \|u\|_{\infty}^2
\, \left( \int \eta^4 \, H^2 \, dx \right) \, \left( \int \eta^4 \,
dx\right)
\right)
\notag \\
&
\leq C_4 (1 + \|u\|_{\infty}^2)
+ C_5 \, \|u\|_{\infty}^2 \left( - 4 f'(t)
+ C_3 ( 1 + \|u\|_{\infty}^2) \right) \, .\end{aligned}$$ Finally, applying Lemma \[l:di\] gives the theorem since, by Lemma \[l:sphere\], $\|u\|_{\infty} \leq \sqrt{2n+1} + \sup_{
B_{\sqrt{2n+1} } } |u (\cdot , 0) |$.
[999]{}
E. Bombieri, E. De Giorgi, and M. Miranda, Una maggiorazione a priori relativa alla ipersuperfici minimali non parametriche, [*Arch. Rational Mech. Anal.*]{}, 32 (1969) 255–267. K. Ecker, Lectures on regularity for mean curvature flow, preprint. K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, [*Annals of Math.*]{}, 130 (1989) 453–471. K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, [*Invent. Math.*]{}, 105 (1991) 547–569. R. Finn, On equations of minimal surface type, [*Annals of Math.*]{}, 60 (1954) 397–416. R. Finn, Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature, [*J. Analyse Math.*]{}, 14 (1965) 265–296. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer Verlag (1983). N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation, Nonlinear functional analysis and its applications, Part 2, [*Proc. of Symposia in Pure Math.*]{} 45 (1986) 81–89.
[^1]: The authors were partially supported by NSF Grants DMS 0104453 and DMS 0104187
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the persistence probability $p(t)$ of the position of a Brownian particle with shape asymmetry in two dimensions. The persistence probability is defined as the probability that a stochastic variable has not changed it’s sign in the given time interval. We explicitly consider two cases – diffusion of a free particle and that of harmonically trapped particle. The later is particularly relevant in experiments which uses trapping and tracking techniques to measure the displacements. We provide analytical expressions of $p(t)$ for both the scenarios and show that in the absence of the shape asymmetry the results reduce to the case of an isotropic particle. The analytical expressions of $p(t)$ are further validated against numerical simulation of the underlying overdamped dynamics. We also illustrate that $p(t)$ can be a measure to determine the shape asymmetry of a colloid and the translational and rotational diffusivities can be estimated from the measured persistence probability. The advantage of this method is that it does not require the tracking of the orientation of the particle.'
author:
- Anirban Ghosh
- Dipanjan Chakraborty
bibliography:
- 'references\_persistence.bib'
title: Persistence in Brownian motion of an ellipsoidal particle in two dimensions
---
Introduction {#sec:intro}
============
Particles that exhibit a shape asymmetry are abundant in nature with sizes ranging from few nanometers to few micrometers. Over the last decade, accelarated by the advancement in particle chemistry, a plethora of such particles with enhanced transport properties have been developed in an attempt to mimic nature. These synthetically engineered colloids with multi-functional properties often find wide ranging applications in photonics, nano and biotechnology, drug delivery and other bio-medical uses. Unlike an isotropic particle, the shape asymmetry leads to different transport properties along the symmetry axes of the particle and any real-life application would require the knowledge of these transport properties. Perhaps, the most crucial of these transport properties are the translational and rotational diffusivities that characterizes their stochastic dynamics. For example, the diffusive dynamics of such particles are completely characterized by the mobility matrix. However, the extraction of the diffusivity from the measured mean-square displacement requires the simultaneous measurement of its translational and orientational degrees of freedom, which might not be always feasible.
In this article, we present an alternative approach to measure the diffusivity of shape asymmetric particle from its position coordinates alone. Our approach does not require the measurement of the symmetry axes of the particle. We choose the simplest asymmetric particle – an ellipsoid and look at its two dimensional Brownian motion. Since the dynamics of the translational and the orientational degrees of freedom are stochastic due to the thermal fluctuations from the bath, the position and the orientation are both random variables in time. We use the stochastic nature of the position to calculate the persistence probability $p(t)$ of the particle. The extraction of the diffusion coefficients along the two symmetry axes of the particle as well the rotational diffusion constant follows from the analytical expression of $p(t)$.
The persistence probability $p(t)$ of a stochastic variable is simply the probability that the variable has not changed sign up to time $t$. In physics, the persistence property has been investigated both theoretically [@derrida1995; @newman1998; @kallabis1999; @toroczkai1999; @sire2000; @constantin2004; @bray2004a; @majumdar1996; @majumdar1996b; @majumdar1999; @majumdar2001; @chakraborty2007; @chakraborty2007c; @chakraborty2008; @chakraborty2009; @chakraborty2012c; @chakraborty2012d; @constantin2005; @dean2001a; @escudero2009; @menon2003; @ray2004a; @krug1997; @singha2005] and experimentally [@Wong:2001dr; @dougherty2002; @Merikoski:2003ju; @Beysens:2006jt; @Soriano:2009br; @Efraim:2011ks; @Takeuchi:2012fe; @takikawa2013] in spatially extended systems that are out of equilibrium. For a more comprehensive review of the persistence probability in spatially extended systems, we invite the readers to look at the recent review by Bray [[*et al.*]{}]{}[@bray2013b] and the brief review by Majumdar [@Majumdar:1999tn] on the subject and the references therein. The persistence probability for such systems decays as a power law $p(t) \sim t^{-\theta}$, with $\theta$ being a non-trivial exponent. This algebraic decay of $p(t)$ has been established for a wide class of non-equilibrium systems that includes the classic random walk problem in finite [@chakraborty2007] and infinite medium [@Majumdar:1999tn; @sire2000; @bray2004a; @chakraborty2012d], critical dynamics [@majumdar1996b; @chakraborty2007c], diffusion in an infinite medium with [@chakraborty2009] and without advection[@majumdar1996; @newman1998], fluctuating interfaces [@derrida1995; @krug1997; @kallabis1999; @toroczkai1999; @constantin2004], disordered systems [@Fisher:1998km; @LeDoussal:1999ik; @chakraborty2008], polymer dynamics [@bhattacharya2007; @chakraborty2012c] and granular media [@Swift:1999kn; @Burkhardt:2000hs]. The estimation of the exponent $\theta$ for a general stochastic process is notoriously difficult and the exact form of $p(t)$ exists in very few cases when the process is Gaussian as well Markovian. For a stochastic process $x(t)$ which is Gaussian as well as Markovian, the non-stationary process can be mapped into a stationary Ornstein-Uhlenbeck process ${\overline}{X}(T)$ via suitable transformations that takes $x \to {\overline}{X}$ and $t \to T$, with the consequence that the correlator $C(T) \equiv \langle {\overline}{X}(T) {\overline}{X}(0)\rangle$ decay exponentially at all times.Following Slepian [@slepian1962], if the stationary correlator $C(T)$ of a stochastic process decays purely exponentially at all times, the persistence probability of $X(T)$ is proportional to $C(T)$ and $p(t)$ can then be constructed back by the inverse time transformation applied to ${\overline}{X}$. In the case when $C(T)$ does not decay exponentially, the exponent $\theta$ can be extracted using the independent interval approximation (IIA) , provided the density of zero crossings remain finite[@majumdar1996]. In the present scenario, as the calculations reveal, the IIA is not required and suitable transformations space and time takes the non-stationary correlation function into a stationary correlator which then be used to calculate $p(t)$.
The rest of the article is organized as follows. In \[sec:free\_ellipsoidal\_particle\] present the results for the two-time correlation function the position of a free Brownian particle with shape asymmetry. The survival probability is determined from this correlation function. In \[sec:harmonically\_trapped\] we carry out a perturbative expansion for the position of an anisotropic Brownian particle trapped in a harmonic potential. The mean-square displacement for the displacements along the two directions and the two-time correlation functions are calculated using the perturbative expansion. Finally, the persistence probability is constructed from this two-time correlation function. A brief conclusion and the relevance of the work is presented in \[sec:conclusion\].
Ellipsoidal Particle in two-dimensions {#sec:free_ellipsoidal_particle}
======================================
We consider an ellipsoidal particle in two dimension with mobilities $\Gamma_x$ and $\Gamma_y$ along the $x$ and $y$ direction respectively and a single rotational mobility $\Gamma_\theta$. The particle is immersed in a bath at a temperature $T$, so that the translational diffusion coefficients along the two directions are given by $D_x={k_{\rm B}}T
\Gamma_x$, $D_y ={k_{\rm B}}T \Gamma_y$ and the rotational diffusion constant $D_\theta ={k_{\rm B}}T \Gamma_\theta$. In a frame fixed to the particle, the translational and the rotational motion of the particle is completely decoupled.However, in the lab-frame, the shape asymmetry of the particle leads to a coupling between the translational and rotational motions of the particle. In the body frame the equations of motion of the particle take the form $$\begin{aligned}
\label{eq:body_frame}
\Gamma_x^{-1}\frac{\partial \tilde{x}}{\partial t}=&F_x \cos
\theta(t) +F_y \sin \theta(t) +\tilde{\eta}_x(t) \nonumber\\
\Gamma_y^{-1}\frac{\partial \tilde{y}}{\partial t}=&F_y \cos
\theta(t) +F_x \sin \theta(t) +\tilde{\eta}_y(t)\nonumber\\
\Gamma_\theta^{-1}\frac{\partial \theta}{\partial t}=& \tau+\tilde{\eta}_\theta, \end{aligned}$$ where $F_x$ and $F_y$ are the forces acting on the particle along the $x$ and $y$ directions and $\tau$ is the torque acting on the particle. The correlations of the thermal fluctuations in the body frame are given by $$\label{eq:noise_body_frame}
\langle \tilde{\eta} \rangle=0 \quad \textrm{and} \quad \langle \tilde{\eta}_i(t) \tilde{\eta}_j(t') \rangle=2 D_i \delta_{ij} \delta(t-t')$$
In the lab frame, the displacements are related to the body frame as, $$\begin{aligned}
\label{eq:lab_body}
\nonumber
\delta x= \cos \theta \delta \tilde{x}-\sin \theta \delta \tilde{y}\\
\delta y =\cos \theta \delta \tilde{y}+\sin \theta \delta \tilde{x}\end{aligned}$$
Using \[eq:body\_frame\], the corresponding Langevin equation in the lab frame is given by, $$\label{eq:langevin}
\frac{\partial x_i}{\partial t}= -\Gamma_{ij} \frac{\partial
U}{\partial x_j} +\eta_i,$$ where $U({\mathbf{r}})$ is the external potential and $\boldsymbol{\Gamma}$ is the mobility tensor given by, $$\label{eq:mobility_tensor}
{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\Gamma}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}=
\begin{pmatrix}
{{\overline}{\Gamma}}+\frac{\D\Gamma}{2} \cos 2\theta & \frac{\Delta
\Gamma}{2} \sin 2 \theta \\
\\
\frac{\D\Gamma}{2} \sin 2 \theta & {{\overline}{\Gamma}}-
\frac{\D\Gamma}{2} \cos 2\theta\\
\end{pmatrix}$$ with ${{\overline}{\Gamma}}=(\Gamma_{\parallel}+\Gamma_{\bot})/2$ and $\D\Gamma
=\Gamma_{\parallel}-\Gamma_{\bot}$. In the component form, the mobility tensor is given by $\Gamma_{ij}={{\overline}{\Gamma}}\delta_{ij}+\frac{\D\Gamma}{2} \Delta
{\boldsymbol{\mathcal{R}}}_{ij}[\theta(t)]$, where the form of $\D
{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}$ is given by $$\label{eq:delta_gamma_matrix}
\D{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}=
\begin{pmatrix}
\cos 2\theta & \sin 2 \theta \\
\\
\sin 2 \theta & -\cos 2\theta\\
\end{pmatrix}$$
Using the correlation of the thermal fluctuations from \[eq:noise\_body\_frame\] and \[eq:body\_frame\], the moments of the stochastic forces are given by, $$\label{eq:fdt}
\la {\boldsymbol{\eta}}\ra =0 \quad \textrm{and} \quad \la
{\boldsymbol{\eta}}(t) {\boldsymbol{\eta}}(t') \ra= 2 {k_{\rm B}}T
\boldsymbol{ \Gamma}[\theta(t)] \delta(t-t')$$
We first look at the case of a free ellipsoidal particle. Setting the external potential to zero, the formal solution to the equation of motion takes the form $$\label{eq:free_particle_solution}
x_i(t)=\int_0^t \eta_i(t') \mathrm{d} t' + x_i(0)$$ The mean-square displacement of the particle, averaged over the orientational noise can be explicitly calculated from the above equation as, $$\begin{aligned}
\label{eq:msd}
\la \Delta x_i^2\ra_{\eta_\theta} &=\int_0^t \mathrm{d} t' \int_0^t {{\@ifnextchar^{\DIfF}{\DIfF^{}}}} t'' \la \eta_i(t')\eta_i(t'')\ra \nonumber\\
&=2 {k_{\rm B}}T \int_0^t \mathrm{d} t' \int_0^t {{\@ifnextchar^{\DIfF}{\DIfF^{}}}} t'' ~~\la \Gamma_{ii}[\theta(t')]\ra_{\eta_\theta}
\delta(t'-t'') \nonumber\\
&=2 {k_{\rm B}}T \int_0^t \mathrm{d} t' ~~\la \Gamma_{ii}[\theta(t')]\ra_{\eta_\theta}
\end{aligned}$$ Using the explicit form of $\Gamma_{xx}$ the mean-square displacement along the $x$-direction reads $$\label{eq:xsq_free}
\la \Delta x_1^2\ra_{\eta_\theta} =2{k_{\rm B}}T \int_0^t {{\@ifnextchar^{\DIfF}{\DIfF^{}}}} t' ~~\left[ {\overline}{\Gamma}+ \frac{\Delta \Gamma}{2} \la \cos \theta(t') \ra_{\eta_\theta}\right]$$ The ensemble average of $\cos \theta(t)$ over the thermal fluctuations in the orientational degrees of freedom can be done explicitly by noting the fact that $\Delta \theta=\theta(t)- \theta_0$ is a Gaussian random variable and consequently the following identity holds: $$\label{eq:identity_1}
\la e^{\pm{\mathbbm{i}}m \Delta \theta(t')}\ra_{\eta_\theta}=
e^{-m^2 D_\theta t'}.$$ Using \[eq:identity\_1\] in \[eq:xsq\_free\], we finally arrive at $$\label{eq:xsq_free_1}
\la \Delta x^2\ra_{\eta_\theta} =2{k_{\rm B}}T ~~\left[ {\overline}{\Gamma} t+ \frac{\Delta \Gamma}{2} \cos 2\theta_0 \left(\frac{1-e^{-4 D_\theta t}}{4 D_\theta}\right)\right]$$ and $$\label{eq:ysq_free_1}
\la \Delta y^2\ra_{\eta_\theta} =2{k_{\rm B}}T ~~\left[ {\overline}{\Gamma} t- \frac{\Delta \Gamma}{2} \sin 2\theta_0 \left(\frac{1-e^{-4 D_\theta t}}{4 D_\theta }\right)\right]$$ The above results are well known [@Han:2006ew; @Grima:2007fo] and have also been experimentally verified. [@Han:2006ew] However, our interest lies in the persistence probability of this system. To calculate that we start with \[eq:free\_particle\_solution\] and choose $x_i(0)=0$. The calculation of the two time correlation function $\la x(t_1) x(t_2) \ra_{\eta_\theta}$ follows the same route as detailed above: $$\la x(t_1) x(t_2) \ra_{\eta_\theta} = \int_0^{t_1} \mathrm{d}t' \int_0^{t_2} \mathrm{d} t'' \la \eta_x(t') \eta_x(t'')\ra_{\eta_\theta}.$$ Taking $t_1>t_2$, the integral evaluates to the following expression for the two time correlation, $$\la x(t_1) x(t_2) \ra_{\eta_\theta} = 2 {k_{\rm B}}T {\overline}{\Gamma} t_2 \left[ 1+ \frac{\Delta \Gamma}{2{\overline}{\Gamma}} \cos \theta_0 \left(\frac{1-e^{-4D_\theta t_2}}{4 D_\theta t_2} \right) \right]
$$ In order to transform the non-stationary correlation into a stationary correlation we first make the transformation ${\overline}{X}(t)=x(t)/\sqrt{\la x^2(t) \ra_{\eta_\theta}}$, and the correlation $\la {\overline}{X}(t_1){\overline}{X}(t_2)\ra_{\eta_\theta}$ reads as $$\label{eq:corr_space}
\la {\overline}{X}(t_1){\overline}{X}(t_2)\ra_{\eta_\theta}=\sqrt{\frac{2 {\overline}{D} t_2}{2 {\overline}{D} t_1}}\sqrt{\frac{1 +\frac{\Delta \Gamma}{2{\overline}{\Gamma}} \cos \theta_0\left(\frac{1-e^{-4D_\theta t_2}}{4 D_\theta t_2} \right)}{1 + \frac{\Delta \Gamma}{2 {\overline}{\Gamma}} \cos \theta_0 \left(\frac{1-e^{-4D_\theta t_1}}{4 D_\theta t_1} \right)}}$$ We now define the transformation in time as $$\label{eq:time_transformation}
e^T=\sqrt{2 {\overline}{D} t \left[1 +\frac{\Delta D}{2{\overline}{D}} \cos \theta_0\left(\frac{1-e^{-4D_\theta t}}{4 D_\theta t} \right)\right]}$$ and \[eq:corr\_space\] takes the simple form of $$\label{eq:corr_space_time}
\la {\overline}{X}(T_1){\overline}{X}(T_2)\ra_{\eta_\theta}=e^{-(T_1-T_2)/2}$$ Following Slepian [@slepian1962], if the correlation function of a stochastic variable $X(T)$ decays exponentially for all times $C_{XX}(T) = e^{-\lambda T}$, then the persistence probability is given by $$\label{eq:per_prob}
P(T) \sim \sin^{-1} e^{-\lambda T}.$$ Asymptotically, $P(T)$ takes the form $P(T) \sim e^{-\lambda
T}$. Consequently, looking at \[eq:corr\_space\_time\] and transforming back in real time $t$, the persistence probability reads as $$\label{eq:per_prob_real_time}
p(t) \sim \frac{1}{\sqrt{2{\overline}{D}
t}}\frac{1}{\sqrt{1+\frac{\Delta D}{2 {\overline}{D}} \cos
\theta_0 \left(\frac{1-e^{-4D_\theta t}}{4 D_\theta t} \right)}}.$$ In the absence of any asymmetry, the expression for $p(t)$ correctly reproduces the persistence probability of that of a random walker. Rearranging \[eq:per\_prob\_real\_time\], the quantity $t^{1/2}p(t)$ can be recast as $$\label{eq:per_prob_real_time_1}
t^{1/2} p(t) \sim \frac{1}{\sqrt{2 {\overline}{D}}}\left[1+\frac{\Delta D}{ {\overline}{D}} \cos \theta_0 \left(\frac{1-e^{-4 \tau}}{8 \tau} \right)\right]^{-1/2}.$$ In the limit of $\D D \to 0$, the persistence probability reduces to that of a random walker $p(t) \sim t^{-1/2}$.
To test \[eq:per\_prob\_real\_time\], we performed numerical integration of the equations of motion using an Euler scheme for discritization. The initial condition was chosen from a Gaussian distribution with a very small width, so that the sign of ${\mathbf{r}}(0)$ is clearly defined. The trajectories was evolved in time with an integration time-step of $\delta t=0.001$. At every instant the the survival of the particle was checked by looking at the sign of ${\mathbf{r}}(t)$. Fraction of trajectories for which the position did not change its sign up to time $t$ gave the survival probability $p(t)$. A total of $10^9$ trajectories were used in estimating the survival probability. A comparison of the measured $p(t)$ with that of the predictions of \[eq:per\_prob\_real\_time\] is shown \[fig:persistence\_fig\_1\] and \[fig:persistence\_fig\_2\]. The comparison in \[fig:persistence\_fig\_1\] clearly shows that the survival probability can pick up the asymmetry in particle shape even when the difference in the diffusivities is as small as $5\%$.
![Plot of $t^{1/2}p(t)$ for different choices of translational diffusivities of the anisotropic particle: $D_{\parallel}=1,
D_{\bot}=0.5$ ([ $\bullet$]{}), $D_{\parallel}=1,
D_{\bot}=0.8$ ([ $\blacksquare$]{}),$D_{\parallel}=1,
D_{\bot}=0.9$ ([${\mathop{\raisebox{-0.275ex}{${\tikz
[x=1.2ex,y=1.85ex,line width=.1ex,line join=round, yshift=-0.285ex]
\draw [draw=mygreen,fill=mygreen] (-0.3,.5) -- (.5,1.) -- (1.3,.5) -- (.5,0) -- (-0.3,.5) -- cycle;}$}}}$]{}) and $D_{\parallel}=1,
D_{\bot}=0.95$ ([ $\blacktriangle$]{}). The rotational diffusion constant and the initial angle $\theta_0$ was fixed at $D_\theta=1$ and $\theta_0=0$,respectively. The solid lines are fit to the data using \[eq:per\_prob\_real\_time\_1\]. The fit yields the overall constant $\mathcal{A}$. The estimated of values of $\mathcal{A}$ from the fit are $0.025132 \pm 0.000014$ for $D_{\parallel}=1,
D_{\bot}=0.5$, $0.025144 \pm 0.000011$ for $D_{\parallel}=1,
D_{\bot}=0.8$, $0.025166 \pm 0.000012$ for $D_{\parallel}=1,
D_{\bot}=0.9$ and $0.025148 \pm 0.000019$ for $D_{\parallel}=1,
D_{\bot}=0.95$. []{data-label="fig:persistence_fig_1"}](persistence_probability_free_particle){width="0.8\linewidth"}
![Plot of $t^{1/2}p(t)$ for different choices of rotational diffusion constant of the anisotropic particle: $D_\theta=0.01$ ([ $\bullet$]{}),$D_\theta=0.1$ ([ $\blacksquare$]{}),$D_\theta=1$ ([ ${\mathop{\raisebox{-0.275ex}{${\tikz
[x=1.2ex,y=1.85ex,line width=.1ex,line join=round, yshift=-0.285ex]
\draw [draw=mygreen,fill=mygreen] (-0.3,.5) -- (.5,1.) -- (1.3,.5) -- (.5,0) -- (-0.3,.5) -- cycle;}$}}}$]{}). The translation diffusion constants in all the three cases were $D_{\parallel}=1$ and $D_{\bot}=0.5$ and the initial orientation was fixed at $\theta_0=0$.[]{data-label="fig:persistence_fig_2"}](persistence_probability_free_particle1){width="0.8\linewidth"}
![Plot of $t^{1/2}p(t)/\mathcal{A}$ for different choices of translational diffusivities of the anisotropic particle: $D_{\parallel}=1,
D_{\bot}=0.5$ ([ $\bullet$]{}), $D_{\parallel}=1,
D_{\bot}=0.8$ ([ $\blacksquare$]{}),$D_{\parallel}=1,
D_{\bot}=0.9$ ([${\mathop{\raisebox{-0.275ex}{${\tikz
[x=1.2ex,y=1.85ex,line width=.1ex,line join=round, yshift=-0.285ex]
\draw [draw=mygreen,fill=mygreen] (-0.3,.5) -- (.5,1.) -- (1.3,.5) -- (.5,0) -- (-0.3,.5) -- cycle;}$}}}$]{}) and $D_{\parallel}=1,
D_{\bot}=0.95$ ([ $\blacktriangle$]{}). The rotational diffusion constant and the initial orientation was fixed at $D_\theta=1$ and $\theta_0=0$, respectively. The solid black line indicates the value of $1/\sqrt{2{\overline}{D}+\Delta D}=1/\sqrt{2
D_\parallel}=1/\sqrt{2}$, whereas the dashed lines indicates the values of $1/\sqrt{2 {\overline}{D}}$. For the choice of the translational diffusivities, the indicated values from top are $1/\sqrt{2
{\overline}{D}} \approx 0.8165, 0.7454, 0.7255$ and $0.7161$.[]{data-label="fig:persistence_fig_3"}](persistence_probability_free_particle3){width="0.8\linewidth"}
![Plot of the dimensionless quantity $\tilde{p}(t)$ (see \[eq:rot\_diff\]) as function of time for different choices of rotational diffusion constant of the anisotropic particle: $D_\theta=0.01$ ([ $\bullet$]{}),$D_\theta=0.1$ ([ $\blacksquare$]{}),$D_\theta=1$ ([ ${\mathop{\raisebox{-0.275ex}{${\tikz
[x=1.2ex,y=1.85ex,line width=.1ex,line join=round, yshift=-0.285ex]
\draw [draw=mygreen,fill=mygreen] (-0.3,.5) -- (.5,1.) -- (1.3,.5) -- (.5,0) -- (-0.3,.5) -- cycle;}$}}}$]{}). The translation diffusion constants in all the three cases were $D_{\parallel}=1$ and $D_{\bot}=0.5$. The initial orientation in all the cases were fixed at $\theta_0=0$. The solid lines are fit to the data using \[eq:rot\_diff\] using $\Delta D/{\overline}{D}$ and $D_\theta$ as fit parameters. The estimated values of these parameters from the fit are compared with the actual values used in the simulation in \[tab:comparison\_table\]. []{data-label="fig:persistence_fig_4"}](persistence_probability_free_particle2){width="0.8\linewidth"}
The process to extract the the diffusion coefficients is as follows. The first step would be to determine the overall constant $\mathcal{A}$ in the expression for the persistence probability. This can be fixed by fitting the data with the form of $p(t)$ given in \[eq:per\_prob\_real\_time\]. This fit yields the value of $\mathcal{A}$. In \[fig:persistence\_fig\_1\] (a), we have shown this fitting for different choices of the diffusivities, with $\mathcal{A}$ as the fit parameter. The value of $\mathcal{A}$ is solely determined by the number of trajectories used to estimate $p(t)$. The values determined from the fit are given in the caption of the figure. An alternative way to determine $\mathcal{A}$ is to measure the persistence probability of an isotropic particle, in which case $p(t) \sim \mathcal{A}/\sqrt{2 D t}$. Once this number is known, we look at the quantity $t^{1/2} p(t)/\mathcal{A}$. In the limit of $ t \to 0$, $t^{1/2} p(t)/\mathcal{A} \to (2 {\overline}{D}+\Delta D)^{1/2}$ and in the limit of $t \to \infty$, $t^{1/2} p(t)/\mathcal{A} \to (2 {\overline}{D})^{1/2}$.
Once we know the two diffusivities, and therefore ${\overline}{D}$, the rotational diffusion constant can be determined from the quantity $\left(\mathcal{A}/\sqrt{2 {\overline}{D} t} p(t)\right)^2$ which goes as $$\label{eq:rot_diff}
\tilde{p}(t)=\left(\frac{\mathcal{A}}{\sqrt{2 {\overline}{D} t} p(t)}\right)^2=1+\left(\frac{\Delta D}{{\overline}{D}}\right)\left(\frac{1-e^{-4 D_\theta
t}}{8D_\theta t}\right)$$ A fit to $\tilde{p}(t)$ with $D_\theta$ as a fit parameter would yield the value of the rotational diffusion coefficient. This is illustrated in \[fig:persistence\_fig\_2\]. In fact, fitting the data for $\tilde{p}$ with $\Delta D/{\overline}{D}$ and $D_\theta$ as fit parameters yields very good estimates for $\Delta D/{\overline}{D}$ and $D_\theta$. A comparison of these values obtained from the fit with that of the actual values is shown in \[tab:comparison\_table\].
[|\*[4]{}[c|]{}]{} & & &\
& 0.01 & $0.6698 \pm 0.0018$ & $0.0117
\pm 0.0002$\
& 0.1 & $0.1146 \pm 0.0009$ & $0.6799
\pm 0.002 $\
& 1.0 & $1.076 \pm 0.06$ & $0.681
\pm
0.0295$\
It should be pointed out, that the values of $\Delta D/{\overline}{D}$ and $D_\theta$ obtained from the fit are sensitive to the value $\mathcal{A}$ and a careful estimation of $\mathcal{A}$ is of paramount importance.
Harmonically trapped ellipsoidal particle {#sec:harmonically_trapped}
=========================================
In experiments, the tracking of colloidal particles are usually done with laser traps and consequently it is pertinent to discuss the scenario where an ellipsoidal particle is trapped in a harmonic trap. In the following, we assume that the harmonic trap is isotropic and there is no preferential direction of alignment. Further, if we suppose a strong confinement, then at late times the deviations from the mean position of the particle is practically zero. Accordingly, the particle rotates freely so that the angular displacements obey Gaussian statistics. The potential confinement has the form $U(x,y)=\kappa (x^2+y^2)/2$ and the corresponding Langevin equation from \[eq:langevin\] take the form
$$\begin{aligned}
\label{eq:langevin_harmonic}
\frac{\partial x}{\partial t}=&-\kappa x \left(
{\overline}{\Gamma}+\frac{1}{2}\Delta \Gamma \cos
\theta(t)\right)-\frac{1}{2}\kappa y \Delta \Gamma \sin \theta(t) +\tilde{\eta}_x(t) \nonumber\\
\frac{\partial y}{\partial t}=&-\frac{1}{2}\kappa x \Delta \Gamma \sin \theta(t)-\kappa y \left(
{\overline}{\Gamma}-\frac{1}{2}\Delta \Gamma \cos
\theta(t)\right) +\tilde{\eta}_y(t)\nonumber\\
\frac{\partial \theta}{\partial t}=& \tilde{\eta}_\theta, \end{aligned}$$
where the correlation of the thermal noise follows \[eq:fdt\].
Purturbative Expansion {#ssec:purturbative_expansion}
----------------------
Defining the vector ${\mathbf{R}}\equiv (x,y)^T$, the equation takes the simple form $$\label{eq:langevin_harmonic_vector}
\overset{\bm .}{{\mathbf{R}}}=-\k \left[{{\overline}{\Gamma}}{\mathds{1}}+ \frac{\D \G}{2} {\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(t)
\right] {\mathbf{R}}(t) +{\boldsymbol{\eta}}(t)$$
To solve the above equation, we use the perturbative expansion $$\label{eq:perturbation_series}
{\mathbf{R}}(t)={\mathbf{R}}_0(t)- \left(\frac{\k \D \G}{2}\right) {\mathbf{R}}_1(t)
+\left(\frac{\k \D \G}{2}\right)^2 {\mathbf{R}}_2(t)+{\mathcal{O}}\left(\frac{\k \D \G}{2}\right)^3$$ Substituting \[eq:perturbation\_series\] in \[eq:langevin\_harmonic\_vector\] and keeping up to the linear order in $\k \D\G/2$ we obtain the equations for ${\mathbf{R}}(t)$ and ${\mathbf{R}}_1(t)$ as $$\label{eq:solution_series}
\begin{split}
\overset{\bm .}{{\mathbf{R}}}_0&=-\k {{\overline}{\Gamma}}{\mathbf{R}}_0(t) +{\boldsymbol{\eta}}(t)\\
\overset{\bm .}{{\mathbf{R}}}_1&=-\k {{\overline}{\Gamma}}{\mathbf{R}}_1(t)
+{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(t){\mathbf{R}}_0(t)\\
\overset{\bm .}{{\mathbf{R}}}_2&=-\k {{\overline}{\Gamma}}{\mathbf{R}}_2(t)
+{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(t){\mathbf{R}}_1(t)\\
\end{split}$$
The solutions for the \[eq:solution\_series\] together with the initial condition ${\mathbf{R}}(0)=0$ take the form $$\label{eq:solution_series_1}
\begin{split}
{\mathbf{R}}_0(t)&=\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-\k {{\overline}{\Gamma}}(t-t')}{\boldsymbol{\eta}}(t)\\
{\mathbf{R}}_1(t)&=\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-\k {{\overline}{\Gamma}}(t-t')}\;{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(t)\;{\mathbf{R}}_0(t)\\
{\mathbf{R}}_2(t)&=\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-\k {{\overline}{\Gamma}}(t-t')}\;{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(t)\;{\mathbf{R}}_1(t)
\end{split}$$
In explicit form, the equal time correlation matrix $R_i(t) R_j(t)$ is then given by $$\label{eq:msd_gen_exp}
\begin{split}
\la R_i(t) R_j(t) \ra_{{\boldsymbol{\eta}},\theta}= \la R_{0,i}(t)
R_{0,j}(t)\ra_{{\boldsymbol{\eta}},\theta} -\left(\frac{\k \D \G}{2}\right) \la
R_{0,i}(t)R_{1,j}(t)\ra_{{\boldsymbol{\eta}},\theta}\\
+\left(\frac{\k \D
\G}{2}\right)^2 \left[\left\la R_{1,i}(t)R_{1,j}(t)\right\ra_{{\boldsymbol{\eta}},\theta}+2
\left\la R_{0,i}(t)R_{2,j}(t)\right\ra_{{\boldsymbol{\eta}},\theta}\right] +{\mathcal{O}}\left(\frac{\k \D \G}{2}\right)^3
\end{split}
$$ where we have used the fact that $\la R_{0,i}R_{1,j} \ra=\la R_{0,j}R_{1,i}\ra$. Further, note that the thermal noise correlation given in \[eq:fdt\] gives an additional factor of $\k \D \G/2$ in the correlation terms $\la R_{\a,i}(t) R_{\beta,j}(t)\ra $, where $\a,\beta$ denotes the order of the perturbation series.
We next proceed to calculate this equal time correlation matrix using the solutions in \[eq:solution\_series\_1\]. The correlation matrix of ${\mathbf{R}}_0(t)$ averaged over the translational and the rotational noise is then given by $$\label{eq:correlation_bfR}
\begin{split}
\la {\mathbf{R}}_0(t) {\mathbf{R}}_0(t) \ra_{{\boldsymbol{\eta}},\theta} =\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t''
e^{-\k {{\overline}{\Gamma}}(t-t')} e^{-\k {{\overline}{\Gamma}}(t-t'')} \la {\boldsymbol{\eta}}(t') {\boldsymbol{\eta}}(t'')
\ra_{{\boldsymbol{\eta}},\theta},
\end{split}$$ where in correlation of the thermal noise is understood as an outer product of the variable $\eta_x$ and $\eta_y$. Using \[eq:fdt\], the calculation is straight forward and the final form of the correlation matrix is given by
$$\label{eq:correlation_bfR_final}
\begin{split}
\la {\mathbf{R}}_0(t) {\mathbf{R}}_0(t) \ra_{{\boldsymbol{\eta}},\theta} &= \frac{{k_{\rm B}}T}{\k} {\mathds{1}}\left(1- e^{-2 \k {{\overline}{\Gamma}}t} \right) \\
&+ \D D \;{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(\theta_0)
\left(\frac{e^{-4 D_\theta t}-e^{-2 \k {{\overline}{\Gamma}}t}}{2\k {{\overline}{\Gamma}}-4 D_\theta}\right)
\end{split}$$
More explicitly, the mean-square displacement along the $x$ and $y$ direction are given by $$\label{eq:msd_x_y}
\begin{split}
\la x^2_0(t)\ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T}{\k}
\left(1- e^{-2 \k {{\overline}{\Gamma}}t} \right)
+ \D D \cos 2\theta_0
\left(\frac{e^{-4 D_\theta t}-e^{-2 \k {{\overline}{\Gamma}}t}}{2\k {{\overline}{\Gamma}}-4
D_\theta}\right) \\
\intertext{and}
\la y^2_0(t)\ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T}{\k}
\left(1- e^{-2 \k {{\overline}{\Gamma}}t} \right)
- \D D \cos 2\theta_0
\left(\frac{e^{-4 D_\theta t}-e^{-2 \k {{\overline}{\Gamma}}t}}{2\k {{\overline}{\Gamma}}-4
D_\theta}\right) \\
\end{split}$$ The cross-correlation function $x_0(t)y_0(t)$ reads $$\label{eq:corr_xy}
\begin{split}
\la x_0(t)y_0(t)\ra_{{\boldsymbol{\eta}},\theta} = \D D \sin 2\theta_0
\left(\frac{e^{-4 D_\theta t}-e^{-2 \k {{\overline}{\Gamma}}t}}{2\k {{\overline}{\Gamma}}-4
D_\theta}\right)
\end{split}$$
![Plot of the mean-square displacement along the $x$ direction of harmonically trapped anisotropic particle for different choices of the stiffness of the harmonic potential as indicated in the legend. The translational diffusivities and the rotational diffusion constant were kept fixed at $D_\parallel=1$, $D_\bot=0.5$ and $D_\theta=0.1$ in all the cases. The initial orientation of the particles were also fixed at $\theta_0=0$. The solid lines are plots of \[eq:msd\_x\_final\] and the dashed lines are plots of \[eq:msd\_x\_approx\] with the appropriate values of $\kappa,D_\parallel,D_\bot$ and $D_\theta$.[]{data-label="fig:msd_x_y_harmonic"}](msd_x_harmonic){width="0.8\linewidth"}
In the limit of $\k \to 0$, \[eq:msd\_x\_y,eq:corr\_xy\] reproduces the correct result of a free diffusion of an anisotropic particle given in \[eq:xsq\_free\_1,eq:ysq\_free\_1\]. On the other hand, for $\D \G \to 0$ \[eq:msd\_x\_y,eq:corr\_xy\] yeilds the correlation matrix for an isotropic Brownian particle in a harmonic trap.
Our next attempt is to look into the correction to the above expression that comes from ${\mathbf{R}}_1(t)$ and ${\mathbf{R}}_2(t)$. For this, we rewrite the solutions for ${\mathbf{R}}_1(t)$ and ${\mathbf{R}}_2(t)$ in explicit form as $$\label{eq:solution_R1_explicit}
\begin{split}
R_{1,i}(t)=\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-\k {{\overline}{\Gamma}}(t-t')} \sum_j
{\boldsymbol{\mathcal{R}}}_{ij}(t')R_{0,j}(t')\\
R_{2,i}(t)=\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-\k {{\overline}{\Gamma}}(t-t')} \sum_j
{\boldsymbol{\mathcal{R}}}_{ij}(t')R_{1,j}(t')
\end{split}$$ where the subscripts are for the two spatial dimensions and can take the values $1$ and $2$. Using \[eq:msd\_gen\_exp\], we proceed to calculate the terms $\la R_{0,i}(t)
R_{0,j}(t)\ra_{{\boldsymbol{\eta}},\theta}$,$\la
R_{1,i}(t) R_{1,j}(t)\ra_{{\boldsymbol{\eta}},\theta}$ and $\la R_{2,i}(t) R_{2,j}(t)\ra_{{\boldsymbol{\eta}},\theta}$. The detailed calculation of the three terms are presented in the \[appendix:appendix\_1,appendix:appendix\_2,appendix:appendix\_3,appendix:appendix\_4\],respectively. In deriving the results presented in the appendices, we have utilized the more general form of the identity given in \[eq:identity\_1\]: $$\label{eq:identity_2}
\la e^{i m \D \h(t')-i n \D
\h(t'')}\ra_{\theta}=e^{-D_\h (m^2 t'+n^2 t''-2 m n \min (t',t''))}$$ Using the above relation, the averages of the trigonometric functions over the rotational noise take the form $$\label{eq:identity_2}
\begin{split}
\la \cos 2[\h(t') -\h(t'')]\ra_\h&=e^{-4 D_\h(t'+t''-2 \min(t',t'') }\\
\la \cos 2[\h(t') +\h(t'')]\ra_\h&=\cos 4 \h_0 e^{-4 D_\h(t'+t''+2
\min(t',t'') }\\
\la \sin 2[\h(t') +\h(t'')]\ra_\h&=\sin 4 \h_0 e^{-4 D_\h(t'+t''+2
\min(t',t'') }\\
\la \sin 2[\h(t') -\h(t'')]\ra_\h&=0\\
\end{split}$$
![Plot of the mean-square displacement along the $y$ direction of harmonically trapped anisotropic particle for different choices of the stiffness of the harmonic potential as indicated in the legend. The translational diffusivities and the rotational diffusion constant were kept fixed at $D_\parallel=1$, $D_\bot=0.5$ and $D_\theta=0.1$ in all the cases. The initial orientation of the particles were also fixed at $\theta_0=0$.The solid lines are plots of \[eq:msd\_y\_final\] and the dashed lines are plots of \[eq:msd\_y\_approx\] with the appropriate values of $\kappa,D_\parallel,D_\bot$ and $D_\theta$.[]{data-label="fig:msd_y_harmonic"}](msd_y_harmonic){width="0.8\linewidth"}
The final form of the expressions is given by $$\label{eq:x0_x1}
\begin{split}
& \la x_0(t) x_1(t) \ra_{{\boldsymbol{\eta}},\h}=\la y_0(t) y_1(t)
\ra_{{\boldsymbol{\eta}},\h}=\\
&\left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \;
\left(\frac{e^{-4 D_\h
t}-e^{-2\k {{\overline}{\Gamma}}t}}{(2\k {{\overline}{\Gamma}}-4 D_\h)} -\frac{e^{-2\k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h} \right)\\
&+2\left(\frac{{k_{\rm B}}T}{\k}\right)\;\left(\frac{\k \D \G}{2}\right)
\left(\frac{1-e^{-2 \k {{\overline}{\Gamma}}t}}{2 \k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4
D_\h)}-\frac{e^{-2 \k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h (2
\k {{\overline}{\Gamma}}+4 D_\h)}\right)\\
\end{split}$$ $$\label{eq:x1_sq_final}
\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}&=\la y^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}=\left( \frac{ {k_{\rm B}}T}{\k} \right)
\left[ \frac{1-e^{-2 \k {{\overline}{\Gamma}}t}}{\k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4
D_\h)}-\frac{t e^{-2 \k {{\overline}{\Gamma}}t}}{4D_\h} \right. \\
&\quad \quad \qquad \qquad \qquad \left. +\k {{\overline}{\Gamma}}\frac{e^{-2 \k
{{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+ 4 D_\h) t}}{4 D^2_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)} \right]
\end{split}$$
$$\label{eq:x0_x2}
\begin{split}
\left\la x_0(t) x_2(t) \right\ra_{{\boldsymbol{\eta}},\theta}&=\left\la y_0(t) y_2(t) \right\ra_{{\boldsymbol{\eta}},\theta}
=\left(\frac{{k_{\rm B}}T}{\k}\right)\left[\frac{1-e^{-2\k
{{\overline}{\Gamma}}t}}{2 \k {{\overline}{\Gamma}}(2\k{{\overline}{\Gamma}}+4 D_\h)}\right.\\
&\left.-\frac{te^{-2\k {{\overline}{\Gamma}}t}
}{4 D_\h} +\frac{2\k {{\overline}{\Gamma}}}{4 D_\h}\left(1-e^{-4 D_\h
t}\right) \right] \\
\end{split}$$
In the limit of $\k \to 0$, both $\la y_1^2(t) \ra =\la x_1^2(t) \ra=0$.
The final expression for the mean-square displacement along the $x$ is given by $$\label{eq:msd_x_final}
\begin{split}
\la x^2(t) \ra_{{\boldsymbol{\eta}},\h}&=\left(\frac{{k_{\rm B}}T}{\k}\right)\left[\left(1-e^{-2 \k {{\overline}{\Gamma}}t}\right) +\left(\frac{\k
\D \G}{2}\right) \cos 2\h_0 \right.\\
&\left( \frac{e^{-4 D_\h t}-e^{-2
\k {{\overline}{\Gamma}}t}}{2 \k {{\overline}{\Gamma}}- 4 D_\h}+\frac{e^{-2 \k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h}\right)\\
&+\left(\frac{\k
\D \G}{2}\right)^2 \left( \frac{1}{4 D_\h^2} e^{-2\k{{\overline}{\Gamma}}t}\left( 1-e^{-4 D_\h t} \right) -\frac{ t}{ D_\h} e^{-2\k
{{\overline}{\Gamma}}t} \right) \\
& \left. +{\mathcal{O}}\left(\frac{\k
\D \G}{2}\right)^3 \right]
\end{split}$$
and that along the $y$-direction is given by $$\label{eq:msd_y_final}
\begin{split}
\la y^2(t) \ra_{{\boldsymbol{\eta}},\h}&=\left(\frac{{k_{\rm B}}T}{\k}\right)\left[\left(1-e^{-2 \k {{\overline}{\Gamma}}t}\right) -\left(\frac{\k
\D \G}{2}\right) \cos 2\h_0 \right.\\
& \left( \frac{e^{-4 D_\h t}-e^{-2
\k {{\overline}{\Gamma}}t}}{2 \k {{\overline}{\Gamma}}- 4 D_\h}+\frac{e^{-2 \k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h}\right)\\
&+\left(\frac{\k
\D \G}{2}\right)^2 \left(\frac{1}{4 D_\h^2} e^{-2\k{{\overline}{\Gamma}}t}\left( 1-e^{-4 D_\h t} \right) -\frac{ t}{ D_\h} e^{-2\k
{{\overline}{\Gamma}}t} \right) \\
&\left.+{\mathcal{O}}\left(\frac{\k \D \G}{2}\right)^3 \right]
\end{split}$$
Mean-square displacement for large rotational diffusion constant
----------------------------------------------------------------
In this section we present an alternate expression for mean-square displacement of an anisotropic particle which is valid for whose rotational diffusion constant is large as compared to the inverse times scales $\k {{\overline}{\Gamma}}$ and $\k \D\G$. In such a scenario, since the particle rotates faster, the mobility of the anisotropic particle is an average mobility over the rotational noise. We start our analysis with \[eq:solution\_langevin\_R\], but we set ${\mathbf{R}}(0)=0$. To proceed further, and in particular to look at the asymptotic limit of the correlations, we define the variable $u=(t-t')/t$. In terms of the new variable $u$, the solution for ${\mathbf{R}}(t)$ takes the form $$\label{eq:solution_langevin_R}
{\mathbf{R}}(t)=t\int_0^1 {\@ifnextchar^{\DIfF}{\DIfF^{}}}u\, e^{-\k {\overline}{\Gamma}\,{\mathds{1}}\, t\, u}
e^{-\frac{\k}{2} \D \Gamma\int_{t(1-u)}^t \,{\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \,
{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta(t'')] } \,{\boldsymbol{\eta}}[t(1-u)]$$ The equal-time correlation is then given by $$\label{eq:correlation_approx}
\begin{split}
\la {\mathbf{R}}(t) {\mathbf{R}}(t) \ra_{{\boldsymbol{\eta}}}&= \int_0^1 {\@ifnextchar^{\DIfF}{\DIfF^{}}}u \int_0^1 {\@ifnextchar^{\DIfF}{\DIfF^{}}}u' e^{-\k
{\overline}{\Gamma}\,{\mathds{1}}\, t\, u} e^{-\k {\overline}{\Gamma}\,{\mathds{1}}\, t\, u'} e^{-\frac{\k}{2} \D \Gamma\int_{t(1-u)}^t \,{\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \,
{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta(t'')] } \\
&e^{-\frac{\k}{2} \D \Gamma\int_{t(1-u')}^t \,{\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \,
{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta(t'')] } \,\,\la
{\boldsymbol{\eta}}[t(1-u)]{\boldsymbol{\eta}}[t(1-u')]\ra_{{\boldsymbol{\eta}}}
\end{split}$$ The correlation of the thermal noise in the transformed variable is $$\label{eq:noise_correlation_u}
\la {\boldsymbol{\eta}}[t(1-u)] {\boldsymbol{\eta}}[t(1-u')] \ra_{{\boldsymbol{\eta}}}= \frac{2{k_{\rm B}}T
}{t}{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\Gamma}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[t(1-u)] \delta(u-u')$$ Substituting the noise correlation into \[eq:correlation\_approx\] and integration over $u'$ we get $$\label{eq:correlation_approx}
\begin{split}
\la {\mathbf{R}}(t) {\mathbf{R}}(t) \ra &= 2 {k_{\rm B}}T t \int_0^1 {\@ifnextchar^{\DIfF}{\DIfF^{}}}u e^{- 2 \k {\overline}{\Gamma}\,{\mathds{1}}\, t\, u} e^{-\k \D \Gamma\int_{t(1-u)}^t \,{\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \,
{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta(t'')] } \times\\
&\phantom{16pt}\;\;\,\left[ {{\overline}{\Gamma}}{\mathds{1}}+ \frac{\D \G }{2} {\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta
(t(1-u))]\right]
\end{split}$$ In the asymptotic limit, the integral is dominated by small values of $u$, the integral in the exponential from $t(1-u)$ to $t$ is vanishingly small and can be set to zero. Further, we set ${\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta(t(1-u))] \approx
{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta(t)]$. Consequently, the correlation matrix averaged over the translational noise take the form $$\label{eq:correlation_approx_1}
\begin{split}
\la {\mathbf{R}}(t) {\mathbf{R}}(t) \ra = 2 {k_{\rm B}}T t \int_0^1 {\@ifnextchar^{\DIfF}{\DIfF^{}}}u e^{- 2 \k {\overline}{\Gamma}\,{\mathds{1}}\, t\, u}
\left[ {{\overline}{\Gamma}}{\mathds{1}}+ \frac{\D \G }{2} {\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta
(t)]\right]
\end{split}$$ and performing the average over the rotational noise and the integral over $u$ we arrive at $$\label{eq:correlation_approx_1}
\begin{split}
\la {\mathbf{R}}(t) {\mathbf{R}}(t) \ra_{{\boldsymbol{\eta}},\theta} = 2 {k_{\rm B}}T t
\left(\frac{1-e^{-2 \k {{\overline}{\Gamma}}t}}{2 \k {{\overline}{\Gamma}}t}\right)
\left( {{\overline}{\Gamma}}{\mathds{1}}+ \frac{\D \G }{2} \la {\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}[\theta(t)]\ra_{\theta}\right)
\end{split}$$ Simplifying the result and using we arrive at $$\label{eq:correlation_approx_2}
\begin{split}
\la {\mathbf{R}}(t) {\mathbf{R}}(t) \ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T }{\k {{\overline}{\Gamma}}}
\left(1-e^{-2 \k {{\overline}{\Gamma}}t}\right)
\left( {{\overline}{\Gamma}}{\mathds{1}}+ \frac{\D \G }{2} \, {\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(\theta_0) \, e^{-4 D_\theta t}\right).
\end{split}$$ The mean-square displacement in the explicit form is given by $$\label{eq:msd_x_approx}
\la \D x^2(t) \ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T }{\k}
\left(1-e^{-2 \k {{\overline}{\Gamma}}t}\right)
\left( 1 + \frac{\D \G }{2{{\overline}{\Gamma}}} \, \cos 2\theta_0 \, e^{-4 D_\theta t}\right).$$
$$\label{eq:msd_y_approx}
\la \D y^2(t) \ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T }{\k}
\left(1-e^{-2 \k {{\overline}{\Gamma}}t}\right)
\left( 1 - \frac{\D \G }{2{{\overline}{\Gamma}}} \, \cos 2\theta_0 \, e^{-4 D_\theta t}\right).$$
and $$\label{eq:msd_xy_approx}
\la \D x(t) \D y(t) \ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T }{\k}
\left(1-e^{-2 \k {{\overline}{\Gamma}}t}\right)
\left( \frac{\D \G }{2{{\overline}{\Gamma}}} \, \sin 2\theta_0 \, e^{-4 D_\theta t}\right).$$ Note that there is striking difference between the \[eq:msd\_x\_approx,eq:msd\_y\_approx\] and that of \[eq:msd\_x\_final,eq:msd\_y\_final\] with respect to the limit of $\kappa \to 0$. While the later expressions correctly reproduces the free diffusion of the anisotropic particle, the limit of $\k \to 0$ in \[eq:correlation\_approx\_2\] yields the correct asymptotic result by setting $e^{-4 D_\h t} \to 0$: $$\label{eq:asymptotic_approx}
\la \D x^2(t)\ra=\la \D y^2(t) \ra =2 {k_{\rm B}}T {{\overline}{\Gamma}}t \; \;\; \textrm{and}
\; \; \;\la \D x(t) \D y(t) \ra_{{\boldsymbol{\eta}},\theta} =0.$$
Persistence Probability {#ssec:persistence_harmonic}
-----------------------
We now turn our attention to the persistence probability of the harmonically trapped ellipsoidal particle. For this, we focus on the two time correlation function $\la x(t_1)x(t_2)
\ra_{{\boldsymbol{\eta}},\theta}$. Using the perturbation series given in \[eq:perturbation\_series\] we have up to order ${\mathcal{O}}(\k\D\G/2)$ $$\label{eq:x_two_time_correlation}
\begin{split}
\la x(t_1)x(t_2)\ra_{{\boldsymbol{\eta}},\theta}&= \la
x_0(t_1)x_0(t_2)\ra_{{\boldsymbol{\eta}},\theta}\\
&-\left(\frac{\k \D\G}{2}\right)\left[
\la x_0(t_1)x_1(t_2)\ra_{{\boldsymbol{\eta}},\theta}+ \la
x_0(t_2)x_1(t_1)\ra_{{\boldsymbol{\eta}},\theta} \right]
\end{split}$$ where $t_1>t_2$. The correlation functions $ \la x_0(t_1)x_1(t_2)\ra_{{\boldsymbol{\eta}},\theta}$ and $ \la x_0(t_2)x_1(t_1)\ra_{{\boldsymbol{\eta}},\theta}$ are equal only in the asymptotic limit, that is for $t_1$ and $t_2 $ large. In this limit, the expression for the two time correlation function takes the form $$\label{eq:x_two_time_correlation_1}
\begin{split}
\la x(t_1)x(t_2)\ra_{{\boldsymbol{\eta}},\theta}&= \la
x_0(t_1)x_0(t_2)\ra_{{\boldsymbol{\eta}},\theta}\\
&-\left(\k \D\G\right)\left[
\la x_0(t_1)x_1(t_2)\ra_{{\boldsymbol{\eta}},\theta}\right]
\end{split}$$ The correlation functions $\la x_0(t_1) x_0(t_2) \ra_{{\boldsymbol{\eta}},\theta}$ and $\la x_0(t_1)x_1(t_2)\ra_{{\boldsymbol{\eta}},\theta}$ are derived in in \[appendix:appendix\_1\] (see \[eq:eq\_1\] ) and in \[appendix:appendix\_4\] (see \[eq:eq\_10\]), respectively. For completeness, we quote the main results here. $$\label{eq:x0_t1_x0_t2}
\begin{split}
\left\la x_0(t_1) x_0(t_2) \right\ra_{{\boldsymbol{\eta}},\theta}&=\frac{{k_{\rm B}}T}{\k} \left[e^{-\k {{\overline}{\Gamma}}|t_1-t_2|} -e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \right]\\
& +\left(\frac{{k_{\rm B}}T}{\k}\right)\;\k\D \G \cos 2\h_0 e^{-\k {{\overline}{\Gamma}}t_1} \left[\frac{e^{(\k {{\overline}{\Gamma}}-4 D_\h)t_2}-e^{-\k {{\overline}{\Gamma}}t_2}}{2\k {{\overline}{\Gamma}}-4 D_\h}\right]
\end{split}$$ $$\label{eq:x0_t1_x1_t2}
\begin{split}
\la x_0(t_1) x_1(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t_1} \;\left(\frac{e^{(\k
{{\overline}{\Gamma}}-4D_\h )t_2} -e^{-\k {{\overline}{\Gamma}}t_2}}{2 \k {{\overline}{\Gamma}}- 4 D_\h}\right.\\
&\left. -\frac{e^{-\k {{\overline}{\Gamma}}t_2}-e^{-(4 D_\h+\k {{\overline}{\Gamma}}) t_2}}{4 D_\h}
\right) +\left(\frac{{k_{\rm B}}T}{\k}\right)\;\left(\frac{\D \G}{2{{\overline}{\Gamma}}}\right)
e^{-\k {{\overline}{\Gamma}}t_1}\\
&\left[\frac{e^{\k {{\overline}{\Gamma}}t_2}-e^{-\k {{\overline}{\Gamma}}t_2}}{(2\k {{\overline}{\Gamma}}+4 D_\h)}
-\frac{2\k{{\overline}{\Gamma}}}{4D_\h}\frac{e^{-\k {{\overline}{\Gamma}}t_2}-e^{-(2 \k {{\overline}{\Gamma}}+4
D_\h) t_2}}
{(\k {{\overline}{\Gamma}}+4 D_\h)}\right]\\
\end{split}$$
![Plot of the survival probability $p(t)$ of a harmonically trapped anisotropic particle for different choices of the rotational diffusion constant and the stiffness of the potential, as indicated alongside each plot. The solid lines are plots of \[eq:persistence\_probability\] with the appropriate values of $\kappa,D_\parallel,D_\bot$ and $D_\theta$. While the rotational diffusion constant and the spring stiffness was varied, the translational diffusivities and the initial angle $\theta_0$ were fixed at values $D_\parallel=1$, $D_\bot=0.5$ and $\theta_0=0$. []{data-label="fig:persistence_harmonic"}](persistence_probability_harmonic_confinement_new){width="\linewidth"}
Note that in calculating the two time correlation function up to an order ${\mathcal{O}}(\k \D \G)$, we will use only the first term appearing in \[eq:x0\_t1\_x1\_t2\]. Looking at \[eq:x\_two\_time\_correlation\_1\] and \[eq:x0\_t1\_x0\_t2,eq:x0\_t1\_x1\_t2\] it is clear that the first term contained in the parenthesis in \[eq:x0\_t1\_x1\_t2\] cancels with the term proportional to $\k \D\ G$ in \[eq:x0\_t1\_x0\_t2\]. The final expression for $\la x(t_1)x(t_2)\ra_{{\boldsymbol{\eta}},\theta}$ reads $$\label{eq:x_two_time_correlation_final}
\begin{split}
\la x(t_1)x(t_2)\ra_{{\boldsymbol{\eta}},\theta}&= \left( \frac{2{k_{\rm B}}T}{\k}\right) e^{-\k {{\overline}{\Gamma}}t_1} \bigg[\sinh \k {{\overline}{\Gamma}}t_2 \\
&+\left( \frac{\k \D \G}{2}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t_2}\left(
\frac{1-e^{-4 D_\h t_2}}{4 D_\h} \right) \bigg]
\end{split}$$
As before, defining the variable $X(t)=x(t)/\sqrt{\la x^2
\ra}_{{\boldsymbol{\eta}},\theta}$, the correlation function of $\la X(t_1) X(t_2)
\ra_{{\boldsymbol{\eta}},\theta}$ is given by $$\label{eq:X_t1_X_t2}
\begin{split}
\la X(t_1) X(t_2) \ra_{{\boldsymbol{\eta}},\theta}&=\\
&\mkern-54mu \frac{e^{-\k {{\overline}{\Gamma}}t_1/2}}{e^{-\k {{\overline}{\Gamma}}t_2/2}}\left[\frac{\sinh \k {{\overline}{\Gamma}}t_2
+\left( \frac{\k \D \G}{2}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t_2}\left(
\frac{1-e^{-4 D_\h t_2}}{4 D_\h} \right)}{\sinh \k {{\overline}{\Gamma}}t_2
+\left( \frac{\k \D \G}{2}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t_2}\left(
\frac{1-e^{-4 D_\h t_2}}{4 D_\h} \right)}\right]^{1/2}
\end{split}$$ Using the transformation $e^T=e^{\k {{\overline}{\Gamma}}t} \left[\sinh \k {{\overline}{\Gamma}}t +\left( \frac{\k \D
\G}{2}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t}\left(
\frac{1-e^{-4 D_\h t}}{4 D_\h} \right) \right]^{1/2}$ for an imaginary time variable $T$, the correlation function $\la
X(T_1) X(T_2) \ra_{{\boldsymbol{\eta}},\theta}$ becomes a stationary correlator : $\la
X(T_1) X(T_2) \ra_{{\boldsymbol{\eta}},\theta} =e^{-(T_1-T_2)/2}$ and the corresponding persistence probability is given by $$\label{eq:persistence_probability}
p(t) \sim \frac{\sqrt{\k}e^{-\k {{\overline}{\Gamma}}t/2}}{\left[\sinh \k {{\overline}{\Gamma}}t +\left( \frac{\k \D
\G}{2}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t}\left(
\frac{1-e^{-4 D_\h t}}{4 D_\h} \right) \right]^{1/2}}$$ In the limit of $\D\G \to 0$ the equation correctly reproduces the persistence of probability of an isotropic particle in the presence of a harmonic trap. [@chakraborty2007] The other limit of $\k \to 0$ reproduces the persistence probability of a free anisotropic particle derived in \[eq:per\_prob\_real\_time\].
Conclusion {#sec:conclusion}
==========
In summary, we have determined the persistence probability of an anisotropic particle in two spatial dimensions, in the presence as well as in the absence of a confining harmonic potential. The two time correlation functions of the position of the particle has been calculated in both cases. In the case of a harmonically confined particle, a purtubative solution has been provided for the correlation functions. The persistence probability is computed from the two-time correlation function using suitable transformations in space and time. The determination of the rotational and the translational diffusion coefficients have been explicitly carried out for an anisotropic particle that undergoes free Brownian motion. Additionally, the analytical results have been confirmed by numerical simulation of the underlying stochastic dynamics.
Calculation of $\mathbf{\la R_{0,i}(t) R_{0,j}(t) \ra}$. {#appendix:appendix_1}
========================================================
$$\label{eq:correlation_bfR_1}
\begin{split}
\la {\mathbf{R}}_0(t) {\mathbf{R}}_0(t) \ra_{{\boldsymbol{\eta}},\theta} =\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t''
e^{-\k {{\overline}{\Gamma}}(t-t')} e^{-\k {{\overline}{\Gamma}}(t-t'')}
\left[ {{\overline}{\Gamma}}{\mathds{1}}+
\frac{\D \G}{2} {\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(t') \right] \delta (t'-t'')
\end{split}$$
$$\label{eq:correlation_bfR_2}
\la {\mathbf{R}}_0(t) {\mathbf{R}}_0(t) \ra_{{\boldsymbol{\eta}},\theta} =2 {k_{\rm B}}T e^{-2 \k {{\overline}{\Gamma}}t} \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{2\k {{\overline}{\Gamma}}t'} \left[ {{\overline}{\Gamma}}{\mathds{1}}+
\frac{\D \G}{2} \left\la {\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(t')\right\ra_{{\boldsymbol{\eta}},\theta} \right]$$
$$\label{eq:correlation_bfR_3}
\begin{split}
\la {\mathbf{R}}_0(t) {\mathbf{R}}_0(t) \ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T}{\k} {\mathds{1}}\left(1- e^{-2 \k {{\overline}{\Gamma}}t} \right)
+ 2 {k_{\rm B}}T e^{-2 \k {{\overline}{\Gamma}}t} \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{2\k {{\overline}{\Gamma}}t'} \;\frac{\D \G}{2} \;{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(\theta_0) \;e^{-4
D_\theta t'}
\end{split}$$
$$\label{eq:correlation_bfR_4}
\begin{split}
\la {\mathbf{R}}_0(t) {\mathbf{R}}_0(t) \ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T}{\k} {\mathds{1}}\left(1- e^{-2 \k {{\overline}{\Gamma}}t} \right)
+ \D D \;{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(\theta_0) \;e^{-2 \k {{\overline}{\Gamma}}t} \left(\frac{e^{(2\k {{\overline}{\Gamma}}-4 D_\theta) t}-1}{2\k {{\overline}{\Gamma}}-4 D_\theta}\right)
\end{split}$$
$$\label{eq:correlation_bfR_4}
\begin{split}
\la {\mathbf{R}}_0(t) {\mathbf{R}}_0(t) \ra_{{\boldsymbol{\eta}},\theta} = \frac{{k_{\rm B}}T}{\k} {\mathds{1}}\left(1- e^{-2 \k {{\overline}{\Gamma}}t} \right)
+ \D D \;{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\overline{{\boldsymbol{\mathcal{R}}}}$} \ht0=\dimexpr\ht0-.15ex\relax \overline{\copy0}}}(\theta_0)
\left(\frac{e^{-4 D_\theta t}-e^{-2 \k {{\overline}{\Gamma}}t}}{2\k {{\overline}{\Gamma}}-4 D_\theta}\right)
\end{split}$$
Calculation of $\la R_{0,i}(t) R_{1,j}(t) \ra$ {#appendix:appendix_2}
==============================================
Calculation of $\la R_{0,i}(t) R_{1,j}(t) \ra$.
$$\label{eq:eq_1}
\begin{split}
\left\la R_{0,i}(t_1) R_{0,j}(t_2) \right\ra_{{\boldsymbol{\eta}}}&=\int_0^{t_1}
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2\; e^{-\k {{\overline}{\Gamma}}(t_1-t'_1)} \;
e^{-\k {{\overline}{\Gamma}}(t_2-t'_2)} \;
\la \eta_i(t'_1)\eta_j(t'_2) \ra_{{\boldsymbol{\eta}}}\\
\left\la R_{0,i}(t_1) R_{0,j}(t_2) \right\ra_{{\boldsymbol{\eta}}}&=2 {k_{\rm B}}T e^{-\k
{{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_1} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2\;
e^{\k {{\overline}{\Gamma}}(t'_1+t'_2)} \;
\left[{{\overline}{\Gamma}}\delta_{ij} +\frac{\D \G}{2} {\boldsymbol{\mathcal{R}}}_{ij}(t'_1)\right] \;\delta(t'_1-t'_2)\\
\left\la R_{0,i}(t_1) R_{0,j}(t_2) \right\ra_{{\boldsymbol{\eta}}}&=2 {k_{\rm B}}T {{\overline}{\Gamma}}\delta_{ij} e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_1} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1
\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2\; e^{\k {{\overline}{\Gamma}}(t'_1+t'_2)}
\;\delta(t'_1-t'_2)\; \\
&+2 {k_{\rm B}}T\;\frac{\D \G}{2} e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_1}
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2\; e^{\k {{\overline}{\Gamma}}(t'_1+t'_2)} {\boldsymbol{\mathcal{R}}}_{ij}(t'_1) \;\delta(t'_1-t'_2)\\
\left\la R_{0,i}(t_1) R_{0,j}(t_2) \right\ra_{{\boldsymbol{\eta}}}&=2 {k_{\rm B}}T {{\overline}{\Gamma}}\delta_{ij} e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{\min(t_1,t_2)} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \;e^{2\k {{\overline}{\Gamma}}t'_1} +2 {k_{\rm B}}T\;\frac{\D \G}{2} e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{\min(t_1,t_2)} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \;e^{2\k {{\overline}{\Gamma}}t'_1}
{\boldsymbol{\mathcal{R}}}_{ij}(t'_1) \\
\left\la R_{0,i}(t_1) R_{0,j}(t_2) \right\ra_{{\boldsymbol{\eta}}}&=\frac{{k_{\rm B}}T}{\k} \delta_{ij} \left[e^{-\k {{\overline}{\Gamma}}|t_1-t_2|} -e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \right] +{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)}
\int_0^{\min(t_1,t_2)} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \;e^{2\k {{\overline}{\Gamma}}t'_1} {\boldsymbol{\mathcal{R}}}_{ij}(t'_1) \\
\end{split}$$
$$\label{eq:eq_2}
\begin{split}
\la R_{0,i}(t) R_{1,j}(t)\ra_{{\boldsymbol{\eta}},\theta}&=\left \la R_{0,i}(t) \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
e^{-\k {{\overline}{\Gamma}}(t-t')} \sum_k {\boldsymbol{\mathcal{R}}}_{jk}(t')R_{0,k}(t')
\right\ra_{{\boldsymbol{\eta}},\theta}\\
\la R_{0,i}(t) R_{1,j}(t)\ra_{{\boldsymbol{\eta}},\theta} &=\left \la \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
e^{-\k {{\overline}{\Gamma}}(t-t')} \sum_k {\boldsymbol{\mathcal{R}}}_{jk}(t')\left\la R_{0,i}(t)
R_{0,k}(t') \right\ra_{{\boldsymbol{\eta}}}
\right\ra_{\theta}
\end{split}$$
Using the final form of $\la R_{0,i}(t_1)R_{0,j}(t_2)\ra$ from \[eq:eq\_1\] and identifying $t_1\equiv t$, $t' \equiv t_2$ with $t'<t$ we get $$\label{eq:eq_3}
\left\la R_{0,i}(t) R_{0,k}(t') \right\ra_{{\boldsymbol{\eta}}}=\frac{{k_{\rm B}}T}{\k} \delta_{ik} \left[e^{-\k {{\overline}{\Gamma}}(t-t')} -e^{-\k {{\overline}{\Gamma}}(t+t')} \right] +{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t+t')}
\int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \;e^{2\k {{\overline}{\Gamma}}t'_1} {\boldsymbol{\mathcal{R}}}_{ik}(t'_1)$$ Substituting \[eq:eq\_3\] in \[eq:eq\_2\] we get $$\label{eq:eq_4}
\begin{split}
\la R_{0,i}(t) R_{1,j}(t)\ra_{{\boldsymbol{\eta}},\theta} &=\left \la \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
e^{-\k {{\overline}{\Gamma}}(t-t')} \sum_k {\boldsymbol{\mathcal{R}}}_{jk}(t')\left[
\frac{{k_{\rm B}}T}{\k} \delta_{ik} \left(e^{-\k {{\overline}{\Gamma}}(t-t')} -e^{-\k {{\overline}{\Gamma}}(t+t')} \right) +{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t+t')}
\int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \;e^{2\k {{\overline}{\Gamma}}t'_1} {\boldsymbol{\mathcal{R}}}_{ik}(t'_1)\right]
\right\ra_{\theta}\\
\la R_{0,i}(t) R_{1,j}(t)\ra_{{\boldsymbol{\eta}},\theta} &=\left(\frac{{k_{\rm B}}T}{\k}\right) {\boldsymbol{\mathcal{R}}}_{ji}(\h_0) e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
e^{-4 D_\h t'} \; \left(e^{2\k {{\overline}{\Gamma}}t'} -1 \right) +{k_{\rm B}}T\;\D \G e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
\int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \;e^{2\k {{\overline}{\Gamma}}t'_1} \left \la \sum_k
{\boldsymbol{\mathcal{R}}}_{jk}(t'){\boldsymbol{\mathcal{R}}}_{ik}(t'_1) \right\ra_{\theta}\\
\la R_{0,i}(t) R_{1,j}(t)\ra_{{\boldsymbol{\eta}},\theta} &=\left(\frac{{k_{\rm B}}T}{\k}\right) {\boldsymbol{\mathcal{R}}}_{ji}(\h_0) e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
\; \left(e^{(2\k {{\overline}{\Gamma}}-4 D_\h) t'} -e^{-4 D_\h t'} \right) +{k_{\rm B}}T\;\D \G e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
\int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \;e^{2\k {{\overline}{\Gamma}}t'_1} \left \la \sum_k
{\boldsymbol{\mathcal{R}}}_{jk}(t'){\boldsymbol{\mathcal{R}}}_{ik}(t'_1) \right\ra_{\theta}
\end{split}$$ For the mean-square displacement along the $x$ and the $y$ direction, the second term in the last line of \[eq:eq\_4\] yeilds $$\label{eq:eq_5}
\begin{split}
\left \la \sum_k
{\boldsymbol{\mathcal{R}}}_{ik}(t'){\boldsymbol{\mathcal{R}}}_{ik}(t'_1) \right\ra_{\theta}=&\la \cos 2\h(t')
\cos 2\h(t'_1) +\sin 2\h (t') \sin 2 \h(t'_1) \ra_\h=\la \cos 2
(\h(t')-\h(t'_1)) \ra_\h\\
\left \la \sum_k
{\boldsymbol{\mathcal{R}}}_{ik}(t'){\boldsymbol{\mathcal{R}}}_{ik}(t'_1) \right\ra_{\theta}&=e^{-4 D_\h (t'-t'_1)}
\end{split}$$ On the other hand for $i \neq j$, the term $ \left \la \sum_k {\boldsymbol{\mathcal{R}}}_{jk}(t'){\boldsymbol{\mathcal{R}}}_{ik}(t'_1)
\right\ra_{\theta}=0$. Using \[eq:eq\_5\] the contribution to the mean-square displacement along the $x$-direction becomes $$\label{eq:eq_6}
\begin{split}
\la x_0(t) x_1(t)\ra_{{\boldsymbol{\eta}},\theta}&=\left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 e^{-2 \k {{\overline}{\Gamma}}t}
\; \left(\frac{e^{(2\k {{\overline}{\Gamma}}-4 D_\h) t}-1}{(2\k {{\overline}{\Gamma}}-4 D_\h)} -\frac{1-e^{-4 D_\h t}}{4 D_\h} \right) +{k_{\rm B}}T\;\D \G e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
\int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \;e^{2\k {{\overline}{\Gamma}}t'_1} e^{-4 D_\h (t'-t'_1)}\\
&=\left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \; \left(\frac{e^{-4 D_\h
t}-e^{-2\k {{\overline}{\Gamma}}t}}{(2\k {{\overline}{\Gamma}}-4 D_\h)} -\frac{e^{-2\k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h} \right)+{k_{\rm B}}T\;\D \G e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
\;e^{-4 D_\h t'} \;\frac{e^{(2\k {{\overline}{\Gamma}}+4 D_\h )t'}-1}{2\k {{\overline}{\Gamma}}+4
D_\h} \\
&=\left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \; \left(\frac{e^{-4 D_\h
t}-e^{-2\k {{\overline}{\Gamma}}t}}{(2\k {{\overline}{\Gamma}}-4 D_\h)} -\frac{e^{-2\k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h} \right)+{k_{\rm B}}T\;\D \G
e^{-2 \k {{\overline}{\Gamma}}t} \left(\frac{e^{2 \k {{\overline}{\Gamma}}t}-1}{2 \k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4
D_\h)}-\frac{1-e^{-4 D_\h t}}{4 D_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)}\right)\\
&=\left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \; \left(\frac{e^{-4 D_\h
t}-e^{-2\k {{\overline}{\Gamma}}t}}{(2\k {{\overline}{\Gamma}}-4 D_\h)} -\frac{e^{-2\k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h} \right)+{k_{\rm B}}T\;\D \G
\left(\frac{1-e^{-2 \k {{\overline}{\Gamma}}t}}{2 \k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4
D_\h)}-\frac{e^{-2 \k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)}\right)\\
\end{split}$$ and that along the $y$-direction takes the form $$\label{eq:eq_y0_y1}
\begin{split}
\la y_0(t)y_1(t) \ra_{{\boldsymbol{\eta}},\theta}=-\left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \; \left(\frac{e^{-4 D_\h
t}-e^{-2\k {{\overline}{\Gamma}}t}}{(2\k {{\overline}{\Gamma}}-4 D_\h)} -\frac{e^{-2\k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h} \right)+{k_{\rm B}}T\;\D \G
\left(\frac{1-e^{-2 \k {{\overline}{\Gamma}}t}}{2 \k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4
D_\h)}-\frac{e^{-2 \k {{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+4 D_\h) t}}{4 D_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)}\right)\\
\end{split}$$
Calculation of $\la R_{1,i}(t) R_{1,j}(t) \ra$ {#appendix:appendix_3}
==============================================
The correlation matrix now takes the form $$\label{eq:correlation_matrix_R1}
\begin{split}
\la R_{1,i}(t) R_{1,j}(t) \ra_{{\boldsymbol{\eta}},\theta}=\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' e^{-\k {{\overline}{\Gamma}}(t-t')}e^{-\k {{\overline}{\Gamma}}(t-t'')}
\left\la \sum_k {\boldsymbol{\mathcal{R}}}_{ik}(t')R_{0,k}(t')
\sum_l {\boldsymbol{\mathcal{R}}}_{jl}(t'')R_{0,l}(t'') \right\ra_{{\boldsymbol{\eta}},\theta}.
\end{split}$$ Rearranging and averaging first over the translational noise we get, $$\label{eq:correlation_matrix_R2}
\begin{split}
\la R_{1,i}(t) R_{1,j}(t) \ra_{{\boldsymbol{\eta}},\theta}=\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' e^{-\k {{\overline}{\Gamma}}(t-t')}e^{-\k {{\overline}{\Gamma}}(t-t'')}
\left\la \sum_{k,l} {\boldsymbol{\mathcal{R}}}_{ik}(t') {\boldsymbol{\mathcal{R}}}_{jl}(t'') \left\la
R_{0,k}(t') R_{0,l}(t'')\right\ra_{{\boldsymbol{\eta}}}
\right\ra_{\theta}\\
\end{split}$$
$$\la R_{1,i}(t) R_{1,j}(t) \ra_{{\boldsymbol{\eta}},\theta}=2 {k_{\rm B}}T \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' e^{-\k {{\overline}{\Gamma}}(t-t')}e^{-\k {{\overline}{\Gamma}}(t-t'')}
\left\la \sum_{k,l} {\boldsymbol{\mathcal{R}}}_{ik}(t') {\boldsymbol{\mathcal{R}}}_{jl}(t'') \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1 \int_0^{t''} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 e^{-\k {{\overline}{\Gamma}}(t'-t'_1)} e^{-\k {{\overline}{\Gamma}}(t''-t'_2)} \left[ {{\overline}{\Gamma}}\delta_{kl}
+\frac{\D \G}{2}\; {\boldsymbol{\mathcal{R}}}_{kl}(t'_1) \right] \delta(t'_1 -t'_2)
\right\ra_{\theta}$$
Integrating over the delta function and ignoring the term proportional to $\D \G$ we get
$$\begin{split}
\la R_{1,i}(t) R_{1,j}(t) \ra_{{\boldsymbol{\eta}},\theta}&=2 {k_{\rm B}}T{{\overline}{\Gamma}}e^{-2
\k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \int_0^{\min(t',t'')} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1e^{2 \k {{\overline}{\Gamma}}t'_1}
\left\la \sum_{k,l} {\boldsymbol{\mathcal{R}}}_{ik}(t') {\boldsymbol{\mathcal{R}}}_{jl}(t'')
\delta_{kl}
\right\ra_{\theta} \\
\la R_{1,i}(t) R_{1,j}(t) \ra_{{\boldsymbol{\eta}},\theta}=&2 {k_{\rm B}}T{{\overline}{\Gamma}}e^{-2
\k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \int_0^{\min(t',t'')} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1e^{2 \k {{\overline}{\Gamma}}t'_1}
\left\la \sum_k {\boldsymbol{\mathcal{R}}}_{ik}(t') {\boldsymbol{\mathcal{R}}}_{jk}(t'')
\right\ra_{\theta}
\end{split}$$
In order to proceed further, we look at $\la x^2_1(t)
\ra_{{\boldsymbol{\eta}},\theta}$ and $\la y^2_1(t) \ra_{{\boldsymbol{\eta}},\h}$ by setting $i=j$ and subsequently using \[eq:eq\_5\] $$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}=2 {k_{\rm B}}T{{\overline}{\Gamma}}e^{-2
\k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \int_0^{\min(t',t'')} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1e^{2 \k {{\overline}{\Gamma}}t'_1}
\left\la \cos 2[\theta(t')-\theta(t'')] \right\ra_{\theta}
\end{split}$$ Substituting for $\la \cos 2[\h(t')-\h(t'')]\ra_\h$ from \[eq:identity\_2\] we get $$\label{eq:x1_sq}
\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}=2 {k_{\rm B}}T{{\overline}{\Gamma}}e^{-2
\k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \int_0^{\min(t',t'')} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_1e^{2 \k {{\overline}{\Gamma}}t'_1}
e^{-4 D_\h \left(t'+t''-2 \min(t',t'')\right)}
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}=2 {k_{\rm B}}T{{\overline}{\Gamma}}e^{-2
\k {{\overline}{\Gamma}}t}\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \frac{e^{2 \k {{\overline}{\Gamma}}\min(t',t'')}-1}{2 \k {{\overline}{\Gamma}}} e^{-4 D_\h \left(t'+t''-2 \min(t',t'')\right)}
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}=\left( \frac{ {k_{\rm B}}T}{\k} \right) e^{-2
\k {{\overline}{\Gamma}}t} \left[\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \left(e^{2 \k {{\overline}{\Gamma}}t''}-1\right) e^{-4 D_\h \left(t'-t''\right)} + \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_{t'}^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \left(e^{2 \k {{\overline}{\Gamma}}t'}-1 \right) e^{-4 D_\h \left(t''-t'\right)} \right]
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}=\left( \frac{ {k_{\rm B}}T}{\k} \right)
e^{-2 \k {{\overline}{\Gamma}}t} \left[\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-4 D_\h t'} \int_0^{t'}
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \left(e^{(2 \k {{\overline}{\Gamma}}+4 D_\h) t''}- e^{4 D_\h
t''}\right) + \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'\left(e^{2 \k {{\overline}{\Gamma}}t'}-1
\right) e^{4 D_\h t'}\int_{t'}^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' e^{-4 D_\h t''}
\right]
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}=\left( \frac{ {k_{\rm B}}T}{\k} \right)
e^{-2 \k {{\overline}{\Gamma}}t} \left[\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-4 D_\h t'} \left[ \frac{e^{(2 \k {{\overline}{\Gamma}}+4 D_\h) t'}-1}{2 \k {{\overline}{\Gamma}}+4 D_\h}- \frac{e^{4 D_\h
t'}-1}{4 D_\h} \right]+ \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'\left(e^{2 \k {{\overline}{\Gamma}}t'}-1
\right) e^{4 D_\h t'} \left[ \frac{e^{-4 D_\h t'}-e^{-4 D_\h
t}}{4D_\h} \right]
\right]
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}&=\left( \frac{ {k_{\rm B}}T}{\k} \right)
e^{-2 \k {{\overline}{\Gamma}}t} \left[\int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \left( \frac{e^{2 \k {{\overline}{\Gamma}}t'}}{2 \k {{\overline}{\Gamma}}+4 D_\h}-\frac{e^{-4 D_\h t'}}{2 \k {{\overline}{\Gamma}}+4 D_\h}- \frac{1-e^{-4 D_\h
t'}}{4 D_\h} \right) \right.\\
&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\left.+ \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'\left(\frac{e^{2 \k
{{\overline}{\Gamma}}t'}-1}{4 D_\h}\right) -e^{-4D_\h t} \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'
\left( \frac{e^{(2\k {{\overline}{\Gamma}}+4 D_\h)t'}-e^{4 D_\h
t'}}{4D_\h} \right)
\right]
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}&=\left( \frac{ {k_{\rm B}}T}{\k} \right)
e^{-2 \k {{\overline}{\Gamma}}t} \left[ \left( \frac{e^{2 \k {{\overline}{\Gamma}}t}-1}{2\k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4 D_\h)}-\frac{1-e^{-4 D_\h
t}}{4 D_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)}-\frac{t}{4D_\h}+ \frac{1-e^{-4 D_\h
t}}{16 D^2_\h} \right) \right.\\
&\quad \quad \quad \left.+ \left(\frac{e^{2 \k
{{\overline}{\Gamma}}t}-1}{4 D_\h 2 \k {{\overline}{\Gamma}}}\right) -\frac{t}{4D_\h}-e^{-4D_\h t}
\left( \frac{e^{(2\k {{\overline}{\Gamma}}+4 D_\h)t}-1}{4D_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)}- \frac{e^{4 D_\h
t}-1}{16D^2_\h}\right)
\right]
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}&=\left( \frac{ {k_{\rm B}}T}{\k} \right)
e^{-2 \k {{\overline}{\Gamma}}t} \left[ \left( \frac{e^{2 \k {{\overline}{\Gamma}}t}-1}{2\k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4 D_\h)}-\frac{1-e^{-4 D_\h
t}}{4 D_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)}-\frac{t}{4D_\h}+ \frac{1-e^{-4 D_\h
t}}{16 D^2_\h} \right) \right.\\
&\quad \quad \quad \left.+ \left(\frac{e^{2 \k
{{\overline}{\Gamma}}t}-1}{4 D_\h 2 \k {{\overline}{\Gamma}}} -\frac{t}{4D_\h}-
\frac{e^{2\k {{\overline}{\Gamma}}t}-1}{4D_\h (2 \k {{\overline}{\Gamma}}+4
D_\h)}-\frac{1-e^{-4 D_\h t}}{4D_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)}- \frac{1-e^{-4 D_\h
t}}{16D^2_\h}\right)
\right]
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}&=\left( \frac{ {k_{\rm B}}T}{\k} \right)
e^{-2 \k {{\overline}{\Gamma}}t} \left[ \frac{e^{2 \k {{\overline}{\Gamma}}t}-1}{\k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4 D_\h)}-\frac{t}{4D_\h}+\k {{\overline}{\Gamma}}\frac{1-e^{-4 D_\h
t}}{4 D^2_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)}
\right]
\end{split}$$
$$\begin{split}
\la x^2_1(t) \ra_{{\boldsymbol{\eta}},\theta}&=\left( \frac{ {k_{\rm B}}T}{\k} \right)
\left[ \frac{1-e^{-2 \k {{\overline}{\Gamma}}t}}{\k {{\overline}{\Gamma}}(2 \k {{\overline}{\Gamma}}+4
D_\h)}-\frac{t e^{-2 \k {{\overline}{\Gamma}}t}}{4D_\h}+\k {{\overline}{\Gamma}}\frac{e^{-2 \k
{{\overline}{\Gamma}}t}-e^{-(2 \k {{\overline}{\Gamma}}+ 4 D_\h) t}}{4 D^2_\h (2 \k {{\overline}{\Gamma}}+4 D_\h)} \right]
\end{split}$$
Calculation of $\la R_{0,i}(t) R_{2,j}(t) \ra$ {#appendix:appendix_4}
==============================================
$$\label{eq:eq_7}
\begin{split}
\la R_{0,i}(t_1) R_{1,j}(t_2) \ra_{{\boldsymbol{\eta}}}&=\left\la R_{0,i}(t_1)
\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 e^{-\k {{\overline}{\Gamma}}(t_2 -t'_2)} \sum_k
{\boldsymbol{\mathcal{R}}}_{jk}(t'_2) R_{0,k}(t'_2) \right\ra_{{\boldsymbol{\eta}}}\\
\la R_{0,i}(t_1) R_{1,j}(t_2) \ra_{{\boldsymbol{\eta}}}&=
\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 e^{-\k {{\overline}{\Gamma}}(t_2 -t'_2)} \sum_k
{\boldsymbol{\mathcal{R}}}_{jk}(t'_2) \left\la R_{0,i}(t_1)R_{0,k}(t'_2)
\right\ra_{{\boldsymbol{\eta}}}\\
\la R_{0,i}(t_1) R_{1,j}(t_2) \ra_{{\boldsymbol{\eta}}}&=
\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 e^{-\k {{\overline}{\Gamma}}(t_2 -t'_2)} \sum_k
{\boldsymbol{\mathcal{R}}}_{jk}(t'_2) \left[\frac{{k_{\rm B}}T}{\k} \delta_{ik} \left[e^{-\k {{\overline}{\Gamma}}(t_1-t'_2)} -e^{-\k {{\overline}{\Gamma}}(t_1+t'_2)} \right] +{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t'_2)}
\int_0^{\min(t_1,t'_2)} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \;e^{2\k {{\overline}{\Gamma}}t''}
{\boldsymbol{\mathcal{R}}}_{ik}(t'')\right]\\
\la R_{0,i}(t_1) R_{1,j}(t_2) \ra_{{\boldsymbol{\eta}}}&= \left(\frac{{k_{\rm B}}T}{\k}\right)\;e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2
\;e^{\k {{\overline}{\Gamma}}t'_2} \;{\boldsymbol{\mathcal{R}}}_{ji}(t'_2)\left(e^{\k {{\overline}{\Gamma}}t'_2} -e^{-\k
{{\overline}{\Gamma}}t'_2} \right) +{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{\min(t_1,t'_2)} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \;e^{2\k {{\overline}{\Gamma}}t''} \sum_k
{\boldsymbol{\mathcal{R}}}_{jk}(t'_2) {\boldsymbol{\mathcal{R}}}_{ik}(t'')\\
\end{split}$$
$$\label{eq:eq_8}
\begin{split}
\la R_{0,i}(t_1) R_{1,j}(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right)\;e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2
\;e^{\k {{\overline}{\Gamma}}t'_2} \;\left\la {\boldsymbol{\mathcal{R}}}_{ji}(t'_2) \right\ra_{\theta}\left(e^{\k {{\overline}{\Gamma}}t'_2} -e^{-\k
{{\overline}{\Gamma}}t'_2} \right) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{\min(t_1,t'_2)} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \;e^{2\k {{\overline}{\Gamma}}t''} \sum_k
\left\la{\boldsymbol{\mathcal{R}}}_{jk}(t'_2) {\boldsymbol{\mathcal{R}}}_{ik}(t'')\right\ra_{\theta}\\
\la R_{0,i}(t_1) R_{1,j}(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) {\boldsymbol{\mathcal{R}}}_{ji}(\h_0)\;e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2
\; e^{-4 D_\h t'_2}\left(e^{2\k {{\overline}{\Gamma}}t'_2} -1 \right) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2\int_0^{\min(t_1,t'_2)} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \;e^{2\k {{\overline}{\Gamma}}t''} \sum_k
\left\la{\boldsymbol{\mathcal{R}}}_{jk}(t'_2) {\boldsymbol{\mathcal{R}}}_{ik}(t'')\right\ra_{\theta}\\
\end{split}$$
$$\label{eq:eq_9}
\begin{split}
\la x_0(t_1) x_1(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \;e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2
\; \left(e^{(2\k {{\overline}{\Gamma}}-4 D_\h) t'_2} -e^{-4 D_\h t'_2} \right) \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad+{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)}\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \int_0^{t'_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \;e^{2\k {{\overline}{\Gamma}}t''} \sum_k
\left\la{\boldsymbol{\mathcal{R}}}_{ik}(t'_2) {\boldsymbol{\mathcal{R}}}_{ik}(t'')\right\ra_{\theta}\\
\la x_0(t_1) x_1(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \;e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2
\; \left(e^{(2\k {{\overline}{\Gamma}}-4 D_\h) t'_2} -e^{-4 D_\h t'_2} \right) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad+{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)}\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \int_0^{t'_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \;e^{2\k {{\overline}{\Gamma}}t''} \sum_k
\left\la \cos 2(\h(t'_2)-\h(t'')) \right\ra_{\theta}\\
\la x_0(t_1) x_1(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \;e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2
\; \left(e^{(2\k {{\overline}{\Gamma}}-4 D_\h) t'_2} -e^{-4 D_\h t'_2} \right) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad+{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)}\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \int_0^{t'_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t''
\;e^{2\k {{\overline}{\Gamma}}t''} e^{-4 D_\h (t'_2+t''-2 \min(t'_2,t''))}\\
\la x_0(t_1) x_1(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \;e^{-\k {{\overline}{\Gamma}}(t_1+t_2)} \; \left(\frac{e^{(2\k {{\overline}{\Gamma}}-4 D_\h) t_2}-1}{2 \k {{\overline}{\Gamma}}- 4 D_\h}
-\frac{1-e^{-4 D_\h t_2}}{4 D_\h} \right)+{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)}\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \int_0^{t'_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t''
\;e^{2\k {{\overline}{\Gamma}}t''} e^{-4 D_\h (t'_2-t'')}\\
\la x_0(t_1) x_1(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t_1} \;\left(\frac{e^{-4
D_\h t_2}-e^{-\k {{\overline}{\Gamma}}t_2}}{2 \k {{\overline}{\Gamma}}- 4 D_\h}
-\frac{e^{-\k {{\overline}{\Gamma}}t_2}-e^{-(4 D_\h+\k {{\overline}{\Gamma}}) t_2}}{4 D_\h} \right)+{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t_1+t_2)}\int_0^{t_2} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \frac{e^{2\k {{\overline}{\Gamma}}t'_2}
-e^{-4 D_\h t'_2}}{2\k {{\overline}{\Gamma}}+4 D_\h}\\
\la x_0(t_1) x_1(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t_1} \;\left(\frac{e^{(\k
{{\overline}{\Gamma}}-4D_\h )t_2}-e^{-\k {{\overline}{\Gamma}}t_2}}{2 \k {{\overline}{\Gamma}}- 4 D_\h}
-\frac{e^{-\k {{\overline}{\Gamma}}t_2}-e^{-(4 D_\h+\k {{\overline}{\Gamma}}) t_2}}{4 D_\h}
\right)\\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad+{k_{\rm B}}T\;\D \G
e^{-\k {{\overline}{\Gamma}}(t_1+t_2)}\left[\frac{e^{2\k {{\overline}{\Gamma}}t_2}-1}{2 \k {{\overline}{\Gamma}}(2\k {{\overline}{\Gamma}}+4 D_\h)}
-\frac{1-e^{-4 D_\h t_2}}{4D_\h(2\k {{\overline}{\Gamma}}+4 D_\h)}\right]\\
\end{split}$$
$$\label{eq:eq_10}
\begin{split}
\la x_0(t_1) x_1(t_2) \ra_{{\boldsymbol{\eta}},\theta}&= \left(\frac{{k_{\rm B}}T}{\k}\right) \cos 2\h_0 \; e^{-\k {{\overline}{\Gamma}}t_1} \;\left(\frac{e^{(\k
{{\overline}{\Gamma}}-4D_\h )t_2}-e^{-\k {{\overline}{\Gamma}}t_2}}{2 \k {{\overline}{\Gamma}}- 4 D_\h}
-\frac{e^{-\k {{\overline}{\Gamma}}t_2}-e^{-(4 D_\h+\k {{\overline}{\Gamma}}) t_2}}{4 D_\h}
\right)\\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad+\left(\frac{{k_{\rm B}}T}{\k}\right)\;\left(\frac{\D \G}{2{{\overline}{\Gamma}}}\right)
e^{-\k {{\overline}{\Gamma}}t_1}\left[\frac{e^{\k {{\overline}{\Gamma}}t_2}-e^{-\k {{\overline}{\Gamma}}t_2}}{(2\k {{\overline}{\Gamma}}+4 D_\h)}
-\frac{2\k{{\overline}{\Gamma}}}{4D_\h}\frac{e^{-\k {{\overline}{\Gamma}}t_2}-e^{-(2 \k {{\overline}{\Gamma}}+4
D_\h) t_2}}
{(\k {{\overline}{\Gamma}}+4 D_\h)}\right]\\
\end{split}$$
$$\label{eq:eq_11}
\begin{split}
\left\la R_{0,i}(t) R_{2,j}(t) \right\ra_{{\boldsymbol{\eta}},\theta}=&\left \la
R_{0,i}(t) \int_0^t {\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-\k {{\overline}{\Gamma}}(t-t')} \sum_k
{\boldsymbol{\mathcal{R}}}_{jk} (t') R_{1,k}(t') \right \ra_{{\boldsymbol{\eta}},\theta}=\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-\k {{\overline}{\Gamma}}(t-t')} \left \la\sum_k
{\boldsymbol{\mathcal{R}}}_{jk} (t') \left \la
R_{0,i}(t) R_{1,k}(t') \right \ra_{{\boldsymbol{\eta}}} \right \ra_\h\\
\end{split}$$
$$\label{eq:eq_12}
\begin{split}
\la R_{0,i}(t) R_{1,k}(t') \ra_{{\boldsymbol{\eta}}}&= \left(\frac{{k_{\rm B}}T}{\k}\right)\;e^{-\k {{\overline}{\Gamma}}(t+t')} \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2
\;e^{\k {{\overline}{\Gamma}}t'_2} \;{\boldsymbol{\mathcal{R}}}_{ki}(t'_2)\left(e^{\k {{\overline}{\Gamma}}t'_2} -e^{-\k
{{\overline}{\Gamma}}t'_2} \right) +{k_{\rm B}}T\;\D \G e^{-\k {{\overline}{\Gamma}}(t+t')} \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'' \;e^{2\k {{\overline}{\Gamma}}t''} \sum_l
{\boldsymbol{\mathcal{R}}}_{kl}(t'_2) {\boldsymbol{\mathcal{R}}}_{il}(t'')\\
\end{split}$$
Neglecting the second term in \[eq:eq\_12\], we have $$\label{eq:eq_13}
\begin{split}
\left\la R_{0,i}(t) R_{2,j}(t) \right\ra_{{\boldsymbol{\eta}},\theta}&=\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-\k {{\overline}{\Gamma}}(t-t')} \left \la\sum_k
{\boldsymbol{\mathcal{R}}}_{jk} (t') \left(\frac{{k_{\rm B}}T}{\k}\right)\;e^{-\k {{\overline}{\Gamma}}(t+t')} \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2
\;e^{\k {{\overline}{\Gamma}}t'_2} \;{\boldsymbol{\mathcal{R}}}_{ki}(t'_2)\left(e^{\k {{\overline}{\Gamma}}t'_2} -e^{-\k
{{\overline}{\Gamma}}t'_2} \right) \right \ra_\h\\
\left\la R_{0,i}(t) R_{2,j}(t) \right\ra_{{\boldsymbol{\eta}},\theta}&=\left(\frac{{k_{\rm B}}T}{\k}\right)\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 e^{-\k {{\overline}{\Gamma}}(t-t')} \;e^{-\k {{\overline}{\Gamma}}(t+t')} \;e^{\k {{\overline}{\Gamma}}t'_2} \left(e^{\k {{\overline}{\Gamma}}t'_2} -e^{-\k
{{\overline}{\Gamma}}t'_2} \right) \left \la\sum_k
{\boldsymbol{\mathcal{R}}}_{jk} (t') \;{\boldsymbol{\mathcal{R}}}_{ki}(t'_2) \right\ra_\h\\
\left\la R_{0,i}(t) R_{2,j}(t) \right\ra_{{\boldsymbol{\eta}},\theta}&=\left(\frac{{k_{\rm B}}T}{\k}\right)e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \left(e^{2\k {{\overline}{\Gamma}}t'_2} -1 \right) \left \la\sum_k
{\boldsymbol{\mathcal{R}}}_{jk} (t') \;{\boldsymbol{\mathcal{R}}}_{ki}(t'_2) \right\ra_\h\\
\end{split}$$
For the mean-square displacement along $x$ and $y$ direction, setting $j=i$ and using \[eq:eq\_5\] we get $$\label{eq:eq_14}
\begin{split}
\left\la x_0(t) x_2(t) \right\ra_{{\boldsymbol{\eta}},\theta}&=\left\la y_0(t) y_2(t) \right\ra_{{\boldsymbol{\eta}},\theta}=\left(\frac{{k_{\rm B}}T}{\k}\right)e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \left(e^{2\k {{\overline}{\Gamma}}t'_2} -1
\right) \left \la \cos 2 \left(\h(t') -\h(t'_2)\right)
\right\ra_\h\\
&=\left(\frac{{k_{\rm B}}T}{\k}\right)e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \left(e^{2\k {{\overline}{\Gamma}}t'_2} -1
\right) e^{-4 D_\h (t'-t'_2)}\\
&=\left(\frac{{k_{\rm B}}T}{\k}\right)e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-4 D_\h t'} \int_0^{t'} {\@ifnextchar^{\DIfF}{\DIfF^{}}}t'_2 \left(e^{(2\k
{{\overline}{\Gamma}}+4 D_\h) t'_2} -e^{4 D_\h t'_2}
\right) \\
&=\left(\frac{{k_{\rm B}}T}{\k}\right)e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' e^{-4 D_\h t'} \left(\frac{e^{(2\k
{{\overline}{\Gamma}}+4 D_\h) t'}-1}{2\k{{\overline}{\Gamma}}+4 D_\h} -\frac{e^{4 D_\h t'}-1}{4 D_\h}
\right) \\
&=\left(\frac{{k_{\rm B}}T}{\k}\right)e^{-2 \k {{\overline}{\Gamma}}t}\int_0^t
{\@ifnextchar^{\DIfF}{\DIfF^{}}}t' \left(\frac{e^{2\k
{{\overline}{\Gamma}}t'}-e^{-4 D_\h t'}}{2\k{{\overline}{\Gamma}}+4 D_\h} -\frac{1-e^{-4 D_\h t'}}{4 D_\h}
\right) \\
&=\left(\frac{{k_{\rm B}}T}{\k}\right)e^{-2 \k {{\overline}{\Gamma}}t}\left(\frac{e^{2\k
{{\overline}{\Gamma}}t}-1}{2 \k {{\overline}{\Gamma}}(2\k{{\overline}{\Gamma}}+4 D_\h)}-\frac{1-e^{-4 D_\h
t}}{4 D_\h(2\k{\overline}+4 D_\h)} -\frac{t}{4 D_\h}-\frac{1-e^{-4 D_\h t'}}{16 D^2_\h}
\right) \\
&=\left(\frac{{k_{\rm B}}T}{\k}\right)\left[\frac{1-e^{-2\k
{{\overline}{\Gamma}}t}}{2 \k {{\overline}{\Gamma}}(2\k{{\overline}{\Gamma}}+4 D_\h)}-\frac{te^{-2\k {{\overline}{\Gamma}}t}
}{4 D_\h}+\frac{2\k {{\overline}{\Gamma}}}{4 D_\h}\left(1-e^{-4 D_\h
t}\right) \right] \\
\end{split}$$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We give an achievable secret key rate of a binary modulated continuous variable quantum key distribution schemes in the collective attack scenario considering quantum channels that impose arbitrary noise on the exchanged signals. Bob performs homodyne measurements on the received states and the two honest parties employ a reverse reconciliation procedure in the classical post-processing step of the protocol.'
author:
- 'Yi-Bo $^1$'
- 'Matthias $^{2,3}$'
- Johannes $^4$
- 'Norbert $^{2,3}$'
title: Security of Binary Modulated Continuous Variable Quantum Key Distribution under Collective Attacks
---
Introduction
============
Quantum key distribution (QKD) is a way to establish a key between two communicating parties, traditionally called Alice and Bob, which is provable secure against any eavesdropping strategy of an technologically unlimited third party Eve. In principle, Alice and Bob can achieve this goal by exchanging nonorthogonal quantum states as signals and using non-commuting measurements on the receiver side. Any eavesdropper needs to interact with these quantum signals to gain information about the sent signal. This inevitably causes a disturbance of the signals and leads to errors in the data that Alice and Bob observe. If the amount of errors lies below a certain threshold, Alice and Bob proceed by post-processing their data: they correct for errors and employ privacy amplification to cut out any residual information that Eve might have with the raw key. In this article, we give a lower bound to the secret key rate of a continuous variable (CV) QKD scheme [@assche05a; @grosshans05a; @heid06a; @heid07a] employing homodyne detection in the collective attack scenario. As the outcomes of Bob’s measurement are continuous, it is convenient to characterize Eve’s interference with the signal states by the first and second moments of Bob’s measurement outcomes. These parameters are usually given in terms of the observed loss and the excess noise of the quantum channel connecting Alice and Bob. The proof technique presented here can be used to compute secret key rates of a binary modulated CV-QKD scheme for arbitrary, in particular non-Gaussian observations, thereby extending the results given in [@heid06a]. This is important from a conceptual point of view, as the optimality of Gaussian attacks [@garcia06a; @navascues06a] has only been shown for CV-schemes using a Gaussian modulated set of coherent states as input [@grosshans03a; @lance05a; @heid07a]. So far, the security of the binary scheme is not fully established yet, even if one limits the eavesdropper to collective attacks. Our analysis presented here is restricted to the asymptotic key limit as the number of exchanged signals $n$ approaches infinity.
We consider the class of collective attacks [@renner05b; @devetak05a], thereby limiting Eve’s possible interaction with the signal states. In this scenario, Eve can only interact with each signal individually, but she can store these quantum states for later usage. In the classical post-processing phase, Alice and Bob exchange information about their shared bit strings over a authenticated classical channel. This information eventually leaks to Eve, who can make use of this additional knowledge to employ optimized measurement on her quantum states. Our starting point of the security estimation presented here is to assume that the quantum state effectively shared by Alice, Bob and Eve, is of product form $\rho_{ABE}^{\otimes n}$. In contrast to that, the most general coherent attacks can introduce correlations between the quantum states describing subsequent signals. However, it is known that this kind of attack does not give any advantage to Eve in the asymptotic key limit, if the local dimension of the involved Hilbert spaces are finite [@renner07a]. Unfortunately, the quantum de Finetti theorem cannot be directly applied here, as one needs to bound the local dimension of Bob’s received states, which generally is infinite dimensional in CV-QKD. Recent work [@Christandl07suba] indicates that there is hope that one can extend this results to the infinite dimensional case.
The experimental feasibility of various CV-QKD schemes using coherent states as input and variations of homodyne detection has already been demonstrated [@grosshans03a; @hirano03a; @lorenz04a; @lance05a; @lorenz06a; @symul07a; @lodewyck07b]. Although promising from a technological point of view as the measurement can operate at high repetition rates, the efficiency of these schemes seems to be limited by the classical post-processing protocol. In general, the performance can be improved by using reverse reconciliation (RR): one reverses the flow of classical information in the error-correction step of the protocol, so that the raw key is based upon Bob’s measurement results [@grosshans03a]. If a practical error-correction procedure with non-ideal efficiency is considered, additional procedures like postselection [@silberhorn02b] might become favorable to increase the efficiency [@heid06a]. Here, we limit ourselves to the idealized scenario of CV-QKD involving noiseless detectors and perfect error-correction. Consequently, we suppose that a RR protocol without postselection procedures is used. The aim is to present the still missing security analysis for a discrete modulated CV-QKD valid in an idealized setting but considering arbitrary noise in the collective attack scenario. It should also be noted that in a typical physical realization, Alice sends an additional phase reference pulse to Bob via Eve’s domain. In general, Eve could interact with this additional mode as well to gain more information about the exchanged signals. As shown by Häseler *et al.* [@haseler08a], a full security proof would have to take the full two mode structure of the signals into account, but also additional measurements have to be done to test the reference-signal structure of the two modes. Here, we present a simplified proof and assume that Bob’s phase reference is prepared locally. Consequently, our signals are single modes.
Typical experiments show that the dominant contribution to the excess noise in CV-QKD is due to the electronic noise of the detectors [@lorenz06a]. Therefore, we expect the channel excess noise relevant in CV-QKD to be relatively low and of the order of a few percent. Our analysis is based on work done by Rigas [@rigas06b], who gave an estimation of the maximal eigenvalues and corresponding eigenstates of a quantum state based on homodyne detection. In our protocol, Alice uses coherent states as signals. If the quantum channel imposes loss onto the signals, but is noiseless otherwise, Bob’s received states $\rho_{B}^{x}$, conditioned on Alice sending the bit-value $x$, are pure coherent states. In contrast to that, Bob will receive mixed conditional states if the quantum channel imposes additional noise upon the signals. Consequently, the maximal eigenvalue of the received states $\rho^{x}_{B}$ will deviate from unity as $1-\tilde{\varepsilon}_{x}$. In this article, we use $\tilde{\varepsilon}_{x}$ together with the overlap of the corresponding eigenstates $\tilde{\varepsilon}_{x}$ as a figure of merit to quantify the amount excess noise present in the quantum channel. These parameters will be connected to the observed measurement outcomes of Alice and Bob in Sec. V. For $\tilde{\varepsilon}_{x}=0$, we retrieve the known results for the lossy channel given in Ref. [@heid06a]. Therefore, we expect our approach to yield positive key rates as long as the noise of the quantum channel and consequently $\tilde{\varepsilon}_{x}$ is small enough.
This article is organized as follows: In the next section, we introduce a binary CV-QKD protocol where Bob is allowed to coarse grain his continuous measurement outcomes to discrete bit-values arbitrarily, which will be used as the raw key. Therefore, we modify the known security analysis for collective attacks to include this additional step in Sec. III. Then, we proceed by computing the secret key rate of a binary CV-QKD protocol with a fixed discretization of the continuous measurement outcomes. This will be done in two steps: in Sec. IV, we give an expression for the secret key rate in terms of maximal eigenvalues and corresponding eigenstates of Bob’s received conditional states. These parameters are then estimated via Bob’s homodyne measurement in the proceeding section. We conclude with a numerical evaluation of the secret key rate in a experimental relevant scenario and a discussion of the results.
The Protocol
============
We consider a prepare-and-measure protocol using continuous variable states and homodyne detection. In general, we allow Bob to discretize his continuous measurement outcomes and to do announcements arbitrarily. However, we also give a description of a concrete protocol as an example with those steps specified. This specific protocol will be used in Sec. VI to evaluate the secret key rate for a typical experiment numerically. Any QKD protocol can be decomposed into two phases. In the first phase Alice prepares quantum states and sends them to Bob, who then performs measurements on them. In the second phase, Alice and Bob use an authenticated two-way channel for classical communication to turn the classical data (knowledge of signals sent, and measurement results) into a secret key.
Quantum phase:
: $\ $\
- Alice sends a sequence of coherent states with amplitude $\alpha$ but randomly selected opposing phase, $|\alpha \rangle $ or $|-\alpha \rangle $, to Bob. Alice stores her choice for signal $i$ in a variable $x_i$ by assigning to the choice $|\alpha \rangle$ the value $x_i=1$, and to $| -
\alpha \rangle$ to $x_i=0$.
- Bob randomly measures each signal with a homodyne measurement corresponding to the $q$ or $p$ quadratures [@silberhorn02b]. We denote Bob’s measurement results as $y_i$ and denote the basis choice by the binary variable $b_i$ (We choose the reference frame such that the signal states are modulated in the $q$ quadratures.
Classical phase:
: $\ $\
- After the quantum phase, Bob announces for each signal the measurement basis.
- Alice and Bob test their correlations by publishing randomly selected data points $x_{i}$ and $y_{i}$. Moreover all of their data (Alice’s modulation and Bob’s full measurement result) that originated from Bob measuring the $p$ quadrature is published and used to check for Eve’s interference.
- Alice and Bob dismiss the data that originated from measuring in the $p$ basis for the remaining key distillation part of the protocol in order to obtain the sifted key.
- Let us denote the string of outcomes pertaining to the sifted key as $\{\vec{x},\vec{y}\}$. From the collection of outcomes $\vec{y}$ Bob computes a string $\vec{u}$ and $\vec{\tilde{y}}$ to that we will refer to as the *announcement* and the *discretization* in the following.
- Bob announces $\vec{u}$ and keeps $\vec{\tilde{y}}$. In general, the announced vector $\vec{u}$ will have continuous entries. Bob could, for example, announce the modulus $|y_{i}|$ of his measurement result, whenever he chose the $q$-quadrature as basis. The discretization $\vec{\tilde{y}}$ is vector with discrete entries from which the secret key will be generated. This could be, for example, the sign of Bob’s outcomes $y_{i}$ whenever he measured the $q$-quadrature.
- Bob sends Alice error correction information to allow her to reconcile her string $\vec{x}$ of the sifted data to the corresponding string $\vec{\tilde{y}}$.
- Alice and Bob do privacy amplification by applying universal-2 hash functions to the string $\vec{\tilde{y}}$, now shared by Alice and Bob. This will effectively shorten the string $\vec{\tilde{y}}$ by $n\tau$ bits of information, where $n$ is number of transmitted signals.
This protocol is equivalent to an entanglement based protocol [@bennett92c]. In step 1, Alice prepares a entangled state $|\Psi \rangle =\frac{1}{\sqrt{2}}(|0\rangle |-\alpha \rangle +|1\rangle|\alpha \rangle )$ and sends the coherent state system to Bob. Then she measures her state in the $|0\rangle$ and $|1\rangle$ basis. Steps 2 to 9 remain the same.
The secret key rate in the infinite key limit
=============================================
Our security analysis follows the one given in Ref. [@renner05b; @devetak05a]. Here, we limit ourselves to the asymptotic key limit as the number of entries $n$ in the raw key $\vec{y}$ tend to infinity. Therefore, we only consider leading terms in $n$ in the formulas. Let $\mathbf{X}$, $\mathbf{Y}$, $\mathbf{\tilde{Y}}$ and $\mathbf{U}$ denote random variables that can take the values $\vec{x}$, $\vec{y}$, $\vec{\tilde{y}}$, $\vec{u}$ as introduced in the preceding section. In step 6 of our protocol, Bob announces $\vec{u}$, so that this information becomes available to both Alice and Eve. The classical information contained in the announcement can be formally embedded in a quantum system $\rho_{\mathbf{U}}$. After the announcement, the system $\rho_{\mathbf{XU}}$ describes Alice’s data and $\rho _{E
\mathbf{U}}$ describes the state Eve holds. Later in the step 8 of the protocol Bob sends error correction information over the public channel to Alice. As Eve can listen to this channel, the information $W$ about the key contained in the error correction becomes available to her. Again, we can formally embed this classical information in a quantum state $\rho_{\mathbf{W}}$. After the error correction, Alice and Bob share $\vec{\tilde{y}}$ and Eve’s knowledge about the exchanged data is summarized in a state $\rho _{E \mathbf{UW}}$. According to Ref. [@renner05a] one has to shrink the raw key by $n \tau=S(\mathbf{\tilde{Y}}:E\mathbf{UW})$ bits of information in the asymptotic key limit, where $S$ denotes the quantum mutual information [@nielsen00a], so that the final key will be secure with high probability. The secret key rate that Alice and Bob finally can obtain is given by $H(\mathbf{\tilde{Y}})-n \tau$, where $H(\mathbf{\tilde{Y}})$ describes the Shannon entropy of $\mathbf{\tilde{Y}}$, which can be evaluated after the channel test. From Ref. [@renner05b] we know that $$n \tau=S(\mathbf{\tilde{Y}}:E\mathbf{UW})\leq
S(\mathbf{\tilde{Y}}:E\mathbf{U})+I(\mathbf{\tilde{Y}}:\mathbf{W}),
\label{Icut}$$ where $I$ denotes the Shannon mutual information [@shannon48a]. Alice has to correct all the errors in her string $\vec{x}$ in step 8 of the protocol. Therefore, Bob sends Alice error correction information. The amount of error correction information necessary for Alice to succeed is given by $$I(\mathbf{\tilde{Y}}:\mathbf{W})=f(e)[H(\mathbf{\tilde{Y}})-I(\mathbf{XU}:\mathbf{\tilde{Y}})], \label{Ierror}$$where $f(e)\geq 1$ denotes the efficiency of the error correction procedure. Alice and Bob know the amount of published error correction information after step 8. In the following, we assume that the error correction is ideal, so that $f(e)=1$. From the Eqs. (\[Icut\],\[Ierror\]) we know that we have to shrink the key in the privacy amplification step by $$n \tau\leq S(\mathbf{\tilde{Y}}:E\mathbf{U})+H(\mathbf{\tilde{Y}})-I(\mathbf{XU}:\mathbf{\tilde{Y}}),$$ bits of information. The length of the final secret key that Alice and Bob can obtain is given by $$\begin{aligned}
\label{e1}
n G &=&H(\mathbf{\tilde{Y}})-n \tau \\
&\geq&I(\mathbf{XU}:\mathbf{\tilde{Y}})-S(E\mathbf{U}:\mathbf{\tilde{Y}}) \nonumber\\
&=&I(\mathbf{X}:\mathbf{\tilde{Y}}|\mathbf{U})-S(E:\mathbf{\tilde{Y}}|\mathbf{U})\nonumber\;.\end{aligned}$$ In the third line we have used the result that $S(UV:W)=S(U:W|V)+S(V:W)$, which also holds for the classical mutual information $I(UV:W)$ in particular. The length of the secret key can be lower bounded as $$\begin{aligned}
n G &=&I(\mathbf{X}:\mathbf{\tilde{Y}}|\mathbf{U})-S(E:\mathbf{\tilde{Y}}|\mathbf{U}) \notag \\
&=& I(\mathbf{X}:\mathbf{\tilde{Y}}|\mathbf{U})-S(E|\mathbf{U})+S(E|\mathbf{U\tilde{Y}}) \notag\\
&\geq&I(\mathbf{X}:\mathbf{\tilde{Y}}|\mathbf{U})-S(\mathbf{Y}:E)\;, \label{e2a}\end{aligned}$$where we have used the definition of the quantum mutual information $S(E:\mathbf{\tilde{Y}}|\mathbf{U})$ in the second line. The third line follows from the concavity of the entropy [@nielsen00a] as we will explain now. After Alice’s and Bob’s measurements, Eve’s knowledge about the exchanged data is summarized in conditional quantum states $\rho_{E}^{\vec{x},\vec{y}}$. Eve’s states conditioned on Bob’s measurement outcomes $y$ are therefore given by $$\rho_{E}^{\vec{y}}=\sum_{\vec{x}}P(\vec{x}|\vec{y})\rho_{E}^{\vec{x},\vec{y}}\;.$$ From the measured outcomes $\vec{y}$, Bob computes the announcement $\vec{u}$ and the discretization $\vec{\tilde{y}}$. This can be modelled by a classical channel described by some given conditional probability distribution $P\left(\vec{y}|\vec{u}\right), \vec{\tilde{y}}$. The state $\rho_{E}^{\vec{u},\vec{\tilde{y}}}$ can therefore be written as $$\label{rhoEuytilde}
\rho_{E}^{\vec{u},\vec{\tilde{y}}}=\sum_{\vec{y}}P\left(\vec{y}|\vec{u},\vec{\tilde{y}}\right) \rho_{E}^{\vec{y}}\;.$$ It follows that the conditional entropy $S(E|\mathbf{U\tilde{Y}})$ can be bounded from below as $$\begin{aligned}
\label{schnick}
S(E|\mathbf{U\tilde{Y}})&=\sum_{\vec{\tilde{y}}}\int d\vec{u} P(\vec{u},\vec{\tilde{y}}) S\left(\rho_{E}^{\vec{u},\vec{\tilde{y}}}\right)\\
&= \sum_{\vec{\tilde{y}}}\int d\vec{u} P(\vec{u},\vec{\tilde{y}}) S\left(\int d\vec{y} P\left(\vec{y}|\vec{u},\vec{\tilde{y}}\right) \rho_{E}^{\vec{y}}\right)\nonumber\\
&\geq \sum_{\vec{\tilde{y}}}\int d\vec{u} P(\vec{u},\vec{\tilde{y}}) \int d\vec{y} P\left(\vec{y}|\vec{u},\vec{\tilde{y}}\right)S(\rho_{E}^{\vec{y}})\nonumber\\
&= \sum_{\vec{y}} P(\vec{y}) S(\rho_{E}^{\vec{y}})=S(E|\mathbf{Y})\nonumber\;,\end{aligned}$$ where we first used Eq. (\[rhoEuytilde\]) and then the concavity of the entropy. Since the conditional entropy $S(E|\mathbf{U})$ obeys $S(E|\mathbf{U})\leq S(E)$ by the concavity of the entropy [@nielsen00a], the last line of Eq.(\[e2a\]) follows with the help of Eq. (\[schnick\]).
The lower bound in Eq. (\[e2a\]) has two terms, one depending on the discretization $\mathbf{\tilde{Y}}$, one independent of it. We expect to be able to find a discretization for arbitrary correlations between Alice and Bob, so that the first term goes to $I(\mathbf{X}:\mathbf{Y})$, e.g. a family of discretizations $\mathbf{\tilde{Y}}_{\Delta}$ that tend to the identity $\mathbf{\tilde{Y}}_{\Delta}\rightarrow\mathbf{Y}$ asymptotically as $\Delta\rightarrow 0$. Here, the parameter $\Delta$ describes the size of the coarse-graining of continuous measurement outcomes to a certain discrete value. In Sec. VI we will give a simple example of a discretization that can achieve the bound $I(\mathbf{X}:\mathbf{Y})$ for particular class of correlations between Alice and Bob without an asymptotic procedure.
In the following, we limit our security analysis to the collective attack scenario and assume that the total state shared by Alice, Bob and Eve has tensor product form $\rho_{ABE}^{\otimes n}$. Thus, the measurement outcomes $x_{i}$ and $y_{i}$ are independently identical distributed, and we can limit ourselves to single letter distributions. Then, Bob computes $\tilde{y}$ and announces values $u$ from his measured value of $y$. Therefore, Eq. (\[e2a\]) can be simplified as $$G \geq I(X:\tilde{Y}|U)-S(Y:E)\;, \label{e2}$$ where we have introduced the single letter random variables $X$, $Y$, $\tilde{Y}$ and $U$ that can take the values $x$, $y$, $\tilde{y}$ and $u$ respectively. The remaining central problem is to find a upper bound to $S(E:Y)$ as the first term is already available from the observed outcomes. Without loss of the generality, we can assume Eve holds the purification of $\rho _{ABE}$. Define the set $\Xi _{ABE}(\rho )$ as a collection of all of the possible pure state $\rho _{ABE}$ that compatible with the observations available from the measurement. The secret key rate is then given by $$\label{keyrate}
G\geq I(X:\tilde{Y}|U)-\underset{\rho _{ABE}\in \Xi _{ABE}(\rho )}{\max }S(Y:E).$$
In this article, we calculate this expression (\[keyrate\]) for the binary modulated CV-QKD scheme introduced in Sec. II. This will be done as follows: first, we will divide the entropy $S(Y:E)$ into three terms. Then we will give an upper bound to each term independently. These bounds can either be directly given by Bob’s observed first and second moments or can be expressed as functions of the maximal eigenvalues and corresponding eigenstates of Eve’s conditional states. We conclude our proof by estimating these parameters via the first and second moments of Bob’s homodyne measurements using the results of Ref. [@rigas06b] combined with an argument based on Schmidt’s decomposition. In the last section we evaluate the expected secret key rate $G$ for typical observations numerically.
Lower bound on the secret key rate
==================================
The central problem of calculating the secret key rate in a reverse reconciliation scheme according to Eq. \[keyrate\] is to find an upper bound for the mutual information $S(Y:E)$ that can be estimated by observable quantities. This will be done in the following. As the mutual information between Alice and Eve is given by $$S(X:E)=S(E)-S(E|X) \;, \notag$$ one can express the quantum mutual information $S(Y:E)$ between Bob and Eve in Eq.(\[keyrate\]) as $$S(Y:E)=S(E|X)+S(X:E)-S(E|Y) \;. \label{SYE}$$ As already mentioned, we will proceed to calculate an upper bound for $S(Y:E)$ by bounding the three terms $S(E|X)$, $S(X:E)$ and $S(E|Y)$ on the right hand side of Eq. (\[SYE\]) individually. As we will see later, we can directly compute an upper bound for $S(E|X)$ from Bob’s observed data. The remaining two terms will be given as functions of the maximal eigenvalues $1-\tilde{\varepsilon}_{x}$ and corresponding eigenstates $|\tilde{\varepsilon}_{x}\rangle $ of Eve’s conditional states $\rho _{E}^{x}$.
In Ref. [@rigas06b], Rigas presented an estimation of the maximal eigenvalue and corresponding eigenstate of an unknown quantum state based on the first and second moments of a homodyne measurement. We use this result to estimate the biggest eigenvalue $1-\tilde{\varepsilon}_{x}$ and corresponding eigenstate $|\tilde{\varepsilon}_{x}\rangle $ of Eve’s conditional states $\rho _{E}^{x}$ via Bob’s measurements. We can express Eve’s conditional states using this notation as $$\label{decompE}
{\rho }_{E}^{x}=(1-\tilde{\varepsilon}_{x})|\tilde{\varepsilon}_{x}\rangle \langle
\tilde{\varepsilon}_{x}|+\tilde{\varepsilon}_{x}\sigma _{E}^{x} \;,$$ where $|\tilde{\varepsilon}_{x}\rangle \langle \tilde{\varepsilon}_{x}|$ have $\sigma _{E}^{x}$ orthogonal support. We will refer to the eigenstate belonging to the maximal eigenvalue as the maximal eigenstate.
In the following, we will assume that the maximal eigenvalues $1-\tilde{\varepsilon}_{x}$ and eigenvectors $|\tilde{\varepsilon}_{x}\rangle$ are given. Section V contains an estimation of these parameters from measurement data and will conclude our approach.
It turns out that an upper bound for Eve’s conditional entropy $S(E|X)$, the first term on the right hand side of Eq. (\[SYE\]), can be obtained by exploiting Gaussian extremality properties [@wolf06a]. The second term is the mutual information between Alice and Eve $S(X:E)$, which can be upper bounded by employing a suitable purification method. The estimation of the third term, the entropy $S(E|Y)$ conditioned on Bob’s measurement outcomes $Y$ is technically more involved and includes a linearization of the respective quantities, so that a bound can be evaluated.
Eve’s entropy $S(E|X)$ conditioned on Alice’s variable $X$
----------------------------------------------------------
For given first and second moments of Bob’s measurement outcomes, we have to find an upper bound for Eve’s conditional entropy $S(E|X)$, which is the first term on the right hand side of Eq. (\[SYE\]). The *a priori* probabilities $P(x)$ are fixed by Alice’s state preparation. In the entanglement based description of the protocol, Alice’s state preparation is equivalent to projection measurement onto her $A$ system of a pure three party state $\rho_{ABE}$. It follows that the combined two party state $\rho_{EB}^x=|\Psi
_{BE}^{x}\rangle\langle\Psi _{BE}^{x}|$ between Eve and Bob conditioned on Alice’s measurement outcome $x$ is pure. Therefore, by Schmidt’s decomposition, we conclude that $S({\rho }_{E}^{x})=S({\rho }_{B}^{x})$ [@nielsen00a]. It is known that the state with maximal entropy $S({\rho }_{B}^{x})$ for fixed first and second moments is Gaussian [@wolf06a; @agarwal71a]. Since $S({\rho }_{E}^{x})=S({\rho }_{B}^{x})$ and $P(x)$ is fixed, one can directly apply the result given in Eqs. (15) and (16) of Ref. [@agarwal71a], so that $$\begin{aligned}
\label{SE|X}
S(E|X)&=\sum_{x}P(x)S(\rho _{E}^{x}) \\
&\leq \frac{1}{2}\sum_{x}[(1+V_{x})\log _{2}(1+V_{x})-V_{x}\log _{2}V_{x}]
\;.\nonumber\end{aligned}$$ The term $$\label{defVx}
V_{x}=\sqrt{V_{Y_{q|x}}^{2}V_{Y_{p|x}}^{2}}-1/2$$ quantifies the amount of excess noise imposed by the quantum channel connecting Alice and Bob. It is a function of Bob’s observed variances $V_{Y_{q}|X}^{2}$ and $V_{Y_{p}|X}^{2}$ of the corresponding quadrature distributions, that are given by $$\begin{aligned}
\label{quadvar}
V_{Y_{q}|X}^{2}&=\mathrm{tr} \left(\rho_{B}^{x} \hat{q}^{2}\right)- \left[\mathrm{tr}\left(\rho_{B}^{x} \hat{q} \right)\right]^{2}\\
V_{Y_{p}|X}^{2}&=\mathrm{tr} \left(\rho_{B}^{x} \hat{p}^{2}\right)- \left[\mathrm{tr}\left(\rho_{B}^{x} \hat{p} \right)\right]^{2}\;,\nonumber\\\end{aligned}$$ and the quadrature operators $\hat{q}$ and $\hat{p}$ are defined as $$\begin{aligned}
\label{quad}
\hat{q}&=\frac{1}{\sqrt2}\left(\hat{a}+\hat{a}^{\dagger}\right)\\
\hat{p}&=\frac{\mathrm{i}}{\sqrt2}\left(\hat{a}-\hat{a}^{\dagger}\right)\;,\nonumber\end{aligned}$$ whereas $\hat{a}$ and $\hat{a}^{\dagger}$ denote the photon annihilation and creation operators.
The mutual information $S(X:E)$ between Alice and Eve
-----------------------------------------------------
Here, we employ methods known from state estimation to calculate the mutual information term $S(X:E)$ between Alice and Eve in Eq. (\[SYE\]). After interacting with the signal states, Eve holds the conditional states $\rho_{E}^{x}$ in her ancilla system, that she wants to distinguish optimally in order to maximize the mutual information $S(X:E)$. If we introduce an auxiliary system $Q$ that contains a purification of the states $\rho_{E}^{0}$ and $\rho_{E}^{1}$, we can give an upper bound for $S(X:E)$: the mutual information can never increase when discarding subsystems, so that $$\label{incmut}
S(X:E)\leq S(X:QE)$$ holds. We choose the purification $Q$, so that the conditional states $\rho_{E}^{x}$ are purified as $|\Psi_{EQ}^{x}\rangle$. There are certainly purifications that would leak too much information to Eve, i.e. if one would supply Eve with a purification of the global state $\rho_{XQE}$. Since Eq.(\[incmut\]) is valid for any purification, we would ideally choose one that minimizes $S(X:QE)$ to make the bound (\[incmut\]) as tight as possible. This problem is closely connected to Uhlmann’s theorem, as we will show now.
It has been shown that the quantum mutual information between a classical register described by the binary variable $X$ and a quantum system $QE$ can be expressed as $$\label{binS}
S(X:QE)=h\left[\frac{1}{2}\left(1-| \langle \Psi_{EQ}^{0}\left|\Psi_{EQ}^{1}\rangle \right|\right)\right],$$ if the conditional states $|\Psi_{EQ}^{x}\rangle$ are pure [@heid06a]. Here, $h$ denotes the binary entropy function $$h(z)=-z \log_{2} z -(1-z) \log_{2}(1-z)\;.$$ Since $S(X:EQ)$ monotonously increases with decreasing overlap $|\langle \Psi _{EQ}^{0}|\Psi _{EQ}^{1}\rangle |$, it is sufficient to find the purification $Q$ that maximizes the overlap $|\langle \Psi _{EQ}^{0}\left|\Psi _{EQ}^{1}\rangle \right|$ to minimize $S(X:EQ)$. The solution to this problem is known as Uhlmann’s theorem [@nielsen00a]: $$\label{uhltheo}
F\left(\rho_{E}^{0},\rho_{E}^{1}\right)=\max_{|\Psi_{EQ}^{0}\rangle,|\Psi_{EQ}^{1}\rangle}|\langle \Psi _{EQ}^{0}|\Psi _{EQ}^{1}\rangle |$$ Here, the Uhlmann fidelity $F\left(\rho_{E}^{0},\rho_{E}^{1}\right)$ is defined as $$\label{uhlfid}
F\left(\rho_{E}^{0},\rho_{E}^{1}\right)=\mathrm{tr_{E}}\left(\sqrt{\sqrt{\rho_{E}^{0}}\rho_{E}^{1}\sqrt{\rho_{E}^{0}}}\right)\;.$$ Therefore, we conclude that the tightest bound obtainable from Eq. (\[incmut\]) to mutual information $S(X:QE)$ for a binary modulated setup is given by Eq. (\[uhltheo\]) and Eq. (\[binS\]) as $$\label{upboundXE}
S(X:E)\leq h\left[\frac{1}{2}\left\{1-F\left(\rho_{E}^{0},\rho_{E}^{1}\right)\right\}\right]$$
In general, the upper bound (\[upboundXE\]) of the mutual information $S(X:E)$ can be calculated, if the Eve’s conditional states $\rho_{E}^{x}$ are known. However, the full information about the states $\rho_{E}^{x}$ is usually not available from measurements. As already mentioned, we base our security analysis on the estimation of the maximal eigenvalues $1-\tilde{\varepsilon}_{x}$ and corresponding eigenstates $|\tilde{\varepsilon}_{x}\rangle$ of Eve’s conditional states $\rho_{E}^{x}$ that we will estimate by Alice and Bob’s observation. Therefore, we proceed by giving an upper bound of $S(X:E)$ as function of these parameters. This can be done by by considering a particular purification $Q$.
Any purification $|\Psi _{EQ}^{x}\rangle $ can be expanded as $$\label{decompQ}
|\Psi _{EQ}^{x}\rangle =\sum_{i}c_{i}^{x}|i_{Q }^{x}\rangle|i_{E}^{x}\rangle \;.$$ Without loss of generality, we can choose the first term in the Schmidt-decomposition (\[decompQ\]) to correspond to the maximal eigenvalue ${c_{0}^{x}}^2:=1-\tilde{\varepsilon}_{x}$. The corresponding eigenstate is then given by Eq. (\[decompE\]) as $|0_{E}^{x}\rangle=|\tilde{\varepsilon}_{x}\rangle$. With the help of expansion (\[decompQ\]), the modulus of the overlap between the two conditional states can be evaluated as $$\left|\langle \Psi _{EQ}^{0}|\Psi _{EQ}^{1}\rangle \right|=\left|\sum_{ij}c_{i}^{0}c_{j}^{1}\langle i _{i}^{0}|j_{Q}^{1}\rangle \langle i
_{E}^{0}|j _{E}^{1}\rangle \right| \;.$$ If one chooses $\langle i _{Q}^{0}|j_{Q}^{1}\rangle =\delta _{ij}e^{i\varphi _{i}}$, where $\delta _{ij}$ is the Kronecker delta function and the phase $\varphi _{i}$ is the negative of the phase of the complex number $\langle i_{E}^{0}|i _{E}^{1}\rangle $, it follows that $$\begin{aligned}
\label{ovE}
\left|\langle \Psi _{EQ}^{0}|\Psi _{EQ}^{1}\rangle \right| &=&\left|\sum_{i}c_{i}^{0}c_{i}^{1}e^{i\varphi _{i}}\langle i_{E}^{0}|i _{E}^{1}\rangle
\right| \\
&\geq &\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}\left|\langle
\tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle \right|\nonumber.\end{aligned}$$
Therefore, we obtain a lower bound on the quantum mutual information $S(X:E)$ using Eq. (\[binS\]) and Eq. (\[ovE\]) as $$\begin{aligned}
\label{SXE}
S(X:E)&\leq S(X:QE)\\&\leq h\left[\frac{1}{2}(1-\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}\gamma )\right]\;,\nonumber\end{aligned}$$ where we introduced $$\label{defgamma}
\gamma :=\left|\langle\tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle \right|\;,$$ as a short hand notation for the overlap of Eve’s maximal eigenstates. In Sec. V we will estimate the values for $\tilde{\varepsilon}_{x}$ and $\gamma$ via Bob’s homodyne measurements.
Eve’s entropy $S(E|Y)$ conditioned on Bob’s measurement outcome $Y$
-------------------------------------------------------------------
The last term of Eq. (\[SYE\]) to be estimated reads $$\label{SE|Y}
S(E|Y)=\int dyP(y)S(\rho _{E}^{y})\;.$$ Prior to Alice’s measurement, the three party state $\rho_{ABE}$ can be assumed to be pure. Since Alice performs a projection measurement on her subsystem, it follows that the combined two party state $%
\rho_{EB}^x=|\Psi _{BE}^{x}\rangle\langle\Psi _{BE}^{x}|$ between Eve and Bob conditioned on Alice’s measurement result is pure. Moreover, Bob performs a projection measurement $|y\rangle \langle y|$ on his subsystem, so that Eve’s state $|\Psi _{E}^{xy}\rangle$ conditioned on Alice’s measurement outcome $x$ and Bob’s outcome $y$ is pure. Eve’s states $\rho _{E}^{y}$ conditioned on Bob’s measurement outcome $y$ can be written as $$\rho _{E}^{y}=P(0|y)|\Psi _{E}^{0y}\rangle \langle \Psi
_{E}^{0y}|+P(1|y)|\Psi _{E}^{1y}\rangle \langle \Psi _{E}^{1y}|\;.$$ From Sec. III.C of Ref. [@heid06a] we know that $$\begin{aligned}
S(\rho _{E}^{y})&=h\left[\frac{1}{2}-\frac{1}{2}\sqrt{1-4P(0|y)P(1|y)(1-|\langle
\Psi _{E}^{0y}|\Psi _{E}^{1y}\rangle |^{2})}\right] \notag \\
&=g\left(P(0|y),\left|\langle \Psi _{E}^{0y}|\Psi _{E}^{1y}\rangle \right|\right), \label{SE|y}\end{aligned}$$ where we have introduced the function $g\left(P(0|y),\left|\langle \Psi _{E}^{0y}|\Psi_{E}^{1y}\rangle\right|\right)$ as a shorthand notation. As we can see, the entropy $S(E|Y)$ to be evaluated is a function of the overlaps $$\label{DefGamma}
\Gamma_{y}=\left|\langle \Psi _{E}^{0y}|\Psi _{E}^{1y}\rangle\right|\;,$$ that depend on the outcomes $y$. Additionally, the probability distributions $P(0|y)$ and $P(y)$ need to be estimated by the channel test. We will proceed to lower bound the entropy $S(E|Y)$ (\[SE|Y\]) by exploiting special properties of the $g$ function given by equation (\[SE|y\]). It can be easily verified that $g\left(P(0|y),\Gamma_{y}\right)$ as a function of the overlaps has the following properties: $$\begin{aligned}
g(P(0|y),1)&=0\label{propg}\\
\frac{\partial g(P(0|y),x)}{\partial x}&\leq 0\label{monoton}\\
\frac{\partial^{2} g(P(0|y),x)}{\partial x^{2}}&\leq 0\label{concave}\end{aligned}$$ We introduce positive and real parameters $\gamma_{y}$ and $\Delta\gamma_y$ such that we can rewrite the overlap $\Gamma_y$ (\[DefGamma\]) as $$\label{decompGamma}
\Gamma_{y}\leq \gamma_{y}+\Delta\gamma_y\;.$$ It follows that for any $0\leq \Gamma_{y}\leq 1$ the inequality $$\begin{aligned}
\label{approxg}
g(P(0|y),\Gamma_{y})&\geq g(P(0|y),\gamma_{y}+\Delta \gamma_{y})\\
&\geq g(P(0|y),\gamma_{y})-\frac{g(P(0|y),\gamma_{y})}{1-\gamma_{y}}\Delta\gamma_{y}\nonumber\end{aligned}$$ holds, as the first line of Eq. (\[approxg\]) follows from the monotonicity (\[monoton\]) and the second line follows from the concavity (\[concave\]) together with property (\[propg\]) if $0\leq \gamma_{y}\leq 1$. Later we will give explicit expressions for the decomposition (\[decompGamma\]), so that these properties can easily be checked. Fig. (\[gfuncpic\]) illustrates Eq. (\[approxg\]) schematically.
![Schematical representation of the function $g\left(P(0|1),\Gamma_{y}\right)$.The validity of Eq. (\[approxg\]) can easily be checked for all $\Gamma_{y}\leq
\gamma_{y}+\Delta\gamma_y$.[]{data-label="gfuncpic"}](gfunc.eps){height="6cm" width="9cm"}
Moreover, the approximation of Eq. (\[approxg\]) can simplified further, if one could find a parameter $\tilde{\gamma}$ independent of $y$ with the properties $\tilde{\gamma}\geq\gamma_{y}$ and $\tilde{\gamma}\leq 1$, as $$\begin{aligned}
\label{approxg2}
g(P(0|y),\gamma_{y})&\geq g\left(P(0|y),\tilde{\gamma}\right)\\
\frac{g\left(P(0|y),\gamma_{y}\right)}{1-\gamma_{y}}&\leq \frac{g\left(P(0|y),\tilde{\gamma}\right)}{1-\tilde{\gamma}}\nonumber\;.\end{aligned}$$ We will see later that setting $\tilde\gamma$ to $\gamma$ as defined in Eq. (\[defgamma\]) satisfies these constraints. The first bound of (\[approxg2\]) is a simple consequence of the monotonicity (\[monoton\]), whereas the second inequality follows from the properties (\[propg\]–\[concave\]). It can easily be verified by realizing that the quantity $\frac{g\left(P(0|y),\Gamma_{y}\right)}{1-\Gamma_{y}}$ is given by the modulus of the gradient of the straight line connecting the points $g\left(P(0|y),\Gamma_{y}\right)$ and $g\left(P(0|y),\Gamma_{y}=1\right)=0$. From Fig. \[gfuncpic\] it is obvious that this modulus increases if one chooses the point $\Gamma_{y}$ to be closer to one. Therefore, the second bound of (\[approxg2\]) is valid for all $\tilde{\gamma}$ satisfying $\gamma_{y} \leq \tilde{\gamma} \leq 1$. Finally, we can estimate the conditional entropy $S(E|Y)$ given by Eq. (\[SE|Y\]) with the help of the expressions (\[approxg\]) and (\[approxg2\]) as $$\begin{aligned}
\label{e6}
S(E|Y)=&\int dy P(y)S(\rho _{E}^{y}) \\
\geq& \int dy P(y)g(P(0|y),\tilde{\gamma}) \nonumber \\
&-\frac{1}{1-\tilde{\gamma} }\int dy P(y)g(P(0|y),\tilde{\gamma})\Delta\gamma_{y} \nonumber\;.\\
=& \int dy P(y)g(P(0|y),\tilde{\gamma})-\Delta S\nonumber\;,\end{aligned}$$ where we introduced the term $\Delta S$ as a shorthand notation.
In the following, we will give explicit expressions for the missing parameters $\gamma_{y}$, $\Delta\gamma_{y}$ and $\tilde{\gamma}$ in order to connect these parameters to quantities that are observable to Alice and Bob. The starting point of this analysis is again noticing that the state $|\Psi _{BE}^{x}\rangle$ that Bob and Eve share conditioned on Alice’s measurement outcome $x$ is pure, so that one can decompose it as $$|\Psi _{BE}^{x}\rangle =\sqrt{(1-\tilde{\varepsilon}_{x})}|\tilde{\beta}_{x}\rangle |\tilde{\varepsilon}_{x}\rangle +\sqrt{\tilde{\varepsilon}_{x}}|\varphi _{EB}^{x}\rangle \;, \label{stateBE}$$ using Schmidt’s decomposition theorem [@nielsen00a]. We have introduced eigenstate $|\tilde{\beta}_{x}\rangle$ of Bob’s conditional density matrix $\rho_{B}^{x}$ corresponding to the maximal eigenvalue $1-\tilde{\varepsilon}_{x}$. All terms orthogonal to $|\tilde{\beta}_{x}\rangle|\tilde{\varepsilon}_{x}\rangle$ are summed up in the term $|\varphi _{EB}^{x}\rangle$, such that $\langle\tilde{\beta}_{x}|\varphi^{x}_{EB}\rangle=0$ and $\langle\tilde{\varepsilon}_{x}|\varphi^{x}_{EB}\rangle=0$. From Eq. (\[stateBE\]), one can construct Eve’s states $|\Psi_{E}^{xy}\rangle$ conditioned on both Alice’s and Bob’s measurement outcomes as $$\label{Exy}
|\Psi _{E}^{xy}\rangle =\frac{\sqrt{(1-\tilde{\varepsilon}_{x})}\langle
y|\tilde{\beta}_{x}\rangle |\tilde{\varepsilon}_{x}\rangle +\sqrt{\tilde{\varepsilon}_{x}%
}\langle y_{B}|\varphi_{EB}^{x}\rangle }{\sqrt{P(y|x)}}\;.$$ by projecting Bob’s system onto $|y\rangle_{B} \langle y|$. The conditional probabilities $P(y|x)$ are given by $$P(y|x)=(1-\tilde{\varepsilon}_{x})|\langle y|\tilde{\beta}_{x}\rangle |^{2}+%
\tilde{\varepsilon}_{x}\left|\langle \varphi_{EB}^{x}|y\rangle_{B}\langle y|\varphi_{EB}^{x}\rangle\right|^{2}\;.$$ By setting $$\label{defayx}
a_{y}^{x}=\frac{\langle y|\tilde{\beta}_{x}\rangle }{\sqrt{P(y|x)}}$$ and $$\label{defbyx}
b_{y}^{x}=\frac{\sqrt{\langle \varphi_{EB}^{x}|y\rangle_{B} \langle y|\varphi_{EB}^{x}\rangle }}{\sqrt{P(y|x)}}\;,$$ we can express Eq.(\[Exy\]) as $$\label{psiexy}
|\Psi _{E}^{xy}\rangle =\sqrt{(1-\tilde{\varepsilon}_{x})}a_{y}^{x}|\tilde{\varepsilon}_{x}\rangle +\sqrt{\tilde{\varepsilon}_{x}}b_{y}^{x}|\varphi_{E}^{xy}\rangle ,$$ where $|\tilde{\varepsilon}_{x}\rangle$ is orthogonal to $|\varphi_{E}^{xy}\rangle$. The normalized states $|\varphi^{xy}_{E}\rangle$ are given by Eqs. (\[Exy\]), (\[defayx\]), (\[defbyx\]) and (\[psiexy\]) as $$|\varphi_{E}^{xy}\rangle=\left(\langle \varphi_{EB}^{x}|y\rangle_{B} \langle y|\varphi_{EB}^{x}\rangle\right)^{-\frac{1}{2}}\langle y_{B}|\varphi_{EB}^{x}\rangle\;.$$ Without loss of generality, we can choose $a_{y}^{x}$ and $b_{y}^{x}$ to be real. Moreover, from expansion (\[psiexy\]) it is obvious that $$\label{approxa}
\sqrt{1-\tilde{\varepsilon}_{x}}a_{y}^{x}\leq 1$$ holds. The overlap $\Gamma_{y}$ is given by Eq. (\[psiexy\]) as $$\begin{aligned}
\label{overlap}
\Gamma_{y}=&\left|\langle \Psi _{E}^{0y}|\Psi _{E}^{1y}\rangle \right|\\
=&\left|\sqrt{(1-\tilde{\varepsilon}
_{0})(1-\tilde{\varepsilon}_{1})}a_{y}^{0}a_{y}^{1}\langle \tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle \right. \notag \\
&+\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}
a_{y}^{0}b_{y}^{1}\langle \tilde{\varepsilon}_{0}|\varphi _{E}^{1y}\rangle \notag\\
&+\sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}b_{y}^{0}a_{y}^{1}\langle
\varphi _{E}^{0y}|\tilde{\varepsilon}_{1}\rangle \notag \\
&\left.+\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}b_{y}^{0}b_{y}^{1}
\langle \varphi _{E}^{0y}|\varphi _{E}^{1y}\rangle \right|\notag \;,\end{aligned}$$ so that Eq. (\[decompGamma\]) follows from (\[overlap\]) by triangle inequality with the parameters $\gamma_{y}$ and $\Delta \gamma_{y}$ defined as $$\begin{aligned}
\label{approxgammay}
\gamma_{y}=&\left|\sqrt{(1-\tilde{\varepsilon}
_{0})(1-\tilde{\varepsilon}_{1})}a_{y}^{0}a_{y}^{1}\langle \tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle\right|\\=&\left|\sqrt{(1-\tilde{\varepsilon}
_{0})(1-\tilde{\varepsilon}_{1})}a_{y}^{0}a_{y}^{1}\right|\gamma\nonumber\\
\Delta\gamma_{y}=&\left|\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}
a_{y}^{0}b_{y}^{1}\langle \tilde{\varepsilon}_{0}|\varphi _{E}^{1y}\rangle \right.\nonumber\\
&+\sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}b_{y}^{0}a_{y}^{1}\langle
\varphi _{E}^{0y}|\tilde{\varepsilon}_{1}\rangle \nonumber \\
&\left.+\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}b_{y}^{0}b_{y}^{1}
\langle \varphi _{E}^{0y}|\varphi _{E}^{1y}\rangle \right|\nonumber\;.\end{aligned}$$ With the help of Eq. (\[approxa\]), the parameter $\gamma_{y}$ can be upper bounded as $$\gamma_{y}=\left|\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}a_{y}^{0}a_{y}^{1}\right|\gamma\leq \gamma \;,$$ so that we can set $$\label{gammaeqtildegamma}
\tilde{\gamma}=\gamma$$ to satisfy $\gamma_{y}\leq \tilde{\gamma}$. Moreover, it can easily be checked that $0\leq\gamma_{y}\leq\gamma\leq 1$ using property (\[approxa\]).
In principle, we have now everything at hand to lower bound the conditional entropy $S(E|Y)$ according to Eq. (\[e6\]). However, as we will see later, we can only estimate the overlap $\gamma=\left|\langle\tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle\right|$ and eigenvalues $1-\tilde{\varepsilon}_{x}$ from Bob’s measurements. As a consequence, the parameter $\Delta\gamma_{y}$ cannot be estimated by the observation and consequently the term $\Delta S$ in Eq. (\[e6\]) cannot computed directly. Since $\Delta S$ is monotone in the parameter $\Delta\gamma_{y}$, it is again possible to lower bound the entropy $S(E|Y)$ by looking for a suitable upper bound for $\Delta\gamma_{y}$ which is a function of Bob’s observable parameters. Here, we estimate the parameter $\Delta\gamma_{y}$ starting from the definitions (\[approxgammay\]) as $$\begin{aligned}
\label{approxDeltagamma}
\Delta\gamma_{y}\leq&\left|\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}a_{y}^{0}b_{y}^{1}\right|\left|\langle \tilde{\varepsilon}_{0}|\varphi _{E}^{1y}\rangle \right|\\
&+\left|\sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}a_{y}^{1}b_{y}^{0}\right|\left|\langle\tilde{\varepsilon}_{1}|\varphi _{E}^{0y}\rangle\right| \nonumber \\
&+\left|\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}b_{y}^{0}b_{y}^{1}\right|\left|\langle \varphi _{E}^{0y}|\varphi _{E}^{1y}\rangle \right|\nonumber\\
\leq&\sqrt{\left(1-\tilde{\varepsilon}_{0}\right)\tilde{\varepsilon}_{1}}a_{y}^{0}b_{y}^{1}\sqrt{1-\left|\langle \tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle \right|^{2}}\nonumber\\
&+\sqrt{\left(1-\tilde{\varepsilon}_{1}\right)\tilde{\varepsilon}_{0}}a_{y}^{1}b_{y}^{0}\sqrt{1-\left|\langle\tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle\right|^{2}}\nonumber\\
&+\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}b_{y}^{0}b_{y}^{1}\nonumber\\
\leq&\sqrt{1-\gamma^{2}}\left(\sqrt{\tilde{\varepsilon}_{1}}b_{y}^{1}+\sqrt{\tilde{\varepsilon}_{0}}b_{y}^{0}\right)+\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}b_{y}^{0}b_{y}^{1}\nonumber\end{aligned}$$ where we first used the triangle inequality. For the second inequality in Eq. (\[approxDeltagamma\]) we used $\left|\langle \varphi _{E}^{0y}|\varphi _{E}^{1y}\rangle \right|\leq 1$ and $$\label{orthovec}
\left|\langle{\Phi}|\tilde{\varepsilon}_{x}\rangle\right|^{2}+ \left|\langle{\Phi}|\varphi_{E}^{xy}\rangle\right|^{2}\leq 1\;,$$ which is valid for any vector $|\Phi\rangle$ by orthogonality of the states $|\tilde{\varepsilon}_{x}\rangle$ and $|\varphi_{E}^{xy}\rangle$. In particular, we used $$\begin{aligned}
\label{orthoeps}
\left|\langle\tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle\right|^{2}+\left|\langle\tilde{\varepsilon}_{0}|\varphi^{1y}_{E}\rangle\right|^{2}&\leq 1\\
\left|\langle\tilde{\varepsilon}_{1}|\tilde{\varepsilon}_{0}\rangle\right|^{2}+\left|\langle\tilde{\varepsilon}_{1}|\varphi^{0y}_{E}\rangle\right|^{2}&\leq 1\nonumber\;,\end{aligned}$$ which follows from Eq. (\[orthovec\]) by setting $|\Phi\rangle=|\varepsilon_{0}\rangle$ and $x=1$ for the first inequality or respectively $|\Phi\rangle=|\varepsilon_{1}\rangle$ and $x=0$ for the last inequality in Eq. (\[orthoeps\]). In the last step of Eq. (\[approxDeltagamma\]), we used the definition (\[defgamma\]) of $\gamma$ and the bound (\[approxa\]).
With the expression (\[approxDeltagamma\]), we can upper bound the term $\Delta S$ of Eq.(\[e6\]) as $$\begin{aligned}
\label{DeltaS}
\Delta S\leq&\sqrt{\frac{1+\gamma}{1-\gamma}}\int dy P(y)g(P(0|y),\gamma)\left(\sqrt{\tilde{\varepsilon}_{0}}b_{y}^{0}+\sqrt{\tilde{\varepsilon}_{1}}b_{y}^{1}\right) \\
&+ \frac{1}{1-\gamma}\int dy P(y)g(P(0|y),\gamma)\left(\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}b_{y}^{0}b_{y}^{1}\right). \nonumber\end{aligned}$$ These integrals can be estimated first applying the completeness relation $\int dy|y\rangle \langle y|=I$ of Bob’s homodyne measurement to the definition (\[defbyx\]). It follows that $$\int dy P(y|x)b_{y}^{x}{}^{2}=\int dy\langle \varphi _{EB}^{x}|y\rangle_{B}
\langle y|\varphi _{EB}^{x}\rangle=1 \;. \label{e4}$$ This condition on the parameters $b_{y}^{x}$ enables us to upper bound the remaining terms in Eq. (\[DeltaS\]) as $$\begin{aligned}
\label{blubber}
&&\int dy P(y)g(P(0|y),\gamma)\sqrt{\tilde{\varepsilon}_{x}}b_{y}^{x} \\
&\leq &\sqrt{\frac{\tilde{\varepsilon}_{x}}{2}\int dy P(y)\frac{g^{2}(P(0|y),\gamma)}{P(x|y)}}, \notag\end{aligned}$$and $$\label{e10}
\int dy P(y)g(P(0|y),\gamma)\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}%
_{1}}b_{y}^{0}b_{y}^{1}\leq \sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}%
_{1}} g\left(\frac{1}{2},\gamma\right)\;.$$ with the help of the Cauchy-Schwarz-Buniakovsky inequality [@gradshteyn94a]. Details of this estimation can be found in Appendix \[extreme\].
Let us summarize our results. We can use Eq. (\[gammaeqtildegamma\]) in Eq. (\[e6\]) to bound the conditional entropy $S(E|Y)$ as $$\label{ichweissnet}
S(E|Y)\geq \int dy P(y)g\left(P(0|y),\gamma\right) - \Delta S \;.$$ It follows from the inequalities (\[DeltaS\]), (\[blubber\]) and (\[e10\]) that the term $\Delta S$ can be upper bounded as $$\Delta S \leq \tilde{\varepsilon}_{0}k_{0}+\tilde{\varepsilon}_{1}k_{1}+\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}\tilde{k}
\;,$$ where we defined parameters $k_{x}$ and $\tilde{k}$ as $$\begin{aligned}
\label{e11}
{k_{x}}&=\sqrt{\frac{1+\gamma}{2(1-\gamma)}\int dy P(y)\frac{g^{2}\left(P(0|y),\gamma\right)}{P(x|y)}} \\
{\tilde{k}}&=\frac{1}{1-\gamma }g\left(\frac{1}{2},\gamma \right)\nonumber \;.\end{aligned}$$ Finally, a lower bound for the conditional entropy $S(E|Y)$ is therefore given by Eqs. (\[ichweissnet\]) and (\[DeltaS\]) as $$\begin{aligned}
\label{e15}
S(E|Y)\geq& \int dy P(y)g\left(P(0|y),\gamma\right)\\
&-\sqrt{\tilde{\varepsilon}_{0}}k_{0}-\sqrt{\tilde{\varepsilon}_{1}}k_{1}-\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}\tilde{k} \;.\nonumber\end{aligned}$$
The mutual information $S(Y:E)$ between Bob and Eve
---------------------------------------------------
We have shown that an upper bound for the mutual information $S(Y:E)$ between Bob and Eve is given by Eq. (\[SYE\]), (\[SE|X\]), (\[SXE\]) and (\[e15\]) as $$\begin{aligned}
\label{e12}
S(Y:E)=& S(E|X)+S(X:E)-S(E|Y) \\
\leq& \frac{1}{2}\sum_{x}[(1+V_{x})\log _{2}(1+V_{x})-V_{x}\log _{2}V_{x}] \nonumber \\
&+ h\left[\frac{1}{2}(1-\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}\gamma )\right] \nonumber\\
&- \int dy P(y)g[P(0|y),\gamma ]\nonumber\\
&+\sqrt{\tilde{\varepsilon}_{0}}k_{0}+\sqrt{\tilde{\varepsilon}_{1}}k_{1}+\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}\tilde{k} \nonumber\\
=& \frac{1}{2}\sum_{x}[(1+V_{x})\log _{2}(1+V_{x})-V_{x}\log _{2}V_{x}] \nonumber \\
&+ s(\tilde{\varepsilon}_{x},\gamma ) \;.\end{aligned}$$ The first term in Eq. (\[e12\]) can be directly computed from Bob’s observed variances (\[quadvar\]) using Eq. (\[defVx\]). Here, we define the function $s(\tilde{\varepsilon}_{x},\gamma )$ to summarize all terms that depend on the maximal eigenvalues $1-\tilde{\varepsilon}_{x}$ and overlap $\gamma$ of the corresponding eigenstates of Eve’s conditional states. The remaining problem is to estimate these parameters via Bob’s homodyne measurement.
Maximal eigenvalue and eigenstate
=================================
We have already shown in the last section that the two party states $|\Psi
_{BE}^{x}\rangle$ conditioned on Alice’s measurement outcome $x$ can be chosen to be pure. Therefore, one can expand these conditional states using the Schmidt-decomposition (\[stateBE\]), so that the state $|\tilde
{\beta}_{x}\rangle|\tilde{\varepsilon}_{x}\rangle $ is orthogonal to $|\varphi_{EB}^{x}\rangle $. From Eq.(\[stateBE\]) it follows that the $\rho_{E}^{x}$ and $\rho_{B}^{x}$ have the same spectrum. Moreover, the eigenvectors of Bob’s and Eve’s system are determined up to a global unitary operation on Eve’s system. According to Eq. (\[e12\]), we need to estimate the modulus of the overlap of Eve’s maximal eigenstates $|\tilde{\varepsilon}_{x}\rangle$ and the maximal eigenvalues $1-\tilde{\varepsilon}_{x}$. These parameters can be estimated from the first and second moments of Bob’s measured data [@rigas06b], as we will see in the following.
Suppose the fidelity between Bob’s received conditional state $\rho_{B}^{x}$ and a pure coherent state $|\overline{\beta}\rangle $ satisfies $$\label{fidelity}
\langle \overline{\beta}_{x}|\rho _{B}^{x}|\overline{\beta}_{x}\rangle=1-\varepsilon_{x}\;.$$ The amplitude $\overline{\beta}_{x}$ is given by the first moments of Bob’s homodyne measurement as $$\begin{aligned}
\label{defbetabar}
\mathrm{Re}(\overline{\beta}_{x})&=\mathrm{tr}(\rho_{B}^{x} \hat{q})\\
\mathrm{Im}(\overline{\beta}_{x})&=\mathrm{tr}(\rho_{B}^{x} \hat{p}) \nonumber \;.\end{aligned}$$ The quadrature operators $\hat{q}$ and $\hat{p}$ are defined in Eq. (\[quad\]). In the following, we will refer to the parameter $\varepsilon_{x}$ as the mixedness of Bob’s conditional states.
It has been shown by Rigas [@rigas06b] that the mixedness $\varepsilon_{x}$ of the conditional states can be upper bounded from the outcomes of a homodyne measurement as $$\label{e16}
\varepsilon _{x}\leq
\frac{1}{2}\left[(V_{Y_{q}|x}^{2}+\frac{1}{2})(V_{Y_{p}|x}^{2}+\frac{1}{2})-1\right]=U_{x},$$ where $V_{Y_{q}|x}$ and $V_{Y_{p}|x}^{2}$ denote the variances of the $q$- and $p$-quadrature distributions (\[quadvar\]) conditioned on Alice’s variable $x$. The proof for the estimation (\[e16\]) is given in Appendix (\[mix\]). Moreover, one can also estimate the overlap $|\langle \tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle |$ of Bob’s maximal conditional eigenstates as $$\label{e17}
c_{l}(\tilde{\varepsilon}_{x},\varepsilon _{x},\kappa)\leq |\langle \tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle |\leq c_{u}(\tilde{\varepsilon}%
_{x},\varepsilon _{x},\kappa) \;,$$ if one assumes that the fidelity (\[fidelity\]) is given. Here, the parameter $\kappa$ is given by the overlap of the coherent states corresponding to the mean values (\[defbetabar\]) as $$\label{defkappa}
\kappa=\left|\langle \overline{\beta}_{0}|\overline{\beta}_{1}\rangle\right| \;.$$ The detailed expression of $c_{l}(\tilde{\varepsilon}_{x},\varepsilon _{x},\kappa)$ and $c_{u}(\tilde{\varepsilon}_{x},\varepsilon _{x},\kappa)$ can be seen in Appendix \[estovereigen\].
The results (\[e16\]) and (\[e17\]) can be used to estimate the maximal eigenvalues and overlap $\gamma$ of the corresponding eigenstates of Eve’s reduced density matrix. From the Schmidt decomposition (\[stateBE\]) it follows that the eigenvalues of Bob’s and Eve’s reduced conditional density matrices are identical, so that $$\tilde{\varepsilon}_{x}\leq\varepsilon_{x}$$ can easily be seen by expanding $\rho_{B}^{x}$ in its eigenbasis. Moreover, Eve’s attack should preserve the inner product [@heid06a], so that $\langle -\alpha |\alpha\rangle =\langle \Psi _{BE}^{0}|\Psi _{BE}^{1}\rangle $. In Appendix \[estimation\] we show that this allows us to bound the overlap $\gamma$ of Eve’s eigenstates as $$d_{l}\leq \gamma \leq d_{u} \;, \label{e18}$$where $$\label{dl}
d_{l}=\frac{|\langle -\alpha |\alpha \rangle |-\sqrt{[\sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}+\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}]^{2}+\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{0}}}{\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}c_{u}(\tilde{\varepsilon}_{x},\varepsilon _{x},\kappa)}$$ and $$\label{du}
d_{u}=\frac{|\langle -\alpha |\alpha \rangle |+\sqrt{[\sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}+\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}]^{2}+\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{0}}}{\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}c_{l}(\tilde{\varepsilon}_{x},\varepsilon _{x},\kappa)}\;.$$ The functions $c_{l}(\tilde{\varepsilon}_{x},\varepsilon _{x},\kappa)$ and $c_{u}(\tilde{\varepsilon}_{x},\varepsilon _{x},\kappa)$ are the extremal values of the overlap $\left|\langle\tilde{\beta}_0|\tilde{\beta}_{1}\rangle\right|$ of Bob’s maximal eigenstates as defined in Eq. (\[e17\]).
If the first and second moments of Bob’s measurement outcomes are fixed, $U_{x}$ is given by Eq. (\[e16\]). Therefore, the parameters $\tilde{\varepsilon}_{x}$ that are compatible with the observed data can vary between $0 \leq \tilde{\varepsilon}_{x}\leq \varepsilon_{x} \leq U_{x}$. In that respect, the quantities $\varepsilon_{x}$ and $\tilde{\varepsilon}_{x}$ are interior parameters that can only be bounded by the value of the observable quantity $U_{x}$. For any given value of $\varepsilon_{x}$, $\tilde{\varepsilon}_{x}$ and $\kappa$, the interval of compatible overlaps $|\langle \tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle |$ according to Eq. (\[e17\]) can be given. This in turn determines the interval of possible overlaps $\gamma$ via Eq. (\[e18\]). The value for $\kappa$ is obtainable from the first moments of Bob’s homodyne measurement, as can be seen from Eq. (\[defbetabar\]). Finally, the secret key rate can be obtained by $$\begin{aligned}
G \geq &I(X:\tilde{Y}|U)-\max_{\begin{array}{c}{0\leq\tilde{\varepsilon}_{x}\leq \varepsilon_{x}\leq U_{x}}\\{d_{l}\leq \gamma \leq d_{u}}
\end{array}}
\{s(\tilde{\varepsilon}_{x},\gamma )\label{final key rate}\\
&-\frac{1}{2}\sum_{x}[(1+V_{x})\log _{2}(1+V_{x})-V_{x}\log _{2}V_{x}]\}\; \nonumber .\end{aligned}$$ The maximum is taken over the interior parameters $\tilde{\varepsilon}_{x}$, $\varepsilon_{x}$ and $\gamma$ satisfying the bounds shown. These interior parameters can vary in intervals that are fixed by the values of $U_{x}$ and $\kappa$ that can be determined from the observation. As the $s$-function (\[e12\]) contains details about Bob’s measured data via the probability distributions $P(y)$ and $P(0|y)$, this additional information must be estimated from the measured data to analyze the secret key rate numerically for a given observation.
Numerical results
=================
The secret key rate (\[final key rate\]) depends on Bob’s observed probability distributions $P(y|x)$ directly via the mutual information term $I(X:\tilde{Y}|U)$ between Alice and Bob and via the term $s(\tilde{\varepsilon}_{x},\gamma)$, as can be seen from Eq. (\[e12\]). The distribution $P(y|x)$ is in principle available from experiments. To evaluate the secret key rate in an example, we simulate data for a typical experimental situation in which we find a Gaussian distribution [@grosshans03a; @lance05a; @lorenz06a]. Therefore, we choose the probability distribution $P(y|x)$ to be parameterized as $$\label{gausscond}
P(y|x)=\frac{1}{\sqrt{2\pi V_{Y_{q}|x}^{2}}}\exp \left[\frac{-(\sqrt{\eta }\alpha
_{x}-y)^{2}}{2V_{Y_{q}|x}^{2}}\right]\;.$$ Here, $\eta$ is the observed channel transmission, the amplitude $\alpha_0=-\alpha_1$ is chosen to be real. In this parameterization, the value of $\kappa$ as defined in Eq. (\[defkappa\]) is given by by the loss of the quantum channel and the overlap of Alice’s input states as $$\kappa=\left|\langle \sqrt{\eta}\alpha|-\sqrt{\eta}\alpha\rangle\right|\;.$$ Furthermore, we assume that Bob observes the same variance (\[quadvar\]) in his measured data for both the $q$- and the $p$- quadratures, so that $$V_{Y_{q}|x}^{2}=V_{Y_{p}|x}^{2}\;.$$ Here, we use the convention for the excess noise $\delta$ given in Ref. [@namiki04a]: $$\delta=\frac{V_{Y_{q}|x}^{2}}{V_{Y_{q}|x, \mathrm{Vac}}^{2}}-1$$ The quantity ${V_{Y_{q}|x,\mathrm{Vac}}^{2}}=\frac{1}{2}$ is the quadrature variance of the vacuum state. As the *a priori* probabilities $p(x)=\frac{1}{2}$ are fixed, the probability distribution $p(y)$ is can easily be evaluated with the help of (\[gausscond\]) and the secret key rate can be evaluated according to (\[final key rate\]). Fig. (\[figkey\]) shows our numerical results for the secret key rate versus the loss $1-\eta$ and different values for the excess noise $\delta$ in this typical scenario.
![Secret key rate versus channel loss for a typical scenario with optimized signal strength. The different lines correspond to an excess noise $\delta$ of $\{0, 0.0004, 0.0008, 0.0012, 0.0016, 0.0020, 0.0024\}$.[]{data-label="figkey"}](keyrateyibo.eps){height="6cm" width="9cm"}
For the simulation, we assume that Bob announces the modulus of his measurement outcomes $y$ as $u=|y|$. The values of $\tilde{y}$ are determined by the map $\tilde{y}=0$ if $y<0$ and $\tilde{y}=1$ otherwise. After the announcement, the conditional mutual information between Alice and Bob is $$\begin{aligned}
I(X:\tilde{Y}|U)&=H(X|U)+H(X|\tilde{Y}U)\\&=H(X)-H(X|Y)\nonumber\\&=I(X:Y)\;.\nonumber\end{aligned}$$ The announcement $u=|y|$ contains no information about the bit-value $x$ for symmetric probability distributions like (\[gausscond\]) as the conditional probability $p(u|x)$ for a particular announcement $u$ is independent of $x$. Therefore it follows that $H(X|U)=H(X)$. The knowledge of Bob’s measured outcome $y$ is obviously equivalent to the knowledge of $u=|y|$ and the sign of $y$, so that we have $H(X|\tilde{Y}U)=H(X|Y)$. Therefore, we can achieve $I(X:\tilde{Y}|U)=I(X:Y)$ with this simple map as long as the probability distribution satisfies the symmetry condition $p(x|u)=1/2$.
For the numerical evaluation we optimize the secret key rate $G$ over the overlap $\langle -\alpha |\alpha \rangle $ of the input states. In the optimization we vary $\alpha $ between zero and 1 with step-width 0.05. For each $\alpha$ we find the maximum of $s(\tilde{\varepsilon}_{x},\gamma)$ over all $\tilde{\varepsilon}_{x}\leq \varepsilon _{x}\leq U_{x}$ and $%
d_{l}\leq \gamma \leq d_{u}$. We find numerically that the maximum of $s(\tilde{\varepsilon}_{x},\gamma)$ is attained at the point $\gamma =d_{l}$.
Fig. (\[figkey\]) shows the results of our simulation. As we can see, the secret key rate is very susceptible to noise, whereas it coincides with the optimal bound given in Ref. [@heid06a] for lossy but noiseless quantum channels. However, one should keep in mind that we only calculated an upper bound for Eve’s knowledge, which we expect not to be tight for finite excess noise. We have bounded all three terms in Eq. (\[SYE\]) separately rather than bounding those terms simultaneously. Furthermore, one might expect to find a different purification for the system $Q$ to make the bound (\[SXE\]) tighter. Finally, we have linearized the conditional entropy $S(E|Y)$ in Section III. B in order to be able to find a bound. However, the error introduced here might be quite large.
Conclusion
==========
We have evaluated a lower bound to the secret key rate for a binary modulated CV-QKD protocol in the collective attack scenario. The analysis can be applied to any given channel noise, as Alice and Bob can estimate the conditional probability distribution $p(y|x)$ of their measurement outcomes arbitrary well in the limit that the number of exchanged signals tends to infinity. For any given probability distribution, the secret key rate can be computed according to Eq. (\[final key rate\]). Although we demonstrate that our approach yields positive secret key rates for the case of small Gaussian excess noise, these results are not satisfying from a practical point of view, as the secret key rates drop quickly with increasing excess noise. Typically, the dominant contribution to the excess noise in CV-QKD experiments originate from noisy detectors. Our numerical results therefore indicate that it is necessary to analyze these kind of schemes in a trusted device scenario, if one wants to drop the assumption of ideal detectors and obtain secret rates of practical relevance. In this scenario, Eve cannot exploit the noise added by the detectors.
There are several options to make the protocol more robust against channel excess noise. One could use more input states in order to test the quantum channel between Alice and Bob more efficiently and consequently limit Eve’s possible interaction with the signal states. If one compares the secret key rates of Fig. (\[figkey\]) with those given in Ref. [@heid07a] which correspond to a protocol using a Gaussian modulated, continuous set of input states and a quantum channel imposing Gaussian noise onto the signal states, one realizes that the robustness of the secret key rate increases by orders of magnitude. An introduction of a postselection step in the protocol can help to increase the performance as well.
The authors want to thank M. Christandl, M. Razavi, H. Häseler, T. Moroder and G. O. Myhr for helpful discussions. Y.-B. Zhao especially wants to thank Z.-F. Han and G.-C. Guo for supporting his visit to the Institute of Quantum Computing and many fruitful discussions on this topic.
This work was supported by the National Fundamental Research Program of China under Grant No 2006CB921900, the National Natural Science Foundation of China under Grants No. 60537020 and 60621064, the Knowledge Innovation Project of the Chinese Academy of Sciences (CAS), the European Union through the IST Integrated Project SECOQC, the NSERC Innovation Platform Quantum Works, the NSERC Discovery Grant and the Spanish Research Directorate, Grant FIS2005-06714.
Cauchy-Schwarz-Buniakowsky inequality {#extreme}
=====================================
The Cauchy-Schwarz-Buniakowsky inequality states [@gradshteyn94a] that for any two integrable functions $f(x)$ and $g(x)$ $$\label{cauchy}
\left(\int_{a}^{b} dy f(y)h(y)\right)^{2}\leq\left(\int_{a}^{b} dy f^{2}(y)\right)\left(\int_{a}^{b} dy h^{2}(y)\right)$$ holds. Application of inequality (\[cauchy\]) to the left hand side of expression (\[blubber\]) yields $$\begin{aligned}
&&\int dy P(y)g[P(0|y),\gamma ]\sqrt{\tilde{\varepsilon}_{x}}b_{y}^{x} \notag
\\
&=&\sqrt{\tilde{\varepsilon}_{x}}\int dy\underset{f(y)}{\underbrace{\sqrt{%
P(y|x)}b_{y}^{x}}}\underset{h(y)}{\underbrace{\{P(y) g[P(0|y),\gamma ]/\sqrt{%
P(y|x)}\}}} \notag \\
&\leq &\sqrt{\tilde{\varepsilon}_{x}}\sqrt{\int dy P(y)g^{2}[P(0|y),\gamma ]%
\frac{P(y)}{P(y|x)}}. \label{whatever}\end{aligned}$$ Since one can rewrite the conditional probability $P(y|x)$ as $%
P(y|x)=P(x|y)P(y)/P(x)$ by using Bayes’ rule and the *a priori* probabilities are given by $P(x)=\frac{1}{2}$, we have $$\label{bayes}
\frac{P(y)}{P(y|x)}=\frac{1}{2P(x|y)}\;$$ and inequality (\[blubber\]) follows from Eq. (\[whatever\]) and Eq. (\[bayes\]).
Similarly, one can evaluate the left hand side of Eq. (\[e10\]) with the condition (\[e4\]) as $$\begin{aligned}
\label{a1}
&&\int dy P(y)g\left[P(0|y)\gamma\right]\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}b_{y}^{0}b_{y}^{1} \\
&=&\sqrt{\tilde{\varepsilon}_{0}}\sqrt{\tilde{\varepsilon}_{1}}\int dy\underset{f(y)}{\underbrace{\sqrt{P(y|1)}b_{y}^{1}}}\underset{h(y)}{\underbrace{\{P(y) g[P(0|y),\gamma ]/\sqrt{P(y|1)}\}}} \notag \\
&\leq &\frac{\sqrt{\tilde{\varepsilon}_{0}\tilde{\varepsilon}_{1}}}{2}\sqrt{\int dy P(y|0){b_{y}^{0}}^{2}\frac{g^{2}\left[P(0|y),\gamma\right]}{P(0|y)P(1|y)}}\;,
\notag\end{aligned}$$ where we used Eq. (\[bayes\]) again in the last step. Furthermore one can show that $$\begin{aligned}
\label{a2}
&\int dy P(y|0){b_{y}^{0}}^{2}\frac{g^2\left[P(0|y),\gamma\right]}{P(0|y)P(1|y)}
\notag \\
&\leq \max_{y}\left\{\frac{g^{2}\left[P(0|y),\gamma\right]}{P(0|y)P(1|y)}\right\} \\
&=4g^{2}\left[\frac{1}{2},\gamma\right] \;. \notag\end{aligned}$$ The first line of Eq. (\[a2\]) again follows from the boundary condition (\[e4\]) for any integrable and bounded function $\frac{g^{2}_{y}(\gamma)}{P(0|y)P(1|y)}$. The second step can be shown by an involved but straight forward calculation. Eq. (\[e10\]) now follows from Eqs. (\[a1\]) and (\[a2\]).
Estimation of the mixedness $\varepsilon_{x}$ via homodyne measurements {#mix}
=======================================================================
In this Appendix, we prove that the parameter $\varepsilon_{x}$ as defined in Eq. (\[fidelity\]) can be estimated via Bob’s homodyne measurements as $$\begin{aligned}
\label{esteps}
\varepsilon_{x} &\leq \frac{1}{2}\left[\left(V^{2}_{y_{q}|x}+\frac{1}{2}\right)\left(V^{2}_{y_{p}|x}+\frac{1}{2}\right)-1\right]=\frac{1}{2}(W-1)\;.\end{aligned}$$ The mixedness $\varepsilon_{x}$ is given by the fidelity between Bob’s received state $\rho_{B}^{x}$ and the pure coherent state $\overline{\beta_{x}}$ as $$\label{lofid}
\langle\overline{\beta}_{x}|\rho_{B}^{x}|\overline{\beta}_{x}\rangle = 1-\varepsilon_{x}\;.$$ The amplitude $\overline{\beta}_{x}$ is given by Eq. (\[defbetabar\]) and we use the convention (\[quad\]) for quadrature operators. The conditional variances $V^{2}_{y_{q}|x}$ and $V^{2}_{y_{p}|x}$ are then given by $$\begin{aligned}
\label{condvar}
V^{2}_{y_{q}|x}&=\mathrm{tr}\left(\rho_{B}^{x}\hat{q}^{2}\right)-\left[\mathrm{tr}\left(\rho_{B}^{x}\hat{q}\right)\right]^{2}\\%=\frac{1}{4}\mathrm{tr}\left[\rho_{B}^{x}\left(2\hat{n}+1+\hat{a}^{2}+\hat{a}^{\dagger 2}\right)\right]\\
V^{2}_{y_{p}|x}&=\mathrm{tr}\left(\rho_{B}^{x}\hat{p}^{2}\right)-\left[\mathrm{tr}\left(\rho_{B}^{x}\hat{p}\right)\right]^{2}\nonumber%=\frac{1}{4}\mathrm{tr}\left[\rho_{B}^{x}\left(2\hat{n}+1-\hat{a}^{2}-\hat{a}^{\dagger 2}\right)\right]\;,\nonumber\;.\end{aligned}$$
Let us introduce a state $\overline{\rho}=\hat{D}(-\overline{\beta}_{x})\rho_{B}^{x}\hat{D}(\overline{\beta}_{x})$ with zero mean values for the quadrature operators (\[quad\]) to simplify the analysis. Here $\hat{D}(\overline{\beta}_{x})$ denotes the displacement operator according to the amplitude $\overline{\beta}_{x}$. Obviously, $$\label{fid2}
\langle\overline{\beta}|\rho_{B}^{x}|\overline{\beta}\rangle=\langle 0|\overline{\rho}|0\rangle=1-\varepsilon_{x}\;$$ holds, whereas $|0\rangle$ denotes the vacuum state. The variances (\[condvar\]) can now be evaluated with the definition (\[quad\]) as $$\begin{aligned}
\label{condvar2}
V^{2}_{y_{q}|x}&=\mathrm{tr}\left(\overline{\rho}\hat{q}^{2}\right)=\frac{1}{2}\mathrm{tr}\left[\overline{\rho}\left(2\hat{n}+1+\hat{a}^{2}+\hat{a}^{\dagger 2}\right)\right]\\
V^{2}_{y_{p}|x}&=\mathrm{tr}\left(\overline{\rho}\hat{p}^{2}\right)\nonumber=\frac{1}{2}\mathrm{tr}\left[\overline{\rho}\left(2\hat{n}+1-\hat{a}^{2}-\hat{a}^{\dagger 2}\right)\right]\nonumber\;,\end{aligned}$$ where we have introduced the photon number operator $\hat{n}=\hat{a}^{\dagger}\hat{a}$ as short hand notation. The quantity $W$ in Eq. (\[esteps\]) now reads $$\begin{aligned}
\label{dabblju}
W=&\frac{1}{4}\mathrm{tr}\left[\overline{\rho}\left(2\hat{n}+2+\hat{a}^{2}+\left(\hat{a}^{\dagger}\right)^{2}\right)\right]\\
&\times\mathrm{tr}\left[\overline\rho\left(2\hat{n}+2-\hat{a}^{2}-\left(\hat{a}^{\dagger}\right)^{2}\right)\right]\nonumber\\
=&\left[\mathrm{tr}\left(\overline{\rho}\hat{n}\right)+1\right]^2-\frac{1}{4}\mathrm{tr}\left[\overline{\rho}\left(\hat{a}^{2}+\hat{a}^{\dagger 2}\right)\right]^{2}\nonumber\;.\end{aligned}$$ We proceed in rewriting the last term in (\[dabblju\]) with the help of Eqs. (\[condvar2\]) in the Fock-basis $\{|n\rangle\}_{n}$ as $$\begin{aligned}
\label{a2a+2}
\mathrm{tr}\left[\overline{\rho}\left(\hat{a}^{2}+\hat{a}^{\dagger 2}\right)\right]=&\sum_{n=0}^{\infty}\sqrt{n+2}\sqrt{n+1} \langle n+2|\overline{\rho}|n\rangle\\
&+\sum_{n=2}^{\infty}\sqrt{n}\sqrt{n-1} \langle n-2|\overline{\rho}|n\rangle\nonumber\\
=&2\sum_{n=0}^{\infty}\sqrt{n+2}\sqrt{n+1} \mathrm{Re}\langle n+2|\overline{\rho}|n\rangle\nonumber\;.\end{aligned}$$ Since $\langle i|\overline{\rho}|j\rangle$ is a positive semidefinite matrix, any principal minor is a positive semidefinite matrix. It follows that $$\label{prinmin}
\langle i|\overline{\rho}|i\rangle \langle j|\overline{\rho}|j\rangle- \left|\langle i|\overline{\rho}|j\rangle\right|^{2} \geq 0\;,$$ as this can be interpreted as the determinant of the 2 by 2 principal minor that arises by only keeping the $i$-th and $j$-th entries. The positivity of this determinant then follows by realizing that the determinant is just the product of the non-negative eigenvalues of the corresponding principal minor [@horn85a]. The result (\[prinmin\]), together with the triangle inequality, can be used to estimate the modulus of Eq. (\[a2a+2\]) as $$\begin{aligned}
\label{moda2a+2}
\left|\mathrm{tr}\left[\overline{\rho}\left(\hat{a}^{2}+\hat{a}^{\dagger 2}\right)\right]\right|\leq& 2\sum_{n=0}^{\infty}\sqrt{n+2}\sqrt{n+1}\sqrt{\langle n|\overline{\rho}|n\rangle}\\
&\times \sqrt{\langle n+2|\overline{\rho}|n+2\rangle}\nonumber\\
\leq& 2\sqrt{\sum_{n=0}^{\infty}(n+1)\langle n|\overline{\rho}|n\rangle}\nonumber\\
&\times \sqrt{\sum_{n=0}^{\infty}(n+2)\langle n+2|\overline{\rho}|n+2\rangle}\nonumber\\
=& 2 \sqrt{\mathrm{tr}\left(\overline{\rho}\hat{n}\right)+1}\sqrt{\mathrm{tr}\left(\overline{\rho}\hat{n}\right)-\langle 1|\overline{\rho}|1\rangle}\;.\end{aligned}$$ The second estimation in Eq. (\[moda2a+2\]) follows from the Cauchy-Schwarz inequality. Inserting this result into Eq. (\[dabblju\]) yields $$\begin{aligned}
\label{dabblju2}
W \geq \left[\mathrm{tr}\left(\overline{\rho}\hat{n}\right)+1\right]\left[1+\langle1|\overline{\rho}|1\rangle\right]\;.\end{aligned}$$ We need to find a lower bound on Eq. (\[dabblju2\]) depending only on $\varepsilon_{x}$. Note that $\mathrm{tr}\left(\overline{\rho} \hat{n}\right)$ can be written as $$\begin{aligned}
\mathrm{tr}\left(\overline{\rho}\hat{n}\right)=&\langle{1}|\overline{\rho}|1\rangle+\sum_{n=2}^{\infty}\langle{n}|\overline{\rho}|n\rangle n\\
\geq& \langle{1}|\overline{\rho}|1\rangle +2\left( \varepsilon_{x} -\langle{1}|\overline{\rho}|1\rangle\right)\;.\nonumber\end{aligned}$$ As it can be seen from Eq. (\[fid2\]), the fidelity of $\overline{\rho}$ with the vacuum is $1-\varepsilon_{x}$. It follows that all matrix elements $\langle n|\overline{\rho}|n \rangle$ for $n\geq 1$ sum up to $\varepsilon_{x}$, so that $\sum_{n=2}^{\infty}\langle{n}|\overline{\rho}|n\rangle n$ is minimal if all $\langle n|\overline{\rho}|n\rangle=0$ except for $\langle 2|\overline{\rho}|2\rangle$, which has then to be equal to $\varepsilon_{x}-\langle 1|\overline{\rho}|1\rangle$ by the summing condition. Therefore, Eq. (\[dabblju2\]) can be estimated as $$\begin{aligned}
\label{dabblju3}
W&\geq \left(1+2\varepsilon_{x}-\langle 1|\overline{\rho}|1\rangle\right)\left(1+\langle 1|\overline{\rho}|1\rangle\right)\\
&=1+2\varepsilon_{x}+\langle 1|\overline{\rho}|1\rangle(2\varepsilon_{x}-\langle 1|\overline{\rho}|1\rangle)\nonumber\;,\end{aligned}$$ As $0 \leq \langle 1|\overline{\rho}|1\rangle\leq \varepsilon_{x}$, the last term of Eq. (\[dabblju3\]) is never negative and equal to zero iff $\langle 1|\overline{\rho}|1\rangle= 0$. It follows that $$\begin{aligned}
W\geq 1+2\varepsilon_{x}\;.\end{aligned}$$ Inserting this result into Eq. (\[esteps\]) concludes the proof.
Estimation to the overlap of Bob’s maximal eigenstates {#estovereigen}
======================================================
In the following, we derive explicit expressions for the bounds to the overlap $|\langle\tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle$ of Bob’s conditional eigenstates to the maximal eigenvalue $1-\tilde{\varepsilon}_{x}$ as given by expression (\[e17\]). Assume that we know the fidelity $$\langle \overline{\beta}_{x}|\rho_{B}^{x}|\overline{\beta}_{x}\rangle = 1-\varepsilon_{x}$$ of Bob’s received state $\rho_{B}^{x}$ with the coherent state $|\overline{\beta_{x}}\rangle$ is given. The amplitude $\overline{\beta_{x}}$ is defined in Eq. (\[defbetabar\]). We can express the conditional states $\rho_{B}^{x}$ in a natural basis of displaced Fock-states $\{|\phi^{x}_{k}\rangle\}=\{D(\overline{\beta}_{x})|k\rangle\}$. Here, the the parameter $k$ labels the photon number. Obviously, $|\overline{\beta}_{x}\rangle=|\phi^{x}_{0}\rangle$ holds. In this basis, $\rho_{B}^{x}$ reads $$\label{reprbobnat}
\rho_{B}^{x}=\left[
\begin{array}{ccc}
a_{00}&a_{01}&... \\
a_{01}^{*}& a_{11}&\\
\vdots&&\ddots
\end{array}
\right]= V^{x}D^{x}{V^{x}}^{\dagger}\;,$$ where $V^{x}$ denotes a unknown unitary matrix and $D^{x}$ is the representation of $\rho_{B}^{x}$ in its eigenbasis. Without loss of generality, we can choose the first element in the $D$-Matrix to correspond to the biggest eigenvalue, so that $$\label{D00}
D^{x}_{00}=1-\tilde{\varepsilon}_{x}\;.$$ From Eq. (\[reprbobnat\]), we know that $$\begin{aligned}
\label{a00}
1-\varepsilon_{x}= a_{00}&=|V_{00}^{x}|^{2}D^{x}_{00}+\sum_{k=1}^{\infty}|V_{0k}^{x}|^2 D^{x}_{kk}\end{aligned}$$ As $V^{x}$ is unitary, it follows that $$\sum_{k=1}^{\infty}|V_{0k}^{x}|^2=1-\left|V_{00}^{x}\right|^{2}\;.$$ Moreover, $D^{x}$ is normalized, so that $$\sum_{k=1}^{\infty} D^{x}_{kk}=1-D^{x}_{00}=\tilde{\varepsilon}_{x}\;,$$ where we used Eq. (\[D00\]). This can be used to bound the infinite sum in Eq. (\[a00\]) as $$\label{maaan}
\sum_{k=1}^{\infty}|V_{0k}^{x}|^2 D^{x}_{kk}\leq \left(1-\left|V_{00}^{x}\right|^{2}\right)\tilde{\varepsilon}_{x}\;,$$ since all terms $|V_{0k}^{x}|^2$ and $D^{x}_{kk}$ appearing in the sum are strictly positive. Therefore, we can bound Eq. (\[a00\]) according to inequality (\[maaan\]) as $$\begin{aligned}
1-\varepsilon_{x}&\leq \left|V_{00}^{x}\right|^{2}(1-\tilde{\varepsilon}_{x})+\left(1-\left|V_{00}^{x}\right|^{2}\right)\tilde{\varepsilon}_{x}\nonumber\\
&=\left|V_{00}^{x}\right|^{2}(1-2\tilde{\varepsilon}_{x})+\tilde{\varepsilon}_{x}\nonumber\;.\end{aligned}$$ It follows that $$\label{lowV}
\left|V_{00}^{x}\right|^{2}\geq\frac{1-\varepsilon_{x}-\tilde{\varepsilon}_{x}}{1-2\tilde{\varepsilon}_{x}}\;.$$ Moreover, one can use Eq. (\[a00\]) to obtain a lower bound on $\left|V_{00}^{x}\right|^{2}$ as $$1-\varepsilon_{x}\geq \left|V_{00}^{x}\right|^{2}(1-\tilde{\varepsilon}_{x})\;,$$ so that $$\label{upV}
\left|V^{x}_{00}\right|^{2}\leq\frac{1-\varepsilon_{x}}{1-\tilde{\varepsilon}_{x}}\;.$$ On the other hand, Bob’s conditional states can be written as $$\begin{aligned}
\label{exp1}
\rho_{B}^{0}&=V^{0}D^{0}{V^{0}}^{\dagger}\\
\rho_{B}^{1}&=U V^{1}D^{1}{V^{1}}^{\dagger} U^{\dagger}\nonumber\;,\end{aligned}$$ where the unitary operation $U$ is given, up to an unimportant unimodular phase, by $U=\hat{D}\left(\overline{\beta}_{1}\right)\hat{D}\left(-\overline{\beta}_{0}\right)$ and $\hat{D}$ denotes the displacement operator. Let us denote the eigenvectors of Bob’s conditional states $\rho_{B}^{x}$ as $\{|\tilde{\beta}_{l}^{x}\rangle\}$ with $|\tilde{\beta}_{x}\rangle$ being the eigenstate corresponding to the biggest eigenvalue $1-\tilde{\varepsilon}_{x}$. With the representation (\[exp1\]), these states can be written as $$\begin{aligned}
|\tilde{\beta}_{0}\rangle&=V^{0}|\phi_{0}^{0}\rangle\\
|\tilde{\beta}_{1}\rangle&=U V^{1}|\phi_{0}^{0}\rangle\nonumber\;,\end{aligned}$$ so that $$\begin{aligned}
\langle \tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle=\sum_{k,j=0}^{\infty}{V^{0}_{k0}}^{*}U_{kl}V_{l0}^{1}\end{aligned}$$ By use of the triangle inequalities, one can construct an upper bound as $$\begin{aligned}
\label{CSu}
\left| \langle\tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle\right|\leq& \left|U_{00}\right|\left|V_{00}^{0}\right|\left|V_{00}^{1}\right|+\left|V_{00}^{0}\right|\left|\sum_{l=1}^{\infty} U_{0l}V_{l0}^{1}\right|\\
&+\left|V_{00}^{1}\right|\left|\sum_{k=1}^{\infty} U_{k0}{V_{k0}^{0}}^{*}\right|+\left|\sum_{k,l=1}^{\infty} {V_{k0}^{0}}^{*}U_{kl}{V_{l0}^{1}}\right|\nonumber\end{aligned}$$ and a lower bound as $$\begin{aligned}
\label{CSl}
\left| \langle\tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle\right|\geq& \left|U_{00}\right|\left|V_{00}^{0}\right|\left|V_{00}^{1}\right|-\left|V_{00}^{0}\right|\left|\sum_{l=1}^{\infty} U_{0l}V_{l0}^{1}\right|\\
&-\left|V_{00}^{1}\right|\left|\sum_{k=1}^{\infty} U_{k0}{V_{k0}^{0}}^{*}\right|-\left|\sum_{k,l=1}^{\infty} {V_{k0}^{0}}^{*}U_{kl}{V_{l0}^{1}}\right|\nonumber\;.\end{aligned}$$ Upper bounds on the sums in Eqs.(\[CSu\]) and (\[CSl\]) can be obtained by using the Cauchy-Schwarz inequality as $$\begin{aligned}
\label{CS2}
\sum_{k=1}^{\infty}\left|U_{k0}\right|\left|V_{k0}^{0}\right|&\leq \sqrt{1-\left|U_{00}\right|^{2}}\sqrt{1-\left|V_{00}^{0}\right|^{2}}\\
\sum_{l=1}^{\infty}\left|U_{0l}\right|\left|V_{l0}^{1}\right|&\leq \sqrt{1-\left|U_{00}\right|^{2}}\sqrt{1-\left|V_{00}^{1}\right|^{2}}\nonumber\\
\left|\sum_{k,l=1}^{\infty} {V_{k0}^{0}}^{*}U_{kl}{V_{l0}^{1}}\right|&\leq \sqrt{1-\left|V_{00}^{0}\right|^{2}}\sqrt{1-\left|V_{00}^{1}\right|^{2}}\nonumber\;.\end{aligned}$$ It is easy to see that $|U_{00}|=|\langle \overline{\beta}_{0}|\overline{\beta}_{1}\rangle|:=\kappa$. Finally, inserting Eqs. (\[lowV\], \[upV\], \[CS2\]) in Eq. (\[CSu\]) yields $$\begin{aligned}
\label{cup}
\left|\langle\tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle\right|\leq& \kappa \sqrt{\frac{1-\varepsilon_{0}}{1-\tilde{\varepsilon}_{0}}}\sqrt{\frac{1-\varepsilon_{1}}{1-\tilde{\varepsilon}_{1}}}\\
&+\sqrt{1-\kappa^{2}}\sqrt{\frac{1-\varepsilon_{0}}{1-\tilde{\varepsilon}_{0}}}\sqrt{\frac{\varepsilon_{1}-\tilde{\varepsilon}_{1}}{1-2\tilde{\varepsilon}_{1}}}\nonumber\\
&+\sqrt{1-\kappa^{2}}\sqrt{\frac{1-\varepsilon_{1}}{1-\tilde{\varepsilon}_{1}}}\sqrt{\frac{\varepsilon_{0}-\tilde{\varepsilon}_{0}}{1-2\tilde{\varepsilon}_{0}}}\nonumber\\
&+\sqrt{\frac{\varepsilon_{1}-\tilde{\varepsilon}_{1}}{1-2\tilde{\varepsilon}_{1}}}\sqrt{\frac{\varepsilon_{0}-\tilde{\varepsilon}_{0}}{1-2\tilde{\varepsilon}_{0}}}\nonumber\;.\end{aligned}$$ Similarly, a lower bound can be obtained by Eqs. (\[lowV\], \[upV\], \[CS2\]) and (\[CSl\]) as $$\begin{aligned}
\label{clow}
\left|\langle\tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle\right|\geq& \kappa \sqrt{\frac{1-\varepsilon_{0}-\tilde{\varepsilon}_{0}}{1-2\tilde{\varepsilon}_{0}}}\sqrt{\frac{1-\varepsilon_{1}-\tilde{\varepsilon}_{1}}{1-2\tilde{\varepsilon}_{1}}}\\
&-\sqrt{1-\kappa^{2}}\sqrt{\frac{1-\varepsilon_{0}}{1-\tilde{\varepsilon}_{0}}}
\sqrt{\frac{\varepsilon_{1}-\tilde{\varepsilon}_{1}}{1-2\tilde{\varepsilon}_{1}}}\nonumber\\
&-\sqrt{1-\kappa^{2}}\sqrt{\frac{1-\varepsilon_{1}}{1-\tilde{\varepsilon}_{1}}}\sqrt{\frac{\varepsilon_{0}-\tilde{\varepsilon}_{0}}{1-2\tilde{\varepsilon}_{0}}}\nonumber\\
&-\sqrt{\frac{\varepsilon_{1}-\tilde{\varepsilon}_{1}}{1-2\tilde{\varepsilon}_{1}}}\sqrt{\frac{\varepsilon_{0}-\tilde{\varepsilon}_{0}}{1-2\tilde{\varepsilon}_{0}}}\nonumber\;.\end{aligned}$$ The explicit expression for $c_{l}\left(\tilde{\varepsilon}_{x},\varepsilon_{x}, \kappa\right)$ is therefore given by Eq. (\[clow\]) and respectively, $c_{u}\left(\tilde{\varepsilon}_{x},\varepsilon_{x}, \kappa\right)$ is given by Eq. (\[cup\]).
Estimation to the overlap of Eve’s maximal eigenstates {#estimation}
======================================================
In the collective attack scenario, Eve’s attack can be modelled by attaching an ancilla system to the signals $|\pm \alpha \rangle$ and performing a unitary operation on the joint system. As any unitary preserves the inner product, the overlap $|\langle \Psi _{BE}^{0}|\Psi_{BE}^{1}\rangle|$ of the states after the interaction is given by input overlap $|\langle -\alpha |\alpha \rangle |$. This can be written as $$\begin{aligned}
\label{innerprod}
|\langle -\alpha |\alpha \rangle | &=&|\langle \Psi _{BE}^{0}|\Psi
_{BE}^{1}\rangle| \\
&=&|\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}\langle
\tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle \langle \tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle \notag \\
&&+\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}\langle \tilde{\beta}_{0}|\langle \tilde{\varepsilon}_{0}|\varphi _{EB}^{1}\rangle \notag \\
&&+\sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}\langle \varphi
_{EB}^{0}|\tilde{\beta}_{1}\rangle |\tilde{\varepsilon}_{1}\rangle \notag \\
&&+\sqrt{\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{0}}\langle \varphi
_{EB}^{0}|\varphi _{EB}^{1}\rangle |. \notag\end{aligned}$$using decomposition (\[stateBE\]), where $|\varphi _{EB}^{x}\rangle $ is orthogonal to $|\tilde{\beta}_{x}\rangle |\tilde{\varepsilon}_{x}\rangle$. This orthogonality can be used to construct the inequalities $$\begin{aligned}
\label{propover}
|\langle\varphi _{EB}^{0}|\varphi _{EB}^{1}\rangle |^{2}+|\langle
\varphi_{EB}^{0}|\tilde{\beta}_{1}\rangle |\tilde{\varepsilon}_{1}\rangle |^{2}&\leq 1 \\
|\langle \varphi _{EB}^{0}|\varphi _{EB}^{1}\rangle |^{2}+|\langle
\tilde{\beta}_{0}|\langle \tilde{\varepsilon}_{0}|\varphi _{EB}^{1}\rangle |^{2}&\leq 1\;.
\notag\end{aligned}$$ We can estimate the last three terms of the right hand side of Eq. (\[innerprod\]) using the triangle inequality and inequalities (\[propover\]) as $$\begin{aligned}
\label{b1}
|&\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}\langle
\tilde{\beta}_{0}|\langle \tilde{\varepsilon}_{0}|\varphi _{EB}^{1}\rangle \\
&+\sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}\langle
\varphi_{EB}^{0}|\tilde{\beta}_{1}\rangle |\tilde{\varepsilon}_{1}\rangle+ \notag \\
&+\sqrt{\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{0}}\langle \varphi
_{EB}^{0}|\varphi_{EB}^{1}\rangle | \notag \\
\leq& \underset{x_{0}}{\underbrace{\sqrt{1-|\langle \varphi
_{EB}^{0}|\varphi_{EB}^{1}\rangle |^{2}}}}\underset{y_{0}}{\underbrace{\left(%
\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}+\sqrt{(1-\tilde{%
\varepsilon}_{1})\tilde{\varepsilon}_{0}}\right)}} \notag \\
&+\underset{x_{1}}{\underbrace{|\langle \varphi
_{EB}^{0}|\varphi_{EB}^{1}\rangle |}}\underset{y_{1}}{\underbrace{\sqrt{%
\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{0}}}} \notag \\
\leq& \sqrt{\lbrack \sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}%
+\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}]^{2}+\tilde{%
\varepsilon}_{1}\tilde{\varepsilon}_{0}}\;. \notag\end{aligned}$$In the sixth line we have use fact the that if $\sum_{i}x_{i}^{2}=1$, $%
\sum_{i}x_{i}y_{i}\leq \sqrt{\sum_{i}y_{i}^{2}}$ holds, which can easily derived from the Cauchy-Schwarz inequality of two vectors in $\mathbbm{R}^{2}$. From Eq. (\[b1\]) and Eq. (\[innerprod\]), we obtain $$\begin{aligned}
&&|\langle \tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle \langle \tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle | \\
&\geq &\frac{|\langle -\alpha |\alpha \rangle |-\sqrt{[\sqrt{(1-\tilde{\varepsilon}_{1})\tilde{\varepsilon}_{0}}+\sqrt{(1-\tilde{\varepsilon}_{0})\tilde{\varepsilon}_{1}}]^{2}+\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{0}}
}{\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}}.\end{aligned}$$ and $$\begin{aligned}
&&|\langle \tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle \langle \tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle | \\
&\leq &\frac{|\langle -\alpha |\alpha \rangle |+\sqrt{[\sqrt{(1-\tilde{%
\varepsilon}_{1})\tilde{\varepsilon}_{0}}+\sqrt{(1-\tilde{\varepsilon}_{0})%
\tilde{\varepsilon}_{1}}]^{2}+\tilde{\varepsilon}_{1}\tilde{\varepsilon}_{0}}%
}{\sqrt{(1-\tilde{\varepsilon}_{0})(1-\tilde{\varepsilon}_{1})}}.\end{aligned}$$Finally, we obtain Eqs. (\[e18\]-\[du\]) by inserting the extremal values for the possible overlaps $\left|\langle\tilde{\beta}_{0}|\tilde{\beta}_{1}\rangle\right|$ of Bob’s maximal eigenstates given by Eq. (\[e17\]) and the definition $\gamma :=|\langle \tilde{\varepsilon}_{0}|\tilde{\varepsilon}_{1}\rangle |$.
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---
abstract: 'By using complex angular momentum techniques, we study the electromagnetic radiation generated by a charged particle falling radially from infinity into a Schwarzschild black hole. We consider both the case of a particle initially at rest and that of a particle projected with a relativistic velocity and we construct complex angular momentum representations and Regge pole approximations of the partial wave expansions defining the Maxwell scalar $\phi_2$ and the energy spectrum $dE/d\omega$ observed at spatial infinity. We show, in particular, that Regge pole approximations involving only one Regge pole provide effective resummations of these partial wave expansions permitting us (i) to reproduce with very good agreement the black hole ringdown without requiring a starting time, (ii) to describe with rather good agreement the tail of the signal and sometimes the pre-ringdown phase, and (iii) to explain the oscillations in the electromagnetic energy spectrum radiated by the charged particle. The present work as well as a previous one concerning the gravitational radiation generated by a massive particle falling into a Schwarzschild black hole \[A. Folacci and M. Ould El Hadj, Phys. Rev. D [**98**]{}, 064052 (2018), arXiv:1807.09056 \[gr-qc\]\] highlight the benefits of studying radiation from black holes in the complex angular momentum framework (they obviously appear when the approximations obtained involve a small number of Regge poles and have a clear physical interpretation) but also to exhibit the limits of this approach (this is the case when it is necessary to take into account background integral contributions).'
author:
- Antoine Folacci
- Mohamed
bibliography:
- 'RP\_RT\_EM.bib'
title: 'Electromagnetic radiation generated by a charged particle falling radially into a Schwarzschild black hole: A complex angular momentum description '
---
Introduction
============
In a recent article [@Folacci:2018sef], we advocated for an alternative description of gravitational radiation from black holes (BHs) based on complex angular momentum (CAM) techniques, i.e., analytic continuation in the CAM plane of partial wave expansions, duality of the ${\cal S}$-matrix, effective resummations involving its Regge poles and the associated residues, Regge trajectories, semiclassical interpretations, etc. In this previous article as well as in more recent works [@Folacci:2019cmc; @Folacci:2019vtt] where we have provided CAM and Regge pole analyses of scattering of scalar, electromagnetic and gravitational waves by a Schwarzschild BH, we have justified the interest of such an approach in the context of BH physics and we shall not return here on this subject. We refer the interested reader to these articles and, more particularly, to the introduction of Ref. [@Folacci:2018sef] as well as to references therein for other works dealing with the CAM approach to BH physics.
In the present article, by using CAM techniques, we shall revisit the problem of the electromagnetic radiation generated by a charged particle falling radially into a Schwarzschild BH. We shall consider both the case of a particle initially at rest and that of a particle projected with a relativistic or an ultra-relativistic velocity and we shall construct CAM representations and Regge pole approximations of the partial wave expansions defining the Maxwell scalar $\phi_2$ and the energy spectrum $dE/d\omega$ observed at spatial infinity. This work extends our previous work concerning the CAM and Regge pole analyses of the gravitational radiation generated by a massive particle falling into a Schwarzschild BH [@Folacci:2018sef]. Both highlight the benefits of working within the CAM framework and strengthen our opinion concerning the interest of the Regge pole approach for describing radiation from BHs.
Problems dealing with the excitation of a BH by a charged particle and the generation of the associated electromagnetic radiation have been considered, since the early 1970s, in the literature (for pioneering works on this subject, see the lectures by Ruffini [@Ruffini:1973pta] in Ref. [@DeWitt:1973uma] and references therein, as well as Refs. [@Ruffini:1972pw; @Ruffini:1972uh; @Tiomno:1972dq; @Cardoso:2003cn] for articles directly relevant to our study) and, currently, an ever-increasing importance is given to them. Indeed, such problems are of great interest with the emergence of multimessenger astronomy which combines the detection and analysis of gravitational waves with those of other types of radiation for a better understanding of our “violent Universe” but also in order to test the BH hypothesis and Einstein’s general relativity in the strong-field regime (see, e.g., Refs. [@Psaltis2008; @Johannsen_2012; @Bambi:2015kza]). In this context, it is particularly interesting to study the electromagnetic partner of the gravitational radiation generated during the accretion of a charged fluid by a BH [@Degollado:2014dfa; @Moreno:2016urq].
Our paper is organized as follows. In Sec. \[SecII\], we first construct the Maxwell scalar $\phi_2$ describing the outgoing electromagnetic radiation at infinity which is generated by a charged particle falling radially into a Schwarzschild BH. To do this, by using Green’s function techniques, we solve in the frequency domain the Regge-Wheeler equation for arbitrary $(\ell,m)$ modes and we proceed to their regularization. We also extract from the multipole expansion of $\phi_2$ the quasinormal ringdown of the BH. In Sec. \[SecIII\], we provide two different CAM representations of the multipolar waveform $\phi_2$: the first one is based on the Poisson summation formula [@MorseFeshbach1953] while the second one is constructed from the Sommerfeld-Watson transformation [@Watson18; @Sommerfeld49; @Newton:1982qc]. From each of them, we extract, as approximations of $\phi_2$, the Fourier transform of a sum over Regge poles and Regge-mode excitation factors. It is important to note that, in order to evaluate numerically these two Regge pole approximations, we need the Regge trajectories (i.e., the curves traced out in the CAM plane by the Regge poles and by the associated residues as a function of the frequency $\omega$). In Sec. \[SecIV\], we numerically compare the multipolar waveform $\phi_2$ constructed by summing over a large number of partial modes (this is particularly necessary for a particle projected with a relativistic or an ultra-relativistic velocity) as well as the associated ringdown with the Regge pole approximations obtained in Sec. \[SecIII\]. This permits us to emphasize the benefits of working with these particular approximations of the Maxwell scalar $\phi_2$. In Sec. \[SecV\], we focus on the electromagnetic energy spectrum $dE/d\omega$ radiated by the charged particle falling into the Schwarzschild BH and we numerically compare it with its CAM representation obtained from the Poisson summation formula. In the Conclusion, we summarize the main results obtained and briefly discuss some possible extensions of our approach.
Throughout this article, we adopt units such that $G = c = \epsilon_0 = \mu_0 = 1$, we use the geometrical conventions of Ref. [@Misner:1974qy] and we perform the numerical calculations using [*Mathematica*]{} [@Mathematica]. We, furthermore, consider that the exterior of the Schwarzschild BH is defined by the line element $ds^2= -f(r) dt^2+ f(r)^{-1}dr^2+ r^2 d\theta^2 + r^2 \sin^2\theta d\varphi^2$ where $f(r)=1-2M/r$ and $M$ is the mass of the BH while $t \in ]-\infty, +\infty[$, $r \in ]2M,+\infty[$, $\theta \in [0,\pi]$ and $\varphi \in [0,2\pi]$ are the usual Schwarzschild coordinates.
Maxwell scalar $\phi_2$ and associated quasinormal ringdown {#SecII}
===========================================================
In this section, we shall construct the Maxwell scalar $\phi_2$ describing the outgoing radiation at infinity due to a charged particle falling radially from infinity into a Schwarzschild BH. Moreover, we shall extract from the multipole expansion of $\phi_2$ the associated ringdown waveform.
Multipole expansion of the Maxwell scalar $\phi_2$ {#SecIIa}
--------------------------------------------------
We consider a charged particle (we denote by $m_0$ its mass and by $q$ its electric charge) falling radially into a Schwarzschild BH. The timelike geodesic followed by such a particle is defined by the coordinates $t_{p}(\tau)$, $r_{p}(\tau)$, $\theta_{p}(\tau)$ and $\varphi_{p}(\tau)$ where $\tau$ is its proper time. Without loss of generality, we can consider that this particle moves in the BH equatorial plane along the positive $x$ axis and in the negative direction, i.e., we assume that $\theta_{p}(\tau)=\pi/2$, $\varphi_{p}(\tau)=0$ and $dr_{p}(\tau)/{d\tau} <0$. The functions $t_{p}(\tau)$, $r_{p}(\tau)$ as well as the function $t_{p}(r)$ can be then obtained from the geodesic equations (see, e.g., Ref. [@Chandrasekhar:1985kt])
\[geodesic\_equations\] $$f(r_{p})\frac{dt_{p}}{d\tau}=\frac{\cal{E}}{m_0}, \label{geodesic_1}$$ and $$\label{geodesic_3}
\left(\frac{dr_{p}}{d\tau}\right)^{2} -\frac{2M}{r_{p}}
=\left(\frac{\cal{E}}{m_0}\right)^{2}-1.$$
Here, $\cal{E}$ is the energy of the particle. It is a constant of motion which can be related to the velocity $v_\infty$ of the particle at infinity and to the associated Lorentz factor $\gamma$ by $$\label{tildeEand gamma}
\frac{\cal{E}}{m_0}=\frac{1}{\sqrt{1-(v_\infty)^2}}=\gamma.$$
The electromagnetic radiation generated by this particle can be described by using the gauge-invariant formalism introduced by Ruffini, Tiomno and Vishveshwara in Ref. [@Ruffini:1972pw] (see also Refs. [@Cunningham:1978zfa; @Cunningham:1979px]) and by working in the framework of the Newman-Penrose formalism (see, e.g., Chap. 8 of Ref. [@Alcubierre:1138167]). We shall therefore focus on the Maxwell scalar $\phi_2$ which can be expressed at spatial infinity as $$\label{phi2_def}
\phi_2 = \frac{1}{2\sqrt{2}}\{\left(E_\theta - i E_\varphi \right) + i \left(B_\theta - i B_\varphi \right)\}$$ where $E_\theta$, $E_\varphi$, $B_\theta$ and $B_\varphi$ denote the components of the electromagnetic field $({\bf E}, {\bf B})$ observed for $r \to \infty $. Here, it is important to note that we have defined $\phi_2$ with respect to the null basis $(l,n,m,m^\ast)$ which is normalized such that the only nonvanishing scalar products involving the vectors of the tetrad are $l^\mu n_\mu = -1$ and $m^\mu m^\ast_\mu=1$ and which is given by (our conventions slightly differ from those of Ref. [@Alcubierre:1138167])
$$\begin{aligned}
\label{NullTetrad}
& & l^\mu=\left(\frac{1}{f(r)},1,0,0 \right), \\
& & n^\mu=\frac{1}{2}\left(1,-\frac{1}{f(r)},0,0\right), \\
& & m^\mu=\frac{1}{\sqrt{2} r}\left(0,0,1, \frac{i}{\sin \theta}\right), \\
& & {m^\ast}^\mu=\frac{1}{\sqrt{2} r}\left(0,0,1, \frac{-i}{\sin \theta}\right).\end{aligned}$$
We recall that the radially infalling particle only excites the even (polar) electromagnetic modes of the Schwarzschild BH and that, in the usual orthonormalized basis $({\bf {\hat{e}}_r},{\bf {\hat{e}}_\theta},{\bf {\hat{e}}_\varphi})$ of the spherical coordinate system, the components of the electric field can be expressed in terms of the gauge-invariant master functions $\psi_{\ell m} (t,r)$ and expanded on the (even) vector spherical harmonics $Y_\theta^{\ell m}(\theta,\varphi)$ and $Y_\varphi^{\ell m}(\theta,\varphi)$ in the form [@Folacci:2018vtf] $$\label{ChampE_even}
{\mathbf E}=\left| \begin{array}{l}
E_r = 0 \\
E_\theta = -\frac{1}{r} \sum\limits_{\ell =1}^{+\infty}\sum\limits_{m=-\ell}^{+\ell} \frac{1}{\ell(\ell+1)}\, \partial_r\psi_{\ell m} \, Y_\theta^{\ell m} \\
E_\varphi = -\frac{1}{r \sin\theta} \sum\limits_{\ell =1}^{+\infty}\sum\limits_{m=-\ell}^{+\ell} \frac{1}{\ell(\ell+1)} \, \partial_r \psi_{\ell m} \, Y_\varphi^{\ell m}, \\
\end{array}
\right.$$ while the magnetic field $\mathbf{B}$ can be obtained from the Maxwell-Faraday equation and its components expressed in terms of those of the electric field. Indeed, for $r \to +\infty$, we have $\partial_t\psi_{\ell m} = - \partial_r\psi_{\ell m}$ and we can write $$\label{Relations_ChampB_even_odd}
{\mathbf B}=\left| \begin{array}{l}
B_r = 0 \\
B_\theta = - E_\varphi \\
B_\varphi = + E_\theta. \\
\end{array}
\right.$$ It should be noted that the vector spherical harmonics appearing in Eq. (\[ChampE\_even\]) are given in terms of the standard scalar spherical harmonics $Y^{\ell m}(\theta,\varphi)$ by $$\label{HSV_even}
Y_\theta^{\ell m} = \frac{\partial}{\partial \theta} Y^{\ell m} \quad \text{and} \quad Y_\varphi^{\ell m}= \frac{\partial}{\partial \varphi} Y^{\ell m}$$ and satisfy the “orthonormalization” relation $$\begin{aligned}
\label{HSV_even_Norm}
& & \int _{{\cal S}^2} d\Omega_2 ~\left[ Y_\theta^{\ell m}(\theta,\varphi) [Y_\theta^{\ell'
m'}(\theta,\varphi)]^* \quad \phantom{\frac{1}{\sin^2 \theta}} \right. \nonumber \\
& & \qquad \left. + \frac{1}{\sin^2 \theta} Y_\varphi^{\ell m}(\theta,\varphi) [Y_\varphi^{\ell'
m'}(\theta,\varphi)]^* \right] \nonumber \\
& & \qquad \qquad = \ell (\ell +1) \delta
_{\ell \ell'}\delta _{m m'}.\end{aligned}$$ Here $d\Omega_2= \sin \theta \, d\theta \, d\varphi$ denotes the area element on the unit sphere ${\cal S}^2$. We also recall that the gauge-invariant master functions $\psi_{\ell m} (t,r)$ appearing in Eq. (\[ChampE\_even\]) can be written in the form $$\label{TF_psi}
\psi_{\ell m} (t,r) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, \psi_{\omega \ell m} (r) e^{-i\omega t}$$ where their Fourier components $\psi_{\omega \ell m} (r) $ satisfy the Regge-Wheeler equation $$\label{ZM EQ_Fourier}
\left[\frac{d^2}{d
r_\ast^2} + \omega^2 - V_\ell (r) \right] \psi_{\omega \ell m} (r) = S_{\omega \ell m} (r).$$ Here, $S_{\omega \ell m} (r)$ is a source term, $r_\ast$ denotes the tortoise coordinate which is defined in terms of the radial Schwarzschild coordinate $r$ by $dr/dr_\ast=f(r)$ and is given by $r_\ast(r)=r+2M \ln[r/(2M)-1]$ while $$\label{pot R-W}
V_\ell(r)=f(r)\left(\frac{\ell(\ell+1)}{r^2}\right)$$ denotes the Regge-Wheeler potential.
As far as the source term $S_{\omega \ell m} (r)$ appearing in the right-hand side (r.h.s.) of the Regge-Wheeler equation (\[ZM EQ\_Fourier\]) is concerned, it depends on the components, in the basis of vector spherical harmonics, of the current associated with the charged particle [@Folacci:2018vtf]. Its expression can be derived from Eqs. (\[geodesic\_equations\]) and (\[tildeEand gamma\]) and we obtain $$\label{SourceRad_gen}
S_{\omega \ell m} (r) =[Y^{\ell m}(\pi/2,0)]^\ast \widetilde{S}_{\omega} (r) e^{+i \omega t_p(r) }$$ where
\[SourceRad\_omR\] $$\begin{aligned}
\label{SourceRad_omR_a}
& & \widetilde{S}_{\omega} (r) =\frac{q}{\sqrt{2 \pi}}f(r) \left[ + i \omega \frac{r}{(\gamma^{2}-1)r + 2M} - \frac{M \gamma}{\sqrt{r} \, \left[(\gamma^{2}-1)r+2M\right]^{3/2}} \right]\end{aligned}$$ and with $$\begin{aligned}
\label{trajectory_Rad}
& & \frac{t_{p}(r)}{2M}=-\frac{2}{3} \left(\frac{r}{2M} \right)^{3/2} -2 \left(\frac{r}{2M} \right)^{1/2} + \ln \left(\frac{\sqrt{\frac{r}{2M}} +1}{\sqrt{\frac{r}{2M}} -1} \right) + \frac{t_{0}}{2M}\end{aligned}$$ for the particle starting at rest from infinity (i.e., for $\gamma=1$) and $$\begin{aligned}
\label{trajectory_RadRel}
& & \frac{t_{p}(r)}{2M}=-\frac{\gamma}{(\gamma^{2}-1)^{3/2}}\sqrt{\left[(\gamma^{2}-1)\frac{r}{2M}\right]
\left[(\gamma^{2}-1)\frac{r}{2M}+1\right]} \nonumber \\
& & \qquad \qquad - \frac{\gamma (2\gamma^{2}-3)}{(\gamma^{2}-1)^{3/2}} \ln\left[\sqrt{(\gamma^{2}-1)\frac{r}{2M}} + \sqrt{(\gamma^{2}-1)\frac{r}{2M}+1}\right] \nonumber \\
& & \qquad \qquad+ \ln\left[\frac{\gamma \sqrt{\frac{r}{2M}}+ \sqrt{(\gamma^{2}-1)\frac{r}{2M}+1}}{ \gamma \sqrt{\frac{r}{2M}}- \sqrt{(\gamma^{2}-1)\frac{r}{2M}+1}}\right] + \frac{t_{0}}{2M}\end{aligned}$$
for a particle projected with a finite kinetic energy at infinity (i.e., for $\gamma > 1$). In Eqs. (\[trajectory\_Rad\]) and (\[trajectory\_RadRel\]), $t_0$ is an arbitrary integration constant.
Regge-Wheeler equation and ${\cal S}$-matrix {#SecIIb}
--------------------------------------------
The Regge-Wheeler equation (\[ZM EQ\_Fourier\]) can be solved by using the machinery of Green’s functions (see, e.g., Ref. [@Breuer:1974uc] for its use in the context of BH physics). [*Mutatis mutandis*]{}, taking into account Eq. (\[SourceRad\_gen\]), the reasoning of Sec. IIC of Ref. [@Folacci:2018cic] permits us to obtain the asymptotic expression, for $r \to +\infty$, of the partial amplitudes $\psi_{\omega\ell m}(r)$. We have
\[Partial\_Response\] $$\label{Partial_Response_a}
\psi_{\omega\ell m}(r)= e^{+i \omega r_\ast(r) } \, \frac{K[\ell,\omega]}{2i\omega A^{(-)}_\ell (\omega)} \,
[Y^{\ell m}(\pi/2,0)]^\ast$$ with $$\begin{aligned}
\label{Partial_Response_b}
& & K[\ell,\omega] = \int_{2M}^{+\infty} \frac{dr'}{f(r')} \,\phi_{\omega, \ell}^\mathrm {in}(r')
\, \widetilde{S}_{\omega} (r') e^{i \omega t_p(r') }.\end{aligned}$$
Here, we have introduced the solution $\phi_{\omega, \ell}^\mathrm {in} (r) $ of the homogeneous Regge-Wheeler equation $$\label{H_RW_equation}
\left[\frac{d^{2}}{dr_{\ast}^{2}}+\omega^{2}-V_{\ell}(r)\right] \phi_{\omega, \ell}^\mathrm {in}= 0$$ which is defined by its behavior at the event horizon $r=2M$ (i.e., for $r_\ast \to -\infty$) and at spatial infinity $r \to +\infty$ (i.e., for $r_\ast \to +\infty$): $$\begin{aligned}
\label{bc_in}
& & \phi_{\omega, \ell}^{\mathrm {in}}(r_{*}) \sim \left\{
\begin{aligned}
&\!\!\displaystyle{e^{-i\omega r_\ast}} \,\, (r_\ast \to -\infty)\\
&\!\!\displaystyle{A^{(-)}_\ell (\omega) e^{-i\omega r_\ast} + A^{(+)}_\ell (\omega) e^{+i\omega r_\ast}} \,\, (r_\ast \to +\infty).
\end{aligned}
\right. \nonumber\\
&&\end{aligned}$$ The coefficients $A^{(-)}_\ell (\omega)$ and $A^{(+)}_\ell (\omega)$ appearing in Eqs. (\[Partial\_Response\]) and (\[bc\_in\]) are complex amplitudes. By evaluating, first for $r_\ast \to - \infty$ and then for $r_\ast \to + \infty$, the Wronskian involving the function $\phi_{\omega\ell}^{\mathrm {in}}$ and its complex conjugate, we can show that they are linked by $$\label{Rel_conserv_Apm}
|A^{(-)}_\ell (\omega)|^2 - |A^{(+)}_\ell (\omega)|^2 = 1.$$ Moreover, with the numerical calculation of the Maxwell scalar $\phi_2$ as well as the study of its properties in mind, it is important to note that
\[Sym\_om\] $$\begin{aligned}
& & \phi_{-\omega, \ell}^{\mathrm {in}} (r)= \left[\phi_{\omega, \ell}^{\mathrm {in}}(r) \right]^\ast, \label{Sym_om_a}\\
& & A^{(\pm )}_\ell (-\omega) = [A^{( \pm)}_\ell (\omega) ]^\ast. \label{Sym_om_b}\end{aligned}$$
It is worth pointing out that the boundary conditions (\[bc\_in\]) for $\phi_{\omega, \ell}^\mathrm {in} (r) $ and therefore the expression (\[Partial\_Response\]) of the partial amplitudes $\psi_{\omega\ell m} (r)$ involve the ${\cal S}$-matrix defined by (see, e.g., Ref. [@DeWitt:2003pm]) $$\begin{aligned}
\label{S_matrix_def}
& & {\cal S}_\ell (\omega) = \left(\,
\begin{aligned}
&\!\!\displaystyle{\qquad\quad 1/A^{(-)}_\ell (\omega)} & \displaystyle{A^{(+)}_\ell (\omega)/A^{(-)}_\ell (\omega)} \\
&\!\!-\displaystyle{[A^{(+)}_\ell (\omega)]^\ast/A^{(-)}_\ell (\omega)} & \displaystyle{1/A^{(-)}_\ell (\omega)}\quad
\end{aligned}
\right). \nonumber\\
&&\end{aligned}$$ Due to Eq. (\[Sym\_om\_b\]), this matrix satisfies the symmetry property ${\cal S}_\ell (-\omega)=\left[ {\cal S}_\ell (\omega) \right]^\ast$ and, due to Eq. (\[Rel\_conserv\_Apm\]), it is in addition unitary, i.e., it satisfies $ {\cal S} {\cal S}^\dag= {\cal S}^\dag {\cal S}= \mathrm{1}$. Here, it is interesting to recall that, in Eq. (\[S\_matrix\_def\]), the term $1/A^{(-)}_\ell (\omega)$ and the term $A^{(+)}_\ell (\omega)/A^{(-)}_\ell (\omega)$ are, respectively, the transmission coefficient $T_\ell(\omega)$ and the reflection coefficient $R^\mathrm{in}_\ell(\omega)$ corresponding to the scattering problem defined by Eq. (\[bc\_in\]). As far as the coefficient $-[A^{(+)}_\ell (\omega)]^\ast/A^{(-)}_\ell (\omega)$ is concerned, it can be considered as the reflection coefficient $R^\mathrm{up}_\ell(\omega)$ involved in the scattering problem defining the modes $\phi^\mathrm{up}_{\omega, \ell} (r)$ [@DeWitt:2003pm].
Compact expression for the multipole expansion of the Maxwell scalar $\phi_2$ {#SecIIc}
-----------------------------------------------------------------------------
We first insert Eq. (\[Partial\_Response\_a\]) into Eq. (\[TF\_psi\]) and we have
\[TF\_psi\_bis\] $$\label{TF_psi_bis_a}
\psi_{\ell m} (t,r) = \psi_\ell (t,r) \, [Y^{\ell m}(\pi/2,0)]^\ast$$ where $$\begin{aligned}
\label{TF_psi_bis_b}
& & \psi_\ell (t,r) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]}\, \frac{K[\ell,\omega]}{2i\omega A^{(-)}_\ell (\omega)}. \nonumber \\
& &\end{aligned}$$
We now substitute Eq. (\[TF\_psi\_bis\]) into Eq. (\[ChampE\_even\]). Furthermore, without loss of generality, we assume that the electromagnetic radiation is observed in a direction lying in the BH equatorial plane and making an angle $\varphi \in [0, \pi]$ with the trajectory of the particle (due to symmetry considerations, we can restrict our study to this interval). By then using the addition theorem for scalar spherical harmonics in the form $$\label{ThAd_HS}
\sum_{m=-\ell}^{+\ell} Y^{\ell m}(\theta,\varphi) [Y^{\ell m}(\pi/2,0)]^\ast = \frac{2\ell +1}{4 \pi} P_\ell (\sin\theta \cos\varphi)$$ where $P_\ell (x)$ denotes the Legendre polynomial of degree $\ell$ [@AS65], we obtain, for $r\to +\infty$, $$\label{hc_def}
r\, E_\theta (t,r,\theta=\pi/2,\varphi)=0$$ and $$\begin{aligned}
\label{hp_def}
& & r\, E_\varphi (t,r,\theta=\pi/2,\varphi)= -\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]} \nonumber \\
& & \qquad \times \left[\frac{1}{4\pi}\sum_{\ell=1}^{+\infty}\frac{2\ell+1}{\ell(\ell+1)}\, \frac{K[\ell,\omega]}{2 A^{(-)}_\ell (\omega)} \, W_\ell (\cos\varphi) \right].\end{aligned}$$ In Eq. (\[hp\_def\]), we have introduced the angular function $$\label{ang_function}
W_\ell (\cos\varphi) = \frac{\partial}{\partial \varphi} P_\ell(\cos\varphi).$$ Finally, taking into account Eq. (\[Relations\_ChampB\_even\_odd\]), we can write by inserting Eqs. (\[hc\_def\]) and (\[hp\_def\]) into Eq. (\[phi2\_def\]) $$\begin{aligned}
\label{phi2_ExpressionDef}
& & \frac{\sqrt{2} \, r}{i}\, \phi_2 (t,r,\theta=\pi/2,\varphi)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]} \nonumber \\
& & \qquad\qquad \times \left[ \frac{1}{4\pi} \sum_{\ell=1}^{+\infty} \frac{2\ell+1}{\ell(\ell+1)} \, \frac{K[\ell,\omega]}{2 A^{(-)}_\ell (\omega)} \, W_\ell (\cos\varphi)\right]\end{aligned}$$ for $r \to + \infty$.
Regularization of the partial waveform amplitudes $\psi_{\omega\ell m}$ and the Maxwell scalar $\phi_2$ {#SecIId}
--------------------------------------------------------------------------------------------------------
To construct the Maxwell scalar $\phi_2$, we need to regularize the partial amplitudes $\psi_{\omega\ell m}(r)$ or, more precisely, $K[\ell, \omega]$. Indeed, the partial waveforms (\[Partial\_Response\]) as integrals over the radial Schwarzschild coordinate are divergent at infinity. This is due to the behavior of the source (\[SourceRad\_omR\]) for $r \to \infty$.
To regularize $K[\ell, \omega]$, we integrate twice by parts and use the homogeneous Regge-Wheeler equation . Then, by dropping intentionally the boundary terms at $r \to \infty$ (regularization), we obtain
$$\begin{aligned}
\label{Partial_Response_c}
& & K[\ell,\omega] = q \, \ell(\ell+1) \frac{\widetilde{K}[\ell,\omega]}{i \omega}\end{aligned}$$
with $$\begin{aligned}
\label{Partial_Response_d}
& & \widetilde{K}[\ell,\omega] = \frac{1}{\sqrt{2\pi}} \int_{2M}^{+\infty} dr' \,\phi_{\omega, \ell}^\mathrm {in}(r')\, \frac{ e^{i \omega t_p(r') }}{r'^{2}}.\end{aligned}$$
Now, by inserting Eqs. and into Eq. , we obtain for the Maxwell scalar $$\begin{aligned}
\label{phi2_ExpressionDef_reg}
& & \frac{\sqrt{2} \, r}{i q}\, \phi_2 (t,r,\theta=\pi/2,\varphi)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]} \nonumber \\
& & \qquad\qquad \times \left[ \sum_{\ell=1}^{+\infty} \frac{2\ell+1}{4\pi} \, \frac{\widetilde{K}[\ell,\omega]}{2 i \omega A^{(-)}_\ell (\omega)} \,W_\ell (\cos\varphi) \right].\end{aligned}$$ It is in addition interesting to note that, by inserting Eqs. and into the partial waveform amplitudes (\[Partial\_Response\_a\]), we can recover the amplitude term derived by Cardoso, Lemos and Yoshida in Ref. [@Cardoso:2003cn] working in the Zerilli gauge [@Zerilli:1971wd; @Zerilli:1974ai].
Moreover, with the numerical calculation of the Maxwell scalar $\phi_2$ as well as the study of its properties in mind, we can observe that
\[Sym\_om\_cd\] $$\widetilde{K}[\ell,-\omega] = \left[\widetilde{K}[\ell,\omega] \right]^\ast \label{Sym_om_c}$$ and $$\widetilde{K}[\ell,-\omega]/A^{(-)}_\ell (-\omega) = \left[ \widetilde{K}[\ell,\omega]/A^{(-)}_\ell (\omega) \right]^\ast \label{Sym_om_d}$$
as a consequence of Eqs. (\[Sym\_om\_a\]) and (\[Sym\_om\_b\]). Due to relation (\[Sym\_om\_d\]), we can see that the term in square brackets in Eq. (\[phi2\_ExpressionDef\_reg\]) satisfies the Hermitian symmetry property and, as a consequence, that the Maxwell scalar $\phi_2$ is a purely imaginary quantity. Similarly, it is worth pointing out that the electromagnetic field is a real quantity.
Two alternative expressions for the multipole expansion of the Maxwell scalar $\phi_2$ {#SecIIe}
--------------------------------------------------------------------------------------
It is important to realize that Eq. (\[phi2\_ExpressionDef\_reg\]) can also be written as $$\begin{aligned}
\label{phi2_ExpressionDef_reg_P}
& & \frac{\sqrt{2} \, r}{i q}\, \phi_2 (t,r,\theta=\pi/2,\varphi)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]} \nonumber \\
& & \qquad\qquad \times \left[ \sum_{\ell=0}^{+\infty} \frac{2\ell+1}{4\pi} \, \frac{\widetilde{K}[\ell,\omega]}{2 i \omega A^{(-)}_\ell (\omega)} \,W_\ell (\cos\varphi) \right].\end{aligned}$$ Indeed, it is possible to start at $\ell=0$ the discrete sum over $\ell$ by noting that $$\label{Mode_Zero}
W_0 (\cos\varphi)=\frac{\partial}{\partial\varphi} P_0(\cos \varphi)= 0$$ and that we have formally $$\label{Formally_ell0}
A_0^{(-)}(\omega)=1 \quad \mathrm{and} \quad \widetilde{K}[0,\omega] \,\, \mathrm{regular}.$$ These last two results are due to the fact that, for $\ell=0$, the solution of the problem (\[H\_RW\_equation\])–(\[bc\_in\]) is $\phi_{\omega, 0}^\mathrm {in} (r) = e^{-i\omega r_\ast}$ because the Regge-Wheeler potential (\[pot R-W\]) vanishes. Of course, in general, it is more natural to work with the multipole expansion (\[phi2\_ExpressionDef\_reg\]) of the Maxwell scalar $\phi_2$ but, in Sec. \[SecIIIc\], we shall take (\[phi2\_ExpressionDef\_reg\_P\]) as a departure point because it will permit us to use the Poisson summation formula in its standard form.
Similarly, it is important to note that Eq. (\[phi2\_ExpressionDef\_reg\_P\]) can be rewritten in the form $$\begin{aligned}
\label{phi2_ExpressionDef_reg_SW}
& & \frac{\sqrt{2} \, r}{i q}\, \phi_2 (t,r,\theta=\pi/2,\varphi)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]} \nonumber \\
& &\, \times \left[ \sum_{\ell=0}^{+\infty} (-1)^{\ell} \frac{2\ell+1}{4\pi} \, \frac{\widetilde{K}[\ell,\omega]}{2 i \omega A^{(-)}_\ell (\omega)} \,W_\ell (-\cos\varphi) \right].\end{aligned}$$ Indeed, we can recover Eq. (\[phi2\_ExpressionDef\_reg\_P\]) from Eq. (\[phi2\_ExpressionDef\_reg\_SW\]) by using the relation [@AS65] $$\label{prop_Pell}
P_\ell (- \cos \varphi) = (-1)^\ell P_\ell (\cos \varphi)$$ in connection with the definition (\[ang\_function\]). In Sec. \[SecIIId\], we shall take Eq. (\[phi2\_ExpressionDef\_reg\_SW\]) as a departure point because it will permit us to use the Sommerfeld-Watson transform in its standard form.
Quasinormal ringdown associated with the Maxwell scalar $\phi_2$ {#SecIIf}
----------------------------------------------------------------
The quasinormal ringdown $\phi^\text{\tiny{QNM}}_2$ generated by the charged particle falling radially from infinity into a Schwarzschild BH can be extracted from Eq. by following, *mutatis mutandis*, the reasoning of Sec. II E of Ref. [@Folacci:2018sef]. We then obtain $$\begin{aligned}
\label{response_QNM}
& & \frac{\sqrt{2} \, r}{iq}\, \phi^\text{\tiny{QNM}}_2 (t,r,\theta=\pi/2,\varphi)=
- 2 \sqrt{2\pi} \, \operatorname{Re} \left[ \, \sum^{+\infty}_{ \ell =1} \sum^{+\infty}_{n =1} \phantom{\frac{\widetilde{K}}{A_{\ell}^{(+)}}} \right. \nonumber \\
& & \,\, \left.\frac{2\ell+1}{4\pi} \, {\cal{B}}_{\ell n} \frac{\widetilde{K}[\ell,\omega_{\ell n}]}{A_{\ell}^{(+)}(\omega_{\ell n})}\, e^{-i \omega_{\ell n}[t-r_\ast(r)]} \, W_\ell (\cos \varphi)\right]\end{aligned}$$ where $$\label{excitation_factor_QNM}
{\cal{B}}_{\ell n}=\left[\frac{1}{2 \omega}\,\,\frac{A_{\ell}^{(+)}(\omega)}{\frac{d}{d \omega}A_{\ell}^{(-)}(\omega)}\right]_{\omega=\omega_{\ell n}}$$ denotes the excitation factor associated with the $(\ell,n)$ quasinormal mode (QNM) of complex frequency $\omega_{\ell n}$. Its expression involves the residue of the function $1/A^{(-)}_\ell (\omega)$ at $\omega=\omega_{\ell n}$. It should be noted that Eq. (\[response\_QNM\]) has been obtained by gathering the contributions of the quasinormal frequencies $\omega_{\ell n}$ and $-\omega_{\ell n}^\ast$ taking into account the relations (\[Sym\_om\_b\]) and (\[Sym\_om\_c\]) which remain valid in the complex $\omega $ plane. As a consequence, the quasinormal ringdown waveform $\phi^\text{\tiny{QNM}}_2$ appears clearly as a purely imaginary quantity.
Let us finally recall that, due to the exponentially divergent behavior of the terms $e^{-i \omega_{\ell n}[t-r_\ast(r)]}$ as $t$ decreases, the ringdown waveform $\phi^\text{\tiny{QNM}}_2$ does not provide physically relevant results at early times. It is therefore necessary to determine, from physical considerations, a starting time $t_\mathrm{start}$ for the BH ringdown. In general, by taking $t_\mathrm{start}=t_p(3M)$, i.e., the moment the particle crosses the photon sphere, we can obtain physically relevant results.
Maxwell scalar $\phi_2$, its CAM representations and its Regge pole approximations {#SecIII}
==================================================================================
In this section, we shall derive two exact CAM representations of the Maxwell scalar $\phi_2$, the first one by using the Poisson summation formula [@MorseFeshbach1953] and the second one by working with the Sommerfeld-Watson transformation [@Watson18; @Sommerfeld49; @Newton:1982qc]. These representations can be written as (the Fourier transform of) a sum over Regge poles plus background integrals along the positive real axis and the imaginary axis of the CAM plane. We shall also consider the Regge pole part of these representations as approximations of the Maxwell scalar $\phi_2$ which can be evaluated numerically from the Regge trajectories followed by the Regge poles and by the excitation factors of the associated Regge modes.
In order to construct the two CAM representations of the Maxwell scalar $\phi_2$ and the associated Regge pole approximations, we shall follow, *mutatis mutandis*, Sec. III of Ref. [@Folacci:2018sef].
Some preliminary remarks concerning analytic extensions in the CAM plane {#SecIIIa}
------------------------------------------------------------------------
![\[RM\_excitation\_factors\_S\_1\] Regge trajectories of the Regge-mode excitation factors ($2M=1$). We consider the Regge modes corresponding to the first three Regge poles of which the behavior has been displayed in Fig. \[TR\_PR\_S\_1\]. The relation (\[EF\_PR\_chgt\_om\]) permits us to describe the Regge trajectories for $\omega<0$ by noting that $\operatorname{Re} [\beta_n(\omega)]$ and $\operatorname{Im} [\beta_n(\omega)]$ are, respectively, odd and even functions of $\omega$.](RM_excitation_factors_S_1)
The CAM machinery permitting us to derive the CAM representations of the multipolar waveform $\phi_2$ requires us to replace in Eqs. (\[phi2\_ExpressionDef\_reg\_P\]) and (\[phi2\_ExpressionDef\_reg\_SW\]) the angular momentum $\ell \in \mathbb{N}$ by the angular momentum $\lambda = \ell +1/2 \in \mathbb{C}$ and therefore to work into the CAM plane. As a consequence, we need to have at our disposal the functions $W_{\lambda -1/2} (\cos \varphi)$, $W_{\lambda -1/2} (-\cos \varphi)$, $A^{(\pm)}_{\lambda-1/2} (\omega)$ and $\widetilde{K}[\lambda-1/2,\omega]$ which are “the” analytic extensions of $W_{\ell} (\cos \varphi)$, $W_{\ell} (-\cos \varphi)$, $A^{(\pm)}_{\ell} (\omega)$ and $\widetilde{K}[\ell,\omega]$ in the complex $\lambda$ plane. We recall that the uniqueness problem for such analytic extensions is a difficult problem. We have briefly discussed it in Sec. IIIA of Ref. [@Folacci:2018sef] (see also Chap. 13 of Ref. [@Newton:1982qc]). Here, in order to construct these analytic extensions, we shall adopt minimal prescriptions that will be justified by the results we shall obtain in Sec. \[SecIV\].
The angular functions $W_{\ell} (\cos \varphi)$, $W_{\ell} (-\cos \varphi)$ are defined from the Legendre polynomial $P_\ell (z)$ \[see Eq. (\[ang\_function\])\] of which the analytic extension usually considered is the hypergeometric function [@AS65] $$\label{Def_ext_LegendreP}
P_{\lambda -1/2} (z) = F(1/2-\lambda,1/2+\lambda;1;(1-z)/2].$$ As a consequence, it is natural to take $$\begin{aligned}
\label{AngFunction_W_ext}
& & W_{\lambda -1/2} (\pm \cos \varphi) \nonumber \\
& & \qquad = \frac{\partial}{\partial \varphi} F(1/2-\lambda,1/2+\lambda;1;(1 \mp \cos \varphi)/2]\end{aligned}$$ and it is worth noting that, due to the properties of the hypergeometric function, we have $$\label{prop_ext_W_a}
W_{-\lambda -1/2} (\pm \cos \varphi) = W_{\lambda -1/2} (\pm \cos \varphi)$$ and $$\label{prop_ext_W_b}
W_{\lambda -1/2} (\pm \cos \varphi) = [W_{\lambda^\ast -1/2} (\pm \cos \varphi)]^\ast.$$ Here, it is crucial to keep in mind that, while the angular functions $W_\ell (\pm \cos \phi)$ are well defined for $\varphi \in [0,\pi]$, this is not the case for their analytic extensions $W_{\lambda -1/2} (\pm \cos \phi)$. Indeed, due to the pathologic behavior of $P_{\lambda -1/2} (z)$ at $z=-1$, $W_{\lambda -1/2} (\cos \phi)$ diverges in the limit $\varphi \to \pi$ and $W_{\lambda -1/2} (-\cos \phi)$ diverges in the limit $\varphi \to 0$. Due to the problems they generate on the Regge pole approximations of $\phi_2$, we shall return to these results later.
Analytic extensions $A^{(\pm)}_{\lambda-1/2} (\omega)$ and $\widetilde{K}[\lambda-1/2,\omega]$ of $A^{(\pm)}_{\ell} (\omega)$ and $\widetilde{K}[\ell,\omega]$ are obtained by assuming that the function $\phi_{\omega, \lambda-1/2}^\mathrm {in} (r) $ and the coefficients $A^{(\pm)}_{\lambda-1/2} (\omega)$ can be defined by the problem (\[H\_RW\_equation\])–(\[bc\_in\]) where now $\ell \in \mathbb{N}$ is replaced by $\lambda-1/2 \in \mathbb{C}$. Such prescription permits us, in particular, to extend in the CAM plane the properties (\[Sym\_om\_a\]), (\[Sym\_om\_b\]), (\[Sym\_om\_c\]) and (\[Sym\_om\_d\]). In the following, we shall therefore consider that
\[Sym\_om\_CAM\_ab\] $$\begin{aligned}
& & \phi_{-\omega, \lambda -1/2}^{\mathrm {in}} (r)= [\phi_{\omega, \lambda^\ast -1/2}^{\mathrm {in}}(r)]^\ast, \label{Sym_om_CAM_a}\\
& & A^{(\pm )}_{\lambda -1/2} (-\omega) = [A^{( \pm)}_{\lambda^\ast -1/2} (\omega) ]^\ast, \label{Sym_om_CAM_b}\end{aligned}$$
and that
\[Sym\_om\_CAM\_cd\] $$\begin{aligned}
& & \widetilde{K}[\lambda -1/2,-\omega] = \left[ \widetilde{K}[\lambda^\ast -1/2,\omega] \right]^\ast, \label{Sym_om_CAM_c} \\
& & \widetilde{K}[\lambda -1/2,-\omega]/A^{(-)}_{\lambda -1/2} (-\omega) = \nonumber \\
& & \qquad\qquad\qquad \left[ \widetilde{K}[\lambda^\ast -1/2,\omega]/A^{(-)}_{\lambda^\ast -1/2} (\omega) \right]^\ast. \label{Sym_om_CAM_d}\end{aligned}$$
Regge poles, Regge modes and associated excitation factors {#SecIIIb}
----------------------------------------------------------
In the next two subsections, contour deformations in the CAM plane will permit us to collect, by using Cauchy’s residue theorem, the contributions from the Regge poles of the ${\cal S}$-matrix or, more precisely, from the poles, in the complex $\lambda$ plane and for $\omega \in \mathbb{R}$, of the matrix ${\cal S}_{\lambda-1/2}(\omega)$. It should be noted that these poles can be defined as the zeros $\lambda_n(\omega)$ with $n=1, 2, 3, \dots $ and $\omega \in \mathbb{R}$ of the coefficient $A^{(-)}_{\lambda-1/2} (\omega)$ \[see Eq. (\[S\_matrix\_def\])\]. They therefore satisfy $$\label{PR_def_Am}
A^{(-)}_{\lambda_n(\omega)-1/2} (\omega)=0.$$
The Regge poles corresponding to electromagnetism in the Schwarzschild BH have been studied in Refs. [@Decanini:2009mu; @Dolan:2009nk]. It should be recalled that, for $\omega >0$, the Regge poles lie in the first and third quadrants of the CAM plane, symmetrically distributed with respect to the origin $O$ of this plane. In this article, due to the use of Fourier transforms, we must be able to locate the Regge poles even for $\omega < 0$. In fact, from the symmetry relation (\[Sym\_om\_CAM\_b\]), we have $$\label{PR_chgt_om}
\lambda_n(-\omega)=[\lambda_n(\omega)]^\ast$$ and we can see immediately that, for $\omega < 0$, the Regge poles lie in the second and fourth quadrants of the CAM plane, symmetrically distributed with respect to the origin $O$ of this plane. Moreover, if we consider the Regge trajectories $\lambda_n(\omega)$ with $\omega \in ]-\infty,+\infty[$, we can observe the migration of the Regge poles. More precisely, as $\omega $ decreases, the Regge poles lying in the first (third) quadrant of the CAM plane migrate in the fourth (second) one.
It should be noted that the solutions of the problem (\[H\_RW\_equation\])–(\[bc\_in\]) with $\ell$ replaced by $\lambda_n(\omega)-1/2 $ are modes that are purely outgoing at infinity and purely ingoing at the horizon. They are the “Regge modes” of the Schwarzschild BH [@Decanini:2009mu; @Dolan:2009nk]. Because of the analogy with the QNMs, it is natural to define excitation factors for these modes. In fact, they will appear in the CAM representations of the Maxwell scalar $\phi_2$. By analogy with the excitation factor associated with the $(\ell,n)$ QNM of complex frequency $\omega_{\ell n}$ \[see Eq. (\[excitation\_factor\_QNM\])\], we define the excitation factor of the Regge mode associated with the Regge pole $\lambda_n(\omega)$ by $$\label{excitation_factor_RP}
\beta_n(\omega)=\left[\frac{1}{2 \omega}\,\,\frac{A_{\lambda -1/2}^{(+)}(\omega)}{\frac{d}{d \lambda}A_{\lambda -1/2}^{(-)}(\omega)}\right]_{\lambda=\lambda_n(\omega)}.$$ Its expression involves the residue of the matrix ${\cal S}_{\lambda-1/2}(\omega)$ \[or, more precisely, of the function $1/A^{(-)}_{\lambda-1/2} (\omega)$\] at $\lambda=\lambda_n(\omega)$. It should be noted that, due to Eq. (\[Sym\_om\_CAM\_b\]), we have $$\label{EF_PR_chgt_om}
\beta_n(-\omega)=-[\beta_n(\omega)]^\ast.$$
We have displayed the Regge trajectories of the first three Regge poles as well as the Regge trajectories of the corresponding excitation factors in Figs. \[TR\_PR\_S\_1\] and \[RM\_excitation\_factors\_S\_1\]. These numerical results have been obtained by using, [*mutatis mutandis*]{}, the methods that have permitted us to obtain, in Refs. [@Folacci:2018vtf; @Folacci:2018cic], for the electromagnetic field and for gravitational waves, the complex quasinormal frequencies of the QNMs and the associated excitation factors (see, e.g., Sec. IV A of Ref. [@Folacci:2018cic]).
It is important to point out that, in Refs. [@Decanini:2002ha; @Decanini:2009mu], we have established a connection between the Regge modes and the (weakly damped) QNMs of the Schwarzschild BH. It will play a central role in the interpretation of our results in Sec. \[SecIV\], and we recall that, for a given $n$, the Regge trajectory $\lambda_n(\omega)$ with $\omega \in \mathbb{R}$ encodes information on the complex quasinormal frequencies $\omega_{\ell n}$ with $\ell=1, 2, 3, \dots$ In fact, the index $n=1, 2, 3, \dots $ not only permits us to distinguish between the different Regge poles but is also associated with the family of quasinormal frequencies generated by the Regge modes.
CAM representation and Regge pole approximation of the Maxwell scalar $\phi_2$ based on the Poisson summation formula {#SecIIIc}
---------------------------------------------------------------------------------------------------------------------
The first CAM representation of the Maxwell scalar $\phi_2$ can be derived from Eq. (\[phi2\_ExpressionDef\_reg\_P\]) by using the Poisson summation formula [@MorseFeshbach1953] as well as Cauchy’s residue theorem. This can be achieved by following, *mutatis mutandis*, the reasoning of Sec. III C of Ref. [@Folacci:2018sef] which has permitted us to construct a CAM representation of the Weyl scalar $\Psi_4$. In fact, it is even possible to avoid repeating in detail this reasoning: indeed, we can note that Eq. (24) of Ref. [@Folacci:2018sef] defining $\Psi_4$ and which is the departure of the reasoning of Sec. IIIC of Ref. [@Folacci:2018sef] and Eq. (\[phi2\_ExpressionDef\_reg\_P\]) of the present article are related by the correspondences
\[PSI4-phi2\] $$\begin{aligned}
r \, \Psi_4 (t,r,\theta=\pi/2,\varphi) & \longleftrightarrow & \frac{\sqrt{2} \, r}{iq} \, \phi_2 (t,r,\theta=\pi/2,\varphi), \nonumber \\
& & \label{PSI4-phi2_a}\\
\frac{i \omega K[\ell,\omega]}{4 A^{(-)}_\ell (\omega)} & \longleftrightarrow & \frac{\widetilde{K}[\ell,\omega]}{2 i \omega A^{(-)}_\ell (\omega)}, \label{PSI4-phi2_b} \\
Z_\ell (\cos \varphi) & \longleftrightarrow & W_\ell (\cos \varphi). \label{PSI4-phi2_c}\end{aligned}$$
As a consequence, Eqs. (48) and (49) of Ref. [@Folacci:2018sef] which provide a CAM representation of the Weyl scalar $\Psi_4$ can be translated to obtain directly a CAM representation of the Maxwell scalar $\phi_2$. We can write
$$\label{CAM_phi2_ExpressionDef_Ptot}
\phi_2 (t,r,\theta=\pi/2,\varphi)= \phi^{\text{\tiny{B}} \, \textit{\tiny{(P)}}}_2 (t,r,\theta=\pi/2,\varphi) + \phi^{\text{\tiny{RP}} \, \textit{\tiny{(P)}}}_2 (t,r,\theta=\pi/2,\varphi)$$
where
\[CAM\_phi2\_ExpressionDef\_P\] $$\begin{aligned}
\label{CAM_phi2_ExpressionDef_P_Background}
& & \frac{\sqrt{2} \, r}{iq}\, \phi^{\text{\tiny{B}} \, \textit{\tiny{(P)}}}_2 (t,r,\theta=\pi/2,\varphi)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]} \left[ \int_{0}^{\infty} d\lambda \, \frac{\lambda}{2\pi} \, \frac{\widetilde{K}[\lambda-1/2,\omega]}{2i\omega A^{(-)}_{\lambda-1/2} (\omega)} \, W_{\lambda -1/2} (\cos \varphi) \right. \nonumber \\
& & \qquad \qquad\qquad \left. -\frac{1}{4\pi} \int_{0}^{+i\infty} d\lambda \, \frac{\lambda e^{i\pi \lambda}}{\cos (\pi \lambda)} \frac{\widetilde{K}[\lambda-1/2,\omega]}{2i\omega A^{(-)}_{\lambda-1/2} (\omega)} \, W_{\lambda -1/2} (\cos \varphi) \right. \nonumber \\
& & \qquad \qquad\qquad \left. -\frac{1}{4\pi} \int_{0}^{-i\infty} d\lambda \, \frac{\lambda e^{-i \pi \lambda}}{\cos (\pi \lambda)} \frac{\widetilde{K}[\lambda-1/2,\omega]}{2i\omega A^{(-)}_{\lambda-1/2} (\omega)} \, W_{\lambda -1/2} (\cos \varphi) \right]\end{aligned}$$ is a background integral contribution and where $$\begin{aligned}
\label{CAM_phi2_ExpressionDef_P_RP}
& & \frac{\sqrt{2} \, r}{iq}\, \phi^{\text{\tiny{RP}} \, \textit{\tiny{(P)}}}_2 (t,r,\theta=\pi/2,\varphi)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]} \nonumber \\
& & \qquad \times \left[ - {\cal H}(\omega) \sum_{n=1}^{+\infty} \frac{\lambda_n(\omega) \beta_n(\omega) e^{i\pi \lambda_n(\omega)}}{\cos [\pi \lambda_n(\omega)]} \, \frac{\widetilde{K}[\lambda_n(\omega)-1/2,\omega]}{2 A^{(+)}_{\lambda_n(\omega)-1/2} (\omega)} \, W_{\lambda_n(\omega) -1/2} (\cos \varphi) \right. \nonumber \\
& & \qquad \quad \left. + {\cal H}(-\omega) \sum_{n=1}^{+\infty} \frac{\lambda_n(\omega) \beta_n(\omega) e^{-i\pi \lambda_n(\omega)}}{\cos [\pi \lambda_n(\omega)]} \, \, \frac{\widetilde{K}[\lambda_n(\omega)-1/2,\omega]}{2 A^{(+)}_{\lambda_n(\omega)-1/2} (\omega)} \, W_{\lambda_n(\omega) -1/2} (\cos \varphi) \right]\end{aligned}$$
is the Fourier transform of a sum over Regge poles. In these expressions, ${\cal H}$ denotes the Heaviside step function and we have introduced the analytic extensions discussed in Sec. \[SecIIIa\] as well as the Regge poles and the associated excitation factors considered in Sec. \[SecIIIb\].
We can again check that $\phi_2$ is a purely imaginary quantity by now considering this new expression. Indeed, due to the relations (\[prop\_ext\_W\_b\]) and (\[Sym\_om\_CAM\_d\]), the first term as well as the sum of the second and third terms within the square brackets on the r.h.s. of Eq. (\[CAM\_phi2\_ExpressionDef\_P\_Background\]) satisfy the Hermitian symmetry property. Such a property is also satisfied by the sum of the two terms within the square bracket on the r.h.s. of Eq. (\[CAM\_phi2\_ExpressionDef\_P\_RP\]) as a consequence of the relations (\[prop\_ext\_W\_b\]), (\[Sym\_om\_CAM\_b\]), (\[Sym\_om\_CAM\_c\]), (\[PR\_chgt\_om\]) and (\[EF\_PR\_chgt\_om\]).
Of course, Eqs. and (\[CAM\_phi2\_ExpressionDef\_P\]) provide an exact representation for the Maxwell scalar $\phi_2$, equivalent to the initial expression . From this CAM representation of $\phi_2$, we can extract the contribution denoted by $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(P)}}}_2$ and given by Eq. (\[CAM\_phi2\_ExpressionDef\_P\_RP\]) which, as a sum over Regge poles, is only an approximation of $\phi_2$. In Sec. \[SecIV\], we shall compare it with the exact expression (\[phi2\_ExpressionDef\]) of $\phi_2$. However, when considering the term $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(P)}}}_2$ alone, we shall encounter some problems due to the pathological behavior of $W_{\lambda_n(\omega) -1/2} (\cos \varphi)$ for $\varphi \to \pi$ (see Sec. \[SecIIIa\]). In fact, both the Regge pole approximation $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(P)}}}_2$ and the background integral contribution $\phi^{\text{\tiny{B}} \, \textit{\tiny{(P)}}}_2$ are divergent in the limit $\varphi \to \pi$ but it is worth pointing out that their sum (\[CAM\_phi2\_ExpressionDef\_Ptot\]) does not present any pathology.
CAM representation and Regge pole approximation of the Maxwell scalar $\phi_2$ based on the Sommerfeld-Watson transform {#SecIIId}
-----------------------------------------------------------------------------------------------------------------------
The second CAM representation of the Maxwell scalar $\phi_2$ can be derived from Eq. (\[phi2\_ExpressionDef\_reg\_SW\]) by using the Sommerfeld-Watson transformation [@Watson18; @Sommerfeld49; @Newton:1982qc] as well as Cauchy’s residue theorem. This can be achieved by following, *mutatis mutandis*, the reasoning of Sec. III D of Ref. [@Folacci:2018sef] which has permitted us to construct a CAM representation of the Weyl scalar $\Psi_4$. Here again, we avoid repeating in detail this reasoning: we note that Eq. (26) of Ref. [@Folacci:2018sef] defining $\Psi_4$ and which is the departure of the reasoning of Sec. III D of Ref. [@Folacci:2018sef] and Eq. (\[phi2\_ExpressionDef\_reg\_SW\]) of the present article are related by the correspondences (\[PSI4-phi2\_a\]), (\[PSI4-phi2\_b\]) and $$\label{PSI4-phi2_d}
Z_\ell (- \cos \varphi) \longleftrightarrow W_\ell (- \cos \varphi).$$ As a consequence, Eqs. (52) and (53) of Ref. [@Folacci:2018sef] which provide a CAM representation of the Weyl scalar $\Psi_4$ permit us to obtain directly a CAM representation of the Maxwell scalar $\phi_2$. We have
$$\label{CAM_phi2_ExpressionDef_SWtot}
\phi_2 (t,r,\theta=\pi/2,\varphi)= \phi^{\text{\tiny{B}} \, \textit{\tiny{(SW)}}}_2 (t,r,\theta=\pi/2,\varphi) + \phi^{\text{\tiny{RP}} \, \textit{\tiny{(SW)}}}_2 (t,r,\theta=\pi/2,\varphi)$$
where
\[CAM\_phi2\_ExpressionDef\_SW\] $$\begin{aligned}
\label{CAM_phi2_ExpressionDef_SW_Background}
& &\frac{\sqrt{2} \, r}{iq}\, \phi^{\text{\tiny{B}} \, \textit{\tiny{(SW)}}}_2 (t,r,\theta=\pi/2,\varphi)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]}
\nonumber\\
& & \qquad\qquad\qquad\qquad \times \left[-\frac{1}{8 \pi} \int_{-i\infty}^{+i\infty} d\lambda \, \frac{\lambda}{\cos (\pi \lambda)} \, \frac{\widetilde{K}[\lambda-1/2,\omega]}{\omega A^{(-)}_{\lambda-1/2} (\omega)} \, W_{\lambda -1/2} (-\cos \varphi) \right]\end{aligned}$$ is a background integral contribution and where $$\begin{aligned}
\label{CAM_phi2_ExpressionDef_SW_RP}
& &\frac{\sqrt{2} \, r}{iq}\, \phi^{\text{\tiny{RP}} \, \textit{\tiny{(SW)}}}_2 (t,r,\theta=\pi/2,\varphi)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} d\omega \, e^{-i\omega[ t-r_\ast(r)]} \nonumber \\
& & \qquad\qquad\qquad\qquad \times \left[ \sum_{n=1}^{+\infty} \frac{\lambda_n(\omega) \beta_n(\omega)}{2i \,\cos[\pi \lambda_n(\omega)]} \, \frac{ \widetilde{K}[\lambda_n(\omega)-1/2,\omega]}{ A^{(+)}_{\lambda_n(\omega)-1/2} (\omega)} \, W_{\lambda_n(\omega) -1/2} (-\cos \varphi) \right]\end{aligned}$$
is the Fourier transform of a sum over Regge poles.
We can again check that $\phi_2$ is a purely imaginary quantity by now considering this last expression. Indeed, due to the relations (\[prop\_ext\_W\_b\]) and (\[Sym\_om\_CAM\_d\]), the term within the square brackets on the r.h.s. of Eq. (\[CAM\_phi2\_ExpressionDef\_SW\_Background\]) satisfies the Hermitian symmetry property. Such a property is also satisfied by the term within the square brackets on the r.h.s. of Eq. (\[CAM\_phi2\_ExpressionDef\_SW\_RP\]) as a consequence of the relations (\[prop\_ext\_W\_b\]), (\[Sym\_om\_CAM\_b\]), (\[Sym\_om\_CAM\_c\]), (\[PR\_chgt\_om\]) and (\[EF\_PR\_chgt\_om\]).
It is important to note that Eq. (\[CAM\_phi2\_ExpressionDef\_SWtot\]) provides an exact expression for the Maxwell scalar $\phi_2$, equivalent to the initial expression (\[phi2\_ExpressionDef\]) and to the expression (\[CAM\_phi2\_ExpressionDef\_P\]) obtained from the Poisson summation formula. From this CAM representation of $\phi_2$, we can extract the contribution denoted by $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(SW)}}}_2$ and given by Eq. (\[CAM\_phi2\_ExpressionDef\_SW\_RP\]) which, as a sum over Regge poles, is only an approximation of $\phi_2$. In Sec. \[SecIV\], we shall compare it with the exact expression (\[phi2\_ExpressionDef\]) of $\phi_2$ and with the Regge pole approximation $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(P)}}}_2$ obtained in Sec. \[SecIIIc\]. However, when considering the term $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(SW)}}}_2$ alone, we shall encounter some problems due to the pathological behavior of $W_{\lambda_n(\omega) -1/2} (-\cos \varphi)$ for $\varphi \to 0$ (see Sec. \[SecIIIa\]). In fact, both the Regge pole approximation $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(SW)}}}_2$ and the background integral contribution $\phi^{\text{\tiny{B}} \, \textit{\tiny{(SW)}}}_2$ are divergent in the limit $\varphi \to 0$ but it is worth pointing out that their sum (\[CAM\_phi2\_ExpressionDef\_SWtot\]) does not present any pathology.
Comparison of the Maxwell scalar $\phi_2$ with its Regge pole approximations {#SecIV}
============================================================================
In this section, we shall construct numerically the multipolar waveform $\phi_2$ given by Eq. (\[phi2\_ExpressionDef\]) by summing over a large number of partial modes. This is particularly necessary for the radially infalling relativistic or ultra-relativistic particle. We shall also construct the associated quasinormal ringdown $\phi^\text{\tiny{QNM}}_2$ given by Eq. (\[response\_QNM\]). We shall then compare these two waveforms with the Regge pole approximations $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(P)}}}_2$ and $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(SW)}}}_2$ respectively given by Eqs. (\[CAM\_phi2\_ExpressionDef\_P\_RP\]) and (\[CAM\_phi2\_ExpressionDef\_SW\_RP\]) and constructed by considering one or two Regge poles. This will allow us to highlight the benefits of working with the Regge pole approximations of $\phi_2$.
Computational methods {#SecIVa}
---------------------
To construct numerically the Maxwell scalar $\phi_2$ as well as its quasinormal and Regge pole approximations, we use, *mutatis mutandis*, the computational methods developed in Refs. [@Folacci:2018vtf; @Folacci:2018cic] which allowed us to describe the electromagnetic field and the gravitational waves generated by a particle plunging from the innermost stable circular orbit into a Schwarzschild BH (see, e.g., Sec. IV A of Ref. [@Folacci:2018vtf]).
Results and comments {#SecIVb}
--------------------
We have compared the multipolar waveform $\phi_2$ and the associated quasinormal ringdown with the Regge pole approximations $\phi^{{\text{\tiny{RP}}}\,\textit{\tiny{(P)}}}_2$ in Figs. \[P\_Exact\_QNM\_CAM\_pis6\_v0\]–\[P\_Exact\_QNM\_CAM\_pis3\_v099\] and with the Regge pole approximation $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(SW)}}}_2$ in Figs. \[SW\_Exact\_QNM\_CAM\_3pis4\_v0\]–\[SW\_Exact\_QNM\_CAM\_5pis6\_v099\]. This has been done for various values of the angle $\varphi \in [0, \pi]$ excluding the cases $\varphi=0$ and $\varphi=\pi$ for which the Maxwell scalar $\phi_2$ vanishes. More precisely, we have considered the case of (i) a particle initially at rest at infinity \[$v_\infty = 0$ ($\gamma=1$)\], (ii) a particle projected with a relativistic velocity at infinity \[we have considered the configurations $v_\infty = 0.75$ ($\gamma\approx 1.51$) and $v_\infty = 0.90$ ($\gamma\approx 2.29$)\], and (iii) a particle projected with an ultra-relativistic velocity at infinity \[$v_\infty = 0.99$ ($\gamma \approx 7.09$)\]. It should be specified that, in order to obtain numerically stable results, the number of partial modes to include in the sum (\[phi2\_ExpressionDef\]) strongly depends on the initial velocity of the particle: the sum over $\ell$ has been truncated at $\ell=13$ for $v_\infty =0$, at $\ell=15$ for $v_\infty =0.75$ and $v_\infty =0.90$, and at $\ell= 19$ for $v_\infty =0.99$. It should be noted that the terminology we used in Ref. [@Folacci:2018sef] to describe the different parts of the multipolar waveform $\Psi_4$ is also adopted for the waveform $\phi_2$: we shall thus designate by “pre-ringdown phase” the early time response of the BH and, as usual, we shall refer to the ringdown phase and to the tail of the signal for those parts of the waveform corresponding respectively to intermediate timescales and to very late times.
In Figs. \[P\_Exact\_QNM\_CAM\_pis6\_v0\]–\[P\_Exact\_QNM\_CAM\_3pis4\_v0\], we have compared the multipolar waveform $\phi_2$ generated by a particle initially at rest at infinity with its Regge pole approximation $\phi^{\text{\tiny{RP}} \, \textit{\tiny{(P)}}}_2$ obtained from the Poisson summation formula. In Figs. \[P\_Exact\_QNM\_CAM\_pis6\_v0\]–\[P\_Exact\_QNM\_CAM\_pis2\_v0\], for $\varphi = \pi/6, \pi/3$ and $\pi/2$, we can observe that the Regge pole approximation constructed from only one Regge pole is in good or very good agreement with the exact waveform, and that an additional Regge pole does not really improve this approximation. More precisely, it is interesting to note that the Regge pole approximation matches the ringdown, describes correctly the pre-ringdown phase and roughly the waveform tail. It is moreover important to note that it provides a description of the ringdown that does not necessitate determining a starting time, in contrast to the ringdown waveform constructed from the QNMs which is exponentially divergent as $t$ decreases. In Fig. \[P\_Exact\_QNM\_CAM\_3pis4\_v0\], for $\varphi = 3\pi/4$, we can observe that the Regge pole approximation is no longer so interesting. Indeed, it only roughly describes the BH response. Here, it should be recall that the Regge pole approximation $\phi^{\text{\tiny{RP}}\,\textit{\tiny {(P)}}}_2 $ diverges for $\varphi \to \pi$ and, as a consequence, for $\varphi=3\pi/4$ (i.e., for a value of $\varphi$ rather close to $\pi$), it would be necessary to consider the background integral contribution $\phi^{\text{\tiny{B}} \, \textit{\tiny{(P)}}}_2$ given by Eq. (\[CAM\_phi2\_ExpressionDef\_P\_Background\]) to correctly describe the multipolar waveform $\phi_2$.
In Figs. \[P\_Exact\_QNM\_CAM\_pis6\_v075\]–\[P\_Exact\_QNM\_CAM\_pis3\_v099\], we have compared, for $\varphi= \pi/6 $ and $\pi/3$, the multipolar waveform $\phi_2$ generated by a particle projected with a relativistic or an ultra-relativistic velocity at infinity with the Regge pole approximation $\phi^{\text{\tiny{RP}}\,\textit{\tiny{(P)}}}_2$ obtained from the Poisson summation formula. We can observe that the whole signal is impressively described by the Regge pole approximation constructed from only one Regge pole and that this approximation is even more efficient in the ultra-relativistic context.
In Fig. \[SW\_Exact\_QNM\_CAM\_3pis4\_v0\], for $\varphi = 3 \pi/4$, we have compared the multipolar waveform $\phi_2$ generated by a particle initially at rest at infinity with the Regge pole approximation $\phi^{\text{\tiny{RP}}\,\textit{\tiny{(SW)}}}_2 $ obtained from the Sommerfeld-Watson transformation. We recall that, while $\phi^{\text{\tiny{RP}}\,\textit{\tiny{(P)}}}_2$ constructed from the Poisson summation formula diverges in the limit $\varphi \to \pi $, the Regge pole approximation $\phi^{\text{\tiny{RP}}\,\textit{\tiny{(SW)}}}_2 $ is regular in the same limit (it only diverges for $\varphi \to 0$). As a consequence, the latter approximation should provide better results than the former one for $\varphi$ close to $\pi$. By comparing Fig. \[SW\_Exact\_QNM\_CAM\_3pis4\_v0\] with Fig. \[P\_Exact\_QNM\_CAM\_3pis4\_v0\], we can see that this seems to be the case if we focus on the ringdown phase of the waveform but that the pre-ringdown phase is not described at all. In fact, here, to correctly describe the waveform $\phi_2$ we should take into account the background integral contribution $\phi^{\text{\tiny{B}}\, \textit{\tiny{(SW)}}}_2$ given by Eq. (\[CAM\_phi2\_ExpressionDef\_SW\_Background\]).
In Figs. \[SW\_Exact\_QNM\_CAM\_5pis6\_v0\]–\[SW\_Exact\_QNM\_CAM\_5pis6\_v099\], for $ \varphi = 5\pi/6 $, we have displayed the multipolar waveform $\phi_2$ generated by a particle initially at rest at infinity and by a particle projected with a relativistic or an ultra-relativistic velocity, and we have compared it with the Regge pole approximation $\Psi^{\text{\tiny {RP}}\,\textit{\tiny {(SW)}}}_4$ obtained from the transformation of Sommerfeld-Watson. Here again, the Regge pole approximation constructed from a single Regge pole does not describe the pre-ringdown phase of the Maxwell scalar $\phi_2$, but it matches a large part of the ringdown phase and approximates the tail rather correctly.
Electromagnetic energy spectrum $dE/d\omega$ and its CAM representation {#SecV}
=======================================================================
In this section, we shall focus on the electromagnetic energy spectrum $dE/d\omega$ observed at infinity which is generated by the charged particle falling radially into the Schwarzschild BH. We shall provide its CAM representation from the Poisson summation formula and Cauchy’s theorem and discuss the interest of this representation and of the corresponding Regge pole approximation.
Total energy radiated by the particle and associated electromagnetic energy spectrum {#SecV_a}
------------------------------------------------------------------------------------
The electromagnetic power ${\cal P}$ radiated at spatial infinity by the charged particle, i.e., the rate $dE/dt$ at which the electromagnetic field generated by this particle carries energy to infinity, can be obtained as the flux of the Poynting vector ${\mathbf R}$ across a spherical surface $S(r)$ with radius $r \to \infty$: we have $$\begin{aligned}
\label{dEdt}
{\cal P}= \frac{dE}{dt} = \lim_{r \to \infty} \int_{S(r)} {\mathbf R} \cdot {\mathbf {dS}}\end{aligned}$$ with ${\mathbf R}={\mathbf E} \wedge {\mathbf B}$ and $\mathbf {dS} = r^2 \sin\theta \, d\theta \, d\varphi \, {\bf {\hat{e}}_r}$. By using Eqs. (\[ChampE\_even\]), (\[Relations\_ChampB\_even\_odd\]) and (\[TF\_psi\_bis\_a\]) as well as the orthonormalization relation (\[HSV\_even\_Norm\]) for the vector spherical harmonics and the addition theorem for scalar spherical harmonics (\[ThAd\_HS\]), we obtain
\[dEdt\] $$\begin{aligned}
\label{dEdt_a}
\frac{dE}{dt}(t) &=&\frac{1}{4\pi} \sum_{\ell m} \frac{2 \ell +1 }{\ell(\ell+1)}\, \Big{|} \partial_t \psi_{\ell} (t,r \to +\infty) \Big{|}^{2}\end{aligned}$$ or, more explicitly, by using Eqs. (\[TF\_psi\_bis\_b\]) and (\[Partial\_Response\_c\]), $$\begin{aligned}
\label{dEdt_b}
& & \frac{dE}{dt}(t) =\frac{q^2}{4\pi}\sum_{\ell = 1}^{+\infty} (2\ell+1)\ell(\ell+1) \nonumber \\
& & \qquad \quad \times \Bigg{|} \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}d\omega\, \frac{e^{-i \omega[t-r_\ast(r)]}}{2i\omega A_{\ell}^{(-)}(\omega)}\widetilde{K}[\ell,\omega]\Bigg{|}^2\end{aligned}$$
with $r \to +\infty$.
The previous result provides, by integration over $t$, the total energy $E$ radiated by the charged particle during its fall in the BH. We have
$$\begin{aligned}
\label{Etot_INTt}
& & E =\int_{-\infty}^{+\infty} dt \, \frac{dE}{dt}(t) \label{Etot_INTt_a}\\
& & \phantom{E} = \frac{q^2}{4\pi}\sum_{\ell = 1}^{+\infty} (2\ell+1)\ell(\ell+1) \nonumber \\
& & \qquad \times \int_{-\infty}^{+\infty} dt \, \Bigg{|} \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}d\omega\, \frac{e^{-i \omega t}}{2i\omega A_{\ell}^{(-)}(\omega)}\widetilde{K}[\ell,\omega]\Bigg{|}^2 \nonumber \\
& & \label{Etot_INTt_b}\end{aligned}$$
\[note that the dependence in $r$ now disappears due to the change of variable $t \to t + r_\ast(r)$\]. We can obtain an alternative expression for the total energy $E$ by applying the Parseval-Plancherel theorem to Eq. (\[Etot\_INTt\_b\]). This gives immediately $$\begin{aligned}
\label{Etot_PP}
& & E = \frac{q^2}{4\pi}\sum_{\ell = 1}^{+\infty} (2\ell+1)\ell(\ell+1) \int_{-\infty}^{+\infty} d\omega \, \Bigg{|} \frac{\widetilde{K}[\ell,\omega]}{2i\omega A_{\ell}^{(-)}(\omega)}\Bigg{|}^2. \nonumber \\
& &\end{aligned}$$ This new form of $E$ permits us to derive the expression of the (total) electromagnetic energy spectrum $dE/d\omega$ radiated by the particle. Indeed, from a physical point of view, it is defined for $\omega \ge 0$ and satisfy $$\label{dEdw_w_def}
E = \int_0^{+\infty} d\omega \, \frac{dE}{d\omega} (\omega).$$ Then, by using Eq. (\[Sym\_om\_d\]) in Eq. (\[Etot\_PP\]), we obtain $$\begin{aligned}
\label{Etot_PP_bis}
& & E = \frac{q^2}{2\pi}\sum_{\ell = 1}^{+\infty} (2\ell+1)\ell(\ell+1) \int_{0}^{+\infty} d\omega \, \Bigg{|} \frac{\widetilde{K}[\ell,\omega]}{2i\omega A_{\ell}^{(-)}(\omega)}\Bigg{|}^2 \nonumber \\
& &\end{aligned}$$ and by comparing Eq. (\[Etot\_PP\_bis\]) with Eq. (\[dEdw\_w\_def\]) we have
\[dEdw\_bis\] $$\label{dEdw_bis_a}
\frac{dE}{d\omega} (\omega) = \sum_{\ell=1}^{+\infty} \frac{dE_\ell}{d\omega}(\omega)$$ where $$\label{dEdw_bis_b}
\frac{dE_\ell}{d\omega}(\omega) = \frac{q^2}{8\pi \omega^2} \times (2\ell+1)\ell(\ell+1)\, \Gamma_\ell(\omega) \, \Big{|}\widetilde{K}[\ell,\omega]\Big{|}^2$$
denotes the partial energy spectrum corresponding to the $\ell$th mode. It is very important to note that in Eq. (\[dEdw\_bis\_b\]), we have chosen to introduce explicitly the greybody factors $$\label{Greybody_factors}
\Gamma_{\ell}(\omega)= \frac{1}{\big{|}A_{\ell}^{(-)}(\omega) \big{|}^2}$$ of the Schwarzschild BH corresponding to the electromagnetic field. It is worth pointing out that we can write $E$ given by Eqs. (\[dEdw\_w\_def\]) and (\[dEdw\_bis\]) in the form
\[Etot\] $$\label{Etot_a}
E = \sum_{\ell=1}^{+\infty} E_\ell$$ where $$\label{Etot_b}
E_\ell= \int_0^{+\infty} d\omega \, \frac{dE_\ell}{d\omega}(\omega)$$
denotes the partial energy radiated in the $\ell$th mode.
Finally, it is important to note that Eq. (\[dEdw\_bis\]) can also be written as $$\label{dEdw_bis2}
\frac{dE}{d\omega}(\omega) =\frac{q^2}{8\pi \omega^2}\sum_{\ell = 0}^{+\infty} (2\ell+1)\ell(\ell+1)\, \Gamma_\ell(\omega) \, \Big{|}\widetilde{K}[\ell,\omega]\Big{|}^2.$$ Indeed, here again, as in Sec. \[SecIIe\], it is possible to start at $\ell=0$ the discrete sum over $\ell$ due to the relations (\[Formally\_ell0\]). In the next subsection, we shall take Eq. (\[dEdw\_bis2\]) as a starting point because it will permit us to use the Poisson summation formula in its standard form.
CAM representation based on the Poisson summation formula {#SecV_b}
---------------------------------------------------------
In order to start the CAM machinery permitting us to derive a CAM representation of the electromagnetic energy spectrum $dE / d\omega$, it is necessary to replace in Eq. (\[dEdw\_bis2\]) the angular momentum $\ell \in \mathbb{N}$ by the angular momentum $\lambda = \ell +1/2 \in \mathbb{C}$ and therefore to have at our disposal the analytic extensions in the complex $\lambda$ plane of all the functions of $\ell$ appearing in Eq. (\[dEdw\_bis2\]). In fact, in Sec. \[SecIIIa\], we have already discussed the construction of the analytic extensions of $A^{(\pm)}_{\ell} (\omega)$ and $\widetilde{K}[\ell,\omega]$. It should be however noted that, here, the situation is a little bit more complicated: indeed, we need the analytic extensions of $\Gamma_{\ell}(\omega)= 1/ \big{|}A_{\ell}^{(-)}(\omega) \big{|}^2$ and $\big{|}\widetilde{K}[\ell,\omega]\big{|}^2$. Fortunately, in Sec. II of Ref. [@Decanini:2011xi] where the absorption problem for a massless scalar field propagating in a Schwarzschild BH has been considered, the analytic extension $\Gamma_{\lambda-1/2}(\omega)$ of the greybody factor $\Gamma_\ell(\omega)$ has been discussed. Here, we shall adopt the same prescription, i.e., we shall assume that $$\label{Gamma_Analytic_Extension}
\Gamma_{\lambda-1/2}(\omega) =\frac{1}{A_{\lambda-1/2}^{(-)}(\omega) \, {[A^{(-)}_{\lambda^\ast -1/2} (\omega) ]}^\ast}.$$ We recall that this particular extension permits us to work with an even function of $\lambda$ which is purely real. (For more details concerning the properties of the greybody factor $\Gamma_{\lambda-1/2}(\omega)$, we refer to Sec. II of Ref. [@Decanini:2011xi].) Furthermore, we shall adopt an analogous prescription for the analytic extension of $\big{|}\widetilde{K}[\ell,\omega]\big{|}^2$ by considering that it is given by $\widetilde{K}[\lambda-1/2,\omega] \, {[ \widetilde{K}[\lambda^\ast -1/2,\omega] ]}^\ast$.
In order to derive a CAM representation of the electromagnetic energy spectrum $dE / d\omega$, the use of CAM techniques requires in addition the determination of the singularities of the analytic extensions in the complex $\lambda$ plane of all the functions of $\ell$ appearing in Eq. (\[dEdw\_bis2\]). Here, the only singularities to consider are the simple poles of the greybody factor $\Gamma_{\lambda-1/2}(\omega)$. In fact, they have been also studied in Sec. II of Ref. [@Decanini:2011xi]. Let us just recall that:
1. The singularities of the function $\Gamma_{\lambda-1/2}(\omega)$ are the Regge poles $\lambda_n(\omega)$, i.e., the zeros of the function $A^{(-)}_{\lambda_n(\omega)-1/2} (\omega)$ \[see Eq. (\[PR\_def\_Am\])\], as well their complex conjugates ${[\lambda_n(\omega)]}^\ast $, i.e., the zeros of the function ${[A^{(-)}_{\lambda^\ast -1/2} (\omega) ]}^\ast$. For $\omega >0$, the Regge poles $\lambda_n(\omega)$ lie in the first and in the third quadrant of the CAM plane, symmetrically distributed with respect to the origin $O$ of this plane and, as a consequence, the Regge poles ${[\lambda_n(\omega)]}^\ast $ lie in the second and in the fourth quadrant of this plane.
2. The residues of the function $\Gamma_{\lambda-1/2}(\omega)$ at the poles $\lambda_n(\omega)$ and ${[\lambda_n(\omega)]}^\ast $ are complex conjugate of each other and we have in particular $$\begin{aligned}
\label{gamma_residues}
\gamma_n(\omega) &=&\text{Res}[\Gamma_\lambda-1/2(\omega)]_{\lambda=\lambda_n(\omega)} \nonumber \\
&=& \frac{1}{\left[\left(\frac{d}{d \lambda}A_{\lambda -1/2}^{(-)}(\omega) \right) {[A^{(-)}_{\lambda^\ast -1/2} (\omega) ]}^\ast\right]_{\lambda=\lambda_n(\omega)}}.\end{aligned}$$
We have now at our disposal all the ingredients permitting us to obtain a CAM representation of the electromagnetic energy spectrum $dE / d\omega$ by using the Poisson summation formula [@MorseFeshbach1953] as well as Cauchy’s residue theorem. In fact, this can be achieved by following, *mutatis mutandis*, the reasoning of Sec. II of Ref. [@Decanini:2011xi] where a CAM representation of the absorption cross section of the Schwarzschild BH has been derived \[we invite the reader to compare Eq. (3) of Ref. [@Decanini:2011xi] with Eq. (\[dEdw\_bis2\]) of the present article\]. Taking into account the previous considerations concerning the greybody factor $\Gamma_{\lambda-1/2}(\omega)$, its poles and the associated residues, we obtain
$$\label{dEdw_CAM_2}
\frac{dE}{d\omega} (\omega)= \frac{dE}{d\omega}^{\text{\tiny{B,Re}}}(\omega) + \frac{dE}{d\omega}^{\text{\tiny{B,Im}}}(\omega) + \frac{dE}{d\omega}^{\text{\tiny{RP}}}(\omega)$$
where
\[dEdw\_CAM\_Background\] $$\label{dEdw_background_integral_Re}
\frac{dE}{d\omega}^{\text{\tiny{B,Re}}} (\omega)= \frac{q^2}{4\pi \omega^2} \int_{0}^{+\infty} d\lambda\, \lambda(\lambda^2-1/4) \Gamma_{\lambda-1/2}(\omega) \Big{|}\widetilde{K}[\lambda-1/2,\omega]\Big{|}^2$$ is a background integral contribution along the real axis, $$\label{dEdw_background_integral_Im}
\frac{dE}{d\omega}^{\text{\tiny{B,Im}}} (\omega) = -\frac{q^2}{4 \pi \omega^2}\int_{0}^{+i\infty} d\lambda \, \lambda(\lambda^2-1/4) \Gamma_{\lambda-1/2}(\omega) \Big{|}\widetilde{K}[\lambda-1/2,\omega]\Big{|}^2 \frac{e^{i \pi \lambda}}{\cos(\lambda \pi)}$$
is a background integral contribution along the imaginary axis and $$\begin{aligned}
\label{dEdw_CAM_PR}
& & \frac{dE}{d\omega}^{\text{\tiny{RP}}} (\omega) = -\frac{q^2}{2 \omega^2} \operatorname{Re} \left(\sum_{n=1}^{+\infty} \frac{e^{i\pi[\lambda_n(\omega)-1/2]}\lambda_n(\omega) \left(\lambda_n(\omega)^2-1/4\right) \, \gamma_n(\omega)}{\sin[\pi(\lambda_n(\omega)-1/2)]} \right. \nonumber \\
& & \left. \qquad\qquad\qquad\qquad\qquad \phantom{\sum_{n=1}^{+\infty}} \times \widetilde{K}[\lambda_n(\omega)-1/2,\omega] \, \left[\widetilde{K}[{[\lambda_n(\omega)]}^\ast-1/2,\omega]\right]^\ast\, \right)\end{aligned}$$
is a sum over the Regge poles lying in the first quadrant of the CAM plane. Of course, Eqs. , and provide an exact CAM representation of the electromagnetic energy spectrum $dE/d\omega$, equivalent to the initial partial wave expansion .
![\[dEdw\_v\_0\_075\_090\_099\_tot\] The total electromagnetic energy spectrum radiated by a charged falling radially into a Schwarzschild BH. The results are given for $v_\infty =0, 0.75, 0.90$ and $0.99$.](dEdw_v_0_075_090_099_tot)
Computational methods {#SecV_c}
---------------------
To construct numerically the electromagnetic energy spectrum radiated by a charged particle falling radially into the Schwarzschild BH and its CAM representation -, we have used the computational methods that have allowed us to obtain numerically the Maxwell scalar $\phi_2$ and its Regge pole approximations in Sec. \[SecIV\]. It should be noted that here, we have in addition evaluated the background integral along the real axis (\[dEdw\_background\_integral\_Re\]) by taking $\lambda \in [0, 25]$ and the background integral along the imaginary axis (\[dEdw\_background\_integral\_Im\]) by taking $\lambda \in [0, 6i]$ (due to the term $e^{i \pi \lambda}/\cos[\lambda \pi]$ in the expression of its integrand, this integral converges rapidly).
![\[dEdw\_Exact\_vs\_CAM\_PR\_Int\_axe\_Re\_v\_0\_090\_099\] The oscillations in the electromagnetic energy spectrum radiated by a charged particle falling radially into a Schwarzschild BH explained by the Regge pole approximation.](dEdw_Exact_vs_CAM_PR_Int_axe_Re_v_0_090_099)
Numerical results and comments {#SecV_d}
------------------------------
We now display and discuss a few results concerning the electromagnetic energy radiated by the charged particle falling radially into a Schwarzschild BH. Here again, as in Sec. \[SecIVb\], we have focused our attention on (i) a particle initially at rest at infinity \[$v_\infty =0$ ($\gamma=1$)\], (ii) a particle projected with a relativistic velocity at infinity \[we have considered the configurations $v_\infty = 0.75$ ($\gamma\approx 1.51$) and $v_\infty = 0.90$ ($\gamma\approx 2.29$)\], and (iii) a particle projected with an ultra-relativistic velocity at infinity \[$v_\infty = 0.99$ ($\gamma \approx 7.09$)\].
In Fig. \[dEdw\_v\_0\_075\_090\_099\], we have displayed some partial electromagnetic energy spectra $dE_\ell/d\omega$ corresponding to the lowest modes. Our results are in perfect agreement with those already obtained in the literature (see Refs. [@Ruffini:1972uh; @Cardoso:2003cn] but note in these articles, the authors used Gaussian units while we consider electromagnetism in the Heaviside system). In Fig. \[dEdw\_v\_0\_075\_090\_099\_tot\], we have displayed the total electromagnetic energy spectrum $dE/d\omega$ for the configurations considered in Fig. \[dEdw\_v\_0\_075\_090\_099\]. It should be noted that, in order to obtain numerically stable results, the number of modes to include in the sum (\[dEdw\_bis\_a\]) strongly depends on the initial velocity of the particle. This clearly appears if we examine the ordinate scales used in the semilog graphs of Fig. \[dEdw\_v\_0\_075\_090\_099\]. In fact, we have truncated the sum over $\ell$ at $\ell=10$ for $v_\infty =0$, at $\ell=15$ for $v_\infty =0.75$ and $v_\infty =0.90$, and at $\ell= 20$ for $v_\infty =0.99$.
In Table \[tab:table1\], we have used Eq. (\[Etot\]) to compute the total energy $E$ radiated by the charged particle for the values $v_\infty =0, 0.75, 0.90$ and $0.99$ of its velocity at infinity. As expected, $E$ increases with $v_\infty$ (see also Refs. [@Ruffini:1972uh; @Cardoso:2003cn]) while the rate of convergence of the series over the partial energies $E_\ell$ which defines it decreases. In other terms, i.e., from a physical point of view, we can observe that for $v_\infty = 0$, the $\ell = 1$ mode radiates the largest amount of energy ($83.20\%$), and that summing over the first five modes, we reach $99.99\%$ of the total electromagnetic energy radiated; on the other hand, for $v_\infty = 0.99$, the $\ell = 1$ mode is responsible for only $16.54\%$ of the total electromagnetic energy radiated while the sum over the first five modes represents only $63.41\%$ of this energy.
In Fig. \[dEdw\_Exact\_vs\_CAM\_v\_0\_075\_090\_099\], we have compared the electromagnetic energy spectrum $dE/d\omega$ given by Eq. (\[dEdw\_bis\]) with its CAM representation (\[dEdw\_CAM\_2\])-(\[dEdw\_CAM\_PR\]). This permits us to emphasize the respective role of the background integrals (\[dEdw\_background\_integral\_Re\]) and (\[dEdw\_background\_integral\_Im\]) and of the Regge pole sum (\[dEdw\_CAM\_PR\]). In particular, we can observe that, for very low frequencies, in order to match the exact energy spectrum, it is necessary to take into account these two background integrals and to consider the first two Regge poles in the Regge pole sum. Out of this frequency regime, the exact energy spectrum can be perfectly described by only considering the background integral along the real axis and a single Regge pole in the Regge pole sum. Here, it is worth pointing out that the Regge pole approximation cannot be used to resum the total electromagnetic energy spectrum because the CAM representation is dominated by the background integrals. However, we can observe in Fig. \[dEdw\_Exact\_vs\_CAM\_PR\_Int\_axe\_Re\_v\_0\_090\_099\] that it is the Regge pole approximation which explains the oscillations appearing in the electromagnetic energy spectrum. Due to the connection existing between the Regge modes and the (weakly damped) QNMs of the Schwarzschild BH [@Decanini:2002ha; @Decanini:2009mu], we can also associate these oscillations with the quasinormal frequencies of the BH.
Conclusion {#Conc}
==========
In this paper, we have revisited the problem of the electromagnetic radiation generated by a charged particle falling radially into a Schwarzschild BH. We have obtained a series of results which highlight the benefits of working within the CAM framework and strengthen our opinion concerning the interest of the Regge pole approach for describing radiation from BHs because they are fairly close to those previously reported in Ref [@Folacci:2018sef] where we discussed an analogous problem in the context of gravitational radiation.
We have described the electromagnetic radiation by the Maxwell scalar $\phi_2$ and we have extracted from its multipole expansion the Fourier transform of a sum over the Regge poles of the BH $\mathcal{S}$-matrix involving, in addition, the excitation factors of the Regge modes. It constitutes an approximation of $\phi_2$ which can be evaluated numerically from the Regge trajectories associated with the Regge poles and their residues. In fact, we have constructed two different Regge pole approximations of $\phi_2$: the first one, which has been obtained from the Poisson summation formula, is given by Eq. and provides very good results (even impressive results for relativistic particles) for observation directions in a large angular sector around the particle trajectory; the second one, which has been derived by using the Sommerfeld-Watson transformation, is given by Eq. and is helpful in a large angular sector around the direction opposite to the particle trajectory. More precisely, it should be noted that, in general, these two Regge pole approximations can reproduce with very good agreement the quasinormal ringdown (it is worth pointing out that, in contrast to the QNM description of the ringdown, the Regge pole description does not require a starting time) as well as with rather good agreement the tail of the signal and that the first approximation even describes the pre-ringdown phase. All our results have been achieved by taking into account only one Regge pole. To understand the interest of this fact, it is important to recall that the partial wave expansion defining $\phi_2$ is a slowly convergent series, especially in the case of a particle projected with a relativistic or an ultra-relativistic velocity into the BH; its Regge pole approximations are efficient resumations which permit us, in addition, to extract the physical information it encodes. It is interesting to recall that, for the analogous problem in the context of gravitational radiation [@Folacci:2018sef], we have obtained rather similar results for the Weyl scalar $\Psi_4$ but that, in this case, taking into account additional Regge poles sometimes improves the Regge pole approximations. It should be finally noted that we have also considered the electromagnetic energy spectrum $dE/d\omega$ (a topic we did not touch on in Ref [@Folacci:2018sef]) and, by using the Poisson summation formula, we have constructed from its multipole expansion its CAM representation given by Eqs. (\[dEdw\_CAM\_2\])–(\[dEdw\_CAM\_PR\]). Unfortunately, here the full CAM representation is necessary to describe the whole electromagnetic energy spectrum but the corresponding Regge pole approximation (\[dEdw\_CAM\_PR\]) is however helpful to understand its oscillations and associate them with QNMs.
In future works, we would like to go beyond the relatively simple problems examined in the present paper and in Ref. [@Folacci:2018sef] by revisiting, using CAM techniques, the problem of the radiation generated by a particle with an arbitrary orbital angular momentum plunging into a Schwarzschild or a Kerr BH. It would be in addition interesting to extract asymptotic expressions from the background integral contributions appearing in the various CAM representations in order to improve the physical interpretation of the results. We would also like to go beyond the case of BHs by considering that of neutron stars and white dwarfs. In this context, the recent CAM analysis of scattering by compact objects [@OuldElHadj:2019kji] could be a natural starting point.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the present contribution, we construct a virtual element (VE) discretization for the problem of miscible displacement of one incompressible fluid by another, described by a time-dependent coupled system of nonlinear partial differential equations. Our work represents a first study to investigate the premises of virtual element methods (VEM) for complex fluid flow problems. We combine the VEM discretization with a time stepping scheme and develop a complete theoretical analysis of the method under the assumption of a regular solution. The scheme is then tested both on a regular and on a more realistic test case. **AMS subject classification**: 65M12, 65M60, 76S05 **Keywords**: virtual element methods, miscible fluid flow, porous media, polygonal meshes'
author:
-
- 'L. Beirão da Veiga[^1], A. Pichler[^2], G.Vacca'
bibliography:
- 'bibliogr.bib'
title: A virtual element method for the miscible displacement of incompressible fluids in porous media
---
Introduction
============
The virtual element method (VEM) was introduced in [@VEMvolley; @hitchhikersguideVEM] (see also [@VEM3Dbasic]) as a generalization of the finite element method (FEM) that allows to use general polygonal and polyhedral meshes. Since its recent birth in 2013, VEM enjoyed a rapid growth in the mathematics and engineering communities. Among the large numbers of papers in the literature, we here cite only [@Brezzi-Marini:2012; @absv_VEM_cahnhilliard; @cangianimanzinisutton_VEMconformingandnonconforming; @VEM_DD_basic; @VEMnavierstokes; @VEMnewtonianflows; @VEMmagnetostatic3D; @VEM_nc_SUPG; @VEMtransmissioneigen; @VEMcurved; @TVEM_Helmholtz_num; @VEMcrank; @VEMtopopt] as representatives, see also the references therein.
In the realm of diffusion problems, virtual elements have been developed for linear model diffusion-convection-reaction equations in primal and mixed form [@VEMvolley; @BBMR_generalsecondorder; @BBMR_generalsecondordermixed; @VEM3Dbasic; @sukumar; @vacca2015virtual; @VEM_SUPG]. It was soon recognized that the flexibility of VEM in terms of meshing could lead to appealing advantages in the presence of complex geometries, such as for discrete fracture network simulation [@VEM_dfn1; @VEM_dfn2; @VEM_dfn3] and, more in general, in the presence of fractures in porous 3D media [@VEM_dfn4; @VEM_fumagalli]. Nevertheless, although in other frameworks (such as solid mechanics) VEM have indeed proven themselves also on tough nonlinear problems, to the best knowledge of the authors, virtual elements have never been developed and tested for more complex diffusion models. Since VEM have indeed been hardly tested (with very promising outcomes) on linear diffusion problems with complex geometries, as often encountered in geophysical flows, developing a VEM also for more complex (and realistic) flow models becomes a key step towards a competitive methodology for applications. In the present contribution, we consider the miscible displacement of one incompressible fluid by another in a reservoir, described by a time-dependent coupled system of nonlinear partial differential equations, that is a basic (but meaningful) model instrumental to applications such as oil recovery and environmental pollution [@russell1983finite; @peaceman2000fundamentals; @ewing1983approximation; @chainais2007convergence; @droniou2]. One must note that, on this and similar models, there already exists a large literature with many competitive schemes, adopting for instance finite elements [@fem1; @fem2; @ewing1983approximation], discontinuous Galerkin methods [@dg1; @dg2; @dg3], and finite volumes [@chainais2007convergence; @fv1; @fv2]. The aim of the present paper is to make a first study on the premises of VEM in this framework (by proposing a numerical scheme, giving a theoretical backbone to it, and finally testing it numerically). We believe that the VEM for this kind of problems could have a value due to its strong robustness with respect to the mesh features and, more heuristically, its potential in terms of flexibility in general terms.
From the mathematical viewpoint, the above model yields a nonlinear time-dependent coupled problem for concentration, velocity and pressure, also with potential issues of stability (at the discrete level) due to possible convection-dominated regimes. We propose a continuous ($H^1$ conforming) approximation for the concentration variable, thus leading to nodal virtual elements [@VEMvolley; @BBMR_generalsecondorder], and an $H({\rm div})$ conforming approximation of the Darcy velocity, thus leading to face virtual elements [@Brezzi-Falk-Marini; @BBMR_generalsecondordermixed]. For the pressure, we adopt a standard piecewise discontinuous polynomial space. Due to the presence of the non-linear coefficients coupling the two set of equations, we make use of projection operators to approximate such terms and of stabilization factors that are suitably chosen. We combine the VEM discretization in space with a simple discretization procedure in time, that is a backward Euler approximation that is explicit in the coefficient terms. As a consequence, the system to be solved at each time step is linear and decoupled, leading to a cheap procedure. Extending the proposed scheme to different time discretization procedures would be, on the basis of the work presented here, quite trivial.
After proposing the method, we develop an error analysis under the assumption of a regular solution. Although such regularity conditions are unrealistic in most cases of interest, we believe that the derived results are still critical in order to give a theoretical backbone to the method. They serve the purpose of showing that the method indeed delivers a solution with the potential to yield the correct approximation order whenever this is feasible (given the approximability of the target solution by the discrete space). No time step size condition is needed in the analysis. Finally, we test the proposed scheme in two different ways. We firstly consider a problem with known regular solution inspired from [@miscibledispl2018hu], in order to validate the convergence properties of the method also in practice and to test some other practical aspect such as the possibility of having different time step sizes for the two different equations. Then, we consider a more realistic test, taken from [@wang2000approximation], in order to have a qualitative comparison with the expected benchmark solution from the literature. In this second test, there is also the risk of overshoots and undershoots in the discrete solution due to strong convection. We here deal with this aspect by introducing in our scheme a modification borrowed from [@FCT], that is recognized in [@smalldiffusion] to be one of the best choices in practice. From the present first theoretical and numerical studies, we believe the VEM is promising and has the possibility to become, after further developments, a competitive scheme for complex flow problems.
The structure of the paper is the following. In Section \[sec:2\], we introduce the continuous problem, in strong and weak form. In Section \[sec:3\], we describe the proposed virtual element discretization, in space and time. In Section \[sec:4\], we develop the theoretical convergence analysis of the scheme. Finally, in Section \[sec:5\], we show the numerical results.
Problem description {#sec:2}
===================
We consider the miscible displacement of one incompressible fluid by another in a porous medium. This problem can be formulated in terms of a system of partial differential equations, where a parabolic diffusion-convection-reaction type equation is nonlinearly coupled with an elliptic system, see also [@peaceman2000fundamentals; @ewing1983approximation; @chainais2007convergence; @droniou2].
We need to introduce some notation and conventions to be adopted throughout the paper. We denote by ${\mathbb{N}}$ and ${\mathbb{N}}_0$ the sets of all natural numbers without and including zero, respectively. Moreover, we employ the standard notation for Sobolev spaces, norms, and seminorms. More precisely, for a given bounded Lipschitz domain $D \subset {\mathbb{R}}^2$, $k \in {\mathbb{N}}\cup \{\infty\}$, and $p \in {\mathbb{N}}$, we define by $W^{k,p}(D)$ the space of all $L^p$ integrable functions over $D$ whose weak derivatives up to order $k$ are again $L^p$ integrable. Sobolev spaces with fractional order can be defined for instance via interpolation theory [@Triebel]. For $p=2$, we write $H^k(D):=W^{k,2}(D)$, and we use $(\cdot,\cdot)_{k,D}$, $|\cdot|_{k,D}$, and $\lVert \cdot \rVert_{k,D}$, to denote the corresponding inner product, seminorm, and norm, respectively. The standard $L^2$ inner product over $D$ is written as $(\cdot,\cdot)_{0,D}$ with corresponding norm $\lVert \cdot \rVert_{0,\Omega}$. Further, $\mathbb{P}_k(D)$ is the space of polynomials up to order $k$, and $[\mathbb{P}_k(D)]^2$ the corresponding vector valued space. Additionally, $|\cdot|$ is the standard Euclidean norm for scalars and vectors. Finally, throughout the paper, $\eta$ denotes a generic constant, possibly varying from one occurrence to the other, but independent of the mesh size and, apart from Theorem \[thm:Cn\_cn\], also independent of the variables.
Continuous Problem
------------------
Let $\Omega \subset {\mathbb{R}}^2$ be a polygonal bounded, convex domain, describing a reservoir of unit thickness. Given a time interval $J:=[0,T]$, for $T>0$, we are interested in finding ${\boldsymbol{u}}={\boldsymbol{u}}({\boldsymbol{x}},t)$, representing the Darcy velocity (volume of fluid flowing cross a unit across-section per unit time), the pressure $p=p({\boldsymbol{x}},t)$ in the fluid mixture, and the concentration $c=c({\boldsymbol{x}},t)$ of one of the fluids (amount of the fluid per unit volume in the fluid mixture), with $({\boldsymbol{x}},t) \in \Omega_T:=\Omega \times J$, such that $$\label{eq:model_problem}
\left\{
\begin{alignedat}{2}
\phi \, \frac{\partial c}{\partial t} + {\boldsymbol{u}}\cdot \nabla c - {\operatorname{\rm div}}(D({\boldsymbol{u}}) \nabla c) &= {q^+}({\widehat{c}}-c) \\
{\operatorname{\rm div}}\, {\boldsymbol{u}}&= G \\
{\boldsymbol{u}}&= -a(c) (\nabla p - {\boldsymbol{\gamma}}(c)),
\end{alignedat}
\right.$$ where $\phi=\phi({\boldsymbol{x}})$ is the porosity of the medium, ${q^+}={q^+}({\boldsymbol{x}},t)$ and ${q^-}={q^-}({\boldsymbol{x}},t)$ are the (non negative) injection and production source terms, respectively, ${\widehat{c}}={\widehat{c}}({\boldsymbol{x}},t)$ is the concentration of the injected fluid, and $$\label{eq:def_G}
G:={q^+}-{q^-}.$$ Moreover, $D({\boldsymbol{u}}) \in {\mathbb{R}}^{2\times 2}$ is the diffusion tensor given by $$\label{eq:def_D}
D({\boldsymbol{u}}):=\phi \left[ d_m I + |{\boldsymbol{u}}|(d_\ell E({\boldsymbol{u}}) + d_t E^\perp({\boldsymbol{u}})) \right],$$ with matrices $$\label{eq:def_E(u)}
E({\boldsymbol{u}}):=\left( \frac{{\boldsymbol{u}}_i {\boldsymbol{u}}_j}{|{\boldsymbol{u}}|^2} \right)_{i,j=1,2}=\frac{{\boldsymbol{u}}{\boldsymbol{u}}^T}{|{\boldsymbol{u}}|^2}, \quad E^\perp({\boldsymbol{u}}):=I-E({\boldsymbol{u}}),$$ and molecular diffusion coefficient $d_m$, longitudinal dispersion coefficient $d_\ell$, and transversal dispersion coefficient $d_t$. Further, $\boldsymbol{\gamma}(c)$ in describes the force density due to gravity (typically written as $\boldsymbol{\gamma}(c)=\gamma_0(c)\boldsymbol{\rho}$ with $\gamma_0(c)$ being the density of the fluid and $\boldsymbol{\rho}$ the gravitational acceleration vector), and $a(c)=a(c,{\boldsymbol{x}})$ is the scalar valued function given by $$a(c):=\frac{k}{\mu(c)},$$ where $k=k({\boldsymbol{x}})$ represents the permeability of the porous rock, and $\mu(c)$ is the viscosity of the fluid mixture, which can be modeled by $$\mu(c)=\mu(0) \left( 1+\left(M^{\frac{1}{4}}-1\right)c \right)^{-4}, \quad \text{ in } [0,1],$$ with mobility ratio $M:=\frac{\mu(0)}{\mu(1)}$. Note that $\mu$ can be set to $\mu(0)$ for $c<0$, and to $\mu(1)$ for $c>1$. We also highlight that, in the literature, $k$ is sometimes assumed to be a tensor. The following analysis can be straightforwardly generalized to that case.
Assuming impermeability of $\partial \Omega$, the system is closed by requiring *no-flow boundary conditions* of the form
$$\label{eq:bdry_cond}
\left\{
\begin{alignedat}{2}
{\boldsymbol{u}}\cdot {\boldsymbol{n}}&=0 \quad \text{ on } \partial \Omega \times J \\
D({\boldsymbol{u}}) \nabla c \cdot {\boldsymbol{n}}&=0 \quad \text{ on } \partial \Omega \times J,
\end{alignedat}
\right.$$
and initial condition $$\label{eq:initial_cond}
c({\boldsymbol{x}},0) = c_0({\boldsymbol{x}}) \quad \text{ in } \Omega,$$ where $0\le c_0({\boldsymbol{x}}) \le 1$ is an initial concentration.
By use of the divergence theorem, the boundary conditions directly imply the following compatibility condition for ${q^+}$ and ${q^-}$: $$\int_{\Omega} {q^+}({\boldsymbol{x}},t) \, {\textup{d}x}= \int_{\Omega} {q^-}({\boldsymbol{x}},t) \, {\textup{d}x},$$ for every $t \in J$.
We highlight that, in the forthcoming theoretical analysis, we will always assume sufficient regularity of the exact solution and the involved functions, such as ${q^+}$, ${q^-}$, ${\widehat{c}}$, *et cetera*, as better motivated in the corresponding section. Moreover, we will make use of the following assumptions.
First of all, we suppose that the functions $a$ and $\phi$ are positive and uniformly bounded from below and above, i.e. there exist positive constants $a_\ast$, $a^\ast$, $\phi_\ast$, and $\phi^\ast$, such that $$\label{ass:a_phi}
a_\ast \le a(z,{\boldsymbol{x}}) \le a^\ast, \qquad
\phi_\ast \le \phi({\boldsymbol{x}}) \le \phi^\ast,$$ for all ${\boldsymbol{x}}\in \Omega$ and $z=z(t)$. For the sake of readability, we define $$A(z)({\boldsymbol{x}}):=a^{-1}(z,{\boldsymbol{x}}).$$ Additionally, we will make use of the following relation of the diffusion and dispersion coefficients, which was observed in laboratory experiments: $$\label{ass:dl_dt_dm}
0 < d_m \le d_t \le d_\ell.$$
Finally, we recall that the source terms ${q^+}$ and ${q^-}$ are, as usual, assumed to be non-negative functions.
Existence of weak solutions to this model problem was shown in [@feng1995existence] for ${\boldsymbol{\gamma}}(c)=0$. An extension of this result to 3D spatial domains, including the presence of ${\boldsymbol{\gamma}}(c)$ and various boundary conditions was discussed in [@chen1999mathematical].
Weak formulation of the continuous problem
------------------------------------------
Here, we fix the basic notation and the functional setting.
To this purpose, given $\Omega$ as above, we first introduce the Sobolev space $${H({\operatorname{\rm div}};\Omega)}:=\{ {\boldsymbol{v}}\in [L^2(\Omega)]^2: \, {\operatorname{\rm div}}{\boldsymbol{v}}\in L^2(\Omega) \}.$$ Then, we define the velocity space $\boldsymbol{V}$, the pressure space $Q$, and the concentration space $Z$ by $$\label{eq:spaces_V_Q_Z}
\begin{split}
\boldsymbol{V}&:=\{ {\boldsymbol{v}}\in {H({\operatorname{\rm div}};\Omega)}: \, {\boldsymbol{v}}\cdot {\boldsymbol{n}}= 0 \, \text{ on } \partial \Omega \} \\
Q&:=L^2_0(\Omega):=\{ \varphi \in L^2(\Omega): \, (\varphi,1)_{0,\Omega}=0 \} \\
Z&:=H^1(\Omega),
\end{split}$$ respectively. These spaces are endowed, respectively, with the following norms: $$\begin{split}
\lVert {\boldsymbol{u}}\rVert_{\boldsymbol{V}}^2:=\lVert {\boldsymbol{u}}\rVert_{0,\Omega}^2 + \lVert {\operatorname{\rm div}}{\boldsymbol{u}}\rVert_{0,\Omega}^2, \qquad
\lVert q \rVert_Q^2:=\lVert q \rVert_{0,\Omega}^2, \qquad
\lVert z \rVert_Z^2:=\lVert z \rVert_{1,\Omega}^2:=\lVert \nabla z \rVert^2_{0,\Omega} + \lVert z \rVert^2_{0,\Omega}.
\end{split}$$ Note that ${\operatorname{\rm div}}\boldsymbol{V}=Q$.
As usual in the framework of parabolic problems, we use the notation $$\label{ansatz:cont_fcts}
{\boldsymbol{u}}(t)(x):={\boldsymbol{u}}(x,t), \qquad p(t)(x):=p(x,t), \qquad c(t)(x):=c(x,t).$$ For $0 \le a \le b$, we further introduce $$\lVert {\boldsymbol{v}}\rVert_{L^2(a,b;\boldsymbol{V})}:=\left(\int_{a}^{b} \lVert {\boldsymbol{v}}(t) \rVert^2_{\boldsymbol{V}} \, {\textup{d}x}\right)^{\frac{1}{2}}, \quad
\lVert {\boldsymbol{v}}\rVert_{L^\infty(a,b;\boldsymbol{V})}:=\underset{t\in [a,b]}{\text{ess sup}} \lVert {\boldsymbol{v}}(t) \rVert_{\boldsymbol{V}};$$ analogously for $p$ and $c$.
Having this, the continuous problem reads as follows: find $c \in L^2(0,T;Z) \cap C^0([0, T]; L^2(\Omega))$, ${\boldsymbol{u}}\in L^2(0,T;\boldsymbol{V})$, and $p \in L^2(0,T;Q)$, such that
$$\label{eq:cont_form}
\left\{
\begin{alignedat}{2}
{\mathcal{M}}\left(\frac{\partial c(t)}{\partial t},z \right) + \left( {\boldsymbol{u}}(t) \cdot \nabla c(t),z \right)_{0,\Omega} + {\mathcal{D}}({\boldsymbol{u}}(t);c(t),z) &=
\left( {q^+}({\widehat{c}}-c)(t),z \right)_{0,\Omega} \\
{\mathcal{A}}(c(t);{\boldsymbol{u}}(t),{\boldsymbol{v}}) + B({\boldsymbol{v}},p(t)) &= ({\boldsymbol{\gamma}}(c(t)),{\boldsymbol{v}})_{0,\Omega} \\
B({\boldsymbol{u}}(t),q) &= - \left( G(t), q \right)_{0,\Omega}
\end{alignedat}
\right.$$
for all ${\boldsymbol{v}}\in \boldsymbol{V}$, $q \in Q$, and $z \in Z$, for almost all $t \in J$ and with initial condition $c(0)=c_0$, where $$\begin{aligned}
{2}
{\mathcal{M}}(c,z)&:=\left(\phi \, c,z\right)_{0,\Omega}, &\qquad {\mathcal{D}}({\boldsymbol{u}};c,z)&:=\left( D({\boldsymbol{u}}) \nabla c,\nabla z \right)_{0,\Omega}, \label{eq:def_A_B_D} \\
{\mathcal{A}}(c;{\boldsymbol{u}},{\boldsymbol{v}})&:=\left( A(c) {\boldsymbol{u}},{\boldsymbol{v}}\right)_{0,\Omega} , & B({\boldsymbol{v}},q)&:=-\left( {\operatorname{\rm div}}{\boldsymbol{v}},q \right)_{0,\Omega}. \nonumber\end{aligned}$$ Note that $c \in L^2(0,T;Z) \cap C^0([0, T]; L^2(\Omega))$ implies $\frac{\partial c}{\partial t} \in L^2(0,T;Z')$, see e.g. [@quarteroni2008numerical Thm. 11.1.1]. For the sake of readability, we suppressed $(t)$ in . From now on, we will use the convention that by writing ${\boldsymbol{u}}$, we mean in fact ${\boldsymbol{u}}(t)$; similarly for the other functions depending on space and time. In general it will be clear from the context whether ${\boldsymbol{u}}$ represents ${\boldsymbol{u}}(t)$ for a fixed $t \in J$, i.e. as a function of space only, or for varying ${\boldsymbol{x}}$ and $t$, as a function of both space and time.
Moreover, we will use the following alternative form for the concentration equation: $$\label{eq:cont_form_altern}
\begin{split}
{\mathcal{M}}\left(\frac{\partial c}{\partial t},z \right) +\Theta({\boldsymbol{u}},c;z)+ {\mathcal{D}}({\boldsymbol{u}};c,z)
= \left( {q^+}\, {\widehat{c}},z \right)_{0,\Omega},
\end{split}$$ where $$\Theta({\boldsymbol{u}},c;z):=\frac{1}{2} \bigg[\left( {\boldsymbol{u}}\cdot \nabla c,z \right)_{0,\Omega} + (({q^+}+{q^-}) \, c, z )_{0,\Omega} - \left( {\boldsymbol{u}}, c \, \nabla z \right)_{0,\Omega} \bigg].$$ This version is obtained from the original one in by rewriting the convective term as $$\left( {\boldsymbol{u}}\cdot \nabla c,z \right)_{0,\Omega}
=\frac{1}{2} \big[\left( {\boldsymbol{u}}\cdot \nabla c,z \right)_{0,\Omega} - (G, c \, z)_{0,\Omega} - \left( {\boldsymbol{u}}, c \, \nabla z \right)_{0,\Omega} \big],$$ where we first integrated by parts, then employed the fact that $\nabla \cdot {\boldsymbol{u}}=G$, together with the definition of $G$ in , and afterwards combined this term with $({q^+}\, c,z)_{0,\Omega}$ from the right hand side of . This representation was inspired by the theory of VEM for general elliptic problems [@cangianimanzinisutton_VEMconformingandnonconforming] and helps to ensure that properties of the continuous bilinear will be preserved after discretization.
In the rest of this section, we summarize some properties of the forms ${\mathcal{M}}(\cdot,\cdot)$, ${\mathcal{A}}(\cdot,\cdot,\cdot)$ and ${\mathcal{D}}(\cdot;\cdot,\cdot)$, all defined in , which will be needed later on. To start with, for ${\mathcal{M}}(\cdot,\cdot)$, it directly holds with the Cauchy-Schwarz inequality and $${\mathcal{M}}(c,z) \le \phi^\ast \lVert c \rVert_{0,\Omega} \lVert z \rVert_{0,\Omega}, \qquad
{\mathcal{M}}(z,z) \ge \phi_\ast \lVert z \rVert_{0,\Omega}^2,$$ for all $c,z\in Z$.
Concerning ${\mathcal{A}}(\cdot;\cdot,\cdot)$, again employing , for all $c\in L^{\infty}(\Omega)$ and ${\boldsymbol{u}},{\boldsymbol{v}}\in [L^2(\Omega)]^2$, we have $${\mathcal{A}}(c;{\boldsymbol{u}},{\boldsymbol{v}}) \le \frac{1}{a_\ast} \lVert {\boldsymbol{u}}\rVert_{0,\Omega} \lVert {\boldsymbol{v}}\rVert_{0,\Omega}.$$ Further, if $c \in L^2(\Omega)$, ${\boldsymbol{u}}\in [L^\infty(\Omega)]^2$ and ${\boldsymbol{v}}\in [L^2(\Omega)]^2$, it holds true that $${\mathcal{A}}(c;{\boldsymbol{u}},{\boldsymbol{v}}) \le \lVert A(c) \rVert_{0,\Omega} \lVert {\boldsymbol{u}}\rVert_{\infty,\Omega} \lVert {\boldsymbol{v}}\rVert_{0,\Omega}.$$ We also have the coercivity bound $$\label{eq:coercivity_A}
{\mathcal{A}}(c;{\boldsymbol{v}},{\boldsymbol{v}}) \ge \frac{1}{a^\ast} \lVert {\boldsymbol{v}}\rVert_{0,\Omega}^2$$ for all $c \in L^\infty(\Omega)$ and ${\boldsymbol{v}}\in [L^2(\Omega)]^2$, from which, after defining the kernel $$\label{eq:cont_kernel}
{\mathcal{K}}:=\{ {\boldsymbol{v}}\in \boldsymbol{V}: \, B({\boldsymbol{v}},q)=0 \quad \forall q \in Q \},$$ coercivity of ${\mathcal{A}}(c;\cdot,\cdot)$ on ${\mathcal{K}}$ in the norm $\lVert \cdot \rVert_{V}$ follows. Regarding ${\mathcal{D}}(\cdot;\cdot,\cdot)$, the following continuity properties can be shown. Firstly, for all ${\boldsymbol{u}}\in [L^\infty(\Omega)]^2$ and $c,z \in H^1(\Omega)$, we have $$\label{eq:cont_D_1}
\begin{split}
{\mathcal{D}}({\boldsymbol{u}};c,z) \le \phi^\ast \left[ d_m + \lVert {\boldsymbol{u}}\rVert_{\infty,\Omega} (d_\ell+d_t)\right] \lVert \nabla c \rVert_{0,\Omega} \lVert \nabla z \rVert_{0,\Omega},
\end{split}$$ which follows directly from the Cauchy-Schwarz inequality, the definition of $D({\boldsymbol{u}})$ in , and the fact that $|E({\boldsymbol{u}}) {\boldsymbol{v}}| \le |{\boldsymbol{v}}|$ and $|E^\perp({\boldsymbol{u}}) {\boldsymbol{v}}| \le |{\boldsymbol{v}}|$ for all ${\boldsymbol{v}}\in {\mathbb{R}}^2$. Moreover, for all ${\boldsymbol{u}}\in [L^2(\Omega)]^2$ and $c,z \in H^1(\Omega)$ with $\nabla c \in L^\infty (\Omega)$, we have the bound $$\label{eq:cont_D_2}
{\mathcal{D}}({\boldsymbol{u}};c,z) \le \lVert D({\boldsymbol{u}}) \rVert_{0,\Omega} \lVert \nabla c \rVert_{\infty,\Omega} \rVert \nabla z \rVert_{0,\Omega}
\le \eta_{\mathcal{D}} (1+\lVert {\boldsymbol{u}}\rVert_{0,\Omega}) \lVert \nabla c \rVert_{\infty,\Omega} \rVert \nabla z \rVert_{0,\Omega} ,$$ with matrix norm $\lVert D({\boldsymbol{u}}) \rVert_{0,\Omega}:=\left(\sum_{i,j=1}^{2} \lVert D_{i,j}({\boldsymbol{u}}) \rVert^2_{0,\Omega} \right)^{\frac{1}{2}}$, and some positive constant $\eta_{\mathcal{D}}$ depending only on $d_m$, $d_{\ell}$, and $d_t$. In addition, coercivity of ${\mathcal{D}}({\boldsymbol{u}};\cdot,\cdot)$ for all ${\boldsymbol{u}}\in [L^\infty(\Omega)]^2$, with respect to $\lVert \cdot \rVert_{0,\Omega}$, follows from $$\label{eq:Du_mu_mu}
\begin{split}
(D({\boldsymbol{u}}) \, {\boldsymbol{\mu}},{\boldsymbol{\mu}})_{0,\Omega} &= (\phi \, d_m \, {\boldsymbol{\mu}},{\boldsymbol{\mu}})_{0,\Omega} + (\phi \, |{\boldsymbol{u}}| \, (d_\ell E({\boldsymbol{u}}) + d_t E^\perp ({\boldsymbol{u}})) \, {\boldsymbol{\mu}},{\boldsymbol{\mu}})_{0,\Omega} \\
&\ge \phi_\ast \, d_m \, \lVert {\boldsymbol{\mu}}\rVert_{0,\Omega}^2 + (\phi \, |{\boldsymbol{u}}| (d_\ell-d_t) E({\boldsymbol{u}}) {\boldsymbol{\mu}},{\boldsymbol{\mu}})_{0,\Omega} + (\phi \, |{\boldsymbol{u}}| d_t \, {\boldsymbol{\mu}},{\boldsymbol{\mu}})_{0,\Omega} \\
&\ge \phi_\ast \left(d_m \, \lVert {\boldsymbol{\mu}}\rVert_{0,\Omega}^2 + d_t \, \lVert |{\boldsymbol{u}}|^{\frac{1}{2}} {\boldsymbol{\mu}}\rVert_{0,\Omega}^2 \right)
\end{split}$$ for all ${\boldsymbol{\mu}}\in [L^2(\Omega)]^2$, where we also employed and .
The virtual element method {#sec:3}
==========================
In this section, we derive a virtual element formulation for the model problem . To this purpose, we firstly fix the concept of polygonal decompositions of $\Omega$ in Section \[subsec:polygonal\_decomp\], and then, we introduce a set of discrete spaces, discrete bilinear forms, and projectors in Section \[subsec:discr\_spaces\_proj\]. Having these ingredients, we state a semidiscrete formulation which is continuous in time and discrete in space in Section \[subsec:semidiscr\_form\]. The fully discrete formulation is the subject of Section \[subsec:fully\_discr\].
Polygonal decompositions {#subsec:polygonal_decomp}
------------------------
Let ${\mathcal T_h}$ be a discretization of $\Omega$ into polygons ${K}$. We denote by ${\mathcal E_h}$ the set of all edges of ${\mathcal T_h}$, and, for a given element ${K}\in {\mathcal T_h}$, by ${\mathcal E^{{K}}}$ the set of edges belonging to ${K}$. Furthermore, ${n_{K}}$ is the number of edges of ${K}$, ${h_{{K}}}$ is the diameter of ${K}$, and $h:=\max_{{K}\in {\mathcal T_h}} {h_{{K}}}$. For a given edge ${e}\in {\mathcal E_h}$, we write ${h_e}$ for its length. Having this, we make the following assumptions on ${\mathcal T_h}$: there exists $\rho_0>0$ such that, for all $h > 0$ and for all ${K}\in{\mathcal T_h}$,
- ${K}$ is star-shaped with respect to a ball of radius $\rho \ge \rho_0 {h_{{K}}}$;
- ${h_e}\ge \rho_0 {h_{{K}}}$ for all ${e}\in {\mathcal E^{{K}}}$.
Note that these two assumptions imply that the number of edges of each element is uniformly bounded. Additionally, we will require quasi-uniformity:
- for all $h>0$ and for all ${K}\in {\mathcal T_h}$, it holds $h_K \ge \rho_1 h$, for some positive uniform constant $\rho_1$.
Given ${\mathcal T_h}$, we define, for all $s>0$, the broken Sobolev spaces on ${\mathcal T_h}$ as $$\label{broken_Sobolev_space}
H^{s}({\mathcal T_h}) := \{v \in L^2(\Omega) \mid v_{|_{K}} \in H^s({K}) \ \forall {K}\in {\mathcal T_h}\},$$ together with the corresponding broken seminorms and norms $$\label{broken_Sobolev_norm}
\vert v \vert^2_{s,{\mathcal T_h}} := \sum_{{K}\in {\mathcal T_h}} \vert v \vert_{s,{K}}^2, \quad \quad \quad \Vert v \Vert^2_{s,{\mathcal T_h}} := \sum_{{K}\in {\mathcal T_h}} \Vert v \Vert_{s,{K}}^2.$$
Both assumptions ([**D1**]{}) and ([**D2**]{}) are standard in the virtual element literature. While condition ([**D1**]{}) is quite critical in the following analysis, assumption ([**D2**]{}) could be possiby avoided by following steps similar to [@beiraolovadinarusso_stabilityVEM; @BrennerGuanSung_someestimatesVEM], at the expense of making the proofs even more lenghty and technical. Finally, assumption ([**D3**]{}) (that can be found also in many FEM papers on the same subject) is only needed to prove bound below.
Discrete spaces and projectors {#subsec:discr_spaces_proj}
------------------------------
Here, we introduce the local discrete VE spaces corresponding to $\boldsymbol{V}$, $Q$ and $Z$ in , a set of local projectors mapping from these VE spaces into spaces made of polynomials, and finally, the related global counterparts.
### Local discrete spaces
Let ${K}\in {\mathcal T_h}$ and let $k \in {\mathbb{N}}_0$ be a given *degree of accuracy*. Then, the local velocity and pressure VE spaces are defined by $$\label{eq:def_local_spaces}
\begin{split}
{\boldsymbol{V}_h({K})}&:=\{ {\boldsymbol{v}}\in {H({\operatorname{\rm div}};{K})}\cap {H({\operatorname{\rm rot}};{K})}: \, {{\boldsymbol{v}}\cdot {\boldsymbol{n}}}_{|_e} \in \mathbb{P}_k(e) \, \forall e \in {\mathcal E^{{K}}}, \\
&\qquad {\operatorname{\rm div}}{\boldsymbol{v}}\in \mathbb{P}_{k}({K}), \, {\operatorname{\rm rot}}{\boldsymbol{v}}\in \mathbb{P}_{k-1}({K}) \}\\
{Q_h({K})}&:=\{ q \in L^2({K}): \, q \in \mathbb{P}_{k}({K}) \}.
\end{split}$$ These spaces are coupled with the [*preliminary*]{} local concentration space $$\label{eq:def_local_spaces_conc}
{\widetilde{Z_h}({K})}:=\{ z \in H^1({K}): \, z_{|_{\partial {K}}} \in C^0(\partial {K}), \, z_{|_e} \in \mathbb{P}_{k+1}(e) \, \forall e \in {\mathcal E^{{K}}}, \, \Delta z \in \mathbb{P}_{k-1}({K}) \}.$$ Moreover, it is important to observe that $[\mathbb{P}_k({K})]^2 \subset {\boldsymbol{V}_h({K})}$ and $\mathbb{P}_{k+1}({K}) \subseteq {\widetilde{Z_h}({K})}$. Associated sets of local degrees of freedom are given as follows:
- for ${\boldsymbol{V}_h({K})}$, a set of degrees of freedom $\{{{\textup{dof}}^{\boldsymbol{V}_h(K)}}_j\}_{j=1}^{{{\textrm{dim}{V}_h(K)}}}$ is defined by $$\label{eq:dofs_VhK}
\begin{split}
1.& \quad \frac{1}{|e|} \int_{e}{\boldsymbol{v}}\cdot {\boldsymbol{n}}\, p_k \, {\textup{d}s}\qquad \forall p_k \in \mathbb{P}_k(e) \quad \forall e \in {\mathcal E^{{K}}}\\
2.& \quad \frac{1}{|{K}|^{\frac{1}{2}}} \int_{K}({\operatorname{\rm div}}{\boldsymbol{v}}) \, p_{k} \, {\textup{d}x}\qquad \forall p_{k} \in \mathbb{P}_{k}({K})/{\mathbb{R}}\\
3.& \quad \frac{1}{|{K}|} \int_{K}{\boldsymbol{v}}\cdot \boldsymbol{x}^\perp \, p_{k-1} \, {\textup{d}x}\qquad \forall p_{k-1} \in \mathbb{P}_{k-1}({K}),
\end{split}$$ with ${\boldsymbol{x}}^\perp:=({\boldsymbol{x}}_2,-{\boldsymbol{x}}_1)^T$, where we assume the coordinates to be centered at the barycenter of the element;
- for ${Q_h({K})}$, we consider $\{{{\textup{dof}}^{Q_h(K)}}_j\}_{j=1}^{{\textrm{dim}{Q_h(K)}}}$ with $$\label{eq:dofs_QhK}
\frac{1}{|{K}|} \int_{K}q \, p_{k} \, {\textup{d}x}\qquad \forall p_{k} \in \mathbb{P}_{k}({K});$$
- for ${\widetilde{Z_h}({K})}$, we take $\{{{\textup{dof}}^{\widetilde{Z_h}(K)}}_j\}_{j=1}^{{\textrm{dim}{\widetilde{Z_h}(K)}}}$ with $$\label{eq:dofs_ZhK}
\begin{split}
1.& \quad \text{ pointwise values at the vertices: } {\boldsymbol{v}}(z) \\
2.& \quad \text{ on each edge $e\in {\mathcal E^{{K}}}$, the values of $z$ at the $k$ internal Gau\ss{}-Lobatto points} \\
3.& \quad \frac{1}{|{K}|} \int_{K}z \, q_{k-1} \, {\textup{d}x}\qquad \forall q_{k-1} \in \mathbb{P}_{k-1}({K}).
\end{split}$$
In all three cases, unisolvency is provided. More precisely, for ${\boldsymbol{V}_h({K})}$, this was proven in e.g. [@HdivHcurlVEM], for ${Q_h({K})}$ it is immediate, and for ${\widetilde{Z_h}({K})}$, see e.g. [@VEMvolley].
We also highlight that ${\boldsymbol{V}_h({K})}$ endowed with mimics the Raviart-Thomas element, but in fact those two elements only coincide in the special case of triangles and $k=0$. An analogous result is true for ${\widetilde{Z_h}({K})}$, when compared to finite elements.
We note that, for $k=0$, one obtains the lowest order local VE spaces. More precisely, in this case, the velocity space ${\boldsymbol{V}_h({K})}$ consists of all rotation free vector fields with constant divergence and edgewise constant normal traces, the pressure space ${Q_h({K})}$ only contains the constant functions, and the concentration space ${\widetilde{Z_h}({K})}$ is made of all harmonic functions that are linear on each edge. This motivates the choice of the present polynomial degrees for the spaces. However, in general, it is also possible to choose a degree of accuracy $k_1$ for ${\boldsymbol{V}_h({K})}$ and ${Q_h({K})}$, and another strictly positive one $k_2$ for ${\widetilde{Z_h}({K})}$; see e.g. [@ewing1983approximation] for FEM. The following analysis can be extended easily to such more general case just by keeping track of the different polynomial degrees.
In order to really have a set of degrees of freedom in the computer code, one clearly needs to choose a basis for the polynomial test spaces appearing in and . We here assume to take the classical choice, that is any monomial basis $\{ m_1,m_2,..,m_\ell \}$ of the polynomial space satisfying $\| m_i \|_{L^\infty} \simeq 1$, $i=1,2,..,\ell$, where the $L^\infty$ norm has to be taken over the corresponding edge or bulk.
### Local projections {#subsec:proj}
For the construction of the method, we will need some tools to deal with VE functions due to the lack of their explicit knowledge in closed form. These tools will be provided in the form of local operators mapping VE functions onto polynomials. To this purpose, following [@VEMvolley; @hitchhikersguideVEM], we introduce the subsequent projectors.
The projector ${\boldsymbol{\Pi^{0,{K}}_{k}}}:\, [L^2({K})]^2 \to [\mathbb{P}_{k}({K})]^2$ is defined as the $L^2$ projector onto vector valued polynomials of degree at most $k$ in each component: Given $\boldsymbol{f} \in [L^2(\Omega)]^2$, $$\label{eq:L2_proj}
({\boldsymbol{\Pi^{0,{K}}_{k}}}\boldsymbol{f}, \boldsymbol{p}_{k})_{0,{K}}=( \boldsymbol{f}, \boldsymbol{p}_{k})_{0,{K}} \quad \forall \boldsymbol{p}_{k} \in [\mathbb{P}_{k}({K})]^2.$$ It can be shown, see [@BBMR_generalsecondorder], that this operator is computable for functions in ${\boldsymbol{V}_h({K})}$ only by knowing their values at the degrees of freedom . Moreover, one has computability also for functions of the form $\nabla {z_h}$ with ${z_h}\in {\widetilde{Z_h}({K})}$. This can be seen by using integration by parts: $$\int_{K}({\boldsymbol{\Pi^{0,{K}}_{k}}}\nabla {z_h}) \cdot \boldsymbol{p}_{k} \, {\textup{d}s}= \int_{K}\nabla {z_h}\cdot \boldsymbol{p}_{k} \, {\textup{d}s}= -\int_{K}{z_h}\, \underbrace{{\operatorname{\rm div}}\boldsymbol{p}_{k}}_{\in \mathbb{P}_{k-1}({K})} \, {\textup{d}s}+ \int_{\partial {K}} {z_h}\, \boldsymbol{p}_{k} \cdot {\boldsymbol{n}}\, {\textup{d}s},$$ for all $\boldsymbol{p}_{k} \in [\mathbb{P}_{k}({K})]^2$, where the right hand side is computable by means of .
The projector ${\Pi^{\nabla,{K}}_{k+1}}: \, H^1({K}) \to \mathbb{P}_{k+1}({K})$ is given, for every $z \in H^1({K})$, by $$\left\{
\begin{split}
(\nabla {\Pi^{\nabla,{K}}_{k+1}}z, \nabla p_k)_{0,{K}} &= (\nabla z, \nabla p_k)_{0,{K}} \quad \forall p_{k+1} \in \mathbb{P}_{k+1}({K}) \\
\frac{1}{|\partial {K}|} \int_{\partial {K}} {\Pi^{\nabla,{K}}_{k+1}}z \, {\textup{d}s}& = \frac{1}{|\partial {K}|} \int_{\partial {K}} z \, {\textup{d}s},
\end{split}
\right.$$ where the second identity is needed to fix the constants. Computability of this mapping for functions in ${\widetilde{Z_h}({K})}$ was shown in [@VEMvolley; @hitchhikersguideVEM].
### Discrete space for concentrations {#subsec:discr_conc_space}
The space introduced in was a preliminary space, useful to introduce the main idea of the construction. Nevertheless, we will here make use of a more advanced space for the discrete concentration variable. Indeed, one can use the operator ${\Pi^{\nabla,{K}}_{k+1}}$ to pinpoint the local enhanced space $$\begin{split}
{Z_h({K})}:=\{z \in H^1({K})&: \, z_{|_{\partial {K}}} \in C^0(\partial {K}), \, z_{|_e} \in \mathbb{P}_{k+1}(e) \, \forall e \in {\mathcal E^{{K}}}, \, \Delta z \in \mathbb{P}_{k+1}({K}), \\
&\int_{K}z \, p_k \, {\textup{d}x}= \int_{K}({\Pi^{\nabla,{K}}_{k+1}}z) \, p_k \, {\textup{d}x}\quad \forall p_k \in \mathbb{P}_{k+1}/\mathbb{P}_{k-1}(K) \},
\end{split}$$ where $\mathbb{P}_{k+1}/\mathbb{P}_{k-1}(K)$ is the space of polynomials in $\mathbb{P}_{k+1}({K})$ which are $L^2({K})$ orthogonal to $\mathbb{P}_{k-1}({K})$. It can be shown that the space ${Z_h({K})}$ has the same dimension and the same degrees of freedom as ${\widetilde{Z_h}({K})}$, see [@equivalentprojectorsforVEM; @bbmr_VEM_generalsecondorderelliptic]. The advantage of the space ${Z_h({K})}$, when compared to ${\widetilde{Z_h}({K})}$, is that *also* the $L^2$ projector ${\Pi^{0,{K}}_{k+1}}:\, L^2({K}) \to \mathbb{P}_{k+1}({K})$ onto polynomials of degree at most $k+1$, defined analogously to , is computable [@hitchhikersguideVEM]
Finally, we state the following approximation result for the three projectors above [@BBMR_generalsecondorder Lemma 5.1]:
\[lem:approx\_properties\] Given ${K}\in {\mathcal T_h}$, let $\psi$ and $\boldsymbol{\psi}$ be sufficiently smooth scalar and vector valued functions, respectively. Then, it holds, for all $k \in {\mathbb{N}}_0$, $$\label{eq:approx_properties}
\begin{split}
\lVert \psi - {\Pi^{0,{K}}_{k}}\psi \rVert_{\ell,{K}} &\le \zeta \, {h_{{K}}}^{s-\ell} \, |\psi|_{s,{K}}, \quad 0 \le \ell \le s \le k+1 \\
\lVert \boldsymbol{\psi} - {\boldsymbol{\Pi^{0,{K}}_{k}}}\boldsymbol{\psi} \rVert_{\ell,{K}} &\le \zeta \, {h_{{K}}}^{s-\ell} \, |\boldsymbol{\psi}|_{s,{K}}, \quad 0 \le \ell \le s \le k+1 \\
\lVert \psi - \Pi^{\nabla,{K}}_k \psi \rVert_{\ell,{K}} &\le \zeta \, {h_{{K}}}^{s-\ell} \, |\psi|_{s,{K}}, \quad 0 \le \ell \le s \le k+1, \, s \ge 1,
\end{split}$$ where $\zeta>0$ only depends on the shape-regularity parameter $\rho_0$ in assumption (**D1**), and $k$.
### Global discrete spaces and projectors
The global discrete spaces are defined via their local counterparts: $$\begin{split}
{\boldsymbol{V}_h}&:=\{ {\boldsymbol{v}}\in \boldsymbol{V}: \, {{\boldsymbol{v}}}_{|_{K}} \in {\boldsymbol{V}_h({K})}\, \forall {K}\in {\mathcal T_h}\} \\
{Q_h}&:=\{ q \in Q: \, q_{|_{K}} \in {Q_h({K})}\, \forall {K}\in {\mathcal T_h}\} \\
{Z_h}&:=\{ z \in Z: \, z_{|_{K}} \in {Z_h({K})}\, \forall {K}\in {\mathcal T_h}\}
\end{split}$$ with the obvious sets of global degrees of freedom.
In addition to the broken Sobolev norm , we introduce, for all ${\boldsymbol{u}_h}\in {\boldsymbol{V}_h}$, $$\lVert {\boldsymbol{u}_h}\rVert_{{\boldsymbol{V}_h}}^2
:=\sum_{{K}\in {\mathcal T_h}} \lVert {\boldsymbol{u}_h}\rVert_{V,{K}}^2
:=\sum_{{K}\in {\mathcal T_h}} \left[\lVert {\boldsymbol{u}_h}\rVert_{0,{K}}^2 + \lVert {\operatorname{\rm div}}{\boldsymbol{u}_h}\rVert_{0,{K}}^2 \right].$$
Moreover, we will denote by ${\boldsymbol{\Pi^{0}_{k}}}$, ${\Pi^{\nabla}_{k+1}}$ and ${\Pi^{0}_{k+1}}$, the global projectors which are defined elementwise as the corresponding local ones in Section \[subsec:proj\] and \[subsec:discr\_conc\_space\].
The sets of global degrees of freedom $\{{{\textup{dof}}^{\boldsymbol{V}_h}}_j\}_{j=1}^{{{\textrm{dim}{V}_h}}}$, $\{{{\textup{dof}}^{Q_h}}_j\}_{j=1}^{{\textrm{dim}{Q_h}}}$, and $\{{{\textup{dof}}^{Z_h}}_j\}_{j=1}^{{\textrm{dim}{Z_h}}}$ are obtained by coupling the local counterparts given in , , and , respectively.
Semidiscrete formulation {#subsec:semidiscr_form}
------------------------
Our aim in this section is to find a semidiscrete formulation for which is continuous in time and discrete in space. To this purpose, we employ the same notation for the numerical approximants ${\boldsymbol{u}_h}$, ${p_h}$, and ${c_h}$, as in for ${\boldsymbol{u}}$, $p$, and $c$, namely $${\boldsymbol{u}_h}(t)(x):={\boldsymbol{u}_h}(x,t), \qquad p_h(t)(x):=p_h(x,t), \qquad {c_h}(t)(x):={c_h}(x,t),$$ where the dependence on $(t)$ will be again suppressed in the sequel.
A semidiscrete variational formulation for can then be written in an abstract way as follows: for almost every $t \in J$, find ${\boldsymbol{u}_h}\in {\boldsymbol{V}_h}$, ${p_h}\in {Q_h}$, and ${c_h}\in {Z_h}$, such that $$\label{eq:semidiscr_var_form}
\left\{
\begin{alignedat}{2}
{\mathcal{M}_h}\left(\frac{\partial {c_h}}{\partial t},{z_h}\right) + {\Theta_h}({\boldsymbol{u}_h},{c_h};{z_h}) + {\mathcal{D}_h}({\boldsymbol{u}_h};{c_h},{z_h}) &= \left( {q^+}\,{\widehat{c}},{z_h}\right)_h \\
{\mathcal{A}_h}({c_h};{\boldsymbol{u}_h},{\boldsymbol{v}_h}) + B({\boldsymbol{v}_h},p_h) &= ({\boldsymbol{\gamma}}({c_h}),{\boldsymbol{v}_h})_h \\
B({\boldsymbol{u}_h},{q_h}) &= - \left( G, {q_h}\right)_{0,\Omega}
\end{alignedat}
\right.$$ for all ${\boldsymbol{v}_h}\in {\boldsymbol{V}_h}$, ${q_h}\in {Q_h}$, and ${z_h}\in {Z_h}$, and the initial condition $${c_h}(0)=c_{0,h} := I_h c_0$$ is satisfied, where $I_h c_0$ is the VEM interpolant of $c_0$ in $Z_h$, and where the involved forms and terms in are specified in the forthcoming lines.
Starting from the continuous problem , by simply replacing the continuous functions by their discrete counterparts, most of the resulting terms cannot be computed any more, owing to the fact that VE functions are not known explicitly in closed form. Thus, these terms need to be substituted by computable versions in the spirit of the VEM philosophy. To this purpose, the following replacements were made:
- The term ${\mathcal{M}}\left(\frac{\partial {c_h}}{\partial t},{z_h}\right)$ in the concentration equation was replaced by $$\label{eq:def_Mh}
{\mathcal{M}_h}\left(\frac{\partial {c_h}}{\partial t},{z_h}\right):=\sum_{{K}\in {\mathcal T_h}} {\mathcal{M}_h^{K}}\left(\frac{\partial {c_h}}{\partial t},{z_h}\right),$$ where the local contributions are given as $$\label{eq:def_Mh_loc}
\begin{split}
{\mathcal{M}_h^{K}}\left({c_h},{z_h}\right)&:=\int_{K}\phi \, ({\Pi^{0,{K}}_{k+1}}{c_h}) \, ({\Pi^{0,{K}}_{k+1}}{z_h}) \, {\textup{d}x}\\
&\qquad+ {\nu_{{\mathcal{M}}}^{K}}(\phi) {S^{K}_{{\mathcal{M}}}}\left((I-{\Pi^{0,{K}}_{k+1}}) {c_h}, (I-{\Pi^{0,{K}}_{k+1}}) {z_h}\right),
\end{split}$$ with ${S^{K}_{{\mathcal{M}}}}(\cdot,\cdot)$ denoting a stabilization term with certain properties and a constant ${\nu_{{\mathcal{M}}}^{K}}(\phi)$, both described in Section \[subsec:stabilizations\] below.
- Next, the term $\Theta({\boldsymbol{u}_h},{c_h};{z_h})$ was substituted by $$\label{eq:def_Theta_h}
{\Theta_h}({\boldsymbol{u}_h},{c_h};{z_h}):=\frac{1}{2} \bigg[ ({\boldsymbol{u}_h}\cdot \nabla {c_h},{z_h})_h + (({q^+}+{q^-})\, {c_h},{z_h})_h - ({\boldsymbol{u}_h}\, {c_h},\nabla {z_h})_h \bigg],$$ where $$\begin{split}
({\boldsymbol{u}_h}\cdot \nabla {c_h},{z_h})_h&:=\sum_{{K}\in {\mathcal T_h}} \int_{K}{\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}\cdot {\boldsymbol{\Pi^{0,{K}}_{k}}}(\nabla {c_h}) \, {\Pi^{0,{K}}_{k+1}}{z_h}\, {\textup{d}x}\\
(({q^+}+{q^-})\, {c_h},{z_h})_h&:=\sum_{{K}\in {\mathcal T_h}} \int_{K}({q^+}+{q^-}) \, {\Pi^{0,{K}}_{k+1}}{c_h}\, {\Pi^{0,{K}}_{k+1}}{z_h}\, {\textup{d}x}\\
({\boldsymbol{u}_h}\, {c_h},\nabla {z_h})_h&:=\sum_{{K}\in {\mathcal T_h}} \int_{K}{\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}\, {\Pi^{0,{K}}_{k+1}}{c_h}\, \cdot {\boldsymbol{\Pi^{0,{K}}_{k}}}(\nabla {z_h}) \, {\textup{d}x}.
\end{split}$$
- Moreover, the term ${\mathcal{D}}({\boldsymbol{u}_h};{c_h},{z_h})$ was replaced by $$\label{eq:def_Dcalh}
{\mathcal{D}_h}\left({\boldsymbol{u}_h};{c_h},{z_h}\right):=\sum_{{K}\in {\mathcal T_h}} {\mathcal{D}_h^{K}}\left({\boldsymbol{u}_h};{c_h},{z_h}\right)$$ with local contributions $$\label{eq:def_Dcalh_loc}
\begin{split}
{\mathcal{D}_h^{K}}\left({\boldsymbol{u}_h};{c_h},{z_h}\right)&:=\int_{K}D({\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}) \, {\boldsymbol{\Pi^{0,{K}}_{k}}}(\nabla {c_h}) \cdot {\boldsymbol{\Pi^{0,{K}}_{k}}}(\nabla {z_h}) \, {\textup{d}x}\\
&\qquad + {\nu_{D}^{K}}({\boldsymbol{u}_h}) \, {S^{K}_{D}}\left((I-{\Pi^{\nabla,{K}}_{k+1}}) {c_h},(I-{\Pi^{\nabla,{K}}_{k+1}}) {z_h}) \right),
\end{split}$$ where ${S^{K}_{D}}(\cdot,\cdot)$ is a stabilization term with certain properties and a constant ${\nu_{D}^{K}}({\boldsymbol{u}_h})$, both described in Section \[subsec:stabilizations\] below.
- Concerning $\left({q^+}\,{\widehat{c}},{z_h}\right)_{0,\Omega}$, this term was approximated by $$\left( {q^+}\,{\widehat{c}},{z_h}\right)_h:=\sum_{{K}\in {\mathcal T_h}} \left[\int_{K}{q^+}\,{\widehat{c}}\, {\Pi^{0,{K}}_{k+1}}{z_h}\, {\textup{d}x}\right].$$
- Regarding the mixed problem, the term ${\mathcal{A}}({c_h};{\boldsymbol{u}_h},{\boldsymbol{v}_h})$ was substituted by $$\label{eq:def_Acalh}
{\mathcal{A}_h}({c_h};{\boldsymbol{u}_h},{\boldsymbol{v}_h}):=\sum_{{K}\in {\mathcal T_h}} {\mathcal{A}_h^{K}}({c_h};{\boldsymbol{u}_h},{\boldsymbol{v}_h})$$ with local forms $$\label{eq:def_Acalh_loc}
\begin{split}
{\mathcal{A}_h^{K}}({c_h};{\boldsymbol{u}_h},{\boldsymbol{v}_h})&:=\int_{K}A({\Pi^{0,{K}}_{k+1}}{c_h}) {\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}\cdot {\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{v}_h}\, {\textup{d}x}\\
&\qquad + {\nu_{{\mathcal{A}}}^{K}}({c_h}) \, {S^{K}_{{\mathcal{A}}}}((I-{\boldsymbol{\Pi^{0,{K}}_{k}}}){\boldsymbol{u}_h},(I-{\boldsymbol{\Pi^{0,{K}}_{k}}}){\boldsymbol{v}_h}),
\end{split}$$ where, similarly as before, ${S^{K}_{{\mathcal{A}}}}(\cdot,\cdot)$ is a stabilization term and ${\nu_{{\mathcal{A}}}^{K}}({c_h})$ a constant, both described in Section \[subsec:stabilizations\] below.
- Finally, the term $({\boldsymbol{\gamma}}({c_h}),{\boldsymbol{v}_h})_{0,\Omega}$ was replaced by $$({\boldsymbol{\gamma}}({c_h}),{\boldsymbol{v}_h})_h:=\sum_{{K}\in {\mathcal T_h}} \left[\int_{K}{\boldsymbol{\gamma}}({\Pi^{0,{K}}_{k+1}}{c_h}) \cdot {\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{v}_h}\, {\textup{d}x}\right].$$
At this point, we highlight that the bilinear form $B(\cdot,\cdot)$ needs not to be substituted since it is computable for VE functions due to the choice of degrees of freedom . Furthermore, the right hand side term $\left( G, {q_h}\right)_{0,\Omega}$ remains unchanged.
Note that we here use the convention that terms which are written in caligraphic letters, such as ${\mathcal{M}_h}$, ${\mathcal{D}_h}$ and ${\mathcal{A}_h}$, include a stabilization term, whereas those in non-caligraphic fashion and those of the form $(\cdot,\cdot)_h$ with subscript $h$ do not. In general, the terms of the type $(\cdot,\cdot)_h$ are approximations of the corresponding (possibly weighted) $L^2$ scalar products $(\cdot,\cdot)_{0,\Omega}$, obtained by introducing projections onto polynomials for all virtual functions, but not for the data terms that are known exactly.
### Construction of the stabilizations {#subsec:stabilizations}
Here, we specify the assumptions on the stabilizations ${S^{K}_{{\mathcal{M}}}}(\cdot,\cdot): \, {Z_h}\times {Z_h}\to {\mathbb{R}}$, ${S^{K}_{D}}(\cdot,\cdot):\, {Z_h}\times {Z_h}\to {\mathbb{R}}$, and ${S^{K}_{{\mathcal{A}}}}(\cdot,\cdot):\, {\boldsymbol{V}_h}\times {\boldsymbol{V}_h}\to {\mathbb{R}}$, in , and , respectively.
We require that these terms represent computable, symmetric, and positive definite bilinear forms that satisfy, for all ${K}\in {\mathcal T_h}$, the following property: there exist positive constants $M_0^{{\mathcal{M}}}$, $M_1^{{\mathcal{M}}}$, $M_0^{{\mathcal{D}}}$, $M_1^{{\mathcal{D}}}$, $M_0^{{\mathcal{A}}}$, $M_1^{{\mathcal{A}}}$, which are independent of $h$ and ${K}$, such that $$\label{eq:stab_assumpt}
\begin{split}
M_0^{{\mathcal{M}}} \lVert {z_h}\rVert_{0,{K}}^2 \le {S^{K}_{{\mathcal{M}}}}({z_h},{z_h}) &\le M_1^{{\mathcal{M}}} \lVert {z_h}\rVert_{0,{K}}^2
\qquad \forall {z_h}\in {Z_h}\cap\ker({\Pi^{0,{K}}_{k+1}}) \\
M_0^{{\mathcal{D}}} \lVert \nabla {z_h}\rVert_{0,{K}}^2 \le {S^{K}_{D}}({z_h},{z_h})
& \le M_1^{{\mathcal{D}}} \lVert \nabla {z_h}\rVert_{0,{K}}^2 \qquad
\forall {z_h}\in {Z_h}\cap\ker({\Pi^{\nabla,{K}}_{k+1}}) \\
M_0^{{\mathcal{A}}} \lVert {\boldsymbol{v}_h}\rVert_{0,{K}}^2 \le {S^{K}_{{\mathcal{A}}}}({\boldsymbol{v}_h},{\boldsymbol{v}_h}) &\le M_1^{{\mathcal{A}}} \lVert {\boldsymbol{v}_h}\rVert_{0,{K}}^2 \qquad
\forall {\boldsymbol{v}_h}\in {\boldsymbol{V}_h}\cap \ker({\boldsymbol{\Pi^{0,{K}}_{k}}}).
\end{split}$$ Note that continuity follows immediately from the properties: $${S^{K}_{{\mathcal{M}}}}({z_h},\widetilde{{z_h}}) \le \left( {S^{K}_{{\mathcal{M}}}}({z_h},{z_h}) \right)^{\frac{1}{2}} \left( {S^{K}_{{\mathcal{M}}}}(\widetilde{{z_h}},\widetilde{{z_h}}) \right)^{\frac{1}{2}} \le M_1^{{\mathcal{M}}} \lVert {z_h}\rVert_{0,{K}} \lVert \widetilde{{z_h}} \rVert_{0,{K}}$$ for all ${z_h}, \widetilde{{z_h}} \in {Z_h}\cap\ker({\Pi^{0,{K}}_{k+1}})$; analogously for the other forms. In practice, under mesh assumptions (**D1**)-(**D2**), one can take the following scaled stabilizations corresponding to the degrees of freedom: $$\label{eq:stabs}
\begin{split}
{S^{K}_{{\mathcal{M}}}}({c_h},{z_h})&=|{K}| \sum_{j=1}^{{\textrm{dim}{Z_h(K)}}} {{\textup{dof}}^{Z_h(K)}}_j({c_h}) \, {{\textup{dof}}^{Z_h(K)}}_j({z_h}) \\
{S^{K}_{D}}({c_h},{z_h})&=\sum_{j=1}^{{\textrm{dim}{Z_h(K)}}} {{\textup{dof}}^{Z_h(K)}}_j({c_h}) \, {{\textup{dof}}^{Z_h(K)}}_j({z_h}) \\
{S^{K}_{{\mathcal{A}}}}({\boldsymbol{u}_h},{\boldsymbol{v}_h})&=|{K}|\sum_{j=1}^{{{\textrm{dim}{V}_h(K)}}} {{\textup{dof}}^{\boldsymbol{V}_h(K)}}_j({\boldsymbol{u}_h}) \, {{\textup{dof}}^{\boldsymbol{V}_h(K)}}_j({\boldsymbol{v}_h}).
\end{split}$$ Regarding the constants appearing in front of the stabilizations in , and , respectively, we pick: $$\label{eq:stab_constants}
{\nu_{{\mathcal{M}}}^{K}}(\phi)=\left|{\Pi^{0,{K}}_{0}}\phi \right|, \quad {\nu_{D}^{K}}({\boldsymbol{u}_h})={\nu_{{\mathcal{M}}}^{K}}(\phi) (d_m+d_t |{\boldsymbol{\Pi^{0,{K}}_{0}}}{\boldsymbol{u}_h}|), \quad
{\nu_{{\mathcal{A}}}^{K}}({c_h})=|A({\Pi^{0,{K}}_{0}}({c_h}))|,$$ where ${\Pi^{0,{K}}_{0}}:\, L^2({K}) \to \mathbb{P}_0({K})$ and ${\boldsymbol{\Pi^{0,{K}}_{0}}}:\, [L^2({K})]^2 \to [\mathbb{P}_0({K})]^2$ are the $L^2$ projectors onto scalar and vector valued constants, respectively.
### Well-posedness of the semidiscrete problem
We first define the constants $${\nu_{{\mathcal{M}}}^-}:=\min_{{K}\in {\mathcal T_h}} {\nu_{{\mathcal{M}}}^{K}}, \qquad {\nu_{{\mathcal{M}}}^+}:=\max_{{K}\in {\mathcal T_h}} {\nu_{{\mathcal{M}}}^{K}}.$$ Analogously, we introduce ${\nu_{{\mathcal{D}}}^-}$, ${\nu_{{\mathcal{D}}}^+}$, ${\nu_{{\mathcal{A}}}^-}$ and ${\nu_{{\mathcal{A}}}^+}$. Recalling and , it is easy to check the following (mesh-uniform) bounds for the above constants: $$\begin{aligned}
& \phi_\ast \le {\nu_{{\mathcal{M}}}^-}\le {\nu_{{\mathcal{M}}}^+}\le \phi^\ast \ , \quad
(a^\ast)^{-1} \le {\nu_{{\mathcal{A}}}^-}\le {\nu_{{\mathcal{A}}}^+}\le a_\ast^{-1} \\
& \phi_\ast d_m \le {\nu_{{\mathcal{D}}}^-}\le {\nu_{{\mathcal{D}}}^+}\le \phi^\ast (d_m + (d_\ell+d_t) \| {\boldsymbol{u}_h}\|_{\infty,\Omega}).
\end{aligned}$$ Then, similarly as for their continuous counterparts, the following continuity and coercivity properties for ${\mathcal{M}_h}(\cdot,\cdot)$ ${\mathcal{D}_h}(\cdot;\cdot,\cdot)$, and ${\mathcal{A}_h}(\cdot;\cdot,\cdot)$, defined in , and , respectively, hold true.
\[lem:properties\_D\_A\] For ${\mathcal{M}_h}(\cdot,\cdot)$, it holds, for all ${c_h},{z_h}\in {Z_h}$, $$\label{eq:cont_coerc_Mcalh}
\begin{split}
{\mathcal{M}_h}({c_h},{z_h}) &\le \max\{\phi^\ast,{\nu_{{\mathcal{M}}}^+}M_1^{{\mathcal{M}}}\} \lVert {c_h}\rVert_{0,\Omega} \lVert {z_h}\rVert_{0,\Omega} \\
{\mathcal{M}_h}({z_h},{z_h}) &\ge \min\{\phi_\ast,{\nu_{{\mathcal{M}}}^-}M_0^{{\mathcal{M}}} \} \lVert {z_h}\rVert_{0,\Omega}^2.
\end{split}$$ Concerning ${\mathcal{D}_h}(\cdot;\cdot,\cdot)$, this form satisfies, for all ${\boldsymbol{u}_h}\in {\boldsymbol{V}_h}$ and ${c_h}, {z_h}\in {Z_h}$, $$\label{eq:cont_coerc_Dcalh}
\begin{split}
{\mathcal{D}_h}({\boldsymbol{u}_h};{c_h},{z_h}) &\le \left[ \phi^\ast \left( d_m + \eta \lVert {\boldsymbol{u}_h}\rVert_{\infty,\Omega} (d_\ell+d_t)\right)
+ {\nu_{{\mathcal{D}}}^+}M_1^{{\mathcal{D}}} \right] |{c_h}|_{1,{\mathcal T_h}} |{z_h}|_{1,{\mathcal T_h}} \\
{\mathcal{D}_h}({\boldsymbol{u}_h};{z_h},{z_h}) &\ge \min\{ \phi_\ast d_m, {\nu_{{\mathcal{D}}}^-}M_0^{{\mathcal{D}}} \} |{z_h}|^2_{1,{\mathcal T_h}}.
\end{split}$$ Regarding ${\mathcal{A}_h}(\cdot;\cdot,\cdot)$, for all ${c_h}\in {Z_h}$ and ${\boldsymbol{u}_h},{\boldsymbol{v}_h}\in {\boldsymbol{V}_h}$, it yields $$\label{eq:cont_coerc_Acalh}
\begin{split}
{\mathcal{A}_h}({c_h};{\boldsymbol{u}_h},{\boldsymbol{v}_h}) &\le \max\left\{ \frac{1}{a_\ast},{\nu_{{\mathcal{A}}}^+}M_1^{{\mathcal{A}}} \right\} \lVert {\boldsymbol{u}_h}\rVert_{0,\Omega} \lVert {\boldsymbol{v}_h}\rVert_{0,\Omega}\\
{\mathcal{A}_h}({c_h};{\boldsymbol{v}_h},{\boldsymbol{v}_h}) &\ge \min\left\{\frac{1}{a^\ast},{\nu_{{\mathcal{A}}}^-}M_0^{{\mathcal{A}}} \right\} \lVert {\boldsymbol{v}_h}\rVert^2_{0,\Omega}.
\end{split}$$ Thus, ${\mathcal{A}_h}({c_h};\cdot,\cdot)$ is coercive on the kernel $$\label{eq:discr_kernel}
{\mathcal{K}_h}:=\{ {\boldsymbol{v}_h}\in {\boldsymbol{V}_h}: \, B({\boldsymbol{v}_h},{q_h})=0 \quad \forall {q_h}\in {Q_h}\} \subset {\mathcal{K}}$$ with respect to $\lVert \cdot \rVert_{{\boldsymbol{V}_h}}$, where ${\mathcal{K}}$ is given in .
The continuity bound in follows directly by using $$\label{eq:splitting_Mh}
{\mathcal{M}_h}({c_h},{z_h}) \le {\mathcal{M}_h}({c_h},{c_h})^{\frac{1}{2}} {\mathcal{M}_h}({z_h},{z_h})^{\frac{1}{2}},$$ and then estimating $$\begin{split}
{\mathcal{M}_h}({c_h},{c_h}) &\le \phi^\ast \lVert {\Pi^{0,{K}}_{k+1}}{c_h}\rVert^2_{0,{K}} + {\nu_{{\mathcal{M}}}^+}M_1^{{\mathcal{M}}} \lVert (I-{\Pi^{0,{K}}_{k+1}}) {c_h}\rVert^2_{0,{K}} \\
&\le \max\{\phi^\ast,{\nu_{{\mathcal{M}}}^+}M_1^{{\mathcal{M}}}\} \left( \lVert {\Pi^{0,{K}}_{k+1}}{c_h}\rVert^2_{0,{K}} + \lVert (I-{\Pi^{0,{K}}_{k+1}}) {c_h}\rVert^2_{0,{K}} \right) \\
&= \max\{\phi^\ast,{\nu_{{\mathcal{M}}}^+}M_1^{{\mathcal{M}}}\} \lVert {c_h}\rVert^2_{0,{K}},
\end{split}$$ where the Pythagorean theorem was applied in the last equality. For the coercivity bound, one can use , , and the Pythagorean theorem.
Regarding the continuity estimate for ${\mathcal{D}_h}(\cdot;\cdot,\cdot)$, by using a splitting of the form , together with an estimate as in , one can deduce at the local level $$\label{eq:estimate_DcalhE}
\begin{split}
{\mathcal{D}_h^{K}}({\boldsymbol{u}_h};{c_h},{c_h}) &\le
\phi^\ast \left( d_m + \eta \lVert {\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}\rVert_{\infty,\Omega} (d_\ell+d_t)\right)
\lVert {\boldsymbol{\Pi^{0,{K}}_{k}}}(\nabla {c_h}) \rVert^2_{0,{K}} \\
& + \left( {\nu_{{\mathcal{D}}}^+}M_1^{{\mathcal{D}}} \right) \lVert \nabla (I-{\Pi^{\nabla,{K}}_{k+1}}){c_h}\rVert^2_{0,{K}} \\
& \le \left[ \phi^\ast \left( d_m + \eta \lVert {\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}\rVert_{\infty,\Omega} (d_\ell+d_t)\right)
+ {\nu_{{\mathcal{D}}}^+}M_1^{{\mathcal{D}}} \right] |{c_h}|_{1,{\mathcal T_h}}^2 .
\end{split}$$ By application of a polynomial inverse estimate [@BrennerScott Lemma 4.5.3], the continuity of the $L^2$ projector, and the Hölder inequality, we further estimate $$\label{eq:cont_Pi_infty}
\lVert {\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}\rVert_{\infty,{K}} \le \eta \, {h_{{K}}}^{-1} \lVert {\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}\rVert_{0,{K}} \le \eta \, {h_{{K}}}^{-1} \lVert {\boldsymbol{u}_h}\rVert_{0,{K}} \le \eta \lVert {\boldsymbol{u}_h}\rVert_{\infty,{K}}.$$ After inserting into , taking the splitting into account, and summing over all elements, the stated bound follows. Concerning the coercivity bound for ${\mathcal{D}_h}(\cdot,\cdot)$, one can proceed similarly as in for the consistency part, and employ for the stabilization term, to obtain elementwise $${\mathcal{D}_h^{K}}({\boldsymbol{u}_h};{z_h},{z_h}) \ge \min\{ \phi_\ast d_m,{\nu_{{\mathcal{D}}}^-}M_0^{{\mathcal{D}}} \} \left[ \lVert {\boldsymbol{\Pi^{0,{K}}_{k}}}\nabla {z_h}\rVert^2_{0,{K}} + \lVert \nabla (I-{\Pi^{\nabla,{K}}_{k+1}}) {z_h}\rVert^2_{0,{K}} \right].$$ We now note that the definitions of ${\Pi^{\nabla,{K}}_{k+1}}$ and ${\boldsymbol{\Pi^{0,{K}}_{k}}}$ easily yield $$\label{eq:I-Pinabla_I-Piz}
\lVert \nabla (I-{\Pi^{\nabla,{K}}_{k+1}}){z_h}\rVert_{0,{K}} \ge \lVert (I-{\boldsymbol{\Pi^{0,{K}}_{k}}})\nabla{z_h}\rVert_{0,{K}} .$$ The estimate then follows with , the Pythagorean theorem and summation over all elements.
The estimates for ${\mathcal{A}_h}(\cdot;\cdot,\cdot)$ are derived in a similar fashion as those for ${\mathcal{M}_h}(\cdot,\cdot)$, using . The coercivity on ${\mathcal{K}_h}$ follows from the fact $${\mathcal{K}_h}\equiv \{ {\boldsymbol{v}_h}\in {\boldsymbol{V}_h}: \, {\operatorname{\rm div}}{\boldsymbol{v}_h}= 0 \} \subset {\mathcal{K}},$$ owing to the definition of ${\boldsymbol{V}_h({K})}$ in .
Well-posedness of problem can be shown by combining the results in [@vacca2015virtual] for parabolic problems with those in [@Brezzi-Falk-Marini; @BBMR_generalsecondorder] for mixed problems, using Lemma \[lem:properties\_D\_A\]. More precisely, in the spirit of the two-step strategy applied in [@ewing1983approximation] for FEM, one can first show that for any given ${c_h}(t) \in L^\infty(\Omega)$, $t \in J$, the mixed problem $$\begin{split}
{\mathcal{A}_h}({c_h};{\boldsymbol{u}_h},{\boldsymbol{v}_h}) + B({\boldsymbol{v}_h},p_h) &= ({\boldsymbol{\gamma}}({c_h}),{\boldsymbol{v}_h})_h \\
B({\boldsymbol{u}_h},{q_h}) &= - \left( G, {q_h}\right)_{0,\Omega}
\end{split}$$ admits a unique solution by applying the techniques in [@Brezzi-Falk-Marini; @BBMR_generalsecondorder], and then, by using the Gronwall lemma and Picard-Lindelöf (see e.g. [@braun1983differential Ch.1.10]), that ${c_h}(t)$ is uniquely determined by the discrete concentration equation $$\begin{split}
{\mathcal{M}_h}\left(\frac{\partial {c_h}}{\partial t},{z_h}\right) + {\Theta_h}({\boldsymbol{u}_h},{c_h};{z_h}) + {\mathcal{D}_h}({\boldsymbol{u}_h};{c_h},{z_h}) = \left( {q^+}\, {\widehat{c}},{z_h}\right)_h,
\end{split}$$ see also [@vacca2015virtual]. We do not write here the details since we focus directly on the fully discrete case, see the next section.
Fully discrete formulation {#subsec:fully_discr}
--------------------------
Here, our goal is to formulate a fully discrete version of .
To start with, we introduce a sequence of time steps ${t_n}=n \tau$, $n=0,\dots,N$, with time step size $\tau$. Next, we define ${\boldsymbol{u}^n}:={\boldsymbol{u}}({t_n})$, ${p^n}:=p({t_n})$, ${c^n}:=c({t_n})$, ${G^n}:=G({t_n})$, ${(q^+)^n}:={q^+}({t_n})$, and ${\widehat{c}^{\, n}}:={\widehat{c}}({t_n})$ as the evaluations of the corresponding functions at time ${t_n}$, $n=0,\dots,N$. Moreover, we denote by ${\boldsymbol{U}^n}\approx {\boldsymbol{u}}_h({t_n})$, ${P^n}\approx p_h({t_n})$ and ${C^n}\approx {c_h}({t_n})$, the approximations of the semidiscrete solutions at those times when using a time integrator method. Among many time discretization schemes, we here make a computationally cheap choice by choosing a backward Euler method that is explicit in the nonlinear terms. The fully discrete system consequently reads as follows:
- for $n=0$: Given ${c_{0,h}}\in {Z_h}$, solve $$\label{eq:fully_discr_model_0}
\begin{split}
{\mathcal{A}_h}({c_{0,h}};{\boldsymbol{U}^n},{\boldsymbol{v}_h}) + B({\boldsymbol{v}_h},{P^n}) &= ({\boldsymbol{\gamma}}({c_{0,h}}),{\boldsymbol{v}_h})_h \\
B({\boldsymbol{U}^n},{q_h}) &= - \left( {G^n}, {q_h}\right)_{0,\Omega}
\end{split}$$ for all ${\boldsymbol{v}_h}\in {\boldsymbol{V}_h}$ and ${q_h}\in {Q_h}$.
- for $n=1,\dots,N$: Solve first the concentration equation for ${C^n}$: $$\label{eq:fully_discr_model_1}
\begin{split}
{\mathcal{M}_h}\left(\frac{{C^n}-{C^{n-1}}}{\tau},{z_h}\right) + {\Theta_h}({\boldsymbol{U}^{n-1}};{C^n},{z_h})+ {\mathcal{D}_h}({\boldsymbol{U}^{n-1}};{C^n},{z_h}) = \left( {(q^+)^n}\,{\widehat{c}^{\, n}},{z_h}\right)_h
\end{split}$$ for all ${z_h}\in {Z_h}$, where $C^0:=c_{0,h}$. Then, solve the mixed problem for ${\boldsymbol{U}^n}$ and ${P^n}$: $$\label{eq:fully_discr_model_2}
\begin{split}
{\mathcal{A}_h}({C^n};{\boldsymbol{U}^n},{\boldsymbol{v}_h}) + B({\boldsymbol{v}_h},{P^n}) &= ({\boldsymbol{\gamma}}({C^n}),{\boldsymbol{v}_h})_h \\
B({\boldsymbol{U}^n},{q_h}) &= - \left( {G^n}, {q_h}\right)_{0,\Omega}
\end{split}$$ for all ${\boldsymbol{v}_h}\in {\boldsymbol{V}_h}$ and ${q_h}\in {Q_h}$.
\[lem:well\_posedness\_semidiscr\] Given $\tau>0$, provided that ${G^n},{(q^+)^n},{P^n},{C^n}\in L^\infty(\Omega)$, ${\boldsymbol{\gamma}}({C^n}) \in [L^2(\Omega)]^2$, and $ {\boldsymbol{U}^n}\in [L^\infty(\Omega)]^2$, for all $n=0,\dots,N$, the formulation - is uniquely solvable.
Similarly as for the semidiscrete case, well-posedness of and follows by using the tools of [@Brezzi-Falk-Marini; @BBMR_generalsecondorder]. Regarding , we first rewrite that equation as $$\label{eq:fully_discr_model_4}
\begin{split}
{\mathcal{M}_h}\left({C^n},{z_h}\right) &+ \tau \left[ {\Theta_h}({\boldsymbol{U}^{n-1}};{C^n},{z_h})+ {\mathcal{D}_h}({\boldsymbol{U}^{n-1}};{C^n},{z_h}) \right] \\
&= \tau \left( {(q^+)^n}\,{\widehat{c}^{\, n}},{z_h}\right)_h + {\mathcal{M}_h}\left({C^{n-1}},{z_h}\right).
\end{split}$$ We observe that all of the term are continuous with respect to the norm $\lVert \cdot \rVert_{1,{\mathcal T_h}}$. More precisely, for ${\mathcal{M}_h}(\cdot,\cdot)$ and ${\mathcal{D}_h}({\boldsymbol{U}^{n-1}};\cdot,\cdot)$, continuity follows from Lemma \[lem:properties\_D\_A\] and the definition of the broken $H^1$ norm. Next, for the term involving ${(q^+)^n}$, we simply apply the Cauchy-Schwarz inequality and the stability of the $L^2$ projector. Finally, for the term with ${\Theta_h}$, we estimate $$\begin{split}
{\Theta_h}({\boldsymbol{U}^{n-1}};{C^n},{z_h}) &=
\frac{1}{2} \left[\left( {\boldsymbol{U}^{n-1}}\cdot \nabla {C^n},{z_h}\right)_{h} + (({q^+}+{q^-})\, {C^n}, {z_h})_{h} - \left( {\boldsymbol{U}^{n-1}}{C^n}, \nabla {z_h}\right)_{h} \right] \\
&\le \eta \left[ \lVert {\boldsymbol{U}^{n-1}}\rVert_{\infty,\Omega} (|{C^n}|_{1,{\mathcal T_h}} + \lVert {C^n}\rVert_{0,\Omega}) + \lVert {q^+}+{q^-}\rVert_{\infty,\Omega} \lVert {C^n}\rVert_{0,\Omega} \right] \lVert {z_h}\rVert_{1,{\mathcal T_h}},
\end{split}$$ where we also employed an inverse inequality as in . Thus, by the Lax-Milgram lemma, it only remains to show that the left hand side of is coercive with respect to $\lVert \cdot \rVert_{1,{\mathcal T_h}}$. This is however a direct consequence of $${\Theta_h}({\boldsymbol{U}^{n-1}};{z_h},{z_h}) = \frac{1}{2} (({q^+}+{q^-})\, {z_h},{z_h})_h \ge 0,$$ owing to the fact that ${q^+}$ and ${q^-}$ are non-negative, and the coercivity bounds and .
Note that both problems and represent linear systems of equations which are decoupled from each other in the sense that, first, given ${\widehat{c}^{\, n}}$ and ${(q^+)^n}$, one can determine ${C^n}$ with knowledge of ${\boldsymbol{U}^{n-1}}$ only, and then one can use ${C^n}$ to compute ${\boldsymbol{U}^n}$ and ${P^n}$. The quantity ${P^n}$ does in fact not influence the calculation of ${C^n}$ directly, but rather takes the role of a Lagrange multiplier and derived variable. This decoupling, combined with the fact that the systems to be solved at each time step are linear, makes the method quite cheap per iteration.
Error analysis for the fully discrete problem {#sec:4}
=============================================
The error analysis is performed in two steps: firstly, we estimate the discretization errors for the velocity and pressure, $\lVert {\boldsymbol{u}^n}-{\boldsymbol{U}^n}\rVert_{0,\Omega}$ and $\lVert {p^n}-{P^n}\rVert_{0,\Omega}$, respectively, and then, in the second step, the concentration error $\lVert {c^n}- {C^n}\rVert_{0,\Omega}$. In the following analysis, we assume all the needed regularity of the exact solution. Although such high regularity will not often be available in practice, the purpose of the following analysis is to give a theoretical backbone to the proposed scheme and to investigate its potential accuracy in the most favorable scenario.
An auxiliary result
-------------------
The subsequent technical lemma will serve as an auxiliary result in the derivation of the error estimates and will be used in several occasions.
\[lem:abstr\_res\_1\] Let $r,s,t \in {\mathbb{N}}_0$. Denote by ${\Pi^{0}_{r}}$ and ${\boldsymbol{\Pi^{0}_{s}}}$, the elementwise defined $L^2$ projectors onto scalar and vector valued polynomials of degree at most $r$ and $s$, respectively. Given a scalar function $\sigma \in H^{{m_r}}({\mathcal T_h})$, $0 \le {m_r}\le r+1$, let $\kappa(\sigma)$ be a tensor valued piecewise Lipschitz continuous function with respect to $\sigma$. Further, let ${\widehat{\sigma}}\in L^2(\Omega)$, and let ${\boldsymbol{\chi}}$ and ${\boldsymbol{\psi}}$ be vector valued functions. We assume that $\kappa(\sigma) \in [L^\infty(\Omega)]^{2 \times 2}$, ${\boldsymbol{\chi}}\in [H^{{m_s}}({\mathcal T_h}) \cap L^\infty(\Omega)]^2$, ${\boldsymbol{\psi}}\in [L^2(\Omega)]^2$, and $\kappa(\sigma) {\boldsymbol{\chi}}\in [H^{{m_t}}({\mathcal T_h})]^2$, for some $0 \le {m_s}\le s+1$ and $0 \le {m_t}\le t+1$. Then, $$\begin{split}
& (\kappa(\sigma){\boldsymbol{\chi}},{\boldsymbol{\psi}})_{0,\Omega} - (\kappa({\Pi^{0}_{r}}{\widehat{\sigma}}) {\boldsymbol{\Pi^{0}_{s}}}{\boldsymbol{\chi}},{\boldsymbol{\Pi^{0}_{t}}}{\boldsymbol{\psi}})_{0,\Omega} \\
& \le \eta \big[h^{{m_t}} |\kappa(\sigma) {\boldsymbol{\chi}}|_{{m_t},{\mathcal T_h}} + h^{{m_s}} |{\boldsymbol{\chi}}|_{{m_s},{\mathcal T_h}} \lVert \kappa(\sigma) \rVert_{\infty,\Omega}
+ (h^{{m_r}} |\sigma|_{{m_r},{\mathcal T_h}} + \lVert \sigma - {\widehat{\sigma}}\rVert_{0,\Omega}) \lVert {\boldsymbol{\chi}}\rVert_{\infty,\Omega} \big] \lVert {\boldsymbol{\psi}}\rVert_{0,\Omega}.
\end{split}$$
We first write $$\label{eq:abstr_lem_proof1}
\begin{split}
& (\kappa(\sigma){\boldsymbol{\chi}},{\boldsymbol{\psi}})_{0,\Omega} - (\kappa({\Pi^{0}_{r}}{\widehat{\sigma}}) {\boldsymbol{\Pi^{0}_{s}}}{\boldsymbol{\chi}},{\boldsymbol{\Pi^{0}_{t}}}{\boldsymbol{\psi}})_{0,\Omega} \\
& = [(\kappa(\sigma){\boldsymbol{\chi}},{\boldsymbol{\psi}})_{0,\Omega} - (\kappa({\Pi^{0}_{r}}\sigma) {\boldsymbol{\Pi^{0}_{s}}}{\boldsymbol{\chi}},{\boldsymbol{\Pi^{0}_{t}}}{\boldsymbol{\psi}})_{0,\Omega} ]
+ ((\kappa({\Pi^{0}_{r}}\sigma) - \kappa({\Pi^{0}_{r}}{\widehat{\sigma}})) {\boldsymbol{\Pi^{0}_{s}}}{\boldsymbol{\chi}},{\boldsymbol{\Pi^{0}_{t}}}{\boldsymbol{\psi}})_{0,\Omega}.
\end{split}$$ Then, for the first part on the right hand side of , we recall that ${\boldsymbol{\Pi^{0}_{t}}}$ is an $L^2$ projection and derive, on each element ${K}\in {\mathcal T_h}$, $$\begin{split}
& \left(\kappa(\sigma){\boldsymbol{\chi}},{\boldsymbol{\psi}}\right)_{0,{K}} - (\kappa({\Pi^{0,{K}}_{r}}\sigma) {\boldsymbol{\Pi^{0,{K}}_{s}}}{\boldsymbol{\chi}},{\boldsymbol{\Pi^{0,{K}}_{t}}}{\boldsymbol{\psi}})_{0,{K}}\\
&=[(\kappa(\sigma){\boldsymbol{\chi}},{\boldsymbol{\psi}})_{0,{K}} - ({\boldsymbol{\Pi^{0,{K}}_{t}}}(\kappa(\sigma){\boldsymbol{\chi}}),{\boldsymbol{\psi}})_{0,{K}}] \\
&\quad +[({\boldsymbol{\Pi^{0,{K}}_{t}}}(\kappa(\sigma){\boldsymbol{\chi}}),{\boldsymbol{\psi}})_{0,{K}} - ({\boldsymbol{\Pi^{0,{K}}_{t}}}(\kappa(\sigma){\boldsymbol{\Pi^{0,{K}}_{s}}}{\boldsymbol{\chi}}),{\boldsymbol{\psi}})_{0,{K}}] \\
&\quad +[({\boldsymbol{\Pi^{0,{K}}_{t}}}(\kappa(\sigma){\boldsymbol{\Pi^{0,{K}}_{s}}}{\boldsymbol{\chi}}),{\boldsymbol{\psi}})_{0,{K}} - ({\boldsymbol{\Pi^{0,{K}}_{t}}}(\kappa({\Pi^{0,{K}}_{r}}\sigma){\boldsymbol{\Pi^{0,{K}}_{s}}}{\boldsymbol{\chi}}),{\boldsymbol{\psi}})_{0,{K}}]\\
&\le \eta \big[ h^{{m_t}} |\kappa(\sigma){\boldsymbol{\chi}}|_{{m_t},{K}} + h^{{m_s}} |{\boldsymbol{\chi}}|_{{m_s},{K}} \lVert \kappa(\sigma) \rVert_{\infty,{K}} + h^{{m_r}} |\sigma|_{{m_r},{K}} \lVert {\boldsymbol{\Pi^{0,{K}}_{s}}}{\boldsymbol{\chi}}\rVert_{\infty,{K}} \big] \lVert {\boldsymbol{\psi}}\rVert_{0,{K}},
\end{split}$$
where in the last step we used Lemma \[lem:approx\_properties\] and the fact that $\kappa$ is Lipschitz continuous with respect to $\sigma$. The term $\lVert {\boldsymbol{\Pi^{0,{K}}_{s}}}{\boldsymbol{\chi}}\rVert_{\infty,{K}}$ is estimated as in . Concerning the second part on the right hand side of , we have, for each ${K}\in {\mathcal T_h}$, $$\begin{split}
((\kappa({\Pi^{0}_{r}}\sigma)-\kappa({\Pi^{0}_{r}}{\widehat{\sigma}})) {\boldsymbol{\Pi^{0}_{s}}}{\boldsymbol{\chi}},{\boldsymbol{\Pi^{0}_{t}}}{\boldsymbol{\psi}})_{0,{K}} &\le \lVert (\kappa({\Pi^{0}_{r}}\sigma)-\kappa({\Pi^{0}_{r}}{\widehat{\sigma}}) \rVert_{0,{K}} \lVert {\boldsymbol{\Pi^{0}_{s}}}{\boldsymbol{\chi}}\rVert_{\infty,{K}} \lVert {\boldsymbol{\Pi^{0}_{t}}}{\boldsymbol{\psi}}\rVert_{0,{K}} \\
&\le \lVert \sigma - {\widehat{\sigma}}\rVert_{0,{K}} \lVert {\boldsymbol{\chi}}\rVert_{\infty,{K}} \lVert {\boldsymbol{\psi}}\rVert_{0,{K}},
\end{split}$$ where we used again the Lipschitz continuity of $\kappa$, the continuity properties of the $L^2$ projectors, and the bound . The assertion of the lemma follows after combining the estimates and summing over all elements.
Note that the above lemma can be easily transferred to the cases where $\sigma$, $\kappa(\sigma)$, $\chi$, and $\psi$ are scalar, and to vector valued ${\boldsymbol{\sigma}}$, ${\boldsymbol{\chi}}$ and scalar $\kappa({\boldsymbol{\sigma}})$, $\psi$.
In the special case of $\chi=1$ and vector valued ${\boldsymbol{\kappa}}$, an adaptation of Lemma \[lem:abstr\_res\_1\] gives $$\label{eq:cor_abstr_res_4}
\begin{split}
({\boldsymbol{\kappa}}(\sigma),{\boldsymbol{\psi}})_{0,\Omega} - ({\boldsymbol{\kappa}}({\Pi^{0}_{r}}{\widehat{\sigma}}),{\boldsymbol{\Pi^{0}_{t}}}{\boldsymbol{\psi}})_{0,\Omega} \le \eta \big[h^{{m_t}} |{\boldsymbol{\kappa}}(\sigma)|_{{m_t},{\mathcal T_h}} + h^{{m_r}} |\sigma|_{{m_r},{\mathcal T_h}} + \lVert \sigma - {\widehat{\sigma}}\rVert_{0,\Omega} \big] \lVert {\boldsymbol{\psi}}\rVert_{0,\Omega}.
\end{split}$$
Bounds on $\lVert {\boldsymbol{u}^n}-{\boldsymbol{U}^n}\rVert_{0,\Omega}$ and $\lVert {p^n}-{P^n}\rVert_{0,\Omega}$
-------------------------------------------------------------------------------------------------------------------
We consider the mixed problem $$\label{eq:mixed_discr_model}
\begin{split}
{\mathcal{A}_h}({C^n};{\boldsymbol{U}^n},{\boldsymbol{v}_h}) + B({\boldsymbol{v}_h},{P^n}) &= ({\boldsymbol{\gamma}}({C^n}),{\boldsymbol{v}_h})_h \\
B({\boldsymbol{U}^n},{q_h}) &= - \left( {G^n}, {q_h}\right)_{0,\Omega} ,
\end{split}$$ where ${C^n}\in {Z_h}$ is the numerical solution of the concentration equation for $n=1,\dots,N$, and $C^0={c_{0,h}}$. The goal is to derive an upper bound for $\lVert {\boldsymbol{u}^n}-{\boldsymbol{U}^n}\rVert_{0,\Omega}$ and $\lVert {p^n}-{P^n}\rVert_{0,\Omega}$ with respect to $\lVert {c^n}-{C^n}\rVert_{0,\Omega}$. For the analysis, we basically follow the ideas of [@Brezzi-Falk-Marini; @BBMR_generalsecondorder] with the major differences that, here, ${\mathcal{A}_h}({C^n};\cdot,\cdot)$ is not consistent with respect to $[\mathbb{P}_k({K})]^2$ due to presence of ${C^n}$, and, additionally, the right hand side of is inhomogeneous.
\[thm:Un\_un\_Pn\_pn\] Given ${C^n}\in {Z_h}$, let $({\boldsymbol{U}^n},{P^n}) \in {\boldsymbol{V}_h}\times {Q_h}$ be the solution to . Let us assume that for the exact solution $({\boldsymbol{u}^n},{p^n},{c^n})$ to at time ${t_n}$, it holds ${\boldsymbol{u}^n}\in [H^{k+1}({\mathcal T_h})]^2$, ${p^n}\in H^{k+1}({\mathcal T_h})$, and ${c^n}\in H^{k+1}({\mathcal T_h})$. Furthermore, we suppose that ${\boldsymbol{\gamma}}(c)$ and $A(c)$ are piecewise Lipschitz continuous functions with respect to $c \in L^2(\Omega)$, and that ${\boldsymbol{\gamma}}({c^n}), A({c^n}){\boldsymbol{u}^n}\in [H^{k+1}({\mathcal T_h})]^2$. Then, the following error estimates hold for all $k \in {\mathbb{N}}_0$: $$\begin{split}
\lVert {\boldsymbol{U}^n}-{\boldsymbol{u}^n}\rVert_{0,\Omega} &\le \lVert {C^n}-{c^n}\rVert_{0,\Omega} \, \zeta_1^n({\boldsymbol{u}^n}) + h^{k+1} \, \zeta_2^n({\boldsymbol{u}^n},{c^n},{\boldsymbol{\gamma}}({c^n}),A({c^n}){\boldsymbol{u}^n}) \\
\lVert {P^n}-{p^n}\rVert_{0,\Omega} &\le \lVert {C^n}-{c^n}\rVert_{0,\Omega} \, \zeta_3^n({\boldsymbol{u}^n}) + h^{k+1} \, \zeta_4^n({\boldsymbol{u}^n},{c^n},{\boldsymbol{\gamma}}({c^n}),A({c^n}){\boldsymbol{u}^n},{p^n}),
\end{split}$$ where $\zeta_1^n$-$\zeta_4^n$ are positive constants independent of $h$ and depending only on the specified functions.
The estimate for $\lVert {\boldsymbol{U}^n}-{\boldsymbol{u}^n}\rVert_{0,\Omega}$ can be obtained as follows.
By using the second equality in , we have ${\operatorname{\rm div}}{\boldsymbol{U}^n}=\Pi^0_k G^n$ (use that ${\operatorname{\rm div}}{\boldsymbol{U}^n}\in \mathbb{P}_k({K})$ for every ${K}\in {\mathcal T_h}$), where we recall that $(\Pi^0_k)_{|_{K}}={\Pi^{0,{K}}_{k}}$. Define now the interpolant ${\boldsymbol{u}^n_I}\in {\boldsymbol{V}_h}$ via the degrees of freedom : $${{\textup{dof}}^{\boldsymbol{V}_h}}_i({\boldsymbol{u}^n_I})={{\textup{dof}}^{\boldsymbol{V}_h}}_i({\boldsymbol{u}^n}), \quad i=1,\dots,{{\textrm{dim}{V}_h}}.$$ Then, it holds [@BBMR_generalsecondorder eq.(28)] $$\label{eq:interpolation_error}
\lVert {\boldsymbol{u}^n}-{\boldsymbol{u}^n_I}\rVert_{0,\Omega} \le \eta \, h^{k+1} \lVert {\boldsymbol{u}^n}\rVert_{k+1,{\mathcal T_h}}.$$ Moreover, one has ${\operatorname{\rm div}}{\boldsymbol{u}^n_I}=\Pi^0_k {G^n}$. Thus, setting ${\boldsymbol{\delta}^n}:={\boldsymbol{U}^n}-{\boldsymbol{u}^n_I}$, it holds that ${\boldsymbol{\delta}^n}\in {\mathcal{K}_h}\subset {\mathcal{K}}$, where ${\mathcal{K}_h}$ and ${\mathcal{K}}$ were defined in and , respectively, and therefore, $\lVert {\boldsymbol{\delta}^n}\rVert_{{\boldsymbol{V}_h}}=\lVert {\boldsymbol{\delta}^n}\rVert_{0,\Omega}$. Owing to the assumptions on $a(\cdot)$ in together with , we have, further using with ${\boldsymbol{v}_h}={\boldsymbol{\delta}^n}\in {\mathcal{K}_h}$ and , $$\label{eq:error_estimate_Un_un}
\begin{split}
\alpha \lVert {\boldsymbol{\delta}^n}\rVert^2_{0,\Omega} &\le {\mathcal{A}_h}({C^n};{\boldsymbol{\delta}^n},{\boldsymbol{\delta}^n}) = {\mathcal{A}_h}({C^n};{\boldsymbol{U}^n},{\boldsymbol{\delta}^n}) - {\mathcal{A}_h}({C^n};{\boldsymbol{u}^n_I},{\boldsymbol{\delta}^n}) \\
&= ({\boldsymbol{\gamma}}({C^n}),{\boldsymbol{\delta}^n})_h - {\mathcal{A}_h}({C^n};{\boldsymbol{u}^n_I},{\boldsymbol{\delta}^n}) \\
&=\left[({\boldsymbol{\gamma}}({C^n}),{\boldsymbol{\delta}^n})_h - ({\boldsymbol{\gamma}}({c^n}),{\boldsymbol{\delta}^n})_{0,\Omega} \right]
+ {\mathcal{A}_h}({C^n};{\boldsymbol{u}^n}-{\boldsymbol{u}^n_I},{\boldsymbol{\delta}^n}) \\
&\quad + \bigg[{\mathcal{A}}({c^n};{\boldsymbol{u}^n},{\boldsymbol{\delta}^n}) - {\mathcal{A}_h}({C^n};{\boldsymbol{u}^n},{\boldsymbol{\delta}^n}) \bigg] \\
&=: T_1 + T_2 + T_3.
\end{split}$$
The terms $T_1$-$T_3$ are bounded as follows:
- term $T_1$: We use equation with ${\boldsymbol{\kappa}}={\boldsymbol{\gamma}}$, $\sigma={c^n}$, ${\widehat{\sigma}}={C^n}$, ${\boldsymbol{\psi}}={\boldsymbol{\delta}^n}$, $r=k+1$, $t=k$, and ${m_r}={m_t}=k+1$, and obtain $$\begin{split}
|T_1| &= |({\boldsymbol{\gamma}}({c^n}),{\boldsymbol{\delta}^n})_{0,\Omega} - ({\boldsymbol{\gamma}}({\Pi^{0}_{k+1}}{C^n}),{\boldsymbol{\Pi^{0}_{k}}}{\boldsymbol{\delta}^n})_{0,\Omega}| \\
&\le \eta \big[h^{k+1} (|{\boldsymbol{\gamma}}({c^n})|_{k+1,{\mathcal T_h}} + |{c^n}|_{k+1,{\mathcal T_h}}) + \lVert {c^n}- {C^n}\rVert_{0,\Omega} \big] \lVert {\boldsymbol{\delta}^n}\rVert_{0,\Omega}.
\end{split}$$
- term $T_2$: Owing to the continuity properties of ${\mathcal{A}_h}(\cdot;\cdot,\cdot)$ and the interpolation error estimate , it holds $$\begin{split}
|T_2|=|{\mathcal{A}_h}({C^n};{\boldsymbol{u}^n}-{\boldsymbol{u}^n_I},{\boldsymbol{\delta}^n})|
\le \eta \lVert {\boldsymbol{u}^n}-{\boldsymbol{u}^n_I}\rVert_{0,\Omega} \lVert {\boldsymbol{\delta}^n}\rVert_{0,\Omega}
\le \eta \, h^{k+1} \lVert {\boldsymbol{u}^n}\rVert_{k+1,{\mathcal T_h}} \lVert {\boldsymbol{\delta}^n}\rVert_{0,\Omega}.
\end{split}$$
- term $T_3$: We have $$\begin{split}
|T_3|&=|{\mathcal{A}}({c^n};{\boldsymbol{u}^n},{\boldsymbol{\delta}^n}) - {\mathcal{A}_h}({C^n};{\boldsymbol{u}^n},{\boldsymbol{\delta}^n})| \\
&\le |(A({c^n}) {\boldsymbol{u}^n}, {\boldsymbol{\delta}^n})_{0,\Omega} - (A({\Pi^{0}_{k+1}}{C^n}) {\boldsymbol{\Pi^{0}_{k}}}{\boldsymbol{u}^n}, {\boldsymbol{\Pi^{0}_{k}}}{\boldsymbol{\delta}^n})_{0,\Omega}| \\
& + \left|\sum_{{K}\in {\mathcal T_h}} {\nu_{{\mathcal{A}}}^{K}}({C^n}) \, {S^{K}_{{\mathcal{A}}}}((I-{\boldsymbol{\Pi^{0,{K}}_{k}}}){\boldsymbol{u}^n},(I-{\boldsymbol{\Pi^{0,{K}}_{k}}}){\boldsymbol{\delta}^n}) \right|\\
&=:T_3^A + T_3^B.
\end{split}$$ For the term $T_3^A$, we use Lemma \[lem:abstr\_res\_1\] with $\kappa=A$, $\sigma={c^n}$, ${\widehat{\sigma}}={C^n}$, ${\boldsymbol{\chi}}={\boldsymbol{u}^n}$, ${\boldsymbol{\psi}}={\boldsymbol{\delta}^n}$, $r=k+1$, $s=t=k$, and ${m_r}={m_s}={m_t}=k+1$, to get $$\begin{split}
T_3^A &\le \eta \bigg[h^{k+1} \big( |A({c^n}) {\boldsymbol{u}^n}|_{k+1,{\mathcal T_h}} + |{\boldsymbol{u}^n}|_{k+1,{\mathcal T_h}} \lVert A({c^n}) \rVert_{\infty,\Omega}
+ |{c^n}|_{k+1,{\mathcal T_h}} \lVert {\boldsymbol{u}^n}\rVert_{\infty,\Omega} \big) \\
&\qquad+ \lVert {c^n}- {C^n}\rVert_{0,\Omega} \lVert {\boldsymbol{u}^n}\rVert_{\infty,\Omega} \bigg] \lVert {\boldsymbol{\delta}^n}\rVert_{0,\Omega}.
\end{split}$$ On the other hand, the term $T_3^B$ can be bounded with , , and Lemma : $$T_3^B \le \eta \, h^{k+1} |{\boldsymbol{u}^n}|_{k+1,{\mathcal T_h}} \lVert {\boldsymbol{\delta}^n}\rVert_{0,\Omega}.$$
After plugging the bounds obtained for $T_1$-$T_3$ into , dividing by $\lVert {\boldsymbol{\delta}^n}\rVert_{0,\Omega}$, using the triangle inequality in the form $$\lVert {\boldsymbol{U}^n}-{\boldsymbol{u}^n}\rVert_{0,\Omega} \le \lVert {\boldsymbol{\delta}^n}\rVert_{0,\Omega} +\lVert {\boldsymbol{u}^n}-{\boldsymbol{u}^n_I}\rVert_{0,\Omega},$$ and employing , the convergence result follows.
The error estimate for the term $\lVert {P^n}-{p^n}\rVert_{0,\Omega}$ follows easily by combining the above ideas with the argument in [@Brezzi-Falk-Marini Theorem 6.1] and is therefore not shown.
Bounds on $\lVert {c^n}-{C^n}\rVert_{0,\Omega}$
-----------------------------------------------
For fixed ${\boldsymbol{u}}(t) \in \boldsymbol{V}$ and $t\in J$, we define the projector ${\mathcal{P}_c }:\, Z \to {Z_h}$ (that to each $c \in Z$ associates ${\mathcal{P}_c }c \in {Z_h}$) by $$\label{eq:proj_Pc}
\begin{split}
{\Gamma_{c,h}}({\boldsymbol{u}}(t);{\mathcal{P}_c }c,{z_h})={\Gamma_c}({\boldsymbol{u}}(t);c,{z_h}),
\end{split}$$ for all ${z_h}\in {Z_h}$, where $$\label{def:Gammalambda_Gammalambdah}
\begin{split}
{\Gamma_{c,h}}({\boldsymbol{u}};c,{z_h})&:={\mathcal{D}_h}({\boldsymbol{u}};c,{z_h}) + {\Theta_h}({\boldsymbol{u}};c,{z_h}) + (c,{z_h})_h \\
{\Gamma_c}({\boldsymbol{u}};c,{z_h})&:={\mathcal{D}}({\boldsymbol{u}};c,{z_h}) + \Theta({\boldsymbol{u}};c,{z_h}) + (c,{z_h})_{0,\Omega},
\end{split}$$ with $$\label{eq:L1}
(c,{z_h})_h:=\sum_{{K}\in {\mathcal T_h}} \int_{K}c \, ({\Pi^{0,{K}}_{k+1}}{z_h}) \, {\textup{d}x}.$$
\[lem:well\_posedness\_Pc\] The projector ${\mathcal{P}_c }:\, Z \to {Z_h}$ given in is well-defined under the assumption that ${\boldsymbol{u}}$, ${q^+}$, and ${q^-}$ are bounded in $L^\infty(\Omega)$ for all $t\in J$.
By the Lax-Milgram lemma, we have to show that the left hand side of defines a continuous and coercive bilinear form and that the right hand side is a continuous functional with respect to $\lVert \cdot \rVert_{1,{\mathcal T_h}}$. Continuity of the latter one is obtained by combining with $$\begin{split}
\Theta({\boldsymbol{u}};c,{z_h}) &+ (c,{z_h})_{0,\Omega}=
\frac{1}{2} \left[\left( {\boldsymbol{u}}\cdot \nabla c,{z_h}\right)_{0,\Omega} + (({q^+}+{q^-}+2) c, {z_h})_{0,\Omega} - \left( {\boldsymbol{u}}\, c, \nabla {z_h}\right)_{0,\Omega} \right] \\
&\le \frac{1}{2} \left[ \lVert {\boldsymbol{u}}\rVert_{\infty,\Omega} (|c|_{1,{\mathcal T_h}} + \lVert c\rVert_{0,\Omega}) + \lVert {q^+}+{q^-}+2 \rVert_{\infty,\Omega} \lVert c \rVert_{0,\Omega} \right] \lVert {z_h}\rVert_{1,{\mathcal T_h}}.
\end{split}$$ By using and performing similar computations as in the proof of Lemma \[lem:well\_posedness\_semidiscr\], continuity of ${\Gamma_{c,h}}$ follows: $$\label{eq:cont_Gammalambdah}
{\Gamma_{c,h}}({\boldsymbol{u}};c,{z_h}) \le \eta \, \zeta({\boldsymbol{u}},{q^+},{q^-}) \lVert c \rVert_{1,{\mathcal T_h}} \lVert {z_h}\rVert_{1,{\mathcal T_h}},$$ where $\zeta$ only depends on the specified functions. Regarding the coercivity of ${\Gamma_{c,h}}$, we first estimate $$\begin{split}
{\Theta_h}({\boldsymbol{u}};{z_h},{z_h}) + ({z_h},{z_h})_h
=\sum_{{K}\in {\mathcal T_h}} \left( \left( \frac{1}{2} ({q^+}+{q^-}) + 1 \right) {\Pi^{0,{K}}_{k+1}}{z_h}, {\Pi^{0,{K}}_{k+1}}{z_h}\right)_{0,{K}} \ge \lVert {\Pi^{0}_{k+1}}{z_h}\rVert^2_{0,\Omega},
\end{split}$$ where we recall that $({\Pi^{0}_{k+1}})_{|_{K}}={\Pi^{0,{K}}_{k+1}}$ for all ${K}\in {\mathcal T_h}$. Then, combining this result with yields $${\Gamma_{c,h}}({\boldsymbol{u}};{z_h},{z_h}) \ge \eta \left[\left| {z_h}\right|^2_{1,{\mathcal T_h}} + \lVert {\Pi^{0,{K}}_{k+1}}{z_h}\rVert^2_{0,\Omega} \right]
\ge \eta \left[\left| {z_h}\right|^2_{1,{\mathcal T_h}} + \lVert \overline{{z_h}} \rVert^2_{0,\Omega} \right],$$ with $\overline{{z_h}}$ denoting the $L^2(\Omega)$ projection of ${z_h}$ onto $\mathbb{P}_0(\Omega)$.
Since $\overline{{z_h}}$ coincides with the average of ${z_h}$, one can use a Poincaré-Friedrichs inequality, see e.g. [@brenner2003poincare], to deduce $$\left| {z_h}\right|^2_{1,{\mathcal T_h}} + \lVert \overline{{z_h}} \rVert^2_{0,\Omega} \ge C_p^{-1} {\mathrm{diam}}(\Omega)^{-1} \lVert {z_h}\rVert^2_{1,{\mathcal T_h}},$$ and consequently the coercivity of ${\Gamma_{c,h}}$.
\[lem:cn\_Pcn\] We assume that ${\boldsymbol{u}}\in [H^{k+1}({\mathcal T_h}) \cap L^\infty(\Omega)]^2$, $c \in H^{k+2}({\mathcal T_h})\cap W^{1,\infty}({\mathcal T_h})$, ${q^+},{q^-}\in L^\infty(\Omega)$, $({q^+}+{q^-}) c \in H^{k+1}({\mathcal T_h})$, ${\boldsymbol{u}}\, c \in [H^{k+1}({\mathcal T_h})]^2$, ${\boldsymbol{u}}\cdot \nabla c \in H^{k+1}({\mathcal T_h})$, and $D({\boldsymbol{u}}) \nabla c \in [H^{k+1}({\mathcal T_h})]^2$ for all $t \in J$. Then, the following error bounds for $c-{\mathcal{P}_c }c$, where ${\mathcal{P}_c }c$ is defined in , hold for all $k \in {\mathbb{N}}_0$: $$\label{eq:bound_rho}
\begin{split}
\lVert c -{\mathcal{P}_c }c \rVert_{1,{\mathcal T_h}} &\le h^{k+1} \, \xi_1(c,{\boldsymbol{u}},{q^+},{q^-},D({\boldsymbol{u}})\nabla c,\nabla c,({q^+}+{q^-}) c, {\boldsymbol{u}}\cdot \nabla c, {\boldsymbol{u}}\, c), \\
\lVert c -{\mathcal{P}_c }c \rVert_{0,\Omega} &\le h^{k+2} \, \xi_0(c,{\boldsymbol{u}},{q^+},{q^-},D({\boldsymbol{u}})\nabla c,\nabla c,({q^+}+{q^-}) c, {\boldsymbol{u}}\cdot \nabla c, {\boldsymbol{u}}\, c),
\end{split}$$ where the constants $\xi_1,\xi_0>0$ only depend on the listed terms and are independent of $h$.
We focus on the error estimate in the broken $H^1$ norm at a fixed time $t \in J$. First, we state the following result. Given $c \in H^{k+2}({\mathcal T_h})$, there exists an interpolant $c_I \in {Z_h}$ such that the following bounds hold true (see for instance [@cangianigeorgulispryersutton_VEMaposteriori; @beiraolovadinarusso_stabilityVEM; @BrennerGuanSung_someestimatesVEM]): $$\label{eq:interpolation_error_cn}
\lVert c-c_I \rVert_{0,\Omega} \le \eta \, h^{k+2} \lVert c \rVert_{k+2,{\mathcal T_h}}, \quad \lVert c-c_I \rVert_{1,{\mathcal T_h}} \le \eta \, h^{k+1} \lVert c \rVert_{k+2,{\mathcal T_h}}.$$ After denoting $\nu:={\mathcal{P}_c }c-c_I$, one obtains with the coercivity of ${\Gamma_{c,h}}$, see the proof of Lemma \[lem:well\_posedness\_Pc\], and the definition of ${\mathcal{P}_c }c$ in , $$\label{eq:estimate_S1_S2_S3}
\begin{split}
M \lVert \nu \rVert^2_{1,{\mathcal T_h}} &\le {\Gamma_{c,h}}({\boldsymbol{u}},\nu,\nu)
= {\Gamma_{c,h}}({\boldsymbol{u}},{\mathcal{P}_c }c,\nu) - {\Gamma_{c,h}}({\boldsymbol{u}},c_I,\nu) \\
&= [{\Gamma_c}({\boldsymbol{u}},c,\nu) - {\Gamma_{c,h}}({\boldsymbol{u}},c,\nu)] + {\Gamma_{c,h}}({\boldsymbol{u}},c-c_I,\nu) \\
&=: S_1 + S_2,
\end{split}$$ for a constant $M >0$. By employing the definitions of ${\Gamma_c}$ and ${\Gamma_{c,h}}$ in , the term $S_1$ is split as follows: $$\begin{split}
S_1 &= [{\mathcal{D}}({\boldsymbol{u}};c,\nu)-{\mathcal{D}_h}({\boldsymbol{u}};c,\nu)] + [\Theta({\boldsymbol{u}};c,\nu)-{\Theta_h}({\boldsymbol{u}};c,\nu)] + [(c,\nu)_{0,\Omega}-(c,\nu)_h] \\
&=:S_1^A + S_1^B + S_1^C.
\end{split}$$
For $S_1^A$, we have $$\begin{split}
S_1^A&= [(D({\boldsymbol{u}})\nabla c,\nabla \nu)_{0,\Omega} - (D({\boldsymbol{\Pi^{0}_{k}}}{\boldsymbol{u}}) \, {\boldsymbol{\Pi^{0}_{k}}}(\nabla c) , {\boldsymbol{\Pi^{0}_{k}}}(\nabla \nu))_{0,\Omega}]\\
&\quad + \sum_{{K}\in {\mathcal T_h}} {\nu_{D}^{K}}({\boldsymbol{u}}) {S^{K}_{D}}((I-{\Pi^{\nabla,{K}}_{k+1}})c,(I-{\Pi^{\nabla,{K}}_{k+1}}) \nu) \\
&\le \eta \, h^{k+1} \left[ |D({\boldsymbol{u}}) \nabla c|_{k+1,{\mathcal T_h}} + |\nabla c|_{k+1,{\mathcal T_h}} (\lVert D({\boldsymbol{u}}) \rVert_{\infty,\Omega}+1) + |{\boldsymbol{u}}|_{k+1,{\mathcal T_h}} \lVert \nabla c \rVert_{\infty,\Omega} \right] |\nu|_{1,{\mathcal T_h}},
\end{split}$$ where in the inequality we applied Lemma \[lem:abstr\_res\_1\] to estimate the first part on the right hand side of $S_1^A$, and made use of the continuity properties of ${S^{K}_{D}}(\cdot,\cdot)$, the trivial continuity property of ${\Pi^{\nabla,{K}}_{k+1}}$ in the $H^1$ seminorm and its approximation properties (stated in Lemma \[lem:approx\_properties\]) to estimate the stabilization term.
Next, for $S_1^B$, we compute $$\begin{split}
S_1^B &= \frac{1}{2} \bigg\{ \left[({\boldsymbol{u}}\cdot \nabla c,\nu)_{0,\Omega} - ({\boldsymbol{\Pi^{0}_{k}}}{\boldsymbol{u}}\cdot {\boldsymbol{\Pi^{0}_{k}}}(\nabla c) ,{\Pi^{0}_{k+1}}\nu)_{0,\Omega} \right] \\
&\qquad + \left[ (({q^+}+{q^-})c,\nu)_{0,\Omega} - (({q^+}+{q^-}) {\Pi^{0}_{k+1}}c, {\Pi^{0}_{k+1}}\nu)_{0,\Omega} \right] \\
&\qquad - \left[({\boldsymbol{u}}\, c,\nabla \nu)_{0,\Omega} - ({\boldsymbol{\Pi^{0}_{k}}}{\boldsymbol{u}}\, {\Pi^{0}_{k+1}}c,{\boldsymbol{\Pi^{0}_{k}}}(\nabla \nu))_{0,\Omega} \right] \bigg\} \\
&\le \eta \, h^{k+1} \big[ |{\boldsymbol{u}}\cdot \nabla c|_{k+1,{\mathcal T_h}} + (|\nabla c|_{k+1,{\mathcal T_h}} + |c|_{k+1,{\mathcal T_h}}) \lVert {\boldsymbol{u}}\rVert_{\infty,\Omega} + |c|_{k+1,{\mathcal T_h}} \lVert {q^+}+ {q^-}\rVert_{\infty,\Omega} \\
& \qquad\qquad +|({q^+}+{q^-}) c|_{k+1,{\mathcal T_h}} + |{\boldsymbol{u}}\, c|_{k+1,{\mathcal T_h}} + |{\boldsymbol{u}}|_{k+1,{\mathcal T_h}} (\lVert c \rVert_{\infty,\Omega} + \lVert \nabla c \rVert_{\infty,\Omega}) \big] \lVert \nu \rVert_{1,{\mathcal T_h}},
\end{split}$$ where in the last inequality we used Lemma \[lem:abstr\_res\_1\] with $\kappa=id$ and $\sigma={\boldsymbol{u}}$ for the first and third term inside the curly bracket, and $\kappa={q^+}+{q^-}$ and $\sigma=1$ for the second one.
Finally, for $S_1^C$, it holds with the definition of the $L^2$ projector and Lemma \[lem:approx\_properties\] $$\begin{split}
S_1^C &= ((I-{\Pi^{0}_{k+1}}) c,\nu)_{0,\Omega} \le \eta \, h^{k+1} |c|_{k+1,{\mathcal T_h}} \lVert \nu \rVert_{0,\Omega}.
\end{split}$$ On the other hand, for $S_2$, we use the continuity of ${\Gamma_{c,h}}$ in , together with the interpolation error estimate , to derive $${\Gamma_{c,h}}({\boldsymbol{u}};c-c_I,\nu) \le \eta \, \zeta({\boldsymbol{u}},{q^+},{q^-}) \lVert c-c_I \rVert_{1,{\mathcal T_h}} \lVert \nu \rVert_{1,{\mathcal T_h}}
\le \eta \,\zeta({\boldsymbol{u}},{q^+},{q^-}) h^{k+1} \lVert c \rVert_{k+2,{\mathcal T_h}} \lVert \nu \rVert_{1,{\mathcal T_h}}.$$ The error bound in the broken $H^1$ norm follows by plugging first the estimates for $S_1^A$, $S_1^B$, and $S_1^C$ into $S_1$, then those obtained for $S_1$ and $S_2$ into , using the definition of the $H^1$ norm, dividing by $\lVert \nu \rVert_{1,{\mathcal T_h}}$, and using the triangle inequality in the form $$\lVert c -{\mathcal{P}_c }c \rVert_{1,{\mathcal T_h}} \le \lVert c -c_I \rVert_{1,{\mathcal T_h}} + \lVert \nu \rVert_{1,{\mathcal T_h}},$$ together with the approximation properties of the interpolant $c_I$.
The $L^2$ error bound can be derived by combining the above arguments with a standard duality argument as in [@vacca2015virtual], also recalling the convexity of $\Omega$; it is omitted here.
By differentiation of in time and use of similar techniques as in the proof of Lemma \[lem:cn\_Pcn\], an analogous result can be obtained for $\frac{\partial}{\partial t}(c-{\mathcal{P}_c }c)$, summarized in the following corollary.
\[cor:bound\_partial\_t\_c\_Pc\] Provided that the continuous data and solution are sufficiently regular in space and time, it holds $$\begin{split}
\left\lVert \frac{\partial}{\partial t} (c -{\mathcal{P}_c }c) \right\rVert_{1,{\mathcal T_h}} \le h^{k+1} \, \xi_{1,t}, \qquad \left\lVert \frac{\partial}{\partial t} (c -{\mathcal{P}_c }c) \right\rVert_{0,\Omega} \le h^{k+2} \, \xi_{0,t},
\end{split}$$ where the constants $\xi_{1,t},\xi_{0,t}>0$ are independent of $h$.
Moreover, we will later on need the two subsequent bounds.
\[lem:partial\_cn\] Under sufficient smoothness of the continuous data and solution, it holds $$\left\lVert \frac{\partial {c^n}}{\partial t} - \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} \right\rVert_{0,\Omega}
\le \tau^{\frac{1}{2}} \left\lVert \frac{\partial^2 c}{\partial s^2} \right\rVert_{L^2({t_{n-1}},{t_n};L^2(\Omega))}
+\tau^{-\frac{1}{2}} h^{k+2} \left(\int_{{t_{n-1}}}^{{t_n}} \xi_{0,t}^2 \, {\textup{d}s}\right)^{\frac{1}{2}},$$ where $\xi_{0,t}$ can be found in Corollary \[cor:bound\_partial\_t\_c\_Pc\].
We estimate $$\begin{split}
\left\lVert \frac{\partial c}{\partial t} - \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} \right\rVert_{0,\Omega}
&\le \left\lVert \frac{\partial {c^n}}{\partial t} - \frac{{c^n}-{c^{n-1}}}{\tau} \right\rVert_{0,\Omega} + \left\lVert \frac{{\mathcal{P}_c }{c^n}-{\mathcal{P}_c }{c^{n-1}}}{\tau} - \frac{{c^n}-{c^{n-1}}}{\tau} \right\rVert_{0,\Omega}\\
&=:(I) + (II).
\end{split}$$ The term $(I)$ can be estimated exactly as for standard finite elements, see for instance [@thomee1984galerkin]: $$\label{eq:term_I}
\begin{split}
&(I)=\left\lVert \frac{\partial {c^n}}{\partial t} - \frac{{c^n}-{c^{n-1}}}{\tau} \right\rVert_{0,\Omega}
\le \int_{{t_{n-1}}}^{{t_n}} \left\lVert \frac{\partial^2 c}{\partial s^2}(s)\right\rVert_{0,\Omega} \, {\textup{d}s}\le \tau^{\frac{1}{2}} \left(\int_{{t_{n-1}}}^{{t_n}} \left\lVert \frac{\partial^2 c}{\partial s^2}(s)\right\rVert^2_{0,\Omega} \, {\textup{d}s}\right)^{\frac{1}{2}},
\end{split}$$ where we also applied the Hölder inequality in the last step. Concerning $(II)$, this term can be bounded as follows, using Corollary \[cor:bound\_partial\_t\_c\_Pc\]: $$\begin{split}
(II) &= \left\lVert \frac{{\mathcal{P}_c }{c^n}-{\mathcal{P}_c }{c^{n-1}}}{\tau} - \frac{{c^n}-{c^{n-1}}}{\tau} \right\rVert_{0,\Omega}
= \frac{1}{\tau}\left\lVert \int_{{t_{n-1}}}^{{t_n}} \frac{\partial}{\partial s} ({\mathcal{P}_c }c-c)(s) \, {\textup{d}s}\right\rVert_{0,\Omega} \\
&\le \tau^{-\frac{1}{2}} \left(\int_{{t_{n-1}}}^{{t_n}} \left\lVert \frac{\partial}{\partial s} ({\mathcal{P}_c }c-c)(s)\right\rVert^2_{0,\Omega} \, {\textup{d}s}\right)^{\frac{1}{2}}
\le \tau^{-\frac{1}{2}} h^{k+2} \left(\int_{{t_{n-1}}}^{{t_n}} \xi_{0,t}^2 \, {\textup{d}s}\right)^{\frac{1}{2}}.
\end{split}$$ The statement of the lemma follows.
\[lem:un\_Uno\] Provided that the continuous data and solution are sufficiently regular in space and time, it holds $$\lVert {\boldsymbol{u}^n}- {\boldsymbol{U}^{n-1}}\rVert_{0,\Omega} \le \tau \left\lVert \frac{\partial {\boldsymbol{u}}}{\partial t} \right\rVert_{L^{\infty}({t_{n-1}},{t_n};L^2(\Omega))} + \lVert {C^{n-1}}-{c^{n-1}}\rVert_{0,\Omega} \, \zeta_1^{n-1} + h^{k+1} \, \zeta_2^{n-1},$$ where $\zeta_1^{n-1}$ and $\zeta_2^{n-1}$ are the constants from Theorem \[thm:Un\_un\_Pn\_pn\].
By using the triangle inequality, one obtains $$\begin{split}
\lVert {\boldsymbol{u}^n}- {\boldsymbol{U}^{n-1}}\rVert_{0,\Omega} &\le \left\lVert {\boldsymbol{u}^n}-{\boldsymbol{u}^{n-1}}\right\rVert_{0,\Omega} + \lVert {\boldsymbol{u}^{n-1}}-{\boldsymbol{U}^{n-1}}\rVert_{0,\Omega}.
\end{split}$$ The first term on the right hand side is estimated by $$\lVert {\boldsymbol{u}^n}-{\boldsymbol{u}^{n-1}}\rVert_{0,\Omega} = \left\lVert \int_{{t_{n-1}}}^{{t_n}} \frac{\partial {\boldsymbol{u}}(s)}{\partial s} \, {\textup{d}s}\right\rVert_{0,\Omega}
\le \tau \left\lVert \frac{\partial {\boldsymbol{u}}}{\partial t} \right\rVert_{L^{\infty}({t_{n-1}},{t_n};L^2(\Omega))},$$ and the second one term is bounded with Theorem \[thm:Un\_un\_Pn\_pn\].
Now, we have all the ingredients to bound $\lVert {c^n}-{C^n}\rVert_{0,\Omega}$.
\[thm:Cn\_cn\] Let the mesh assumptions (**D1**)-(**D3**) be satisfied. Then, provided that the continuous data and solutions are sufficiently regular, it yields $$\lVert {c^n}-{C^n}\rVert_{0,\Omega} \le \eta \left[ \lVert {c_{0,h}}-c^0 \rVert_{0,\Omega} + h^{k+1} \,\varphi_1 + \tau \: \varphi_2 \right],$$ where the regularity terms $\varphi_1, \varphi_2$ and the positive constant $\eta$ now depend on ${\boldsymbol{u}}$, $c$, ${q^+}$, ${q^-}$, ${\widehat{c}}$, $\frac{\partial {\boldsymbol{u}}}{\partial t}$, $\frac{\partial^2 {\boldsymbol{u}}}{\partial t^2}$, $\frac{\partial c}{\partial t}$, and $\frac{\partial^2 c}{\partial t^2}$ (and products of these functions).
To start with, we write $${C^n}-{c^n}= ({C^n}- {\mathcal{P}_c }{c^n}) + ({\mathcal{P}_c }{c^n}- {c^n})=:{\vartheta^n}+{\rho^n}.$$ Equation gives a bound on ${\rho^n}$. In order to deal with ${\vartheta^n}$, we use the continuous concentration equation with $z={\vartheta^n}$, the fully discretized version with ${z_h}={\vartheta^n}$, and the definition of the projector ${\mathcal{P}_c }{c^n}$ in with ${z_h}={\vartheta^n}$: $$\label{eq:rec_rel_theta}
\begin{split}
&{\mathcal{M}_h}\left( \frac{{\vartheta^n}-{\vartheta^{n-1}}}{\tau}, {\vartheta^n}\right)
+ {\mathcal{D}_h}({\boldsymbol{U}^{n-1}};{\vartheta^n},{\vartheta^n})\\
&=\left[ {\mathcal{M}}\left(\frac{\partial {c^n}}{\partial t},{\vartheta^n}\right)_{0,\Omega} - {\mathcal{M}_h}\left( \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau},{\vartheta^n}\right) \right] \\
&\quad + \left[ {\Theta_h}({\boldsymbol{u}^n};{\mathcal{P}_c }{c^n},{\vartheta^n}) - {\Theta_h}({\boldsymbol{U}^{n-1}};{C^n},{\vartheta^n}) \right]
+ [{\mathcal{D}_h}\left({\boldsymbol{u}^n};{\mathcal{P}_c }{c^n}, {\vartheta^n}\right) - {\mathcal{D}_h}\left({\boldsymbol{U}^{n-1}};{\mathcal{P}_c }{c^n}, {\vartheta^n}\right)] \\
&\quad + \left[ ({\mathcal{P}_c }{c^n},{\vartheta^n})_h-({c^n},{\vartheta^n})_{0,\Omega} \right] + \left[ ({(q^+)^n}{\widehat{c}^{\, n}},{\vartheta^n})_h - ({(q^+)^n}{\widehat{c}^{\, n}},{\vartheta^n})_{0,\Omega} \right]\\
&\quad =:R_1 + R_2 + R_3 + R_4 + R_5.
\end{split}$$ Owing to the coercivity properties in , the second term on the left hand side of can be estimated by $$\label{eq:estimate:Dh}
{\mathcal{D}_h}({\boldsymbol{U}^{n-1}};{\vartheta^n},{\vartheta^n}) \ge D_\ast \left| {\vartheta^n}\right|^2_{1,{\mathcal T_h}},$$ with some constant $D_\ast>0$ independent of $h$ and ${\boldsymbol{U}^{n-1}}$.
The terms $R_1$-$R_5$ on the right hand side of are estimated as follows:
- term $R_1$: Using the definition of ${\mathcal{M}_h}(\cdot,\cdot)$ in together with yields $$\begin{split}
R_1 &= {\mathcal{M}}\left(\frac{\partial {c^n}}{\partial t},{\vartheta^n}\right)_{0,\Omega} - {\mathcal{M}_h}\left( \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau},{\vartheta^n}\right) \\
&= \bigg[ \left(\phi \, \frac{\partial {c^n}}{\partial t}, {\vartheta^n}\right)_{0,\Omega} - \left({\Pi^{0}_{k+1}}\left(\phi \, {\Pi^{0}_{k+1}}\left( \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} \right)\right), {\vartheta^n}\right)_{0,\Omega} \\
& \qquad - \sum_{{K}\in {\mathcal T_h}} {\nu_{{\mathcal{M}}}^{K}}(\phi) {S^{K}_{{\mathcal{M}}}}\left((I-{\Pi^{0,{K}}_{k+1}}) \left( \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} \right), (I-{\Pi^{0,{K}}_{k+1}}) {\vartheta^n}\right) \bigg] \\
&\le \eta \bigg[\left\lVert \phi \, \frac{\partial {c^n}}{\partial t} - {\Pi^{0}_{k+1}}\left(\phi \, {\Pi^{0}_{k+1}}\left( \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} \right)\right) \right\rVert_{0,\Omega} \\
&\qquad+ \left\lVert (I-{\Pi^{0}_{k+1}}) \left( \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} \right) \right\rVert_{0,\Omega} \bigg] \lVert {\vartheta^n}\rVert_{0,\Omega} \\
&=:\eta [R_1^A + R_1^B] \lVert {\vartheta^n}\rVert_{0,\Omega}.
\end{split}$$ The term $R_1^A$ is estimated by using the continuity of the $L^2$ projector, the assumption on $\phi$, and the approximation properties in Lemma : $$\begin{split}
R_1^A &\le \left\lVert (I-{\Pi^{0}_{k+1}}) \left(\phi \, \frac{\partial {c^n}}{\partial t}\right) \right\rVert_{0,\Omega}
+ \left\lVert {\Pi^{0}_{k+1}}\left(\phi \, \frac{\partial {c^n}}{\partial t} - \phi \, {\Pi^{0}_{k+1}}\left(\frac{\partial {c^n}}{\partial t}\right) \right) \right\rVert_{0,\Omega} \\
& \qquad + \left\lVert {\Pi^{0}_{k+1}}\left( \phi \, {\Pi^{0}_{k+1}}\left(\frac{\partial {c^n}}{\partial t} - \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} \right)\right) \right\rVert_{0,\Omega} \\
&\le \eta \bigg[ h^{k+2} \left(\left| \phi \frac{\partial {c^n}}{\partial t} \right|_{k+2,{\mathcal T_h}} + \left|\frac{\partial {c^n}}{\partial t} \right|_{k+2,{\mathcal T_h}} \right) + \left\lVert \frac{\partial {c^n}}{\partial t} - \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} \right\rVert_{0,\Omega} \bigg].
\end{split}$$ Next, we bound $R_1^B$ with similar tools as for $R_1^A$: $$\begin{split}
R_1^B &\le \left\lVert (I-{\Pi^{0}_{k+1}}) \left( \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} - \frac{\partial {c^n}}{\partial t} \right) \right\rVert_{0,\Omega} + \left\lVert (I-{\Pi^{0}_{k+1}}) \frac{\partial {c^n}}{\partial t} \right\rVert_{0,\Omega} \\
&\quad \le \left\lVert \frac{{\mathcal{P}_c }{c^n}- {\mathcal{P}_c }{c^{n-1}}}{\tau} - \frac{\partial {c^n}}{\partial t} \right\rVert_{0,\Omega} + \eta \, h^{k+2} \left| \frac{\partial {c^n}}{\partial t} \right|_{k+2,{\mathcal T_h}}.
\end{split}$$ Thus, we deduce with Lemma \[lem:partial\_cn\] $$\label{eq:def_R1}
\begin{split}
R_1 &\le \eta \bigg[ h^{k+2} \left(\left| \phi \frac{\partial {c^n}}{\partial t} \right|_{k+2,{\mathcal T_h}} + \left|\frac{\partial {c^n}}{\partial t} \right|_{k+2,{\mathcal T_h}} \right) +\tau^{-\frac{1}{2}} h^{k+2} \left(\int_{{t_{n-1}}}^{{t_n}} \xi_{0,t}^2 \, {\textup{d}s}\right)^{\frac{1}{2}} \\
&\qquad + \tau^{\frac{1}{2}} \left\lVert \frac{\partial^2 c}{\partial s^2} \right\rVert_{L^2({t_{n-1}},{t_n};L^2(\Omega))} \bigg] \lVert {\vartheta^n}\rVert_{0,\Omega} \\
&=:\bigg[ h^{k+2} R_1^{n,1} + \tau^{-\frac{1}{2}} h^{k+2} R_1^{n,2} + \tau^{\frac{1}{2}} R_1^{n,3} \bigg] \lVert{\vartheta^n}\rVert_{0,\Omega},
\end{split}$$ with the obvious definitions for the regularity terms $R_1^{n,1}$, $R_1^{n,2}$, and $R_1^{n,3}$.
- term $R_2$: By the definition of ${\Theta_h}(\cdot;\cdot,\cdot)$ in , the identity ${\vartheta^n}={C^n}-{\mathcal{P}_c }{c^n}$, and the fact that ${(q^+)^n}$ and ${(q^-)^n}$ are non-negative, it holds $$\begin{split}
&{\Theta_h}({\boldsymbol{u}^n};{\mathcal{P}_c }{c^n},{\vartheta^n}) - {\Theta_h}({\boldsymbol{U}^{n-1}};{C^n},{\vartheta^n})\\
& = \frac{1}{2} \left[ \left({\boldsymbol{u}^n}\cdot \nabla {\mathcal{P}_c }{c^n},{\vartheta^n}\right)_h - \left({\boldsymbol{U}^{n-1}}\cdot \nabla {C^n},{\vartheta^n}\right)_h \right]
-\frac{1}{2} \left( ({q^+}+{q^-}) {\vartheta^n},{\vartheta^n}\right)_{0,\Omega }\\
& - \frac{1}{2} \left[ \left( {\boldsymbol{u}^n}{\mathcal{P}_c }{c^n},\nabla {\vartheta^n}\right)_h - \left( {\boldsymbol{U}^{n-1}}{C^n},\nabla {\vartheta^n}\right)_h \right] \\
& \le \frac{1}{2} \left[ \left({\boldsymbol{u}^n}\cdot \nabla {\mathcal{P}_c }{c^n},{\vartheta^n}\right)_h - \left({\boldsymbol{U}^{n-1}}\cdot \nabla {C^n},{\vartheta^n}\right)_h - \left( {\boldsymbol{u}^n}{\mathcal{P}_c }{c^n},\nabla {\vartheta^n}\right)_h + \left( {\boldsymbol{U}^{n-1}}{C^n},\nabla {\vartheta^n}\right)_h \right] .
\end{split}$$ The above equation, after adding zero in the form $$\begin{split}
0&=({\boldsymbol{U}^{n-1}}\cdot \nabla {\vartheta^n},{\vartheta^n})_{h}
-({\boldsymbol{U}^{n-1}}\cdot \nabla {\vartheta^n},{\vartheta^n})_{h}\\
&=({\boldsymbol{U}^{n-1}}\cdot \nabla {C^n},{\vartheta^n})_h
-({\boldsymbol{U}^{n-1}}\cdot \nabla {\mathcal{P}_c }{c^n},{\vartheta^n})_h
-({\boldsymbol{U}^{n-1}}\cdot \nabla {\vartheta^n},{C^n})_h
+({\boldsymbol{U}^{n-1}}\cdot \nabla {\vartheta^n},{\mathcal{P}_c }{c^n})_h
\end{split}$$ to the right hand side, can be equivalently expressed as $$\begin{split}
&{\Theta_h}({\boldsymbol{u}^n};{\mathcal{P}_c }{c^n},{\vartheta^n}) - {\Theta_h}({\boldsymbol{U}^{n-1}};{C^n},{\vartheta^n})\\
& \le \frac{1}{2} \left[ \left(({\boldsymbol{u}^n}-{\boldsymbol{U}^{n-1}}) \cdot \nabla {\mathcal{P}_c }{c^n},{\vartheta^n}\right)_h - \left( ({\boldsymbol{u}^n}-{\boldsymbol{U}^{n-1}}) {\mathcal{P}_c }{c^n},\nabla {\vartheta^n}\right)_h\right] =:R_2^A + R_2^B.
\end{split}$$ For $R_2^A$, we estimate $$\begin{split}
R_2^A=\frac{1}{2} \left(({\boldsymbol{u}^n}-{\boldsymbol{U}^{n-1}}) \nabla {\mathcal{P}_c }{c^n},{\vartheta^n}\right)_h \le \frac{1}{2} \lVert {\boldsymbol{u}^n}-{\boldsymbol{U}^{n-1}}\rVert_{0,\Omega} \lVert {\boldsymbol{\Pi^{0}_{k}}}\nabla {\mathcal{P}_c }{c^n}\rVert_{\infty,\Omega} \lVert {\vartheta^n}\rVert_{0,\Omega}.
\end{split}$$ We now use an inverse estimate [@BrennerScott Lemma 4.5.3], the continuity of ${\boldsymbol{\Pi^{0,{K}}_{k}}}$, a triangle inequality, the assumption that ${\mathcal T_h}$ is quasi-regular, and Lemma \[lem:cn\_Pcn\], to deduce, for every ${K}\in {\mathcal T_h}$, $$\label{eq:bound_Pi_nabla_Pc}
\begin{split}
&\lVert {\boldsymbol{\Pi^{0,{K}}_{k}}}\nabla {\mathcal{P}_c }{c^n}\rVert_{\infty,{K}} \le \eta \, {h_{{K}}}^{-1} \lVert {\boldsymbol{\Pi^{0,{K}}_{k}}}\nabla {\mathcal{P}_c }{c^n}\rVert_{0,{K}} \le \eta \, {h_{{K}}}^{-1} \lVert \nabla {\mathcal{P}_c }{c^n}\rVert_{0,{K}} \\
& \quad \le \eta \, {h_{{K}}}^{-1} \left( \lVert \nabla {\mathcal{P}_c }{c^n}- \nabla {c^n}\rVert_{0,{K}} + \lVert \nabla {c^n}\rVert_{0,{K}} \right) \\
&\quad \le \eta \, \left(h^{-1} \lVert \nabla {\mathcal{P}_c }{c^n}- \nabla {c^n}\rVert_{0,{\mathcal T_h}} + \lVert \nabla {c^n}\rVert_{\infty,{K}} \right)
\le \eta,
\end{split}$$ Recalling Lemma \[lem:un\_Uno\], the definitions of ${\vartheta^{n-1}}$ and ${\rho^{n-1}}$, and Lemma \[lem:cn\_Pcn\], we get $$\label{eq:bound_bun_bUno}
\begin{split}
&\lVert {\boldsymbol{u}^n}- {\boldsymbol{U}^{n-1}}\rVert_{0,\Omega} \\
&\quad \le \tau \left\lVert {\partial {\boldsymbol{u}}}/{\partial t} \right\rVert_{L^{\infty}({t_{n-1}},{t_n};L^2(\Omega))} + (\lVert {\vartheta^{n-1}}\rVert_{0,\Omega} + \lVert {\rho^{n-1}}\rVert_{0,\Omega}) \zeta_1^{n-1} + h^{k+1} \, \zeta_2^{n-1} \\
&\quad \le \tau \left\lVert {\partial {\boldsymbol{u}}}/{\partial t} \right\rVert_{L^{\infty}({t_{n-1}},{t_n};L^2(\Omega))} + (\lVert {\vartheta^{n-1}}\rVert_{0,\Omega} + h^{k+2} \xi_0^{n-1}) \zeta_1^{n-1} + h^{k+1} \, \zeta_2^{n-1},
\end{split}$$ thus implying $$R_2^A \le \eta \bigg[h^{k+1} R_2^{n,1} + \tau R_2^{n,2} + \lVert {\vartheta^{n-1}}\rVert_{0,\Omega} R_2^{n,3}\bigg] \lVert {\vartheta^n}\rVert_{0,\Omega},$$ with the obvious definitions for the regularity terms $R_2^{n,1}$, $R_2^{n,2}$, and $R_2^{n,3}$.
The term $R_2^B$ can be bounded analogously to $R_2^A$, giving $$R_2^B=\frac{1}{2} \left( ({\boldsymbol{U}^{n-1}}-{\boldsymbol{u}^n}) {\mathcal{P}_c }{c^n},\nabla {\vartheta^n}\right)_h
\le \eta \, \lVert {\boldsymbol{u}^n}-{\boldsymbol{U}^{n-1}}\rVert_{0,\Omega} |{\vartheta^n}|_{1,{\mathcal T_h}} .$$ Using again the bound , one obtains $$R_2^B \le \eta \bigg[h^{k+1} R_2^{n,1} + \tau R_2^{n,2} + \lVert {\vartheta^{n-1}}\rVert_{0,\Omega} R_2^{n,3}\bigg] |{\vartheta^n}|_{1,{\mathcal T_h}}.$$ Thus, $$R_2 \le \eta \bigg[h^{k+1} R_2^{n,1} + \tau R_2^{n,2} + \lVert {\vartheta^{n-1}}\rVert_{0,\Omega} R_2^{n,3}\bigg] \left( \lVert {\vartheta^n}\rVert_{0,\Omega} + |{\vartheta^n}|_{1,{\mathcal T_h}} \right).$$
- term $R_3$: We use the definition of ${\mathcal{D}_h}(\cdot;\cdot,\cdot)$ in , a standard Hölder inequality in the spirit of , the estimate , the scaling properties of the stabilization in , and the Lipschitz continuity of $D(\cdot;\cdot,\cdot)$ and ${\nu_{D}^{K}}$ in , to deduce $$\begin{split}
R_3 &= {\mathcal{D}_h}\left({\boldsymbol{u}^n};{\mathcal{P}_c }{c^n}, {\vartheta^n}\right) - {\mathcal{D}_h}\left({\boldsymbol{U}^{n-1}};{\mathcal{P}_c }{c^n}, {\vartheta^n}\right) \\
&= ( (D({\boldsymbol{\Pi^{0}_{k}}}{\boldsymbol{u}^n})-D({\boldsymbol{\Pi^{0}_{k}}}{\boldsymbol{U}^{n-1}})) \, {\boldsymbol{\Pi^{0}_{k}}}(\nabla {\mathcal{P}_c }{c^n}) \cdot {\boldsymbol{\Pi^{0}_{k}}}(\nabla {\vartheta^n}) )_{0,\Omega} \\
&\quad + \sum_{{K}\in {\mathcal T_h}} ({\nu_{D}^{K}}({\boldsymbol{u}^n})-{\nu_{D}^{K}}({\boldsymbol{U}^{n-1}})) {S^{K}_{D}}\left((I-{\Pi^{\nabla,{K}}_{k+1}}) {\mathcal{P}_c }{c^n},(I-{\Pi^{\nabla,{K}}_{k+1}}) {\vartheta^n}) \right) \\
&\le \eta \lVert {\boldsymbol{u}^n}-{\boldsymbol{U}^{n-1}}\rVert_{0,\Omega} |{\vartheta^n}|_{1,{\mathcal T_h}}.
\end{split}$$ Hence, with we have $$R_3 \le \eta \bigg[h^{k+1} R_2^{n,1} + \tau R_2^{n,2} + \lVert {\vartheta^{n-1}}\rVert_{0,\Omega} R_2^{n,3} \bigg] |{\vartheta^n}|_{1,{\mathcal T_h}}.$$
- term $R_4$: The use of Lemma \[lem:cn\_Pcn\] yields $$\begin{split}
R_4 &= - [({c^n},{\vartheta^n})_{0,\Omega} - ({\mathcal{P}_c }{c^n},{\vartheta^n})_h]
= -[((I-{\Pi^{0}_{k+1}}) {c^n},{\vartheta^n})_{0,\Omega} + ({\Pi^{0}_{k+1}}({c^n}-{\mathcal{P}_c }{c^n}), {\vartheta^n})_{0,\Omega}] \\
&\le \eta \, h^{k+2} \left[|{c^n}|_{k+2,{\mathcal T_h}} + \xi_0^n \right] \lVert {\vartheta^n}\rVert_{0,\Omega} =:\eta \, h^{k+2} R_4^{n,1} \lVert {\vartheta^n}\rVert_{0,\Omega},
\end{split}$$ with the obvious definition of $R_4^{n,1}$.
- term $R_5$: The approximation properties in Lemma \[lem:approx\_properties\] yield $$\begin{split}
R_5 = - \left((I-{\Pi^{0}_{k+1}}) ({(q^+)^n}{\widehat{c}^{\, n}}), {\vartheta^n}\right)_{0,\Omega} \le \eta \, h^{k+2} \, |{(q^+)^n}{\widehat{c}^{\, n}}|_{k+2,{\mathcal T_h}} \lVert{\vartheta^n}\rVert_{0,\Omega} =:\eta \, h^{k+2} R_5^{n,1} \lVert{\vartheta^n}\rVert_{0,\Omega},
\end{split}$$ with the obvious definition of $R_5^{n,1}$.
We now insert and the bounds on $R_1$-$R_5$ into . Afterwards, we observe that all regularity terms $\{R_J^{n,i}\}$ above only depend on the continuous solution and can be assumed to be bounded uniformly in $h$. We only keep track of the terms $R_1^{n,2}$ and $R_1^{n,3}$. This yields $$\label{eq:ev_eq_proof}
\begin{split}
&\frac{1}{\tau} {\mathcal{M}_h}\left({\vartheta^n}-{\vartheta^{n-1}},{\vartheta^n}\right) + D_\ast \left| {\vartheta^n}\right|^2_{1,{\mathcal T_h}} \\
&\le \lVert {\vartheta^{n-1}}\rVert_{0,\Omega} \lVert {\vartheta^n}\rVert_{0,\Omega} \,\omega^n_1 + \lVert {\vartheta^{n-1}}\rVert_{0,\Omega} |{\vartheta^n}|_{1,{\mathcal T_h}} \omega^n_2 + \lVert {\vartheta^n}\rVert_{0,\Omega} \omega^n_3 + |{\vartheta^n}|_{1,{\mathcal T_h}} \omega^n_4 \\
&= \lVert {\vartheta^n}\rVert_{0,\Omega} \left[\omega^n_3 + \lVert {\vartheta^{n-1}}\rVert_{0,\Omega} \,\omega^n_1 \right] +
|{\vartheta^n}|_{1,{\mathcal T_h}} \left[ \omega^n_4 + \lVert {\vartheta^{n-1}}\rVert_{0,\Omega} \,\omega^n_2 \right],
\end{split}$$ with the positive scalars $$\label{eq:omega_i}
\begin{split}
\omega_i^n \le \eta, \quad i=1,2, \quad
\omega^n_3\le \eta \left( \tau + h^{k+1} + \tau^{-\frac{1}{2}} h^{k+2} R_1^{n,2} + \tau^{\frac{1}{2}} R_1^{n,3} \right), \quad
\omega^n_4\le \eta \left(\tau + h^{k+1} \right).
\end{split}$$ Next, we introduce, for all $w_h \in {Z_h}$, the discrete norm $$\label{eq:discr_norm}
\lVert w_h \rVert^2_{0,h}:={\mathcal{M}_h}(w_h,w_h).$$ Owing to Lemma \[lem:properties\_D\_A\], there exist positive constants $c_\ast$ and $c^\ast$, such that, for all $w_h \in {Z_h}$, it holds $$\label{eq:discr_norm_equiv}
c_\ast \lVert w_h \rVert_{0,h} \le \lVert w_h \rVert_{0,\Omega} \le c^\ast \lVert w_h \rVert_{0,h}.$$ Reshaping , and employing and , then gives $$\label{eq:ev_eq_proof_2}
\begin{split}
&\lVert {\vartheta^n}\rVert^2_{0,h} + \tau D_\ast \left| {\vartheta^n}\right|^2_{1,{\mathcal T_h}} \\
&\le {\mathcal{M}_h}({\vartheta^{n-1}},{\vartheta^n})
+ \tau \lVert {\vartheta^n}\rVert_{0,h} \left[c^\ast \omega^n_3 + \lVert {\vartheta^{n-1}}\rVert_{0,h} (c^\ast)^2 \omega^n_1 \right]
+ \tau |{\vartheta^n}|_{1,{\mathcal T_h}} \left[ \omega^n_4 + \lVert {\vartheta^{n-1}}\rVert_{0,h} \, c^\ast \omega^n_2 \right]\\
&=:T_1+T_2+T_3
\end{split}$$ The terms $T_1$ and $T_2$ are bounded as follows: $$\label{eq:T1_T2}
\begin{split}
T_1+T_2 &\le
\lVert {\vartheta^n}\rVert_{0,h} \left[ (1+\tau \eta) \lVert {\vartheta^{n-1}}\rVert_{0,h} + \tau c^\ast \omega^n_3 \right] \\
&\le \frac{1}{2} \left( \lVert {\vartheta^n}\rVert^2_{0,h} + \left[ (1+\tau \eta) \lVert {\vartheta^{n-1}}\rVert_{0,h} + \tau c^\ast \omega^n_3 \right]^2 \right),
\end{split}$$ where we used and in the first step. The term $T_3$ is bounded as follows $$\label{eq:T3}
\begin{split}
T_3 &\le \tau D_\ast |{\vartheta^n}|^2_{1,{\mathcal T_h}} + \frac{\tau}{4 D_\ast} \left[ \omega^n_4 + \lVert {\vartheta^{n-1}}\rVert_{0,h} \, c^\ast \omega^n_2 \right]^2 \\
&\le \tau D_\ast |{\vartheta^n}|^2_{1,{\mathcal T_h}} + \frac{\tau}{2} \eta \left[ (\omega^n_4)^2 + \lVert {\vartheta^{n-1}}\rVert^2_{0,h} \right]^2.
\end{split}$$ Next, we plug and into , cancel the terms $\tau D_\ast |{\vartheta^n}|^2_{1,{\mathcal T_h}}$ and manipulate the resulting inequality, to obtain $$\begin{split}
\lVert {\vartheta^n}\rVert^2_{0,h} \le \left[ (1+\tau \eta) \lVert {\vartheta^{n-1}}\rVert_{0,h} + \tau c^\ast \omega^n_3 \right]^2 + \tau \eta \left[ (\omega^n_4)^2 + \lVert {\vartheta^{n-1}}\rVert^2_{0,h} \right]^2.
\end{split}$$ Moreover, we estimate $$\begin{split}
&\left[ (1+\tau \eta) \lVert {\vartheta^{n-1}}\rVert_{0,h} + \tau c^\ast \omega^n_3 \right]^2 \\
&\quad=(1+\tau \eta)^2 \lVert {\vartheta^{n-1}}\rVert^2_{0,h}
+2 \tau^{\frac{1}{2}} \lVert {\vartheta^{n-1}}\rVert_{0,h} \tau^{\frac{1}{2}} (1+\tau \eta) c^\ast \omega^n_3 + \tau^2 (c^\ast)^2 (\omega^n_3)^2 \\
&\quad \le \left[ (1+\tau \eta)^2 + \tau \right] \lVert {\vartheta^{n-1}}\rVert^2_{0,h} + \left[ \tau(1+\tau \eta)^2 + \tau^2 \right] (c^\ast)^2 (\omega^n_3)^2 \\
&\quad \le \left( 1+\tau \eta \right) \lVert {\vartheta^{n-1}}\rVert^2_{0,h} + \tau \eta (\omega^n_3)^2.
\end{split}$$ Hence, $$\lVert {\vartheta^n}\rVert^2_{0,h} \le (1+\tau \eta) \lVert {\vartheta^{n-1}}\rVert^2_{0,h} + \tau \eta \left[ (\omega^n_3)^2 + (\omega^n_4)^2 \right].$$ Defining $$\gamma^n:=(\omega^n_3)^2 + (\omega^n_4)^2$$ and solving the recursion then leads to $$\begin{split}
\lVert {\vartheta^n}\rVert^2_{0,h} \le (1+\tau \eta)^n \lVert {\vartheta^0}\rVert^2_{0,h} + \tau \eta \sum_{j=1}^n \gamma^j
\le \eta \lVert {\vartheta^0}\rVert^2_{0,h} + \tau \eta \sum_{j=1}^n \gamma^j,
\end{split}$$ where we recall that $n \le T/\tau$ with $T$ the final time instant. With the estimate in the $L^2$ norm is a direct consequence: $$\label{eq:thetan_est1}
\begin{split}
\lVert {\vartheta^n}\rVert^2_{0,\Omega} \le \eta \lVert {\vartheta^0}\rVert^2_{0,\Omega} + \tau \eta \sum_{j=1}^n \gamma^j.
\end{split}$$ The initial term $\lVert {\vartheta^0}\rVert_{0,\Omega}^2$ is estimated by $$\label{eq:L3}
\lVert {\vartheta^0}\rVert_{0,\Omega} = \lVert {c_{0,h}}- {\mathcal{P}_c }c^0 \rVert_{0,\Omega} \le \lVert {c_{0,h}}- c^0 \rVert_{0,\Omega} + \lVert c^0 - {\mathcal{P}_c }c^0 \rVert_{0,\Omega} \le \lVert {c_{0,h}}- c^0 \rVert_{0,\Omega} + h^{k+2} \, \xi_0^0,$$ where we applied Lemma \[lem:cn\_Pcn\]. Moreover, using , the fact that $\sum_{j=1}^n \tau \le T$, and the definitions of $R_1^{j,2}$ and $R_1^{j,3}$ in , after some simple manipulations, we obtain $$\label{eq:thetan_est2}
\begin{split}
\tau \eta \sum_{j=1}^n \gamma_j &\le \eta \left( \sum_{j=1}^n \tau (\omega^j_3)^2 + \sum_{j=1}^n \tau (\omega_4^j)^2 \right) \\
&\le \eta \left[ \sum_{j=1}^n \tau (\tau+h^{k+1})^2 + (h^{k+2})^2 \sum_{j=1}^n (R_1^{j,2})^2 + \tau^2 \sum_{j=1}^{n} (R_1^{j,3})^2 \right] \\
&\le \eta \left[ (\tau+h^{k+1})^2 + (h^{k+2})^2 \sum_{j=1}^n (R_1^{j,2})^2 + \tau^2 \sum_{j=1}^{n} (R_1^{j,3})^2 \right] \\
&\le \eta \left[ (\tau+h^{k+1})^2 + (h^{k+2})^2 \int_{0}^{{t_n}} \xi_{0,t}^2 \, {\textup{d}s}+ \tau^2 \int_{0}^{{t_n}} \left\lVert \frac{\partial^2 c}{\partial s^2}(s)\right\rVert^2_{0,\Omega} \, {\textup{d}s}\right].
\end{split}$$ The assertion of the theorem follows by combining with and .
Numerical experiments {#sec:5}
=====================
In this section, we demonstrate the performance of the method on the basis of numerical experiments, focusing on the lowest order case $k=0$. To this purpose, we first consider an ideal test case (*Example 1*), and then a more realistic one (*Example 2*). The aim of the first test is to validate (also numerically) the convergence of the method on a problem with regular known solution, whereas those of the second test is to check the method’s performance on a well-known benchmark that mimics a more realistic situation.
*Example 1:* Here, we study a generalized version of , given by $$\label{eq:generalized_problem}
\left\{
\begin{alignedat}{2}
\phi \, \frac{\partial c}{\partial t} + {\boldsymbol{u}}\cdot \nabla c - {\operatorname{\rm div}}(D({\boldsymbol{u}}) \nabla c) &= f \\
{\operatorname{\rm div}}\, {\boldsymbol{u}}&= g \\
{\boldsymbol{u}}&= -a(c) (\nabla p - {\boldsymbol{\gamma}}(c)),
\end{alignedat}
\right.$$ endowed with the boundary and initial conditions in and , respectively. We fix $\Omega=(0,1)^2$ and pick the same choice of parameters as in [@miscibledispl2018hu], namely $T=0.01$, $\phi=1$, $D({\boldsymbol{u}})=|{\boldsymbol{u}}|+0.02$, $d_m=0.02$, $d_{\ell}=d_t=1$, $c_0=0$, ${\boldsymbol{\gamma}}(c)=0$, and $a(c)=(c+2)^{-1}$, where $f$ and $g$ are taken in accordance with the analytical solutions $$\label{eq:Hu1_fcts}
\begin{split}
c(x,y,t)&=t^2 \left[x^2(x-1)^2 + y^2(y-1)^2\right] \\
{\boldsymbol{u}}(x,y,t)&=2t^2 \begin{pmatrix} x(x-1)(2x-1) \\ y(y-1)(2y-1) \end{pmatrix} \\
p(x,y,t)&=-\frac{1}{2} c^2 -2c + \frac{17}{6300} t^4 + \frac{2}{15}t^2.
\end{split}$$ Plots of the exact solution at the final time $T$ are shown in Figures \[fig:Hu1\_c\_p\] and \[fig:Hu1\_u\].
![Exact concentration $c$ (left) and pressure $p$ (right) of example 1, given by , at the final time $T=0.01$.[]{data-label="fig:Hu1_c_p"}](Hu1_c.png){width="\textwidth"}
![Exact concentration $c$ (left) and pressure $p$ (right) of example 1, given by , at the final time $T=0.01$.[]{data-label="fig:Hu1_c_p"}](Hu1_p.png){width="\textwidth"}
![Exact vector field ${\boldsymbol{u}}$ of example 1, given by , at the final time $T=0.01$.[]{data-label="fig:Hu1_u"}](Hu1_u.png){width="60.00000%"}
We employ a sequence of regular Cartesian meshes and Voronoi meshes, as portrayed in Figure \[fig:meshes\]. In addition to the current version, we also test the method when replacing the stabilization terms in , , and by alternative ones: $$\begin{split}
{\nu_{{\mathcal{M}}}^{K}}(\phi) {S^{K}_{{\mathcal{M}}}}\left((I-{\Pi^{0,{K}}_{k+1}}) {c_h}, (I-{\Pi^{0,{K}}_{k+1}}) {z_h}\right) \quad &\rightsquigarrow \quad \widetilde{{S^{K}_{{\mathcal{M}}}}}\left((I-{\Pi^{0,{K}}_{k+1}}) {c_h}, (I-{\Pi^{0,{K}}_{k+1}}) {z_h}\right) \\
{\nu_{D}^{K}}({\boldsymbol{u}_h}) \, {S^{K}_{D}}\left((I-{\Pi^{\nabla,{K}}_{k+1}}) {c_h},(I-{\Pi^{\nabla,{K}}_{k+1}}) {z_h}) \right) \quad &\rightsquigarrow \quad \widetilde{{S^{K}_{D}}}\left({\boldsymbol{u}_h};(I-{\Pi^{\nabla,{K}}_{k+1}}) {c_h},(I-{\Pi^{\nabla,{K}}_{k+1}}) {z_h}) \right)\\
{\nu_{{\mathcal{A}}}^{K}}({c_h}) \, {S^{K}_{{\mathcal{A}}}}((I-{\boldsymbol{\Pi^{0,{K}}_{k}}}){\boldsymbol{u}_h},(I-{\boldsymbol{\Pi^{0,{K}}_{k}}}){\boldsymbol{v}_h}) \quad &\rightsquigarrow \quad \widetilde{{S^{K}_{{\mathcal{A}}}}}({c_h};(I-{\boldsymbol{\Pi^{0,{K}}_{k}}}){\boldsymbol{u}_h},(I-{\boldsymbol{\Pi^{0,{K}}_{k}}}){\boldsymbol{v}_h}).
\end{split}$$ The alternative (diagonal) stabilizations are given by $$\label{eq:stabs_new}
\begin{split}
\widetilde{{S^{K}_{{\mathcal{M}}}}}({c_h},{z_h})&=|{K}| \sum_{j=1}^{{\textrm{dim}{Z_h(K)}}} d^{\mathcal{M}}_j \, {{\textup{dof}}^{Z_h(K)}}_j({c_h}) \, {{\textup{dof}}^{Z_h(K)}}_j({z_h}) \\
\widetilde{{S^{K}_{D}}}({c_h},{z_h})&=\sum_{j=1}^{{\textrm{dim}{Z_h(K)}}} d^{D}_j \, {{\textup{dof}}^{Z_h(K)}}_j({c_h}) \, {{\textup{dof}}^{Z_h(K)}}_j({z_h}) \\
\widetilde{{S^{K}_{{\mathcal{A}}}}}({\boldsymbol{u}_h},{\boldsymbol{v}_h})&=|{K}|\sum_{j=1}^{{{\textrm{dim}{V}_h(K)}}} d^{\mathcal{A}}_j \, {{\textup{dof}}^{\boldsymbol{V}_h(K)}}_j({\boldsymbol{u}_h}) \, {{\textup{dof}}^{\boldsymbol{V}_h(K)}}_j({\boldsymbol{v}_h}),
\end{split}$$ with $$\label{eq:stabs_new_coeff}
\begin{split}
d^{\mathcal{M}}_j \,&:=\max\left\{ \frac{1}{ |{K}|} \,\int_{K}\phi \, ({\Pi^{0,{K}}_{k+1}}{\varphi_j^{K}})^2 \, {\textup{d}x},\, \sigma {\nu_{{\mathcal{M}}}^{K}}(\phi) \right\} \\
d^{D}_j \,&:=\max\left\{\int_{K}D({\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{u}_h}) \, |{\boldsymbol{\Pi^{0,{K}}_{k}}}(\nabla {\varphi_j^{K}})|^2 \, {\textup{d}x}, \, \sigma {\nu_{D}^{K}}({\boldsymbol{u}_h}) \right\} \\
d^{\mathcal{A}}_j \,&:=\max\left\{\frac{1}{ |{K}|} \, \int_{K}A({\Pi^{0,{K}}_{k+1}}{c_h}) \, |{\boldsymbol{\Pi^{0,{K}}_{k}}}{\boldsymbol{\psi}_j^{K}}|^2 \, {\textup{d}x}, \, \sigma {\nu_{{\mathcal{A}}}^{K}}({c_h}) \right\},
\end{split}$$ where $\{{\varphi_j^{K}}\}_{\ell=1}^{\textrm{dim}{Z_h({K})}}$ and $\{{\boldsymbol{\psi}_j^{K}}\}_{\ell=1}^{{{\textrm{dim}{V}_h(K)}}}$ denote the local canonical basis functions for ${Z_h({K})}$ and ${\boldsymbol{V}_h({K})}$, and $\sigma>0$ is a safety parameter. In the forthcoming experiments, we set $\sigma=1e-3$. We highlight that these stabilizations are in fact modifications of the so-called *D-recipe*, which was introduced in [@VEM3Dbasic] and has already been successfully applied in some variants to other model problems, such as the Helmholtz problem [@TVEM_Helmholtz_num]. The first entry inside the max is simply the “diagonal part” of the consistency term of the local approximate forms in , , and , respectively, whereas the second terms correspond to the original stabilizations associated to the degrees of freedom in multiplied by $\sigma$, which acts as a positivity safeguard. Importantly, it is easy to check that the error analysis can be easily extended to the new choice of stabilizations.
![Meshes: regular 8x8 Cartesian mesh (left); Voronoi mesh with 64 elements (right).[]{data-label="fig:meshes"}](quadr_mesh.pdf){width="75.00000%"}
![Meshes: regular 8x8 Cartesian mesh (left); Voronoi mesh with 64 elements (right).[]{data-label="fig:meshes"}](voro_mesh.pdf){width="75.00000%"}
Due to the virtuality of the basis functions, we measure the following relative $L^2$ errors: $$\frac{\lVert c-\Pi^0_1 {C^n}\rVert_{0,\Omega}}{\lVert c \rVert_{0,\Omega}}, \quad
\frac{\lVert {\boldsymbol{u}}-\boldsymbol{\Pi}^0_0 {\boldsymbol{U}^n}\rVert_{0,\Omega}}{\lVert {\boldsymbol{U}^n}\rVert_{0,\Omega}}, \quad
\frac{\lVert p-\Pi^0_0 {P^n}\rVert_{0,\Omega}}{\lVert p \rVert_{0,\Omega}},$$ where ${C^n}$, ${\boldsymbol{U}^n}$, and ${P^n}$ are the numerical solutions at the final time $T$.
The relative $L^2$ discretization errors for the concentration are plotted in Figure \[fig:Ex1:c\] in terms of the mesh size $h$ for both families of meshes and both variants of stabilizations. In order to better underline the expected linear convergence of the method both in $h$ and $\tau$ (see Theorem \[thm:Cn\_cn\], recalling that $k=0$), the time step $\tau$ is chosen proportional to $h$. In other words, starting with the coarsest mesh and $\tau=T/5$, each subsequent case is obtained by dividing both $h$ (adopting a finer mesh) and $\tau$ by a factor of 2. Analogous plots are shown for the velocity and pressure variable errors in Figures \[fig:Ex1:u\] and \[fig:Ex1:p\]. In all cases, the linear convergence rates are in accordance with Theorem \[thm:Un\_un\_Pn\_pn\] and Theorem \[thm:Cn\_cn\]. For the pressure discretization error, since the initial meshes are very coarse, we observe some pre-asymptotic regime when employing the original stabilizations in . This effect, however, is not present for the alternative stabilizations in . Both variants lead to similar results for the concentration and velocity errors.
![Relative $L^2$ errors for the concentration in example 1 at the final time $T$ on regular Cartesian meshes (left) and Voronoi meshes (right). The original stabilization and the D-recipe stabilization are employed.[]{data-label="fig:Ex1:c"}](c_Cart.pdf){width="\textwidth"}
![Relative $L^2$ errors for the concentration in example 1 at the final time $T$ on regular Cartesian meshes (left) and Voronoi meshes (right). The original stabilization and the D-recipe stabilization are employed.[]{data-label="fig:Ex1:c"}](c_Voro.pdf){width="\textwidth"}
![Relative $L^2$ errors for the velocity field in example 1 at the final time $T$ on regular Cartesian meshes (left) and Voronoi meshes (right). The original stabilization and the D-recipe stabilization are employed.[]{data-label="fig:Ex1:u"}](u_Cart.pdf){width="\textwidth"}
![Relative $L^2$ errors for the velocity field in example 1 at the final time $T$ on regular Cartesian meshes (left) and Voronoi meshes (right). The original stabilization and the D-recipe stabilization are employed.[]{data-label="fig:Ex1:u"}](u_Voro.pdf){width="\textwidth"}
![Relative $L^2$ errors for the pressure in example 1 at the final time $T$ on regular Cartesian meshes (left) and Voronoi meshes (right). The original stabilization and the D-recipe stabilization are employed.[]{data-label="fig:Ex1:p"}](p_Cart.pdf){width="\textwidth"}
![Relative $L^2$ errors for the pressure in example 1 at the final time $T$ on regular Cartesian meshes (left) and Voronoi meshes (right). The original stabilization and the D-recipe stabilization are employed.[]{data-label="fig:Ex1:p"}](p_Voro.pdf){width="\textwidth"}
Since the concentration often evolves more rapidly than the velocity and pressure, it could be worth to consider a cheaper variant of the discrete scheme - where the discrete velocity-pressure pair is updated only every R time steps (with $R \in {\mathbb N}$). This leads to a smaller number of linear system resolutions (possibly with a small reduction in accuracy) since only system is solved at every time step, while is solved only every R steps. In order to test this, we tried to run the same test above and compare the original version with the cheaper version with $R=5$. The difference in error was only at the fourth meaningful digit; we do not plot the graphs since these would completely overlap the ones of the original method.
*Example 2:* Next, we investigate the behavior of the method for Test 1 and Test 2 in [@wang2000approximation; @chainais2007convergence].
The problem is given in the form with boundary conditions and initial condition over the spatial domain $\Omega=(0,1000)^2$ ft$^2$. Moreover, $T=3600$ days and $\tau=36$ days. At the upper right corner, i.e. at $[1000,1000]$, fluid with concentration ${\widehat{c}}=1.0$ is injected with rate ${q^+}=30$ ft$^2$/day, whereas at the lower left corner, i.e. at $[0,0]$, material is absorbed with rate ${q^-}=30$ ft$^2$/day. Both wells are henceforth treated as Dirac masses, which is admissible at the discrete level since the discrete functions are piecewise regular (which can be interpreted as an approximation of the Dirac delta by a localized function with support within the corner element and unitary integral). Furthermore, the following choices for the parameters are picked: $\phi=0.1$, $d_{\ell}=50$, $d_t=5$, $c_0=0$, ${\boldsymbol{\gamma}}(c)=0$, and $a(c)=80(1+(M^{\frac{1}{4}}-1)c)^4$, where $$\text{Test A}: d_m=10, M=1; \qquad\qquad
\text{Test B}: d_m=0, M=41.$$
Whereas $a(c)$ is constant for Test A, it changes rapidly across the fluid interface for Test B (which is in fact not covered by the theoretical analysis since $d_m=0$, but is interesting to study numerically) resulting in a much faster propagation of the fluid concentration front along the diagonal direction ($d_\ell \gg d_t$). This effect is known as *macroscopic fingering phenomenon*[@ewing1983mathematics].
For this example, we used a regular 25x25 Cartesian mesh and we employed the more sophisticated stabilization in . Since Test B is highly convection-dominated, pure application of our method leads to local disturbances in the form of *overshoots* and *undershoots* of the numerical solution for the concentration, typical in the context of convection-dominated problem. To this purpose, for this test case, we employ the flux-corrected transport (FCT) algorithm with linearization [@FCT1; @FCT]. The FCT scheme with linearization for convection-dominated flow problems operates in two steps: (1) advance the solution in time by a low-order overly diffusive scheme to suppress spurious oscillations, (2) correct the solution using (linear) antidiffusive fluxes. In that way the computed solution does not show spurious oscillations and layers are not smeared.
Due to the fact that no analytical solutions are available for Test A and Test B, we plot the numerical solutions (and the corresponding contour plots) for the concentration after 3 and 10 years. These times correspond to $n=30$ and $n=100$, respectively. For visualization of the results, since the numerical solution is virtual but the nodal values are known, we simply add, inside each square, the barycenter with associated mean value of the nodal values, then create a triangulation based upon these points, and finally interpolate the function values linearly inside each triangle. In Figures \[fig:Ex2\_testA\_3\] and \[fig:Ex2\_testA\_10\], the results for Test A are portrayed, and in Figures \[fig:Ex2\_testB\_3\] and \[fig:Ex2\_testB\_10\], those for Test B. The results are similar to those obtained in [@wang2000approximation; @chainais2007convergence].
![Numerical solution for the concentration (left) and contour plot (right) after 3 years in Test A.[]{data-label="fig:Ex2_testA_3"}](testA_misc1_three.png){width="\textwidth"}
![Numerical solution for the concentration (left) and contour plot (right) after 3 years in Test A.[]{data-label="fig:Ex2_testA_3"}](testA_misc2_three.png){width="\textwidth"}
![Numerical solution for the concentration (left) and contour plot (right) after 10 years in Test A.[]{data-label="fig:Ex2_testA_10"}](testA_misc1_final.png){width="\textwidth"}
![Numerical solution for the concentration (left) and contour plot (right) after 10 years in Test A.[]{data-label="fig:Ex2_testA_10"}](testA_misc2_final.png){width="\textwidth"}
![Numerical solution for the concentration (left) and contour plot (right) after 3 years in Test B.[]{data-label="fig:Ex2_testB_3"}](testB_misc1_three.png){width="\textwidth"}
![Numerical solution for the concentration (left) and contour plot (right) after 3 years in Test B.[]{data-label="fig:Ex2_testB_3"}](testB_misc2_three.png){width="\textwidth"}
![Numerical solution for the concentration (left) and contour plot (right) after 10 years in Test B.[]{data-label="fig:Ex2_testB_10"}](testB_misc1_final.png){width="\textwidth"}
![Numerical solution for the concentration (left) and contour plot (right) after 10 years in Test B.[]{data-label="fig:Ex2_testB_10"}](testB_misc2_final.png){width="\textwidth"}
Acknowledgements {#acknowledgements .unnumbered}
================
The first (L.B.d.V.) and last (G.V.) authors where partially supported by the European Research Council through the H2020 Consolidator Grant (grant no. 681162) CAVE, Challenges and Advancements in Virtual Elements. This support is gratefully acknowledged. The second author (A.P.) has been funded by the Austrian Science Fund (FWF) through the project P 29197-N32, and by the Doctoral Program (DK) through the FWF Project W1245.
[^1]: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, 20125 Milano-Bicocca, Italy ([email protected], [email protected])
[^2]: Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria ([email protected])
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper studies the coherent acceleration of ions interacting with two electrostatic waves in a uniform magnetic field ${\vec{B}}_0$. It generalizes an earlier analysis of waves propagating perpendicularly to ${\vec{B}}_0$ to include the effect of wavenumbers along ${\vec{B}}_0$. The Lie transformation technique is used to develop a perturbation theory describing the ion motion, and results are compared with numerical solutions of the complete equations of motion. Coherent energization occurs when the Doppler-shifted wave frequencies differ by nearly an integer multiple of the ion cyclotron frequency. When the difference in the parallel wavenumbers of the two waves is increased the coherent energization of ions is limited to a small part of the phase space. The energization of ions and its dependence on wave parameters is discussed.'
author:
- 'D. J. Strozzi'
- 'A. K. Ram'
- 'A. Bers'
bibliography:
- '035307PHP.bib'
title: Coherent acceleration of magnetized ions by electrostatic waves with arbitrary wavenumbers
---
Introduction
============
The motion of charged particles in the presence of electromagnetic waves is a rich dynamical system that has been studied for a variety of cases. Important physical applications of this problem occur in laboratory and space plasmas, such as for high-temperature (collisionless) plasma heating and current drive and the transverse energization of ions for times short compared to collisional times. A particular area of interest is the nonlinear heating of ions (as opposed to linear mechanisms such as Landau and cyclotron damping) by electrostatic waves propagating through a plasma in a uniform magnetic field ${\vec{B}}_0$.
For a single electrostatic wave propagating across ${\vec{B}}_0$, the stochastic heating of ions by waves with frequency ${\omega}\gg{\omega}_{ci}$ but ${\omega}\neq N{\omega}_{ci}$ (where $N$ is an integer and ${\omega}_{ci}\equiv qB_0/M$ is the ion cyclotron frequency) was studied by Karney and Bers [@karnbers; @karney1]. It was found that ions with speeds across ${\vec{B}}_0$ less than the phase velocity of the wave ${\omega}/k_\perp$ (that is, $k_\perp\rho_i\gtrsim{\omega}/{\omega}_{ci}$) exhibit regular motion and do not gain energy. However, for wave amplitudes above a threshold amplitude, the ions are stochastically heated if their speeds are inside a region with a lower bound near ${\omega}/k_\perp$. The stochastic “webs” generated by a single perpendicularly propagating wave with frequency ${\omega}=N{\omega}_{ci}$ also lead to stochastic ion heating [@zaslavsky_weakchaos; @fukuyama; @chia1].
For a single wave propagating obliquely to ${\vec{B}}_0$ it was found that ions could also be stochastically heated [@smithpof; @smithprl]. It has recently been shown that single and multiple drift-Alfvén waves with ${\omega}<{\omega}_{ci}$ can induce stochastic ion heating [@chenliu], which may account for certain experimental observations [@mcchesney]. For two waves propagating obliquely to ${\vec{B}}_0$, the threshold wave amplitudes needed for stochastic motion can be significantly lowered [@benkadda]. There is still a lower bound for the stochastic region of phase space.
For two perpendicular waves that satisfy the resonance condition ${\omega}_1-{\omega}_2=N{\omega}_{ci}$ Ram [*et al.* ]{}discovered numerically [@ram_agu_97] that coherent (as opposed to chaotic) energization can bring ions from low energies into the stochastic domain. Bénisti [*et al.* ]{}[@benisti1] then showed that this coherent energization was described by perturbation analysis using Lie transformation methods. The coherent energization was also shown by Ram [*et al.* ]{}[@abhayjgr] to be described by a multiple time scale analysis, and invoked to explain the energization of hydrogen and oxygen ions from Earth’s upper auroral ionosphere into the magnetosphere. For two non-collinear, perpendicularly propagating waves, the coherent energization was found to persist as long as the angle between them was less than $30^\circ$ [@ram_agu_98].
Coherent acceleration by electrostatic waves with ${\omega}_1-{\omega}_2=N{\omega}_{ci}$ can only occur when both wave frequencies are larger than ${\omega}_{ci}$. This process is most interesting for cases where ions with energy well below the stochastic region ($k_\perp\rho_i\ll{\omega}/{\omega}_{ci}$) are accelerated into it. Most of the work on coherent acceleration has focused on waves with frequencies much higher than ${\omega}_{ci}$. In magnetic fusion experiments and in the Earth’s ionosphere, lower-hybrid waves fit this description (${\omega}_{lh}\sim{\omega}_{pi}\gg{\omega}_{ci}$, ${\omega}_{lh}=$ lower-hybrid frequency, ${\omega}_{pi}=$ ion plasma frequency).
In this paper we study the interaction of ions with electrostatic waves ranging in frequency from lower-hybrid frequencies down to a few multiples of ${\omega}_{ci}$. The analysis of Bénisti [*et al.* ]{}[@benisti1] is generalized to include nonzero wavenumbers along ${\vec{B}}_0$. We develop a perturbation theory using the Lie transformation method and find conditions for which coherent acceleration persists. We also discuss the dependence of the range of energization and period of coherent oscillations on wave parameters.
The Hamiltonian formulation of the problem is given in Section II. An analytic perturbation theory for the coherent motion based on the Lie transformation technique is described in Section III. Section IV compares the results for the perturbation theory with numerical results obtained from the complete dynamical equations. The scalings of coherent energization and the period of oscillation, for perpendicularly propagating waves, are obtained. Section V discusses the case of obliquely propagating waves and compares the results with those for two perpendicularly propagating waves.
Equations of Motion
===================
The nonrelativistic equation of motion of an ion in the presence of a uniform magnetic field ${\vec{B}}_0=B_0{\hat{z}}$ in a plasma and interacting with two electrostatic waves is $$M\frac{d^2{\vec{x}}}{dt^2}=q\sum_{i=1}^2 \Phi_i{\vec{k}}_i\sin({\vec{k}}_i\cdot{\vec{x}}-{\omega}_it+{\alpha}_i) + q{\vec{v}}\times{\vec{B}}_0$$ where $\Phi_i$ is the electrostatic potential amplitude, ${\vec{k}}_i$ is the wavevector, ${\omega}_i$ is the wave frequency, and ${\alpha}_i$ is the phase of the $i^\textrm{th}$ wave. We normalize times to the inverse of the ion cyclotron frequency ${\omega}_{ci}$, distances to the inverse of $k_{1x}$, and masses to the ion mass $M$. We restrict our attention to the case where both ${\vec{k}}_i$’s lie in the $x-z$ plane. Let $\nu_i\equiv{\omega}_i/{\omega}_{ci}$ and ${\epsilon}_i\equiv({\omega}_{Bi}/{\omega}_{ci})^2$, where ${\omega}_{Bi}\equiv (qk_{1x}^2\Phi_i/M)^{1/2}$ is the bounce frequency in the $i^\textrm{th}$ wave. The Hamiltonian for this system is $$h(\vec{x},\vec{p},t) = {\tfrac{1}{2}}(\vec{p}-\vec{A})^2 + \sum_i{\epsilon}_i\cos(\vec{k}_i\cdot\vec{x} - \nu_i t + {\alpha}_i)$$ where $\vec{A}=B_0x{\hat{y}}$ is the vector potential, and ${\vec{p}}=m{\vec{v}}+q{\vec{A}}{\rightarrow}{\vec{v}}+x{\hat{y}}$ is the (nondimensional) canonical momentum.
Since $h$ is independent of $y$, we can eliminate the $y$ degree of freedom by making a Galilean transformation to a frame moving in the ${\hat{y}}$ direction with speed $p_{y0}=v_{y0}+x_0$ (the subscript 0 refers to a quantity’s initial value). Following Ref. [@Goldstein], the generating function for the canonical transformation from $(y,p_y)$ to $(y',p_y')$ is $F_2=(y-p_{y0}t)(p_y'+p_{y0})$. Then $y'=y-p_{y0}t$ and $p_y'=p_y-p_{y0}$, so that $p_{y0}'=0$. The transformed Hamiltonian (to within a constant) is $$h(x,z,p_x,p_y',p_z,t)={\tfrac{1}{2}}{\left[}p_x^2+p_y'^2+(x-p_{y0})^2+p_z^2{\right]}+ \sum_i{\epsilon}_i\cos(\vec{k}_i\cdot\vec{x} - \nu_i t + {\alpha}_i)$$ Since ${\partial}h'/{\partial}y'=0$, $p_y'$ is independent of time so that $p_y'=0$. This eliminates the $y'$ degree of freedom from the dynamics. Replacing $x$ by $x'=x-p_{y0}$ and $p_z$ by $v_z$ gives $$h'(x',z,p_x,v_z,t)={\tfrac{1}{2}}(p_x^2+x'^2+v_z^2) + \sum_i{\epsilon}_i\cos(k_{ix}x' + k_{iz}z - \nu_it + {\alpha}_i)$$ where ${\alpha}_i+k_{ix}p_{y0}$ is replaced by ${\alpha}_i$.
In a frame moving with velocity $u{\hat{z}}$ the Hamiltonian remains unchanged except that the wave frequencies are Doppler shifted: $\nu_i{\rightarrow}\nu_i-k_{iz}u$. Without loss of generality we assume that $v_{z0}=0$ and consider the effects of $v_{z0}$ on oblique propagation in Section \[sec:oblique\].
We transform $(x',p_x)$ to action-angle coordinates $(\phi,I)$ using the generating function $F_1={\frac{1}{2}}x'^2\cot\phi$. Then $I={\frac{1}{2}}(v_x^2+x'^2)={\frac{1}{2}}(v_x^2+v_y^2)$ is the perpendicular kinetic energy, and $\phi=\arctan(x'/v_x)=\arctan(-v_y/v_x)$ is the gyrophase. The transformed Hamiltonian is $$\label{eqn:fullham}
H(\phi,z,I,v_z,t) = I+{\tfrac{1}{2}}v_z^2+\sum_i{\epsilon}_i\cos(k_{ix}\rho\sin\phi+k_{iz}z-\nu_it+{\alpha}_i)$$ where $\rho=\sqrt{2I}$ is the ion gyroradius.
Perturbation Analysis of Coherent Motion {#sec:pert}
========================================
In general, the equations of motion obtained from (\[eqn:fullham\]) cannot be solved analytically. Consequently, we resort to numerical solutions to provide an insight into the dynamics of ions in two electrostatic waves. Figure \[fig:rvst\_perp\_full\] shows the time evolution of $\rho$ for three ions having the same initial $\rho_0$, but different $\phi_0$, interacting with two waves of frequencies $\nu_1=40.37$ and $\nu_2=39.37$, and amplitudes ${\epsilon}_1={\epsilon}_2=4$. (All the numerical solutions of ordinary differential equations have been carried out using the Bulirsch-Stoer algorithm described in Ref. [@numrec].) There are two distinct kinds of motion: the slow, smooth, “coherent” oscillations at lower $\rho$, and the irregular, “stochastic” motion at higher $\rho$. Superimposed on the coherent motion are small-amplitude, high-frequency fluctuations. Figure \[fig:rvst\_perp\_off\] shows the orbits for the same parameters as Fig. \[fig:rvst\_perp\_full\] except that $\nu_2=39.369$ and the initial conditions are different. This demonstrates that the coherent acceleration from low to high energies occurs only when $\nu_1-\nu_2$ is an integer.
Our interest is to provide an analytical description of the coherent dynamics without going into details of the stochastic region, other than to note its existence for $\rho \approx\min(\nu_i)$ [@karney1; @karnbers]. We assume that the waves are perturbing the cyclotron motion of the ions and express $$H=H_0+H_1$$ where $$\label{eqn:H0H1}
H_0=I+{\tfrac{1}{2}}v_z^2 \quad\mathrm{and}\quad H_1=\sum_i{\epsilon}_i\cos(k_{ix}\rho\sin\phi+k_{iz}z-\nu_it+{\alpha}_i).$$ An approximate analytical description of the ion motion in the coherent regime is obtained by using the Lie perturbation technique [@cary; @lichtlieb] with the ordering parameter ${\epsilon}\ ({\epsilon}\sim{\epsilon}_1\sim{\epsilon}_2$). We assume that $\nu_i\notin\mathcal{Z}$ but $(\nu_1-\nu_2)=N\in\mathcal{Z}$. For $\nu_i\in\mathcal{Z}$ a web structure is formed in phase space and has been discussed elsewhere for a single wave [@fukuyama] and for two waves propagating across ${\vec{B}}_0$ [@benisti2]. For the case of a single wave the stochastic web structure also has a lower bound [@benisti_pla].
From the Lie perturbation analysis (Appendix \[app:lie\]) the Hamiltonian that describes the coherent ion motion to $O({\epsilon}^2)$ is $${{\bar{H}}}({{\bar{\phi}}},{{\bar{z}}},{{\bar{I}}},{{\bar{v}}}_z,t) = {{\bar{I}}}+{\tfrac{1}{2}}{{\bar{v}}}_z^2+{{\bar{H}}}_2$$ where $$\begin{aligned}
{{\bar{H}}}_2 &=& S_0({{\bar{I}}},{{\bar{v}}}_z)+S_-({{\bar{I}}},{{\bar{v}}}_z)\cos(N({{\bar{\phi}}}-t)+{\Delta}k_z{{\bar{z}}}+{\alpha}_1-{\alpha}_2)
\\S_0 &=& S_{0x} + S_{0z} \label{eqn:S0}
\\S_{0x} &=&-\frac{1}{2{{\bar{\rho}}}}\sum_i k_{ix}{\epsilon}_i^2\frac{m}{m-\mu_i}J_{m,i}J_{m,i}'
\\S_{0z} &=& {\tfrac{1}{4}}\sum_i k_{iz}^2{\epsilon}_i^2 \frac{J_{m,i}^2}{(m-\mu_i)^2} \label{eqn:S0z}
\\S_- &=& S_{-x} + S_{-z} \label{eqn:Sm}
\\S_{-x} &=& -\frac{{\epsilon}_1{\epsilon}_2}{4{{\bar{\rho}}}(m-\mu_1)}(k_{1x}(m-N)J_{m,1}'J_{m-N,2}+k_{2x}mJ_{m,1}J_{m-N,2}')
\\ \nonumber && -\frac{{\epsilon}_1{\epsilon}_2}{4{{\bar{\rho}}}(m-\mu_2)}(k_{1x}mJ_{m,2}J_{m+N,2}'+k_{2x}(m+N)J_{m,2}'J_{m+N,1})
\\S_{-z} &=& {\tfrac{1}{4}}k_{1z}k_{2z}{\epsilon}_1{\epsilon}_2 {\left(}\frac{J_{m,1}J_{m-N,2}}{(m-\mu_1)^2} + \frac{J_{m,2}J_{m+N,1}}{(m-\mu_2)^2} {\right)}\label{eqn:Smz}\end{aligned}$$ ${\Delta}k_z=(k_{1z}-k_{2z}), \mu_i=\nu_i-k_{iz}{{\bar{v}}}_z,$ and $m$ is summed from $-\infty$ to $+\infty$. $J_{m,i}\equiv J_m(k_{ix}{{\bar{\rho}}})$ is the Bessel function of the first kind and $f'(\xi)=df/d\xi$. The barred coordinates are related to the original coordinates by a near-identity transformation: $({{\bar{\phi}}},{{\bar{z}}},{{\bar{I}}},{{\bar{v}}}_z)=(\phi,z,I,v_z)+O({\epsilon})$ (Appendix \[app:lie\]). For instance, the relation between $I$ and ${{\bar{I}}}$ is: $$\label{eqn:ItoIbar}
I \approx {{\bar{I}}}-{\epsilon}_i\sum_m\frac{mJ_{m,i}}{m-\mu_i}\cos(m{{\bar{\phi}}}+k_{iz}{{\bar{z}}}-\nu_it+{\alpha}_i)$$ The Hamiltonian ${{\bar{H}}}$ is a generalization to oblique waves of the results obtained in [@benisti1] for collinear perpendicularly propagating waves. In the limit $k_{iz}{\rightarrow}0$ the above reduces to the description in Ref. [@benisti1]. A nonzero ${\alpha}_1-{\alpha}_2$ is equivalent to a shift in the initial ${{\bar{\phi}}}_0$ so that, without loss of generality, we can set ${\alpha}_1={\alpha}_2=0$.
Our perturbation analysis assumes there are no resonances at $O({\epsilon})$. Such resonances occur if $\nu_i$ is an integer, where our present analysis breaks down.
The explicit time dependence in ${{\bar{H}}}$ can be eliminated by transforming from ${{\bar{\phi}}}$ to ${{\bar{\psi}}}={{\bar{\phi}}}-t$ using the generating function $F_2=\tilde{I}({{\bar{\phi}}}-t)$. The transformed Hamiltonian is: $$\label{eqn:Htil}
{\tilde{H}}({{\bar{\psi}}},{{\bar{z}}},{{\bar{I}}},{{\bar{v}}}_z) = {\tfrac{1}{2}}{{\bar{v}}}_z^2 + S_0({{\bar{I}}},{{\bar{v}}}_z) + S_-({{\bar{I}}},{{\bar{v}}}_z)\cos(N{{\bar{\psi}}}+{\Delta}k_z{{\bar{z}}})$$ where ${{\bar{I}}}={\tilde{I}}$ has replaced ${\tilde{I}}$ (${{\bar{I}}}$ and ${{\bar{\psi}}}$ are canonically conjugate). Since ${\tilde{H}}$ does not depend explicitly on time it is a constant of the motion.
Using Hamilton’s equations for $\dot{{{\bar{I}}}}$ and $\dot{{{\bar{v}}}}_z$, we find a second constant of the motion: $$\label{eqn:vzconst} \frac{d}{dt}{\left(}{{\bar{v}}}_z-\frac{{\Delta}k_z}{N}{{\bar{I}}}{\right)}=0$$ Thus, the system is integrable and the dynamics described by ${\tilde{H}}$ are not stochastic. Along an orbit, ${{\bar{v}}}_z$ is a function of ${{\bar{I}}}$ and initial conditions only: $$\label{eqn:vzlie}
{{\bar{v}}}_z = v_{z0}+\frac{{\Delta}k_z}{N}({{\bar{I}}}-I_0).$$ Therefore, $S_0$ and $S_-$ are functions just of ${{\bar{I}}}$. Since $|\cos{x}|\leq1$ $$\label{eqn:Hbound}
H_-\leq \tilde{H} \leq H_+, \qquad H_\pm({{\bar{I}}})={\tfrac{1}{2}}{{\bar{v}}}_z^2 + S_0({{\bar{I}}})\pm |S_-({{\bar{I}}})|$$ For an initial condition with a given value of ${\tilde{H}}$, ${{\bar{\rho}}}$ varies between the two points where ${\tilde{H}}$ equals $H_+({{\bar{\rho}}})$ or $H_-({{\bar{\rho}}})$. We refer to $H_\pm$ as the potential barriers, since they delimit the allowed and forbidden regions of phase space.
Figure \[fig:rvst\_perp\_coh\] shows the orbits generated by the second-order Hamiltonian (\[eqn:Htil\]) for the same parameters as in Fig. \[fig:rvst\_perp\_full\]. Our perturbation analysis accurately captures the coherent motion of the full system except near the stochastic region $\rho\approx\min(\nu_i)$ where our perturbation theory breaks down. Below this region, $\rho$ and ${{\bar{\rho}}}$ differ by small fluctuations that are accounted for, to $O({\epsilon})$, by the transformation (\[eqn:ItoIbar\]).
Coherent Motion for Perpendicular Waves
=======================================
Using the Hamiltonian (\[eqn:Htil\]) we now analyze the ion motion for two perpendicularly propagating waves. Figures \[fig:rvst\_perp\_full\] and \[fig:rvst\_perp\_coh\] show the complete and coherent motion, respectively, for two perpendicularly propagating waves ($k_{iz}=0$). Figure \[fig:hvsr\_perp\] displays $H_+$ and $H_-$ from (\[eqn:Hbound\]) for the same parameters as in Figs. \[fig:rvst\_perp\_full\] and \[fig:rvst\_perp\_coh\], and the values of ${\tilde{H}}$ for the three initial conditions. The coherent analysis correctly predicts that particle 3 in Fig. \[fig:rvst\_perp\_full\] will not make it into the chaotic regime because it is reflected by the bump in $H_-$.
If we multiply ${\epsilon}_1$ and ${\epsilon}_2$ by the same factor $a$ then ${\tilde{H}}$ in (\[eqn:Htil\]) is multiplied by $a^2$ (note that for perpendicularly propagating waves, ${\tfrac{1}{2}}{{\bar{v}}}_z^2$ is a constant and can be eliminated from $ {\tilde{H}}$). Since a rescaling of the Hamiltonian is equivalent to a rescaling of time, rescaling both ${\epsilon}_i$’s does not affect the range of motion in ${{\bar{\rho}}}$ but rescales the period by $1/a^2$. For ${\epsilon}_1\sim{\epsilon}_2$, this means the period scales like $1/{\epsilon}_1^2$. This reflects the fact that the coherent motion is second-order in the wave field amplitudes. It also shows that in certain physical situations, at sufficiently large amplitudes, the effects of collisions on the coherent energization can be made negligible.
The range of coherent motion in $\rho$ scales linearly with the wave frequencies. In Fig. \[fig:hvsr\_nu\] we plot $H_\pm/H_+(\xi=0)$ versus $\xi\equiv{{\bar{\rho}}}/\nu_1$ for two values of $\nu_1$ with $N=1$. Note that the potential barriers do not change significantly with $\nu_1$. Figure \[fig:xirng\_nu\] shows that, as a function of $\nu_1$, the average $\xi_{min}$ and $\xi_{max}$ have a small variation ($\xi_{min}$ and $\xi_{max}$ are the maximum and minimum $\xi$ attained by an ion undergoing coherent motion and occur when the ion reaches the barriers $H_\pm$). The average is over ions with the same initial $\xi_0=0.4$ and different Waves with higher frequencies can therefore produce coherent energization to higher energies. Since the lower bound of the stochastic region is ions with the same initial $\xi_0$ and $\phi_0$ either will or will not reach the stochastic region regardless of the wave frequencies. For $\nu_1$ near an integer $\xi_{max}$ is about 20% higher than when $\nu_1$ is near a half-integer (this is not shown explicitly in Fig. \[fig:xirng\_nu\]).
The period of oscillation in ${{\bar{I}}}$ (see Fig. \[fig:rvst\_perp\_coh\]) can be estimated from the equation of motion for $\dot{{{\bar{I}}}}$: $$\label{eqn:idot} \dot{{{\bar{I}}}}=-\frac{{\partial}{\tilde{H}}}{{\partial}{{\bar{\psi}}}}=NS_-\sin(N{{\bar{\psi}}})$$ An orbit’s turning point typically occurs when ${{\bar{\psi}}}=n\pi/N$, i.e., when it hits one of the barriers $H_\pm$. Therefore, approximately, the period of oscillation $\tau$ is given by $\tau \approx 2\pi/(N\langle\dot{{{\bar{\psi}}}}\rangle)$, where $\langle\rangle$ denotes the average over one period. From the asymptotic forms of $S_0$ and $S_-$ for $\nu_1\sim\nu_2\gg1$ (Appendix \[app:asym\]) we find that $${\tilde{H}}\approx \nu_1^{-2}h_a(\xi,{{\bar{\psi}}})$$ where $h_a$ depends on $\nu_i$ only through $\xi$. Then $$\label{eqn:psidot} \dot{{{\bar{\psi}}}} = \frac{{\partial}{\tilde{H}}}{{\partial}{{\bar{I}}}} \approx \nu_1^{-2}\frac{{\partial}h_a}{{\partial}{{\bar{I}}}}=\nu_1^{-4}\frac{1}{\xi}\frac{{\partial}h_a}{{\partial}\xi}$$ Thus, $\tau \sim \nu_1^4$. Waves of lower frequency accelerate ions much more rapidly than those with higher frequency and thus may also be made less sensitive to the effects of collisions. Figure \[fig:period\] compares this scaling with the periods of two actual orbits obtained from ${\tilde{H}}$.
Coherent Motion for Oblique Waves {#sec:oblique}
=================================
In this section we describe the motion of ions when the waves have nonzero parallel wavenumber $k_{iz}$. This couples the parallel dynamics to the perpendicular motion.
For ions with initial $v_{z0}=0$ interacting with a *single* oblique wave, the motion is stochastic when [@smithpof]: $$\sqrt{|J_{n_0}(\rho)|}+\sqrt{|J_{n_0+1}(\rho)|} \geq \frac{1}{2k_z\sqrt{{\epsilon}}}$$ where $n_0$ is the greatest integer less than $\nu$. For $n_0\gg1$, the lower bound of the stochastic region is $$\rho\approx n_0+\frac{0.15}{{\epsilon}k_z^2}n_0^{2/3}-1.1n_0^{1/3}$$ As for a single perpendicular wave, the lower bound in $\rho$ is roughly the wave frequency and decreases with ${\epsilon}$. The stochastic region in $v_z$ extends from $v_z\approx0$ to $v_z\approx2\nu$.
For *two* waves Eq. (\[eqn:vzlie\]) shows that ${{\bar{v}}}_z$ changes only when ${\Delta}k_z\neq0$. Thus, the cases ${\Delta}k_z=0$ and ${\Delta}k_z\neq0$ lead to different dynamics and are treated separately.
Equal Parallel Wavenumbers: ${\Delta}k_z=0$
-------------------------------------------
Figures \[fig:rvst\_dkz0\] and \[fig:vzvst\_dkz0\] show the time evolution of $\rho$ and $v_z$ for two waves propagating at an angle of $45^\circ$ ($k_{iz}=k_{ix}=1)$ to ${\vec{B}}_0$. As in the case of two perpendicularly propagating waves, there is coherent change in $\rho$. During this coherent evolution $v_z$ has small-amplitude fluctuations around its initial value. In the region where the motion in $\rho$ becomes stochastic so does the motion in $v_z$. The stochastic region in $v_z$ agrees with the estimate given above. Since ${{\bar{v}}}_z$ is a constant during the coherent motion, the fluctuations in $v_z$ are due to the transformation between ${{\bar{v}}}_z$ and $v_z$.
The main effect of equal parallel wavenumbers is to slightly decrease the range of coherent motion from what it is for perpendicularly propagating waves, thus inhibiting some ions from reaching the stochastic region. Numerical studies indicate that $|S_{0x}/S_{0z}|$ and $|S_{-x}/S_{-z}|$ are both unity for $\xi<1$ but approach 0 as $\xi{\rightarrow}1$. This raises the bump in $H_-$ as $k_{iz}$ is increased. Consequently, more ions are reflected by $H_-$ and the range of coherent motion in $\rho$ is slightly lowered. This is evident from Fig. \[fig:hvsr\_k1z\], which shows $H_\pm/H_+({{\bar{\rho}}}=0)$ for $k_{1z}=0.1$ and 1. Figure \[fig:xirng\_k1z\] shows the range of motion $\xi_{min},\xi_{max}$ for different $k_{1z}$. Increasing $k_{iz}$ slightly lowers $\xi_{max}$ since the enhanced bump in $H_-$ reflects more ions. Generally then, significant coherent energization and access to the stochastic region is obtainable with oblique waves provided that ${\Delta}k_z = 0$, while the normalized $k_z$ may be large.
Unequal Parallel Wavenumbers: ${\Delta}k_z\neq0$
------------------------------------------------
When the parallel wavenumbers of the two waves are different, the coherent motion of the ions changes drastically. In this case $v_z$ undergoes coherent motion and the term ${\tfrac{1}{2}}{{\bar{v}}}_z^2$ in ${\tilde{H}}$ (\[eqn:Htil\]) is no longer a constant. This limits the range in $\rho$ as ${\Delta}k_z$ is increased. Figures \[fig:rvst\_dkzp001\] and \[fig:vzvst\_dkzp001\] show the time evolution of $\rho$ and $v_z$ for the exact orbits obtained from (\[eqn:fullham\]) with $k_{1z}=0.001$ and $k_{2z}=0$. These figures illustrate the limits in $\rho$.
Figure \[fig:hvsr\_dkzp001\] shows the variation of $H_\pm$ and ${\tfrac{1}{2}}{{\bar{v}}}_z^2$ as functions of ${{\bar{\rho}}}$. For ${{\bar{\rho}}}$ far from ${{\bar{\rho}}}_0=17$, $H_+-H_-=2|S_-|\ll{\tfrac{1}{2}}{{\bar{v}}}_z^2$ so that $H_+\approx H_-$. Figure \[fig:xirng\_dkz\] shows the limitation on the range of coherent motion in $\xi$ for ${\Delta}k_z\neq0$.
The coherent motion in $v_z$ has the effect of detuning the waves from exact resonance. The resonance condition for an ion with $v_z\neq0$ is $$\label{eqn:rescond}
R \equiv \nu_{10}-\nu_{20}-({\Delta}k_z) v_z\in\mathcal{Z}$$ where $\nu_{10}$ and $\nu_{20}$ are the wave frequencies in the laboratory frame, and $\nu_i=\nu_{i0}-k_{iz}v_z$. ${\tilde{H}}$ describes the ion’s motion as long as $R$ is close to an integer. For ${\Delta}k_z\neq0$, $v_z$ changes coherently. Condition (\[eqn:rescond\]) is not satisfied for all times, and the resonant interaction becomes less effective. The coherent change in $v_z$ thus limits itself, which keeps $R$ close to an integer. Since the coherent changes in ${{\bar{\rho}}}$ and ${{\bar{v}}}_z$ are linked via (\[eqn:vzlie\]), the coherent change in ${{\bar{\rho}}}$ is also small.
Consider a distribution of ions with different initial $v_{z0}$ interacting with two waves of frequencies $\nu_{10}$ and $\nu_{20}$. For ${\Delta}k_z=0$, all ions will be in resonance with the waves provided $\nu_{10}-\nu_{20}\in\mathcal{Z}$. For ${\Delta}k_z\neq0$, the resonance condition (\[eqn:rescond\]) implies that only ions with certain $v_{z0}$, namely $$v_{z0} \approx \frac{\nu_{10}-\nu_{20}-n}{{\Delta}k_z}, \qquad n\in\mathcal{Z}$$ are initially in resonance. As $v_z$ changes coherently, they fall out of resonance.
This situation is analogous to the case of two perpendicularly propagating waves when the wave frequencies do not differ by an integer [@benisti1]. Following Section IV.C of [@benisti1], the approximate Hamiltonian, correct to second order in wave amplitudes, that describes the coherent motion is $$\label{eqn:hoff}
{\tilde{H}}_{off}=-\frac{{\Delta}\nu}{N}{{\bar{I}}}+ {\tilde{H}}$$ where $(\nu_1-\nu_2)=N+{\Delta}\nu$ and $|{\Delta}\nu|\ll 1$. In this case the barriers $H_\pm$ are given by $$\label{eqn:Hpm_off}
H_\pm=-\frac{{\Delta}\nu}{N}{{\bar{I}}}+{\tfrac{1}{2}}{{\bar{v}}}_z^2+S_0\pm|S_-|$$ The first term in (\[eqn:hoff\]) limits the coherent motion, and plays a similar role to ${\tfrac{1}{2}}{{\bar{v}}}_z^2$. Figure \[fig:xirng\_dnu\] shows the range of motion in $\xi$ as a function of ${\Delta}\nu$. the largest range of coherent motion occurs for ${\Delta}\nu$ slightly different from 0, which allows $-({\Delta}\nu/N){{\bar{I}}}$ to partly cancel $S_0$ in
As the wave frequencies are increased, the range in ${\Delta}k_z$ and ${\Delta}\nu$ for which there is appreciable coherent motion becomes much narrower. Let $\xi_a({\Delta}k_z)$ be either the upper or lower bound of coherent motion in $\xi$ for wave frequencies $\nu_{1a}$ and $\nu_{2a}=\nu_{1a}-N$. The asymptotic forms in Appendix \[app:asym\] indicate that for two different frequencies $\nu_{1b}$ and , $$\label{eqn:dkzscal}
\xi_b({\Delta}k_z) \approx \xi_a{\left(}{\left(}\frac{\nu_{1b}}{\nu_{1a}}{\right)}^3 {\Delta}k_z {\right)}$$ Suppose $\xi_a$ is large for $k_1\leq{\Delta}k_z\leq k_2$, and that $\nu_{1b}=4\nu_{1a}$. Then $\xi_b$ is large only for $k_1/64\leq{\Delta}k_z\leq k_2/64$. Coherent motion occurs over a smaller range of ${\Delta}k_z$ when the wave frequencies are larger. Similarly, $$\label{eqn:dnuscal}
\xi_b({\Delta}\nu) \approx \xi_a{\left(}{\left(}\frac{\nu_{1b}}{\nu_{1a}}{\right)}^4 {\Delta}\nu {\right)}$$ Figures \[fig:xirng\_dkz\] and \[fig:xirng\_dnu\] demonstrate the range of coherent motion versus ${\Delta}k_z$ and ${\Delta}\nu$, respectively, and validate the scalings in (\[eqn:dkzscal\]) and (\[eqn:dnuscal\]) with wave frequency. As the wave frequencies are increased, ${\Delta}k_z$ and ${\Delta}\nu$ must be much smaller for ions to be energized to the stochastic region. Hence, just as in the cases of perpendicular propagation or ${\Delta}k_z=0$, for nonzero but small ${\Delta}k_z$ energization by waves with low frequencies is more advantageous than by waves with high frequencies.
Conclusions
===========
We have shown that two electrostatic waves propagating obliquely to an ambient magnetic field can coherently energize ions when their Doppler-shifted frequencies differ by a multiple of the ion cyclotron frequency. A second-order Hamiltonian, derived using the Lie perturbation technique, accurately describes the coherent motion and agrees well with numerical simulations of the complete dynamical equations. The energization of ions occurs regardless of the angle of wave propagation provided the parallel wavenumbers of the two waves are approximately equal. If the parallel wavenumbers are equal, there is no coherent acceleration along ${\vec{B}}_0$ but considerable stochastic energization both along and across ${\vec{B}}_0$. Moreover, the perpendicular coherent motion is quite similar to the case of perpendicularly propagating waves. There is a small amount of coherent acceleration along ${\vec{B}}_0$ when the parallel wavenumbers differ, but this causes the resonance condition to be violated. A difference between the parallel wavenumbers is similar to the difference between $({\omega}_1-{\omega}_2)/{\omega}_{ci}$ and the nearest integer.
There is no threshold ion energy or wave amplitude required for the coherent acceleration. The change in the ion gyroradius is linear in the wave frequencies and independent of wave amplitude. The period of coherent motion is inversely proportional to the square of the wave amplitudes and is proportional to the fourth power of the wave frequency ${\omega}$ (${\omega}\sim{\omega}_1\sim{\omega}_2$). Furthermore, the deviation from resonance ${\Delta}{\omega}={\omega}_1-{\omega}_2-N{\omega}_{ci}$ for which appreciable coherent acceleration occurs scales like ${\omega}^{-4}$, while the range in ${\Delta}k_z=k_{1z}-k_{2z}$ for coherent motion scales like ${\omega}^{-3}$. This implies that for lower-frequency waves coherent ion acceleration is faster and less sensitive to small changes in wave parameters.
Coherent ion energization occurs for two waves with appropriately chosen frequencies. An experiment is being constructed that will be able to test the theoretical predictions of this paper [@choueiri_aps_00]. Coherent acceleration could also occur for a broadband spectrum of waves extending over at least two ${\omega}_{ci}$ in frequency. Such a situation can occur naturally in the Earth’s ionosphere [@abhayjgr]. Detailed analyses of a broad spectrum of waves, and of the effects of weak collisions, remain to be carried out in future work.
The authors thank Prof. A. Brizard for helpful discussions on the Lie transformation technique. We also appreciate enlightening comments from Dr. D. Bénisti about his work on this problem. We thank R. Spektor for discussing his work with us and exploring possible experimental realizations of this process. This work was supported by DOE Contract , DOE/NSF Contract , NSF Contract , and Princeton University Subcontract . DJS was partly supported by an NDSEG Graduate Fellowship.
Lie Perturbation Method for Two Oblique Waves {#app:lie}
=============================================
We develop the Lie perturbation method following Refs. [@lichtlieb] and [@cary] and follow the notation in Section 2.5 of Ref. [@lichtlieb].
The Lie method provides a Hamiltonian ${{\bar{H}}}$ that describes just the coherent motion, and a change of coordinates that accounts for the incoherent fluctuations. The physical variables $x=(q,p)$ are governed by the full Hamiltonian $H(x)$, and the new coordinates ${{\bar{x}}}=({{\bar{q}}},{{\bar{p}}})$ are governed by ${{\bar{H}}}({{\bar{x}}})$. ${{\bar{x}}}$ depends on $x$ and a parameter ${\epsilon}$ which orders the perturbation via $$\label{eqn:lietrans}
\frac{{\partial}{{\bar{x}}}}{{\partial}{\epsilon}}=[{{\bar{x}}},w({{\bar{x}}},t)]_{{\bar{x}}}, \qquad {{\bar{x}}}({\epsilon}=0)=x$$ where $[f,g]_x=\sum_i[({\partial}f/{\partial}q_i)({\partial}g/{\partial}p_i)-({\partial}f/{\partial}p_i)({\partial}g/{\partial}q_i)]$ is the Poisson bracket. The old coordinates enter only as a condition for ${\epsilon}=0$, which ensures that the transformation for any $w$ is canonical and near-identity.
The operator $T$ relates the representation of a physical quantity $f$ in the two coordinate systems by $f({{\bar{x}}})=(Tf)(x)$. In particular, $f({{\bar{x}}})={{\bar{x}}}$ gives ${{\bar{x}}}=Tx$. $T$ satisfies $$\label{eqn:dTde}
\frac{{\partial}T}{{\partial}{\epsilon}}f(x)=-T[w(x,t),f(x)]_x$$ ${{\bar{H}}}$ is given by $$\label{eqn:Hbar}
{{\bar{H}}}({{\bar{x}}}) = T^{-1}H(x)+T^{-1}\int_0^{\epsilon}d{\epsilon}'T({\epsilon}')\frac{{\partial}w(x,t)}{{\partial}t}$$ The second term is not needed for an autonomous system.
We expand $w,H,T,$ and ${{\bar{H}}}$ in powers of ${\epsilon}$ and equate terms at each order in ${\epsilon}$. Collecting terms in (\[eqn:Hbar\]) at each order in ${\epsilon}$ gives equations for $w_i$. Upon carrying out the perturbation expansion to second order in ${\epsilon}$, we find $$\begin{aligned}
D_0w_1 &=& {{\bar{H}}}_1-H_1 \label{eqn:D0eqn1}
\\ D_0w_2 &=& 2({{\bar{H}}}_2-H_2)-[w_1,{{\bar{H}}}_1+H_1] \label{eqn:D0eqn2}\end{aligned}$$ $D_0f\equiv{\partial}_tf+[f,H_0]$ is the time derivative along the unperturbed trajectories. All expressions here are functions of the same set of coordinates. For simplicity we use $x$ for this purpose, but the final expression for ${{\bar{H}}}$ governs the evolution of ${{\bar{x}}}$. Clearly, ${{\bar{H}}}_0=H_0$.
For $T$ to be a near-identity operator, $w$ must remain small. We choose ${{\bar{H}}}_i$ in the right-hand side of (\[eqn:D0eqn1\]) and (\[eqn:D0eqn2\]) to eliminate any terms that would violate this condition. Such terms are referred to as “resonant” terms.
For the two-wave problem, $H_0$ and $H_1$ are given in (\[eqn:H0H1\]), while $H_i=0$ for $i\geq2$. Using a Bessel-function identity (see p. 361 of [@absteg]), we obtain $$H_1 = \sum_i\sum_{m=-\infty}^\infty {\epsilon}_i J_{m,i}\cos\psi_{mi}$$ where $J_{m,i}\equiv J_m(k_{ix}\rho)$ and $\psi_{mi}\equiv m\phi+k_{iz}z-\nu_it+{\alpha}_i$. Then from (\[eqn:D0eqn1\]) $$\label{eqn:dw1}
({\partial}_t+{\partial}_\phi+v_z{\partial}_z)w_1= \bar{H}_1-\sum_{i,m}{\epsilon}_iJ_{m,i}\cos\psi_{mi}$$
The unperturbed orbits are $\phi=t+\phi_0,z=z_0,v_z=0,I=I_0$ (in a frame where the ion’s initial $v_{z0}=0$). Along these orbits there are no resonant terms on the right-hand side of (\[eqn:dw1\]), so we choose ${{\bar{H}}}_1=0$. Then $$\label{eqn:w1}
w_1 = -\sum_{i,m}\frac{{\epsilon}_iJ_{m,i}}{m+k_{iz}v_z-\nu_i}\sin\psi_{mi}$$ Since $H_2$ and ${{\bar{H}}}_1$ are zero, (\[eqn:D0eqn2\]) leads to $$({\partial}_t+{\partial}_\phi+v_z{\partial}_z)w_2 = 2{{\bar{H}}}_2 - [w_1,H_1]$$ From (\[eqn:w1\]) $$\begin{split}
[w_1,H_1]=\sum_{i,j,m,n} \Big\{ &\frac{{\epsilon}_i{\epsilon}_j}{2\rho}\frac{1}{m-\mu_i}(-k_{jx}mJ_{m,i}J_{n,j}'+k_{ix}nJ_{m,i}'J_{n,j})\cos{\Gamma}_+
\\&+\frac{{\epsilon}_i{\epsilon}_j}{2\rho}\frac{1}{m-\mu_i}(-k_{jx}mJ_{m,i}J_{n,j}'-k_{ix}nJ_{m,i}'J_{n,j})\cos{\Gamma}_-
\\&+{\tfrac{1}{2}}{\epsilon}_i{\epsilon}_jk_{iz}k_{jz}\frac{J_{m,i}J_{n,j}}{(m-\mu_i)^2}(-\cos{\Gamma}_+ +\cos{\Gamma}_-) \Big\}
\end{split}$$ where ${\Gamma}_\pm\equiv \psi_{mi} \pm \psi_{nj}$. Along the unperturbed orbits, ${\Gamma}_\pm=\{m-\nu_i\pm(n-\nu_j)\}t$ + const. Some terms are resonant when $i=j$ regardless of the $\nu_i$’s. Other terms are resonant when either $2\nu_i, N_+\equiv(\nu_1+\nu_2)$, or $N\equiv(\nu_1-\nu_2)$ is an integer. We construct ${{\bar{H}}}_2$ to cancel these terms: $$\begin{split}
{{\bar{H}}}_2 = S_0(I,v_z) &+ {\delta}_-S_-(I,v_z)\cos((\nu_1-\nu_2)(\phi-t)+(k_{1z}-k_{2z})z+{\alpha}_1-{\alpha}_2)
\\ &+ {\delta}_+S_+(I,v_z)\cos((\nu_1+\nu_2)(\phi-t)+(k_{1z}+k_{2z})z+{\alpha}_1+{\alpha}_2)
\\ &+ \sum_i{\delta}_iS_i(I,v_z)\cos(2\nu_i(\phi-t)+2k_{iz}z+2{\alpha}_i)
\end{split}$$ where ${\delta}_-,{\delta}_+,$ and ${\delta}_i$ are unity when, respectively, $N,N_+,$ and $2\nu_i$ are integers and 0 otherwise. Equations (\[eqn:S0\]) and (\[eqn:Sm\]) give $S_0$ and $S_-$, and $$\begin{aligned}
S_+ =& -\sum_m \Big\{ \frac{{\epsilon}_1{\epsilon}_2}{4\rho(m-\mu_1)} (k_{1x}(m-N_+)J_{m,1}'J_{-m+N_+,2}+k_{2x}mJ_{m,1}J_{-m+N_+,2}') \notag
\\ & + {\tfrac{1}{4}}k_{1z}k_{2z}{\epsilon}_1{\epsilon}_2 \frac{J_{m,1}J_{-m+N_+,2}}{(m-\mu_1)^2} \notag
\\ & + \textrm{the same with subscripts 1 and 2 switched} \Big\}
\\S_i =& -\sum_m \Big\{ \frac{{\epsilon}_i^2}{4\rho(m-\mu_i)} (mk_{ix}J_{m,i}J_{-m+2\nu_i,i}'+(m-2\nu_i)k_{ix}J_{m,i}'J_{-m+2\nu_i,i}) \notag
\\ & + {\tfrac{1}{4}}{\epsilon}_i^2k_{iz}^2\frac{J_{m,i}J_{-m+2\nu_i,i}}{(m-\mu_i)^2} \Big\}\end{aligned}$$
The coherent Hamiltonian is $${{\bar{H}}}({{\bar{x}}},t) = H_0({{\bar{x}}}) + {{\bar{H}}}_2({{\bar{x}}},t)$$ Using ${{\bar{\psi}}}={{\bar{\phi}}}-t$ as the coordinate conjugate to ${{\bar{I}}}$, the transformed Hamiltonian is $$\label{eqn:htilapp}
{\tilde{H}}({{\bar{\psi}}},{{\bar{z}}},{{\bar{I}}},{{\bar{v}}}_z) = {\tfrac{1}{2}}{{\bar{v}}}_z^2 + {{\bar{H}}}_2$$ ${\tilde{H}}$ is a constant of the motion.
When only one resonance condition is satisfied, we find a second constant of the motion besides ${\tilde{H}}$, which relates ${{\bar{I}}}$ and ${{\bar{v}}}_z$ (when only $N$ is an integer, this constant is ${{\bar{v}}}_z-({\Delta}k_z/N){{\bar{I}}}$). The dynamical system described by (\[eqn:htilapp\]) is thus completely integrable. When any two resonance conditions are satisfied it is easy to see that $\nu_1$ and $\nu_2$ must both be half-integers. Then all four resonance conditions are satisfied. It does not appear that, in this case, there exists a second constant of the motion. The dynamics described by ${\tilde{H}}$ could be stochastic.
To find the transformation relating $x$ and ${{\bar{x}}}$, we expand (\[eqn:dTde\]) and use ${{\bar{x}}}=Tx$. To first order in ${\epsilon}$ we obtain $${{\bar{x}}}= Tx \approx x - {\epsilon}[w_1(x,t),x]_x + O({\epsilon}^2)$$ As desired, the coordinate change is near-identity. The relation between $I$ and ${{\bar{I}}}$ is given in (\[eqn:ItoIbar\]).
Asymptotic forms for $S_0$ and $S_-$ {#app:asym}
====================================
Here we derive the asymptotic forms for the terms in ${\tilde{H}}$, given in (\[eqn:Htil\]), using results in Refs. [@absteg; @gradryz; @chia1; @spektor_iepc01]. For $k_{1x}=k_{2x}=1$, let $$\begin{aligned}
S_{0x} &=& \sum_{i=1}^2 {\epsilon}_i^2 s_x(\rho,\mu_i,0)
\\S_{-x} &=& {\epsilon}_1{\epsilon}_2(s_x(\rho,\mu_1,N)+s_x(\rho,\mu_2,-N))
\\S_{0z} &=& \sum_{i=1}^2 k_{iz}^2{\epsilon}_i^2 s_z(\rho,\mu_i,0)
\\S_{-z} &=& {\epsilon}_1{\epsilon}_2k_{1z}k_{2z}(s_z(\rho,\mu_1,N)+s_z(\rho,\mu_2,-N))\end{aligned}$$ where $$\begin{aligned}
s_x(\rho,\mu,n) &=& (-)^n\frac{\pi}{8}\csc\mu\pi(J_{\mu+1}J_{-(\mu+1)+n}-J_{\mu-1}J_{-(\mu-1)+n})
\\s_z(\rho,\mu,n) &=& (-)^{n+1}\frac{\pi}{4}\frac{{\partial}}{{\partial}\mu}{\left[}\csc\mu\pi J_\mu J_{-\mu+n} {\right]}\end{aligned}$$ and $J_\mu=J_\mu(\rho)$.
Bessel functions of negative order are replaced with $$J_{-\mu}\approx-\sin(\mu\pi)Y_\mu$$ where $Y_\mu$ is the Bessel function of the second kind. Using the asymptotic forms for $J_\mu$ and $Y_\mu$ [@absteg], we obtain $$\begin{aligned}
{\sigma}(\mu,\rho,n) &\equiv& J_\mu(\rho) Y_{\mu-n}(\rho) \sim \beta e^{\gamma}\\ \beta &=& -\frac{1}{\pi}(\mu(\mu-n)\tanh{\alpha}_0\tanh{\alpha}_n)^{-1/2}
\\ {\gamma}&=& \mu(\tanh{\alpha}_0-{\alpha}_0)-(\mu-n)(\tanh{\alpha}_n-{\alpha}_n)
\\ {\textrm{sech }}{\alpha}_n &=& \frac{\rho}{\mu-n}\end{aligned}$$
For $N+1\ll\mu$ all the $n/\mu$’s are small, and to leading order in $n/\mu$ we find $$\begin{aligned}
\beta &\approx& -\frac{1}{\mu\pi}(1-\xi^2)^{-1/2}
\\ {\gamma}&\approx& -n{\alpha}_0
\\ {\sigma}&\approx& -\frac{1}{\mu\pi}{\sigma}_0(\xi,n)
\\ {\sigma}_0(\xi,n) &\equiv& (1-\xi^2)^{-1/2}{\left(}\frac{\xi}{1+\sqrt{1-\xi^2}}{\right)}^n\end{aligned}$$
Thus, $$\begin{aligned}
s_z &\approx& \frac{\pi}{4}\frac{{\partial}}{{\partial}\mu}{\sigma}(\mu,\rho,n)
\\ &\approx& \frac{f(\xi,n)}{\mu^2}
\\ f(\xi,n) &\equiv& \frac{1}{4}{\left(}{\sigma}_0+\xi\frac{{\partial}{\sigma}_0}{{\partial}\xi}{\right)}\end{aligned}$$ Similarly, $$\begin{aligned}
s_x &\approx& \frac{\pi}{8}({\sigma}(\mu+1,\rho,n)-{\sigma}(\mu-1,\rho,n))
\\ &\approx& \frac{\pi}{8}({\sigma}_1({\epsilon})-{\sigma}_1(-{\epsilon}))
\\ {\sigma}_1({\epsilon}) &=& \frac{{\sigma}_0(\xi/(1+{\epsilon}),n)}{\mu(1+{\epsilon})}\end{aligned}$$ where ${\epsilon}\equiv1/\mu$ is small for $\mu\gg1$. Expanding ${\sigma}_1$ to leading order in ${\epsilon}$, $$\begin{aligned}
s_x &\approx& \frac{\pi}{4}{\epsilon}{\sigma}_1'(0)
\\ &=& \frac{f(\xi,n)}{\mu^2}\end{aligned}$$ Thus, $$s_x \approx s_z \sim \frac{1}{\mu^2}$$ $S_0$ and $S_-$ both scale like $1/\nu_1^2$ when $\nu_1\approx\nu_2$.
![\[fig:rvst\_perp\_full\] $\rho$ versus $t$ for three ions interacting with two perpendicular waves from the full Hamiltonian $H$ (\[eqn:fullham\]). Quantities in all figures are given in terms of the normalized units defined in the text. The initial $\rho_0=15.95\ (\xi_0=\rho_0/\nu_1=0.4)$ for all three ions while their phases are $\phi_0=(-0.3,0.2,0.4)\pi$ for ions labelled 1, 2, and 3, respectively. The parameters for the two waves are: $\nu_1=40.37$, and $\nu_2=\nu_1-1$.](035307PHP1.eps)
![\[fig:rvst\_perp\_off\] $\rho$ versus $t$ for the same parameters as in Fig. \[fig:rvst\_perp\_full\] except that $\rho_0=15.95,30,45$ and ](035307PHP2.eps)
![\[fig:rvst\_perp\_coh\] ${{\bar{\rho}}}$ versus $t$ from the coherent Hamiltonian ${\tilde{H}}$ (\[eqn:Htil\]) for the same parameters as in Fig. \[fig:rvst\_perp\_full\]. SR indicates the stochastic region for the full Hamiltonian.](035307PHP3.eps)
![\[fig:hvsr\_perp\] $H_+$ and $H_-$ versus ${{\bar{\rho}}}$ for the same parameters as in Fig. \[fig:rvst\_perp\_full\]. The initial values of ${\tilde{H}}$ for the three ions in Fig. \[fig:rvst\_perp\_full\] are marked by the open circles.](035307PHP4.eps)
![\[fig:hvsr\_nu\] $H_\pm/H_+(\xi=0)$ versus $\xi\equiv{{\bar{\rho}}}/\nu_1$ for $N=1$ and $\nu_1=10.37$ (solid line) and $\nu_1=70.37$ (dashed line). ${\epsilon}_1={\epsilon}_2=$arbitrary, $k_{1x}=k_{2x}=1$, and $k_{1z}=k_{2z}=0$.](035307PHP5.eps)
![\[fig:xirng\_nu\] Average $\xi_{min}$ and $\xi_{max}$ versus $\nu_1$ for perpendicularly propagating waves based on the barriers $H_\pm$. Parameters are as in Fig. \[fig:hvsr\_nu\] except that $\nu_1=(3.37,4.37,...,80.37), \xi_0=0.4$, and the average is over $\phi_0=(0,0.05,...,1)\pi$.](035307PHP6.eps)
![\[fig:period\] The period of coherent oscillation $\tau$ versus $\nu_1$. The other wave parameters are the same as in Fig. \[fig:rvst\_perp\_full\]. The initial conditions are $\xi_0=0.4$ and $\phi_0=(0.2,0.6)\pi$. The open circles and stars are the periods obtained from integrations of the dynamics given by ${\tilde{H}}$. The solid lines are proportional to $\nu_1^4$ with the constant of proportionality chosen to match the period at $\nu_1=40.37$. The vertical axis is scaled so that $\nu_1^4$ is a straight line.](035307PHP7.eps)
![\[fig:rvst\_dkz0\] $\rho$ versus $t$ for the same parameters as in Fig. \[fig:rvst\_perp\_full\] except that $k_{1z}=k_{2z}=1$.](035307PHP8.eps)
![\[fig:vzvst\_dkz0\] $v_z$ versus $t$ for the same parameters as in Fig. \[fig:rvst\_dkz0\].](035307PHP9.eps)
![\[fig:hvsr\_k1z\] $H_\pm/H_+({{\bar{\rho}}}=0)$ versus ${{\bar{\rho}}}$ for $k_{1z}=k_{2z}=0.1$ (dots) and 1 (dashes). Other parameters are as in Fig. \[fig:rvst\_dkz0\]. The curves for $k_{1z}=0$ are very close to those for $k_{1z}=0.1$.](035307PHP10.eps)
![\[fig:xirng\_k1z\] Average $\xi_{min}$ and $\xi_{max}$ versus $k_{1z}$ for $k_{2z}=k_{1z}$ from $H_\pm$. $\nu_1=40.37$ and the other parameters are as in Fig. \[fig:xirng\_nu\].](035307PHP11.eps)
![\[fig:rvst\_dkzp001\] $\rho$ versus $t$ for the same parameters as in Fig. \[fig:rvst\_perp\_full\] except that $k_{1z}=0.001$ and $k_{2z}=0$.](035307PHP12.eps)
![\[fig:vzvst\_dkzp001\] $v_z$ versus $t$ for the same parameters as in Fig. \[fig:rvst\_dkzp001\]. ](035307PHP13.eps)
![\[fig:hvsr\_dkzp001\] $H_\pm$ and ${\tfrac{1}{2}}{{\bar{v}}}_z^2$ versus ${{\bar{\rho}}}$ for the same parameters as in Fig. \[fig:rvst\_dkzp001\]. ](035307PHP14.eps)
![\[fig:xirng\_dkz\] Average $\xi_{min}$ and $\xi_{max}$ versus ${\Delta}k_z$ from $H_\pm$ for $k_{1z}=0.1,\ k_{2z}=k_{1z}-{\Delta}k_z,\ \nu_1=10.37$ (dotted) or 40.37 (dashed), and the other wave parameters as in Fig. \[fig:rvst\_perp\_full\]. Initial conditions and averaging are as in Fig. \[fig:xirng\_nu\]. The abscissa for $\nu_1=40.37$ has been rescaled by $(10.37/40.37)^3$. If the scaling in (\[eqn:dkzscal\]) were exact, the dotted and dashed lines would coincide.](035307PHP15.eps)
![\[fig:xirng\_dnu\] Average $\xi_{min}$ and $\xi_{max}$ versus ${\Delta}\nu$ from $H_\pm$ for $\nu_2=\nu_1-1-{\Delta}\nu,\ \nu_1=10.37$ (dotted) or 40.37 (dashed) and the other wave parameters as in Fig. \[fig:rvst\_perp\_full\]. Initial conditions and averaging are as in Fig. \[fig:xirng\_nu\]. The abscissa for $\nu_1=40.37$ has been rescaled by $(10.37/40.37)^4$. If the scaling in (\[eqn:dnuscal\]) were exact, the dotted and dashed lines would coincide.](035307PHP16.eps)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study arithmetic degree of a dominant rational self-map on a smooth projective variety over a function field of characteristic zero. We see that the notion of arithmetic degree and some related problems over function fields are interpreted into geometric ones. We give another proof of the theorem that the arithmetic degree at any point is smaller than or equal to the dynamical degree. We give a sufficient condition for an arithmetic degree to coincide with the dynamical degree, and prove that any self-map has so many points whose arithmetic degrees are equal to the dynamical degree. We study dominant rational self-maps on projective spaces in detail.'
address:
- 'Graduate school of Mathematical Sciences, the University of Tokyo, Komaba, Tokyo, 153-8914, Japan'
- 'Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan'
- 'Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan'
author:
- Yohsuke Matsuzawa
- Kaoru Sano
- Takahiro Shibata
title: Arithmetic degrees for dynamical systems over function fields of characteristic zero
---
Introduction {#sec1}
============
Let $K$ be a field where heights $h_X$ on smooth projective varieties $X$ can be defined (e.g. a number field or a function field). Given a dominant rational self-map $f: X \dashrightarrow X$ on a smooth projective variety $X$ and a rational point $P \in X(K)$, it is important to study how the height $h_X(f^m(P))$ varies as $m$ grows. As a quantity representing the growth rate of $h_X(f^m(P))$, Kawaguchi and Silverman defined arithmetic degree of $f$ at $P$: $$\alpha_f(P)=\lim_{m \to \infty} \max \{ h_X(f^m(P)), 1 \}^{1/m}$$ (cf. [@KaSi2] or Definition \[def3.2\] (ii)). On the other hand, there is another invariant for self-maps, the (first) dynamical degree of $f$: $$\delta_f = \lim_{m \to \infty} ((f^m)^*H \cdot H^{\dim X-1})^{1/m},$$ where $H$ is an ample Cartier divisor on $X$ (cf. Definition \[def3.2\] (i)). It seems important and interesting to investigate relations between those two quantities. Especially, it is natural to ask when $\alpha_f(P)$ coincides with $\delta_f$. Kawaguchi and Silverman conjectured some properties of arithmetic degree and a sufficient condition for a rational point $P$ on which the arithmetic degree at $P$ coincides with the dynamical degree (cf. [@KaSi2 Conjecture 6]). These conjectures have been verified in some special cases. For details, see Conjecture \[conj3.4\], Remark \[rem3.5\] and Remark \[rem3.7\].
In this article, we study arithmetic degrees over function fields in one variable over an algebraically closed field of characteristic zero. The advantage of considering height theory over function fields is that we can interpret the height of a rational point into the degree of the curve corresponding to the rational point. A variety $X$ over a function field $K$ of a curve $C$ can be seen as the generic fiber of a fibration $\pi: \mathcal X \to C$ over $C$, and then a $K$-rational point $P$ of $X$ corresponds to a section $\sigma$ of $\pi$. In this situation, the height $h_X(P)$ of $P$ is equal to $\deg(\sigma^*\mathcal H)$, where $\mathcal H$ is a $\pi$-ample Cartier divisor on $\mathcal X$. So arithmetic degree is also described by the degrees of divisors. We will use this geometric interpretation to deduce the results in this article.
A fundamental relation on those two types of degree is the following inequality: $$\alpha_f(P) \leq \delta_f.$$ This inequality was proved over any field where height functions can be defined by Kawaguchi–Silverman and Matsuzawa (cf. [[@KaSi2 Theorem 4]]{} and [[@Mat]]{}). We will prove it over function fields of characteristic 0 by using the geometric interpretation of height (Theorem \[thm4\]). Our proof works only over function fields, but it seems simple and short.
After we obtain the fundamental inequality, it is natural to ask when the arithmetic degree of a given rational point attains the dynamical degree. Over $\overline{\mathbb Q}$, Kawaguchi and Silverman predicts a sufficient condition for a rational point to have the arithmetic degree which is equal to the dynamical degree (Conjecture \[conj3.4\] (iv)). However, over $\overline{k(t)}$, this conjecture does not hold in general (Example \[ex3.4.2\]). We give a sufficient condition for the arithmetic degree of a rational point to attain the dynamical degree as a geometric condition of the corresponding section (Theorem \[thm\_suff\]). For a dynamical system on a projective space, we will give some other sufficient conditions (Theorem \[thm5.0.1\] and Theorem \[thm5.2\]).
It is also important to investigate whether there exists a rational point whose arithmetic degree attains the dynamical degree. Over $\overline{k(t)}$ with $k$ an uncountable algebraically closed field of characteristic 0, we will prove the result that there are densely many points having pairwise disjoint orbits such that the arithmetic degree of them attain the dynamical degree (Theorem \[thm\_orbit\]). Over number fields, this theorem has been proved only for some particular cases (cf. [@KaSi1 Theorem 3] and [@MSS Theorem 1.7]). More strongly, for a self-map on a smooth projective rational variety, we can take such points over a fixed function field (Theorem \[thm5.4\]).
- - Throughout this article, $k$ denotes an algebraically closed field of characteristic zero, and $\overline{k(t)}$ denotes the algebraic closure of the rational function field of one variable over $k$.
- For a rational map $f : X \dashrightarrow Y$, $I_f$ denotes the indeterminacy locus of $f$.
- A *curve* simply means a smooth projective variety of dimension 1 unless otherwise stated.
- For any $\mathbb R$-valued function $h(x)$, we set $h^+(x) = \max \{ h(x), 1 \}$.
- Let $f$, $g$ and $h$ be $\mathbb R$-valued functions on a domain $S$. The equality $f = g + O(h)$ means that there is a positive constant $C$ such that $|f(x)-g(x)| \leq C |h(x)|$ for every $x \in S$. The equality $f=g + O(1)$ means that there is a positive constant $C'$ such that $|f(x)-g(x)| \leq C'$ for every $x \in S$.
The authors would like to thank Professors Osamu Fujino and Shu Kawaguchi for reading a manuscript of this paper and giving useful comments. The first author wold like to thank Professor Tomohide Terasoma for attending his seminar and giving valuable comments.
Height functions for varieties over function fields {#sec2}
===================================================
In this section, we define the (Weil) height functions on projective varieties over $\overline{k(t)}$, and see that there is another description of height in terms of the degree of a divisor on a curve. Basic facts of (Weil) height functions over function fields is explained for example in [@Lan Chapter 3, §3] and [@HiSi B.10]. So we omit most of the proofs.
First, we define the height functions on projective spaces over function fields.
\[def2.1\] Let $C$ be a (smooth projective) curve over $k$ and $p \in C$ a closed point. We define a valuation $v_p: K(C)^\times \to \mathbb Z$ as $$v_p(f) = (\mathrm{the\ multiplicity\ of\ }f \mathrm{\ at\ }p).$$ This is nothing but the discrete valuation associated to the discrete valuation ring $\mathcal O_{C, p}$. Set $v_p(0)=\infty$.
\[def2.2\] Let $C$ be a curve over $k$. Take $P \in \mathbb P^n(K(C))$. Represent $P$ by homogeneous coordinates as $P=(f_0: f_1: \cdots : f_n)$, where $f_i \in K(C)$. We define the *height function on $\mathbb P^n(K(C))$ relative to $K(C)$* as $$h_{K(C)}(P)= \sum_{p \in C} -\min \{v_p(f_0), \ldots, v_p(f_n)\}.$$
\[rem2.3\] (i) Since $\# \{p \in C | v_p(f) \neq 0 \}$ is finite for any $f \in K(C)^\times$, $\sum_{p \in C} -\min \{v_p(f_0), \ldots, v_p(f_n)\}$ is in fact a finite sum.
\(ii) $h_{K(C)}(P)$ is independent of the representation of $P$. For, if we represent $P$ as $P= (gf_0 : \cdots : gf_n)$, where $g \in K(C)^\times$, then $$\begin{aligned}
\sum_p -\min_i v_p(gf_i) &= \sum_p -\min_i(v_p(g)+v_p(f_i)) \\
&= \sum_p -v_p(g)+ \sum_p -\min v_p(f_i) \\
&= \sum_p -\min_i v_p(f_i),\end{aligned}$$ since $\sum_p -v_p(g) = - \deg ( (g) ) = 0$.
Next, we give the definition of the (absolute) height function on $\mathbb P^n(\overline{k(t)})$ which is essentially compatible with the previous definition of height functions over function fields.
\[lem2.5\] Let $\phi: C' \to C$ be a finite surjective morphism of two curves. Set $K=K(C), K'=K(C')$. Take $P \in \mathbb P^n(K)$. Then $$\frac{1}{[K: k(t)]} h_K(P)= \frac{1}{[K': k(t)]} h_{K'}(P).$$
We define the (absolute) height function on $\mathbb P^n(\overline{k(t)})$ as follows:
\[def2.6\] Take $P \in \mathbb P^n(\overline{k(t)})$. We define the *height function on $\mathbb P^n(\overline{k(t)})$* as $$h(P)= \frac{1}{[K: k(t)]} h_K(P),$$ where $K$ is a finite extension field of $k(t)$ such that $P \in \mathbb P^n(K)$. $h(P)$ is independent of the choice of $K$ by Lemma \[lem2.5\].
Using the height functions on projective spaces, we define the height function on a projective variety $X$ associated to a Cartier divisor. We prepare the following lemma.
\[lem2.8\] Let $X$ be a projective variety over $\overline{k(t)}$ and $D$ a base point free Cartier divisor on $X$.
- Let $\phi: X \to \mathbb P^n$, $\psi: X \to \mathbb P^n$ be two morphisms associated to the complete linear system $|D|$. Then $h \circ \phi = h \circ \psi + O(1)$.
- Let $\phi_1: X \to \mathbb P^{n_1}$, $\phi_2: X \to \mathbb P^{n_2}$ be morphisms associated to very ample Cartier divisors $A_1, A_2$ on $X$, respectively. Let $\phi: X \to \mathbb P^n$ be a morphism associated to $A_1 + A_2$. Then $h \circ \phi = h \circ \phi_1 + h \circ \phi_2+ O(1)$.
Using Lemma \[lem2.8\], we can define the height function associated to a Cartier divisor up to the difference of a bounded function.
\[def2.9\] Let $X$ be a projective variety over $\overline{k(t)}$.
\(i) Let $A$ be a base point free Cartier divisor on $X$. We define a *height function on $X$ associated to $A$* as $$h_A = h \circ \phi_A,$$ where $\phi_A$ is a morphism associated to $|A|$. By Lemma \[lem2.8\] (i), $h_A$ is well-defined up to a bounded function.
\(ii) Let $D$ be a Cartier divisor on $X$. We define a *height function on $X$ associated to $D$* as $$h_D= h \circ \phi_A - h \circ \phi_B,$$ where $A, B$ are base point free Cartier divisors such that $D \sim A-B$ and $\phi_A, \phi_B$ are morphisms associated to $|A|, |B|$, respectively.
Note that we can always take such $A$ and $B$. Assume that we have other base point free Cartier divisors $A', B'$ such that $D \sim A' - B'$. Then $A+B' \sim A'+B$. Lemma \[lem2.8\] (i) implies that $h_{A+B'}=h_{A'+B} + O(1)$. Moreover, by Lemma \[lem2.8\] (ii), $h_{A+B'}
=h_A+ h_{B'}+ O(1)$ and $h_{A'+B}=h_{A'}+h_B+O(1)$. So $h_A - h_B= h_{A'}- h_{B'}+O(1)$. Hence $h_D$ is well-defined up to a bounded function.
\[prop2.9.1\] Let $X$ be a projective variety over $\overline{k(t)}$.
- Let $D, D'$ be Cartier divisors on $X$ and $n, n'$ be integers. Then $$h_{nD+n'D'}= nh_D + n'h_{D'}+O(1).$$
- Let $f: X \to Y$ be a morphism to a projective variety $Y$ and $E$ a Cartier divisor on $Y$. Then $$h_{f^*E}=h_E \circ f+ O(1).$$
Next, we give another description of height in terms of the degree of a divisor on a curve.
\[lem2.10\] Let $X, S$ be integral schemes and $f: X \to S$ a morphism of finite type. Let $\eta$ be the generic point of $S$ and $X_\eta$ the generic fiber of $f$. For a rational section $\sigma: S \dashrightarrow X$ of $f$ (i.e. a rational map $\sigma: S \dashrightarrow X$ such that $f \circ \sigma=\mathrm{id}_S$), $P \in X_\eta(k(\eta))$ denotes the rational point obtained by the base change of $\sigma$ along $k(\eta) \to S$. Then the mapping $\sigma \mapsto P$ gives a one-to-one correspondence between the set of rational sections of $f$ and $X_\eta(k(\eta))$.
This lemma follows from an elementary scheme-theoretic argument. When $X$ is a projective variety and $S=C$ is a (smooth projective) curve, Lemma \[lem2.10\] is reduced to the following.
\[prop2.11\] Let $C$ be a curve over $k$ and set $K=K(C)$.
- Let $\pi: X \to C$ be a surjective morphism from a projective variety $X$ to $C$ and $X_\eta$ the generic fiber of $\pi$. Then $X_\eta(K)$ corresponds one-to-one to the set of sections of $\pi$.
- Let $Y_k$ be a projective variety over $k$ and set $Y_K=Y_k \times_k K$. Then $Y_K(K)$ corresponds one-to-one to the set of $k$-morphisms from $C$ to $Y_k$.
\(i) Note that any rational map $\varphi: C \dashrightarrow X$ is in fact a morphism since $\operatorname{codim}I_\varphi \geq 2$. So the assertion follows from Lemma \[lem2.10\].
\(ii) Apply (i) to the projection $\operatorname{pr}_C: Y_k \times_k C \to C$. Note that the sections of $\operatorname{pr}_C$ correspond one-to-one to the $k$-morphisms from $C$ to $Y_k$.
Here is another description of height by the degree of divisors for projective spaces.
\[prop2.12\] Let $C$ be a curve and set $K=K(C)$. Take $P \in \mathbb P^n(K)$ and let $g: C \to \mathbb P^n_k$ be the corresponding morphism. Then $$h(P)= \frac{1}{[K:k(t)]}\deg(g^*\mathcal O(1)).$$
For $P=(f_0: \cdots: f_n) \in \mathbb P^n(K)$, $g$ is represented as $g(x)=(f_0(x): \cdots: f_n(x))$. Let $H_i=(x_i)$ be the hyperplane of $\mathbb P^n_k$ associated to the $i$-th coordinate. Assume that $g(C) \subset H_i$ for some $i$. Then $f_i=0$, so $$h_K(P)= \sum_{p \in C} -\min_{j \neq i} v_p(f_j).$$ On the other hand, $\deg(g^*\mathcal O_{\mathbb P^n}(1)) =
\deg(g^*\mathcal O_{H_i}(1))$. Therefore we can replace $\mathbb P^n_k$ by $H_i \cong \mathbb P^{n-1}_k$. If $g$ is a constant mapping, then $f_i \in k$ for all $i$ and $h_K(P)=\deg(g^*\mathcal O(1))=0$. So we may assume that $g(C)$ is not contained $H_i$ for all $i$.
Set $U=\operatorname{Spec}A = C \setminus g^{-1}(H_0)$ and $f_i=(g|_U)^*(x_i/x_0) \in A$. Then $P=(1: f_1: \cdots: f_n)$. So $$\begin{aligned}
h_K(P) &= \sum_{p \in C} -\min \{v_p(1), v_p(f_1), \ldots, v_p(f_n) \} \\
&= \sum_{p \in C \setminus U} -\min \{v_p(1), v_p(f_1), \ldots, v_p(f_n) \} \\
&= \sum_{p \in C \setminus U} -\min_{1 \leq i \leq n} v_p(f_i) \\
&= \sum_{p \in C \setminus U} -\min_{1 \leq i \leq n} v_p(g^*(x_i/x_0)) \\
&= \sum_{p \in C \setminus U} -\min_{1 \leq i \leq n} v_p(g^*H_i)
+ \sum_{p \in C \setminus U} v_p(g^*H_0), \\\end{aligned}$$ where the second equality holds because $v_p(f_i) \geq 0$ for $p \in U$ and the third equality holds because $\displaystyle \min_{1 \leq i \leq n} v_p(f_i) <0$ for $p \in C \setminus U$. Obviously $$\sum_{p \in C \setminus U} v_p(g^*H_0) =
\sum_{p \in C} v_p(g^*H_0) = \deg(g^*H_0).$$ For $p \in C \setminus U$, $v_p(g^*H_i) >0$ is equivalent to $g(p) \in H_i$. Since $g(p) \in H_0$, $g(p) \not\in H_i$ for some $i>0$ and then $v_p(g^*H_i)=0$. Hence $\displaystyle \min_{1 \leq i \leq n}v_p(g^*H_i)=0$ for all $p \in C \setminus U$. As a consequence, $$h(P)=\frac{1}{[K:k(t)]} \deg(g^*H_0) = \frac{1}{[K:k(t)]} \deg(g^*\mathcal O(1)).$$
By Proposition \[prop2.12\], we can see that $h(P) \geq 0$ for any $P \in \mathbb P^n(K)$. Furthermore, for a rational point $P \in \mathbb P^n(K)$ corresponding to a morphism $g: C \to \mathbb P_k^n$, $P \in \mathbb P^n(k)$ if and only if $g$ is a constant map. So we obtain the following.
\[prop2.12.1\]
- - $h(P) \geq 0$ for any $P \in \mathbb P^n(\overline{k(t)})$.
- For $P \in \mathbb P^n(\overline{k(t)})$, $h(P) = 0$ if and only if $P \in \mathbb P^n(k)$.
We give a description of height by the degree of divisors for a projective variety over $\overline{k(t)}$ which has a model over a curve or, more strongly, over $k$.
\[defn2.12.2\] Let $X$ be a projective variety over $\overline{k(t)}$ and $H$ an ample Cartier divisor on $X$. We define a function $\tilde h_H : X(\overline{k(t)}) \to \mathbb R_{\geq 0}$ as follows. Fix a model $(X_C \overset{\pi}{\to} C, H_C)$ of $(X, H)$ over a curve $C$, that is, a projective variety $X_C$ over $k$ with a surjection $\pi:X_C \to C$ whose geometric generic fiber is $X$, and a $\pi$-ample Cartier divisor $H_C$ on $X_C$ such that $(X \to X_C)^*H_C \sim H$. For any $P \in X(\overline{k(t)})$, take a curve $C_1$ with $K(C_1) \supset K(C)$ and the section $\sigma_1$ of $\pi_{C_1}: X_C \times_C C_1 \to C_1$ corresponding to $P$, and set $H_{C_1}=(X_C \times_C C_1 \to X_C)^*H_C$ and $$\tilde h_H(P)= \frac{1}{[K(C_1):k(t)]}\deg(\sigma_1^*H_{C_1}).$$
\[prop2.12.3\] Notation is as in Definition \[defn2.12.2\]. Then $\tilde h_H$ is a well-defined height function associated to $H$.
Take any point $P \in X(\overline{k(t)})$. Take curves $C_i$ with $K(C_i)\supset K(C)$ and the sections $\sigma_i$ of $\pi_{C_i}: X_{C_i}=X_C \times_C C_i \to C_i$ for $i=1,2$. To see the well-definedness of $\tilde h_H$, we may assume that $K(C_2) \supset K(C_1)$. $$\xymatrix{
C_2 \ar[d]^{\sigma_2} \ar[r]^{\phi_2} & C_1 \ar[d]^{\sigma_1} \\
X_{C_2} \ar[d]^{\pi_2} \ar[r]^{\psi_2} & X_{C_1} \ar[d]^{\pi_1} \ar[r]^{\psi_1}
& X_C \ar[d]^{\pi} \\
C_2 \ar[r]^{\phi_2} & C_1 \ar[r]^{\phi_1} & C
}$$ Set $H_{C_1}=\psi_1^*H_C$ and $H_{C_2}=\psi_2^*H_{C_1}$. Then $$\begin{aligned}
\frac{\deg(\sigma_2^*H_{C_2})}{[K(C_2):k(t)]}
&= \frac{\deg(\phi_2^*\sigma_1^*H_{C_1})}{[K(C_2):K(C_1)][K(C_1):k(t)]}\\
&= \frac{\deg(\sigma_1^*H_{C_1})}{[K(C_1):k(t)]}.\end{aligned}$$ So it follows that $\tilde h_H(P)$ is well-defined.
Take a sufficiently large integer $N$ such that there is a morphism $\iota_C: X_C \to \mathbb P^n_k \times C$ over $C$ with $\iota_C^*\mathcal O_{\mathbb P^n_C}(1) \sim_\pi NH_C$. Take a Cartier divisor $D_C$ on $C$ such that $\iota_C^*\mathcal O_{\mathbb P^n_C}(1) \sim NH_C + \pi^*D_C$. Let $\iota:X \to \mathbb P^n_{\overline{k(t)}}$ be the base change of $\iota_C$ by $\operatorname{Spec}\overline{k(t)} \to C$. Then the function $\frac{1}{N}h \circ \iota$ is a height function associated to $H$. $\iota(P) \in \mathbb P^n(\overline{k(t)})$ corresponds to the morphism $\operatorname{pr}_{\mathbb P^n_k} \circ \iota_{C_1} \circ \sigma_1: C_1 \to \mathbb P^n_k$, where $\iota_{C_1}$ be the base change of $\iota_C$ by $C_1 \to C$. We compute $$\begin{aligned}
\frac{1}{N}h(\iota(P))
&= \frac{\deg((\operatorname{pr}_{\mathbb P^n_k} \circ \iota_{C_1} \circ \sigma_1)^*
\mathcal O_{\mathbb P^n_k} (1))}{N[K(C_1):k(t)]}
\ \ \ \mathrm{(by\ Proposition\ \ref{prop2.12})} \\
&= \frac{\deg(\sigma_1^* (NH_{C_1}+\psi_1^*\pi^*D_C))}{N[K(C_1):k(t)]} \\
&= \frac{\deg(\sigma_1^*(NH_{C_1}))+[K(C_1):K(C)]\deg(D_C)}{N[K(C_1):k(t)]} \\
&= \tilde h_H(P)
+ \frac{\deg(D_C)}{N[K(C):k(t)]}.\end{aligned}$$ So $\tilde h_H$ is a height function associated to $H$.
Arithmetic degrees for dynamical systems over function fields {#sec3}
=============================================================
\[def3.1\] A *dynamical system over a field $K$* is a pair $(X, f)$ of a smooth projective variety $X$ over $K$ and a dominant rational self-map $f: X \dashrightarrow X$ over $K$.
Let $(X, f)$ be a dynamical system over a field $K$. Set $$X_f = \{ P \in X(K) | f^m(P) \not\in I_f
\mathrm{\ for\ every\ }m \geq 0\}.$$ For $P \in X_f$, set $$O_f(P) = \{ f^m(P) | m \in \mathbb Z_{\geq 0} \},$$ which we call the (*forward*) *$f$-orbit of $P$*.
Let $H$ be an ample divisor on $X$. The (*first*) *dynamical degree of $f$* is the number $$\delta_f = \lim_{m \to \infty}((f^m)^*H \cdot H^{\dim X-1})^{1/m} \ \in [1, \infty).$$ It is known that the limit converges and $\delta_f$ is independent of the choice of $H$.
\[def3.2\] Let $(X,f)$ be a dynamical system over an algebraically closed field $K$ where heights are well-defined (e.g. $\overline{k(t)}$ or $\overline{\mathbb Q}$). Take an ample Cartier divisor $H$ on $X$ and a rational point $P \in X_f$. The *arithmetic degree of $f$ at $P$* is defined as $$\alpha_f(P) = \lim_{m \to \infty} h_H^+(f^m(P))^{1/m},$$ where $h_H$ is a height function associated to $H$ and $h_H^+(P)= \max \{ h_H(P), 1 \}$. Note that we do not know whether the limit converges (cf. Conjecture \[conj3.4\] (i)). Similarly, $\overline \alpha_f(P), \underline \alpha_f(P)$ are defined as $$\overline \alpha_f(P) = \limsup_{m \to \infty} h_H^+(f^m(P))^{1/m},$$ $$\underline \alpha_f(P) = \liminf_{m \to \infty} h_H^+(f^m(P))^{1/m}.$$
\[prop3.3\] In the notation of Definition \[def3.2\], $\alpha_f(P)$, $\overline \alpha_f(P)$ and $\underline \alpha_f(P)$ are independent of the choices of $H$ and $h_H$.
It is obvious that $\alpha_f(P)$, $\overline \alpha_f(P)$ and $\underline \alpha_f(P)$ are independent of the choice of $h_H$ for a fixed ample Cartier divisor $H$.
Let $H'$ be another ample Cartier divisor on $X$. Take a sufficiently large integer $N$ such that $NH-H'$ is ample. Then $$\begin{aligned}
\lim_{m \to \infty} h_H^+(f^m(P))^{1/m}
&= \lim_{m \to \infty} (\frac{1}{N}h_{NH}^+(f^m(P)))^{1/m} \\
&= \lim_{m \to \infty} h_{NH}^+(f^m(P))^{1/m} \\
&= \lim_{m \to \infty} \max \{ h_{NH-H'}(f^m(P))+ h_{H'}(f^m(P)), 1 \}^{1/m} \\
&\geq \lim_{m \to \infty} h_{H'}^+(f^m(P))^{1/m}, \end{aligned}$$ where we take $h_{NH-H'}$ as a non-negative function. Similarly, $$\lim_{m \to \infty} h_{H'}^+(f^m(P))^{1/m}
\geq \lim_{m \to \infty} h_H^+(f^m(P))^{1/m}.$$ So $\alpha_f(P)$ is independent of $H$. The proofs for $\overline \alpha_f(P)$ and $\underline \alpha_f(P)$ are similar.
Now we introduce the following conjecture (over $\overline{\mathbb Q}$) given by Kawaguchi and Silverman (see [@KaSi2]).
\[conj3.4\] Let $(X, f)$ be a dynamical system over $\overline{\mathbb Q}$.
- The limit defining $\alpha_f(P)$ exists for every $P \in X_f$.
- $\alpha_f(P)$ is an algebraic integer for every $P \in X_f$.
- $\{ \alpha_f(P) | P \in X_f \}$ is a finite set.
- Take $P \in X_f$. Assume that $O_f(P)$ is Zariski dense in $X$. Then $\alpha_f(P) = \delta_f$.
\[rem3.5\] In the case when $f$ is a morphism (see [@KaSi3 Theorem 2]), Kawaguchi and Silverman showed that (i), (ii), and (iii) of Conjecture \[conj3.4\] hold even over global fields of characteristic zero.
\[rem3.7\] It is known that Conjecture \[conj3.4\] holds in several cases (cf. [@MSS Remark 1.3]).
As the number field case, we consider:
\[prob3.4.1\] Let $(X, f)$ be a dynamical system over $\overline{k(t)}$. Take a point $P \in X_f$. When the equality $\alpha_f(P)=\delta_f$ holds?
The following examples show that the function field case of Conjecture \[conj3.4\] (iv) is not true.
\[ex3.4.2\] (i) Let $f: \mathbb P_{\overline{k(t)}}^1 \to \mathbb P_{\overline{k(t)}}^1$ be a surjective endomorphism with $\delta_f >1$. Take a $k$-valued non-preperiodic point $P \in \mathbb P^1(k)$. Then $O_f(P)$ is Zariski dense in $\mathbb P_{\overline{k(t)}}^1$, but $\alpha_f(P)=1 < \delta_f$.
\(ii) Define $f: \mathbb A_{\overline{k(t)}}^2 \to \mathbb A_{\overline{k(t)}}^2$ as $f(x,y)=(x^2,y^3)$. Then $f$ naturally extends to the morphism $f: \mathbb P_{\overline{k(t)}}^2 \to \mathbb P_{\overline{k(t)}}^2$ and $\delta_f=3$. Take a point $P=(t,2) \in \mathbb A^2(k(t))$. Then $f^m(P)=(t^{2^m}, 2^{3^m})$ and $$\alpha_f(P)=\lim_{m \to \infty} \max \{\deg(t^{2^m}), \deg(2^{3^m}) \}^{1/m}
= \lim_{m \to \infty} (2^m)^{1/m} = 2.$$
We show that $O_f(P)=\{(t^{2^m}, 2^{3^m})\}_{m=0}^\infty$ is dense in $\mathbb P_{\overline{k(t)}}^2$. It is enough to show that $O_f(P)$ is dense in $\mathbb A_{k(t)}^2$. Suppose $O_f(P)$ is contained in the zero locus of a polynomial $\phi(t,x,y) \in k(t)[x,y]$. Multiplying $\phi$ with a polynomial in $k[t]$, we may assume that $\phi \in k[t,x,y]$. Set $\phi(t,x,y)=\phi_r(t,y)x^r + \phi_{r-1}x^{r-1}+ \cdots + \phi_0(t,y)$, $\phi_r(t,y) \neq 0$. By assumption, $\phi(t,t^{2^m},2^{3^m})=0$ as a polynomial in $k[t]$. Since $$\deg(t^{2^mr})
> \deg(\phi(t,t^{2^m},2^{3^m})-\phi_r(t,2^{3^m})t^{2^mr})
= \deg(-\phi_r(t,2^{3^m})t^{2^mr})$$ for sufficiently large $m$, it follows that $\phi_r(t,2^{3^m})t^{2^mr}=0$ as a polynomial in $k[t]$ for sufficiently large $m$. Therefore $\phi_r(t,y)=0$ as a polynomial in $k[t,y]$, which is a contradiction. So $O_f(P)$ is Zariski dense in $\mathbb P_{\overline{k(t)}}^2$.
In the rest of this article, we will find some other conditions for an arithmetic degree to coincide with the dynamical degree.
A fundamental inequality {#sec4}
========================
There is a fundamental inequality between arithmetic degrees and dynamical degrees:
\[thm4\] $K$ denotes an algebraically closed field where heights are well-defined. Let $(X, f)$ be a dynamical system over $K$. Then $$\overline \alpha_f(P) \leq \delta_f$$ holds for any $P \in X_f$.
\[rem4.0\] This inequality was stated by Kawaguchi and Silverman (see [@KaSi2 Theorem 4]), and a correct proof of it was given by Matsuzawa (see [@Mat]).
We will give another proof of the inequality over $\overline{k(t)}$.
\[thm4.4\] Let $(X, f)$ be a dynamical system over $\overline{k(t)}$. Then the inequality $$\overline \alpha_f(P) \leq \delta_f$$ holds for any $P \in X_f$.
To prove Theorem \[thm4.4\], we prepare some lemmas. To begin with, we define a model of a dynamical system over $\overline{k(t)}$.
\[def3.1.1\] Let $(X, f)$ be a dynamical system over $\overline{k(t)}$. A *model of the dynamical system $(X, f)$ over a curve $C$* is a pair $(X_C \overset{\pi}{\to} C, f)$ of a surjective morphism $\pi: X_C \to C$ from a smooth projective $k$-variety $X_C$ to $C$ and a dominant rational self-map $f_C: X_C \dashrightarrow X_C$ over $C$ such that $X_C \times_C \overline{k(t)} = X$ and the base change of $f_C$ along $\operatorname{Spec}\overline{k(t)} \to C$ is equal to $f$.
\[lem4.1\] Let $(X, f)$ be a dynamical system over $\overline{k(t)}$. Then there exists a model of $(X, f)$ over a curve $C$.
Such a model is obtained by resolution of singularities.
\[lem4.1.2\] Let $f: X \dashrightarrow Y$ and $g: Y \dashrightarrow Z$ be dominant rational maps of smooth projective varieties. Take a Cartier divisor $H$ on $Z$ and a curve $C$ on $X$.
- If $C \not\subset I_f$, $f(C) \not\subset I_g$ and $H$ is nef, then $$((g \circ f)^*H \cdot C) \leq (f^*g^*H \cdot C).$$
- If $C \cap I_f = \varnothing$ and $f(C) \cap I_g = \varnothing$, then $$((g \circ f)^*H \cdot C) = (f^*g^*H \cdot C).$$
Since a nef divisor is the limit of a sequence of ample divisors, we may assume that $H$ is ample. $$\xymatrix{
X'' \ar[d]_{\mu'} \ar[dr]^{\tilde h} \\
X' \ar[d]_\mu \ar@{-->}[r]^h \ar[dr]^{\tilde f} & Y' \ar[d]^\nu \ar[dr]^{\tilde g} \\
X \ar@{-->}[r]^f & Y \ar@{-->}[r]^g & Z
}$$ In the above diagram, $\tilde f: X' \to Y$ (resp. $\tilde g: Y' \to Z$) is an elimination of indeterminacy of $f$ (resp. $g$) by blowing up smooth centers in $I_f$ (resp. $I_g$), $h = \nu^{-1} \circ f \circ \mu$, and $\tilde h: X'' \to Y'$ is an elimination of indeterminacy of $h$ by blowing up smooth centers in $I_h$. Then $$\begin{aligned}
&(g \circ f)^*H \cdot C \leq f^*g^*H \cdot C\ \ \cdots \mathrm{(1)}\\
&\iff (\mu \circ \mu')_* (\tilde g \circ \tilde h)^*H \cdot C
\leq \mu_* \tilde f^* \nu_* \tilde g^*H \cdot C \\
&\iff \mu_* \mu'_* \tilde h^* \tilde g^* H \cdot C
\leq \mu_* \tilde f^* \nu_* \tilde g^*H \cdot C.\end{aligned}$$ Here $\mu_* \tilde f^* \nu_* \tilde g^*H
=\mu_* \mu'_* \mu'^*\tilde f^* \nu_* \tilde g^*H
=\mu_* \mu'_* \tilde h^* \nu^* \nu_* \tilde g^*H$. Set $$E = \nu^* \nu_* \tilde g^* H- \tilde g^*H,$$ then (1) is equivalent to the inequality $$\mu_* \mu'_* \tilde h^* E \cdot C \geq 0.$$ By negativity lemma (cf. [@KoMo Lemma 3.39]), $E$ is an effective and $\nu$-exceptional divisor. Take a curve $C'$ on $X'$ such that $\mu(C')=C$ and a curve $C''$ on $X''$ such that $\mu'(C'')=C'$.
$\nu(\tilde h(C''))= \tilde f(\mu'(C''))= \tilde f(C')= f(C) \not\subset I_g$, so $\tilde h(C'') \not\subset \operatorname{Exc}(\nu)$. In particular, $\tilde h(C'') \not\subset \operatorname{Supp}E$ and then $C'' \not\subset \operatorname{Supp}\tilde h^*E$. Hence $$C=\mu(\mu'(C'')) \not\subset \mu(\mu'(\operatorname{Supp}\tilde h^*E))
= \operatorname{Supp}\mu_* \mu'_* \tilde h^*E.$$ This implies (1).
\(ii) is obvious since $f|_C: C \to f(C)$ and $g|_{f(C)}: f(C) \to Y$ are morphisms.
The following lemma is a variant of Lemma \[lem4.1.2\].
\[lem4.2\] Let $f: X \dashrightarrow Y$ be a dominant rational map of smooth projective varieties and $g: C \to X$ a morphism from a curve $C$. Take a Cartier divisor $H$ on $Y$.
- If $g(C) \not\subset I_f$ and $H$ is nef, then $$\deg((f \circ g)^*H) \leq \deg(g^* f^*H).$$
- If $g(C) \cap I_f = \varnothing$, then $$\deg((f \circ g)^*H) = \deg(g^* f^*H).$$
\(i) Since a nef divisor is the limit of a sequence of ample divisors, we may assume that $H$ is ample. $$\xymatrix{
& X' \ar[d]^\mu \ar[dr]^{\tilde f} \\
C \ar[ur]^{\tilde g} \ar[r]^g & X \ar@{-->}[r]^f & Y
}$$ In the above diagram, $\tilde f: X' \to Y$ is an elimination of indeterminacy of $f$ by blowing up smooth centers in $I_f$, and we can define the composition $\tilde g = \mu \circ g$ by the assumption that $g(C) \not\subset I_f$. Moreover it is a morphism.
We compute $$\begin{aligned}
\deg(g^*f^*H) - \deg((f \circ g)^*H)
&= \deg(\tilde g^* \mu^* \mu_* \tilde f^* H - \tilde g^* \tilde f^*H) \\
&= \deg(\tilde g^*E), \end{aligned}$$ where we set $E= \mu^*\mu_* \tilde f^*H - \tilde f^*H$. By negativity lemma(cf. [@KoMo Lemma 3.39]), $E$ is an effective and $\mu$-exceptional divisor on $X'$. Moreover $\tilde g(C) \not\subset \operatorname{Supp}E$ since $\mu(\tilde g(C)) = g(C) \not\subset I_f$ and $\mu(E) \subset I_f$. So $\deg(\tilde g^*E) \geq 0$.
\(ii) is obvious since both $g: C \to g(C)$ and $f|_{g(C)}: g(C) \to Y$ are morphisms.
\[lem4.3\] Let $X$ be a smooth projective variety with an ample Cartier divisor $H$. $\overline{\operatorname{Eff}}(X) \subset N^1(X)_{\mathbb R}$ denotes the pseudo-effective cone of $X$. Take a 1-cycle $Z \in N_1(X)_{\mathbb R}$. Then there is a constant $M>0$ such that $$(E \cdot Z) \leq M(E \cdot H^{\dim X-1})$$ holds for any $E \in \overline{\operatorname{Eff}}(X)$.
Note that $(E \cdot H^{\dim X-1})>0$ for any $E \in \overline{\operatorname{Eff}}(X) \setminus \{0\}$ (see [@KaSi2 Lemma 20]). We define a function $f: \overline{\operatorname{Eff}}(X) \setminus \{0\} \to \mathbb R$ as $$f(E)=\frac{(E \cdot Z)}{(E \cdot H^{\dim X-1})}.$$ Take a norm $||\cdot||$ on $N^1(X)_{\mathbb R}$ and set $S=\{E \in \overline{\operatorname{Eff}}(X) |\ ||E||=1 \}$. Then we can take an upper bound $M>0$ of $f|_S$ since $S$ is compact. But $f$ satisfies $f(cE)=f(E)$ for $E \in N^1(X)_{\mathbb R}$ and $c>0$, so $M$ is in fact an upper bound of $f$. This implies the claim.
\[lem4.3.1\] Let $(X, f)$ be a dynamical system over $\overline{k(t)}$ with a model $(X_C \overset{\pi}{\to} C, f_C)$ over a curve $C$ or $k$. Then $\delta_f=\delta_{f_C}$.
We define the $k$-th dynamical degree and the $k$-th relative dynamical degree: $$\lambda_k(f_C)= \lim_{m \to \infty} ((f_C^m)^* H_C^k \cdot H_C^{n+1-k})^{1/m},$$ $$\lambda_k(f_C|\pi)=\lim_{m \to \infty} ((f_C^m)^*H_C^k \cdot H_C^{n-k} \cdot F)^{1/m}.$$ Note that $\lambda_1(f_C)=\delta_{f_C}$.
Set $n= \dim X$. Take an ample divisor $H_C$ on $X_C$ and a general fiber $F$ of $\pi$. Fix an integer $m>0$. Take an elimination of indeterminacy of $f_C^m$: $$\xymatrix{
&\Gamma_C \ar[ld]_{p_C} \ar[rd]^{g_C}&\\
X_C \ar@{-->}[rr]_{f_C^m}&&X_C
}$$ Pulling it back along $\overline{k(t)} \to C$, we get the following diagram: $$\xymatrix{
X \ar[r]^{\eta}&X_C\\
\Gamma \ar[u]^p \ar[d]_g \ar[r]^{\eta_{\Gamma}}&\Gamma_C \ar[u]_{p_C} \ar[d]^{g_C}\\
X \ar[r]_{\eta}&X_C
}$$ Set $H=\eta^*H_C$. We can show that $g^*\eta^*=\eta_\Gamma^* g_C^*$ and $p_*\eta_\Gamma^*=\eta^*p_{C*}$. So we have $$(f^m)^*H = p_*g^*\eta^*H_C = p_*\eta_\Gamma^* g_C^*H_C
= \eta^* p_{C*} g_C^* H_C = \eta^*(f_C^m)^*H_C.$$ So $((f^m)^*H \cdot H^{n-1})= (\eta^*(f_C^m)^*H_C \cdot (\eta^*H_C)^{n-1})$. Hence $((f^m)^*H \cdot H^{n-1})$ is equal to the coefficient of the monomial $t_1 \cdots t_n$ for the numerical polynomial $$\chi(X, t_1 \eta^*(f_C^m)^*H_C + t_2 \eta^*H_C + \cdots + t_n \eta^*H_C).$$ For any Cartier divisor $D$ on $X_C$ and a general fiber $F$ of $\pi$, the equality $\chi(X, \eta^*D)=\chi(F, D|_F)$ holds. So $$\chi(X, t_1 \eta^*(f_C^m)^*H_C + t_2 \eta^*H_C + \cdots + t_n \eta^*H_C)$$ $$= \chi(F, t_1 (f_C^m)^*H_C|_F + t_2 H_C|_F + \cdots + t_n H_C|_F).$$ Hence we have $(\eta^*(f_C^m)^*H_C \cdot (\eta^*H_C)^{n-1})
= ((f_C^m)^*H_C|_F \cdot (H_C|_F)^{n-1})
= ((f_C^m)^*H_C \cdot H_C^{n-1} \cdot F)$, and so $\lambda_1(f)= \lambda_1(f_C|\pi)$.
On the other hand, by [@Tru Theorem 1.4], $$\begin{aligned}
\lambda_1(f_C)
&=\max \{ \lambda_1 (f_C|\pi)\lambda_0(\operatorname{id}_{C}),
\lambda_0 (f_C|\pi) \lambda_1(\operatorname{id}_{C}) \} \\
&= \max \{ \lambda_1(f_C|\pi) , 1 \} \\
&=\lambda_{1}(f_C|\pi).\end{aligned}$$ Note that $\lambda_q(\mathrm{id}_C)=1$ for all $q$ and $\lambda_0(f_C|\pi)=1$ by definition. So $$\delta_f =\lambda_1(f)= \lambda_1(f_C|\pi)=\lambda_1(f_C)=\delta_{f_C}.$$
Take $P \in X_f$. Put $n = \dim X$. By Lemma \[lem4.1\], we can take a model $(X_C \overset{\pi}{\to} C, f_C)$ over a curve $C$. We may assume that $P$ corresponds to a section $\sigma: C \to X_C$ of $\pi$.
Take an ample Cartier divisor $H_C$ on $X_C$ and set $H=(X \to X_C)^*H_C$. By Lemma \[lem4.3.1\], $$\delta_f= \delta_{f_C} = \lim_{m \to \infty} ((f_C^m)^*H_C \cdot H_C^n)^{1/m}.$$ On the other hand, by Proposition \[prop2.12.3\], $$\begin{aligned}
\overline \alpha_f(P)
&= \limsup_{m \to \infty} \tilde h^+_H(f^m(P))^{1/m} \\
&= \limsup_{m \to \infty} \deg^+((f_C^m \circ \sigma)^*H_C)^{1/m}. \end{aligned}$$ Note that $\operatorname{Im}(\sigma) \not\subset I_{f_C^m}$ since $P \not\in I_{f^m}$. By Lemma \[lem4.2\] (i), $$\deg((f_C^m \circ \sigma)^*H_C)
\leq \deg(\sigma^*(f_C^m)^*H_C) = ((f_C^m)^*H_C \cdot \sigma_*C).$$ It is obvious that $(f_C^m)^*H_C \in \overline{\operatorname{Eff}}(X_C)$ for every $m$. So, by Lemma \[lem4.3\], there is a constant $M>0$ such that the inequality $$((f_C^m)^*H_C \cdot \sigma_*C) \leq M((f_C^m)^*H_C \cdot H_C^n)$$ holds for every $m$. Therefore we have $$\begin{aligned}
\overline \alpha_f(P)
&\leq \limsup_{m \to \infty} ((f_C^m)^*H_C \cdot \sigma_*C)^{1/m} \\
&\leq \limsup_{m \to \infty} (M((f_C^m)^*H_C \cdot H_C^n))^{1/m} \\
&= \limsup_{m \to \infty} ((f_C^m)^*H_C \cdot H_C^n)^{1/m} \\
&= \delta_{f_C} \\
&= \delta_f. \end{aligned}$$
A sufficient condition {#sec_suff}
======================
Let $(X,f)$ be a dynamical system over $\overline{k(t)}$. In this section, we give a sufficient condition of a rational point $P \in X_f$ whose arithmetic degree attains the dynamical degree.
\[thm\_suff\] Let $(X,f)$ be a dynamical system over $\overline{k(t)}$ and $(X_C \overset{\pi}{\to} C, f_C)$ a model over a curve $C$. Take a rational point $P \in X_f$ corresponding to a section $\sigma:C \to X_C$ of $\pi$. Assume that
- $\sigma(C) \cap I_{f_C^m} = \varnothing$ for every $m \geq 1$ and
- $(E \cdot \sigma(C)) >0$ for any $E \in \overline{\operatorname{Eff}}(X) \setminus \{ 0\}$.
Then $\alpha_f(P)$ exists and $\alpha_f(P)=\delta_f$.
We prepare the following lemma.
\[lem\_norm\] Let $X$ be a smooth projective variety and $Z \subset X$ a 1-cycle such that $(E \cdot Z) >0$ for any $E \in \overline{\operatorname{Eff}}(X) \setminus \{0\}$. We define a non-negative function $||\cdot ||_Z: N^1(X)_{\mathbb R} \to \mathbb R$ as $$||v||_Z = \inf \{ (v_1 \cdot Z)+ (v_2 \cdot Z) | v=v_1 - v_2, \ v_1, v_2\
\mathrm{are\ effective\ classes} \}.$$
- $||v||_Z=(v \cdot Z)$ for any effective class $v \in N^1(X)_\mathbb R$.
- $||\cdot ||_Z$ is a norm on $N^1(X)_{\mathbb R}$.
\(i) For effective classes $v_1, v_2$ such that $v=v_1 - v_2$, we have $(v \cdot Z)= (v_1 \cdot Z) - (v_2 \cdot Z) \leq (v_1 \cdot Z) + (v_2 \cdot Z)$. So $||v||_Z=(v \cdot Z)$.
\(ii) It is easy to see that
- $||cv||_Z=|c| \cdot ||v||_Z$ for any $c \in \mathbb R$ and $v \in N^1(X)_\mathbb R$ and
- $||v+w||_Z \leq ||v||_Z + ||w||_Z$ for any $v, w \in N^1(X)_\mathbb R$.
Take $v \in N^1(X)_\mathbb R$ and assume that $||v||_Z=0$. Then we have $$\{ v_n^+ \}_n, \{ v_n^- \}_n \subset N^1(X)_\mathbb R$$ such that $v=v_n^+ - v_n^-$ for every $n$ and $$\lim_{n \to \infty} ((v_n^+ \cdot Z)+(v_n^- \cdot Z))=0.$$ So $\lim_{n \to \infty} (v_n^{\pm} \cdot Z) =0$. Since $(w \cdot Z)>0$ for any $w \in \overline{\operatorname{Eff}}(X) \setminus \{0\}$, it follows that $\lim_{n \to \infty} v_n^{\pm}=0$. Therefore $v=\lim_{n \to \infty}(v_n^+ - v_n^-)=0$. So $||\cdot||_Z$ satisfies the conditions of norm.
Set $n=\dim X$. We have $$\begin{aligned}
\delta_f
&= \delta_{f_C}\ \ \ \mathrm{(by\ Lemma\ \ref{lem4.3.1})} \\
&= \lim_{m \to \infty} ((f_C^m)^*H_C \cdot H_C^n)^{1/m} \\
&= \lim_{m \to \infty} ||(f_C^m)^*H_C||_{H_C^n}^{1/m}.\ \ \
\mathrm{(by\ Lemma\ \ref{lem_norm}\ (i))}\end{aligned}$$ Note that $||\cdot ||_{H_C^n}$ is a norm since $(E \cdot H_C^n) >0$ for every $E \in \overline{\operatorname{Eff}}(X) \setminus \{0\}$ (cf. [@KaSi2 Lemma 20]). We obtain $$\begin{aligned}
\delta_f
&=\lim_{m \to \infty} ||(f_C^m)^*H_C||_{\sigma(C)}^{1/m}\ \ \
\mathrm{(since\ }||\cdot ||_{H_C^n}\ \mathrm{is\ equivalent\ to\ }||\cdot ||_{\sigma(C)})\\
&=\lim_{m \to \infty} ((f_C^m)^*H_C \cdot \sigma(C))^{1/m} \ \ \
\mathrm{(by\ Lemma\ \ref{lem_norm}\ (i))} \\
&=\lim_{m \to \infty} \deg^+(\sigma^*(f_C^m)^*H_C)^{1/m} \\
&=\lim_{m \to \infty} \deg^+((f_C^m \circ \sigma)^*H_C)^{1/m}\ \ \
\mathrm{(by\ Lemma\ \ref{lem4.2}\ (ii))} \\
&= \lim_{m \to \infty} \tilde h_H^+(f^m(P))^{1/m}\ \ \
\mathrm{(by\ Proposition\ \ref{prop2.12.3})} \\
&=\alpha_f(P).\end{aligned}$$
Arithmetic degrees for projective spaces {#sec5}
========================================
In this section, we study arithmetic degrees for dynamical systems on projective spaces.
At first, we give some sufficient conditions for a rational point at which the arithmetic degree attains the dynamical degree.
\[lem5.0\] Let $C$ be a curve. Set $X= \mathbb P_k^n \times C$. Take an pseudo-effective Cartier divisor $E$ on $X$ and a general fiber $F$ of $\pi= \operatorname{pr}_C$. Then $\mathcal O_X(E) \equiv \mathcal O_X(d) \otimes \mathcal O_X(eF)$ for some $d, e \in \mathbb Z_{\geq 0}$.
It is sufficient to prove the claim for effective divisors, so we may assume that $E$ is effective. Since $\operatorname{Pic}(X)$ is generated by $\mathcal O_X(1)$ and $\pi^* \operatorname{Pic}(C)$, there are an integer $d$ and a divisor $D_C$ on $C$ such that $\mathcal O_X(E) \cong \mathcal O_X(d) \otimes \pi^* \mathcal O_C(D_C)$. Set $e=\deg D_C$. Then $\mathcal O_X(E) \equiv
\mathcal O_X(d) \otimes \mathcal O_X(eF)$.
Since $E|_F$ is effective and $\mathcal O_F(E|_F) \cong \mathcal O_{\mathbb P_{\mathbb C}^n}(d)$, $d \geq 0$. By projection formula, $$\pi_*(\mathcal O_X(d) \otimes \pi^* \mathcal O_C(D_C))
\cong \pi_*(\mathcal O_X(d)) \otimes \mathcal O_C(D_C)
\cong S^d(\mathcal O_C^{\oplus n+1}) \otimes \mathcal O_C(D_C).$$ Since $H^0(C,S^d(\mathcal O_C^{\oplus n+1}) \otimes \mathcal O_C(D_C))=
H^0(X, \mathcal O_X(d) \otimes \pi^* \mathcal O_C(D_C)) \neq 0$, $D_C$ is effective. So $e=\deg D_C \geq 0$.
\[thm5.0.1\] Let $(X= \mathbb P_{\overline{k(t)}}^n, f)$ be a dynamical system over $\overline{k(t)}$ and $(X_C= \mathbb P_k^n \times C \overset{\operatorname{pr}_{C}}{\to} C, f_C)$ a model of $(X, f)$ over a curve $C$. Take a morphism $g: C \to \mathbb P^n_k$ corresponding to a rational point $P_g \in X_f$ and set $\sigma_g= (g, \mathrm{id}_C): C \to X_C$.
- Assume that $g$ is non-constant and $\operatorname{Im}(\sigma_g) \cap I_{f_C^m}=\varnothing$ for every $m \geq 1$. Then $\alpha_f(P_g)$ exists and $\alpha_f(P_g)= \delta_f$.
- Assume the following conditions.
- For every $m \geq 0$, $\operatorname{pr}_{\mathbb P_k^n} \circ f_C^m \circ \sigma_g$ is non-constant.
- There is a sequence of positive integers $m_1 < m_2 < \ldots$ such that $\operatorname{Im}(f_C^{m_k} \circ \sigma_g) \cap I_{f_C^{m_{k+1}-m_k}}=\varnothing$ for every $k \geq 1$ and $\lim_{k \to \infty} (m_k/m_{k+1})=0$.
- The limit $\alpha_f(P_g)$ exists.
Then the equality $\alpha_f(P_g)=\delta_f$ holds.
\(i) Let $F$ be a general fiber of $\operatorname{pr}_C$. Then $$\overline{\operatorname{Eff}}(X_C)
=\mathbb R_{\geq 0} \mathcal O_X(F) + \mathbb R_{\geq 0} \mathcal O_X(1)$$ by Lemma \[lem5.0\]. It is obvious that $\sigma_g$ satisfies the assuption of Theorem \[thm\_suff\]. So (i) follows from Theorem \[thm\_suff\].
\(ii) For $m \geq 1$, set $(f_C^m)^*\mathcal O_{X_C}(1) \equiv \mathcal O_{X_C}(d_m) \otimes
\mathcal O_{X_C}(e_mF)$. Then $d_m, e_m \geq 0$ by Lemma \[lem5.0\]. It is clear that $(f^m)^*\mathcal O_X(1)
\equiv \mathcal O_X(d_m)$, and so $$\delta_f= \lim_{m \to \infty} ((f^m)^*\mathcal O_X(1) \cdot
\mathcal O_X(1)^{n-1})^{1/m}=\lim_{m \to \infty} d_m^{1/m}.$$
Set $b_m= \deg((f_C^m \circ \sigma_g)^*\mathcal O_{X_C}(1))$. By ($*$), $b_m \geq 1$ for every $m$. So we have $$\alpha_f(P_g)=\lim_{m \to \infty} \max \{b_m, 1\}^{1/m}
= \lim_{m \to \infty} b_m^{1/m}.$$ For $k \geq 1$, set $l_k=m_{k+1}-m_k$. We compute $$\begin{aligned}
b_{m_{k+1}}
&= \deg((f_C^{m_{k+1}} \circ \sigma_g)^*\mathcal O_{X_C}(1)) \\
&= \deg(f_C^{l_k} \circ f_C^{m_k} \circ \sigma_g)^*\mathcal O_{X_C}(1))\\
&= \deg(f_C^{m_k} \circ \sigma_g)^*(f_C^{l_k})^*\mathcal O_{X_C}(1))\ \ \
\mathrm{(by\ }(**)\ \mathrm{and\ Lemma\ \ref{lem4.2}\ (ii))} \\
&= \deg((f_C^{m_k} \circ \sigma_g)^*\mathcal O_{X_C}(d_{l_k})
\otimes \mathcal O_{X_C}(e_{l_k}F)) \\
&\geq \deg((f_C^{m_k} \circ \sigma_g)^*\mathcal O_{X_C}(d_{l_k})) \\
&=d_{l_k} b_{m_k} \\
&\geq d_{l_k}.\end{aligned}$$ Note that $\lim_{k \to \infty} l_k =\infty$ by the assumption that $\lim_{k \to \infty} (m_k/m_{k+1})=0$. Hence $$\begin{aligned}
\alpha_f(P_g)
&= \lim_{m \to \infty} (b_m)^\frac{1}{m}\ \ \
\mathrm{(by\ Proposition\ \ref{prop2.12.3})} \\
&= \lim_{k \to \infty} (b_{m_{k+1}})^\frac{1}{m_{k+1}} \\
&\geq \lim_{k \to \infty} (d_{l_k})^{\frac{1}{l_k} \cdot (1-\frac{m_k}{m_{k+1}})} \\
&= \delta_f.\end{aligned}$$ Combining with Theorem \[thm4\], it follows that $\alpha_f(P_g)=\delta_f$.
Next, we show that a sufficiently general morphism $g: C \to \mathbb P_k^n$ of a given sufficiently large degree corresponds to a rational point whose arithmetic degree attains the dynamical degree.
\[def5.1\] Let $C$ be a curve of genus $g(C)$ over $k$ and $d, n$ positive integers. $\operatorname{Mor}_d(C, \mathbb P_k^n)$ denotes the set of morphisms $g: C \to \mathbb P_k^n$ such that $\deg(g^*\mathcal O(1)) = d$.
$\operatorname{Mor}_d(C, \mathbb P_k^n)$ has a structure of $k$-variety with the evaluation $e:\operatorname{Mor}_d(C, \mathbb P_k^n) \times C \to \mathbb P_k^n$ which maps $(g,p)$ to $g(p)$. Moreover, if $\operatorname{Mor}_d(C, \mathbb P_k^n)$ is non-empty, we have $$\dim \operatorname{Mor}_d(C, \mathbb P_k^n) \geq (n+1)d+ n(1-g(C))$$ (cf. [@KoMo 1.1]).
\[thm5.2\] Let $(X= \mathbb P_{\overline{k(t)}}^n, f)$ be a dynamical system over $\overline{k(t)}$ and $(X_C= \mathbb P_k \times C \overset{\operatorname{pr}_C}{\to} C, f_C)$ a model of $(X, f)$ over a curve $C$ of genus $g(C)$. Take a positive integer $d$ satisfying $d > \frac{n(g(C)-1)}{n+1}.$ $P_g \in X(\overline{k(t)})$ denotes the rational point corresponding to $g \in \operatorname{Mor}(C, \mathbb P_k^n)$. Then $\alpha_f(P_g)$ exists and $\alpha_f(P_g) = \delta_f$ for a sufficiently general $g \in \operatorname{Mor}_d(C, \mathbb P_k^n)$.
Let $M \subset \operatorname{Mor}_d(C, \mathbb P_k^n)$ be an irreducible component of maximal dimension. Then $\dim M>0$ by assumption. Set $\Phi =(e, \mathrm{id}_C): M \times C \to X_C$, where $e$ is the evaluation. For any $g \in M$ and $\rho \in \operatorname{Aut}(\mathbb P_k^n)$, we have $\deg(g^* \rho^* \mathcal O_{\mathbb P^n}(1))
= \deg(g^* \mathcal O_{\mathbb P^n}(1))= d$, so $\operatorname{Aut}(\mathbb P_k^n)$ acts on $M$. Fix $g_0 \in M$. For any $(x, p) \in X_C$, we can take $\rho \in \operatorname{Aut}(\mathbb P_k^n)$ such that $\rho(g_0(p))= x$. Then $\Phi(\rho \circ g_0, p)= ((\rho \circ g_0)(p), p)= (x, p)$. So it follows that $\Phi$ is surjective.
For every $m \geq 1$, we compute $$\begin{aligned}
\dim \Phi^{-1}(I_{f_C^m})
&\leq (\dim(M \times C) - \dim X_C) + \dim I_{f_C^m} \\
&\leq (\dim M + 1 - \dim X_C) + \dim X_C -2 \\
&= \dim M -1.\end{aligned}$$ Hence $\operatorname{pr}_M(\Phi^{-1}(I_{f_C^m})) \subset M$ is a proper subset of $M$ for every $m \geq 1$.
For $g \in M$, $\sigma_g=(g, \mathrm{id}_C): C \to X_C$ denotes the corresponding section of $\operatorname{pr}_C$. For $g \in M$, we have $$\begin{aligned}
\sigma_g(C) \cap I_{f_C^m} = \varnothing
&\iff \Phi(\{g\} \times C) \cap I_{f_C^m} = \varnothing \\
&\iff \{g\} \times C \cap \Phi^{-1}(I_{f_C^m}) = \varnothing \\
&\iff g \not\in \operatorname{pr}_M(\Phi^{-1}(I_{f_C^m})). \end{aligned}$$ Set $(f_C^m)^*\mathcal O_{X_C}(1) \equiv \mathcal O_{X_C}(d_m) \otimes
\mathcal O_{X_C}(e_mF)$, where $F$ is a general fiber of $\operatorname{pr}_C$. Then $d_m, e_m \geq 0$ by Lemma \[lem5.0\]. Take $g \in M \setminus \bigcup_{m \geq 1} \operatorname{pr}_M(\Phi^{-1}(I_{f_C^m}))$. We compute $$\begin{aligned}
\underline \alpha_f(P_g)
&= \liminf_{m \to \infty} \deg ((f_C^m \circ \sigma_g)^*\mathcal O_{X_C}(1))_+^{1/m} \ \ \
\mathrm{(by\ Proposition\ \ref{prop2.12.3})}\\
&= \liminf_{m \to \infty} \deg (\sigma_g^*(f_C^m)^*\mathcal O_{X_C}(1))_+^{1/m}\ \ \
\mathrm{(by\ Lemma\ \ref{lem4.2}\ (ii))} \\
&= \liminf_{m \to \infty} (\mathcal O_{X_C}(d_m) \otimes
\mathcal O_{X_C}(e_mF) \cdot \sigma_{g*}C)_+^{1/m} \\
&\geq \liminf_{m \to \infty} (\mathcal O_{X_C}(d_m) \cdot \sigma_{g*} C)_+^{1/m} \\
&= \liminf_{m \to \infty} (d_m (\mathcal O_{X_C}(1) \cdot \sigma_{g*}C))_+^{1/m} \\
&= \liminf_{m \to \infty} d_m^{1/m} \\
&= \delta_f.\end{aligned}$$ Note that $(\mathcal O_{X_C}(1) \cdot \sigma_{g*}C)
= (\mathcal O_{\mathbb P_k^n}(1) \cdot g_*C) >0$ since $d= \deg(g) >0$. Combining with Theorem \[thm4.4\], it follows that $\alpha_f(P_g)$ exists and $\alpha_f(P_g)=\delta_f$.
Construction of orbits {#sec_orbit}
======================
In this section, we consider a problem on the existence of the rational points at which the arithmetic degree attains the dynamical degree.
Over $\overline{\mathbb Q}$, the following problem is studied in some papers (cf. [@KaSi1 Theorem 3] and [@MSS Theorem 1.7]).
\[prob5.3\] Let $(X, f)$ be a dynamical system over $\overline{\mathbb Q}$. Is there a subset $S \subset X_f$ such that
- $\alpha_f(P)$ exists and $\alpha_f(P)=\delta_f$ for every $P \in S$,
- $O_f(P) \cap O_f(Q) = \varnothing$ if $P, Q \in S$ and $P \neq Q$, and
- $S$ is a Zariski dense subset of $X$.
or not?
We give an affirmative answer for any dynamical system over $\overline{k(t)}$, where $k$ is an uncountable algebraically closed field of characteristic 0.
\[thm\_orbit\] Assume that $k$ is an uncountable algebraically closed field of characteristic 0. Let $(X, f)$ be a dynamical system over $\overline{k(t)}$. Then there exists a subset $S \subset X_f$ such that
- $\alpha_f(P)$ exists and $\alpha_f(P)=\delta_f$ for every $P \in S$,
- $O_f(P) \cap O_f(Q) = \varnothing$ if $P, Q \in S$ and $P \neq Q$, and
- $S$ is a Zariski dense subset of $X$.
\[lem\_points\] Let $k$ be an uncountable algebraically closed field and $X$ an algebraic scheme of positive dimension over $k$. Let $Z_1, Z_2, \ldots \subset X$ be proper closed subsets of $X$. Then $\bigcup_i Z_i \neq X$ and there exists a countable set $M$ of $k$-valued points of $X \setminus \bigcup_i Z_i$ such that $M$ is Zariski dense in $X$.
Replacing $X$ by an affine open subset, we may assume that $X$ is affine. By Noether’s normalization lemma, there is a finite cover $\phi: X \to \mathbb A_k^n$. Replacing $X$ and $Z_1, Z_2, \ldots$ by $\mathbb A_k^n$ and $\phi(Z_1), \phi(Z_2), \ldots$, we may assume that $X=\mathbb A_k^n$.
We prove the claim by induction on $n$. Assume that $n=1$. Then $Z_i(k)$ is a finite set for every $i$ and $\mathbb A_k^1(k)=k$ is uncountable, so $\bigcup_i Z_i \neq \mathbb A_k^1$. We take an infinite subset $M$ of $\mathbb A_k^1(k) \setminus \bigcup_i Z_i(k)$. Then $M$ is a Zariski dense subset of $\mathbb A_k^1$.
Assume that the claim holds for $\mathbb A_k^1, \mathbb A_k^2,\ldots, \mathbb A_k^{n-1}$. Define $p: \mathbb A_k^n \to \mathbb A_k^{n-1}$ and $q: \mathbb A_k^n \to \mathbb A_k^1$ as $p(x_1, \ldots ,x_n)=(x_1,\ldots ,x_{n-1})$ and $q(x_1, \ldots ,x_n)=x_n$. Let $\{ Z_j'\}_j$ (resp. $\{ Z_k''\}_k$) be the members of $\{ Z_i \}_i$ such that $p(Z_j') \neq \mathbb A_k^{n-1}$ (resp. $p(Z_k'') = \mathbb A_k^{n-1}$). Let $W_k \subset \mathbb A_k^{n-1}$ be the set of points $w \in \mathbb A_k^{n-1}$ such that the fiber $(p|_{Z_k''})^{-1}(w)= p^{-1}(w) \cap Z_k''$ of $p|_{Z_k''}$ over $w$ has positive dimension. Then $W_k$ is a proper closed subset of $\mathbb A_k^{n-1}$. By induction hypothesis, $\bigcup_j \phi(Z_j') \cup \bigcup_k W_k \neq \mathbb A_k^{n-1}$ and we can take a countable subset $M' \subset \mathbb A_k^{n-1}(k) \setminus (\bigcup_j \phi(Z_j')(k) \cup \bigcup_k W_k(k))$ such that $M'=\{ a_m\}_{m=1}^\infty$ is Zariski dense in $\mathbb A_k^{n-1}$. For every $m$ and $k$, $p^{-1}(a_m) \cap Z_k'' \neq \mathbb A_k^1$ since $a_m \not\in W_k$. So $\bigcup_{m,k} (p^{-1}(a_m) \cap Z_k'') \not\subset \mathbb A_k^1$ and we can take a countable subset $M'' \subset \mathbb A_k^1(k) \setminus \bigcup_{m,k} (p^{-1}(a_m)(k) \cap Z_k''(k))$ such that $M''$ is Zariski dense in $\mathbb A_k^1$, by induction hypothesis. Set $M=M' \times M'' \subset \mathbb A_k^{n-1} \times \mathbb A_k^1$. Then it is clear that $M$ satisfies the claim.
Take a model $(X_{C_0} \overset{\pi_{C_0}}{\to} C_0, f_{C_0})$ of $(X, f)$ over a curve $C_0$. For any curve $C$ with a finite morphism $C \to C_0$, $(X_C \overset{\pi_C}{\to} C, f_C)$ denotes the pull-back of $(X_{C_0} \overset{\pi_{C_0}}{\to} C_0, f_{C_0})$ by $C \to C_0$ and $\psi_C: X_C \to X_{C_0}$ denote the projection. For a section $\sigma:C \to X$ and a finite morphism $C' \to C$ of curves, $(\sigma)_{C'}: C' \to X \times_C C'$ denotes the pull-back of $\sigma$ by $C' \to C$.
By Lemma \[lem\_points\], we can take a countable subset $M=\{a_i\}_{i=1}^\infty \subset X_{C_0}$ such that
- $M$ is Zariski dense in $X_{C_0}$ and
- $a_i \not\in I_{f_{C_0}^m}$ for every $m \geq 1$ and $i \geq 1$.
We will construct rational points $P_1, P_2, \ldots \in X$ inductively. Let $C_k \to C_{k-1} \to \cdots \to C_1 \to C_0$ be a sequence of finite morphisms of curves and $P_i \in X$ a rational point corresponding to a section $\sigma_i: C_i \to X_{C_i}$ of $\pi_{C_i}$ for each $1 \leq i \leq k$. Assume that $P_1, \ldots, P_k \in X$ satisfy the following condition $(*)_k$:
- $P_i \in X_f$ for $1 \leq i \leq k$,
- $\alpha_f(P_i)=\delta_f$ for $1 \leq i \leq k$,
- $O_f(P_i) \cap O_f(P_j) = \varnothing$ if $1 \leq i, j \leq k$ and $i \neq j$, and
- $a_i \in \operatorname{Im}(\psi_{C_i} \circ \sigma_i)$ for $1 \leq i \leq k$.
Set $n=\dim X$. Note that $X_{C_k}$ is smooth outside a finite union of fibers of $\pi_{C_k}$.
Let $p_k: X_k \to X_{C_k}$ be a resolution of $(X_{C_k})_{\mathrm{red}}$ whose exceptional locus is contained in a finite union of fibers of $\pi_{C_k}$. By blowing up a point in $(p_k \circ \psi_{C_k})^{-1}(a_{k+1})$, we may assume that $(p_k \circ \psi_{C_k})^{-1}(a_{k+1})$ has codimension 1. We take a very ample divisor $H$ on $X_k$ and suitable members $H_1, \ldots, H_n \in |H|$, and set $C_{k+1}=H_1 \cap \cdots \cap H_n$. Let $\iota: C_{k+1} \to X_k$ denote the inclusion. We can choose $H_1, \ldots, H_n$ as satisfying
- $C_{k+1}$ is a smooth and irreducible curve satisfying $\operatorname{Im}(\pi_{C_k} \circ p_k \circ \iota)=C_k$,
- $C_{k+1} \not\subset p_k^{-1} (f_{C_k}^m)^{-1}(I_{f_{C_k}})$ for every $m \geq 0$,
- $C_{k+1} \cap I_{f_k^m} = \varnothing$ for every $m \geq 1$,
- $C_{k+1} \not\subset (f_k^{m'})^{-1}(\operatorname{Im}(f_k^m \circ p_k^{-1} \circ (\sigma_i)_{C_k}))$ for every $m, m'$ and $1 \leq i \leq k$, and
- $a_{k+1} \in \operatorname{Im}(\psi_{C_k} \circ p_k \circ \iota)$.
Set $\phi=\pi_{C_k} \circ p_k \circ \iota: C_{k+1} \to C_k$. Then we obtain the following diagram: $$\xymatrix{
C_{k+1} \ar@(d, ul)[ddr]_{\mathrm{id}} \ar@(r, ul)[drr]^{p_k \circ \iota} \ar[dr]^{\sigma_{k+1}}
& & \\
& X_{C_{k+1}} \ar[r]^\psi \ar[d]^{\pi_{C_{k+1}}} & X_{C_k} \ar[d]^{\pi_{C_k}}\\
& C_{k+1} \ar[r]^{\phi} & C_k
}$$ Here $X_{C_{k+1}}=X_{C_k} \times_{C_k} C_{k+1}$ and $\sigma_{k+1}$ is the unique morphism which makes the above diagram commutative. Let $P_{k+1} \in X$ be the rational point of $X$ corresponding to $\sigma_{k+1}$. By (II), $\operatorname{Im}(\sigma_{k+1}) \not\subset (f_{C_{k+1}}^m)^{-1}(I_{f_{C_{k+1}}})$ for every $m \geq 0$. Hence $\operatorname{Im}(f_{C_{k+1}}^m \circ \sigma_{k+1}) \not\subset I_{f_{C_{k+1}}}$ and so $f^m(P_{k+1}) \not\in I_f$ for every $m \geq 0$. Therefore $P_{k+1} \in X_f$.
Let $p_{k+1}: X_{k+1} \to X_{C_{k+1}}$ be a resolution of $(X_{C_{k+1}})_{\mathrm{red}}$ whose exceptional locus is over a finite union of fibers of $\pi_{C_{k+1}}$ and $\theta=p_k^{-1} \circ \psi \circ p_{k+1}$ becomes a morphism. Then we obtain the following diagram: $$\xymatrix{
X_{k+1} \ar[r]^\theta \ar[d]^{p_{k+1}} & X_k \ar[d]^{p_k} & \\
X_{C_{k+1}} \ar[r]^\psi \ar[d]^{\pi_{C_{k+1}}} & X_{C_k} \ar[r]^{\psi_k} \ar[d]^{\pi_{C_k}}
& X_{C_0} \ar[d]^{\pi_{C_0}} \\
C_{k+1} \ar[r]^{\phi} \ar@(ul, dl)[u]^{\sigma_{k+1}} \ar[ur]_{p_k \circ \iota} & C_k \ar[r] & C_0
}$$ Set $f_k=p_k^{-1} \circ f_{C_k} \circ p_k$, $f_{k+1}=p_{k+1}^{-1} \circ f_{C_{k+1}} \circ p_{k+1}$ and $\sigma'_{k+1}= p_{k+1}^{-1} \circ \sigma_{k+1}$. Fix a positive integer $m$. Then it follows that $p_k \circ \theta \circ f_{k+1}^m \circ \sigma_{k+1}'
= p_k \circ f_k^m \circ \iota$. Since $p_k$ is birational, we have $\theta \circ f_{k+1}^m \circ \sigma_{k+1}' =f_k^m \circ \iota$. Take an ample divisor $A$ on $X_{k+1}$ such that $A-\theta^*H$ is ample. We compute $$\begin{aligned}
\deg (f_{k+1}^m \circ \sigma_{k+1}')^*A
&\geq \deg (f_{k+1}^m \circ \sigma_{k+1}')^* \theta^*H \\
&= \deg (\theta \circ f_{k+1}^m \circ \sigma_{k+1}')^*H \\
&= \deg (f_k^m \circ \iota)^*H.\end{aligned}$$ By (III) and Lemma \[lem4.2\] (ii), we have $$\deg (f_k^m \circ \iota)^*H=\deg \iota^*(f_k^m)^*H= ((f_k^m)^*H \cdot H^{n-1}).$$ Now $(X_{k+1} \xrightarrow{\pi_{C_{k+1}} \circ p_{k+1}} C_{k+1}, f_{k+1})$ is a model of $(X, f)$ and $\sigma_{k+1}'$ is a section of $\pi_{C_{k+1}} \circ p_{k+1}$ corresponding to $P_{k+1}$. Therefore $$\begin{aligned}
\underline \alpha_f(P_{k+1})
&= \liminf_{m \to \infty} \deg((f_{k+1}^m \circ \sigma_{k+1}')^*A)^{1/m}\ \ \
\mathrm{(by\ Proposition\ \ref{prop2.12.3})} \\
&\geq \liminf_{m \to \infty} ((f_k^m)^*H \cdot H^{n-1})^{1/m} \\
&=\delta_{f_k} = \delta_f.\end{aligned}$$ So $\alpha_f(P_{k+1})$ exists and $\alpha_f(P_{k+1})=\delta_f$.
Fix $i \in \{0, \ldots, k \}$ and $m, m' \geq 0$. By (IV), $\operatorname{Im}(f_k^{m'} \circ \iota) \neq \operatorname{Im}(f_k^m \circ p_k^{-1} \circ (\sigma_i)_{C_k})
=\operatorname{Im}(f_k^m \circ p_k^{-1} \circ (\sigma_i)_{C_k} \circ \phi)$. Since $p_k$ is birational and the images of both $f_k^{m'} \circ \iota$ and $f_k^m \circ p_k^{-1} \circ (\sigma_i)_{C_k} \circ \phi$ intersects the isomorphic locus of $p_k$, we have $$\operatorname{Im}(p_k \circ f_k^{m'} \circ \iota)
\neq \operatorname{Im}(p_k \circ f_k^m \circ p_k^{-1} \circ (\sigma_i)_{C_k} \circ \phi).$$ On the other hand, $$p_k \circ f_k^{m'} \circ \iota = \psi \circ f_{C_{k+1}}^{m'} \circ \sigma_{k+1}$$ and $$p_k \circ f_k^m \circ p_k^{-1} \circ (\sigma_i)_{C_k} \circ \phi
= \psi \circ f_{C_{k+1}}^{m} \circ (\sigma_i)_{C_{k+1}}.$$ So $\operatorname{Im}(\psi \circ f_{C_{k+1}}^{m'} \circ \sigma_{k+1}) \neq
\operatorname{Im}(\psi \circ f_{C_{k+1}}^{m} \circ (\sigma_i)_{C_{k+1}})$, and so $f_{C_{k+1}}^{m'} \circ \sigma_{k+1} \neq f_{C_{k+1}}^{m} \circ (\sigma_i)_{C_{k+1}}$. This means that $f^{m'}(P_{k+1}) \neq f^m(P_i)$. Hence $O_f(P_{k+1}) \cap O_f(P_i) = \varnothing$.
Set $\psi_{k+1}=\psi_k \circ \psi$. Then $\psi_{k+1} \circ \sigma_{k+1}= \psi_k \circ p_k \circ \iota$. By (V), $a_i \in \operatorname{Im}(\psi_{k+1} \circ \sigma_{k+1})$. As a consequence, $P_1, \ldots, P_{k+1}$ satisfies $(*)_{k+1}$.
Continuing this process, we obtain a subset $S= \{P_1, P_2, \ldots \} \subset X_f$, a sequence $\cdots \to C_1 \to C_0$ of finite morphisms of curves, and sections $\sigma_i: C_i \to X_{C_i}$ corresponding to $P_i$ for each $i \geq 1$ such that
- $\alpha_f(P_i)=\delta_f$ for every $i$,
- $O_f(P_i) \cap O_f(P_j) = \varnothing$ if $i \neq j$, and
- $a_i \in \operatorname{Im}(\psi_{C_i} \circ \sigma_i)$ for every $i$.
So it is enough to show that $S$ is a Zariski dense subset of $X$. Let $Z \subset X$ be a proper closed subset of $X$. We take a finite cover $C \to C_0$ such that $Z$ lifts to a proper closed subset $Z_C \subset X_C= X_{C_0} \times_{C_0} C$. Since $\psi_C(Z_C)$ is a proper closed subset of $X_{C_0}$, $a_i \not\in \psi_C(Z_C)$ for some $i$. Take a curve $C'$ with finite morphisms $C' \to C$ and $C' \to C_i$ which makes the following diagram commutative: $$\xymatrix{
&C \ar[rd]&\\
C' \ar[ru] \ar[rd]&&C_{0}\\
&C_i \ar[ru]&
}$$ Set $Z_{C'} = Z_C \times_C C' \subset X_{C'}$. Since $a_i \in \operatorname{Im}(\psi_{C'} \circ (\sigma_i)_{C'})$ and $a_i \not\in \psi_C(Z_C)=\psi_{C'}(Z_{C'})$, $\operatorname{Im}((\sigma_i)_{C'}) \not\subset Z_{C'}$. So $P_i \not\in Z$. Therefore $S$ is a Zariski dense subset of $X$.
Theorem \[thm\_orbit\] includes the following result.
\[cor\_existence\] Assume that $k$ is an uncountable algebraically closed field of characteristic 0. Let $(X, f)$ be a dynamical system over $\overline{k(t)}$. Then there exists a rational point $P \in X_f$ such that $\alpha_f(P)=\delta_f$.
For a dynamical system on a rational variety, we can take a subset $S$ as in the statement of Theorem \[thm\_orbit\] over a fixed function field.
\[thm5.4\] Assume that $k$ is an uncountable algebraically closed field of characteristic 0. Let $(X, f)$ be a dynamical system over $\overline{k(t)}$ such that $X$ is rational. Then there exists a subset $S \subset X_f$ such that
- There exists a function field $K$ of a curve over $k$ and a model $X_K$ of $X$ over $K$ such that all points in $S$ are defined over $K$,
- $\alpha_f(P)=\delta_f$ for every $P \in S$,
- $S$ is a Zariski dense subset of $X$, and
- $O_f(P) \cap O_f(Q) = \varnothing$ if $P, Q \in S$ and $P \neq Q$.
We need a result in [@MSS].
\[Theorem:BirationalInvariance\] Let $f \colon X \dashrightarrow X$ and $g \colon Y \dashrightarrow Y$ be dominant rational self-maps on smooth projective varieties and $\phi \colon Y \dashrightarrow X$ a birational map such that $\phi \circ g= f \circ \phi$. Let $V \subset Y$ be an open subset such that $\phi|_V : V \to \phi(V)$ is an isomorphism. Then $\overline \alpha_g(Q)=\overline \alpha_f(\phi(Q))$ and $\underline \alpha_g(Q)=\underline \alpha_f(\phi(Q))$ for any $Q \in Y_g \cap \phi^{-1}(X_f)$ satisfying $O_g(Q) \subset V$.
Let $\phi: Y= \mathbb P^n_{\overline{k(t)}} \dashrightarrow X$ be a birational map. Set $g= \phi^{-1} \circ f \circ \phi$. Take open subsets $U \subset X$ and $V \subset Y$ such that $\phi|_V: V \to U$ is isomorphic. We can take a curve $C$, a model $(X_C \overset{\operatorname{pr}_C}{\to} C, f_C)$ (resp. $(Y_C=\mathbb P_{k}^n \times C \overset{\operatorname{pr}_C}{\to} C, g_C)$) of $(X, f)$ (resp. $(Y, g)$), a lift $U_C$ (resp. $V_C$) of $U$ (resp. $V$), and a birational map $\phi_C: Y_C \dashrightarrow X_C$ such that $\phi_C|_{V_C}: V_C \to U_C$ is isomorphic.
Set $Z_C=Y_C \setminus V_C$. By Lemma \[lem\_points\], we can take a countable subset $M=\{a_i=(b_i, c_i)\}_{i=1}^\infty \subset V_C$ such that
- $M$ is Zariski dense in $Y_C$,
- $a_i \not\in (g_C^m)^{-1}(I_{g_C} \cup Z_C) \cup \phi_C^{-1} (f_C^m)^{-1}(I_{f_C})$ for every $m \geq 0$, and
- $b_i \not\in \operatorname{pr}_{\mathbb P_{k}^n}(I_{g_C^m})$ for every $m \geq 1$ and $i \geq 1$.
Note that $\operatorname{pr}_{\mathbb P_{k}^n}(I_{g_C^m})$ is a proper closed subset of $\mathbb P_{k}^n$ for every $m$ because $\dim I_{g_C^m} \leq n-1$. For $m \geq 1$, let $J_m \subset \operatorname{pr}_{\mathbb P_{k}^n}(I_{g^m})$ be the closed subset over which the fibers of $\operatorname{pr}_{\mathbb P_{k}^n}|_{I_{g^m}}$ have positive dimensions. Then $\operatorname{codim}_{\mathbb P_{k}^n} J_m \geq 2$. We take a point $q \in C$ such that $q \neq c_i$ for all $i$ and $f_C|_{F_q}: F_q \dashrightarrow F_q$ is well-defined and dominant, where $F_q$ denotes the fiber of $\operatorname{pr}_C: Y_C \to C$ over $q$.
Assume that we have sections $\tau_1, \ldots, \tau_k$ of $\operatorname{pr}_C$ corresponding to $Q_1, \ldots, Q_k \in Y$ which satisfy the condition $(*)_k$:
- $Q_i \in Y_g \cap \phi^{-1}(X_f)$ such that $O_g(Q_i) \subset V$ for $1 \leq i \leq k$,
- $\alpha_g(Q_i)=\delta_g$ for $1 \leq i \leq k$,
- $O_g(Q_i) \cap O_g(Q_j) = \varnothing$ if $1 \leq i, j \leq k$ and $i \neq j$, and
- $a_i \in \operatorname{Im}(\tau_i)$ for $1 \leq i \leq k$.
We take general hyperplanes $H_1, \ldots, H_{n-1}$ of $\mathbb P_{k}^n$. Then we have a line $L=H_1 \cap \cdots \cap H_{n-1} \subset \mathbb P_{k}^n$. We can choose $H_1, \ldots, H_{n-1}$ as satisfying
- $L \not\subset \operatorname{pr}_{\mathbb P_{k}^n}(I_{g_C^m})$ and $L \cap J_m= \varnothing$ for every $m \geq 1$,
- $L \times \{q\} \not\subset
(g_C^{m'}|_{F_q})^{-1}(\operatorname{Im}(g_C^m \circ \tau_i)\cap F_q)$ for every $m, m' \geq 0$ and $1 \leq i \leq k$, and
- $b_{k+1} \in L$.
Note that $\operatorname{Im}(g_C^m \circ \tau_i)\cap F_q$ is a point since $g_C^m \circ \tau_i$ is a section of $\operatorname{pr}_C$. By (I), $I_{g_C^m} \cap (L \times C)
\subset Y$ is a finite set. Set $\bigcup_{m \geq 1} (I_{g_C^m} \cap (L \times C))
=\{ (x_j, y_j) \}_j$. We can construct a finite cover $\phi: C \to L$ satisfying
- $\phi(c_{k+1})=b_{k+1}$,
- $\phi(y_j) \neq x_j$ for every $j$, and
- $(\phi(q), q) \not\in (g^{m'}|_{F_q})^{-1}(\operatorname{Im}(g^m \circ \tau_i)\cap F_q)$ for every $m, m'$ and $1 \leq i \leq k$,
by composing a fixed finite morphism $C \to L$ with a suitable automorphism on $L$.
Set $\tau_{k+1}: C \overset{(\phi, \mathrm{id}_C)}{\to} L \times C
\hookrightarrow X_C$ and let $Q_{k+1} \in \mathbb P^n(K)$ be the corresponding rational point. Then $a_{k+1} \in \operatorname{Im}(\tau_{k+1})$ by (1). Since $a_{k+1} \not\in (g_C^m)^{-1}(I_{g_C} \cup Z_C) \cup \phi_C^{-1} (f_C^m)^{-1}(I_{f_C})$, $\operatorname{Im}(g_C^m \circ \tau_{k+1}) \not\subset I_{g_C} \cup Z_C$ and $\operatorname{Im}(f_C^m \circ \phi_C \circ \tau_{k+1})
\not\in I_{f_C}$ for every $m \geq 0$. Hence $Q_{k+1} \in Y_g \cap \phi^{-1}(X_f)$ such that $O_g(Q_{k+1}) \subset V$. Further, $\operatorname{Im}(\tau_{k+1}) \cap I_{f^m}=\varnothing$ for every $m$ by (2). Therefore $\alpha_g(Q_{k+1})$ exists and $\alpha_g(Q_{k+1})=\delta_g$ by Theorem \[thm5.0.1\] (i). Moreover, $(g_C^{m'} \circ \tau_{k+1})(q) \neq (g^m \circ \tau_i)(q)$ for every $m, m'$ and $1 \leq i \leq k$ by (3). In particular, it follows that $g_C^{m'} \circ \tau_{k+1} \neq g_C^m \circ \tau_i$ for every $m, m'$ and $1 \leq i \leq k$. This means that $O_g(Q_{k+1}) \cap O_g(Q_i) = \varnothing$ for $1 \leq i \leq k$. As a consequence, $\{ \tau_1, \ldots, \tau_k, \tau_{k+1} \}$ satisfies $(*)_{k+1}$.
Continuing this process, we obtain morphisms $\tau_1, \tau_2, \ldots $ and a subset $T= \{Q_1, Q_2, \ldots \} \subset V \cap Y_g$. Set $P_i = \phi(Q_i)$ and $S=\{ P_1, P_2, \ldots \}$. Then $P_i$ corresponds to a section $\sigma_i=\phi_{C} \circ \tau_i$. Since $Q_i \in Y_g \cap \phi^{-1}(X_f)$ and $O_g(Q_i) \subset V$, $\alpha_g(Q_i)=\alpha_f(P_i)$ by Theorem \[Theorem:BirationalInvariance\]. So we have $\alpha_f(P_i)=\delta_f$. For $i, j$ with $i \neq j$, $O_g(Q_i) \cap O_g(Q_j) = \varnothing$ implies that $O_f(P_i) \cap O_f(P_j) = \varnothing$. Since $p_i \in \operatorname{Im}(\tau_i)$ for every $i$, $\bigcup_i \operatorname{Im}(\tau_i)$ is Zariski dense in $Y_C$ and so $\bigcup_i \operatorname{Im}(\sigma_i)$ is Zariski dense in $X_C$. So $S$ is Zariski dense in $X$. Therefore $S$ satisfies the claim.
[KoMo]{}
V. Guedj, Ergodic properties of rational mappings with large topological degree, Ann. of Math. [**161**]{} (2005), 1589–1607.
M. Hindry, J. H. Silverman, *Diophantine Geometry: An Introduction*, Springer-Verlag, New York, 2000.
S. Kawaguchi, J. H. Silverman, Examples of dynamical degree equals arithmetic degree, Michigan Math. J. [**63**]{} (2014), no. 1, 41–63.
S. Kawaguchi, J. H. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. [**713**]{} (2016), 21–48.
S. Kawaguchi, J. H. Silverman, Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties, Trans. Amer. Math. Soc. [**368**]{} (2016), no. 7, 5009–5035.
J. Kollár, S. Mori, *Birational geometry of algebraic varieties*, Cambridge Univ. Press, 1998.
S. Lang, *Fundamentals of Diophantine Geometry*, Springer-Verlag, New York, 1983.
Y. Matsuzawa, On upper bounds of arithmetic degrees, preprint.
Y. Matsuzawa, K. Sano, T. Shibata, Arithmetic degrees and dynamical degrees of endomorphisms on surfaces, preprint.
J. H. Silverman, Arithmetic and dynamical degrees on abelian varieties, preprint.
T. T. Truong, (Relative) dynamical degrees of rational maps over an algebraic closed field, preprint.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We discuss the possibility for matter effects in the three-neutrino oscillations of the atmospheric $\nu_e$ ($\bar{\nu}_e$) and $\nu_\mu$ ($\bar{\nu}_\mu$), driven by one neutrino mass squared difference, $|\Delta m^2_{31}| \gg \Delta m^2_{21}$, to be observable under appropriate conditions. We derive predictions for the Nadir angle ($\theta_n$) dependence of the ratio $N_{\mu}/N_e$ of the rates of the $\mu-$like and $e-$like multi-GeV events which is particularly sensitive to the Earth matter effects in the atmospheric neutrino oscillations, and thus to the values of $\sin^2\theta_{13}$ and $\sin^2\theta_{23}$, and also to the type of neutrino mass spectrum.'
address:
- 'Departamento de Física Teórica and IFIC, Universidad de Valencia-CSIC, 46100 Burjassot, Valencia, Spain'
- 'Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA'
- 'Department of Physics and Astronomy, Vanderbilt University, Nashville TN 37235, USA'
author:
- 'Sergio Palomares-Ruiz[^1], José Bernabéu'
title: 'Atmospheric neutrinos and $\nu$-mass hierarchy'
---
Introduction
============
Present evidence for neutrino masses and mixings can be summarized as: 1) the atmospheric $|\Delta m^{2}_{31}|$ is associated with a mixing $\theta_{23}$ close to maximal [@SKatm00]; 2) the solar $\Delta
m^{2}_{21}$ prefers the LMA-MSW solution [@solar; @KamLAND]; 3) CHOOZ reactor data [@CHOOZ] give severe limits for $|U_{e
3}|$. Here, we discuss that contrary to a wide spread belief, Earth effects on the propagation of atmospheric neutrinos can become observable, in detectors with lepton charge-discrimination [@mantle; @core] and in water-Čerenkov ones [@th13cerenkov], even if $|U_{e 3}|$ is small, yet non-vanishing. This fact could allow to determine the sign of $\Delta
m^{2}_{31}$ and obtain more stringent constraints on the values of $\theta_{13}$ and $\theta_{23}$. Getting more precise information about the value of these mixing angles and determining the type of the neutrino mass spectrum (with normal or inverted hierarchy) with a higher precision is of fundamental importance for the progress in the studies of neutrino mixing.
We study here the possibilities to obtain this type of information using the atmospheric neutrino data that can be provided by present and future water-Čerenkov detectors. For baselines $L$ smaller than the Earth diameter, appropiate for atmospheric neutrinos, $\frac{\Delta m^{2}_{21}}{4 E} L \equiv \Delta_{21} \ll 1$, so that we will neglect the (1,2)-oscillating phase in vacuum against the (2,3)-one. This is a very good aproximation for the presently best favored LMA-I solution to the solar neutrino problem [@solar].
The Earth matter effects, which can resonantly enhance the $\nu_{\mu}
\rightarrow \nu_e$ and $\nu_{e} \rightarrow \nu_{\mu(\tau)}$ transitions, lead to the reduction of the rate of the multi-GeV $\mu-$like events and to the increase of the rate of the multi-GeV $e-$like events in a water-Čerenkov detector with respect to the case of absence of these transitions (see, e.g., [@core; @SP3198]). Correspondingly, as observables which are sensitive to the Earth matter effects, we consider the Nadir-angle distribution of the ratio $N_{\mu}/N_{e}$, where $N_{\mu}$ and $N_e$ are the multi-GeV $\mu-$like and $e$-like numbers of events, respectively.
The matter-induced neutrino spectrum
====================================
If $V$ is the effective neutrino potential, in going from $\nu$ to $\overline{\nu}$, there are matter-induced CP- and CPT- odd effects associated with the change $V \rightarrow - V$. The effects here discussed depend on the interference between the different flavors and on the relative sign between $2 E V$ and $\Delta m^{2}_{31}$. For atmospheric neutrinos, an appreciable interference will be present if and only if there are appreciable matter effects, i.e., the “connecting” mixing $U_{e 3}$ between the $\nu_e$-flavor and the $\nu_3$ mass eigenstate does not vanish.
For $s_{13} \equiv \sin{\theta_{13}} = 0$, matter effects lead to a breaking of the (1,2)-degeneracy such that $\tilde{\nu}_2$ coincides with $\nu_e$. The net effect is that $\nu^{m}_{1}$ and $\nu^{m}_{3}$ lead to the atmospheric $\nu_{\mu} \rightarrow \nu_{\tau}$ indicated by SK, likewise in vacuum the (2,3)-mixing does. The $\nu_e$-flavor decouples in matter, even if there was a large mixing in the (1,2)-system. No matter effects would then be expected when starting with $\nu_{\mu}$, i.e., there would be no chances to distinguish the type of mass hierarchy by these means. However, for small $s_{13}$, even if the effects on the spectrum are expected to be small, there could be a substantial mixing of $\nu_e$ with $\nu^{m}_{3}$. This would lead to a resonant MSW behaviour in the case of neutrinos crossing only the Earth mantle and to new resonant effects (NOLR) [@SP3198] in the case of neutrinos crossing also the Earth core. But still $\langle \nu^{m}_{1} | \nu_e \rangle = 0$. This vanishing mixing in matter is responsible for the absence of fundamental CP-violating effects, even if there are three non-degenerate mass eigenstates in matter. In vacuum, the absence of genuine CP-odd probabilities was due to the degeneracy $\Delta_{21} =
0$.
Atmospheric neutrino oscillations in the Earth
==============================================
The fluxes of atmospheric $\nu_{e,\mu}$ of energy $E$, which reach the detector after crossing the Earth along a given trajectory specified by the value of $\theta_{n}$, $\Phi_{\nu_{e,\mu}}(E,\theta_{n})$, are given by the following expressions in the case of the three-neutrino oscillations under discussion [@SP3198]:
$$\Phi_{\nu_e}(E,\theta_{n}) \cong
\Phi^{0}_{\nu_e}~\left ( 1 +
[s^2_{23}r - 1]~P_{2\nu}\right )
\label{Phie}$$
$$\begin{aligned}
\label{Phimu}
\Phi_{\nu_{\mu}}(E,\theta_{n}) \hspace{-3mm} & \cong \hspace{-2.5mm} &
\Phi^{0}_{\nu_{\mu}} \left( 1 + s^4_{23}~ [(s^2_{23}~r)^{-1} -
1]~P_{2\nu} \right. \nonumber \\
& & \left. - 2c^2_{23}s^2_{23}~\left [ 1 - Re~( e^{-i\kappa}
A_{2\nu}) \right ] \right) \end{aligned}$$
where $P_{2\nu} \equiv P_{2\nu}(\Delta m^2_{31},
\theta_{13};E,\theta_{n})$ is the probability of two-neutrino oscillations in the Earth, $\kappa$ and $A_{2\nu}$ are known phase and two-neutrino transition probability amplitude, $\Phi^{0}_{\nu_{e(\mu)}}$ is the $\nu_{e(\mu)}$ flux in the absence of neutrino oscillations and
$$r \equiv r(E,\theta_{n}) \equiv
\frac{\Phi^{0}_{\nu_{\mu}}(E,\theta_{n})}
{\Phi^{0}_{\nu_{e}}(E,\theta_{n})}~~.
\label{r}$$
The predicted ratio [@flux] for atmospheric sub-GeV neutrinos is $r \cong (2.0 - 2.5)$, whereas $r \cong (2.6 - 4.5)$ for multi-GeV atmospheric neutrinos. If $s^2_{23} = 0.5$, the possible effects of the $\nu_{\mu} \rightarrow \nu_{e}$ and $\nu_{e} \rightarrow \nu_{\mu
(\tau)}$ transitions on the sub-GeV $e-$like events would be strongly suppressed even if these transitions were maximally enhanced by the Earth matter effects. On the other hand, $r > 2$ for the multi-GeV sample, and matter effects can show up. Thus, in the case under study, the effects of the $\nu_{\mu} \rightarrow \nu_{e}$ and $\nu_{e} \rightarrow \nu_{\mu (\tau)}$ oscillations, increase with the increase of $s^2_{23}$, are considerably larger in the multi-GeV samples of events than in the sub-GeV ones and in the multi-GeV case, they lead to an increase of the rate of $e-$like events and to a slight decrease of the $\mu-$like event rate. This discussion suggests that the quantity most sensitive to the effects of the oscillations of interest should be the ratio of the $\mu-$like and $e-$like multi-GeV events, $N_{\mu}/N_{e}$.
If $s_{13} \ne 0$, the Earth matter effects can resonantly enhance $P_{2\nu}$ for $\Delta m^2_{31} > 0$ and $\bar{P}_{2\nu}$ if $\Delta
m^2_{31} < 0$ [@mantle]. Due to the difference of cross sections for neutrinos and antineutrinos, approximately 2/3 of the total rate of the $\mu-$like and $e-$like multi-GeV atmospheric neutrino events in a water-Čerenkov detector, i.e., $\sim 2N_{\mu}/3$ and $\sim 2N_e/3$, are due to neutrinos $\nu_{\mu}$ and $\nu_e$, respectively, while the remaining $\sim 1/3$ of the multi-GeV event rates, i.e., $\sim N_{\mu}/3$ and $\sim N_e/3$, are produced by antineutrinos $\bar{\nu}_{\mu}$ and $\bar{\nu}_e$. This implies that the Earth matter effects in the multi-GeV samples of $\mu-$like and $e-$like events will be larger if $\Delta m^2_{31} > 0$ (normal hierarchy), than if $\Delta m^2_{31} < 0$ (inverted hierarchy). Thus, the ratio $N_{\mu}/N_e$ of the multi-GeV $\mu-$like and $e-$like event rates measured in water-Čerenkov detectors is sensitive, in principle, to the type of the neutrino mass spectrum [@th13cerenkov].
Results
=======
In Fig. 1 we show the predicted dependences on $\cos\theta_n$ of the ratios of the multi-GeV $\mu-$ and $e-$ like events, integrated over the neutrino energy from the interval $E = (2 - 10)$ GeV, for different cases (see figure).
For $\cos\theta_n \ltap 0.2$ and $|\Delta m^2_{31}| = 3\times
10^{-3}~{\rm eV^2}$, the oscillations of the atmospheric $\nu_e$ and $\bar{\nu}_e$ with energies in the multi-GeV range $E \sim
(2 - 10)$ GeV, are suppressed. If $\Delta m^2_{31} > 0$, for instance, the Earth matter effects suppress the antineutrino
=8.5cm
[ Figure 1. The dependence on $\cos\theta_n$ of the ratios of the multi-GeV $\mu-$ and $e-$ like events, integrated over the neutrino energy in the interval $E = (2 - 10)$ GeV, in the cases i) of two-neutrino $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillations in vacuum and no $\nu_e$ oscillations, $N^{2\nu}_{\mu}/N^{0}_{e}$ (solid lines), ii) three-neutrino oscillations in vacuum of $\nu_{\mu}$ and $\nu_e$, $(N^{3\nu}_{\mu}/N^{3\nu}_{e})_{vac}$ (dash-dotted lines), iii) three-neutrino oscillations of $\nu_{\mu}$ and $\nu_e$ and in the Earth and neutrino mass spectrum with normal hierarchy $(N^{3\nu}_{\mu}/N^{3\nu}_{e})_{\rm NH}$ (dashed lines), or with inverted hierarchy, $(N^{3\nu}_{\mu}/N^{3\nu}_{e})_{\rm IH}$ (dotted lines). The results shown are for $|\Delta m^2_{31}| = 3\times
10^{-3}~{\rm eV^2}$, $\sin^2\theta_{23} = 0.36~(\rm upper~panels);~0.50
~(\rm middle~panels);~0.64$ (lower panels), and $\sin^22\theta_{13} = 0.05$ (left panels); $0.10 (\rm
right~panels)$.]{}
oscillation probability $\bar{P}_{2\nu}$, but can enhance the neutrino mixing in matter. However, since the neutrino path in the Earth mantle is relatively short, $P_{2\nu} \ll 1$.
At $\cos\theta_n \gtap 0.4$, the Earth matter effects in the oscillations of the atmospheric $\nu_{\mu}$ and $\nu_e$ can generate noticeable differences between $N^{2\nu}_{\mu}/N^{0}_{e}$ (or $(N^{3\nu}_{\mu}/N^{3\nu}_{e})_{vac}$) and $(N^{3\nu}_{\mu}/N^{3\nu}_{e})_{\rm NH(IH)}$, as well as between $(N^{3\nu}_{\mu}/N^{3\nu}_{e})_{\rm NH}$ and $(N^{3\nu}_{\mu}/N^{3\nu}_{e})_{\rm IH}$.
Conclusions
===========
We have studied the possibility to obtain evidences for Earth matter enhanced atmospheric neutrino oscillations involving, in particular, the $\nu_e$, from the analysis of the $\mu-$like and $e-$like multi-GeV event data that can be provided by present and future water-Čerenkov detectors. We have seen that such evidences could give also important quantitative information on the values of $\sin^2\theta_{13}$ and $\sin^2\theta_{23}$ and on the sign of $\Delta
m^2_{31}$.
[**Acknowledgments**]{} — This work is supported by the Spanish Grant FPA2002-00612 of the MCT, by the Spanish MCD and SPR in part by NASA Grant ATP02-0000-0151.
[9]{} M. Shiozawa, talk given at “Neutrino’02”, May 25 - 30, 2002, Munich, Germany.
Y. Fukuda [*et al.*]{}, Phys. Rev. Lett. [**86**]{} (2001) 5651 and 5656; Q. R. Ahmad [*et al.*]{}, Phys. Rev. Lett. [**87**]{} (2001) 071301; and Phys. Rev. Lett. [**89**]{} (2002) 011302 and 011301. K. Eguchi [*et al.*]{}, Phys. Rev. Lett. [**90**]{} (2003) 021802. M. Apollonio [*et al.*]{}, Phys. Lett. B [**466**]{} (1999) 415. M. C. Bañuls, G. Barenboim and J. Bernabéu, Phys. Lett. B [**513**]{}, 391 (2001). J. Bernabéu and S. Palomares-Ruiz, hep-ph/0112002; and Nucl. Phys. Proc. Suppl. [**110**]{}, 339 (2002), hep-ph/0201090. J. Bernabéu [*et al.*]{}, Phys. Lett. B [**531**]{}, 90 (2002) J. Bernabéu, S. Palomares-Ruiz, S. T. Petcov Nucl. Phys. B [**669**]{} (2003) 255. S. T. Petcov, Phys. Lett. B [**434**]{} (1998) 321, (E) [*ibid.*]{} B [**444**]{} (1998) 584; S. T. Petcov, Nucl. Phys. B (Proc. Suppl.) [**77**]{} (1999) 93, hep-ph/9809587; M. V. Chizhov, M. Maris and S. T. Petcov, hep-ph/9810501; M. V. Chizhov and S. T. Petcov, Phys. Rev. D [**63**]{} (2001) 073003. M. Honda [*et al.*]{}, Phys. Rev. D [**52**]{}, 4985 (1995); V. Agraval [*et al.*]{}, Phys. Rev. D [**53**]{}, 1314 (1996); G. Fiorentini, V. A. Naumov and F. L. Villante, Phys. Lett. B [**578**]{} (2000) 27.
[^1]: Talk presented at TAUP 2003, Seattle, USA, 5-9 September 2003.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Blind source separation techniques are used to reanalyse two exoplanetary transit lightcurves of the exoplanet HD189733b recorded with the IR camera IRAC on board the Spitzer Space Telescope at 3.6$\mu$m during the “cold” era. These observations, together with observations at other IR wavelengths, are crucial to characterise the atmosphere of the planet HD189733b. Previous analyses of the same datasets reported discrepant results, hence the necessity of the reanalyses. The method we used here is based on the Independent Component Analysis (ICA) statistical technique, which ensures a high degree of objectivity. The use of ICA to detrend single photometric observations in a self-consistent way is novel in the literature. The advantage of our reanalyses over previous work is that we do not have to make any assumptions on the structure of the unknown instrumental systematics. Such “admission of ignorance” may result in larger error bars than reported in the literature, up to a factor $1.6$. This is a worthwhile trade-off for much higher objectivity, necessary for trustworthy claims. Our main results are (1) improved and robust values of orbital and stellar parameters, (2) new measurements of the transit depths at 3.6$\mu$m, (3) consistency between the parameters estimated from the two observations, (4) repeatability of the measurement within the photometric level of $\sim 2 \times 10^{-4}$ in the IR, (5) no evidence of stellar variability at the same photometric level within 1 year.'
author:
- 'G. Morello, I. P. Waldmann, G. Tinetti'
- 'G. Peres'
- 'G. Micela'
- 'I. D. Howarth'
title: A new look at Spitzer primary transit observations of the exoplanet HD189733b
---
Introduction
============
Observations of exoplanetary transits are a powerful tool to investigate the nature of planets around other stars. Transits are revealed through periodic drops in the apparent stellar brightness, due to the interposition of a planet between the star and the observer. The shape of an exoplanetary transit lightcurve depends on the geometry of the star-planet-observer system and the spatial distribution of the stellar emission at the wavelength at which observations are taken [@ma02]. By solving the inverse problem, it is possible to characterise fully the planet’s orbit (Period, $P$; semimajor axis, $a$; inclination, $i$; eccentricity, $e$; and argument of periastron, $\omega$), and to measure its radius, $r_p$ [@sea03; @kip08; @ma02]. Knowledge of the inclination enables determination of the mass of the planet, $m_p$, if $m_p \sin{i}$ is known from radial-velocity measurements.
Multiwavelength transit observations can be used to characterise the atmospheres of exoplanets, through differences in the transit depths, typically at the level of one part in $\sim 10^{4}$ in stellar flux for giant planets [@brown01; @sea00; @tin07b]. For this purpose, the diagnostic parameter is the wavelength-dependent factor $p=r_p/R_s$, i.e. the ratio between the planetary and the stellar radii (or its square, related to the transit depth).
The exoplanet HD189733b is one of the most extensively studied hot Jupiters: the brightness of its star allows spectroscopic characterisation of the planet’s atmosphere.
The 3.6$\mu$m transit depth for the exoplanet HD189733b has been debated in the literature. Different analyses of the same dataset, including two simultaneous Spitzer/IRAC observations at 3.6$\mu$m and 5.8$\mu$m, have been used to infer the presence of water vapour in the atmosphere of HD189733b [@bea08; @tin07], or to reject this hypothesis [@des09]. Another analysis of this dataset is reported by [@ehr07], but we do not comment further their results, as they were not conclusive, because of the very large error bars. [@des11] reported the analysis of a second Spitzer/IRAC dataset at 3.6$\mu$m using the same techniques. Their new estimates of the planet’s parameters were significantly different from those reported previously by the same authors [@des09]; the discrepancies were attributed by the authors to variations in the star.
Although stellar activity may significantly affect estimates of exoplanetary parameters from transit lightcurves [@bal12; @ber11], the method used to retrieve the signal of the planet also plays a critical role. The analyses mentioned above were all based on parametric corrections of the instrumental systematics, and are thus, to some degree, subjective. Recently, non-parametric methods have been proposed to decorrelate the transit signals from the astrophysical and instrumental noise, and ensure a higher degree of objectivity. [@wal12; @wal13] suggested algorithms based on Independent Component Analysis (ICA) to extract information of an exoplanetary atmosphere from Hubble/NICMOS and Spitzer/IRS spectrophotometric datasets.
In this paper we adopt a similar approach to detrend the transit signal from photometric observations by using Point Spread Functions (PSFs) covering multiple pixels on the detector. We apply this technique to re-analyse the two observations of primary transits of HD189733b recorded with Spitzer/IRAC at 3.6$\mu$m (channel 1 of IRAC) in the “cold Spitzer” era. We present a series of tests to assess the robustness of the method and the error bars of the parameters estimated. Critically, by comparing the results obtained for the two measurements, we discuss the level of repeatibility of transit measurements in the IR, limited by the absolute photometric accuracy of the instrument and possible stellar activity effects. We discuss the reliability of our results for orbital and stellar parameters in the light of previous multiple 8$\mu$m observations [@agol10].
Data analysis {#sec:analysis}
=============
Observations
------------
The two Spitzer observations of HD189733 b discussed here were performed on 2006 October 31 (ID 30590), and 2007 November 25 (ID 40732).
The first observation consists of 1936 exposures using IRAC’s stellar mode (full-array), taken over 4.5 hr; 1.8 hr on the primary transit of the planet, 1.6 hr before, and 1.1 hr after transit. The reset time is 8.4 s. During the observation, the centroid of the star HD189733 was stable to within one pixel.
The second observation was of 1920 exposures using IRAC’s sub-array mode, over 4.5 hr; 1.8 hr were spent on the primary transit of the planet, 1.7 hr before, and 1 hr after transit. The interval between consecutive exposures is 8.4 s. Each exposure consists of 64 reads at high speed cadence of 0.1 s. Only for the observation ID 40732, we replaced the 64 reads of each exposure with their mean values, in order to have a manageable number of data points, to reduce the random scatter, and to have the same sampling of the observation ID 30590. During the observation, the centroid of the star HD189733 was again stable to within one pixel.\
Independent Component Analysis in the context of exoplanetary transits lightcurves {#ssec:ica}
----------------------------------------------------------------------------------
Independent Component Analysis (ICA) consists of a transformation from a set of recorded signals to an equivalent set of maximally independent components. The underlying assumptions are that:
1. each recorded signal is a linear combination of the same source signals;
2. the source signals are mutually independent.
We can express this model as: $$\label{eqn:ica_system}
\begin{array}{c}
x_1 = a_{1,1} s_1 + a_{1,2} s_2 + ... + a_{1,n} s_n \\
x_2 = a_{2,1} s_1 + a_{2,2} s_2 + ... + a_{2,n} s_n \\
\vdots \\
x_n = a_{n,1} s_1 + a_{n,2} s_2 + ... + a_{n,n} s_n
\end{array}$$ where $x_i$, $i=1 \dots n$, are the recorded signals, $s_j$, $j=1 \dots n$, are the source signals, and $a_{i,j}$ are numerical coefficients. Eq. \[eqn:ica\_system\] can be written in matrix form as: $$\label{eqn:ica_direct}
\textbf{x} = \textbf{A} \textbf{s}$$ where $\textbf{x}$ is the column vector containing the recorded signals, $\textbf{s}$ is the column vector containing the source signals, and $\textbf{A}$ is the matrix of the coefficients, the so-called “mixing matrix”.
The aim of ICA is the ‘blind’ separation of the source signals from the observations, i.e., without any additional information (except the assumed mutual independence of the source signals). In other words, the ICA algorithms search for the matrix $\textbf{W}$ that transforms the recorded signals such that the mutual statistical independence is maximised: $$\textbf{W} \textbf{x} = \textbf{W} \textbf{A} \textbf{s}$$ If the assumptions are valid, then $\textbf{W} \textbf{A} = \textbf{D}$, where $\textbf{D}$ is a diagonal matrix, so that: $$\label{eqn:ica_inversion}
\textbf{W} \textbf{x} = \textbf{D} \textbf{s}$$ The diagonal matrix $\textbf{D}$ means that the extracted signals can be rescaled without changing the mutual independence.\
To maximise said independence, several approaches and implementations have been proposed [@hyv01; @tic08]. We used the MULTICOMBI algorithm [@tic08], which optimally mixes EFICA and WASOBI, based on maximising the nongaussianity of the extracted signals [@kol06], and their temporal decorrelations [@yer00], respectively.
In this work, the observed signals are lightcurves of a star, recorded for a time interval that includes an exoplanetary transit event. These lightcurves contain at least three independent contributing signals:
- the astrophysical signal;
- the signal of instrumental systematics;
- stochastic noise.
It is possible, in principle, to decompose further the astrophysical and instrumental systematics signals. The former is the sum of the transit signal, the astrophysical background, possible stellar activity signals, etc.; the latter is the sum of different effects from different parts of the instrumentation. All these signals are expected to be independent from each other as they have different origins. By contrast, their linear combinations (i.e. the observed lightcurves) are clearly not mutually independent. It is worth stressing that to disentangle effectively all these signals we need, at least, the number of available lightcurves to be equal to the number of signals. Therefore, we need lightcurves recorded with the same instrument (since lightcurves recorded with different instruments have different systematics plus the astrophysical signals, so that the number of source signals is greater than the number of lightcurves). In principle, using lightcurves recorded at different times with the same instruments should not work, since the systematics have the same origins; but the relevant signals are not necessarily in phase, and so may differ by more than a simple scaling factor. Additionally, further differences might be present due to stellar variability. However, the transit signal, being common to all the lightcurves, is potentially detrendable. A successful extraction of a transit signal from a time series spanning several exoplanetary transit events, conveniently split into sub-lightcurves, is described in [@wal12].\
The advantage of spectroscopic observations over photometry is the provision of simultaneous lightcurves at different wavelengths with largely common instrumental systematics. The transit signals at each wavelength can be obtained by subtracting proper systematics models from the lightcurves (an accurate direct extraction of the transit is impossible due to the limb darkening effect). By using this technique, [@wal13] have extracted an infrared transmission spectrum of HD189733b between 1.51 $\mu$m and 2.43 $\mu$m, from a Hubble/NICMOS dataset.
ICA using pixel-lightcurves {#ssec:the_method}
---------------------------
The main novelty of the algorithms we use here is their ability to detrend the transit signal from a single photometric observation of just one primary transit. This is possible because, even if stars can be approximated by point sources, the instrument is purposely de-focused to spread the PSF over several detector pixels, and the position of the target star on the detectors is stable to within one pixel. During an observation, there are several pixels detecting the same astrophysical signals at any time, but with different scaling factors, depending on their received flux, their quantum efficiency, and the instrument PSF.\
We performed an ICA decomposition over several pixel-lightcurves, i.e. the time series from individual pixels, in order to extract the transit signal and other independent signal components (stellar or instrumental in nature).\
Once a set of independent components has been obtained from a selected set of pixel-lightcurves, different approaches to obtain the transit signal can be considered.
*: direct identification of the transit component*\
In principle, if one of the independent components extracted has the morphology of the transit signal, we assume that one to be the transit signal, multiplied by an undetermined scaling factor. We renormalise the signal by the mean value calculated on the out-of-transit part, so that the out-of-transit level is unity.\
Method 1 is not applicable to the extraction of accurate transit signals from spectroscopically resolved observations of a primary transit at different wavelengths, because of the wavelength dependence of stellar limb darkening. This is not a problem in our case, because all the pixels record the same wavelengths.\
*: non-transit-components subtraction*\
Another approach to estimating the transit signal is to remove all the other effects from an observed lightcurve, i.e. by subtracting all the components other than the transit one, properly scaled. The scaling factors can be determined by fitting a linear combination of the components, plus a constant term, to the out-of-transit part of the lightcurve [^1]. The coefficients of the linear combination and the constant are the free parameters to fit.\
Instead of fitting the non-transit-components on the pixel-lightcurves, and then subtracting, we performed these processes on the spatially integrated lightcurves, obtained by summing all the individual pixel-lightcurves. The integrated lightcurves are much less noisy than the individual pixel-curves.\
Transit lightcurve fitting and error bars {#ssec:curvefit}
-----------------------------------------
After the extractions of the detrended and normalised transit time series, we modelled them by using the [@ma02] analytical formulae. We can compute the observed flux as a function $F(p,z)$, where $p=r_p/R_s$ is the ratio between the planetary and the stellar radii, and $z=d/R_s$ is the distance between the centres of their disks projected onto the sky divided by the stellar radius. The relative distance $z$ is a function of time, determined by the orbital parameters.\
We assumed the orbital period $P$, zero eccentricity, and a quadratic limb darkening model [@how11]. The values of the fixed parameters are reported in Tab. \[tab1\].
[cc]{} $P$ & $2.218573 \ days$\
$e$ & $0$\
$\gamma_1$ & $7.82118 \times 10^{-2}$\
$\gamma_2$ & $2.00656 \times 10^{-1}$\
\
We first determined the centers of the transit ephemeris by fitting some symmetric models with all the other parameters fixed. Recent papers [@col10; @tri10] report a small but non-zero eccentricity ($e \simeq 4 \cdot 10^{-3}$), but we verified this would affect our estimates of the other parameters by a negligible fraction of their error bars.\
We then performed a fit with three free parameters:
1. the ratio of planetary to stellar radii, $p = r_p/R_s$;
2. the orbital semimajor axis (in units of the stellar radius), $a_0 = a/R_s$;
3. the orbital inclination, $i$.
We chose these as free parameters, because:
- there is a large range of values published in the literature;
- they largely determine the shape of the transit signal;
- they do not show strong cross-correlations.
For completeness, and for comparisons with the literature, in the final results we report also the transit depth, $p^2$, the impact parameter, $b$, and the duration of the transit, $T$, where $$b = a_0 \cos{i}$$ $$T = \frac{P \sqrt{1-b^2}}{ \pi a_0}$$
We used a Nelder-Mead optimisation algorithm [@lag98], to obtain first estimates of the parameters of a model. To confirm/improve these estimates and to determine error bars, we ran an Adaptive Metropolis algorithm with delayed rejection [@haa06] for 20,000 iterations, starting from the optimal values initially determined, in order to sample the probability distributions of the fitted parameters. The updated best estimates and error bars of the parameters are the means and the standard deviations of the sampled distributions (approximately gaussians), respectively. No burn-in is required, because of the optimal starting points of the chains.\
The variance of the likelihood function is initialised as the variance of the residuals obtained for the first model and then sampled together with the other free parameters ($\sigma_0^2$). In this way, we take into account both white and the autocorrelated noise present in the detrended time series, but we ignore possible systematic errors due to the preliminary ICA deconvolution. The ICA errors can be represented as an additional uncertainty, $\sigma_{ICA}$, on each point in the time series. The likelihood’s variance, $\sigma_{like}^2$, becomes: $$\sigma_{like}^2 = \sigma_0^2 + \sigma_{ICA}^2$$ We tested that resampling the parameters’ chains with $\sigma_{like}^2$ does not affect their best values, while the total error bars of the single parameters, $\sigma_{par}$, increase with respect to the previous estimates (without the ICA errors), $\sigma_{par,0}$, as: $$\sigma_{par} = \sigma_{par,0} \frac{ \sigma_{like}}{ \sigma_{0}} = \sigma_{par,0} \sqrt{ \frac{ \sigma_{0}^2 + \sigma_{ICA}^2 }{ \sigma_{0}^2}}$$ A measure of the uncertainties on the independent components extracted by ICA is given by the Interference-to-Signal-Ratio matrix, $\textbf{ISR}$, i.e. a $n \times n$ matrix, where $n$ is the number of signals. The $\textbf{ISR}_{ij}$ element estimates the relative remaining presence of the $j^{th}$ component in the $i^{th}$ one. Then, $$\textbf{ISR}_i = \sum_{j=1, \ j \ne i}^{n} \textbf{ISR}_{ij}$$ estimates the relative remaining presence of all the other components in the $i^{th}$ one.\
If the $i^{th}$ component represents the transit signal, and if we estimate the transit signal through *method 1*, we can identify: $$\sigma_{ICA}^2 = f^2 \textbf{ISR}_i$$ $f$ being the scaling factor used.\
If the $i^{th}$ component represents the transit signal, but we estimate it through *method 2*, $\sigma_{ICA}$ has to contain a weighted sum of the **ISR**s of the non-transit components removed, plus the discrepancies of the fit to the out-of-transit phases: $$\label{eqn:sigmaica2}
\sigma_{ICA}^2 = f^2 \left ( \sum_{j=1}^{m} o_j^2 \textbf{ISR}_j + \sigma_{ntc-fit}^2 \right )$$ $o_j$ being the coefficients of the non-transit components, $m$ the number of components considered, $\sigma_{ntc-fit}$ the standard deviation of the residuals from the reference lightcurve (out of the transit), and $f$ the normalising factor for the model-subtracted lightcurve.\
The MULTICOMBI code produces two Interference-to-Signal-Ratio matrices, $\textbf{ISR}^{EF}$, associated with the algorithm EFICA, and the $\textbf{ISR}^{WA}$, associated with the algorithm WASOBI. We estimated the global $\textbf{ISR}$ as their average: $$\textbf{ISR} = \frac{ \textbf{ISR}^{EF} + \textbf{ISR}^{WA}}{2}$$ This is a conservative estimate, given that, according to [@tic08], the MULTICOMBI $\textbf{ISR}$ slightly outperforms the best of $\textbf{ISR}^{EF}$ and $\textbf{ISR}^{WA}$ (then it could be smaller), but these estimates are entirely reliable only under certain assumptions on the signals which may be not satisfied in these cases. Here we take them as worst-case estimates.
Application to observations {#ssec:applyID30590}
---------------------------
Here, we describe the main steps of the analyses performed on each of the two observations (ID 30590 and ID 40732), which include some tests of robustness. We now discuss only results obtained with method 2, as they are much more stable; results obtained with method 1 are reported in Appendix \[sec:app2\], along with a critical comparison of the two methods.
### Choice of the pixels {#ssec:pixel_choice}
The first step in the analysis is the choice of the pixel-lightcurves to analyse. This is determined by:
- the instrument point response function (PRF), i.e. the measured intensity profile of the star on the detector [^2];
- the noise level of the detector;
- the effective number of significant components to disentangle.
The number of significant components is not known a priori. The ICA code extracts a number of components equal to the number of lightcurves that it receives as input. Apart from the collective behaviour common to all the pixel-lightcurves, each pixel introduces an individual signature. Only if the individual signatures are negligible compared to the collective behaviour are we able to select enough lightcurves to disentangle the significant components. The PRF and the noise level of the detector limit the number of pixels containing potentially useful astrophysical information.
In practice, we considered several arrays of pixels with the stellar centroid at their centers, having dimensions $3 \times 3$, $5 \times 5$, $7 \times 7$, $9 \times 9$, and $11 \times 11$ pixels. Fig. \[fig1\] shows the “integral lightcurves”, obtained by summing the contributions from the various pixels. We looked for outliers in the time series, i.e. points discrepant more than 5$\sigma$ from a first transit-lightcurve model (fitted on the original data), and we replaced those outliers with the averages of the points immediately before and after. We found only one outlier in observation ID 30590, and nine in ID 40732. Although the observed lightcurves are two primary transits of the same exoplanet, observed at the same wavelength through the same instrument, they appear very different, mainly because of the different observing strategies. In particular, observation ID 40732 seems to be much less affected by systematics, and less noisy.
The integral lightcurves from the various arrays of pixels look very similar in shape, but have different absolute intensities, as expected. The mean intensities of the integral $3 \times 3$, $5 \times 5$, $7 \times 7$ and $9 \times 9$ lightcurves are respectively $\sim 83 \%$, $\sim 92 \%$, $\sim 96 \%$ and $\sim 98 \%$ of the mean intensity of the integral $11 \times 11$ lightcurve. We are not interested in absolute photometry, but only in relative variations of the intensity, therefore it is not important whether the PRF is totally contained in the square used for the analysis or not, provided it contains enough information to detrend the transit signal. Larger arrays include pixels which add noise with little or no astrophysical information. We concluded that the $3 \times 3$ or the $5 \times 5$ arrays were the optimal choices. However, we tested all the pixel arrays, to assess the robustness of the results.
We binned the transit time series by replacing groups of nine consecutive points with their mean values, in order to reduce the computational time required to sample the parameters’ distributions in the [@ma02] model (see Sec. \[ssec:curvefit\]). We checked that in select cases this approach does not affect the parameter estimates.
The best values of $p$, $a_0$, and $i$ are stable, within the error bars, with respect to the choice of the set of pixel-lightcurves used to detrend the signals. The discrepancies between the extracted signals and the relative fits are the biggest for the $3 \times 3$ array; for larger arrays they are smaller, and are either all at the same level (ID 30590), or slightly decrease with the size of the array (ID 40732). Our interpretation of this is that the $5 \times 5$ and larger arrays contain the same amount of useful information, while in the $3 \times 3$ array something is missed. The ICA errors confirm this hypothesis, being the smallest for the $7 \times 7$ (ID 30590) and $5 \times 5$ (ID 40732) arrays. Higher values for larger arrays were expected, but do not differ significantly. We conclude that the choice of the array size is not crucial.
### Choice of the components {#ssec:components_ID30590}
In Sec. \[ssec:pixel\_choice\], we corrected the observed lightcurves by subtracting all the non-transit components (see Sec. \[ssec:the\_method\]). Here, we show how to identify the most significant components, and how many should be considered. We generally expect that some components are related to collective behaviours, common to all the pixels, and others to individual pixels’ signatures and/or noisy mixtures of the sources. By inspection, a few of the components clearly present time structures, while others are random scattered time series.\
We report results from the $5 \times 5$ array only, as it is the smallest array containing all the astrophysical and instrumental information.\
To evaluate the impact of each component in the out-of-transit data, we found the best fits of the single components (plus additive constants) to that part of the integral lightcurve, and calculated the means and standard deviations of the residuals. In this way, we established a ranking of importance of the components, based on the minimisation of the discrepancies between their fits and the integral lightcurve, out of transit. We then computed other best fits by using the $n$ most important components, according to that ranking, with $n$ from $1$ to $24$. The best coefficients for the components and the additive constants were determined through the Nelder-Mead optimisation algorithm.\
Fig. \[fig2\] reports the standard deviations of the residuals of the single-component fits, normalised to the out-of-transit level; Fig. \[fig3\] reports analogous fits obtained using more components. Note that figures related to different observations are not reported with the same scale, because they have very different accuracies.
Most systematics are contained in one major component, but the use of multiple components increases the detrending accuracy.
We computed twenty-four estimates of the transit signal by removing the $n$ most significant non-transit components from the integral lightcurve. We binned these over nine points, as in Sec. \[ssec:pixel\_choice\], and fitted [@ma02] models to these curves. The standard deviations of the residuals between each curve and the corresponding best model of [@ma02] are reported in Fig. \[fig4\], and confirm that the use of multiple components for detrending improves the results.\
ICA separation errors are plotted in Fig. \[fig5\], showing the same trends.
Given these tests, we expect to have good estimates of the transit signals by removing the first few most significant components, but some improvements can be made by removing more components, up to a saturation point. The best values of the parameters $p$, $a_0$, and $i$, for each estimated transit signal, are shown in Fig. \[fig12\].
The dispersions in the parameters are fully contained in the intervals previously estimated by using the signals with all non-transit components removed (see Appendix \[sec:app0\], Tab. \[tab6\], column 2), except for values from the transit signal from observation ID 40732 with only one component removed.
Results
=======
Fig. \[fig6\] shows the normalised transit signals extracted using the $5 \times 5$ arrays, considering all the independent components; the relative best lightcurve fits to the binned and detrended data; and the residuals. The standard deviations of the residuals are $\sigma_0^{ID 30590} = 6.4 \times 10^{-4}$ and $\sigma_0^{ID 40732} =1.45 \times 10^{-4}$. Note that the signal extracted from observation ID 40732 has a dispersion smaller by a factor $\sim 4.4$.\
Fig. \[fig7\] illustrates the sampled distributions of the parameters $p$, $a_0$, and $i$, from the transit signal extracted by observation ID 40732 (see Sec. \[ssec:curvefit\]). Similar distributions, but with larger dispersions, were obtained for the other transit signal.
Tab. \[tab2\] reports the starting values, sampled means, and standard deviations of the parameters obtained.
[cccc]{} & $Starting \ value$ & $Mean$ & $Standard \ deviation$\
ID 30590 & & &\
$p$ & $0.15471$ & $0.15470$ & $3.6 \cdot 10^{-4}$\
$a_0$ & $9.05$ & $9.06$ & $0.09$\
$i$ & $85.93$ & $85.94$ & $0.10$\
ID 40732 & & &\
$p$ & $0.15534$ & $0.15534$ & $8 \cdot 10^{-5}$\
$a_0$ & $8.92$ & $8.92$ & $0.02$\
$i$ & $85.78$ & $85.78$ & $0.02$\
Note that the starting values agree very well with the sampled means. The likelihood variances without the ICA contribute, calculated as detailed in Sec. \[ssec:curvefit\], are equal to the variances of the residuals: $\sigma_{0}^{ID 30590} = \left ( 6.5 \pm 0.3 \right ) \times 10^{-4}$, and $\sigma_{0}^{ID 40732} = \left ( 1.46 \pm 0.07 \right ) \times 10^{-4}$.
Tab. \[tab3\] gives the final results for the parameters $p$, $a_0$, $i$, $p^2$, $b$, and $T$.\
[ccc]{} & ID 30590 & ID 40732\
$p$ & $0.1547 \pm 0.0005$ & $0.15534 \pm 0.00011$\
$a_0$ & $9.05 \pm 0.16$ & $8.92 \pm 0.03$\
$i$ & $85.93 \pm 0.15$ & $85.78 \pm 0.03$\
$p^2$ & $0.02394 \pm 0.00017$ & $0.02413 \pm 0.00003$\
$b$ & $0.64 \pm 0.03$ & $0.657 \pm 0.005$\
$T$ & $5170 \pm 200 \ s$ & $5157 \pm 34 \ s$\
Combining observations {#ssec:combo}
----------------------
The parameter estimates determined from observation ID 40732 are much more accurate than those from ID 30590. Assuming that the orbital parameters were the same along the two observations, as expected because of the stability of the planetary orbit, we computed a chain for ID 30590, for $p$ only, with $a_0$ and $i$ fixed to the best values estimated from ID 40732. In this way, we can make a direct comparison of $p$ between the two observations, and avoid possible correlations with the other parameters. Even if $a_0$ and $i$ were badly determined, due to an inaccurate stellar model being assumed (e.g. wrong limb darkening coefficients, star spots, or faculae), they would introduce a systematic error on $p$ that would be equal for both observations. Thus variations of $p$ (or $p^2$), obtained while keeping all other parameters fixed, are a more objective measurement of the stellar variations. Results are reported in Tab. \[tab4\]; note that $\sigma_{0}$ is unchanged. Fig. \[fig8\] shows the discrepancies between the detrended signal and the model.
[cc]{} $p$ & $0.15507$\
$\sigma_p^0$ & $2.7 \cdot 10^{-4}$\
$\sigma_{0}$ & $6.5 \cdot 10^{-4}$\
Including the ICA errors we found: $$\begin{array}{c}
p = 0.1551 \pm 0.0004 \\
p^2 = 0.02405 \pm 0.00014
\end{array}$$ Fig. \[fig9\] compares the original estimates for $p$ and $p^2$, with those obtained by keeping $a_0$ and $i$ fixed.
We note that:
- the best value from ID 30590 with $a_0$ and $i$ fixed agrees better with result from ID 40732;
- the new estimate from ID 30590 is consistent with the previous one, but with a (slightly) smaller error bar.
Discussion
==========
Comparison of the two observations
----------------------------------
The planetary, orbital, and stellar parameters derived separately from the two observations are all consistent within 1$\sigma$. In particular, the duration of the transit is extremely stable between the two observations. This is not surprising, because its measure is almost insensitive to calibration errors and stellar activity; all the other parameters are much more affected by these sources of noise. Furthermore, these other parameters are strongly correlated; e.g. a non-optimal estimate of the impact parameter $b$ will result in an imprecise transit depth $p$, etc. Fig. \[fig10\] shows the differences between the transit signals extracted for the two observations.
The standard deviation of the differences is $\sim 6.8 \times 10^{-4}$, which is comparable with the standard deviation of the discrepancies between the signal from observation ID 30590 and the relative model fit ($\sigma_{0}^{ID 30590} = \left ( 6.5 \pm 0.3 \right ) \times 10^{-4}$); the discrepancies between the signal from observation ID 40732 and the corresponding model fit are negligible. Hence there is no evidence of physical variations in the transit signal from one observation to the other one.\
The results of Sec. \[ssec:combo\] reinforce our claim of non detectable stellar activity variations.\
We conclude that the two observations lead to consistent results, but the second constrains the orbital and stellar parameters much better, and allows the estimate of the transit depth for the first one to be refined.
Comparison with observations at $8 \mu m$
-----------------------------------------
[@agol10] report a detailed study of seven primary transits and seven secondary eclipses of HD189733b, observed with Spitzer/IRAC at 8$\mu$m (channel 4 of IRAC). Their measured orbital parameters differ from ours by less than the joint 1-$\sigma$ uncertainties. Fig. \[fig11\] includes a comparison of the parameters $a_0$, $i$, and $b$, obtained in this paper with their values. Given the number of primary transits and secondary eclipses they analysed, and the small impact of the limb darkening effect at 8$\mu$m, this is a robust confirmation of the validity of our results at 3.6$\mu$m. We suggest the use of these parameters for future observations at other wavelengths.\
[@agol10] found variations in the transit depth with a range of $\sim 2 \times 10^{-4}$ on $p^2$. We could not detect such a difference between the two observations analysed at 3.6$\mu$m, as it is comparable with the first error bar.
Comparison with previous analyses of the same observations
----------------------------------------------------------
Our results are consistent, at $1 \sigma$ level, with those of [@bea08], [@des09] for ID 30590, and [@des11] for ID 40732. However, our results afford a substantial agreement (within 1$\sigma$) between the transit parameters determined from the two observations, while previous analyses by [@des09] and [@des11] claimed significant variations of all parameters (e.g., discrepancy $> 4\sigma$ for transit depths). [@des11] suggested stellar activity as a possible explanations for those differences. Our results do not support such conclusions, and we find that any possible stellar-activity variations are within the error bars. Our error bars from the observation ID 40732 are of the same order (for transit depth) or even smaller (for orbital parameters) than in [@des11], while those from the observation ID 30590 are larger by a factor $\sim 1.6$ with respect to the error bars in [@des09]. The factor $\sim 1.6$ comes from adding the ICA errors to the parameter error bars derived from the extracted signals. [@des09; @des11] applied parametric corrections to detrend the transit signals from other disturbances, without attributing any uncertainties to the detrending processes. The fact that we obtained smaller error bars from the observation ID 40732, even including the contributions from the detrending process, indicates that, in that context, our blind extraction performed better than their parameterisation. Orbital parameters determined by [@bea08] and [@des09] for observation ID 30590 are not consistent with those for observation ID 40732 obtained by [@des11], the results presented here, or the 8$\mu$m observations by [@agol10]. Given that the second measurement was superior in quality, and given the agreement with observations at another wavelength, we conclude that the parameters presented in this paper are more robust than those reported by [@bea08], or by [@des09] using the same data.
We note that [@bea08] used the same impact parameter at 3.6$\mu$m and 5.8$\mu$m, while [@des09] used similar, but not identical, values. Given the conclusions obtained in this paper about the orbital parameters, we suggest that the transit depth at 5.8$\mu$m be recalculated accordingly. A re-analysis of the observation at 5.8$\mu$m, then the differences between the transit depths at the two wavelengths, which were used to infer about the atmosphere of the planet, should not be strongly affected by this bias, at least in the first case. However, because their conclusions were controversial, a re-analysis of the observation at $5.8 \mu m$, with more precise orbital parameters and possibly non parametric technique, as done here, is needed.\
Conclusions
===========
We have introduced a blind signal-source separation method, based on ICA, to analyse photometric data of transiting exoplanets, with a high degree of objectivity; a novel aspect is the use of pixel-lightcurves, rather than multiple observations.\
We have applied the method to a reanalysis of two Spitzer/IRAC datasets at 3.6$\mu$m, which previous analyses found to give discrepant results, and obtained consistent transit parameters from the two observations.\
We suggest that the large scatter of results reported in the literature arises from:
- use of arbitrary parametric methods to detrend the transit signals, neglecting the relevant uncertainties;
- correlations between parameters in the lightcurve fit.
We found, for observation ID 40732, values for the orbital parameters that are in excellent agreement with those found by [@agol10], based on Spitzer/IRAC observations at $8 \mu m$. By applying these values to observation ID 30590, we improved the accuracy of the inferred transit depth, and strengthened the consistency between the two observations.
G. Morello was partly funded by Erasmus (LLP), “Borse di Studio finalizzate alla ricerca e Assegni finanziati da Programmi Comunitari, decreto 3505/2012” of Università degli Studi di Palermo, Perren/Impact (CJ4M/CJ0T). G. Tinetti is a Royal Society URF. Part of this work was supported by STFC, and ASI-INAF agreement I/022/12/0.
Partial results {#sec:app0}
===============
Observation ID 30590
--------------------
Tab. \[tab5\] reports the best values of the parameters for the transit signals extracted from different arrays of pixels, the standard deviations of the residuals between the signals and the best transit models, and the standard deviations attributed to the ICA separation.\
[cccccc]{} & $3 \times 3$ & $5 \times 5$ & $7 \times 7$ & $9 \times 9$ & $11 \times 11$\
$p$ & $0.1549$ & $0.1547$ & $0.1546$ & $0.1547$ & $0.1547$\
$a_0$ & $9.02$ & $9.05$ & $9.07$ & $9.06$ & $9.07$\
$i$ & $85.90$ & $85.93$ & $85.95$ & $85.94$ & $85.95$\
$\sigma_{0}$ & $7.1 \cdot 10^{-4}$ & $6.5 \cdot 10^{-4}$ & $6.5 \cdot 10^{-4}$ & $6.5 \cdot 10^{-4}$ & $6.5 \cdot 10^{-4}$\
$\sigma_{ICA}$ & $7.8 \cdot 10^{-4}$ & $7.6 \cdot 10^{-4}$ & $7.4 \cdot 10^{-4}$ & $8.2 \cdot 10^{-4}$ & $8.3 \cdot 10^{-4}$\
$\sigma_p$ & $0.00058$ & $0.00055$ & $0.00054$ & $0.00057$ & $0.00058$\
$\sigma_{a_0}$ & $0.17$ & $0.16$ & $0.16$ & $0.17$ & $0.17$\
$\sigma_i$ & $0.15$ & $0.15$ & $0.14$ & $0.15$ & $0.15$\
Fig. \[fig12\] shows the best values of the parameters $p$, $a_0$, and $i$, respectively, for the transit signals extracted removing the $n$ most significant components from the integral $5 \times 5$ lightcurve, binned by nine points.
Observation ID 40732
--------------------
Tab. \[tab6\] reports the best values of the parameters for the transit signals extracted from different arrays of pixels, the standard deviations of the residuals between the signals and the best transit models, and the standard deviations attributed to the ICA separation.
[cccccc]{} & $3 \times 3$ & $5 \times 5$ & $7 \times 7$ & $9 \times 9$ & $11 \times 11$\
$p$ & $0.15546$ & $0.15534$ & $0.15533$ & $0.15533$ & $0.15528$\
$a_0$ & $8.93$ & $8.92$ & $8.93$ & $8.93$ & $8.94$\
$i$ & $85.79$ & $85.78$ & $85.79$ & $85.79$ & $85.79$\
$\sigma_{0}$ & $1.62 \cdot 10^{-4}$ & $1.46 \cdot 10^{-4}$ & $1.45 \cdot 10^{-4}$ & $1.45 \cdot 10^{-4}$ & $1.41 \cdot 10^{-4}$\
$\sigma_{ICA}$ & $1.70 \cdot 10^{-4}$ & $1.45 \cdot 10^{-4}$ & $1.60 \cdot 10^{-4}$ & $1.53 \cdot 10^{-4}$ & $1.65 \cdot 10^{-4}$\
$\sigma_p$ & $0.00013$ & $0.00011$ & $0.00012$ & $0.00011$ & $0.00012$\
$\sigma_{a_0}$ & $0.03$ & $0.03$ & $0.03$ & $0.03$ & $0.03$\
$\sigma_i$ & $0.03$ & $0.03$ & $0.03$ & $0.03$ & $0.03$\
Fig. \[fig13\] reports the best values of the parameters $p$, $a_0$, and $i$, respectively, for the transit signals extracted removing the $n$ most significant components from the integral $5 \times 5$ lightcurve, binned by nine points.
Subdatasets {#sec:app1}
===========
An important test to verify the robustness of the analyses is to apply the same techniques to subdatasets. They clearly share the same phenomena, but recorded for different time intervals, largely overlapping. If the technique is able to separate the source components, the detrended transit signals from different subdatasets should be essentially equivalent, otherwise there is a problem with at least one of them. A critical factor could be the time length of a subdataset compared to the timescales of the source signals; for this reason, the separation performed using longer subdatasets or the whole dataset, might be more reliable, unless they strengthen some trends or introduce bad data, for example if they are not well calibrated, or affected by spurious events.\
Observation ID 30590
--------------------
We considered twenty-eight subdatasets, obtained combining seven different starting and four ending times, disposed with regular cadence of $\sim$14 minutes (see Fig. \[fig14\]).
As before, we used the $5 \times 5$ array, and we applied method 2, by removing all the independent components from the integral lightcurve. Fig. \[fig15\] shows the best values of the parameters $p$, $a_0$, and $i$, estimated using each subdataset.\
We can point out some correlations between the best values and both the start and the end points of the subdatasets. The overall scatters are compatible with the ranges determined before. Tab. \[tab7\] reports the estimated ranges of the parameters with the scatters observed by the subdatasets, either by including or by rejecting the two shortest subdatasets.
[cccc]{} Parameters & Estimated values & Overall scatters by subdatasets & With rejections\
$p$ & $0.1547 \pm 0.0005$ & $0.1543 \div 0.1557$ & $0.1543 \div 0.1550$\
$a_0$ & $9.05 \pm 0.16$ & $9.05 \div 9.15$ & $9.05 \div 9.11$\
$i$ & $85.93 \pm 0.15$ & $85.92 \div 86.09$ & $85.92 \div 86.01$\
Observation ID 40732
--------------------
We considered thirty-two subdatasets, obtained combining eight different starting times and four ending times, disposed with regular cadence of $\sim$14 minutes (see Fig. \[fig16\]).
As usual, we used the $5 \times 5$ array, and we applied method 2, and removed all the independent components from the integral lightcurve. Fig. \[fig17\] shows the best values of the parameters $p$, $a_0$, and $i$, estimated using each subdataset.
Again, there are some correlations between the best values and the extremes of the subdatasets, but the overall scatters are compatible with the ranges previously estimated. Tab. \[tab8\] reports the estimated ranges of the parameters with the scatters observed by the subdatasets:\
[ccc]{} Parameters & Estimated values & Overall scatters by subdatasets\
$p$ & $0.15534 \pm 0.00011$ & $0.15510 \div 0.15534$\
$a_0$ & $8.92 \pm 0.03$ & $8.92 \div 8.96$\
$i$ & $85.78 \pm 0.03$ & $85.77 \div 85.82$\
Method 1: direct identification of the transit component {#sec:app2}
========================================================
Tab. \[tab9\] reports the results obtained by applying method 1 and method 2 on both observations, using the whole datasets, and the $5 \times 5$ arrays.
[ccc]{} ID 30590 & Method 1 & Method 2\
$p$ & $0.1547 \pm 0.0019$ & $0.1547 \pm 0.0005$\
$a_0$ & $9.1 \pm 0.5$ & $9.05 \pm 0.16$\
$i$ & $85.9 \pm 0.5$ & $85.93 \pm 0.15$\
$p^2$ & $0.0239 \pm 0.0006$ & $0.02394 \pm 0.00017$\
$b$ & $0.64 \pm 0.11$ & $0.64 \pm 0.03$\
$T$ & $5160 \pm 900 \ s$ & $5170 \pm 200 \ s$\
[ccc]{} ID 40732 & Method 1 & Method 2\
$p$ & $0.1553 \pm 0.0004$ & $0.15534 \pm 0.00011$\
$a_0$ & $8.96 \pm 0.10$ & $8.92 \pm 0.03$\
$i$ & $85.81 \pm 0.11$ & $85.78 \pm 0.03$\
$p^2$ & $0.02413 \pm 0.00012$ & $0.02413 \pm 0.00003$\
$b$ & $0.654 \pm 0.019$ & $0.657 \pm 0.005$\
$T$ & $5156 \pm 124 \ s$ & $5157 \pm 34 \ s$\
It is straightforward to note that the best values are almost coincident, but the uncertainties derived with method 1 are larger by a factor $\sim 3 \div 4$. The differences are due to the ICA contributions to the error bars.\
We also observed that, in these cases, the transit signals estimated with method 2 tend to the ones obtained by method 1, when increasing the number of non-transit-components removed; this is shown in Fig. \[fig18\].
However, the larger error bars provided by the ICA terms are justyfied by the scatters obtained by using different arrays of pixels and different subdatasets. We do not report the results in detail, but we summarise the main facts observed:
- In some cases, the transit component is clearly corrupted, discouraging a quantitative analysis;
- The scatters of the transit parameters obtained by using different subdatasets are comparable with the error bars estimated (the arrays of pixels play a minor role, but more important than if using method 2);
- For longer subdatasets, which are expected to allow better extractions of the independent components, the results obtained with methods 1 and 2 tend to agree.
Agol, E., Cowan, N. B., Knutson, H. A., Deming, D., Steffen, J. H., Henry, G. W., & Charbonneau, D. 2010, , 721, 1861 Ballerini, P., Micela, G., Lanza, A. F., & Pagano, I. 2012, , 539, A140 Beaulieu, J. P., Carey, S., Ribas, I., & Tinetti, G. 2008, , 677, 1343 Berta, Z. K., Charbonneau, D., Bean, J., Irwin, J., Burke, C. J., Désert, J. M., Nutzman, P., & Falco, E. E. 2011, , 736, 12 Brown, T. M. 2001, , 553, 1006 Collier Cameron, A., Bruce, V. A., Miller, G. R. M., Triaud, A. H. M. J., & Queloz, D. 2010, , 403, 151 Désert, J. M., Lecavelier des Etangs, A., Hébrard, G., Sing, D. K., Ehrenreich, D., Ferlet, R., & Vidal-Madjar, A. 2009, , 699, 478 Désert, J. M., Sing, D. K., Vidal-Madjar, A., Hébrard, G., Ehrenreich, D., Lecavelier des Etangs, A., Parmentier, V., Ferlet, R., & Henry, G. W. 2011, , 526, A12 Ehrenreich, D., Guillaume, H., Lecavelier des Etangs, A., Sing, D. K., Désert, J. M., Bouchy, F., Ferlet, R., & Vidal-Madjar, A. 2007, , 668, 179 Haario, H., Laine, M., Mira, A., & Saksman, E. 2006, Statistics and Computing, 16, 339 Holman, M. J., & Murray, N. W. 2005, Science, 307, 1288 Howarth, I. D. 2011, , 418, 1165 Hyvärinen, A., Karhunen, J., & Oja, E. 2001, John Wiley & Sons, Inc., Independent Component Analysis, ISBN: 0-471-40540-X Kipping, D. M. 2008, , 389, 1383 Koldovský, Z., Tichavský, P., & Oja, E. 2006, IEEE Transations on Neural Networks, 17, 1265 Lagarias, J. C., Reeds, J. A., Wright, M. H., & Wright, P. E. 1998, SIAM Journal of Optimization, 9, 112 Mandel, K., & Agol, E. 2002, , 580, L171 Seager, S., & Mallén-Ornelas, G. 2003, , 585, 1038 Seager, S., & Sasselov, D. D. 2000, , 537, 916 Tichavský, P., Koldovský, Z., Yeredor, A., Gómez-Herrero, G., & Doron, E. 2008, IEEE Transations on Neural Networks, 19, 421 Tinetti, G., Vidal-Madjar, A., Liang, M. C., Beaulieu, J. P., Yung, Y., Carey, S., Barber, R. J., Tennyson, J., Ribas, I., Allard, N., Ballester, G. E., Sing, D. K., & Selsis, F. 2007, Nature, 448, 169 Tinetti, G., Liang, M., Vidal-Madjar, A., Ehrenreich, D., Lecavelier Des Etangs, A, & Yung, Y. L. 2007, , 654, L99 Triaud, A. H. M. J., Collier Cameron, A., Queloz, D., Anderson, D. R., Gillon, M., Hebb, L., Hellier, C., Loeillet, B., Maxted, P. F. L., Mayor, M., Pepe, F., Pollacco, D., Ségransan, D., Smalley, B., Udry, S., West, R. G., & Wheatley, P. J. 2010, , 524, A25 Winn, J. N. 2011, arXiv:1001.2010v4 \[astro-ph.EP\] Waldmann, I. P. 2012, , 747, 12 Waldmann, I. P., Tinetti, G., Deroo, P., Hollis, M. D. J., Yurchenko, S. N., & Tennyson, J. 2013, , 766, 7 Yeredor, A. 2000, IEEE Signal Processing Letters, 7, 197
[^1]: The out-of-transit limits do not have to be known exactly. They can be chosen in a way to be sure of not including part of the transit while fitting, relying on parameters reported in previous papers and on the lightcurves themselves. The results should not be affected by this choice, but it is worth checking this point.
[^2]: Note that the PRF is, in principle, slightly different to the PSF: the PSF is the intensity profile incident on the detector, while the PRF is the measured intensity profile (including the detector response).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Topological Stone-Wales defect in carbon nanotubes plays a central role in plastic deformation, chemical functionalization, and superstructure formation. Here, we systematically investigate the formation kinetics of such defects within density functional approach coupled with the transition state theory. We find that both the formation and activation energies depend critically on the nanotube chairality, diameter, and defect orientation. The microscopic origin of the observed dependence is explained with curvature induced rehybridization in nanotube. Surprisingly, the kinetic barrier follows an empirical Br[ø]{}nsted-Evans-Polanyi type correlation with the corresponding formation energy, and can be understood in terms of overlap between energy-coordinate parabolas representing the structures with and without the defect. Further, we propose a possible route to substantially decrease the kinetic activation barrier. Such accelerated rates of defect formation are desirable in many novel electronic, mechanical and chemical applications, and also facilitate the formation of three-dimensional nanotube superstructures.'
author:
- Mukul Kabir
- 'Krystyn J. Van Vliet'
title: 'Kinetics of Topological Stone-Wales Defect Formation in Single Walled Carbon Nanotubes'
---
Introduction
============
Defects including atomic impurities, vacancies, or topological junctions and kinks may be present in as-prepared carbon nanotubes (CNTs) [@doi:10.1021/ar010166k; @CNTDefectChapter] and graphene.[@doi:10.1021/nn102598m] In particular, Stone-Wales (SW) defects are important topological defect class, [@Stone1986501; @Wales.Nature.394] with analogy to dislocation dipoles in bulk materials. Individual SW defects have been identified experimentally in fullerene, [@Nat.Mater.7.790] CNTs, [@NatureNanotech.2.358; @PhysRevB.83.245420; @doi:10.1021/ja100760m] and graphene. [@doi:10.1021/nn102598m; @doi:10.1021/nl801386m]
The presence of SW defects markedly alters the chemical and mechanical properties of CNTs. Moreover, abundance of these SW defects is desirable for many novel chemical, and electronic applications requiring CNT network formation. [@doi:10.1021/nl015541g; @PhysRevB.66.245403; @PhysRevLett.92.075504; @NatMatKrasheninnikov; @doi:10.1021/cr050569o; @doi:10.1021/nn700143q; @ANIE:ANIE200705053; @Scientific.Report.2] For example, the SW transformation is found to be the microscopic unit process for nanojunction formation. [@doi:10.1021/nl015541g; @PhysRevB.66.245403; @PhysRevLett.92.075504; @NatMatKrasheninnikov] Additionally, the chemical reactivity and electronic properties of CNTs [@doi:10.1021/cr050569o] and graphene [@doi:10.1021/nl802234n] are modified by presence of SW defects, which act as anchor sites for chemical functionalization and are thus desirable in that context. It has also been posited that plastic deformation of CNTs is mediated via spontaneous formation and migration of topological SW defects under external tension. [@NatureNanotech.2.358; @PhysRevLett.81.4656; @PhysRevB.57.R4277] Thus, the thermodynamics and kinetics of the SW defect pose multiple implications for understanding and use of CNTs.
These properties of the SW defect are already established for graphene. The thermodynamic formation energy and the concurrent kinetic barrier for SW formation are very high at $\sim$ 5 and 10 eV, respectively, in graphene. [@doi:10.1021/nn102598m; @PhysRevB.80.033407] However once formed, the high reverse kinetic barrier ($\sim$5 eV) implies defect stability over a wide temperature range. In contrast to graphene, it could be anticipated that both the formation energy and kinetic barrier for CNTs will depend explicitly on intrinsic structural parameters including the CNT diameter, and the relative orientation of SW defects. However, these dependences have not yet been determined. In this Letter, we analyze both the thermodynamics and kinetics of SW formation in single-wall CNTs. We address systematically how these quantities depend on nanotube chirality and diameter, as well as relative defect orientation. We also correlate the thermodynamic formation energy to the activation barrier, and provide a microscopic description. Further, we show that the SW activation barrier can be modified substantially by substitutional heteroatom doping. Such doping accelerates the SW formation by six to twenty orders of magnitude at relevant temperatures and, in turn, assists nanotube welding [@doi:10.1021/nl015541g; @PhysRevB.66.245403; @PhysRevLett.92.075504; @NatMatKrasheninnikov] and formation of three-dimensional nanotube superstructures as has been observed experimentally. [@doi:10.1021/nn700143q; @ANIE:ANIE200705053; @Scientific.Report.2]
Computational Details
=====================
Here we couple density functional theory (DFT) and the climbing-image nudged elastic band (CINEB) method that correctly predicts the first-order transition state. [@PhysRevB.47.558; @PhysRevB.54.11169; @Henkelman-1] The DFT calculations are carried out within the projector augmented wave potential [@PhysRevB.50.17953] implemented in the Vienna [*ab initio*]{} simulation package [@PhysRevB.47.558; @PhysRevB.54.11169]. A plane-wave cutoff of 500 eV, and Perdew-Burke-Ernzerhof generalized gradient approximation are used. [@PhysRevLett.77.3865] The first Brillouin zone is sampled with a Monkhorst-Pack grid [@PhysRevB.13.5188] of 1$\times$1$\times$4. The position of all atoms are relaxed until the forces are less than 0.01 eV/Å. The tube axis is along the $Z$-direction, and a vacuum space of more than 10 Å along the $X$ and $Y$ directions is used to eliminate the image interaction. Minimum energy path for SW defect formation were sampled using the CINEB method. [@Henkelman-1] In CINEB, a set of intermediate structures (images) are distributed along the reaction path connecting optimized pristine and defected carbon nanotubes (CNTs). The images are connected via an elastic spring, and each intermediate image is fully relaxed in the hyperspace perpendicular to the reaction coordinate. The nature of the transition states have been confirmed via the phonon calculation, where one and only one imaginary frequency confirms the transition state to be of first-order.
The SW defect produces a long-range strain field, and the resultant dislocation-dislocation interaction may affect formation and activation energies. We indeed find that these energies depend on the CNT length, and converge at tube length of $\sim$2.5 nm, which we fixed throughout the present calculations (Table S1 in Supporting Information). On a CNT surface, the SW defect can have three possible orientations: $\theta =$ $\chi$ and $\pi/3 \pm \chi$, where $\chi$ is the chiral angle. [@zhou:1222] Of these three sets, two inequivalent orientations are shown in Fig. \[fig:SWf\](a)-(b) for zigzag, and Fig. \[fig:SWf\](c)-(d) for armchair nanotubes. The formation energy is calculated as $E_f=E_{\rm SW} - E_{\rm P}$, where $E_{\rm SW}$ and $E_{\rm P}$ are the energies of the CNT with and without the defect, respectively. The activation barrier is calculated as the energy difference between the first-order transition state and the pristine CNT. Although few attempts have been made to calculate the formation energy alone, [@zhou:1222; @dumitrica:2775] there is no systematic investigation that facilitates comparisons or general conclusions. Moreover, prior attempts either suffered from inadequate chemical accuracy for CÐC bonding (classical many-body potential or tight-binding approach) [@PhysRevLett.81.4656; @PhysRevB.57.R4277; @zhou:1222] or assumed insufficient structural models, [@dumitrica:2775] and did not consider the plausible orientational contribution.
Results and Discussion
======================
[**Formation energy, nanotube curvature, and defect orientation.**]{} The calculated SW defect formation energy depends strongly on the nanotube diameter $d$, and the orientation of SW dislocation dipole $\theta$. Figure \[fig:SWf\](e) illustrates a systematic variation of $E_f(d, \theta)$ obtained with our present calculations, from which we observe two distinct trends. First, the calculated $E_f(d, \theta)$ increases monotonically with increasing $d$ for any particular $\theta$, and converges toward the value for two-dimensional graphene $E_f^G$. Applying the identical theoretical approach for graphene, $E_f^G$ is calculated to be 4.96 eV, which agrees well with previous calculations. [@PhysRevB.80.033407] Note that the SW defect formation energy (1.57 eV) for C$_{60}$ fullerene (diameter $\sim$ 7 Å) is comparable with that of a nanotube with similar diameter. [@PhysRevB.84.205404] Interestingly, we find that for any particular CNT with diameter $d$, the calculated $E_f(d, \theta)$ increases monotonically with the angle $\theta$ made by the rotating C–C bond with the tube axis in the pristine structure \[Fig. \[fig:SWf\](e)\]. For zigzag and armchair nanotubes, we find the formation energy to follow $E_f[(n,0), \theta=0] < E_f[(n,0), \theta=\pi/3]$ and $E_f[(n,n), \theta=\pi/6] < E_f[(n,n), \theta=\pi/2]$ order. Since $E_f(d, \theta)$ is the energy of the defected structure relative to the corresponding nanotube without the defect, the dependence on $d$ and $\theta$ can be explained qualitatively by the curvature induced rehybridization for the defected structure (see Supporting Information). [@Haddon17091993; @Dumitrica2002182] The Coulomb repulsion inside the nanotube increases with increasing curvature, leading to significant rehybridization between $\pi$ and $\sigma$ orbitals. Thus, the true hybridization in CNTs is intermediate between $sp^2$ and $sp^3$, i.e., $sp^{2+\tau}$ with $\tau \in$ \[0,1\] is the degree of rehybridization. With increasing diameter (decreasing curvature), $\tau$ decreases rapidly and approaches zero, and the hybridization state of the affected bond is increasingly $sp^2$-like (Table S2 in Supporting Information). Thus, relative to the pristine structure, the energy of the defected CNT shifts toward higher energy with increasing $d$ (Fig. S3 in Supporting Information). Therefore, $E_f(d, \theta)$ increases with increasing $d$, and approaches to $E_f^G$ of two-dimensional $sp^2$-graphene. Similarly, for a given $d$, the $\theta$ dependence can be explained by considering the local environment of the rotated C–C bond for the defected structure. With increasing $\theta$, the local curvature of the rotated C–C bond \[shown in Fig. \[fig:SWf\](a)-(d)\] decreases, and thus the degree of rehybridization $\tau$ decreases. Therefore, the energy of the defected structure increases with increasing $\theta$, as compared to the corresponding pristine structure (Fig. S4 in Supporting Information). Alternatively, the $\theta$-dependence can also be explained qualitatively by comparing the rotating C–C bond lengths for nanotubes with and without the defect, and we find the former to be shorter (Table S3 in Supporting Information). The difference $\Delta b$ ($= b_{P}-b_{SW}$) is larger for larger $\theta$: $\Delta b(\theta=0)<\Delta b(\theta=\pi/3$) for zigzag configurations, and $\Delta b(\theta=\pi/6)<\Delta b(\theta=\pi/2$) for armchair configurations. Thus, the defected structure with $\theta=0$ ($\theta=\pi/6$) is lower in energy, due to comparatively higher rehybridization, than the corresponding $\theta=\pi/3$ ($\theta=\pi/2$) structure.
![Calculated activation barriers $E_a(d, \theta)$ show strong dependence on the chiral vectors (tube diameter $d$) and the defect orientation $\theta$. Such variation for zigzagg nanotubes is stronger compared to armchair counterparts. The SW activation barrier approaches that of the two-dimensional graphene (9.26 eV) with increasing tube diameter. The zero point energy $\Delta$ZPE correction lowers the activation barrier, which is found to be less than 0.25 eV for all cases.[]{data-label="fig:SWa"}](figure2.jpg)
[**Activation barrier, nanotube curvature, and defect orientation.**]{} Thermodynamic quantities such as formation energy are insufficient to answer key questions of interest in CNT structural transformations. For example, how long does it take to form a metastable SW defect? Once formed, how long will such defects persist? This information related to formation kinetics under defined external conditions is important to understand processes including mechanical deformation and CNT nanojunction or superstructure formation. There are few estimates of this kinetic barrier to date, and none of which we are aware that considered potential orientation dependence on this barrier. Reported estimates have included incorrect descriptions of chemical bonding, and/or adopted methodologies to locate the first-order transition state that are now generally considered inadequate. [@PhysRevB.57.R4277; @dumitrica:2775] Here, we locate the (first-order) transition state via DFT-CINEB methods described in Supporting Information, and subsequently calculate the corresponding activation barrier $E_a$.
The calculated $E_a (d, \theta)$ for varied tube diameter and inequivalent defect orientations are shown in Fig. \[fig:SWa\] for zigzag and armchair nanotubes. The overall qualitative trend of $E_a (d, \theta)$ is similar to that observed above for the formation energy: $E_a (d, \theta)$ increases with $d$, and shows a similar $\theta$ dependence (Fig. \[fig:SWa\]). For all cases considered, the calculated $E_a(d, \theta)$ converges to the graphene value $E_a^G$ (9.26 eV) with increasing $d$, and this convergence occurs at larger $d$ than for the formation energy \[Fig. \[fig:SWf\](e) and Fig. \[fig:SWa\]\]. We calculated the activation energy for graphene $E_a^G$ with the identical theoretical approach, and also allowed the defect induced buckling perpendicular to the graphene plane. The present value for $E_a^G$ is in excellent agreement with previous calculations for graphene. [@0957-4484-24-43-435707] Similar to the trends observed for $E_f (d, \theta)$, for all tube diameters $E_a (\theta = \pi/3) > E_a (\theta = 0)$ for zigzag nanotubes and $E_a (\theta = \pi/2) > E_a (\theta = \pi/6)$ for armchair nanotubes. The complete $d$ and $\theta$ dependence of $E_a(d, \theta)$ can again be explained via curvature induced rehybridization. It is important to note that while the kinetic barrier of SW formation is very high (4–9 eV; Fig. \[fig:SWa\]), the reverse barrier \[$E_a(d, \theta) - E_f(d, \theta)$\] ranges between 4 and 5.5 eV for the nanotubes studied herein. This significant reverse activation barrier implies the (meta)stability of SW defects over a wide temperature range.
![Correlation between the thermodynamic formation energy and the kinetic barrier for $d > 0.5$ nm, which follows the linear Br[ø]{}nsted-Evans-Polanyi empirical rule. [@Bronsted1928; @Evans1938] The solid line is a linear fit, and the observed linear correlation can be explained by two overlapping parabolas representing the pristine and defected nanotube (inset).[]{data-label="fig:correlation"}](figure3.jpg)
[**Correlation between ${\bm E_f}$ and ${\bm E_a}$.**]{} This systematic study enabled investigation of possible and generalized correlations between the (thermodynamic) SW formation energy and the kinetic activation barrier. Indeed, we find an empirical Br[ø]{}nsted-Evans-Polanyi type linear relationship: $E_a = k_1 + k_2 E_f$, where $k$’s are empirical constants (Fig. \[fig:correlation\]). [@Bronsted1928; @Evans1938] Although we have demonstrated above the capacity to locate transition states and calculate associated kinetic barriers in Fig. \[fig:SWa\], that approach is computationally demanding and thus intractable for all possible combinations of CNTs and SW defect configurations. With this observed Br[ø]{}nsted-Evans-Polanyi correlation, we propose a reasonable estimate of activation energies that can be obtained for any nanotube via only knowledge of the formation energy (that is relatively easier to compute or measure). We find $E_a = (4.19 \pm 0.15)$ eV + $(1.05 \pm 0.04)E_f$ to be a good fit for the calculated values (Fig. \[fig:correlation\]). Such linear correlation between $E_a$ and $E_f$ can be understood qualitatively by two overlapping parabolas (inset of Fig. \[fig:correlation\]) representing the structures in the absence and presence of the defect. [@AriehBook] In this model, if one or both of the parabolas shift in energy such that the energy difference between the minima (formation energy $E_f$) increases (decreases), the corresponding activation barrier concurrently increases (decreases).
[**Manipulation of activation barrier.**]{} With this improved understanding of the relative thermodynamic and kinetic barriers of SW defects in CNTs, we next consider whether the considerable kinetic barrier for SW formation could be reduced significantly. Such a reduction that would promote SW defect formation is desirable in many novel electronic, mechanical and chemical applications, including the formation of CNT assemblies and superstructures. [@PhysRevLett.92.075504; @doi:10.1021/cr050569o; @Scientific.Report.2] It is known that applied uniaxial tension reduces the activation barrier. [@PhysRevB.57.R4277; @dumitrica:2775] Although that correlation explains the mechanical response of nanotubes, that approach to barrier reduction is not practically feasible for most applications. Thus, here we assess other plausible ways to manipulate the activation barrier, and find that substitutional heteroatom doping (with elements B, N, or S) strongly modulates the activation barrier (Table \[table1\]). Regardless of the type of CNT (defined by chirality and diameter) and orientation of the defect, we find that the activation barrier is reduced substantially ($\Delta E_a$ $\sim$ 1.3 - 4.6 eV) due to heteroatom doping at the active bond (Table \[table1\]). Doping with sulfur reduces the barrier most significantly, by 25–60% depending on the tube type and defect orientation. The reduction in activation barrier due to substitutional heteroatom doping can be qualitatively explained by bond weakening around the active site. This has been explained in detail earlier for fullerene. [@PhysRevB.84.205404] Due to the weaker C–X bonds in X@CNT (X = B, N, S) compared to the C–C bonds in updoped CNTs, the SW rotation becomes easier for heteroatom doped CNTs. Such B, N and S-doped CNTs have been synthesized experimentally, and are proposed as metal-free electrocatalysts for oxygen reduction reactions. [@Stephan09121994; @Gong06022009; @C3TA12647A] These dopants have also been found to facilitate the formation of novel three-dimensional CNT covalent networks, [@doi:10.1021/nn700143q; @ANIE:ANIE200705053; @Scientific.Report.2] and our determination that SW defects are favored with such doping is consistent with such doping also favoring network formation.
[rC[0.8cm]{}C[0.8cm]{}R[1.2cm]{}C[0.8cm]{}C[0.8cm]{}R[1.2cm]{}]{} CNT & & $\nu \times 10^{13}$ & &$\nu \times 10^{13}$\
& $E_f$ & $E_a$ & (Hz) & $E_f$ & $E_a$ & (Hz)\
& &\
(10,0) & 2.88 & 7.71 & 229.3 & 3.32 & 8.44 & 93.8\
B@(10,0) & 2.01 & 5.77 & 195.3 & 3.37 & 6.37 & 17.3\
N@(10,0) & 2.69 & 6.40 & 44.3 & 2.55 & 7.07 & 31.2\
S@(10,0) & 1.35 & 3.10 & 10.8 & 3.42 & 5.15 & 6.8\
& &\
(6,6) & 3.15 & 7.84 & 19.1 & 3.72 & 8.70 & 42.9\
B@(6,6) & 2.52 & 5.63 & 31.9 & 3.96 & 7.24 & 6.0\
N@(6,6) & 2.57 & 5.56 & 16.3 & 2.87 & 7.03 & 21.6\
S@(6,6) & 1.92 & 3.25 & 7.1 & 3.79 & 6.59 & 4.6\
\[table1\]
The rate of SW defect formation can be estimated from the activation energy using a simple Arrhenius expression, $\Gamma = \nu \exp(-E_a/k_BT)$, where the prefactor $\nu$ is related to the vibrational frequency (Table \[table1\]), $k_B$ is the Boltzmann constant, and $T$ is absolute temperature. Thus, with heteroatom doping the rate of SW activation becomes $\sim \exp(\Delta E_a/k_BT)$ times faster, as compared to the undoped case. For example, the rate of activation becomes six to 20 orders of magnitude faster for (10,0)-CNT due to heteroatom doping at temperatures relevant to fusion and chemical vapor deposition growth (1000 K). Thus, the reduction in activation barrier would promote CNT fusion via ion/electron irradiation, as the fusion proceeds via a series of SW bond rotations. [@doi:10.1021/nl015541g; @PhysRevB.66.245403; @PhysRevLett.92.075504; @NatMatKrasheninnikov] Moreover, the doping centers act as the SW nucleation center. This would be expected to facilitate covalent superstructure formation, which has been observed in recent experiments. [@doi:10.1021/nn700143q; @ANIE:ANIE200705053; @Scientific.Report.2] The present calculation indeed supports these experimental observations, and further indicate that S-doping should be more effective in this regard because the reduction in activation barrier is much larger (Table \[table1\]). It is important to note here that due to the accelerated formation kinetics and increased thermodynamic concentration, the chemistry of SW defects should be accounted for accurately in such heteroatom-doped CNTs developed for catalytic applications. [@Stephan09121994; @Gong06022009; @C3TA12647A]
Conclusions
===========
In summary, we have studied the thermodynamic and kinetic properties of important topological Stone-Wales defects in single-wall carbon nanotubes, via density functional theory coupled with nudged elastic band identification of transition states. Calculated formation and activation barriers depend systematically on the tube chirality (and thus on tube diameter) and defect orientation. The microscopic origin of such dependence is attributable to curvature induced rehybridization. Generally, both the formation and activation energies increase with increasing (decreasing) tube diameter (curvature), and approach the respective values for two-dimensional graphene (Fig. \[fig:SWf\] and Fig. \[fig:SWa\]). The (kinetic) activation barrier is correlated with the (thermodynamic) formation energy, and follows the linear Br[ø]{}nsted-Evans-Polanyi relation (Fig. \[fig:correlation\]). Thus, the kinetic barrier for SW nucleation can now be estimated from knowledge of the formation energy, the calculation of which is less demanding computationally. Further, we propose that the activation barrier can be manipulated substantially by heteroatom doping (Table \[table1\]) to increase the defect formation rate by up to 20 orders of magnitude at temperatures relevant to CNT fusion and chemical vapor deposition-based superstructure growth. This computational finding explains the recent experimental observations that heteroatom doping favors CNT nanojunction and superstructure formation. [@doi:10.1021/nn700143q; @ANIE:ANIE200705053; @Scientific.Report.2] Further, we propose that sulfur is a more effective dopant than nitrogen and boron for applications such as CNT fusion and superstructure formation that proceed via Stone-Wales bond rotation. The present findings can guide future experiments that seek to promote covalent CNT assembly.
Convergence of defect formation and activation energy on the tube length, and curvature dependent rehybridization and its effect on the formation and activation energy have been described and analyzed in the Supporting Information. Representative structures for heteroatom doped CNTs are also shown. MK acknowledges helpful discussion with A. Warshel, and a grant from the Department of Science and Technology, India under the Ramanujan Fellowship. Computational resources included the supercomputing facility at the Centre for Development of Advanced Computing, Pune, and at the Inter University Accelerator Centre, Delhi. The authors acknowledge use of computational resources supported by the National Research Foundation of Singapore through the BioSystems and Micromechanics Interdisciplinary Research Group of the Singapore-MIT Alliance for Research and Technology.
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Supporting Information
======================
Defect orientation and tube-length
----------------------------------
The SW defects on the CNT surface can be generated at any of the three inequivalent sets of C–C bonds. Thus, for a particular CNT, there are three possible orientations of SW defect: $\theta =$ $\pi/3 - \chi$, $\chi$, and $\pi/3 + \chi$, where $\chi$ is the chiral angle. Out of these three sets, two of them are equivalent for both zigzag ($\chi = 0$) and armchair ($\chi=\pi/6$) nanotubes. The two inequivalent orientations are shown in the manuscript \[Fig. 1 (a)-(b) for zigzag, and Fig. 1 (c)-(d) for armchair nanotubes\]. Before any discussion of formation energy $E_f$, and the kinetic barrier $E_a$ for SW defect, one should investigate their dependance on the tube length in the simulation box, which dictates the defect-defect interaction mediated via long-range strain field created by the defect itself. We calculated these quantities with varied tube length within the simulation cell, with $(6,m)$ CNTs as test cases (Table \[table1\]). We find that both $E_f$ and $E_a$ strongly depend on the tube length, and these calculated properties converge at lengths of 25.65 and 24.68 Å for (6,0) and (6,6) nanotubes, respectively. Thus, throughout the present calculations, we consider these lengths for zigzag and armchair tubes, respectively, which minimizes defect-defect interaction.
[cR[1cm]{}C[0.8cm]{}C[0.8cm]{}C[0.8cm]{}C[0.8cm]{}]{} ($n,m$) & Length & &\
& (Å) & $E_f$ & $E_a$ & $E_f$ & $E_a$\
(6,0) & & &\
& 17.10 & 2.01 & 6.38 & 3.05 & 7.35\
& 25.65 & 1.74 & 6.32 & 2.73 & 7.20\
& 34.20 & 1.72 & 6.35 & 2.72 & 7.14\
(6,6) & & &\
& 9.87 & 3.31 & 7.91 & 4.40 & 8.95\
& 14.80 & 3.15 & 7.83 & 4.00 & 8.91\
& 19.74 & 3.15 & 7.89 & 3.72 & 8.70\
& 24.68 & 3.10 & 7.82 & 3.65 & 8.75\
\[table1\]
Phonon calculation
------------------
Phonons are calculated using the finite difference method, in which we considered 16 atoms around the rotating bond that are mostly affected by the C–C bond rotation as highlighted in Fig. \[fig:phonon\]. All the transition states have been confirmed via one-and-only-one imaginary frequency, which indicate that these are indeed first-order transition states. These phonons are used to calculate the zero point energy correction to the formation energy and kinetic activation barrier. The prefactor to the reaction rate within the harmonic transition state theory has been also calculated using these phonons: $\nu = \Pi_i^{3N} \nu_i^{\rm P}/\Pi_i^{(3N-1)} \nu_i^{\rm TS}$, where $\nu_i^{\rm P} (\nu_i^{\rm TS})$ are the normal mode frequencies corresponding to pristine (transition state) structure.
![Phonons are calculated by considering only the atoms around the rotating bond, which are affected by the C–C bond rotation. These atoms are highlighted in red color.[]{data-label="fig:phonon"}](phonon.jpg)
![(Color online) Schematic orbitals in graphene and nanotubes. (a) The $\pi$ orbital in planar graphene is orthogonal to the $\sigma$ bonds. (b) The $\pi$ orbital is no longer perpendicular to the $\sigma$ bonds on a curved CNT surface, as the $\sigma$ bonds are tilted down by an angle $\vartheta$ relative to the tangential direction of the tube, and the $\pi$ orbital bending by an angle $\delta$, respect to the normal drawn on tube surface. []{data-label="fig:cartoon"}](Supple-C.jpg)
Curvature and rehybridzation
----------------------------
The carbon network on graphene is planar, and thus forms $sp^2$ hybridization with orthogonal $\sigma$ and $\pi$ orbitals \[Fig. \[fig:cartoon\](a)\]. In contrast, the carbon atoms on the CNT surface lie on a curved surface, and thus the $\sigma$ bonds are pyramidalized, and the $\pi$ orbitals bend \[Fig. \[fig:cartoon\](b)\]. Therefore, unlike in planar graphene, the $\sigma$ and $\pi$ orbitals are no longer perpendicular to each other. As a consequence the parts of the of the $\pi$ orbitals outside and inside rearrange due to Coulomb repulsion, and the outer contribution is much larger than the inner one \[Fig. \[fig:cartoon\](b)\]. These lead to mixing of $\sigma$ and $\pi$ orbitals, which is known as rehybridization, and crucially depends on the nanotube diameter and chirality. The rehybridization leads to bonding which is in between $sp^2$ and $sp^3$, and can be recognized as $sp^{2+\tau}$ hybridization (where $\tau$ lies within 0 and 1, depending on the tube diameter and chirality). [@Haddon17091993; @Dumitrica2002182]
[R[0.8cm]{}C[0.8cm]{}C[0.8cm]{}R[1.6cm]{}R[0.8cm]{}C[0.8cm]{}]{}\
CNT & $d$ (Å) & $\tau$ & CNT & $d$ (Å) & $\tau$\
(4,0) & 3.14 & 0.151 & (4,4) & 5.44 & 0.051\
(5,0) & 3.93 & 0.097 & (5,5) & 6.80 & 0.033\
(6,0) & 4.71 & 0.068 & (6,6) & 8.16 & 0.023\
(7,0) & 5.50 & 0.050 & (7,7) & 9.53 & 0.017\
(8,0) & 6.29 & 0.038 & (8,8) & 10.89 & 0.013\
(9,0) & 7.07 & 0.030 & (9,9) & 12.25 & 0.010\
(10,0) & 7.86 & 0.025 & (10,10) & 13.61 & 0.008\
\[table:tau\]
These facts can be mathematically accounted for within $\pi$ orbital axis vector construction, where it is assumed that the wave function is still separable in terms of $\sigma$ and $\pi$ orbitals. Assuming the $\sigma$ bonds are tilted down by an angle $\vartheta$ (pyramidalization angle) relative to the tangential direction of the tube. This introduces mixing of $p_z$ orbital with the $\sigma$ network. Under the orthogonality condition, the $\pi$ states on the curved nanotube surface can be written as, [@Haddon17091993; @Dumitrica2002182] $$| h_{\pi}\rangle = \frac{1}{\sqrt{1+\lambda^2}}(|s\rangle + \lambda |p_z\rangle),$$ where $\lambda$ depends only on the pyramidalization angle $\vartheta$ as $\lambda = (1-3\sin^2\vartheta)/2\sin^2\vartheta$. Rehybridizied states have new wave functions, where $\pi$ orbital consists both $\sigma$ and $s$ orbitals. One can estimate the degree of rehybridization $\tau$ depending on the tube diameter $d$, and chirality $(n,m)$. Let $\delta$ be the bending angle of $\pi$ orbital relative to the normal drawn on the tube surface, and presuming that the angles between the $\sigma$ bonds and the $\pi$ orbitals are equal due to symmetry, one can show that $\delta$ depends on tube diameter, and chirality. [@PhysRevB.64.113402; @rehybridization] For a zigzag nanotube, $$\tan\delta = \frac{\sin^2\frac{\pi}{2n}}{\frac{\sqrt{3}\pi}{6n}+\sqrt{\frac{\pi^2}{12n^2}+\sin^2\frac{\pi}{2n}}},$$ and for an armchair nanotube, $$\tan\delta = \frac{\tan\frac{\pi}{3n}\left( 2\sqrt{\frac{\pi^2}{12n^2}+\sin^2\frac{\pi}{6n}} - \tan\frac{\pi}{6n} \right)}{2\sqrt{\frac{\pi^2}{12n^2}+\sin^2\frac{\pi}{6n}} + \tan\frac{\pi}{3n}}$$
Finally, one can derive an analytical expression for the degree of rehybridization $\tau$ in $sp^{2+\tau}$ for both zigzag and armchair nanotubes, $$\tau_{\rm zigzag} = \frac{4(1+3\sin^2\delta)}{3(1+2\sin^2\delta)} \frac{\sin^4 \frac{\pi}{2n}}{\frac{\pi^2}{12n^2}+\sin^2\frac{\pi}{2n}},$$
$$\tau_{{\rm armchair}} = \frac{2(1+3\sin^2(\delta-\frac{\pi}{3n}))}{3(1+2\sin^2(\delta-\frac{\pi}{3n}))} \frac{\sin^2\frac{\pi}{3n}+2\sin^4\frac{\pi}{6n}}{\frac{\pi^2}{12n^2}+\sin^2\frac{\pi}{6n}}$$
![The formation energy $E_f$ is calculated as the energy of the defected structure relative to the pristine tube. With the increase in tube diameter $d$, the curvature induced rehybridization decreases, and thus the true hybridization becomes more and more $sp^2$ like, and converge to pure $sp^2$ for flat graphene. Thus, with increasing $d$ the defected structure lie higher in energy compared to the corresponding pristine tube, and therefore increasing the formation energy. Similarly, the activation energy also increases with increasing $d$.[]{data-label="fig:formation"}](Supple-A.jpg)
![For a particular CNT with diameter $d$, the rehybridization $\tau$ decreases with increasing $\theta$, the angle relative to the tube axis that is created by the rotating C–C bond in the pristine structure. For the defected structure, the local curvature of the rotated C–C bond increases with increasing $\theta$. Thus, $\tau$ decreases, which in turn increases $E_f$ and the corresponding $E_a$.[]{data-label="fig:orientation"}](Supple-B.jpg)
![Representative structures for heteroatom doped CNT. Here we show (a) undefected, (b) transition-state, and (c) SW defected geometries for S@(10,0) nanotube. The red ball is the substitutional S atom.[]{data-label="fig:doping"}](heteroatom.jpg)
Calculated degree of rehybridization $\tau$ is shown in the Table \[table:tau\] for both zigzag and armchair nanotubes, and is clear that with the increase in tube diameter $d$ (i.e., with decreasing curvature) $\tau$ decreases monotonically, and will be zero for planar graphene. Thus, with increasing diameter the true hybridization becomes more and more $sp^2$-like, as it has been discussed in the manuscript. This explains the $d$ dependance of both $E_f(d,\theta)$ and $E_a(d,\theta)$, which are calculated as the energy of the defected structure or the transition-state, respectively, relative to the pristine tube. Figure \[fig:formation\] explains the $d$ dependance of $E_f(d,\theta)$. With increasing $d$, the rehybridization $\tau$ of the rotated C-C bond in the defected structure decreases, and thus the defected structure is pushed toward higher energy relative to the pristine structure. Thus, increasing the energy difference between the structure with and without the defect explains the observed increasing $E_f(d,\theta)$ with increasing $d$. Similarly, the $d$ dependance of $E_a(d,\theta)$ can be explained by considering the first-order transition state.
The above explanation does not account for the local curvature of the rotating/rotated C–C bond, as it was assumed that all three $\sigma$ bonds are equal and are all are tilted down equally. Ot was also assumed that the $\pi$ orbitals form equal angles with the $\sigma$ bonds. However, this is not the case, specially for the tubes with smaller diameter. This fact is evident from the bond length analysis shown in Table \[table:bond\]. Depending on the orientation of the rotating C–C bond, bond length differs reflecting the effect of curvature induced rehybridization. However, this difference decreases with increasing tube diameter. The theta dependence can be explained by considering these effect of these features on the local curvature of the rotated C–C bond. With increasing $\theta$ the local curvature of rotated C–C bond decreases, and consequently the rehybridization $\tau$ decreases (shown in Fig. \[fig:orientation\]). Therefore, both the formation energy and activation barrier increase with increasing $\theta$ for both zigzag and armchair nanotubes.
Substitutional doping
---------------------
Substitutional heteroatom (such as B, N, and S) doped CNTs have been experimentally synthesized. The topological SW defect activation is much easier in these CNTs. Representative undefected, transition-state and SW defected structures are shown for S@(10,0) nanotube in Fig. \[fig:doping\]. In this case one could calculate two different formation energies. The energy requirement for heteroatom substitution, i.e., the formation energy of the substitutional defect. [@reviewer] However, we are not interested in this formation energy, as our goal is to calculate the SW defect formation energy once we already have doped CNT. The SW defected formation energy is calculated as, $E_f({\rm X@CNT}) = E_{{\rm SW}}({\rm X@CNT}) - E_{\rm P}(\rm X@CNT)$, where X=B, N, or S. The reduction in SW activation barrier for X@CNTs can be understood by bond weakening around the active site. The corresponding X–C bonds are much weaker than the C–C bonds. For example, the S–C bond strength (2.73 eV) is much weaker than C–C bonds (5.18 eV) for S@(10,0) nanotube. Thus, the bond rotations become much easier in X@CNTs.
@ifundefined
[6]{}
Haddon, R. C. Chemistry of the Fullerenes: The Manifestation of Strain in a Class of Continuous Aromatic Molecules. *Science* **1993**, *261*, 1545–1550 Dumitrica, T.; Landis, C. M.; Jakobson, B. I. Curvature-Induced Polarization in Carbon Nanoshells. *Chem. Phys. Lett.* **2002**, *360*, 182 – 188 Kleiner, A.; Eggert, S. Curvature, Hybridization, and STM Images of Carbon Nanotubes. *Phys. Rev. B* **2001**, *64*, 113402 Yu, O.; Jing-Cui, P.; Hui, W.; Zhi-Hua, P. The Rehybridization of Electronic Orbitals in Carbon Nanotubes. *Chin. Phys. B* **2008**, *17*, 3123–3129 Garcia, A. G.; Baltazar, S. E.; Castro, A. H. R.; Robles, J. F. P.; Rubio, A. Influence of S and P Doping in a Graphene Sheet. *J. Comput. Theor. Nanosci.* **2008**, *5*, 2221–2229
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we explore the theory of communities of practice in the context of a physics college course and in particular the classroom environment of an advanced laboratory. We introduce the idea of elements of a classroom community being able to provide students with the opportunity to have an accelerated trajectory towards being a more central participant of the community of practice of physicists. This opportunity is a result of structural features of the course and a primary instructional choice which result in the development of a learning community with several elements that encourage students to engage in more authentic practices of a physicist. A jump in accountable disciplinary knowledge is also explored as a motivation for enculturation into the community of practice of physicists. In the advanced laboratory what students are being assessed on as counting as physics is significantly different and so they need to assimilate in order to succeed.'
author:
- 'Paul W. Irving'
- 'Eleanor C. Sayre'
title: Conditions for building a community of practice in an advanced physics laboratory
---
Introduction
============
The development of a professional identity is a fundamental part of student development[@Pierrakos2009]. An appropriate subject-specific identity is a strong influence on students’ persistence in a discipline [@Barton2000; @Chinn2002; @Cleaves2005; @Shanahan2007]. There is a strong relationship between the development of a professional, subject-specific identity and participation in a related community [@Hunter2006; @Bonnar2007; @Lave1991]; in fact, professional identity and community participation are inextricably and symbiotically linked[@Lave1991; @Del-Castillo2003a; @Li2011].
Laboratory work in particular is generally seen as an opportunity for students to learn problem solving and develop their understanding of physics as well as to understand how the science community works and to eventually be able to take part in the community themselves.[@Danielsson2007].
In this paper, we claim that structural and programmatic features of a junior-level Advanced Laboratory course (“AdLab”) at Kansas State University, supported by instructor strategies, promote students’ enculturation into the physics community of practice by fostering a classroom learning community engaged in bench research. We support our claims with ethnographic interviews and with observations of AdLab students.
Communities of Practice framework
=================================
We use a communities of practice framework to describe how students develop a classroom community in AdLab. Communities of practice have three key characteristics: the individuals within form a group, either co-located or distributed[@Wenger1998; @Coakes2006]; the group has common goals or shared enterprise[@Wenger1998; @Barab2002]; the group shares and develops knowledge focused on a common practice [@Wenger1998; @Barab2002]. This final characteristic can be extended to include the sharing of mutually defined practices, beliefs, values and history.
An individual participates in several, overlapping communities of practice. A physics student involved in a research group might also be the goalie on a sports team, for example. That same research group might be part of a larger collaboration and, at the same time, members of the research group are also members of the physics department. Because one individual participates in several, overlapping communities, it is important to study how “more expansive networks”[@Nespor1994; @Bonnar2007] affect individuals’ participation. Active participants in different communities of practice have opportunities to learn the knowledge, rituals, and histories valued within each community[@Charney2007; @Hsu2010; @Lave1991; @Lopatto2009]. Inasmuch as the different communities overlap, knowledge and practices learned in one community affects practices in another[@Li2011; @Furman2006; @Aschbacher2010]. Conversely, when communities have different values, individual members may have difficulty importing practices from one community to another[@Aikenhead1996; @Costa1995].
In this paper, we are primarily concerned with two overlapping communities: the community which develops within AdLab, and the generalized physics community to which students are aspiring members. These two communities share many goals and norms; AdLab is part of students’ training to become physicists; some (but not all) of the practices in AdLab are common to the professional practice of physics. Of course, the students in these two communities are also members of other communities, but we do not focus on those aspects of their identity in this paper.
Duration
--------
Frequently, communities of practice evolve and grow for extended periods[@Wenger1998] and may involve many participants over time. In these communities, new-comers are socialized into the community of practice through mutual engagement with and support of old-timers. Through low-level but authentic practices, these peripheral participants are slowly inducted into the knowledge and skills of a particular practice. Over time, they develop more understanding, knowledge, and skills, becoming central participants and eventually mentoring their own peripheral participants[@Lave1991; @Barab2002].
Students in the process of moving from being a peripheral participant to central participant are referred to as having a trajectory towards being a central member of a community. Being on a trajectory within a community of practice is generally considered a slow induction process [@Lave1991]. In the AdLab course students are exposed to a greater number of the authentic practices of members of the community of practice of physicists. We believe that different classroom communities of practice provide different levels of authentic practice and therefore the opportunity for students to accelerate their own trajectory towards becoming a central member of a discipline based community.
Other communities of practice have shorter duration, such as the length of a semester, and may have fewer members. Classrooms as communities of practice are well-studied[@Borasi1998; @Schoenfeld1992; @Lemke1990; @Berland2011]. In these shorter-term, temporally-bounded communities[@Nathan2005], we discard the idea of new-comers and old-timers in favor of the more general idea of peripheral and central participants. Legitimate peripheral participants may sit on the outskirts of classroom discussion, learning discourse and norms[@Cobb2001] as they gradually become enculturated[@Li2011]. Conversely, central participants may speak frequently in discussion, be more active in setting norms, or interact with more participants.
Learning
--------
Learning physics is a primary objective in a physics classroom. In a community of practice, learning can conceptualized using situated cognition [@Danielsson2007], participation theory [@Rogoff1996; @Goertzen2011], and socially constructed knowledge or understanding [@Lampert1990; @Lave1991], or as a process of becoming a member of a community [@Hsu2010; @Lopatto2009]. Under these models, learning physics is not merely about learning the contents of physics textbooks, but also about learning ways to participate in the cultural enterprise of professional physicists.
Tension between scientist and classroom practices
-------------------------------------------------
If courses like AdLab are to prepare students to be physicists – to become more central participants in the physicist community of practice – then those students should engage in legitimate peripheral activities in the physicist community. Though physics classrooms and the larger physicist community share many of the same norms and practices, they differ in several key respects[@Bruffee1993; @Cockrell2000; @Duschl2002; @Squire2003]. For example, traditional teaching laboratories tend to emphasize reproducing prior results rather than creating new knowledge[@Hogan2001]. Introductory physics classes tend to promote students solving many problems weekly while professional physicists work in large teams over multiple years to solve single problems.
To counteract this disconnect between school science practices and professional ones, the teacher can take on the role of a broker, acting as a go-between the two communities and guiding the classroom community closer to that of the practicing physics community. She can promote classroom norms and allow activities that are legitimate activities of physicists[@Demaree2009]. More advanced coursework is more likely to enact norms and practices that are more like those in the larger professional community, as many faculty are more likely to treat advanced students as junior physicists.
Instructional Context
=====================
At Kansas State University, AdLab is traditionally taken by sophomores and juniors, both physics majors and physics minors. It meets twice weekly for three hours each meeting; experiments usually take two to three weeks to complete. Class time is almost entirely devoted to laboratory work, with student presentations once during the semester. The students produce an individual laboratory report for each experiment. The experiments include common topics in modern physics such as the Lifetime of the $\mu$ meson and Microwave Optics. Like many upper-level laboratory classes, each experimental set-up has only one set of equipment. Students rotate through the experiments, and each student will perform a subset of the total number of experiments available.
The advanced laboratory is described as the following in the course catalogue: “The completion of experiments of current and/or historical interest in contemporary physics. Students develop skills in and knowledge of measurement techniques using digital and analog instruments. Various data analysis techniques are used.”
There were 18 students enrolled in the lab at the beginning of the semester and 17 finished the semester; students were organized into six groups. Group members stayed together for the first three experiments and then switched some members for the final three experiments.
Structural and Instructional features
-------------------------------------
Within AdLab, there are several reasons for the development of a classroom community. We find four structural features:
Paucity of instructor time:
: There are six groups working on six different experiments, each of which is complicated and prone to conceptual, experimental, or equipment difficulties. There is one instructor. She simply does not have enough time to spend with each group. When students need help, they must frequently turn to other sources.
All in the room together:
: All groups work in the same room at the same time. Because they are in close proximity to each other, there are more chances for interaction between groups.
Experiments long and hard:
: The experiments last two or three weeks, and involve many complicated or finicky equipment, difficult error propagation techniques, or conceptual complexity. This has two implications for community formation: students need to seek out resources to help with troubleshooting their own experiments, and (at any given time) they have time available to help their peers troubleshoot a different experiment.
Same experiments at different times:
: Because groups cycle through experiments, pockets of localized expertise develop. When a new group starts on an experiment, the last group to perform that experiment has direct, localized expertise about performing it.
Additionally, we find one primary instructional choice which supports the development of a community of practice within AdLab. The instructor of the class, recognizing the structural constraints above, deliberately encourages the sharing and developing of knowledge and understanding between lab groups.
Elements of classroom community
-------------------------------
These four structural features, supported by the instructional choice, work in concert to promote the development of a classroom community of practice. This classroom community of practice has several elements as a result of the structural features and the instructional choice that are not typical of a classroom community. We will refer to these elements as enculturation elements as these elements encourage some of the authentic practices of physicists.
Classroom norms and expectations:
: The students have a greater control over the norms that are negotiated within the classroom. These norms are negotiated over time but result in a more collaborative learning environment and in norms that are more similar to those of professional physicists. The same is true for expectations as students expectations of what counts as physics changes over time due to the jump in ADK.
Distributed expertise:
: The students become experts in different experiments which encourages collaboration when groups experience problems with specific experiments.
Community involvement:
: The students collaborate and socialize between groups a significant proportion of their time within the AdLab environment.
Many central players:
: The socializing and collaboration is not focused on one particular group and is instead distributed throughout all the groups over the length of the AdLab course.
Instructor is not sole mediator:
: As the community developed the students began to perceive the instructor as not the sole mediator of learning.
We believe that all four of the structural features are necessary for these enculturation elements to develop. If there were enough instructor time, then students would be more likely to turn to the instructor(s) for help, even if the other three features were present. If the students were not working in the same room at the same time (as happened in the previous laboratory course), the barriers to intergroup interaction would be larger because students would have to seek each other out outside of class, and they would not have the equipment in front of them as they discussed the experiments. If the experiments were too simple, the students would not need much help, and if the experiments were too short, they would not have enough time to visit with their colleagues. Finally, if they all performed the same experiment at the same time, they would all develop expertise at about the same rate, so it would be more difficult for more localized pockets to develop. Also, if all groups work on the same experiments at the same time, they are likely to develop similar difficulties at similar times, encouraging the instructors to do mini-lectures on specific kinds of troubleshooting and discouraging inter-group discussion.
Methods
=======
The research presented in this paper is part of a ongoing ethnographic research project on the identity development of undergraduate physics students.
As a methodology, ethnography originates in anthropology[@Pirie1997b; @Garfinkel1967] and is commonly used to understand community life[@Marcus1998; @Barab2002]. Ethnography is generally concerned with the sociocultural features of an environment, including how people interact and their discursive practices [@Brown2004; @Pirie1997b]. In educational settings, it is used to investigate “classroom culture”, characterizing various relationships and events[@Collins2004; @Brown2004].
Data Sources
------------
Ethnography typically draws its data from a number of sources in order to get a more complete picture of the culture of the classroom but also in an attempt to overcome some of the weaknesses of subjectivity through triangulating multiple viewpoints [@Ernest1997; @Emerson1993; @Barab2002; @Barton2000; @Hunter2006; @Case2011]. Our data are drawn from diverse sources to triangulate multiple viewpoints on student experiences in Adlab.
The primary data for this analysis comes from observations of students participating in AdLab. Lab groups of three students were observed twice a week for three hour class sessions. This paper focuses on data from the first two weeks of the semester and the last two weeks of the semester. We follow three separate groups at both times. One of the groups (Group A) remained the same for the whole semester. Group B changed one member at the halfway point. In Group C, only one group member remaining the same. Figure \[fig:groupmembers\] shows group membership and changes over time[^1].
{width="\figwide"}
\[fig:groupmembers\]
The secondary data for this paper comes from semi-structured interviews with students who were recruited from upper-level physics courses in electromagnetism, mechanics, modern laboratory and AdLab as part of an ongoing identity study. Only data from AdLab students are included in this analysis, including interviews from before, during, and after their time in the course. We developed a 45 minute semi-structured interview protocol drawing on identity formation, epistemological sophistication, and metacognition literature that also focused on asking the students to describe their AdLab experiences. The interviews were video-taped and transcribed for analysis.
For supporting evidence, we conducted discussions with the course instructor about her goals for the course and how her instructional choices supported them. We also collected course artifacts such as instruction manuals and the syllabus.
Analysis methods
----------------
Starting with a macro-level of analysis, we looked at each class period and referred to our field notes in order to identify “activity segments”: all activities whether whole class, particular lab group or individual that occurred during each laboratory session [@Kelly1997]. With this index of activities we mapped the events of the classroom over time[@Green1981; @Kelly1997]. This mapping process allowed for analysis both on a topical level and a sequential level and the identification of thematic content.
One theme that emerged from our data was that the different student groups doing separate experiments began to talk to each other more frequently and with higher quality interactions as the course progressed. Another theme that emerged was that both the students and the instructor felt that the physics material and scientific practices in AdLab were closer than previous laboratory classes to ongoing research of practicing physicists. (Other themes emerged; they are not the focus of this paper and will not be discussed here.) We selected these themes for further study and analysis to help us understand how community of practice develops in the advanced laboratory community and how the AdLab experience affects the professional development of students in the course.
The micro-ethnographic analysis began by first identifying interactions between different groups of students. The geometry of the AdLab room helped us identify cross-group conversations: while working on a given experiment, a lab group tends to stay clustered around the equipment. We point the camera at the equipment. When a student from another group chats with our group of interest, he or she tends to physically visit the group of interest.
After all of the interactions had been identified, we began to look at the context and content of the inter-group interactions. We considered the pre- and post-context of the interaction, student discourse (content, tone of voice, volume of speech, and rhythm of turn taking), and body language of the interaction to interpret how the participants frame the interaction. Framing refers to the resources the students bring to bear for a particular interaction[@Sayre2008; @Irving2013b].
Once the different ways of framing inter-group interactions were identified, we then purposefully sampled specific episodes which represented significant evidence of each type of frame. This analysis of interactions with this micro-ethnographic approach allowed for a correlation to how these interactions related to the development of a community of practice within the advanced laboratory classroom. In order to provide further evidence for our claim that a community of practice developed we also quantitatively assessed how the number of interactions between groups changed over time and how the amount of time spent having interactions also changed over time. As we developed the themes and our observational evidence for it, we triangulated and refined the theme using data from the semi-structured interviews. Were students aware that a classroom community developed? We also consulted with the instructor to investigate how her instructional goals might shape the course.
Observations of community development
=====================================
The following section focuses on the observational evidence of the community of practice developing in the AdLab learning environment. Through ethnographic analysis of the emerging theme of development of a community of practice we identified the following episodes which highlight either how the structural features or instructor choices helped this community to form. The episodes where also interpreted to show how the community developed over time from its initiation in the first week. These changes are indicated by the change in negotiated norms and discourse that the students use within the AdLab learning environment.
### Episode 1: Typical first experiment interaction (the brief me on the experiment interaction).
This episode occurs during the first week of the AdLab course during each groups first experiment. It is the second day of Group C (Larry, Bob, Matt) working on the “E/M Hoag” experiment. This is an experiment that uses a cathode-ray tube to measure the charge to mass ratio for an electron by sending electrons down a tube with a known magnetic field supplied by a solenoid. The group struggled on the first day to get the experiment successfully set-up to allow for the taking of experimental data but by the time of this episode on the second day they are just at the point where they are successfully taking data. Carl from another group walks by Group C and spots them sitting closely together staring at a screen and decides to ask them how their experiment is going. This episode occurs due to the two of the four structural features being in place: *all in the room together* and *same experiments at different times*.
Carl:
: Whats going on over here?
Bob:
: We’re just getting numbers now
Matt:
: (wearily) Lots of numbers
Bob:
: (sarcastic tone) Very technical
Carl:
: Just looking at that thing, it looks ancient
Bob:
: (getting more excited) It’s funny sometimes the voltage will drop by hundreds of volts and to fix it you turn it off and turn it back on (makes a “can you believe this” face) also this knob broke off so we use a screwdriver to turn it, this knob doesn’t even exist anymore.
Carl:
: Nice...I think I’ll avoid this one
This was a very typical interaction at the start of the semester as students took note of what other experiments different groups where doing and enquired as to the level of difficulty that they involved. The students are aware that they have to do one of the experiments in the room next and because they are *all in the room together* and are doing *same experiments at different times* it allows them the opportunity to discuss the different experiments with their colleagues.
The briefness of this exchange is also typical of the first week of the semester. The AdLab community of practice had not fully negotiated the norm associated with the amount of time these enquires about experiments could last. In the first week these exchanges where all tentative and brief in nature and the students kept to their own group the majority of their time in lab as evidenced by the results in Table 1.1.
Another regularity of the beginning of the semester was the superficial nature in which the Bob talks about the problems with the experiment. His problem is not with the theory behind the experiment or the setting up of the equipment (both of which his group and him had significant trouble with). Instead his focus is on the machinery being dated and problematic. During the first experiment groups would often have intergroup conversations about the difficulty associated with particular experiments but did so superficially. This could be attributed to the development aspect of the bounded community of practice. The norm for how such conversations should occur had not been fully negotiated yet.
This episode indicates the need for the classroom structural features of *all in the room together* and *same experiments at different times* to be present in order for intergroup interactions to occur. These interactions are vital to the development of a community of practice. This episode also indicates that during the first experiment the development process was still occurring and the norms for the community had not yet been negotiated.
### Episode 2: The “brief me on the experiment” interaction in week 8.
This episode occurs in week 8 when Larry from Group C has now changed groups and is currently working with Abbey and Roy on the Microwave Optics experiment. It is the last day for all groups on their respective experiments and they are all in the process of deciding what experiment to do next. Liam from another group approaches Larry and asks him about the “E/M Hoag” experiment which he completed as his first experiment. Essentially this is a repeat of the “whats going on over here?” interaction that is described in episode 1.
Although the types of interactions progressed from just asking how an experiment is, the “whats going on over here?” interaction continued regularly but the quality of the interaction increased over time. As before this episode occurs due to the structural features of *all in the room together* and *same experiments at different times* but also *experiments long and hard* as students try to preempt troubleshooting before the experiment begins by asking more detailed questions about the experiment to help with their decision making process.
Liam:
: Did you do that one before? (pointing in the direction of a laboratory bench)
Larry:
: The rubidium? (pause) Oh, “E/M Hoag”, yeah
Liam:
: How was that, like for, for theory?
Larry:
: (enthusiastically) Basically I combined the theory and derivation, I just talked about, so we’ve got this device, how can you get a measurement for E over M for the solenoid, you know for the magnetic field and everything, so in talking about how the field was created inside the solenoid and how that affected the path of the electron I felt that covered the theory.
Larry continues to answer several more questions about the experiment before Liam is satisfied with whether he should recommend doing the experiment next to his group.
This episode indicates the change in quality of the intergroup interactions as the community norms have been negotiated at this point in the semester. It is now a large part of the community of practice to have long detailed discussions about the experiments and to that enquiry about specific details of an experiment is okay and revealing specific experiment based expertise to other group members is also okay. The groups are becoming more collaborative. This is evidence of the evolving nature of the community of practice as collaboration becomes more frequent and constructive once the norms of the community have been negotiated.
### Episode 3: Social interactions.
Episodes 1 and 2 focused on the “brief me on the experiment” interaction. This was not the only type of interaction that occurred in the AdLab community of practice. Social interactions were also infrequent to begin with but as with the previous interactions became more prevalent once the community had negotiated its norms that related to what was acceptable as a social interaction in this community. These interactions ranged from the frivolity of cracking jokes to discussions about topics that would be considered off topic but often inspired by some aspect of the experiment they are engaged in.
In the following episode Matt and Larry are no longer working in the same group but are, for their corresponding experiments, working in close proximity. It is week 9 and Matt has just completed the experiment that Larry is now working on: “Microwave Optics”. In this experiment students are expected to demonstrate the wave nature of light in a number of interference, diffraction and reflection experiments using microwaves. Matt is currently working on Scanning Tunneling Microscope with his group. This episode demonstrates the camaraderie and social aspect of the community of practice that evolved over time.
Matt:
: (concerned tone) Are the microwaves on?
Larry:
: Well they are going this way (indicates the direction he thinks the waves are going).
Matt:
: They’re reflecting onto your crotch.
Larry:
: (laughs) Oh yeah your right, oops, I was like I’ll make sure that Percy and Matt are not in the line of fire, I forgot to make sure I wasn’t in the line of fire....thanks for your concern about my crotch.
Matt:
: (smiling) Your welcome.
This episode can be interpreted in two different ways. Firstly it has an obvious component of Matt playing Larry’s set-up of the experimental equipment for humor by referencing the rays reflecting on his crotch. Humor can have a large effect on community building and is a form of discourse that can emphasize membership. The getting of a joke can illustrate that “you are one of us”, just as missing the humor behind a joke can result in alienation from a community. This is a joke situated within the lab community and presence of such social interactions indicates the development of a community or practice.
The second interpretation is that because of the structural features of the classroom *all in the room together, experiments long and hard* and *same experiments at different times* this interaction is able to occur. If Matt had not completed the “Microwave Optics” lab previously; was not in the room with Larry; had not built up the content expertise and had the time to pay attention to what Larry was doing then he may not have the ability to say anything about Larry’s setup. Incidentally there is also an affective element to this interaction as well, Matt is genuinely concerned that Larry is doing something wrong that might have negative effects on Larry in some capacity, even though it is communicated through humor. It indicates an element of the affective nature of communities in that members will look out for their fellow members.
### Episode 4: Experiment Specific Experts.
This episode focuses on the *experiments long and hard* structural component. Oliver, Toby and Laura are working on the “Millikan Oil Drop” experiment. For this experiment the students are attempting to measure the charge on the electron by measuring the charge on small oil droplets and relating this charge to being a multiple of some quantized charge unit. It is week 8 of the AdLab class and this is the groups second day working on the experiment. Toby has been inputting the results the group have been getting so far into his laptop and both he and Oliver are confused about how the equation related to the experiment needs to interpreted with their results and decides to ask for help from Tom who completed the “Millikan Oil Drop” experiment the previous week.
Oliver:
: So which is the first plate? Is that the bottom here?
Toby:
: Lets ask someone, hey Tom, I have a question for you (Tom walks over)
Tom:
: This is d and this is the equation here (points at a point on Toby’s screen).
Tom proceeds to spend at least the next 5 minutes explaining his interpretation of their results so far.
As the students began to have a history with experiments the amount of “can you help me” interactions increased dramatically as evidenced in both the observational data and in the interviews. The students would make reference to not being able to complete a given experiment if it was not for another group helping them out at a critical juncture. The helping of other students is a clear indication of a community developing with students building up experiment centric expertise and then sharing this expertise due to all of the highlighted structural features of the learning environment but especially *paucity of instructor time* and *experiments long and hard*. At the beginning of the AdLab students would rely on the instructor to help them out and this was often limited to a sort of take a ticket for instructor time set-up. By the end of the AdLab students who had now developed experiment centric expertise where now being asked for help and would freely oblige often spending upwards of 30 minutes helping another group.
### Episode 5: AdLab Based Discourse.
As mentioned in episode 3 and in the section on community of practice a big part of being integrated into a community is to begin to appropriate the discourse of the community. If the misinterpretation of jokes can lead to alienation so can the inability to communicate in the language of the community. The following episode occurs as Sally, Danny and Mike are on their second day of working with the nuclear magnetic resonance (NMR) spectrometer. The NMR has multiple possible experiments designed for use with the equipment, some of which are reliant on obtaining the Free Induction Decays (FID) signal on an oscilloscope. Oliver, Toby, and Laura had previously completed the NMR experiment and are working on the “Millikan Oil Drop” experiment, which is not located next to (but is within sight of) the NMR set-up. Laura had just borrowed an ruler off Sally. While she is returning it, Laura relays a message from Oliver to Sally’s group.
Laura:
: Thanks Sally, Oliver says nice FID signal
Sally:
: (laughs) thanks
Although brief, this example gives a great sense of the development of the AdLab community of practice as by this time period of the community, experiment specific discourse has become ubiquitous amongst those who have carried out certain experiments. It wasn’t just nice signal, it was nice “FID” signal. The students began to develop and appropriate the language of the community and use it within the classroom.
Another element of community development that is in evidence in this exchange is the fact that Oliver feels comfortable to comment on another groups experiment and how well they are doing. The groups moved from a beginning point where they where insulated groups occasionally discussing how hard an experiment was to the point where they are freely discussing, socializing and evaluating each others work on a regular basis.
### Episode 6: Instructional Choice.
The final episode focuses on the other crucial element present for the AdLab community of practice to develop and that is the instructional choice of the instructor. Toby, Oliver, and Laura are working on the “Millikan Oil Drop” Experiment as described in episode 4. It is the first session of the new experiment and the instructor comes over to quiz them on how their first tentative steps to setting up the experiment is going. In the “Millikan Oil Drop” experiment there is a choice of several oil atomizers that can be employed in the setup of the experiment and this is the focal point of the initial discussions. The instructor due to the structural constraints of the lab at this point in the semester is not aware of which atomizer has been working best and invites over a member of the previous group that has carried out the experiment (Tom) to discuss with Toby, Oliver, and Laura the expertise he has developed.
Instructor:
: Did you guys do this last?
Tom:
: My group did it last (volunteering quickly from other end of room)
Instructor:
: Good, do you have any tips for them?
Tom:
: (Tom walks over) Um, get used to taking it apart and cleaning it
Instructor:
: Okay, keep cleaning it a lot
Tom:
: Yeah do that a lot, if you get a build up, if there is a big white blotch, the top which is actually the bottom of the T.V. screen
Instructor:
: It’s labeled top but it says bottom because its inverted right?
Tom:
: Yeah, if you get a big blotch there you can probably, its a build up of oil, you have a little thing to dab it out, dab and then dab it on a paper towel
Instructor:
: Use this to dab it out?
Tom:
: Yeah
Instructor:
: Oh thats nice, so you don’t have to take it all apart?
Tom:
: Yeah, yeah
Instructor:
: Okay
Tom:
: That’s an easier way of getting rid of some of the excessive stuff
Instructor:
: Okay thats a good tip
Tom:
: Em, thats about it
Instructor:
: Okay
Tom:
: (enthusiastically) It’s fun if you get it to work
By inviting Tom over the instructor is sublimely negotiating the norm that it is okay to consult with other groups especially those who have previously completed the concerned experiment for help and advice. This encouragement of the development and sharing of knowledge and resources is a deliberate choice by the instructor due to the structural features of AdLab learning environment.
Episodes 1-6 have been presented above to demonstrate how the structural features and a instructional choice on behalf of the instructor encouraged intergroup cooperation and collaboration that has helped to develop a classroom community in the AdLab course with enculturation elements. The episodes emphasize the importance of there structural features and how they are connected to the development of specific elements of our classroom community of practice like the negotiation of norms or distributed expertise . The above episodes are a tiny minority of episodes that could have been chosen as evidence of the development of a learning community with these enculturation elements. In the next section we present quantitative evidence of how often groups interacted as further evidence of the many central participants and level of community involvement and collaboration elements of the learning community.
Quantitative analysis of community talk
---------------------------------------
Time Period of Semester Group A Group B Group C
------------------------- --------- --------- ---------
Initial 1.8% 0.9% 5.1%
Final 12.4% 8.0% 17.3%
: Inter-group interactions at the beginning and end of the semester. Numbers are percent of total time spent talking to other groups in the second experiment of the semester (Initial) and penultimate lab of the semester (Final).
\[tab:intergroup\]
Table \[tab:intergroup\] presents the percentage of laboratory time that the three groups observed spent interacting with another group in the laboratory environment at two different time periods. “Initial” refers to each groups percentage interactions with other groups during their second experiment of the semester. An experiment typically lasted 4 classroom sessions over a two week period which would be approximately 12 hours class time. The “final” is the percentage interactions with other groups for their second-to-last experiment of the semester. By the penultimate experiment of the semester group A has remained static in its membership while group B and C have changed members after their third experiment as indicated in Figure \[fig:groupmembers\] An intergroup interaction was coded in one of three ways. The first was if a member of another group came over to the group being observed and interacted with them. The second was if a member of the group being observed left that group to go interact with another group. The third was if groups initiated a conversation or joined a conversation with another group while being physically adjacent to their experimental set-up. The total time spent interacting with other groups by the 3 previously described methods was combined to calculate the total amount of time spent interacting with other groups.
The results indicate that the difference in time spent interacting with other groups between the two time periods “initial” and “final” for all three groups is significantly different. The amount of interactions that each group had with other groups at the start of the semester are substantially less than the amount of interactions at the end of the semester. The consistency in the difference between amount time interacting with other groups between the “initial” and “final” time periods across all groups allows a claim that more classroom discourse was occurring between groups by the end of the semester. This is compelling argument that a community of practice did develop over time in the advanced laboratory community. There are differences between the increase in interactions between groups especially in the case of group B which as a group did not increase in the amount they interacted with other groups as greatly as the other two groups. Although group membership and personality may account for the difference it is worth noting that group B’s final experiment was the NMR set-up. The NMR experiment was new to the advanced laboratory learning environment and the groups that had completed the experiment prior to group B all struggled with it. This resulted in the instructor spending more time with the group than was typical and preempting problems that the group may have sought solutions for from prior groups. Overall though these results demonstrate that all 3 groups became further involved in the community as the semester progressed.
Interview reflections on community development
==============================================
As part of the longitudinal study examining how upper-level physics students develop an identity as a physicist we conducted semi-structured interviews on a regular basis for the majority of this group of students. One of these sets of interviews was conducted at the 10 week point into the AdLab semester. As part of this interview we enquired about the students experiences in the AdLab. An important theme to emerge from the interview data is that the students also noticed several of the structural features that promote community development. In the following sections we discuss extracts from the interviews that pertain to specific structural or community building factors.
### Extract 1: Paucity if instructor time.
In extract 1 the interviewer asks Matt what he thought of the approach to instruction that was taken in the AdLab environment. From observations by the investigators in AdLab sessions they noticed that in the beginning of the semester there was often a queue for the instructors attention but that this became a less prominent feature of the classroom as time passed. We wanted to know if the students were aware of this and how did they feel about a perhaps perceived lack of access to the instructor.
Matt:
: It was pretty well taught but there was a lot of people in there so we couldn’t get a lot of one on one time, when we needed help...so two of our experiments, the first two, we where the first group doing so we couldn’t ask anyone else about them, but the other ones, when we couldn’t consult with (Instructor) we went to the people that had already done that experiment and they were usually able to figure what it is we were missing or what went wrong when we were setting up like that.
Matt specifically references the amount of people in the room and the lack of one on one time when help was required. This is Matt noticing the structural feature of *paucity of instructor time* and indicating that this was something he found problematic at first. This was resolved once the other groups in the lab and himself had build up experiment specific expertise and began to consult with each other. The consulting with each other and experiment specific expertise are further evidence of the structural features *all in the room together, experiments long and hard* and *same experiments at different times* although Matt is not being as explicit about the last 3 features.
### Extract 2: Experiments long and hard
Extract 2 covers all four of the structural features again but Toby’s reflections refer more explicitly to the *experiments long and hard* aspect of the structure. Toby is answering the same question as Matt did in extract 1 in regards to what he thought of the approach to instruction and describes spending time working with other students. The interviewer follows up by asking Toby specifically about collaboration and working with other students.
Int:
: How did you collaborate with the other people and what did you get from the other people in advanced lab?
Toby:
: If we ever had a problem, like we had a problem, with the Zeeman experiment. We couldn’t quite figure out how we where supposed to set it up, so we went to Mike, asked him and he showed us how he did it. For NMR (referring to the group currently doing that experiment), they weren’t quite sure what they where doing so they had Oliver and me come over. Mainly Oliver but I helped a little bit. We did the “E/M Hoag” \[experiment\]. For the “E/M Hoag” we had to derive the equation we needed and we went to eh Larry and Roy and we were able to look at their work and see what they did and once we saw where they started it wasn’t particularly hard to get it. So we basically drew on their experience, everyone seemed to draw on the experience of the experiments everyone else had when starting.
Toby’s description of the give and take of assistance between groups over several experiments indicates the growth of a community of practice. The *paucity of instructor time* is referenced in Toby’s description of going to another group when a problem arose as opposed to the instructor. The *experiments long and hard* is indicated by Toby seeking out other groups to help with equipment set or derivations and the other groups had both the expertise and the time to help them out and reciprocally Toby had the time to help other groups when they had similar problems. Doing the *same experiments at different times* allows the experiment specific expertise to develop.
### Extract 3: Community Development
A portion of each interview was aimed at examining how students perceived what they where getting out of their advanced laboratory experiences. For the most part this involved students describing how the experience had helped them understand the material but some questions where directed at asking what they thought particular elements of the course design where for. In extract 3 the interviewer asks Tom what he thought the purpose of the presentations that each student had to perform once a semester where for.
Int:
: So what do you think the point is behind the presentations?
Tom:
: So we have to present things in real life, we have to talk to people....it also strengthens our knowledge of the experiments and builds a community in the class, you get to talk to other people.
Tom thinks that the presentations are a part of the course in order to foster real world experiences or in other words develop some authentic physicist practices. Students identifying aspects of the course that they perceived as contributing to their preparedness for future endeavors in the interviews was common. It was also common that students made reference to collaborating or working with other groups as indicated in extract 2 and 3 as Tom does by identifying explicitly that the goal of the presentation activity is community development driven.
### Extract 4: Development over time
As with extracts 1 and 2 part of the semi-structured interview focused on collaboration with other groups and students. In the description of bounded communities of practice earlier in the paper we described that they did not just occur when you put a group of people together in a room. A development process has to occur and norms have to be negotiated. In extract 4, Tom reflects that he did not ask other groups about labs in the beginning but that this changed over time.
Int:
: So did you ask other people about labs often?
Tom:
: At the start not really. I kind of just kept to my group, except, well with the other groups that I knew I made jokes with, I’d hear things and just make jokes. I’m doing it more now, other people are talking to me as well about labs
It was indicated in section 5 that this process of isolated groups becoming more interactive over time was observed both in the quantitative and qualitative observational results. Tom reflecting on the process is further evidence that the community developed over time. The next section will also reflect on interview data but will focus on the other element of the AdLab being classified as a crucible course: the steep rise in ADK.
All of the above extracts provide further evidence that the enculturation elements developed within the classroom community over time and that students where aware of some of these elements.
Students’ descriptions of AdLab as a jump in accountable disciplinary knowledge
-------------------------------------------------------------------------------
Another feature of the AdLab learning environment is that there is a quite observable jump in accountable disciplinary knowledge[@Stevens2008] (ADK) from the previous courses that the students would have taken. Accountable disciplinary knowledge is “what counts” as doing physics: the kinds of activities, problems, and discourse that people engage in when they are participating in a physics community of practice. For example, doing well at the introductory physics level often entails solving 15 end-of-chapter problems weekly in a few hours alone and doing well at the upper-division undergraduate level entails solving a few problems weekly in 15 hours with peers. This difference in “what counts” as doing physics well constitutes a jump in accountable disciplinary knowledge between introductory and upper-division physics. Evidence of an ADK jump in AdLab is very striking in students’ descriptions of the course after participating in it for one semester.
Tom:
: The labs are more complex and more interesting. A lot less hand holding. There more enjoyable and they are actually looking at phenomena that I am interested in…its more about us discovering the phenomena…it feels like more of a professional setting than most of my other courses.
Matt:
: we have been investigating actual atomic structures or how to find the mass of an electron…previous labs would be a lot more cut and dry. Here’s the procedure. Follow it. You’ll get the results, easily, these ones where more of, heres the procedure. Most of it usually. Follow it and try and understand whats going on cause if you don’t you won’t know if what your getting is any good…the real feeling of being a physicist was trying to understand all that stuff that we get from it.
Laura:
: I really had to do a lot of work on my own and I wasn’t really expecting that…I thought maybe the lab write ups would be a little bit more prescribed and not so quite, its kind of like, these are your objectives, this is how the machine works, do it, and thats good.
Toby:
: Yeah the subject matter itself changed but thats to be expected for a higher level class…obviously they are trying to get you to really think about the subject matter. To understand the subject matter at a deeper level than just in EP labs. They want you to see it happen in advanced lab. They want you to see it happen and understand why its happening, by figuring it out yourself rather than being told. I mean we don’t want to create people who can just rattle of equations without understanding what those equations really mean. You want people who actually understand what those equations really mean…this time we have a lot more freedom in the time that it takes to do it. You know we have some constraints because the other groups have to use the equipment as well but we can come in on our own and do it. The freedom was nice even if it was the result of having more work.
Several of the students perceive that a lot of what they were doing in the AdLab environment and how they participated in it where more like authentic practices of physicists. The students also clearly perceived a jump in the level of the material and what was expected of them in the AdLab classroom. Changes in expectations are obvious from students noting that there was a lot more freedom and the labs where more prescribed and that they were expected to gain an understanding of the material and not just get a set of data.
Relating Communities of Practice to Accountable Disciplinary Knowledge: Crucible Courses
========================================================================================
The combination of quantitative and qualitative results presented in this paper clearly indicate that a classroom community of practice developed in the AdLab learning environment with certain enculturation elements. It is also obvious from the students reflections on the course that there was was a significant jump in ADK from their previous experiences due to significant changes in the structural and programmatic features of the AdLab learning environment. This combination of jump in ADK and enculturation elements of the course resulted in students being offered the opportunity to accelerate their own trajectory to being more of a central participant of the physicists community of practice. This emphasis on enculturation from both ADK and structure has resulted us in labeling the AdLab course as a possible “crucible course”.
We describe crucible courses as the first courses in which students work on difficult physics problems surrounded primarily by other physics students, are treated by their professors as junior physicists, and take on identities as part of a community of physics students. In our prior work, we have identified crucible courses: those associated with large changes in ADK and developments of physics identity. Both students [@Irving2013] and researchers[@Sayre2008] seem to know these courses “when they see them” [@Jacobellis1964]. The courses are typically intermediate level (taken by sophomores or juniors) and are among the first courses populated predominately by physics majors and minors. They have smaller enrollments and foster a greater sense of community within the class. They may be theory courses or laboratory courses, but in either case the expectations of students and their perceptions of the stakes are substantially higher than in previous courses. This is a working definition and we intend to further investigate what are the key elements of a crucible course. We believe an emphasis on enculturation is a key feature of a crucible course.
To discuss further this enculturation process we must first examine communities of practice and how we interpret they fit into the college environment. In alignment with previous researchers [@Schoenfeld1992; @Duschl2002; @Demaree2009; @Lemke1990; @Berland2011] we believe it is applicable to view the classroom community as a community of practice. If that is the case as a student you will occupy many communities of practice concurrently within the college environment while also being a member of several other communities outside of the college context. In fact the majority of students waking hours during their time in college will not be spent in the classroom [@Li2011]. The combination of these memberships to a variety of communities of practice will all have influences on each other and can help in the development of a physics identity both in obvious and less obvious ways.
Students are on trajectories to developing an identity as a physicist when they enter a physics classroom. Once they enter a physics classroom they are developing a relationship with physics that may turn into a physics identity. They may not intend on becoming a physicist, it might not even be their major but when they enter a physics classroom they engage in a variation of the practices of becoming a physicist. That is the nature of the a classroom being a community of practice and so in essence any physics classroom is a sub-community of the community of physicists.
Discussion
==========
All classroom communities of practice are different and these differences may be trivial or may be extensive. Different classroom communities offer different levels of exposure to the authentic expectations, practices, content knowledge and discourses of the discipline of physics. Therefore the differences between classroom communities of practice can result in students moving along their trajectories to being a member of the physicists community of practice at different accelerations. To clarify the classrooms would offer the opportunity but it is up to the students to participate either peripherally or centrally.
The AdLab classroom is an example of a community of practice that offers the opportunity to have an accelerated trajectory towards being a central participant of the community of practice of physicists. It introduces students to the expectations, practices, content knowledge and discourses that more closely resemble those of the physicists community. This is achieved by having the students collaborating as a group and with other groups on long and hard physics experiments that are generally more modern in setting in an environment that echoes what students might perceive as a research environment. Being a central participant in this environment will accelerate ones trajectory to being a more central participant of the community or practice of physicists.
We do not think that all classroom communities of practice should offer opportunities of accelerated trajectories. An accelerated trajectory classroom in introductory physics would be inappropriate. It has been indicated [@Borasi1998] that there are already great shifts being expected of students in introductory classes as teachers try to move students away from being socialized to memorize, practice and recite and move towards being comfortable with constructivist and social constructivist perspectives. Also the norms of college can be very different from the norms of school and again the norms of actual practitioners of physics.
We have argued that AdLab develops into this a community of practice very effectively due the factors of: paucity of instructor time; all in the room together; experiments long and hard, do some of the same experiments at different times; instructor supports the development of a community of practice. Of the above claims all of them have been discussed extensively in the results except for *All in the room together*. This claim comes from the assertion that in the previous semester some of the same students took the modern laboratory course which is set up so that each group of students attempted the same experiment each week and so no community of practice could develop except amongst each separate group of 3 or 2 students. When asked in interviews whether they had discussed the laboratory they where trying to complete with other members of the modern laboratory class the answer was generally no, although they often did work with their group outside of class.
As mentioned previously the development of a community in community of practice literature is not commonly discussed but to us is a key feature of bounded communities of learning. A community of practice in a classroom does not form on the condition of putting students in a room together although it may result in one eventually. In our case a classroom community of practice developed that had several elements: classroom norms and expectations; distributed expertise; community involvement; many central players and the instructor is not sole mediator. We argue that these elements developed as a result of the presence of all structural features of the classroom previously mentioned and the instructors choice to emphasize collaboration. We also argue that these elements are an important part of the AdLab course offering the opportunity for a student to accelerate ones trajectory towards being a more central participant of the physicists community of practice.
The classroom community did not start with the enculturation elements. Students’ ways of participating change as they learn the norms and practices of the classroom community of practice, which includes developing a shared discourse with their fellow community members of students and instructors [@Bielaczyc1999]. The students also have to figure out the boundary constraints [@Barab2002] of this new community of practice due to it being a bounded community. Norming is one of the five stages of group development[@Tuckman1965] and although not necessarily relatable to the communities of practice theory does indicate that a classroom has to go through some development before it becomes the finalized version of the learning community. We believe for the AdLab classroom the features previously mentioned are the reason why it developed into a classroom community with several enculturation elements and the majority (if not all) of the students participating centrally.
A big jump in ADK from course to course can be difficult for students as often what they think doing physics means has changed from what it has meant in the past. It could be argued that the community of practice developing is a support mechanism for the students in order to deal with the jump in ADK. A big jump in ADK without a community developing could result in greater losses in retention and persistence as students struggle to deal with the changes in norms and expectations. Added to this is that cultural practices of professional scientists are always adapted to fit the realities of the classroom and to suit the teachers values/goals [@Hogan2001b; @Squire2003]. When designing curricula or courses careful consideration should be given to the expectations, practices, content knowledge and discourses of the community of physicists that are being incorporated into the design. A realization must be made that what we ask of the students is not just different content but a different level of content that attached to it has a different set of norms and expectations.
Adding structural and instructional features to a course that encourages the development of an effective community of practice may be one way of equipping students to deal with such transitions.
Conclusion
==========
The advanced laboratory community of practice was identified as a community of practice that provides the opportunity to accelerate a students trajectory to becoming a member of the physicists community of practice. It was also identified as a community of practice that forms quickly and has several enculturation elements due to several reasons: paucity of instructor time; all in the room together; experiments long and hard, do some of the same experiments at different times; instructor encourages the sharing and co-development of knowledge and understanding.
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[^1]: All names are pseudonyms.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The singular seesaw mechanism can naturally explain the atmospheric neutrino deficit by the maximal oscillation between $\nu_{\mu_L}$ and $\nu_{\mu_R}$. This mechanism can also induce three different scales of neutrino mass squared differences, which can explain the neutrino deficits of three independent experiments (solar, atmospheric, and LSND) by neutrino oscillations. In this paper we show that the realistic mixing angles among neutrinos can be obtained by introducing the hierarchy in the Dirac neutrino mass. In the case where Majorana neutrino mass matrix has rank 2, the solar neutrino deficit is explained by the vacuum oscillation between $\nu_e$ and $\nu_\tau$. We also consider the case where Majorana neutrino mass matrix has rank 1. In this case, the mater enhanced Mikheyev-Smirnov-Wolfenstein solar neutrino solution is prefered as the solution of the solar neutrino deficit.'
address:
- |
Department of Physics, Tokyo Institute of Technology\
Oh-okayama, Meguro, Tokyo 152-0033, Japan
- |
Faculty of Engineering, Mie University,\
Mie, 514-0008, Japan
- 'Theory Group, KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan'
author:
- 'Yuichi Chikira[^1]'
- 'Naoyuki Haba[^2]'
- 'Yukihiro Mimura[^3]'
title: |
hep-ph/9808254\
TIT-HEP-397\
KEK-TH-582\
The Singular Seesaw Mechanism with Hierarchical Dirac Neutrino Mass
---
\#1[[$\backslash$\#1]{}]{}
4.60.Pq, 14.60.St
Introduction
============
According to the recent Super-Kamiokande experiment [@superk], their atmospheric neutrino data indicate the oscillation between $\nu_\mu$ and $\nu_\tau$ or sterile neutrinos with the maximal mixing $$\sin^2 2\theta_{\mu x} \sim 1,$$ where $x$ represents $\tau$ or sterile neutrinos. The neutrino mass squared difference $\Delta m^2_{\rm atm}$ is of order $10^{-3} [{\rm eV}^2]$.
It is well-known that other two independent experiments also imply neutrino oscillations. One is the solar neutrino experiment. This experiment implies the oscillation between $\nu_e$ and other neutrinos, and there are three possible solutions, namely, large or small mixing angle Mikheyev-Smirnov-Wolfenstein (MSW) solution [@MSW], and the vacuum oscillation solution [@vac]. The small (large) angle MSW solution suggests [@solar] $$\sin^2 2\theta_{e x} \sim 10^{-2} \;\; (0.4 {\rm -} 1)$$ with mass squared difference of order $ 10^{-5} [{\rm eV}^2]$, and the vacuum oscillation solution suggests $$\sin^2 2\theta_{e x} \sim 0.75{\rm -}1
\label{vacuum-mixing}$$ with the mass squared difference of order $ 10^{-10} [{\rm eV}^2]$. The vacuum oscillation solution is now the most suitable solution from the electron energy spectrum of the recent Super-Kamiokande experimental data [@Suzuki], although the small angle MSW solution has been regarded as the most realistic candidate. On the other hand, LSND experiment measures oscillation between $\bar\nu_\mu$ and $\bar \nu_e$ [@LSND] with a short base line experiment. Although the confirmation of the LSND results still awaits future experiments,[^4] their results indicate the small mixing angle $$\sin^2 2\theta_{\mu e} \sim 10^{-2}$$ with mass squared difference $\Delta m^2_{\rm LSND} \sim 1 [{\rm eV}^2]$.
The most interesting mechanism which can naturally explain the smallness of neutrino masses is the so-called seesaw mechanism [@seesaw]. The general mass matrix of neutrinos above $SU(2)_L$ breaking is given by $${\cal M} = \left( \begin{array}{cc} 0 & m_D \\ m_D^T & M_R \end{array} \right),$$ where $m_D$ and $M_R$ represent Dirac and Majorana $3 \times 3$ flavor space mass matrices, respectively. In the case of $m_D \ll M_R$, there appear three light neutrinos with mass matrix $${\cal M}_{light} = - m_D M_R^{-1} m_D^T .$$ This is the essence of the seesaw mechanism. It is worth noting that here it is assumed that there exists inverse matrix of $M_R^{-1}$, that is, $\det M_R \neq 0$. The singular seesaw mechanism [@singular2; @singular1], which is also called “partially broken seesaw mechanism” [@FY][^5], is just the case of $\det M_R = 0$. Then, some light right-handed neutrinos are not integrated out, and behave as sterile neutrinos. It turns out that mixings between the survived sterile neutrinos and active neutrinos are large in general because of the pseudo-Dirac texture [@pD]. We can use this mechanism to explain the large mixing of the atmospheric neutrino experiment. If nature adopts four (or more) neutrino oscillations, the singular seesaw mechanism supplies one of the most attractive models.
The authors of Ref.[@singular2] discussed this singular seesaw mechanism in the case that there is no hierarchy in the Dirac mass matrix $m_D$ and Majorana mass matrix $M_R$. They did not take the small mixing of the LSND into account. In this paper, we study the singular seesaw mechanism by introducing the hierarchy in the Dirac mass matrix $m_D$ in order to explain the small mixing of the LSND experiment. We will also study whether the hierarchical Dirac mass can induce not only the small mixing of the LSND experiment but also the small mixing of the MSW solar neutrino solution.
This paper is organized as follows: In section II, we will review the singular seesaw mechanism briefly. In section III, we introduce hierarchical Dirac mass matrix, and determine the order of parameters. We show that the vacuum oscillation solution is prefered as the solution of the solar neutrino deficit in the case where Majorana neutrino mass matrix has rank 2 and that the MSW solution is prefered in the case where Majorana neutrino mass matrix has rank 1. In section IV, we give summary and discussions.
Singular Seesaw Mechanism
=========================
At first, we explain the pseudo-Dirac mass texture [@pD]. In the one generation the neutrino mass term above $SU(2)_L$ breaking is given by $$\label{massmatrix1}
-{\cal L} = \frac{1}{2}\left( \begin{array}{cc}\nu & \nu^C\end{array}\right)
\left( \begin{array}{cc}0 & m \\ m & M \end{array}\right)
\left( \begin{array}{c} \nu \\ \nu^C \end{array} \right),$$ where $\nu$ and $\nu^C$ represent (two component) left- and right-handed neutrinos, respectively. Here we consider the case of $M \ll m$. In this case the mass matrix (\[massmatrix1\]) realizes large mixing angle of $\sin^2 2\theta = \frac{m^2}{m^2 + M^2/4} \sim 1$ between $\nu$ and $\nu^C$. The eigenvalues of this mass matrix are $\pm m + M/2$, and the neutrino mass squared difference is $\Delta m^2 = 2mM$ [@pD]. This mass term is almost Dirac but not exact, so it is called pseudo-Dirac texture, which can naturally induce the maximal mixing. The mass term in the opposite case of $M \gg m$ is that of ordinary seesaw mechanism.
Now let us take three generations into consideration. We take $m$ and $M$ as $3 \times 3$ matrices $m_D$ and $M_M$, respectively in Eq.(\[massmatrix1\]). The right-handed Majorana neutrino mass matrix $M_M$ is assumed to be rank $2$ (or $1$). In this case two (one) neutrinos become light by the ordinary seesaw mechanism, and remaining one (two) neutrino has the pseudo-Dirac mass texture. For example, in the rank-2 case, we can obtain the eigenvalues of four light neutrinos [@singular2; @singular1] as $$\beta m, \quad \beta m, \quad{\rm and}\: \pm m + \beta m,$$ where $\beta = m/M$, in the case of no hierarchy in the mass matrices $m_D$ and $M_M$. It is interesting that the two lighter neutrinos’ masses and the mass splitting for the pseudo-Dirac neutrinos are the same scale. Then three mass squared differences form geometric series as $$\Delta m^2 = \beta^2 m^2, \quad \beta m^2, \quad {\rm and}\:\: m^2,$$ and are favorable to explain three known neutrino oscillation modes, namely, solar neutrinos(MSW solution), atmospheric neutrinos and LSND [@singular1]. Furthermore, since the middle scale of the mass squared difference for atmospheric neutrinos corresponds to pseudo-Dirac neutrinos, its maximal mixing realizes naturally. However, they can not explain neither the small mixing angle of LSND nor the small angle MSW solution if there is no mass hierarchy in $m_D$ and $M_M$. We will see that the hierarchical Dirac mass matrices lead to different series of mass squared difference, which is suitable for vacuum oscillation solution for solar neutrino deficit rather than MSW solutions.
The flaws of the singular seesaw mechanism is that the Dirac neutrino mass $m$ needs to be too small (about 1\[eV\]), and it fails the motivation of original seesaw mechanism. When we incorporate the singular seesaw mechanism into phenomenological models, we need some extra mechanism to apply the small Dirac neutrino mass. Its smallness will be realized, for example, by the non-renormalizable interactions [@Langacker], though we do not mention the detail in this paper.
Singular Seesaw Mechanism with Hierarchical Dirac Neutrino Mass Matrix
======================================================================
We introduce the hierarchy in the Dirac neutrino mass matrix as $$\label{mD}
m_D = \left( \begin{array}{ccc}
\epsilon' m_{11} & \epsilon' m_{12} & \epsilon' m_{13} \\
\epsilon m_{21} & \epsilon m_{22} & \epsilon m_{23} \\
m_{31} & m_{32} & m_{33}
\end{array} \right) .$$ We can take the hierarchical parameter $\epsilon$ and $\epsilon'$ as $\epsilon' \le \epsilon < 1$, when we do not order the left-handed indices with naming of neutrino flavors. In this paper, we do not mention the hierarchical structure with respect to right-handed indices. The mass term of neutrinos is given by $$-{\cal L} = m_{Dij}\nu_i\nu_j^C + \frac{1}{2}
M_{Mij}\nu_i^C\nu_j^C .$$ We analyze this model in two cases, where the rank of Majorana mass matrix $M_M$ is 1 or 2.
At first, we study the case where Majorana mass $M_M$ has rank 2 as $M_M={\rm diag}(M_1,M_2,0)$. After integrating out the heavy neutrinos[^6], light neutrinos have masses as $$-{\cal L} = -\frac{1}{2}\left( \frac{m_{Di1} m_{Dj1}}{M_{1}} +
\frac{m_{Di2} m_{Dj2}}{M_{2}} \right) \nu_i \nu_j + m_{Di3}\nu_i\nu_3^C.$$ The mass matrix for $(\nu_1,\nu_2,\nu_3,\nu_3^C) \equiv
(\alpha,\beta,\gamma,s)$ is given by $${\cal M} \sim \left( \begin{array}{cccc}
-\epsilon'^2 \beta & -\epsilon \epsilon' \beta & -\epsilon' \beta & \epsilon'\\
-\epsilon \epsilon' \beta & -\epsilon^2 \beta & -\epsilon \beta & \epsilon \\
-\epsilon' \beta & -\epsilon \beta & -\beta & 1 \\
\epsilon' & \epsilon & 1 & 0
\end{array}\right) m.$$ This matrix is diagonalized as $$U^\dagger {\cal M} U \sim
{\rm diag} (\epsilon'^2 \beta m,\ \epsilon^2 \beta m,\
(1-\beta) m,\ -(1+\beta)m ),
\label{rank-2:diag}$$ where $$U \sim \left( \begin{array}{cccc}
1 & -{\epsilon'}/{\epsilon} & \epsilon' & -\epsilon' \\
{\epsilon'}/{\epsilon} & 1 & \epsilon & -\epsilon \\
-\epsilon' & -\epsilon & 1 & -1 \\
\epsilon' \beta & -\epsilon \beta & 1 & 1
\end{array} \right).$$ Now we estimate the probability of the neutrino oscillation, which is given by $$P(\nu_\alpha \rightarrow \nu_\beta) = \delta_{\alpha \beta} -
4 \sum_{i<j} U_{\alpha i} U^*_{\beta i} U^*_{\alpha j} U_{\beta j}
\sin^2 \frac{\Delta m_{ij}^2}{4E}L ,$$ where we neglect $CP$ phase for simplicity. The oscillation amplitude between $\alpha$ and $\beta$ is given by $$-4 \sum_{i<j} U_{\alpha i}U^*_{\beta i} U^*_{\alpha j} U_{\beta j}.$$ From Eq.(\[rank-2:diag\]), we can obtain three scales of mass squared differences of $\Delta m^2_{12} \sim \epsilon^4 \beta^2 m^2$, $\Delta m^2_{34} \sim \beta m^2$, and $\Delta m^2_{13} \sim \Delta m^2_{14} \sim \Delta m^2_{23} \sim \Delta m^2_{24}
\sim m^2$. We list the amplitudes corresponding to these three oscillation in Table I. The oscillation between $\gamma \leftrightarrow s$ gives a large mixing, which is expected to correspond to atmospheric neutrino oscillation. Then, we fix $$\beta m^2 \sim 10^{-3} {\rm [eV^2]}.$$ The oscillation with $\Delta m^2 \sim m^2$ may correspond to LSND data, so we fix $$m^2 \sim 1 {\rm [eV^2]}.$$ Then there remain two patterns whether $(\alpha, \beta)$ is assigned as $(e, \tau)$ or $( \tau , e)$. Let us consider both possibilities here.
(1-1)
: In the case of $( \alpha, \beta) = (e, \tau)$, $\epsilon'$ must be of order $10^{-1}$ from the small mixing of LSND data. Then the oscillation with $\Delta m^2 \sim \epsilon^4 \beta^2 m^2
\sim 10^{-6} \epsilon^4 {\rm [eV^2]}$ should correspond to the solar neutrino oscillation. $(i)$: For the mass squared difference of MSW solution, we must choose the parameter $\epsilon$ to be close to 1. Then, it turns out that the $\nu_\mu$-$\nu_\tau$ mixing is large with $\Delta m^2 \sim 1[{\rm eV}^2]$. This oscillation leads to contradiction with the atmospheric neutrino data. Therefore we can not obtain the MSW solution in this pattern. $(ii)$: On the other hand, the vacuum oscillation solution can be realized when $\epsilon = O(10^{-1})$. We can realize the large mixing of Eq.(\[vacuum-mixing\]) because the corresponding mixing angle is of order $(\epsilon'/\epsilon)^2$.
(1-2)
: In the case of $(\alpha, \beta) = (\tau, e)$, $\epsilon$ must be of order $10^{-1}$ from the small mixing of LSND data. In this pattern, the mass squared difference corresponding to the solar neutrino oscillation should be of order $10^{-10}$\[eV$^2$\]. Therefore, only vacuum oscillation solution can be allowed. To explain the large mixing of the solution, $\epsilon'$ must be satisfy $\epsilon \sim 10^{-1}$. This is the same parameters as the case of $(ii)$ in [**(1-1)**]{}.
Next, let us consider the case where Majorana mass $M_M$ has rank 1 as $M_M = {\rm diag}(M,0,0)$. The mass term of neutrinos is given by $$-{\cal L} = m_{Dij} \nu_i \nu_j^C + \frac{1}{2}M \nu_1^C \nu_1^C.$$ After integrating $\nu_1^C$ out, light neutrinos $(\nu_1, \nu_2, \nu_3, \nu_2^C, \nu_3^C) \equiv
(\alpha, \beta, \gamma, s_1, s_2)$ have masses as $${\cal M} \sim \left( \begin{array}{ccccc}
-\epsilon'^2 \beta & -\epsilon' \epsilon \beta & -\epsilon' \beta
& \epsilon' & \epsilon' \\
-\epsilon \epsilon' \beta & -\epsilon^2 \beta & -\epsilon \beta &
\epsilon & \epsilon \\
-\epsilon' \beta & -\epsilon \beta & -\beta & 1 & 1 \\
\epsilon' & \epsilon & 1 & 0 & 0 \\
\epsilon' & \epsilon & 1 & 0 & 0 \\ \end{array} \right)m.$$ This matrix can be diagonalized as $$U^\dagger {\cal M} U \sim
{\rm diag} ( \epsilon'^2 \beta m,\ (\epsilon - \epsilon^2 \beta) m, \
-(\epsilon + \epsilon^2 \beta) m,\ (1 - \beta) m, \
-(1 + \beta) m ),$$ where $$U \sim \left( \begin{array}{ccccc}
1 & -{\epsilon'}/{\epsilon} & -{\epsilon'}/{\epsilon} &
\epsilon' & \epsilon' \\
{\epsilon'}/{\epsilon} & 1 & 1 & \epsilon & \epsilon \\
\epsilon' & -\epsilon & -\epsilon & 1 & 1 \\
\epsilon' \beta & -1 & 1 & 1 & -1 \\
\epsilon' \beta & 1 & -1 & 1 & -1
\end{array} \right).$$ There are four scales of mass squared differences as $\Delta m^2 \sim \epsilon^3 \beta m^2$, $\beta m^2$, $\epsilon^2 m^2$ and $m^2$. The oscillation amplitudes corresponding to these oscillation modes are listed in Table II. The atmospheric neutrino oscillation can be regarded as $\gamma \leftrightarrow s_1,s_2$. Therefore, the mass squared difference $\beta m^2$ must be of order $10^{-3}$\[eV$^2$\]. Then, the mass squared difference corresponding to the solar neutrino oscillation should be $\epsilon^3 \beta m^2 {\rm [eV^2]}
\sim 10^{-3} \epsilon^3 {\rm [eV^2]}$. There are two candidates for the mass squared differences of LSND, namely, $\Delta m^2_{\rm LSND} \sim \epsilon^2 m^2$ or $\Delta m^2_{\rm LSND} \sim
m^2$. Here we consider the both possibilities.
(2-1)
: In the case of $(\alpha, \beta) = (e, \tau)$, $\epsilon'$ must be of order $10^{-1}$ from the small mixing of LSND data. As for the solar neutrinos, the vacuum oscillation solution is excluded because the parameter $\epsilon$ can not be smaller than $\epsilon' \sim 10^{-1}$. Then we consider the parameter $\epsilon = O(10^{-1})$ in order to obtain the suitable mass squared difference for the MSW solution. In this case $\epsilon'/\epsilon$ tend to becomes close to $1$, and in this case, the $\nu_e \leftrightarrow
\nu_\tau$ oscillation with $\Delta m^2 \sim
\epsilon^2 m^2$ becomes large mixing. However, the large mixing of the order of $(\epsilon'/\epsilon)^2 \sim 1$ with $\Delta m^2 > 10^{-3} {\rm [eV^2]}$ is excluded from CHOOZ experiment [@chooz], and we should choose the mixing $(\epsilon'/\epsilon)^2$ to be smaller than $O(10^{-1})$. This choice of parameters leads the solar neutrino solution to the small angle MSW solution. Therefore, $(\epsilon'/\epsilon)^2$ appears to be of order $10^{-2}$. In order to get such a $(\epsilon'/\epsilon)^2$, we need delicate tuning of parameters.
(2-2)
: In the case of $(\alpha, \beta) = (\tau, e)$, $\epsilon'$($\epsilon$) must be of order $10^{-1}$ from small mixing amplitude of LSND with $\Delta m^2_{\rm LSND} \sim \epsilon^2 m^2$ ($\Delta m^2_{\rm LSND} \sim m^2$). For the same reason of [**(2-1)**]{}, the vacuum oscillation solution is excluded, and the parameter $\epsilon$ should be chosen to be of order $10^{-1}$ for the MSW solution. Though the large angle MSW solution through $\nu_e \leftrightarrow \nu_s$ oscillation mode seems to be possible, it is not allowed at the 99% C.L. [@LMA] in two flavor analysis. Therefore, in this case, we can not help but consider other oscillation mode, namely, $\nu_e \leftrightarrow \nu_\tau$, as the solution for solar neutrino deficit. As we mentioned in [**(2-1)**]{}, the small angle MSW solution through $\nu_e \leftrightarrow \nu_\tau$ oscillation mode seems to be possible. However, since the mixing of $\nu_e$ and $\nu_s$ is large, we need the detail analysis of three generation mixing in this case.
Conclusion
==========
The recent atmospheric neutrino data of Super-Kamiokande suggests the maximal mixing between $\nu_{\mu}$ and other neutrinos. The singular seesaw mechanism is one of the most interesting scenario that can naturally explain this large mixing angle between $\nu_{\mu_L}$ and $\nu_{\mu_R}$. This mechanism can also induce three independent mass squared differences, which are suitable for the solutions of the solar and atmospheric neutrino anomalies, and the LSND data. The original scenario in Ref.[@singular2] can not explain neither the small mixing angle of the LSND data nor small angle solution of MSW. Thus, we introduce the hierarchy in the Dirac neutrino mass matrix, and reanalyzed the singular seesaw mechanism. As the results, we can obtain the small mixing solutions of the LSND and MSW as follows.
In the case of rank-2 Majorana mass, the Dirac mass matrix should be the form of $$\left( \begin{array}{ccc}
\epsilon m_{ee} & \epsilon m_{e\mu} & \epsilon m_{e\tau} \\
m_{\mu e} & m_{\mu \mu} & m_{\mu \tau} \\
\epsilon m_{\tau e} & \epsilon m_{\tau \mu} & \epsilon m_{\tau \tau} \\
\end{array} \right),$$ where dimensionless parameter $\epsilon$ is of order $10^{-1}$ and $m_{\alpha \beta}\sim 1$\[eV\]. The non-zero elements of Majorana mass should be of order 1\[keV\]. It is important that the solar neutrino deficit can be explained by the vacuum oscillation between $\nu_e$ and $\nu_\tau$, in contrast to the original framework of Ref.[@singular2].
In the case of rank-1 Majorana mass, the small angle MSW solution is suitable for the solar neutrino oscillation. The Dirac mass matrix should be $$\left( \begin{array}{ccc}
\epsilon m_{ee} & \epsilon m_{e\mu} & \epsilon m_{e\tau} \\
m_{\mu e} & m_{\mu \mu} & m_{\mu \tau} \\
\epsilon' m_{\tau e} & \epsilon' m_{\tau \mu} & \epsilon' m_{\tau \tau}
\\
\end{array} \right),$$ where $\epsilon$ is of order $10^{-1}$ and $\epsilon'$ should satisfy the condition $(\epsilon'/\epsilon)^2 < 10^{-1}$. There is an extra oscillation mode $\Delta m^2 \sim 10^{-2} {\rm [eV^2]}$ or $\Delta m^2 \sim 10^2 {\rm [eV^2]}$.
Finally, we would like to comment about Big Bang nucleosynthesis (BBN) in the sterile scenario. The ratio of deuterium to hydrogen and the abundance of $^4$He are determined by the ratio of neutrons to protons at the time when the weak interaction freeze out. The effective number of light neutrino flavors $N_{\nu}$ contribute to the energy density, which influences the expansion rate. Thus, we can obtain the upper limit of $N_{\nu}$ from the BBN constraint [@Hata]. Although the standard BBN scenario shows $N_{\nu} \leq 3.6$ [@Hata], the large lepton number asymmetry in the early universe may allow $N_{\nu}=4$ [@Foot], which corresponds to the rank-2 case in this paper. On the other hand, rank-1 case induces $N_{\nu}=5$, since $\nu_{\tau}$-$\nu_s$ mixing is large. Thus this case needs extra mechanism which suppress the effective number of neutrinos in the early universe[^7].
N.H would like to thank Professor S. Raby and Dr. K. Tobe for helpful discussions. Y.M. wishes to thank Institute for Cosmic Ray Research, University of Tokyo, in which most of this research was carried out. This work was partially supported by JSPS Research Fellowships for Young Scientists (Y.C. and Y.M.).
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H. Minakata and H. Nunokawa, Phys. Rev. [**D45**]{}, 3316 (1992). P. Langacker, Phys. Rev. [**D58**]{}, 093017 (1998). CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. [**B420**]{}, 397 (1998). J. N. Bahcall, P. I. Krastev, and A. Yu. Smirnov, hep-ph/9807216. See for example, N. Hata, G. Steigman, S. Bludman, and P. Langacker, Phys. Rev. [**D55**]{}, 540 (1997). See for example, R. Foot and R. R. Volkas, Phys. Rev. [**D56**]{}, 6653 (1997). A. Geiser, hep-ph/9810493.
--------------------------------- ----------------------------------------------- ----------------------------- -----------------------
$\Delta m^2 \sim \epsilon^4 \beta^2 m^2$ $\Delta m^2 \sim \beta m^2$ $\Delta m^2 \sim m^2$
$\alpha \leftrightarrow \beta$ $\left( \frac{\epsilon'}{\epsilon} \right)^2$ $\epsilon^2 \epsilon'^2$ $\epsilon'^2$
$\alpha \leftrightarrow \gamma$ $\epsilon'^2$ $\epsilon'^2$ $\epsilon'^2$
$\alpha \leftrightarrow s$ $\epsilon'^2 \beta^2$ $\epsilon'^2$ $\epsilon'^2 \beta$
$\beta \leftrightarrow \gamma$ $\epsilon'^2$ $\epsilon^2$ $\epsilon^2$
$\beta \leftrightarrow s$ $\epsilon'^2 \beta^2$ $\epsilon^2$ $\epsilon^2 \beta$
$\gamma \leftrightarrow s$ $\epsilon'^2 \epsilon^2 \beta^2$ $1$ $\epsilon^2 \beta$
--------------------------------- ----------------------------------------------- ----------------------------- -----------------------
: Oscillation amplitudes in the case of rank-2
---------------------------------- ---------------------------------------- ----------------------------- ------------------------------------- --------------------------------
$\Delta m^2 \sim \epsilon^3 \beta m^2$ $\Delta m^2 \sim \beta m^2$ $\Delta m^2 \sim \epsilon^2 m^2$ $\Delta m^2 \sim m^2$
$\alpha \leftrightarrow \beta$ $(\frac{\epsilon'}{\epsilon})^2$ $\epsilon^2 \epsilon'^2$ $(\frac{\epsilon'}{\epsilon})^2$ $\epsilon'^2$
$\alpha \leftrightarrow \gamma$ $\epsilon'^2$ $\epsilon'^2$ $\epsilon'^2$ $\epsilon'^2$
$\alpha \leftrightarrow s_1,s_2$ $(\frac{\epsilon'}{\epsilon})^2$ $\epsilon'^2$ $\frac{\epsilon'^2}{\epsilon}\beta$ $\frac{\epsilon'^2}{\epsilon}$
$\beta \leftrightarrow \gamma$ $\epsilon^2$ $\epsilon^2$ $\epsilon'^2$ $\epsilon^2$
$\beta \leftrightarrow s_1,s_2$ $1$ $\epsilon^2$ $\frac{\epsilon'^2}{\epsilon}\beta$ $\epsilon$
$\gamma \leftrightarrow s_1,s_2$ $\epsilon^2$ $1$ $\epsilon \epsilon' \beta$ $\epsilon$
---------------------------------- ---------------------------------------- ----------------------------- ------------------------------------- --------------------------------
: Oscillation amplitudes in the case of rank-1
[^1]: e-mail: [email protected]
[^2]: e-mail: [email protected]
[^3]: e-mail: [email protected]
[^4]: Recent measurements in the KARMEN detector exclude part of the LSND allowed region [@KARMEN].
[^5]: In this paper we call this mechanism “the singular seesaw mechanism”.
[^6]: This heavy neutrino mass $M$ turns out to be 1\[keV\]-1\[MeV\]. We integrate out the heavy neutrinos here though the scale is lower than the momentum scales of neutrino experiments (e.g., about 1\[GeV\] for atmospheric neutrinos). This integrating out method is used simply because we display our results clearly. Since the heavy neutrinos have very small mixing with light neutrinos, there is considerable validity to our results below.
[^7]: One possibility is to consider the large uncertainty of the systematic error in $^4$He abundance [@Gaiser].
|
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\
[ **FIRST INTEGRALS OF LINEAR DIFFERENTIAL SYSTEMS**]{}
\
**V.N.Gorbuzov$\!{}^{*},$ A.F. Pranevich$\!{}^{**}$**
\
*$\!{}^{*}\!\!$Department of Mathematics and Computer Science, Yanka Kupala Grodno State University,*
\
*Ozeshko 22, Grodno, 230023, Belarus*
\
E-mail: [email protected]
\
*$\!{}^{**}\!\!$Department of Economics and Management, Yanka Kupala Grodno State University,*
\
*Ozeshko 22, Grodno, 230023, Belarus*
\
E-mail: [email protected]
\
[**Abstract**]{}
\
We investigate the problem of the existence of first integrals for multidimensional and ordinary linear differential systems with constant coefficients. The spectral method of the first integrals basis construction for these systems of linear differential equations is developed.\
: linear system of equations in total differentials, ordinary linear differential system, partial integral, first integral.\
: 34A30\
[**Contents**]{}\
[ **1. ${\mathbb R}\!$-differentiable integrals of ${\mathbb R}\!$-linear systems in total differentials**]{} 2\
1.1. ${\mathbb R}\!$-linear homogeneous systems in total differentials 2\
1.1.1. ${\mathbb R}\!$-linear partial integral 2\
1.1.2. Autonomous ${\mathbb R}\!$-differentiable first integrals 3\
1.1.3. Nonautonomous ${\mathbb R}\!$-differentiable first integrals 7\
1.2. ${{\mathbb R}}\!$-linear nonhomogeneous systems in total differentials 8\
[ **2. First integrals of linear real systems in total differentials**]{} 9\
2.1. Linear real homogeneous systems in total differentials 9\
2.1.1. Partial integrals 9\
2.1.2. Autonomous first integrals 10\
2.1.3. Nonautonomous first integrals 20\
2.2. Linear real nonhomogeneous systems in total differentials 22\
[ **3. First integrals of linear systems of ordinary differential equations**]{} 26\
3.1. Linear homogeneous systems of ordinary differential equations 26\
3.1.1. Autonomous first integrals 26\
3.1.2. Nonautonomous first integrals 30\
3.2. Linear nonhomogeneous systems of ordinary differential equations 32\
[ **References**]{} 34
\
**1. ${\mathbb R}\!$-differentiable integrals of ${{\mathbb R}}\!$-linear systems in total differentials**
\
\
We consider the system of equations in total differentials\
$
dw = X_1(w)dz+X_2(w)d\,\overline{z}\,,
$ (1.1)\
where $w\!=\mbox{colon}(w_1,\ldots,w_n)\!\in\! {\mathbb C}^n, \,
z\!=\mbox{colon}(z_1,\ldots,z_m)\!\in\! {\mathbb C}^m; \
dw\!=\mbox{colon}(dw_1,\ldots,dw_n),\!\!$ $dz=\mbox{colon}(dz_1,\ldots,dz_m),$ and $d\,\overline{z}=\mbox{colon}(d\,\overline{z}_1,\ldots,d\,\overline{z}_m)$ are vector columns; $\overline{z}_j$ is complex conjugate of $z_j;$ the entries of the matrices $X_1(w)=\|X_{\tau j}(w)\|$ and $X_2(w)=\|X_{\tau, m+j}(w)\|,$ $\tau=1,\ldots, n,\ j=1,\ldots, m$ are the ${{\mathbb R}}\!$-linear functions \[1, p. 21\]\
$
\displaystyle
X_{{}_{\scriptstyle \tau k}}\colon w\to
\sum\limits_{\xi=1}^{n}
\bigl( a_{{}_{\scriptstyle \tau k\xi}} w_{{}_{\scriptstyle \xi}}+
a_{{}_{\scriptstyle \tau k,n+\xi}}
\overline{w}_{{}_{\scriptstyle \xi}}\bigr)
\ \ \text{for all}\ w\in {\mathbb C}^n,
\ \ \ k=1,\ldots, 2m,
\ \ \theta=1,\ldots, n,
\hfill
$\
with constant coefficients $a_{{}_{\scriptstyle \tau k\varrho}}\in {\mathbb C},\
\varrho=1,\ldots, 2n,\ k=1,\ldots,2m, \ \tau=1,\ldots,n.$ We assume that the linear differential operators\
$
\displaystyle
{\frak x}_j(w) = \sum\limits_{\xi=1}^{n}
\bigl(X_{\xi j}(w)\partial_{{}_{\scriptstyle w_\xi}} +
\overline{X}_{\xi,m+j}(w)\partial_{{}_{\scriptstyle \overline{w}_\xi}}
\bigr)
\ \ \ \text{for all}\ w\in {\mathbb C}^n,
\ \ j=1,\ldots, m,
$ (1.2)\
and\
$
\displaystyle
{\frak x}_{m+j}(w) = \sum\limits_{\xi=1}^{n}
\bigl(X_{\xi,m+j}(w)\partial_{{}_{\scriptstyle w_\xi}} +
\overline{X}_{\xi j}(w)\partial_{{}_{\scriptstyle \overline{w}_\xi}}
\bigr)
\ \ \ \text{for all}\ w\in {\mathbb C}^n,
\ \ j=1,\ldots, m,
$ (1.3)\
induced by this system are related by the Frobenius conditions \[2; 3\]. These conditions are represented via Poisson brackets as the system of identities\
$
\bigl[ {\frak x}_{k}(w), {\frak x}_{l}(w)\bigr] = {\frak O}
\ \ \ \text{for all}\ w\in {\mathbb C}^n,
\ \ k=1,\ldots, 2m,\ l=1,\ldots, 2m,
$ (1.4)\
i.e., system (1.1) is completely solvable \[4; 5, pp. 15 – 25\].
A general integral of the completely solvable system in total differentials (1.1) is $n$ functionally independent ${\mathbb R}\!$-differentiable first integrals of (1.1). The completely solvable differential system (1.1) has also $n-m$ autonomous ${\mathbb R}\!$-differentiable first integrals (see \[6\]).
In this paper we study Darboux’s problem of finding first integrals in case that partial integrals are known \[7\]. Using method of partial integrals \[5; 8; 9\], we obtain the spectral method for building first integrals of linear differential systems \[5, pp. 239 – 272; 10 – 17\].
The problems related to those studied in the present paper were intensively investigated by many people. For example, see for systems of ordinary differential equations \[7; 12; 16 – 54\], for partial differential systems \[5; 8; 10; 15; 55 – 59\], for systems of equations in total differentials $[2 - 6; 8; 9; 11; 13; 14; 16; 32; 60 - 63],$ and this list is very far from being complete.\
. The ${{\mathbb R}}\!$-linear function\
$
\displaystyle
p\colon w\to
\sum\limits_{\xi=1}^{n}
\bigl(
b_{{}_{\scriptstyle \xi}} w_{{}_{\scriptstyle \xi}} +
b_{{}_{\scriptstyle n+\xi}}
\overline{w}_{{}_{\scriptstyle \xi}}\bigr)
\ \ \text{for all}\ \,w\in {\mathbb C}^n
\quad
(b_{\varrho}\in {\mathbb C},\ \varrho=1,\ldots, 2n)
\hfill
$\
is a [*partial integral*]{} of the system in total differentials (1.1) iff\
$
{\frak x}_{{}_{\scriptstyle k}} p(w)= p(w)\lambda^k
\ \ \text{for all}\ w\in {\mathbb C}^n,
\quad
\lambda^k\in {\mathbb C}, \ k=1,\ldots, 2m.
\hfill
$\
This system of identities is equivalent to the linear homogeneous system\
$
\bigl(A_{k} - \lambda^{k} E\bigr)\,b= 0,
\ \ \ k=1,\ldots, 2m,
$ (1.5)\
where $A_{j}= \|a_{1j}\ldots a_{nj}\,
\overline{a}_{1,m+j}\ldots \overline{a}_{n,m+j}\|$ and $A_{m+j}= \|a_{1,m+j}\ldots a_{n,m+j}
\overline{a}_{1j}\ldots\overline{a}_{nj}\|\!$ are the $\!\!2n\!\times\! 2n\!$-matrices with $\!a_{\tau k}\!\!=\!\mbox{colon}(\!a_{\tau k1},\ldots,a_{\tau k,2n}\!),
\overline{a}_{\tau k}\!\!=\!
\mbox{colon}(\overline{a}_{\tau k,n+1},\ldots,\overline{a}_{\tau k,2n},
\overline{a}_{\tau k1},\ldots,\overline{a}_{\tau kn}\!)\!,\!\!$ $\tau =1,\ldots,n,\ k=1,\ldots, 2m, \ j=1,\ldots,m,\
E$ is the $2n\times 2n$ identity matrix, and $b=\mbox{colon}(b_1,\ldots,b_{2n})$ is a vector column.
The Frobenius conditions (1.4) for system (1.1) are equivalent \[60, p. 73\]\
$
A_{k}A_{l} = A_{l}A_{k},
\ \ \
k=1,\ldots,2m,
\ \ l=1,\ldots, 2m.
\hfill
$\
Then there exists a relation \[64, pp. 193 – 194; 65\] between eigenvectors and eigenvalues of the matrices $A_{k},\ k=1,\ldots, 2m.$
[**Lemma 1.1**]{}. (1.1).
[*Proof*]{}. If $\nu$ is a common eigenvector of the matrices $A_{k},\ k=1,\ldots, 2m,$ then $\nu$ is a solution to system (1.5), where $\lambda^{k}$ is an eigenvalue of the matrix $A_{k}$ corresponding to the eigenvector $\nu.$ We obtain $
{\frak x}_{k} (\nu\gamma) = \lambda^{k} \nu\gamma$ for all $w\in {\mathbb C}^n,\ k=1,\ldots,2m.$ Therefore the ${\mathbb R}\!$-linear function $p$ is a partial integral of the system in total differential (1.1).
------------------------------------------------------------------------
[**1.1.2. Autonomous ${\mathbb R}\!$-differentiable first integrals**]{}\
. [ *Let $\nu^{\theta},\, \theta=1,\ldots,2m+1$ be common eigenvectors of the matrices $A_{k},$ $k=1,\ldots, 2m.$ Then the system [(1.1)]{} has the ${\mathbb R}\!$-differentiable autonomous first integral\
$
\displaystyle
F\colon w\to
\prod\limits_{\theta=1}^{2m+1}
\bigl(\nu^{\theta}\gamma\bigr)^{h_{\theta}}$ for all $w\in\Omega,
\ \ \ \Omega\subset {\rm D}(F),
$\
where $h_{1},\ldots,h_{2m+1}$ is a nontrivial solution to the system $
\sum\limits_{\theta=1}^{2m+1}\,\lambda_{\theta}^{k}\,h_{\theta} =0,
\ k=1,\ldots,2m,
$ and $\lambda_{\theta}^{k}$ are the eigenvalues of the matrices $A_{k},\ k=1,\ldots, 2m,$ corresponding to the common eigenvectors $\nu^{\theta},\ \theta=1,\ldots, 2m+1.$* ]{}
[*Proof*]{}. Suppose $\nu^{\theta}$ are common eigenvectors of the matrices $A_{k}$ corresponding to the eigenvalues $\lambda_{\theta}^{k},\ k=1,\ldots, 2m,\ \theta=1,\ldots,2m+1,$ respectively. By lemma 1.1, it follows that the ${\mathbb R}\!$-linear functions $w\to\nu^{\theta}\gamma$ for all $w\in {\mathbb C}^n,\ \theta=1,\ldots, 2m+1$ are partial integrals of the system of equations in total differentials (1.1). Hence,\
$
{\frak x}_{{}_{\scriptstyle k}}\,\nu^{\theta}\gamma =
\lambda_{\theta}^{k}\,\nu^{\theta}\gamma
$ for all $w\in {\mathbb C}^n,
\ \
k=1,\ldots, 2m,
\quad \theta=1,\ldots, 2m+1.
$ (1.7)\
We form the function\
$
\displaystyle
F\colon w\to
\prod\limits_{\theta=1}^{2m+1}
\bigl(\nu^{\theta}\gamma\bigr)^{h_{\theta}}
$ for all $ w\in\Omega,
\quad
\Omega\subset {\mathbb C}^n,
\hfill
$\
where $\Omega$ is a domain (open arcwise connected set) in ${\mathbb C}^n$ and $h_{\theta},\,\theta=1,\ldots, 2m+1$ are complex numbers with $\sum\limits_{\theta=1}^{2m+1}|h_{\theta}|\ne 0.$ The Lie derivative of $F$ by virtue of (1.1) is equal to\
$
\displaystyle
{\frak x}_{{}_{\scriptstyle k}} F(w)=
\prod\limits_{\theta=1}^{2m+1}
\bigl(\nu^{\theta}\gamma\bigr)^{h_{\theta}-1}\,
\sum\limits_{\theta=1}^{2m+1} h_{\theta}\,
\prod\limits_{l=1,l\ne \theta}^{2m+1}(\nu^{l}\gamma) \
{\frak x}_{{}_{\scriptstyle k}}\,\nu^{\theta}\gamma
$ for all $w\in\Omega,
\quad k=1,\ldots, 2m.
\hfill
$\
Using (1.7), we get\
$
\displaystyle
{\frak x}_{{}_{\scriptstyle k}}F(w)=
\sum\limits_{\theta=1}^{2m+1}\lambda_{\theta}^{k}h_{\theta}\,F(w)
$ for all $w\in\Omega,
\quad k=1,\ldots, 2m.
\hfill
$\
If $\sum\limits_{\theta=1}^{2m+1}\lambda_{\theta}^{k}h_{\theta}=0,\
k=1,\ldots 2m,$ then the function (1.6) is an autonomous ${\mathbb R}\!$-differentiable first integral of the system in total differentials (1.1).
------------------------------------------------------------------------
[**Corollary 1.1**]{}.
For example, the ${\mathbb R}\!$-linear autonomous system of equations in total differentials\
$
dw_1=
(2w_1-i(w_2+\overline{w}_1)+(1-i)\overline{w}_2)dz
+(w_1+(2-i)(w_2+\overline{w}_1)+(1-i)\overline{w}_2)d\,\overline{z},
\hfill
$\
(1.8)\
$
dw_2=
{}-((2+i)(w_1+\overline{w}_2)+iw_2)dz
-(i(w_1+\overline{w}_2)+(i-1)w_2)d\,\overline{z}
\hfill
$\
has the commuting matrices\
$
A_1 =
\left\|\!
\begin{array}{rrcc}
2 & -2-i\, & 2+i & 0
\\
-i & -i\, & 1+i & i
\\
-i & 0\, & 1 & i
\\
1-i & -2-i\, & 2+i & 1+i
\end{array}\!
\right\|
$ and $
A_2 =
\left\|\!
\begin{array}{crcc}
1 & -i\, & i & 0
\\
2-i & 1-i\, & 1+i & -2+i
\\
2-i & 0\, & 2 & -2+i
\\
1-i & -i\, & i & i
\end{array}\!
\right\|.
\hfill
$\
Therefore the system of equations in total differentials (1.8) is completely solvable.
The matrices $A_1$ and $A_2$ have the eigenvalues $\lambda_{1}^1=1+i,
\ \lambda_{2}^{1}={}-i,\
\lambda_{3}^{1}=1,
\ \lambda_{4}^1=2,
$ and $
\lambda_{1}^2=i,
\
\lambda_{2}^{2}=1-i,
\
\lambda_{3}^{2}=2,
\
\lambda_{4}^2=1
$ corresponding to the eigenvectors $\nu^1=(0,1,1,1),$ $\nu^2\!=\!(1,1,0,1),\ \nu^3\!=\!(0,1,1,0),\ \nu^4\!=\!(1,0,0,1),$ respectively.
The solution to the linear homogeneous system\
$
\left\{\!\!
\begin{array}{l}
(1+i)h_1-ih_2+h_3=0,
\\[1ex]
ih_1+(1-i)h_2+2h_3=0
\end{array}
\right.
\iff
\left\{\!\!
\begin{array}{l}
h_1={}-(1+i)h_3,
\\[1ex]
h_2={}-(2+i)h_3
\end{array}
\right.
\hfill
$\
is $h_1=1+i,\ h_2=2+i, \ h_3={}-1.$
The ${\mathbb R}\!$-differentiable function (by Theorem 1.1)\
$
F\colon w\to
\dfrac{(w_2+\overline{w}_1+\overline{w}_2)^{1+i}
(w_1+w_2+\overline{w}_2)^{2+i}}{w_2+\overline{w}_1}$ for all $w\in \Omega,
$ (1.9)\
where a domain $\Omega\subset \{w\colon w_2+\overline{w}_1\ne 0\},$ is an autonomous first integral of the system (1.8).
The ${\mathbb R}\!$-differentiable first integral (1.9) is an autonomous general integral of the completely solvable system of equations in total differentials (1.8).
From the entire set of ordinary differential systems induced by the completely solvable system of equations in total differentials (1.1), we extract system\
$
dw_\tau= X_{\tau\zeta}(w)\,dz_{\zeta}+
X_{\tau,m+\zeta}(w)\,d\,\overline{z}_{\zeta},
\ \ \ \tau=1,\ldots, n,
\quad
\zeta\in \{1,\ldots,m\}
\hfill (1.1.\zeta)
$\
such that the matrix $A_{\zeta}$ has the smallest number of elementary divisors \[64, p. 147\].
[**Definition 1.1**]{}. [*Let $\nu^{0l}$ be an eigenvector of the matrix $A_{\zeta}$ corresponding to the eigenvalue $\lambda_{l}^{\zeta}$ with elementary divisor of multiplicity $s_{l}.$ A non-zero vector $\nu^{\eta l}\in {\mathbb C}^{2n}$ is called a ***generalized eigenvector of order*** [$\eta$]{} for $\lambda_{l}^{\zeta}$ if and only if\
$
(A_{\zeta}-\lambda_{l}^{\zeta} E)\,\nu^{\eta}=\eta\cdot \nu^{\eta-1},
\quad
\eta=1,\ldots, s_{l}-1,
\hfill (1.10)
$\
where $E$ is the $2n\times 2n$ identity matrix.* ]{}
Using Lemma 1.1 and (1.10), we obtain\
$
{\frak x}_{\zeta}\,\nu^{0l}\gamma =
\lambda_{l}^{\zeta}\,\nu^{0l}\gamma,
\ \ \ \
{\frak x}_{\zeta}\,\nu^{\eta l}\gamma =
\lambda_{l}^{\zeta}\,\nu^{\eta l}\gamma +
\eta\,\nu^{\eta-1{,}\,l}\gamma$ for all $w\in {\mathbb C}^n,
\ \ \eta=1,\ldots, s_{l}-1.
$ (1.11)\
The following lemmas are needed for the sequel.
[**Lemma 1.2**]{}. [*Let $\nu^{\,0l}$ be a common eigenvector of the matrices $A_k$ corresponding to the eigenvalues $\lambda^k_{l}, \ k=1,\ldots,2m,$ respectively. Let $\nu^{\,\eta l},\ \eta =1,\ldots, s_l-1$ be generalized eigenvectors of the matrix $A_\zeta$ corresponding to the eigenvalue $\lambda^{\zeta}_{l}$ with elementary divisor of multiplicity $s_l\ (s_{l}{\geqslant}2).$ If the system $(1.1.\zeta)$ hasn’t the first integrals\
$
\displaystyle
F_{k\eta l}^{\,\zeta}\colon w\to
{\frak x}_k\, \Psi_{\eta l}^{\zeta}(w)$ for all $w\in \Omega,
\ \ k=1,\ldots, 2m, \ \, k\ne\zeta,
\ \ \ \eta =1,\ldots, s_l-1,
$\
then\
$
{\frak x}_{\zeta}\, \Psi_{\eta l}^{\zeta}(w)=
\left[\!\!
\begin{array}{lll}
1\! & \text{for all}\ \ w\in \Omega, & \eta =1,
\\[1ex]
0\! & \text{for all}\ \ w\in \Omega, & \eta =2,\ldots, s_{l}-1,
\end{array}
\right.
$\
$
{\frak x}_{k}\, \Psi_{\eta l}^{\zeta}(w)=\mu_{\eta l}^{k\zeta}={\rm const}$ for all $w\in \Omega,
\ \ k=1,\ldots, 2m,\ \ k\ne \zeta,
\ \ \ \eta =1,\ldots, s_l-1,
\hfill
$\
where $\Psi_{\eta l}^{\zeta}\colon \Omega\to {\mathbb C},\ \eta =1,\ldots, s_l-1$ is a solution to the system\
$
\nu^{\,\eta l}\gamma=
{\displaystyle \sum\limits_{\delta=1}^{\eta} }
\binom{\eta -1}{\delta-1}\Psi_{\delta l}^{\zeta}(w)\cdot \nu^{\,\eta-\delta,l}\gamma,
\quad
\eta=1,\ldots, s_l-1,
\quad
\Omega\subset \{w\colon \nu^{0l}\gamma\ne 0\}.
$\
*]{} . The system (1.14) has the determinant $(\nu^{0l}\gamma)^{s_{l}-1}.$ Therefore there exists the solution $\Psi_{\eta l}^{\zeta}, \eta=1,\ldots, s_l-1$ on a domain $\Omega\subset \{w\colon \nu^{0l}\gamma\ne 0\}$ of the system (1.14).
The proof of the lemma is by induction on $\eta.$
For $\eta=1$ and $\eta=2,$ the assertion (1.13) follows from (1.11).
Assume that (1.13) for $\eta=1,\ldots,\varepsilon-1$ is true. Using (1.11) and (1.14), we get\
$
\displaystyle
{\frak x}_{\zeta}\,\nu^{\varepsilon l}\gamma =
\lambda_{l}^{\zeta}\sum\limits_{\delta=1}^{\varepsilon}
{\textstyle\binom{\varepsilon -1}{\delta-1}}\,\Psi_{\delta l}^{\zeta}(w)\,
\nu^{\varepsilon-\delta{,}\,l}\gamma +
(\varepsilon -1)\sum\limits_{\delta=1}^{\varepsilon -1}
{\textstyle\binom{\varepsilon -2}{\delta-1}}\,\Psi_{\delta l}^{\zeta}(w)\,
\nu^{\varepsilon-\delta-1{,}\,l}\gamma \ +
\hfill
$\
$
\displaystyle
+\ \nu^{\varepsilon -1{,}\,l}\gamma \,+\,
\nu^{0l}\gamma\ {\frak x}_{\zeta}\Psi_{\varepsilon l}^{\zeta}(w)$ for all $w\in \Omega.
\hfill
$\
Combining (1.14) for $\eta=\varepsilon-1$ and $\eta=\varepsilon,$ (1.11) for $\eta=\varepsilon,$ and $\nu^{0l}\gamma\not\equiv 0$ in ${{\mathbb C}}^n,$ we obtain ${\frak x}_{\zeta}\,\Psi_{\varepsilon l}^{\zeta}(w)=0$ for all $w\in\Omega.$ So by the principle of mathematical induction, the statement (1.13) is true for every $\eta=1,\ldots, s_{l}-1$ and $\zeta\in \{1,\ldots, m\}.$
Taking into account (1.4) and (1.12), we have the statement (1.13) is true for $k\ne \zeta.$
------------------------------------------------------------------------
[**Lemma 1.3**]{} [*Under the conditions of Lemma [1.2]{}, we have\
$
{\frak x}_{k}\,\nu^{\,\eta l}\gamma=
{\displaystyle \sum\limits_{\delta=0}^{\eta}}
\binom{\eta}{\delta}\,\mu_{\delta l}^{ k\zeta}\cdot\nu^{\,\eta-\delta, l}\gamma$ for all $w\in \Omega,
\ \ \ k=1,\ldots, 2m,
\ \ \eta=1,\ldots, s_{l}-1,
$\
where $\mu_{0l}^{k\zeta}=\lambda^k_{l}, \ \mu_{\eta l}^{k\zeta}={\frak x}_{k}\,\Psi_{\eta l}^{\zeta}(w),
\ \eta=1,\ldots, s_{l}-1, \ k=1,\ldots, 2m.$* ]{}
[*Proof*]{}. The proof of Lemma 1.3 is by induction on $\eta.$
Let $\eta=1.$ Using (1.14), we get\
$
\nu^{\,1 l}\gamma=
\Psi_{1 l}^{\zeta}(w)\cdot\nu^{0 l}\gamma$ for all $w\in \Omega,
\ \ \ \Omega\subset \{w\colon \nu^{0l}\gamma\ne 0\}.
$\
Then\
$
{\frak x}_{k}\nu^{\,1l}\gamma=
{\frak x}_{k}\Psi_{1 l}^{\zeta}(w)\cdot\nu^{0l}\gamma +
\Psi_{1 l}^{\zeta}(w)\cdot {\frak x}_{k}\, \nu^{0l}\gamma$ for all $w\in \Omega,
\ \ \ k=1,\ldots, 2m.
\hfill
$\
Taking into account Lemma 1.1, Lemma 1.2 and (1.16),we obtain\
$
{\frak x}_{k}\nu^{\,1 l}\gamma=
\mu_{1 l}^{k\zeta}\cdot\nu^{0l}\gamma +
\mu_{0 l}^{k\zeta}\,\Psi_{1l }^{\zeta}(w)\cdot \nu^{0l}\gamma=
\mu_{0l}^{k\zeta}\cdot\nu^{1l}\gamma+
\mu_{1l}^{k\zeta}\cdot \nu^{0l}\gamma,
\quad
k=1,\ldots, 2m,
\hfill
$\
where $\mu_{0 l}^{k\zeta}=\lambda^k_l,\
{\frak x}_{k}\,\Psi_{1 l}^{\zeta}(w)=\mu_{1l}^{k\zeta}, \ k=1,\ldots, 2m.$ So (1.15) for $\eta=1$ is true.
Suppose that the assertion of the lemma is valid for $\eta=1,\ldots, \varepsilon-1.$ From (1.14), we get\
$
{\frak x}_{k}\nu^{\varepsilon l}\gamma=
{\displaystyle \sum\limits_{\delta=1}^{\varepsilon} }
\binom{\varepsilon-1}{\delta-1}\,{\frak x}_{k}\Psi_{\delta l}^{\zeta}(w)\,\nu^{\,\varepsilon-\delta, l}\gamma +
{\displaystyle \sum\limits_{\delta=1}^{\varepsilon} }
\binom{\varepsilon-1}{\delta-1}\,\Psi_{\delta l}^{\zeta}(w)\, {\frak x}_{k}\nu^{\,\varepsilon-\delta,l}\gamma$ for all $w\in \Omega,
\, k\!=\!1,\ldots, 2m.
\hfill
$\
By the induction hypothesis, we have\
$
{\frak x}_{k}\nu^{\varepsilon l}\gamma =
{\displaystyle \sum\limits_{\delta=1}^{\varepsilon} }
\binom{\varepsilon-1}{\delta-1}\,\mu_{\delta l}^{k\zeta}\cdot \nu^{\,\varepsilon-\delta,l}\gamma +
{\displaystyle \sum\limits_{\delta=1}^{\varepsilon} }
\binom{\varepsilon-1}{\delta-1}\,\Psi_{\delta l}^{\zeta}(w)\cdot
{\displaystyle \sum\limits_{\varkappa=0}^{\varepsilon-\delta} }
\binom{\varepsilon-\delta}{\varkappa}\,\mu_{\varkappa l}^{k\zeta}\cdot \nu^{\,\varepsilon-\delta-\varkappa,l}\gamma =
\hfill
$\
$
={\displaystyle \sum\limits_{\delta=1}^{\varepsilon} }
\binom{\varepsilon-1}{\delta-1}\,\mu_{\delta l}^{k\zeta}\cdot \nu^{\,\varepsilon-\delta,l}\gamma +
{\displaystyle \sum\limits_{\alpha=0}^{\varepsilon-1} }
\binom{\varepsilon-1}{\alpha}\,\mu_{\alpha l}^{k\zeta}\cdot
{\displaystyle \sum\limits_{\beta=1}^{\varepsilon-\alpha} }
\binom{\varepsilon-\alpha-1}{\beta-1}\,\Psi_{\beta l}^{\zeta}(w)\cdot
\nu^{(\varepsilon-\alpha)-\beta,l}\gamma=
\hfill
$\
$
={\displaystyle \sum\limits_{\delta=1}^{\varepsilon} }
\binom{\varepsilon-1}{\delta-1}\,\mu_{\delta l}^{k\zeta}\cdot \nu^{\,\varepsilon-\delta, l}\gamma +
{\displaystyle \sum\limits_{\alpha=0}^{\varepsilon-1} }
\binom{\varepsilon-1}{\alpha}\,\mu_{\alpha l}^{k\zeta}\cdot \nu^{\varepsilon-\alpha,l}\gamma=
\hfill
$\
$
={\displaystyle \sum\limits_{\delta=1}^{\varepsilon-1} }
\binom{\varepsilon-1}{\delta-1}\,\mu_{\delta l}^{k\zeta}\cdot \nu^{\,\varepsilon-\delta,l}\gamma +
\mu_{\varepsilon l}^{k\zeta}\cdot \nu^{0l}\gamma +\mu_{0l}^{k\zeta}\cdot \nu^{\varepsilon l}\gamma +
{\displaystyle \sum\limits_{\delta=1}^{\varepsilon-1} }
\binom{\varepsilon-1}{\delta}\,\mu_{\delta l}^{k\zeta}\cdot \nu^{\varepsilon-\delta,l}\gamma=
\hfill
$\
$
=\mu_{0l}^{k\zeta}\cdot \nu^{\varepsilon l}\gamma +
{\displaystyle \sum\limits_{\delta=1}^{\varepsilon-1} }
\binom{\varepsilon}{\delta}\,\mu_{\delta l}^{k\zeta}\cdot \nu^{\,\varepsilon-\delta,l}\gamma +
\mu_{\varepsilon l}^{k\zeta}\cdot \nu^{0l}\gamma =
{\displaystyle \sum\limits_{\delta=0}^{\varepsilon} }
\binom{\varepsilon}{\delta}\,\mu_{\delta l}^{k\zeta}\cdot \nu^{\varepsilon-\delta,l}\gamma,
\ \ \ k=1,\ldots, 2m.
\hfill
$\
Thus by the principle of induction, the statement (1.15) is true for $\eta=1,\ldots, s_{l}-1.\!\!$
------------------------------------------------------------------------
[**Theorem 1.2**]{}. [ *Let the assumptions of Lemma [1.2]{} with $l=1,\ldots r\ \Bigl(\, \sum\limits_{l=1}^{r}s_{l}{\geqslant}m+1\Bigr)$ hold. Then the completely solvable system [(1.1)]{} has the autonomous first integral\
$
\displaystyle
F\colon w\to
\prod\limits_{\xi=1}^{\alpha}\bigl(\nu^{0\xi} \gamma\bigr)^{h_{0 \xi}}
\exp\sum\limits_{q=1}^{\varepsilon_{\xi}}\,h_{q\xi} \Psi_{q\xi}^{\zeta}(w)$ for all $w\in \Omega,
\quad \Omega\subset {\rm D}(F),
$\
where $\sum\limits_{\xi=1}^{\alpha}\varepsilon_{\xi}=2m-\alpha+1, \
\varepsilon_{\xi}{\leqslant}s_{\xi}\!-\!1,\, \xi=1,\ldots,\alpha,\, \alpha{\leqslant}r,\!$ and $h_{q\xi},\, q\!=\!0,\ldots,\varepsilon_{\xi},\, \xi\!=\!1,\ldots, \alpha$ is a nontrivial solution to the linear homogeneous algebraic system of equations\
$
\displaystyle
\sum\limits_{\xi=1}^{\alpha}
\bigl(\lambda_{\xi}^{k}\,h_{0\xi}
+ \sum\limits_{q=1}^{\varepsilon_{\xi}}
\mu_{q\xi}^{k\zeta}\,h_{q\xi}\big) = 0,
\ \ k=1,\ldots, 2m.
\hfill
$\
*]{} . The Lie derivative of (1.17) by virtue of (1.1) is equal to\
$
\displaystyle
{\frak x}_{k}\,F(w) =
\sum\limits_{\xi=1}^{\alpha}
\bigl(\lambda_{\xi}^{k}h_{0\xi} +
\sum\limits_{q=1}^{\varepsilon_{\xi}}\mu_{q\xi}^{k\zeta}h_{q\xi}\bigr)F(w)$ for all $w\in\Omega,
\quad
k=1,\ldots, 2m.
\hfill
$\
If $
\sum\limits_{\xi=1}^{\alpha}
\bigl(\lambda_{\xi}^{k}h_{0\xi} +
\sum\limits_{q=1}^{\varepsilon_{\xi}}
\mu_{q\xi}^{k\zeta}h_{q\xi}\bigr) = 0,\ k=1,\ldots, 2m,$ then the ${\mathbb R}\!$-differentiable function (1.17) is an autonomous first integral of the completely solvable system (1.1).
------------------------------------------------------------------------
As an example, the completely solvable ${\mathbb R}\!$-linear system of equations in total differentials\
$
\begin{array}{l}
dw_1=
((1+i)w_1+iw_2-\overline{w}_1-\overline{w}_2)dz
+(w_1+iw_2-\overline{w}_1-\overline{w}_2)d\,\overline{z},
\\[1ex]
dw_2=
(w_2+\overline{w}_1+\overline{w}_2)dz
+((1-i)w_2+\overline{w}_1+\overline{w}_2)d\,\overline{z},
\\[1ex]
dw_3=
({}-w_1+w_2+w_3-i\,\overline{w}_2)dz
+({}-w_1+w_2+(1-i)w_3-i\,\overline{w}_2)d\,\overline{z}
\end{array}
$ (1.18)\
has two eigenvalues: $\!\lambda_1^1\!=\!1+i$ with elementary divisor $(\lambda^1\!-\!1\!-\!i)^3\!\!$ and $\lambda_2^1\!=\!1\!$ with elementary divisor $(\lambda^1\!-\!1)^3.$ The $\lambda_1^1\!=\!1+i$ corresponding to the eigenvector $\nu^{01}\!=\!(1, 1, 0, 0, 0, 0)\!$ and to the generalized eigenvectors $\!\nu^{11}\!=\!(0, 0, 0, 0, 1, 0), \, \nu^{21}\!=\!(0, 1, 0, 0, 0, 1).\!\!$ The eigenvalue $\!\lambda_2^1\!=\!1$ corresponding to the $\nu^{02}\!=(0, 0, 0, 1, 1, 0),\, \nu^{12}\!=(0, 1, 0, 0, 0, 0), \, \nu^{22}\!=(0, 0, 1, 0, 1, 0).$
The scalar functions (see (1.14))\
$
\Psi_{11}^1\colon w\to
\dfrac{\overline{w}_2}{w_1+w_2}\,,
\quad
\Psi_{21}^1\colon w\to
\dfrac{(w_1+w_2)(w_2+\overline{w}_3)-\overline{w}_2^{\,2}}{(w_1+w_2)^2}$ for all $w\in\Omega,
\hfill
$\
where $\Omega\subset \{w\colon w_1+w_2\ne 0\}\subset {\mathbb C}^3.$ The ${\mathbb R}\!$-differentiable first integrals (by Theorem 1.2)\
$
F_{1}\colon w\to \Psi_{21}^1(w)$ for all $w\in \Omega
$ (1.19)\
and\
$
F_{2}\colon w\to
\dfrac{\overline{w}_1+\overline{w}_2}{w_1+w_2} \
\exp\Bigl(i\,\dfrac{\overline{w}_2}{w_1+w_2}\Bigr)$ for all $w\in\Omega
$ (1.20)\
are an autonomous general integral of the system of equations in total differentials (1.18).\
\
[*Proof*]{}. Using Lemma 1.1, we obtain $
{\frak X}_{k}F(z,w) =0$ for all $(z,w)\in {\mathbb C}^{m+n},
\ k=1,\ldots, 2m,$ where the linear nonautonomous differential operators\
$
{\frak X}_j(z,w)=\partial_{z_j}+{\frak x}_j(w),
\
{\frak X}_{m+j}(z,w)=
\partial_{{}_{\scriptstyle\overline{z}_j}}+ {\frak x}_{m+j}(w)$ for all $(z,w)\!\in\! {\mathbb C}^{m+n},
\, j\!=\!1,\ldots, m.{\rule{0.7em}{0.7em}}\hfill
$\
Consider the system (1.8). Using the eigenvector $\nu^1=(0,1,1,1)$ corresponding to the eigenvalues $\lambda_1^1=1+i$ and $\lambda_1^2=i,$ we can build the first integral (by Theorem 1.3)\
$
F\colon (z,w)\to
(w_2+\overline{w}_1+\overline{w}_2)\exp({}-(1+i)z-i\,\overline{z}\,)$ for all $(z,w)\in {\mathbb C}^3.
$ (1.21)\
The first integrals (1.9) and (1.21) are a general integral of the system (1.8).
[**Theorem 1.4.**]{} [ *Suppose the system [(1.1)]{} satisfies the conditions of Lemma [1.2.]{} Then the completely solvable system [(1.1)]{} has the ${\mathbb R}\!$-differentiable first integrals\
$
\displaystyle
F_{\eta}\colon (z,w)\to
\Psi_{\eta l}^{\zeta}(w) -
\sum_{j=1}^{m}\bigl(
\mu_{\eta l}^{j\zeta}z_{j}+\mu_{\eta l}^{m+j,\zeta}\overline{z}_{j}\bigr)$ for all $(z,w)\in G,
\ \ \eta=1,\ldots, s_{l}-1,
$\
where a domain $G\subset {\mathbb C}^{m+n},$ the functions $\Psi_{\eta l}^{\zeta}$ are the solution to system [(1.14)]{}, the numbers $\mu_{\eta l}^{k\zeta}={\frak x}_{k}\Psi_{\eta l}^{\zeta}(w),\
\eta=1,\ldots, s_{l}-1,\ k=1,\ldots, 2m.$* ]{}
[*Proof.*]{} The Lie derivative of (1.22) by virtue of (1.1) is\
$
{\frak X}_k F_{\eta}(z,w) =
{}-\mu_{\eta l}^{k\zeta} + {\frak x}_k \Psi_{\eta l}^{\zeta}(w)$ for all $(z,w)\in G,
\ \ k=1,\ldots, 2m,
\ \ \eta=1,\ldots, s_{l}-1.
\hfill
$\
Taking into account Lemma 1.2, we get the functions (1.22) are first integrals of (1.1).
------------------------------------------------------------------------
For example, the system (1.18) has the numbers $\mu_{11}^{11}=1, \ \mu_{11}^{21}=1,$ and the first integral\
$
F\colon (z,w)\to
\dfrac{\overline{w}_2}{w_1+w_2} -z -\overline{z}$ for all $(z,w)\in{{\mathbb C}}\times \Omega
$ (by Theorem 1.4).\
The ${\mathbb R}\!$-differentiable first integrals (1.19), (1.20), and $F$ are a general integral on a domain ${{\mathbb C}}\times \Omega$ of the system (1.18), where a domain $\Omega\subset \{w\colon w_1+w_2\ne 0\}.$\
\
Let us consider a nonhomogeneous system of equations in total differentials\
$
\displaystyle
dw =\sum\limits_{j=1}^m\bigl((B_j\,\gamma+f_j(z))\,dz_j+
(B_{m+j}\,\gamma+f_{m+j}(z))\,d\,\overline{z}_j\bigr)
$ (1.23)\
corresponding to the ${{\mathbb R}}\!$-linear homogeneous system (1.1), where the matrices $B_1,\ldots, B_{2m}$ are transpose of $A_1,\ldots, A_{2m},$ respectively, and the vector functions\
$
f_{k}\colon z\to {\rm colon}(f_{k1}(z),\ldots, f_{kn}(z))$ for all $z\in V,
\quad
k=1,\ldots, 2m
\hfill
$\
are continuously ${\mathbb R}\!$-differentiable on a domain $V\subset {\mathbb C}^{m}.$
The Frobenius conditions for the total solvability of system of equations in total differentials (1.23) are the relations (1.4) and\
$
\partial_{z_j}f^{\zeta}(z)+B_{\zeta}f^j(z)=
\partial_{z_\zeta}f^{j}(z)+B_{j}f^{\zeta}(z)$ for all $z\in V,
\quad
j,\zeta=1,\ldots, m,
\hfill
$\
$
\partial_{\overline{z}_j}f^{m+\zeta}(z)+B_{m+\zeta}f^{m+j}(z)=
\partial_{\overline{z}_\zeta}f^{m+j}(z)+B_{m+j}f^{m+\zeta}(z)$ for all $z\in V,
\ \
j,\zeta=1,\ldots, m,
\hfill
$\
$
\partial_{z_j}f^{m+\zeta}(z)+B_{m+\zeta}f^{j}(z)=
\partial_{\overline{z}_\zeta}f^{j}(z)+B_{j}f^{m+\zeta}(z)$ for all $z\in V,
\quad
j,\zeta=1,\ldots, m,
\hfill
$\
where the vector functions\
$
f^{j}\colon z\to {\rm colon}(f_{j1}(z),\ldots, f_{jn}(z),\overline{f}_{m+j,1}(z),\ldots, \overline{f}_{m+j,n}(z))$ for all $z\in V,
\quad
j=1,\ldots, m,
\hfill
$\
$
f^{m+j}\colon z\to {\rm colon}(f_{m+j,1}(z),\ldots, f_{m+j,n}(z),\overline{f}_{j1}(z),\ldots, \overline{f}_{jn}(z))$ for all $z\in V,
\ j=1,\ldots, m.
\hfill
$\
.
\
**2. First integrals of linear real systems in total differentials**
\
\
Consider an autonomous system of equations in total differentials\
$
dx = A(x)\,dt,
$ (2.1)\
where $x=(x_{1},\ldots,x_{n})\!\in {\mathbb R}^n,\, t=(t_{1},\ldots,t_{m})\!\in {\mathbb R}^{m},$ the entries of the matrix $\!A(x)\!=\!\| a_{{}_{\scriptstyle ij}}(x)\|\!$ (with $n$ rows and $m$ columns) are linear homogeneous functions\
$
\displaystyle
a_{{}_{\scriptstyle ij}}\colon x\to
\sum\limits_{\xi=1}^{n}
a_{{}_{\scriptstyle ij\xi}} x_{{}_{\scriptstyle \xi}}$ for all $x\in {\mathbb R}^{n}
\ \ \ (a_{{}_{\scriptstyle ij\xi}}\in {\mathbb R}, \
\xi=1,\ldots, n,\ j=1,\ldots, m,\ i=1,\ldots, n),
\hfill
$\
and $dx=\mbox{colon}(dx_{1},\ldots,dx_{n})$ and $dt=\mbox{colon}(dt_{1},\ldots,dt_{m})$ are vector columns.
Assume that this system is completely solvable. The Frobenius condition \[5, p. 19\] for the total solvability of system (2.1) in terms of the Poisson brackets is given by the relations\
$
[{\frak p}_{j}(x), {\frak p}_{\zeta}(x)] = 0$ for all $x\in {\mathbb R}^{n},
\quad
j=1,\ldots,m,
\ \ \zeta=1,\ldots, m,
$ (2.2)\
where the linear autonomous differential operators\
$
\displaystyle
{\frak p}_{j}(x)=
\sum\limits_{i=1}^{n}
a_{{}_{\scriptstyle ij}}(x)
\partial_{{}_{\scriptstyle x_{{}_{\scriptsize i}}}}$ for all $x\in {\mathbb R}^{n},
\quad j=1,\ldots, m.
\hfill
$\
The Frobenius conditions (2.2) for system (2.1) are equivalent \[60, p. 73\]:\
$
A_{j}A_{\zeta} = A_{\zeta}A_{j},
\quad
j=1,\ldots,m,
\ \ \zeta =1,\ldots, m,
\hfill
$\
where $A_{j}= \bigl\| a_{{}_{\scriptstyle \xi j i}} \bigr\|, \ j=1,\ldots,m$ are real $n\times n$ matrices.
[**2.1.1. Partial integrals**]{}. The complex-valued linear homogeneous function\
$
\displaystyle
p \colon x\to
\sum\limits_{\xi=1}^{n} b_{\xi} x_{\xi}$ for all $x\in {\mathbb R}^{n}
\quad (b_{\xi}\in {\mathbb C}, \ \xi=1,\ldots, n)
\hfill
$\
is a [*partial integral*]{} of the system in total differentials (2.1) iff\
$
{\frak p}_{j}\,p(x) = \lambda^{j}\,p(x)$ for all $x\in {\mathbb R}^{n},
\ \ \ \lambda^{j}\in {\mathbb C},
\ \ j=1,\ldots, m.
\hfill
$\
This system of identities is equivalent to the linear homogeneous system of equations $\bigl(A_{j} - \lambda^{j} E\bigr)b = 0,\, j\!=\!1,\ldots, m,$ where $E$ is the $n\times n$ identity matrix, $b\!=\!\mbox{colon}(b_{1},\ldots,b_{n}).$
The proof of the following statement is similar to that of Lemma 1.1.
[**Lemma 2.1**]{}. (2.1).
The following properties are needed for the sequel.
[**Property 2.1**]{} (\[11\]). [*Suppose $\nu={\stackrel{*}{\nu}}+\widetilde{\nu}\,i\ \,
({\stackrel{*}{\nu}}={\rm Re}\,\nu,\ \widetilde{\nu}={\rm Im}\,\nu)$ is a common eigenvector of the matrices $A_{j}$ corresponding to the eigenvalues $\lambda^j={\stackrel{*}{\lambda}}{}^j+\widetilde{\lambda}{}^j\,i\
({\stackrel{*}{\lambda}}{}^j={\rm Re}\,\lambda^j,\ \widetilde{\lambda}{}^j={\rm Im}\,\lambda{}^j),$ $j=1,\ldots, m,$ respectively. Then the real-valued function\
$
P\colon x\to ({\stackrel{*}{\nu}}x)^2\, +\, (\widetilde{\nu}x)^2$ for all $x\in {\mathbb R}^n
\hfill
$\
is a partial integral of the system of equations in total differentials [(2.1)]{} and\
$
{\frak p}_j P(x) =
2\,{\stackrel{*}{\lambda}}{}^j\, P(x)$ for all $x\in {\mathbb R}^n,
\quad
j= 1,\ldots,m.
\hfill
$\
*]{}
[**Property 2.2**]{} (\[11\]). [*Let $\nu={\stackrel{*}{\nu}}+\widetilde{\nu}\,i$ be a common eigenvector of the matrices $A_{j}$ corresponding to the eigenvalues $\lambda^j={\stackrel{*}{\lambda}}{}^j+\widetilde{\lambda}{}^j\,i,\ j=1,\ldots, m,$ respectively. Then\
$
{\frak p}_j\,{\rm arctg}\,\dfrac{\widetilde{\nu}x}{{\stackrel{*}{\nu}}x}\,=\,\widetilde{\lambda}{}^j$ for all $x\in {\mathscr X},
\quad j=1,\ldots, m,
\quad {\mathscr X}\subset \{x\colon {\stackrel{*}{\nu}}x\ne 0\}.
\hfill
$* ]{}\
. The proof of the following assertions is similar to those of Theorem 1.1 and Corollary 1.1.
[**Theorem 2.1**]{}. [ *Suppose $\nu^{k}$ are real common eigenvectors of the matrices $A_{j}$ corresponding to the eigenvalues $\lambda_{k}^{j},\ j=1,\ldots, m, \ k=1,\ldots, m+1,$ respectively. Then the system of equations in total differentials [(2.1)]{} has the autonomous first integral\
$
\displaystyle
F\colon x\to
\prod\limits_{k=1}^{m+1}\bigl|\nu^{k} x\bigr|^{h_{k}}$ for all $x\in {\mathscr X},
\quad {\mathscr X}\subset {\rm D}(F),
\hfill
$\
where $h_{1},\ldots,h_{m+1}$ is a real nontrivial solution to the system $
\sum\limits_{k=1}^{m+1}\!\lambda_{k}^{j}h_{k} = 0,
\, j=1,\ldots, m.
$* ]{}
[**Corollary 2.1**]{}. [*Let $\nu^{k}$ be real common eigenvectors of the matrices $A_{j}$ corresponding to the eigenvalues $\lambda_{k}^{j},\ j=1,\ldots, m, \ k=1,\ldots, m+1,$ respectively. Then an autonomous first integral of the system of equations in total differentials [(2.1)]{} is the function\
$
\displaystyle
F_{12\ldots m(m+1)}\colon x\to
\prod\limits_{k=1}^{m}
\bigl|\nu^{k} x\bigr|^{{}^{\scriptstyle {}-\triangle_{k}}}
\bigl|\nu^{m+1} x\bigr|^{{}^{\scriptstyle\triangle}}$ for all $x\in {\mathscr X},
\quad
{\mathscr X}\subset {\rm D}(F_{12\ldots m(m+1)}),
\hfill
$\
where the determinants $\triangle_{k},\, k=1,\ldots, m$ are obtained by replacing the $k\!$-th column of the determinant $\triangle = \big|\lambda_{k}^{j}\big|$ by ${\rm colon}\left(\lambda_{m+1}^{1},\ldots,\lambda_{m+1}^{m}\right),$ respectively.* ]{}
As an example, the linear autonomous system of equations in total differentials\
$
\begin{array}{ll}
dx_1 = {}-x_1\,dt_2, &
dx_2 = 2(x_3+x_4)\,dt_1 + x_2\,dt_2,
\\[1.25ex]
dx_3 = x_2\,dt_1 + x_4\,dt_2,\quad &
dx_4 = x_2\,dt_1 + x_3\,dt_2
\end{array}
\hfill (2.3)
$\
has the commuting matrices\
$
A_1 =
\left\|
\begin{array}{cccc}
0\, & 0\, & 0\, & 0
\\
0\, & 0\, & 1\, & 1
\\
0\, & 2\, & 0\, & 0
\\
0\, & 2\, & 0\, & 0
\end{array}
\right\|
$ and $
A_2 =
\left\|\!
\begin{array}{rccc}
{}-1\, & 0\, & 0\, & 0
\\
0\, & 1\, & 0\, & 0
\\
0\, & 0\, & 0\, & 1
\\
0\, & 0\, & 1\, & 0
\end{array}
\right\|.
\hfill
$\
Therefore the system of equations in total differentials (2.3) is completely solvable.
The matrices $A_1$ and $A_2$ have the eigenvalues $\lambda_1^1\!=\!-2,\, \lambda_2^1=\lambda_3^1=0,\, \lambda_4^1=2,\!$ and $\lambda_1^2\!=\!1,$ $\lambda_2^2=\lambda_3^2=-1, \, \lambda_4^2=1$ corresponding to the eigenvectors $
\nu^{1}\! = (0, -1, 1, 1),\,
\nu^{2}\! = (1, 0, 0, 0),$$\nu^{3} = (0, 0, 1,{}-1),\
\nu^{4} = (0, 1, 1, 1),
$ respectively.
The determinants\
$
\triangle =
\left|\!\!
\begin{array}{rr}
-2 & 0
\\
1 & -1
\end{array}
\!\!\right| = 2,
\quad
\triangle_{11} =
\left|\!\!
\begin{array}{rr}
0 & 0
\\
-1 & -1
\end{array}
\!\!\right| = 0,
\quad
\triangle_{21} =
\left|\!\!
\begin{array}{rr}
-2 & 0
\\
1 & -1
\end{array}
\!\!\right| = 2,
\hfill
$\
$
\triangle_{12} =
\left|\!\!
\begin{array}{rr}
2 & 0
\\
1 & -1
\end{array}
\!\!\right| ={}-2,
\qquad
\triangle_{22} =
\left|\!\!
\begin{array}{rr}
-2 & 2
\\
1 & 1
\end{array}
\!\!\right| ={}-4.
\hfill
$\
By Corollary 2.1 we have that the functionally independent functions\
$
F_{{}_{\scriptstyle 123}}\colon x\to \dfrac{(x_3-x_4)^2}{x_1^2}$ for all $x\in {\mathscr X}
\hfill (2.4)
$\
and\
$
F_{{}_{\scriptstyle 124}}\colon x\to
x_1^4\,(x_2^2-(x_3+x_4)^2\,)^2$ for all $x\in {\mathbb R}^4
\hfill (2.5)
$\
are first integrals of the system (2.3), where a domain ${\mathscr X}\subset \{x\colon x_1\ne 0\}.$
[**Theorem 2.2.**]{} [*Let $\nu^{k} = {\stackrel{*}{\nu}}{}^{\,k}+\widetilde{\nu}{}^{\,k}\,i$ [(]{}this set hasn’t complex conjugate vectors[)]{} be common complex eigenvectors of the matrices $A_{j}$ corresponding to the eigenvalues $\lambda_{k}^{j} = {\stackrel{*}{\lambda}}{}_{k}^{j} +
\widetilde{\lambda}{}_{k}^{j}\,i$ $j=1,\ldots, m,\ k=1,\ldots, s,$ $s{\leqslant}(m+1)/2,$ respectively. Let $\nu^{\theta}$ be common real eigenvectors of $A_{j}$ corresponding to the eigenvalues $\lambda_{\theta}^{j},\ j=1,\ldots, m,\ \theta=s+1,\ldots, m+1-s.$ Then the system of equations in total differentials [(2.1)]{} has the autonomous first integral\
$
\displaystyle
F\colon x\to
\prod\limits_{k=1}^{s}
(P_{k}(x))^{{\stackrel{*}{h}_{k}}}
\exp\Bigl({}-2\,\widetilde{h}_{k}\,\varphi_{k}(x)\Bigr)
\prod\limits_{\theta=s+1}^{m+1-s}
\bigl|\nu^{\theta} x\bigr|^{h_{\theta}}$ for all $x\in {\mathscr X},
\hfill (2.6)
$\
where a domain ${\mathscr X}\subset {\rm D}(F),$ the functions\
$
P_{k}\colon x\to
\bigl({\stackrel{*}{\nu}}{}^{\,k} x\bigr)^{2} +
\bigl(\widetilde{\nu}{}^{\,k} x\bigr)^{2}\!$ for all $x\in {\mathbb R}^n,
\
\varphi_{k}\colon x\to
{\rm arctg}\,\dfrac{\widetilde{\nu}{}^{\,k} x}
{{\stackrel{*}{\nu}}{}^{\,k} x}$ for all $x\in {\mathscr X},
\, k=1,\ldots, s,
\hfill
$\
and $\stackrel{*}{h}_{k},\, \widetilde{h}_{k},\, k=1,\ldots, s,\
h_{\theta},\, \theta=s+1,\ldots, m+1-s$ is a real nontrivial solution to\
$
\displaystyle
2\, \sum\limits_{k=1}^{s}
\bigl(\,{\stackrel{*}{\lambda}}{}_{k}^{j}\,
{\stackrel{*}{h}}_{k} -
\widetilde{\lambda}{}_{k}^{j}\,
\widetilde{h}_{k}\bigr) \ +\
\sum\limits_{\theta=s+1}^{m+1-s}
\lambda_{\theta}^{j}h_{\theta}=0,
\quad
j=1,\ldots, m.
\hfill
$\
*]{} . Taking into account Property 2.1 and Lemma 2.1, we obtain\
$
{\frak p}_{j}
\bigl( ({\stackrel{*}{\nu}}{}^{\,k}x)^2 +(\widetilde{\nu}{}^{\,k} x)^2\bigr) =
2\,{\stackrel{*}{\lambda}}{}_{k}^{j}\,
\bigl( ({\stackrel{*}{\nu}}{}^{\,k}x)^2 +(\widetilde{\nu}{}^{\,k} x)^2\bigr),
\ \ j=1,\ldots, m,
\quad
k=1,\ldots, s,
\hfill
$\
(2.7)\
$
{\frak p}_{j}\,\nu^{\theta} x\, =\,
\lambda_{\theta}^{j}\, \nu^{\theta} x$ for all $x\in {\mathbb R}^n,
\ \ \ j=1,\ldots, m,
\quad \theta=s+1,\ldots, m+1-s.
\hfill
$\
The Lie derivative of (2.6) by virtue of (2.1) is equal to\
$
\displaystyle
{\frak p}_{j} F(x) =
\biggl(\
\prod\limits_{k=1}^{s}
\bigl(P_{k}(x)\bigr)^{{\stackrel{*}{h}_{k}}-1}
\exp\Bigl({}-2\,\widetilde{h}_{k}\,\varphi_{k}(x)\Bigr)
\, \sum\limits_{k=1}^{s}{\stackrel{*}{h}}_{k}\,
\prod\limits_{l=1, l\ne k}^{s} P_{l}(x)\cdot
{\frak p}_{j}P_{k}(x) \ +
\hfill
$\
$
\displaystyle
+ \ \prod\limits_{k=1}^{s}
\bigl(P_{k}(x)\bigr)^{{\stackrel{*}{h}}_{k}}
\exp\Bigl({}-2\,\widetilde{h}_{k}\,\varphi_{k}(x)\Bigr)\,
\sum\limits_{k=1}^{s}
{\frak p}_{j}
\Bigl({}-2\,\widetilde{h}_{k}\,\varphi_{k}(x)\Bigr)\biggr)
\prod\limits_{\theta=s+1}^{m+1-s}
\bigl|\nu^{\theta} x\bigr|^{h_{\theta}} \ +
\hfill
$\
$
\displaystyle
+\ \prod\limits_{k=1}^{s}
\bigl(P_{k}(x)\bigr)^{{\stackrel{*}{h}}_{k}}
\exp\Bigl({}-2\,\widetilde{h}_{k}\,\varphi_{k}(x)\Bigr)\,
\prod\limits_{\theta=s+1}^{m+1-s}
\bigl|\nu^{\theta} x\bigr|^{h_{\theta}-1}\ \cdot
\hfill
$\
$
\displaystyle
\cdot\
\sum\limits_{\theta=s+1}^{m+1-s}
\mbox{sgn}\,\bigl(\nu^{\theta} x\bigr)\,h_{\theta}\,
\prod\limits_{l=s+1,l\ne \theta}^{m+1-s}
\bigl|\nu^{l} x\bigr|\cdot
{\frak p}_{j}\bigl(\nu^{\theta} x\bigr)$ for all $x\in {\mathscr X},
\quad j=1,\ldots, m.
\hfill
$\
Using Property 2.2 and (2.7), we get\
$
\displaystyle
{\frak p}_{j}F(x) =
\biggl(\,
\sum\limits_{k=1}^{s}
2\bigl({\stackrel{*}{\lambda}}{}_{k}^{j}\, {\stackrel{*}{h}}_{k} -
\widetilde{\lambda}{}_{k}^{j}\,
\widetilde{h}_{k}\bigr) +
\sum\limits_{\theta=s+1}^{m+1-s}\lambda_{\theta}^{j}h_{\theta}
\biggr) F(x)$ for all $x\in {\mathscr X},
\quad
j=1,\ldots, m.
\hfill
$\
If $
2\sum\limits_{k=1}^{s}
\bigl({\stackrel{*}{\lambda}}{}_{k}^{j}\, {\stackrel{*}{h}}_{k} -
\widetilde{\lambda}{}_{k}^{j}\, \widetilde{h}_{k}\bigr) +
\sum\limits_{\theta=s+1}^{m+1-s}
\lambda_{\theta}^{j}h_{\theta} = 0,
\ j=1,\ldots, m,$ then the function (2.6) is an autonomous first integral of the system of equations in total differentials [(2.1)]{}.
------------------------------------------------------------------------
For example, the completely solvable linear system of total differential equations\
$
dx_1 = x_1\,dt_1 + x_2\,dt_2,
\quad \
dx_2 = x_2\,dt_1 - x_1\,dt_2,
\quad \
dx_3 = x_3\,dt_1 - x_3\,dt_2
\hfill (2.8)
$\
has the eigenvalues $\lambda_1^1 =\lambda_2^1 =\lambda_3^1 = 1,\
\vspace{0.35ex}
\lambda_1^2 = -\,i, \ \lambda_2^2 = i,\ \lambda_3^2 = -\,1
$ corresponding to the eigenvectors $
\nu^{1}\! = (1, i, 0),
\, \nu^{2}\! =(1,-i, 0),
\, \nu^{3}\! = (0, 0, 1),
$ respectively. According to Theorem 2.2, we can construct the autonomous first integral of the system (2.8):\
$
F\colon x\to
\dfrac{x_{1}^{2} + x_{2}^{2}}{x_{3}^{2}}\
\exp\Bigl( 2\,{\rm arctg}\, \dfrac{x_2}{x_1}\, \Bigr)$ for all $x\in {\mathscr X},
\quad
{\mathscr X}\subset \{ x\colon x_1\ne 0,\ x_3\ne 0\}.
\hfill (2.9)
$\
. [*Suppose $\nu^{\tau} ={\stackrel{*}{\nu}}{}^{\,\tau}+
\widetilde{\nu}{}^{\,\tau}\,i, \
\nu^{s+\tau} ={\stackrel{*}{\nu}}{}^{\,\tau} -
\widetilde{\nu}{}^{\,\tau}\,i,
\ \tau=1,\ldots, s,\ s{\leqslant}m/2,$ and $\nu^{2s+1} ={\stackrel{*}{\nu}}{}^{\,2s+1} +
\widetilde{\nu}{}^{\,2s+1}\,i$ are common complex eigenvectors of the matrices $A_{j}$ corresponding to the eigenvalues $
\lambda_{\tau}^{j}\! =\!{\stackrel{*}{\lambda}}{}_{\tau}^{j}\, +
\widetilde{\lambda}{}_{\tau}^{j}\,i,\
\lambda_{s+\tau}^{j}\! =\!
{\stackrel{*}{\lambda}}{}_{\tau}^{j}\,-
\widetilde{\lambda}{}_{\tau}^{j}\,i, \,
\tau\!=\!1,\ldots, s,\
\lambda_{2s+1}^{j}=
{\stackrel{*}{\lambda}}{}_{2s+1}^{j} +
\widetilde{\lambda}{}_{2s+1}^{j}\,i,$ $j=1,\ldots, m,$ respectively. Let $\nu^{\theta}$ be common real eigenvectors of $A_{j}$ corresponding to the eigenvalues $\lambda_{\theta}^{j},\ j=1,\ldots, m,\ \theta=2s+2,\ldots, m+1,$ respectively. Then the system of equations in total differentials [(2.1)]{} has the autonomous first integrals\
$
\displaystyle
F_{1}\colon x\to
\prod\limits_{k=1}^{s}
\bigl(P_{k}(x)\bigr)^{{\stackrel{*}{h}}_{k}+{\stackrel{*}{h}}_{s+k}}
\exp\Bigl({}-2\,\bigl(\widetilde{h}_{k}-
\widetilde{h}_{s+k}\bigr)\varphi_{k}(x)\Bigr) \cdot
\hfill
$\
\
$
\displaystyle
\cdot
\bigl(P_{2s+1}(x)\bigr)^{{\stackrel{*}{h}}_{2s+1}}
\exp\Bigl({}-2\,\widetilde{h}_{2s+1}\,\varphi_{2s+1}(x)\Bigr)
\prod\limits_{\theta=2s+2}^{m+1}
\bigl(\nu^{\theta} x\bigr)^{2\,{\stackrel{*}{h}}_{\theta}}$ for all $x\in {\mathscr X}
\hfill
$\
and\
$
\displaystyle
F_{2}\colon x\to
\prod\limits_{k=1}^{s}
\bigl(P_{k}(x)\bigr)^{
\widetilde{h}_{k}+\widetilde{h}_{s+k}}
\exp\Bigl(2\,\bigl({\stackrel{*}{h}}_{k} -
{\stackrel{*}{h}}_{s+k}\bigr)\varphi_{k}(x) \Bigr) \cdot
\hfill
$\
\
$
\displaystyle
\cdot
\bigl(P_{2s+1}(x)\bigr)^{\widetilde{h}_{2s+1}}
\exp\Bigl(2\,{\stackrel{*}{h}}_{2s+1}\,\varphi_{2s+1}(x)\Bigr)
\prod\limits_{\theta=2s+2}^{m+1}
\bigl(\nu^{\theta} x\bigr)^{2\,\widetilde{h}_{\theta}}$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset {\rm D}(F_1)\cap{\rm D}(F_2),$ the functions\
$
P_{k}\colon x\to
({\stackrel{*}{\nu}}{}^{\,k} x)^{2} +
(\widetilde{\nu}{}^{\,k} x)^{2}$ for all $x\in {\mathbb R}^n,
\quad
k=1,\ldots, s, 2s+1,
\hfill
$\
$
\varphi_{k}\colon x\to
{\rm arctg}\,\dfrac{\widetilde{\nu}{}^{\,k} x}{{\stackrel{*}{\nu}}{}^{\,k} x}$ for all $x\in {\mathscr X},
\quad
k=1,\ldots, s, 2s+1,
\hfill
$\
and $h_{k} = {\stackrel{*}{h}}_{k} + \widetilde{h}_{k}\,i,\, k=1,\ldots, m+1$ is a nontrivial solution to $\sum\limits_{k=1}^{m+1}{\lambda}_{k}^{j}\,h_{k}=0,\, j=1,\ldots, m.$* ]{}
[*Proof.*]{} We form two complex-valued functions\
$
\displaystyle
{\stackrel{*}{F}}\colon x\to
\prod\limits_{k=1}^{2s}\bigl(\nu^{k} x\bigl)^{h_{k}}
\bigl(\nu^{2s+1} x\bigr)^{h_{2s+1}}
\prod\limits_{\theta=2s+2}^{m+1}
\bigl(\nu^{\theta} x\bigr)^{h_{\theta}}$ for all $x\in {\mathscr X}
\hfill
$\
and\
$
\displaystyle
\stackrel{**}{F}\colon x\to
\prod\limits_{k=1}^{2s}\bigl( \nu^{k} x\bigr)^{l_{k}}
\bigl(\overline{\nu^{2s+1}} x\bigr)^{l_{2s+1}}
\prod\limits_{\theta=2s+2}^{m+1}
\bigl(\nu^{\theta} x\bigr)^{l_{\theta}}$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset {\mathbb R}^n$ and $h_{k},\ l_{k},\, k=1,\ldots, m+1$ are complex numbers.
Using Lemma 2.1, we get\
$
\displaystyle
{\frak p}_{j}\,{\stackrel{*}{F}}(x) =
\sum\limits_{k=1}^{m+1}
\lambda_{k}^{j}\,h_{k}\,{\stackrel{*}{F}}(x)$ for all $x\in {\mathscr X},
\quad j=1,\ldots, m,
\hfill
$\
$
\displaystyle
{\frak p}_{j}\,{\stackrel{**}{F}}(x) =
\biggl(\ \sum\limits_{k=1}^{2s}
\lambda_{k}^{j}\,l_{k} +
\overline{\lambda_{2s+1}^{j}}\, l_{2s+1} +
\sum\limits_{\theta=2s+2}^{m+1}
\lambda_{\theta}^{j}\,l_{\theta}
\biggr){\stackrel{**}{F}}(x)$ for all $x\in {\mathscr X},
\quad j=1,\ldots, m.
\hfill
$\
Let $h_{k}\! =\!{\stackrel{*}{h}}_{k}+
\widetilde{h}_{k}\,i,\, k\!=\!1,\ldots, m+1$ be a nontrivial solution to $\sum\limits_{k=1}^{m+1}\lambda_{k}^{j}h_{k} = 0, \, j\!=\!1,\ldots, m.$ Then $l_{k}={\stackrel{*}{h}}_{s+k} -\widetilde{h}_{s+k}\,i, \
\vspace{0.5ex}
l_{s+k} ={\stackrel{*}{h}}_{k} - \widetilde{h}_{k}\,i,\,
k\!=\!1,\ldots, s, \
l_{2s+1} ={\stackrel{*}{h}}_{2s+1} - \widetilde{h}_{2s+1}\,i,\
l_{\theta} ={\stackrel{*}{h}}_{\theta} - \widetilde{h}_{\theta}\,i,$ $\theta=2s+2,\ldots, m+1$ is a solution to $
\sum\limits_{k=1}^{2s}\lambda_{k}^{j}\,l_{k} +
\overline{\lambda_{2s+1}^{j}}\,l_{2s+1} +
\sum\limits_{\theta=2s+2}^{m+1}
\lambda_{\theta}^{j}\,l_{\theta} = 0,
\, j\!=\!1,\ldots, m$ and the functions ${\stackrel{*}{F}}\colon {\mathscr X}\to {\mathbb C},\
{\stackrel{**}{F}}\colon {\mathscr X}\to {\mathbb C}$ are first integrals of the system (2.1).
Since $F_1 = {\stackrel{*}{F}}\,{\stackrel{**}{F}}$ and $F_{2} = \bigl({\stackrel{**}{F}}/{\stackrel{*}{F}}\bigr)^{i},$ we see that the functions (2.10) and (211) are autonomous first integrals of the system of equations in total differentials (2.1).
------------------------------------------------------------------------
In particular, the completely solvable linear system of total differential equations\
$
dx_1 = x_1\,dt_1 + x_3\,dt_2,
\qquad
dx_2 = {}-x_2\,dt_1 + x_4\,dt_2,
\hfill
$\
(2.12)\
$
dx_3 = x_3\,dt_1 - x_1\,dt_2,
\qquad
dx_4 = {}-x_4\,dt_1 - x_2\,dt_2
\hfill
$\
has the eigenvalues $
\lambda_1^1=\lambda_2^1={}-1,\
\lambda_3^1=\lambda_4^1=1,\
\lambda_1^2=\lambda_3^2={}-i,\ \lambda_2^2=\lambda_4^2=i$ corresponding to the linearly independent eigenvectors $\nu^{1} = (0, {}-i, 0, 1),
\ \nu^{2} = (0, i,0, 1),
\ \nu^{3} = ({}-i,0, 1, 0),$ and $\nu^{4} = (i, 0,1, 0),$ respectively. The functions (by Theorem 2.3)\
$
F_1\colon x\to \dfrac{x_1 x_2 + x_3 x_4}{x_1 x_4 - x_2 x_3}$ for all $x\in {\mathscr X},
$ $
F_2\colon x\to (x_1^2+ x_3^2)(x_2^2+ x_4^2)$ for all $x\in {\mathbb R}^4
\hfill (2.13)
$\
are first integrals of the system (2.12), where a domain ${\mathscr X}\subset \{x\colon x_1 x_4 - x_2 x_3\ne 0\}\subset {\mathbb R}^4.$
From the entire set of ordinary differential systems induced by the completely solvable system of equations in total differentials (2.1), we extract system\
$
dx=A^{\zeta}(x)\,dt_{\zeta},
\quad
A^{\zeta}(x)=\mbox{colon}(a_{1\zeta}(x),\ldots,a_{n\zeta}(x))$ for all $x\in {\mathbb R}^n,
\hfill (2.1.\zeta)
$\
such that the matrix $A_{\zeta}$ has the smallest number of elementary divisors.
[**Definition 2.1**]{}. [*Let $\nu^{0l}$ be an eigenvector of the matrix $A_{\zeta}$ corresponding to the eigenvalue $\lambda^{\zeta}_{l}$ with elementary divisor of multiplicity $s_l.$ A non-zero vector $\nu^{\theta l}\in {\mathbb C}^n$ is called a ***generalized eigenvector of order*** [$\theta$]{} for $\lambda^{\zeta}_{l}$ if and only if\
$
(A_{\zeta}-\lambda^{\zeta}_{l} E)\,\nu^{\theta l}=\theta \cdot \nu^{\theta-1,l},
\quad
\theta =1,\ldots, s_l-1,
\hfill (2.14)
$\
where $E$ is the $n\times n$ identity matrix.* ]{}
Using Lemma 2.1 and (2.14), we obtain\
$
{\frak p}_{\zeta}\,\nu^{0l}x =
\lambda^{\zeta}_{l}\,\nu^{0l}x,
\ \ \ \
{\frak p}_{\zeta}\,\nu^{\,\theta l}x =
\lambda^{\zeta}_{l}\,\nu^{\,\theta l}x+\theta\,\nu^{\,\theta -1,l}x$ for all $x\in {\mathbb R}^n,
\ \ \ \theta=1,\ldots, s_l-1.
$ (2.15)\
The following lemmas are needed for the sequel.
[**Lemma 2.2**]{}. [*Let $\nu^{\,0l}$ be a common eigenvector of the matrices $A_j$ corresponding to the eigenvalues $\lambda^j_{l}, \ j=1,\ldots,m,$ respectively. Let $\nu^{\,\theta l},\ \theta =1,\ldots, s_l-1$ be generalized eigenvectors of the matrix $A_\zeta$ corresponding to the eigenvalue $\lambda^{\zeta}_{l}$ with elementary divisor of multiplicity $s_l\ (s_{l}{\geqslant}2).$ If the system $(2.1.\zeta)$ hasn’t the first integrals\
$
\displaystyle
F_{j\theta l}^{\,\zeta}\colon x\to
{\frak p}_j\, v_{\theta l}^{\zeta}(x)$ for all $x\in {\mathscr X},
\ \ j=1,\ldots, m, \ \, j\ne\zeta,
\ \ \ \theta =1,\ldots, s_l-1,
\hfill (2.16)
$\
then\
$
{\frak p}_{\zeta}\,v_{\theta l}^{\zeta}(x)=
\left[\!\!
\begin{array}{lll}
1\! & \text{for all}\ \ x\in {\mathscr X}, & \theta =1,
\\[1ex]
0\! & \text{for all}\ \ x\in {\mathscr X} & \theta =2,\ldots, s_{l}-1,
\end{array}
\right.
\hfill
$\
$
{\frak p}_{j}\,v_{\theta l}^{\zeta}(x)=\mu_{\theta l}^{j\zeta}={\rm const}$ for all $x\in {\mathscr X},
\ \ j=1,\ldots, m,\ \ j\ne \zeta,
\ \ \ \theta =1,\ldots, s_l-1,
\hfill
$\
where $v_{\theta l}^{\zeta}\colon {\mathscr X}\to {\mathbb R},\ \theta =1,\ldots, s_l-1$ is a solution to the system\
$
\nu^{\,\theta l}x=
{\displaystyle \sum\limits_{\delta=1}^{\theta} }
\binom{\theta -1}{\delta-1}v_{\delta l}^{\zeta}(x)\cdot \nu^{\,\theta-\delta,l}x,
\quad
\theta=1,\ldots, s_l-1,
\quad
{\mathscr X}\subset \{x\colon \nu^{0l}x\ne 0\}.
\hfill (2.17)
$\
*]{}
[**Theorem 2.4**]{}. [ *Let the assumptions of Lemma [2.2]{} with $l=1,\ldots r\ \Bigl(\, \sum\limits_{l=1}^{r}s_{l}{\geqslant}m+1\Bigr)$ hold. Then the completely solvable system [(2.1)]{} has the autonomous first integral\
$
\displaystyle
F\colon x\to
\prod\limits_{\xi=1}^{k}\bigl(\nu^{0\xi} x\bigr)^{h_{0 \xi}}
\exp\sum\limits_{q=1}^{\varepsilon_{\xi}}\,h_{q\xi} v_{q\xi}^{\zeta}(x)$ for all $x\in {\mathscr X},
\quad {\mathscr X}\subset {\rm D}(F),
\hfill
$\
where $\sum\limits_{\xi=1}^{k}\varepsilon_{\xi}=m-k+1, \,
\varepsilon_{\xi}{\leqslant}s_{\xi}-1, \, \xi=1,\ldots, k,\, k{\leqslant}r,$ and $h_{q\xi},\, q=0,\ldots,\varepsilon_{\xi}, \, \xi=1,\ldots, k$ is a nontrivial solution to the linear homogeneous algebraic system of equations\
$
\displaystyle
\sum\limits_{\xi=1}^{k}
\bigl(\lambda_{\xi}^{j}\,h_{0\xi}
+ \sum\limits_{q=1}^{\varepsilon_{\xi}}
\mu_{q\xi}^{j\zeta}\,h_{q\xi}\big) = 0,
\ \ j=1,\ldots, m.
\hfill
$\
*]{} The completely solvable linear homogeneous system of total differential equations\
$
\begin{array}{l}
dx_1 = x_2\,dt_1 + (2x_1-x_3)\,dt_2,
\\[1ex]
dx_2 = (2x_2-x_3-x_4)\,dt_1 + ({}-x_1+2x_2+x_4)\,dt_2,
\\[1ex]
dx_3 = (x_1-x_4)\,dt_1 + ({}-x_1+3x_3+x_4)\,dt_2,
\\[1ex]
dx_4 = ({}-x_1+2x_3+2x_4)\,dt_1 + (x_2-3x_3+x_4)\,dt_2
\end{array}
\hfill (2.18)
$\
has the eigenvalue $\lambda_1^1=1$ with elementary divisor $(\lambda^1-1)^4$ corresponding to the eigenvector $\nu^{0}=({}-1,1,{}-1,0)$ and to the generalized eigenvectors $\nu^{1}=(1,0,{}-1,{}-1),\ \nu^{2}=(1,{}-1,3,0),$ $\nu^{3}=({}-3,0,9,9).$ The functions (see (2.17))\
$
v_{11}^1\colon x\to
\dfrac{x_1-x_3-x_4}{{}-x_1+x_2-x_3}$ for all $x\in {\mathscr X},
\hfill
$\
$
v_{21}^1\colon x\to
\dfrac{({}-x_1+x_2-x_3)(x_1-x_2+3x_3)-(x_1-x_3-x_4)^2}
{({}-x_1+x_2-x_3)^{2}}$ for all $x\in {\mathscr X},
\hfill (2.19)
$\
$
v_{31}^1\colon x\to
\dfrac{1}{({}-x_1+x_2-x_3)^{3}}\,
\bigl(({}-3x_1+9x_3+9x_4)({}-x_1+x_2-x_3)^2\, -
\hfill
$\
$
-\,3({}-x_1+x_2-x_3)(x_1-x_3-x_4)(x_1-x_2+3x_3) +
2(x_1-x_3-x_4)^3\,\bigr)$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \{ x\colon x_1-x_2+x_3\ne 0\}.$
Autonomous first integrals of the system (2.18) are the functions (by Theorem 2.4)\
$
F_{1}\colon x\to v_{21}^{1}(x),
\ \
F_{2}\colon x\to ({}-x_{1}+x_{2}-x_{3})^{2}
\exp\bigl( {}-2v_{11}^{1}(x)-v_{31}^{1}(x)\bigr)$ for all $x\in {\mathscr X}.
\hfill (2.20)
$\
In the complex case, we shall have two logical possibilities:
1\. Any function from the set $\!V\!=\!\{\nu^{0\xi}x, v_{q\xi}^{\zeta}(x)\colon\! q\!=\!1,\ldots,\varepsilon_{\xi},\, \xi\!=\!1,\ldots, k,
\sum\limits_{\xi=1}^{k}\!\varepsilon_{\xi}\!=\!m\!-\!k\!+\!1\}\!\!$ has the complex conjugate function in the set $V.$
2\. At least one function from the set $\!V\!$ has not the complex conjugate function in the $\!V.\!\!$
[*Case*]{} 1. The completely solvable system (2.1) has the autonomous first integral\
$
\displaystyle
F\colon x\to
\prod\limits_{\xi=1}^{k_1}
\bigl(\bigl({\stackrel{*}{\nu}}{}^{0\xi} x\bigr)^{2} +
\bigl(\,\widetilde{\nu}\,{}^{0\xi} x\bigr)^{2}\,
\bigr)^{{\stackrel{*}{h}}_{0\xi}}\,
\exp\Bigl({}-2\,\widetilde{h}_{0\xi}\
{\rm arctg}\,\dfrac{\widetilde{\nu}\,{}^{0\xi} x}
{{\stackrel{*}{\nu}}{}^{0\xi} x} \ +
\hfill
$\
$
\displaystyle
+ \ 2\sum\limits_{q=1}^{\varepsilon_{\xi}}
\bigl(\,{\stackrel{*}{h}}_{q\xi}\,{\stackrel{*}{v}}{}_{q\xi}^{\,\zeta}(x) -
\widetilde{h}_{q\xi}\,\widetilde{v}{}_{q\xi}^{\,\,\zeta}(x)
\bigr)\Bigr)
\prod\limits_{\theta=1}^{k_2}
\bigl|\nu^{0\theta} x\bigr|^{h_{0\theta}}
\exp\sum\limits_{q=1}^{\varepsilon_{\theta}}
h_{q\theta}\,v_{q\theta}^{\,\zeta}(x)$ for all $x\in {\mathscr X},
\ \ {\mathscr X}\subset {\rm D}(F),
\hfill
$\
where $\stackrel{*}{h}_{q\xi},\ \widetilde{h}_{q\xi}, q=0,\ldots, \varepsilon_\xi,\ \xi=1,\ldots, k_1,\
h_{q\theta},\ q=0,\ldots, \varepsilon_\theta,\ \theta=1,\ldots, k_2$ is a real nontrivial solution to the linear homogeneous algebraic system of equations\
$
\displaystyle
2\sum\limits_{\xi=1}^{k_1}
\Bigl( \bigl(
{\stackrel{*}{\lambda}}{}_{\xi}^{j}\,{\stackrel{*}{h}}_{0\xi} -
\widetilde{\lambda}{}_{\xi}^{j}\,\widetilde{h}_{0\xi}
\bigr) +
\sum\limits_{q=1}^{\varepsilon_{\xi}}
\bigl(
{\stackrel{*}{\mu}}{}_{q\xi}^{\,j\zeta}\,{\stackrel{*}{h}}_{q\xi} -
\widetilde{\mu}{}_{q\xi}^{\,j\zeta}\,\widetilde{h}_{q\xi}
\bigr)
\Bigr) +
\sum\limits_{\theta=1}^{k_2}
\Bigl(
\lambda_{\theta}^{j}\,h_{0\theta} +
\sum\limits_{q=1}^{\varepsilon_{\theta}}
\mu_{q\theta}^{j\zeta}\,h_{q\theta} \Bigr)\! =\! 0,
\ \ j=1,\ldots, m.
\hfill
$\
Here $\nu^{0\xi}={\stackrel{*}{\nu}}{}^{0\xi} +\widetilde{\nu}\,{}^{0\xi}\,i$ are complex common eigenvectors of the matrices $A_j$ corresponding to the eigenvalues $\lambda_{\xi}^{j} ={\stackrel{*}{\lambda}}{}_{\xi}^{j} +\widetilde{\lambda}{}_{\xi}^{j}\,i,\
j=1,\ldots, m,\ \xi=1,\ldots, k_1,$ respectively; $\nu^{0\theta}$ are real common eigenvectors of $A_j$ corresponding to the eigenvalues $\lambda_{\theta}^{j}, \ j=1,\ldots, m,\ \theta=1,\ldots, k_2,$ respectively; the functions $v_{q\xi}^{\,\zeta}\!=\!{\stackrel{*}{v}}{}_{q\xi}^{\,\zeta}\! +\!\widetilde{v}{}_{q\xi}^{\,\,\zeta}\,i,
v_{q\theta}^{\,\zeta}\!$ is the solution to the system (2.17); the numbers\
$
{\stackrel{*}{\mu}}{}_{q\xi}^{\,j\zeta} =
{\frak p}_{j}\,{\rm Re}\,v_{q\xi}^{\zeta}={\frak p}_{j}\,{\stackrel{*}{v}}{}_{q\xi}^{\,\zeta},
\ \ \ \
\widetilde{\mu}{}_{q\xi}^{\,j\zeta} =
{\frak p}_{j}\,{\rm Im}\,v_{q\xi}^{\zeta}={\frak p}_{j}\,\widetilde{v}{}_{q\xi}^{\,\,\zeta},
\ \ \ q=1,\ldots, \varepsilon_\xi,
\ \ \xi=1,\ldots, k_1,
\hfill
$\
$
{\mu}_{q\theta}^{j\zeta}={\frak p}_{j}\, v_{q\theta}^{\zeta},
\quad
q=1,\ldots,\varepsilon_\theta,
\ \ \theta=1,\ldots, k_2,
\ \ \ j=1,\ldots, m;
\hfill
$\
the numbers $\varepsilon_{\xi},\ \varepsilon_{\theta}$ such that $
2\sum\limits_{\xi=1}^{k_1}\varepsilon_{\xi}
+ \sum\limits_{\theta=1}^{k_2}
\varepsilon_{\theta} = m-2k_{1}-k_{2}+1$ with $
2k_1\!+\!k_2{\leqslant}r,\, \varepsilon_{\xi}{\leqslant}s_{\xi}\!-\!1,$ $\xi=1,\ldots, k_1,\,
\varepsilon_{\theta}{\leqslant}s_{\theta}\!-\!1,\, \theta=1,\ldots, k_2,$ where $k_1\!$ is a number of complex common eigenvectors [(]{}this set hasn’t complex conjugate vectors[)]{} of the matrices $A_j$ and $k_2$ is a number of real common eigenvectors of the matrices $A_j, \ j=1,\ldots, m.$
For example, the completely solvable system of total differential equations\
$
dx_1=(3,-4,4,1,0,2)x\,dt_1+
(0,-4,2,1,-1,1)x\,dt_2
-(3,-2,4,3,0,2)x\,dt_3,
$\
$
dx_2={}-(1,-3,3,0,2,3)x\,dt_1 +
(1,3,0,0,1,-1)x\,dt_2 + (2,-3,3,3,-1,2)x\,dt_3,
\hfill (2.21)
$\
$
dx_3={}-(3,-5,5,1,2,4)x\,dt_1-
(0,-6,2,1,-2,1)xdt_2+(3,-3,5,4,0,2)x\,dt_3,
$\
$
dx_4= (3,-6,4,4,-1,5)x\,dt_1+ (2,-6,2,3,-4,2)x\,dt_2 -(3,-2,6,4,1,1)x\,dt_3,
\hfill
$\
$
dx_5=(5,-5,8,3,3,6)x\,dt_1 +
(1,-6,3,2,-2,2)x\,dt_2 - (3,-3,6,4,1,2)x\,dt_3,
$\
$
dx_6=-(2,-5,4,3,-1,2)x\,dt_1-(2,-4,3,3,-2,2)x\,dt_2
+(2,-1,4,2,1,0) x\,dt_3
$\
has the eigenvalue $\lambda_{1}^{1}=1+2i$ with elementary divisor $\left(\lambda^1-1-2i\right)^3$ corresponding to the eigenvector $\nu^{0}=(1, 0, 1+i, 1, i, 1)$ and to the generalized eigenvectors $\nu^{1}=(1, 1+i, 0, 0, i, i),$ $\nu^{2}=(2+2i, 0, 2+2i, 0, 2i, 2i).$ The scalar functions of the vector argument\
$
{\stackrel{*}{v}}{}_{11}^{\,1}\colon x\to
\bigl( (x_1+x_2)(x_1+x_3+x_4+x_6)\,+\,(x_3+x_5)(x_2+x_5+x_6)\bigr) / P(x),
\hfill
$\
$
\widetilde{v}{}_{11}^{\,1}\colon x\to
\bigl( (x_1+x_3+x_4+x_6)(x_2+x_5+x_6)-(x_1+x_2)(x_3+x_5)\bigr) / P(x),
\hfill
$\
$
{\stackrel{*}{v}}{}_{21}^{\,1}\colon x\to
\bigl(
\bigl( (x_1+x_3+x_4+x_6)(x_2+x_5+x_6)-(x_1+x_2)(x_3+x_5)\bigr)^2 \ +
\hfill
$\
$
+\ 2\,P(x)\bigl(
(x_1+x_3)(x_1+x_3+x_4+x_6)+(x_3+x_5)(x_1+x_3+x_5+x_6)
\bigr)\ -
\hfill
$\
$
- \, \bigl(
(x_1+x_2)(x_1+x_3+x_4+x_6)\,+\,(x_3+x_5)(x_2+x_5+x_6)
\bigr)^2 \bigr) / P(x),
\hfill (2.22)
$\
$
\widetilde{v}{}_{21}^{\,1}\colon x\to
2\bigl( P(x)
\bigl( (x_1+x_3+x_4+x_6)(x_1+x_3+x_5+x_6) - (x_1+x_3)(x_3+x_5) \bigr)\ +
\hfill
$\
$
+\ \bigl((x_3+x_5)(x_1+x_2) - (x_1+x_3+x_4+x_6)(x_2+x_5+x_6) \bigr)\, \cdot
\hfill
$\
$
\cdot
\bigl((x_1+x_2)(x_1+x_3+x_4+x_6)+(x_3+x_5)(x_2+x_5+x_6)
\bigr)\bigr)/ P(x)$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \{ x\colon x_1+x_3+x_4+x_6\ne 0\},$ the polynomial\
$
P\colon x\to (x_1+x_3+x_4+x_6)^2+(x_3+x_5)^2$ for all $x\in {\mathbb R}^6.
\hfill (2.23)
$\
Autonomous first integrals of the system (2.21) are the functions\
$
F_1\colon x\to P(x)\exp\bigl({}-4\varphi(x)
+6\,{\stackrel{*}{v}}{}_{11}^{\,1}(x)+2\,\widetilde{v}{}_{11}^{\,1}(x)\bigr)$ for all $x\in {\mathscr X},
\hfill (2.24)
$\
$
F_2\colon x\to
P^2(x)\exp\bigl({}-2\varphi(x)+
{\stackrel{*}{v}}{}_{21}^{\,1}(x) - \widetilde{v}{}_{21}^{\,1}(x)\bigr)$ for all $x\in {\mathscr X},
\hfill (2.25)
$\
$
F_3\colon x\to
2\,\widetilde{v}{}_{11}^{\,1}(x) - 2\,{\stackrel{*}{v}}{}_{21}^{\,1}(x)
- \widetilde{v}{}_{21}^{\,1}(x)$ for all $x\in {\mathscr X},
\hfill (2.26)
$\
where\
$
\varphi\colon x\to \
{\rm arctg}\,\dfrac{x_3+x_5}{x_1+x_3+x_4+x_6}$ for all $x\in {\mathscr X}.
\hfill (2.27)
$\
2A. A common complex eigenvector of the matrices $\!A_{j}, j\!=\!1,\ldots, m\!$ hasn’t the complex conjugate vector. First integrals of the completely solvable system (2.1) are the functions\
$
\displaystyle
F_{1}\colon x\to
\prod\limits_{\xi=1}^{k_1}
\bigl(P_{\xi}(x)\bigr)^{{\stackrel{*}{h}}_{0\xi}+
{\stackrel{*}{h}}_{0,(k_1+\xi)}}
\exp\Bigl({}-2\bigl(\,\widetilde{h}_{0\xi}-
\widetilde{h}_{0,(k_1+\xi)}\bigr)\varphi_{\xi}(x)\ +
\hfill
$\
$
\displaystyle
+\ 2\sum\limits_{q=1}^{\varepsilon_{\xi}}
\Bigl(
\bigl(\,
{\stackrel{*}{h}}_{q\xi}+{\stackrel{*}{h}}_{q,(k_1+\xi)}
\bigr)\,{\stackrel{*}{v}}{}_{q\xi}^{\,\zeta}(x) +
\bigl(\,
\widetilde{h}_{q,(k_1+\xi)}-\widetilde{h}_{q\xi}
\bigr)\,\widetilde{v}{}_{q\xi}^{\,\zeta}(x)
\Bigr)\Bigr) \bigl(P_{2k_1+1}(x)\bigr)^{{\stackrel{*}{h}}_{0,(2k_1+1)}}\
\cdot
\hfill
$\
$
\displaystyle
\cdot\,
\exp\Bigl({}-2\,
\widetilde{h}_{0,(2k_1+1)}\,\varphi_{2k_1+1}(x)\Bigr)\,\cdot
\prod\limits_{\theta=1}^{k_2}
\bigl(\nu^{0\theta} x\bigr)^{2\,{\stackrel{*}{h}}_{0\theta}}\,
\exp\Bigl(\, 2\sum\limits_{q=1}^{\varepsilon_{\theta}}
{\stackrel{*}{h}}_{q\theta}\,v_{q\theta}^{\,\zeta}(x)\Bigr)$ for all $x\in {\mathscr X},
\hfill
$\
$
\displaystyle
F_{2}\colon x\to
\prod\limits_{\xi=1}^{k_1}
\bigl(P_{\xi}(x)\bigr)^{\widetilde{h}_{0\xi}+\widetilde{h}_{0,(k_1+\xi)}}
\exp\Big( 2\bigl(\, {\stackrel{*}{h}}_{0\xi}-
{\stackrel{*}{h}}_{0,(k_1+\xi)}\bigr)\,\varphi_{\xi}(x) \ +
\hfill
$\
$
\displaystyle
+ \ 2\sum\limits_{q=1}^{\varepsilon_{\xi}}
\Bigl(
\bigl(\,\widetilde{h}_{q\xi}+\widetilde{h}_{q,(k_1+\xi)}\bigr)\,{\stackrel{*}{v}}{}_{q\xi}^{\,\zeta}(x) +
\bigl(\,
{\stackrel{*}{h}}_{q\xi}-{\stackrel{*}{h}}_{q,(k_1+\xi)}
\bigr)\,\widetilde{v}{}_{q\xi}^{\,\zeta}(x)
\Bigr)
\Bigr) \bigl( P_{2k_1+1}(x)\bigr)^{\widetilde{h}_{0,(2k_1+1)}}\ \cdot
\hfill
$\
$
\displaystyle
\cdot
\exp\Bigl(\,2\,{\stackrel{*}{h}}_{0,(2k_1+1)}\,\varphi_{2k_1+1}(x)
\Bigr)\cdot
\prod\limits_{\theta=1}^{k_2}
\bigl( \nu^{0\theta} x \bigr)^
{2\,\widetilde{h}_{0\theta}}
\exp\Bigl(\, 2\sum\limits_{q=1}^{\varepsilon_{\theta}}\widetilde{h}_{q\theta}\,v_{q\theta}^{\,\zeta}(x)\Bigr)$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset {\rm D}(F_1)\cap {\rm D}(F_2),$ the functions\
$
P_{\xi}\colon x \to
({\stackrel{*}{\nu}}{}^{\,0\xi} x)^{2} +
(\,\widetilde{\nu}{}^{\,0\xi}x)^{2}$ for all $x\in {\mathbb R}^n,
\quad
\xi=1,\ldots, k_1, 2k_1+1,
\hfill
$\
$
\varphi_{\xi}\colon x\to\
{\rm arctg}\,\dfrac{\widetilde{\nu}{}^{\,0\xi} x}
{{\stackrel{*}{\nu}}{}^{\,0\xi} x}$ for all $x\in {\mathscr X},
\quad
\xi=1,\ldots, k_1, 2k_1+1,
\hfill
$\
and $h_{q\xi}={\stackrel{*}{h}}_{q\xi}+\widetilde{h}_{q\xi}\,i,\
h_{q\theta}={\stackrel{*}{h}}_{q\theta}+\widetilde{h}_{q\theta}\,i$ is a nontrivial solution to the system\
$
\displaystyle
\sum\limits_{\xi=1}^{2k_1}
\Bigl(
\lambda_{\xi}^{j}h_{0\xi} +
\sum\limits_{q=1}^{\varepsilon_{\xi}}
\mu_{q\xi}^{\,j\zeta}h_{q\xi}
\Bigr) +
\lambda_{2k_1+1}^{j}h_{0,(2k_1+1)} +
\sum\limits_{\theta=1}^{k_2}
\Bigl(
\lambda_{\theta}^{j}h_{0\theta} +\!\!
\sum\limits_{q=1}^{\varepsilon_{\theta}}
\mu_{q\theta}^{j\zeta}h_{q\theta}
\Bigr) = 0,
\ \
j=1,\ldots, m.
\hfill
$\
Here $\nu^{0\xi}={\stackrel{*}{\nu}}{}^{\,0\xi}\, +\widetilde{\nu}{}^{\,0\xi}\,i, \
\nu^{0,(k_1+\xi)}=\overline{\nu^{0\xi}},\
\nu^{0,(2k_1+1)}={\stackrel{*}{\nu}}{}^{\,0,(2k_1+1)}\,+\widetilde{\nu}{}^{\,\, 0,(2k_1+1)}\,i$ are complex eigenvectors of the matrices $\!\!A_j\!\!$ corresponding to the eigenvalues $\!\lambda_{\xi}^{j}\!\!=\!{\stackrel{*}{\lambda}}{}_{\xi}^{j}\! +\!\widetilde{\lambda}{}_{\xi}^{j}\,i,
\lambda_{k_1+\xi}^{j}\!\!=\!\overline{\lambda_{\xi}^{j}},$ $\xi=1,\ldots, k_1,\ \lambda_{2k_1+1}^{j}={\stackrel{*}{\lambda}}{}_{2k_1+1}^{j}\,+
\widetilde{\lambda}{}_{2k_1+1}^{j}\,i,\ j=1,\ldots, m,$ respectively; $\nu^{0\theta}$ are real common eigenvectors of $A_j$ corresponding to the eigenvalues $\lambda_{\theta}^{j},\ j=1,\ldots,m,\ \theta=1,\ldots, k_2,$ respectively; the functions $v_{q\xi}^{\,\zeta}={\stackrel{*}{v}}{}_{q\xi}^{\,\zeta} +\widetilde{v}{}_{q\xi}^{\,\,\zeta}\,i,\
v_{q\theta}^{\,\zeta}$ is the solution to the system (2.17); the numbers\
$
{\mu}_{q\xi}^{j\zeta}={\frak p}_{j}\,v_{q\xi}^{\,\zeta}(x),
\ \ \ \
{\stackrel{*}{\mu}}{}_{q\xi}^{\,j\zeta} =\, {\rm Re}\,{\mu}_{q\xi}^{j\zeta},
\ \ \
\widetilde{\mu}{}_{q\xi}^{\,j\zeta} =\,{\rm Im}\,{\mu}_{q\xi}^{j\zeta},
\ \ \
q=1,\ldots,\varepsilon_\xi,
\ \
\xi=1,\ldots, 2k_1,
\hfill
$\
$
\mu_{q\theta}^{j\zeta} ={\frak p}_{j}\,v_{q\theta}^{\,\zeta}(x),
\ \ \
q=1,\ldots,\varepsilon_\theta,
\ \
\theta=1,\ldots, k_2,
\ \ \ j=1,\ldots,m;
\hfill
$\
the numbers $\varepsilon_{\xi},\ \varepsilon_{\theta}$ such that $
2\sum\limits_{\xi=1}^{k_1} \varepsilon_{\xi} +
\sum\limits_{\theta=1}^{k_2} \varepsilon_{\theta} =m-2k_1-k_2$ with $2k_1+1+k_2 {\leqslant}r,\ \varepsilon_{\xi}{\leqslant}s_{\xi}-1,$ $\xi=1,\ldots, k_1,\ \varepsilon_{\theta}{\leqslant}s_{\theta}-1,\, \theta=1,\ldots, k_2,$ where $k_1\!$ is a number of complex common eigenvectors [(]{}this set hasn’t complex conjugate vectors[)]{} of the matrices $A_j$ and $k_2$ is a number of real common eigenvectors of the matrices $A_j, \ j=1,\ldots, m.$
As an example, the completely solvable system of equations in total differentials\
$
dx_1 \!=\! (1,-2,2,0,1,1)xdt_1 + (0,2,0,0,1,1)xdt_2 +
(3,0,0,0,-1,-1)xdt_3 + (1,-2,4,0,2,2)xdt_4,
$\
$
dx_2\! =\! (0,2,-2,0,-2,-2)xdt_1 - (1,3,0,0,1,1)xdt_2 +
(-1,2,0,0,1,1)xdt_3 - (2,1,4,0,4,4)xdt_4,
$\
$
dx_3 = (0,3,-2,0,-2,-2)x\,dt_1 - (1,3,1,0,2,2)x\,dt_2\ +
$\
$
+ \ (-2,-1,2,0,1,1)x\,dt_3 - (3,-2,7,0,5,5)x\,dt_4,
\qquad (2.28)
$\
$\!\!
dx_4\! =\! x\bigl(\! (0,\!-4,0,2,\!-2,2)dt_1 \!+ (2,2,0,1,0,4)dt_2 +
(1,2,\!-2,1,\!-1,\!-1)dt_3 + (3,\!-4,10,2,7,7)dt_4\!\bigr),\!\!
$\
$
dx_5 \!=\! (2,-3,4,2,2,4)xdt_1 + (3,3,2,2,1,4)xdt_2 +
(2,1,-1,0,0,-1)xdt_3 + (3,-2,9,0,7,5)xdt_4,
$\
$\!
dx_6 \!=\! x\bigl(\!-(1,\!-3,2,2,\!-1,1)dt_1\! - (2,1,2,2,1,4)dt_2\! +
(\!-1,\!-1,1,0,1,2)dt_3\! - (\!-1,\!-4,5,0,4,2)dt_4\!\bigr)\!\!
$\
has the eigenvalue $\lambda_{1}^{1}=1+i$ with elementary divisor $(\lambda^1-1-i)^2$ corresponding to the eigenvector $\nu^{01}=(1, 1+i, 0, 0, i, i)$ and to the generalized eigenvector $\nu^{11}=(1+i, 0, 1+i, 0, i, i)$ and the simple eigenvalue $\lambda_{2}^1=2i$ with common eigenvector $\nu^{02}=(1, 0, 1+i, 1,i, 1).$
The real-valued scalar functions\
$
{\stackrel{*}{v}}{}_{11}^{\,1} \colon x\to
\dfrac{(x_1+x_2)(x_1+x_3)+(x_2+x_5+x_6)(x_1+x_3+x_5+x_6)}{P_1(x)}$ for all $x\in {\mathscr X},
\hfill
$\
(2.29)\
$
\widetilde{v}{}_{11}^{\,1} \colon x\to
\dfrac{(x_1+x_2)(x_1+x_3+x_5+x_6)-(x_1+x_3)(x_2+x_5+x_6)}{P_1(x)}$ for all $x\in {\mathscr X},
\hfill
$\
where $P_1\colon x\to (x_1+x_2)^2+(x_2+x_5+x_6)^2$ for all $x\in {\mathbb R}^6.$
The autonomous general integral of the system (2.28) is the functions\
$
F_1\colon x\to P_1(x)\bigl(P_2(x)\bigr)^2
\exp\bigl({}-10\varphi_1(x) + 8\,{\stackrel{*}{v}}{}_{11}^{\,1}(x) +
6\,\widetilde{v}{}_{11}^{\,1}(x)\bigr)$ for all $x\in {\mathscr X}
\hfill (2.30)
$\
and\
$
F_2\colon x\to
\bigl(P_1(x)\bigr)^3 \exp\bigl({}-10\varphi_1(x) -
4\varphi_2(x) + 12\,{\stackrel{*}{v}}{}_{11}^{\,1}(x) +
14\,\widetilde{v}{}_{11}^{\,1}(x)\bigr)$ for all $x\in {\mathscr X},
\hfill (2.31)
$\
where a domain ${\mathscr X}\subset \{ x\colon x_1+x_2\ne 0,\ x_1+x_3+x_4+x_6\ne 0\},$ the scalar functions\
$
P_2\colon x\to (x_1+x_3+x_4+x_6)^2+(x_3+x_5)^2$ for all $x\in {\mathbb R}^6,
\hfill
$\
$
\varphi_1\colon x\to \
{\rm arctg}\,\dfrac{x_2+x_5+x_6}{x_1+x_2}\,,
\quad
\varphi_2\colon x\to \
{\rm arctg}\,\dfrac{x_3+x_5}{x_1+x_3+x_4+x_6}$ for all $x\in {\mathscr X}.
\hfill
$\
2B. A function $\!v_{l\gamma}^{\,\zeta},\, \gamma\!\in\! \{1,\ldots,k_1\},\, l\!\in\! \{1,\ldots,\varepsilon_{\gamma}\}\!$ hasn’t the complex conjugate function. Autonomous first integrals of the completely solvable system (2.1) are the functions\
$
\displaystyle
F_{1}\colon x\to
\prod\limits_{\xi=1}^{k_1}
\bigl(P_{\xi}(x)\bigr)^{{\stackrel{*}{h}}_{0\xi}+
{\stackrel{*}{h}}_{0,(k_1+\xi)}}
\exp\Bigl({}-2\bigl(\,\widetilde{h}_{0\xi}-
\widetilde{h}_{0,(k_1+\xi)}\bigr)\,\varphi_{\xi}(x) \ +
\hfill
$\
$
\displaystyle
+\ 2\sum\limits_{q=1}^{\varepsilon_{\xi}}
(1-\delta_{ql}\delta_{\xi\gamma})\Bigl(
\bigl(\,
{\stackrel{*}{h}}_{q\xi}+{\stackrel{*}{h}}_{q,(k_1+\xi)}
\bigr)\,{\stackrel{*}{v}}{}_{q\xi}^{\,\zeta}(x) +
\bigl(\,
\widetilde{h}_{q,(k_1+\xi)}-\widetilde{h}_{q\xi}
\bigr)\,\widetilde{v}{}_{q\xi}^{\,\zeta}(x)
\Bigr) \ +
$
\
$
\displaystyle
+ \ 2\,\Bigl(\,{\stackrel{*}{h}}_{l\gamma}\,
{\stackrel{*}{v}}{}_{l\gamma}^{\,\zeta}(x) -
\widetilde{h}_{l\gamma}\,
\widetilde{v}{}_{l\gamma}^{\,\zeta}(x)\Bigr)
\Bigr)
\prod\limits_{\theta=1}^{k_2}
\bigl(\nu^{0\theta} x \bigr)^{2\,{\stackrel{*}{h}}_{0\theta}}
\exp\biggl( 2\sum\limits_{q=1}^{\varepsilon_{\theta}}
{\stackrel{*}{h}}_{q\theta}\, v_{q\theta}^{\,\zeta}(x)\biggr)$ for all $x\in {\mathscr X},
\hfill
$\
and\
$
\displaystyle
F_{2}\colon x\to
\prod\limits_{\xi=1}^{k_1}
\bigl(P_{\xi}(x)\bigr)^{\widetilde{h}_{0\xi}+
\widetilde{h}_{0,(k_1+\xi)}}
\exp\Bigl( 2\bigl(\, {\stackrel{*}{h}}_{0\xi}-
{\stackrel{*}{h}}_{0,(k_1+\xi)}\bigr)\,\varphi_{\xi}(x)\ +
\hfill
$\
$
\displaystyle
+\ 2\sum\limits_{q=1}^{\varepsilon_{\xi}}
(1-\delta_{ql}\delta_{\xi\gamma})\Bigl(
\bigl(\,
\widetilde{h}_{q\xi} +
\widetilde{h}_{q,(k_1+\xi)}\bigr)\,
{\stackrel{*}{v}}{}_{q\xi}^{\,\zeta}(x) +
\bigl(\,
{\stackrel{*}{h}}_{q\xi}-{\stackrel{*}{h}}_{q,(k_1+\xi)}
\bigr)\,
\widetilde{v}{}_{q\xi}^{\,\zeta}(x)
\Bigr) \ +
$
\
$
\displaystyle
+\ 2\bigl(\, {\stackrel{*}{h}}_{l\gamma}\,
\widetilde{v}{}_{l\gamma}^{\,\zeta}(x) +
\widetilde{h}{}_{l\gamma}\,
{\stackrel{*}{v}}{}_{l\gamma}^{\,\zeta}(x)\bigr)
\Bigr)
\prod\limits_{\theta=1}^{k_2}
\bigl(\nu^{0\theta}x\bigr)^{2\,\widetilde{h}_{0\theta}}
\exp\biggl( 2\sum\limits_{q=1}^{\varepsilon_{\theta}}
\widetilde{h}_{q\theta}\,v_{q\theta}^{\,\zeta}(x)\biggr)$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset {\rm D}(F_1)\cap {\rm D}(F_2),\ \delta$ is the Kronecker delta, the functions\
$
P_{\xi}\colon\! x \to
({\stackrel{*}{\nu}}{}^{\,0\xi} x)^{2} +
(\,\widetilde{\nu}{}^{\,0\xi} x)^{2}\!$ for all $\!x\in {\mathbb R}^n,
\
\varphi_{\xi}\colon x\to
{\rm arctg}\,\dfrac{\widetilde{\nu}{}^{\,0\xi} x}
{{\stackrel{*}{\nu}}{}^{\,0\xi} x}\!$ for all $\!x\!\in\! {\mathscr X},\,
\xi\!=\!1,\ldots,k_1,
\hfill
$\
and $h_{q\xi}={\stackrel{*}{h}}_{q\xi}+\widetilde{h}_{q\xi}\,i,\
h_{q\theta}={\stackrel{*}{h}}_{q\theta}+\widetilde{h}_{q\theta}\,i$ is a nontrivial solution to the linear system\
$
\displaystyle
\sum\limits_{\xi=1}^{2k_1}
\Bigl(
\lambda_{\xi}^{j}\,h_{0\xi}+
\sum\limits_{q=1}^{\varepsilon_{\xi}}
\mu_{q\xi}^{j\zeta}\,h_{q\xi}\Bigr) -
\mu_{l,(k_1+\gamma)}^{j\zeta}h_{l,(k_1+\gamma)} +
\sum\limits_{\theta=1}^{k_2}
\Bigl(
\lambda_{\theta}^{j}\,h_{0\theta} +
\sum\limits_{q=1}^{\varepsilon_{\theta}}
\mu_{q\theta}^{j\zeta}\,h_{q\theta}\Bigr) \!= 0,
\, j=1,\ldots, m.
$
\
Here $\nu^{0\xi}={\stackrel{*}{\nu}}{}^{\,0\xi}\, +\widetilde{\nu}{}^{\,0\xi}\,i, \
\nu^{0,(k_1+\xi)}=\overline{\nu^{0\xi}}$ are complex common eigenvectors of the matrices $A_j$ corresponding to the eigenvalues $\lambda_{\xi}^{j}={\stackrel{*}{\lambda}}{}_{\xi}^{j} +\widetilde{\lambda}{}_{\xi}^{j}\,i,\
\lambda_{k_1+\xi}^{j}=\overline{\lambda_{\xi}^{j}},\ j=1,\ldots, m,\ \xi=1,\ldots, k_1,$ respectively; $\nu^{0\theta}$ are real common eigenvectors of the matrices $A_j$ corresponding to the eigenvalues $\lambda_{\theta}^{j},\ j=1,\ldots,m,\ \theta=1,\ldots, k_2,$ respectively; the functions $v_{q\xi}^{\,\zeta}={\stackrel{*}{v}}{}_{q\xi}^{\,\zeta} +\widetilde{v}{}_{q\xi}^{\,\,\zeta}\,i,\
v_{q\theta}^{\,\zeta}$ is the solution to the system (2.17); the numbers\
$
{\mu}_{q\xi}^{j\zeta}={\frak p}_{j}\,v_{q\xi}^{\,\zeta}(x),
\ \ \ \
{\stackrel{*}{\mu}}{}_{q\xi}^{\,j\zeta} =\, {\rm Re}\,{\mu}_{q\xi}^{j\zeta},
\ \ \
\widetilde{\mu}{}_{q\xi}^{\,j\zeta} =\,{\rm Im}\,{\mu}_{q\xi}^{j\zeta},
\ \ \
q=1,\ldots,\varepsilon_\xi,
\ \
\xi=1,\ldots, 2k_1,
\hfill
$\
$
\mu_{q\theta}^{j\zeta} ={\frak p}_{j}\,v_{q\theta}^{\,\zeta}(x),
\ \ \
q=1,\ldots,\varepsilon_\theta,
\ \
\theta=1,\ldots, k_2,
\ \ \ j=1,\ldots,m;
\hfill
$\
the numbers $\varepsilon_{\xi},\ \varepsilon_{\theta}$ such that $
2\sum\limits_{\xi=1}^{k_1} \varepsilon_{\xi} +
\sum\limits_{\theta=1}^{k_2} \varepsilon_{\theta} =m-2k_{1}-k_{2}+2
$ with $2k_1+k_2 {\leqslant}r, \ \varepsilon_{\xi}{\leqslant}s_{\xi}-1,$ $\xi=1,\ldots, k_1,\ \varepsilon_{\theta}{\leqslant}s_{\theta}-1, \ \theta=1,\ldots, k_2,$ where $k_1\!$ is a number of complex common eigenvectors [(]{}this set hasn’t complex conjugate vectors[)]{} of the matrices $A_j$ and $k_2$ is a number of real common eigenvectors of the matrices $A_j, \ j=1,\ldots, m.$
In particular, the completely solvable system of total differential equations\
$
dx_1 = (3,{}-4,4,1,0,2) x\,dt_1 + (0,{}-4,2,1,{}-1,1) x\,dt_2,
\hfill
$\
$
dx_2 = ({}-1,3,{}-3,0,{}-2,{}-3) x\,dt_1 + (1,3,0,0,1,{}-1) x\,dt_2,
\hfill
$\
$
dx_3 = ({}-3,5,{}-5,{}-1,{}-2,{}-4) x\,dt_1 +
(0,6,{}-2,{}-1,2,{}-1) x\,dt_2,
\hfill (2.32)
$\
$
dx_4 = (3,{}-6,4,4,{}-1,5,) x\,dt_1 + (2,{}-6,2,3,{}-4,2) x\,dt_2,
\hfill
$\
$
dx_5 = (5,{}-5,8,3,3,6) x\,dt_1 + (1,{}-6,3,2,{}-2,2) x\,dt_2,
\hfill
$\
$
dx_6 =
({}-2,5,{}-4,{}-3,1,{}-2) x\,dt_1 + ({}-2,4,{}-3,{}-3,2,{}-2) x\,dt_2
\hfill
$\
has the eigenvalue $\lambda_{1}^{1}\!=\!1+2i$ with elementary divisor $\!(\lambda^1\!-1-2i)^3\!$ corresponding to the eigenvector $\nu^{01}\!=\!(1, 0, 1+i, 1, i, 1)$ and to the generalized eigenvectors $\!\nu^{11}\!=\!(1, 1+i, 0, 0, i, i),$ $\nu^{21}\!=\!(2+2i, 0, 2+2i, 0, 2i, 2i).\!$ Autonomous first integrals of the system (2.32) are the functions\
$
F_1\colon x\to
P(x)\exp\bigl({} -\varphi(x)- \widetilde{v}{}_{11}^{\,1}(x)\bigr)$ for all $x\in {\mathscr X},
\hfill (2.33)
$\
$
F_2\colon x\to
P(x)\exp\bigl({}-2\varphi(x) + 2\,{\stackrel{*}{v}}{}_{11}^{\,1}(x)\bigr)$ for all $x\in {\mathscr X},
\hfill (2.34)
$\
$
F_3\colon x\to
P^2(x)\exp\big({}-2\varphi(x)- \widetilde{v}{}_{21}^{\,1}(x)\big)$ for all $x\in {\mathscr X},
\hfill (2.35)
$\
and\
$
F_4\colon x\to {\stackrel{*}{v}}{}_{21}^{\,1}(x)$ for all $x\in {\mathscr X},
\quad {\mathscr X}\subset \{ x\colon x_1+x_3+x_4+x_6\ne 0\},
\hfill (2.36)
$\
where the functions $\widetilde{v}{}_{11}^{\,1},\ {\stackrel{*}{v}}{}_{11}^{\,1},\
\widetilde{v}{}_{21}^{\,1},\ {\stackrel{*}{v}}{}_{21}^{\,1},\ P,$ and $\varphi$ are given by (2.22), (2.23), and (2.27).
[**Remark 2.1**]{} Consider the completely solvable system of equations in total differentials\
$
dx_1 = (2x_1+x_3)\,dt_1 +(2x_1+3x_2+3x_3)\,dt_2,
\quad
dx_2 = (x_1+x_2+x_3)\,dt_1 +2x_2\,dt_2,
\hfill
$\
(2.37)\
$
dx_3 = {}-x_1\,dt_1 -(3x_2+x_3)\,dt_2,
\quad
dx_4 = x_4\,dt_1 + (x_2+x_3-x_4)\,dt_2.
\hfill
$\
The matrices $A_1$ and $A_2$ of (2.37) have the elementary divisors $(\lambda^1-1)^2,\ \lambda^1-1,\ \lambda^1-1,$ and $\lambda^2-2,\ \lambda^2-2, \ (\lambda^2+1)^2,$ respectively. Using the common eigenvector $\nu^{01}=(1,0,1,0),$ generalized eigenvector $\nu^{11}=(0,1,0,0)$ of $A_1$ and the common eigenvector $\nu^{03}=(0,1,1,0),$ generalized eigenvector $\nu^{13}=(0,0,0,1)$ of the matrix $A_2,$ we can build the functions\
$
v_{11}^{1}\colon x\to \dfrac{x_2}{x_1+x_3}\,,
\quad
v_{13}^{2}\colon x\to \dfrac{x_4}{x_2+x_3}$ for all $x\in {\mathscr X},
\ \ {\mathscr X}\subset \{ x\colon x_1+x_3\ne 0, x_2+x_3\ne 0\}.
\hfill
$\
The system (2.37) has the autonomous first integrals on a domain ${\mathscr X}\colon$\
$
F_1\colon (t,x)\to (x_1+x_3)(x_2+x_3)^2\exp ({}-3v_{11}^{1}(x)),
\quad
F_2\colon (t,x)\to \dfrac{x_2+x_3}{x_1+x_3}\,\exp (3v_{13}^{2}(x)).
\hfill
$\
[**2.1.3. Nonautonomous first integrals**]{}.\
. [ *Suppose $\nu$ is a real common eigenvector of the matrices $A_{j}$ corresponding to the eigenvalues $\lambda^{j},\ j=1,\ldots, m,$ respectively. Then the scalar function\
$
\displaystyle
F\colon (t,x)\to
(\nu x) \exp\biggl({}-\sum\limits_{j=1}^{m}\lambda^{j}\,t_{j}\biggr)$ for all $(t,x)\in {\mathbb R}^{n+m}
\hfill
$\
is a first integral of the linear system of equations in total differentials [(2.1)]{}.* ]{}
Consider the system (2.3). Using the eigenvalues $\lambda_1^1={}-2,\ \lambda_1^2=1$ corresponding to the common eigenvector $\nu^1=(0,-1,1,1)$ and the eigenvalues $\lambda_2^1=0,\ \lambda_2^2={}-1$ corresponding to the eigenvector $\nu^2=(1,0,0,0),$ we can build the first integrals of the system (2.3):\
$
F_1\colon (t,x)\to ({}-x_2+x_3+x_4)\exp( 2t_1-t_2),
\ \ \
F_2\colon (t,x)\to x_1\exp t_2$ for all $(t,x)\in {\mathbb R}^6.
\hfill
$\
The functions $F_1,\ F_2,$ (2.4), and (2.5) are the general integral of the system (2.3).
[**Corollary 2.2**]{}. [ *Let $\nu={\stackrel{*}{\nu}}+\widetilde{\nu}\,i\
({\stackrel{*}{\nu}}={\rm Re}\,\nu,\ \widetilde{\nu}={\rm Im}\,\nu)$ be a common complex eigenvector of the matrices $A_j$ corresponding to the eigenvalues $\lambda^j={\stackrel{*}{\lambda}}{}^{j}+\widetilde{\lambda}{}^{j}\,i\
({\stackrel{*}{\lambda}}{}^{j}={\rm Re}\,\lambda^j,\ \widetilde{\lambda}{}^j={\rm Im}\,\lambda^j),$ $j=1,\ldots, m,$ respectively. Then the system [(2.1)]{} has the first integrals\
$
\displaystyle
F_1\colon (t,x)\to
\bigl( ({\stackrel{*}{\nu}}x)^{2} + (\widetilde{\nu}x)^{2}\bigr)
\exp\biggl({}-2\sum\limits_{j=1}^{m}
{\stackrel{*}{\lambda}}{}^{j}\, t_{j}\biggr)$ for all $(t,x)\in {\mathbb R}^{n+m}
\hfill
$\
and\
$
\displaystyle
F_2\colon (t,x)\to\
{\rm arctg}\,\dfrac{\widetilde{\nu}x}{{\stackrel{*}{\nu}x}}\ - \
\sum\limits_{j=1}^{m} \widetilde{\lambda}{}^{j}\, t_{j}$ for all $(t,x)\in {\mathscr D},
\quad {\mathscr D}\subset {\mathbb R}^{n+m}.
\hfill
$* ]{}\
For example, the completely solvable system (2.8) has the eigenvector $\nu^{1}=(1, i, 0)$ corresponding to the eigenvalues $\lambda_1^1=1, \lambda_1^2={}-i$ and the first integrals (by Corollary 2.2)\
$
F_1\colon (t,x)\to (x_1^2+x_2^2)\exp({}-2t_1)$ for all $(t,x)\in {\mathbb R}^5,
\hfill
$\
$
F_2\colon (t,x)\to \
{\rm arctg}\,\dfrac{x_2}{x_1} +t_2$ for all $(t,x)\in {\mathbb R}^2\times {\mathscr X},
\quad {\mathscr X}\subset \{ x\colon x_1\ne 0,\ x_3\ne 0\}.
\hfill
$\
The functionally independent first integrals (2.9), $F_1,$ and $F_2$ are the general integral on a domain ${\mathbb R}^2\times {\mathscr X}$ for the system of equations in total differentials (2.8).
The completely solvable system (2.12) has the eigenvector $\nu^{1}\!=\! (0, -i, 0, 1)$ corresponding to the eigenvalues $\lambda_1^1={}-1$ and $\lambda_1^2={}-i.$ The first integrals (2.13) and (by Corollary 2.2)\
$
F_1\colon (t,x)\to (x_2^2+x_4^2)\exp(2t_1)$ for all $(t,x)\in {\mathbb R}^6,
\hfill
$\
$
F_2\colon (t,x)\to \
{\rm arctg}\,\dfrac{x_2}{x_4} - t_2$ for all $(t,x)\in {\mathbb R}^2\times {\mathscr X},
\ \ \
{\mathscr X}\subset \{x\colon x_1x_4-x_2x_3\ne 0,\, x_4\ne 0\},
\hfill
$\
are the general integral on a domain ${\mathbb R}^2\times {\mathscr X}$ of the linear system (2.12).
[**Theorem 2.6**]{}. [*Let $\nu^{\,0}$ be a real common eigenvector of the matrices $A_j$ corresponding to the eigenvalues $\lambda^j, \ j=1,\ldots,m,$ respectively. Let $\nu^{\,\theta},\ \theta =1,\ldots, s-1$ be real generalized eigenvectors of the matrix $A_\zeta$ corresponding to the eigenvalue $\lambda^{\zeta}$ with elementary divisor of multiplicity $s{\geqslant}2.$ Then the completely solvable system [(2.1)]{} has the first integrals\
$
\displaystyle
F_q\colon (t, x)\to
v_{q}^{\zeta}(x)\ -\
\sum\limits_{j=1}^{m}\mu_{q}^{j\zeta}\, t_{j}$ for all $(t,x)\in {\mathbb R}^{m}\times {\mathscr X},
\ \ \ q=1,\ldots, s-1,
\hfill
$\
where the set of functions $v_{q}^{\zeta}\colon {\mathscr X}\to {\mathbb R}$ is the solution to the system [(2.17)]{}, the numbers $\mu_{q}^{j\zeta} = {\frak p}_{j}\,v_{q}^{\zeta}(x),\
q=1,\ldots, s-1,\ j=1,\ldots, m,$ and a domain ${\mathscr X}\subset \{x\colon \nu^0x\ne 0\}.$* ]{}
The completely solvable system (2.18) has $\mu_{1}^{11}=1, \ \mu_{3}^{11}=0,\ \mu_{1}^{21}={}-1, \ \mu_{3}^{21}=6,$ and the first integrals (by Theorem 2.6)\
$
F_1\colon (t,x)\to v_{11}^{\,1}(x) -t_1+t_2$ for all $(t,x)\in {\mathbb R}^2\times {\mathscr X},
\hfill
$\
$
F_2\colon (t,x)\to v_{31}^{\,1}(x) -6t_2$ for all $(t,x)\in {\mathbb R}^2\times {\mathscr X},
\quad {\mathscr X}\subset \{x\colon x_1-x_2+x_3\ne 0\},
\hfill
$\
where the functions $v_{11}^{1},\ v_{31}^{1}$ are given by (2.19). The functionally independent first integrals (2.20), $F_1,$ and $F_2$ are the general integral on a domain ${\mathbb R}^2\times {\mathscr X}$ of the system (2.18).
[**Corollary 2.3**]{}. [*Let $\nu^{\,0}$ be a common eigenvector of the matrices $A_j$ corresponding to the eigenvalues $\lambda^j, \ j=1,\ldots,m.$ Suppose $\nu^{\,\theta},\ \theta =1,\ldots, s-1$ are generalized eigenvectors of $A_\zeta$ corresponding to the complex eigenvalue $\lambda^{\zeta}\ ({\rm Im}\,\lambda^{\zeta}\ne 0)$ with elementary divisor of multiplicity $s{\geqslant}2.$ Then the completely solvable system [(2.1)]{} has the first integrals\
$
\displaystyle
F_{1 q}\colon (t,x)\to
{\stackrel{*}{v}}{}_{q}^{\zeta}(x) -
\sum\limits_{j=1}^{m}\,
{\stackrel{*}{\mu}}\!{}_{q}^{\,j\zeta}\, t_{j}$ for all $(t,x)\in {\mathbb R}^{m}\times {\mathscr X},
\quad q=1,\ldots, s-1,
\hfill
$\
and\
$
\displaystyle
F_{2 q}\colon (t, x)\to
\widetilde{v}{}_{q}^{\,\zeta}(x) -
\sum\limits_{j=1}^{m}\,
\widetilde{\mu}{}_{q}^{\,j\zeta}\, t_{j}$ for all $(t,x)\in {\mathbb R}^{m}\times {\mathscr X},
\quad q=1,\ldots, s-1,
\ \ \ \ {\mathscr X}\subset {\mathbb R}^{n},
\hfill
$\
where the set of functions $v_{q}^{\zeta}\colon x\to {\stackrel{*}{v}}{}_{q}^{\,\zeta}(x) + \widetilde{v}{}_{q}^{\,\zeta}(x)\,i$ for all $x\in {\mathscr X}$ is the solution to the system [(2.17)]{}, the numbers ${\stackrel{*}{\mu}}{}_{q}^{\,j\zeta}= {\frak p}_j\,{\stackrel{*}{v}}{}_{q}^{\,\zeta}(x), \
\widetilde{\mu}{}_{q}^{\,j\zeta}={\frak p}_j\,\widetilde{v}{}_{q}^{\,\zeta}(x), \
j=1,\ldots, m,\ q=1,\ldots, s-1.$* ]{}
The completely solvable system (2.21) has the first integrals (by Corollary 2.3)\
$
\displaystyle
F_{11}\colon (t,x)\to
{\stackrel{*}{v}}{}_{11}^{\,1}(x) -t_1-t_2,
\ \ \,
F_{21}\colon (t,x)\to
\widetilde{v}{}_{11}^{\,1}(x) +t_2-t_3$ for all $(t,x)\in {\mathbb R}^3\times {\mathscr X},
\hfill
$\
and\
$
F_{12}\colon (t,x)\to
{\stackrel{*}{v}}{}_{21}^{\,1}(x) -2t_3$ for all $(t,x)\in {\mathbb R}^3\times {\mathscr X},
\quad
{\mathscr X}\subset \{x\colon x_1+x_3+x_4+x_6\ne 0\},
\hfill
$\
where ${\stackrel{*}{v}}{}_{11}^{\,1},\ \widetilde{v}{}_{11}^{\,1},$ and ${\stackrel{*}{v}}{}_{21}^{\,1}$ are given by (2.22). The general integral for the linear system (2.21) is the functionally independent first integrals (2.24), (2.25), (2.26), $F_{11},\ F_{21},$ and $F_{12}.$
For example, the system (2.28) has the eigenvector $\nu^{01}=(1,1+i,0,0,i,i)$ corresponding to the eigenvalues $\lambda_1^1=1+i,\ \lambda_1^2={}-1,\ \lambda_1^3=2,\ \lambda_1^4=-1+2i$ and the first integrals\
$
F_1\colon (t,x)\to
\bigl( (x_1+x_2)^2+(x_2+x_5+x_6)^2\bigr)
\exp({}-2t_1+2t_2-4t_3+2t_4)$ (by Corollary 2.2),\
$
F_2\colon (t,x)\to\
{\rm arctg}\,\dfrac{x_2+x_5+x_6}{x_1+x_2} -t_1-2t_4$ for all $(t,x)\in{{\mathbb R}}^{4}\times {\mathscr X},
\hfill
$\
where ${\mathscr X}\subset \{x\colon x_1+x_2\ne 0,\ x_1+x_3+x_4+x_6\ne 0\}\subset {\mathbb R}^6.$ Using Corollary 2.3, we get\
$
F_3\colon (t,x)\to
{\stackrel{*}{v}}{}_{11}^{\,1}(x)-t_1+t_3-t_4$ for all $(t,x)\in {\mathbb R}^{4}\times {\mathscr X}
\hfill
$\
and\
$
F_4\colon (t,x)\to
\widetilde{v}{}_{11}^{\,1}(x)-t_2-t_4$ for all $(t,x)\in {\mathbb R}^{4}\times {\mathscr X},
\hfill
$\
where ${\stackrel{*}{v}}{}_{11}^{\,1}$ and $\widetilde{v}{}_{11}^{\,1}$ are given by (2.29). The first integrals (2.30), (2.31), $F_1,\ldots, F_4$ are the general integral on a domain ${\mathbb R}^4\times {\mathscr X}$ for the completely solvable linear system (2.28).
Consider the completely solvable linear system of equations in total differentials (2.32). According to Corollary 2.3, we can construct the first integrals\
$
F_{11}\colon (t,x)\to
{\stackrel{*}{v}}{}_{11}^{\,1}(x)-t_1-t_2,
\ \ \
F_{21}\colon (t,x)\to
\widetilde{v}{}_{11}^{\,1}(x)+t_2$ for all $(t,x)\in {\mathbb R}^2\times {\mathscr X},
\hfill
$\
where the functions ${\stackrel{*}{v}}{}_{11}^{\,1}$ and $\widetilde{v}{}_{11}^{\,1}$ are given by (2.22), a domain ${\mathscr X}\subset \{x\colon x_1+x_3+x_4+x_6\ne 0\}.$
The functionally independent first integrals (2.33), (2.34), (2.35), (2.36), $F_{11},$ and $F_{21}$ are the general integral on a domain ${\mathbb R}^2\times {\mathscr X}$ for the completely solvable system (2.32).
[**Remark 2.2**]{}. The system of equations in total differentials\
$
dx_1 = {}-(x_1+x_2+3x_3)\,dt_1 -3(x_1+2x_3)\,dt_2,
\hfill (2.38)
$\
$
dx_2 = x_2\,dt_1 -2x_2\,dt_2,
\quad
dx_3 = (x_2+2x_3)\,dt_1 + 3x_3\,dt_2
\hfill
$\
is not completely solvable (this system has the defect \[5, p. 54\] of order 1):\
$
[{\frak p}_1(x),{\frak p}_2(x)]=[{}-(x_1+x_2+3x_3)\,\partial_{x_1}+x_2\,\partial_{x_2}+(x_2+2x_3)\,\partial_{x_3},
\hfill
$\
$
{}-3(x_1+2x_3)\,\partial_{x_1}-2x_2\,\partial_{x_2}+3x_3\,\partial_{x_3}]=
{}-5x_2\partial_{x_1}+5x_2\partial_{x_3}\equiv {\frak p}_3(x)$ for all $x\in {\mathbb R}^3,
\hfill
$\
$
[{\frak p}_1(x),{\frak p}_3(x)]={}-{\frak p}_3(x),
\quad
[{\frak p}_2(x),{\frak p}_3(x)]={}-5\,{\frak p}_3(x)$ for all $x\in {\mathbb R}^3.
\hfill
$\
The system (2.38) has two common eigenvectors $\nu^{1}\!=\!(1,0,1)$ and $\nu^{2}\!=\!(0,1,0)$ corresponding to the eigenvalues $\lambda_1^1\!=\!{}-1,\,
\lambda_1^2=\!{}-3,$ and $\lambda_2^1=1,\ \lambda_2^2=\!{}-2.$ The first integrals\
$
F_{1}\colon (t,x)\to (x_1+x_3)\exp(t_1+3t_2),
\ \
F_{2}\colon (t,x)\to x_2\exp(2t_2-t_1)$ for all $(t,x)\in {\mathbb R}^5
\hfill
$\
are a general integral of the linear system of equations in total differentials (2.38).\
\
Consider now a nonhomogeneous system of equations in total differentials\
$
\displaystyle
dx =\sum\limits_{j=1}^m(B_j\,x+f_j(t))\,dt_j
$ (2.39)\
corresponding to the linear homogeneous system (2.1), where the matrices $B_1,\ldots, B_m$ are transpose of $A_1,\ldots, A_m,$ respectively, and the vector functions\
$
f_j\colon t\to {\rm colon}(f_{j1}(t),\ldots, f_{jn}(t))$ for all $t\in {\mathcal T},
\quad
j=1,\ldots, m
\hfill
$\
are continuously differentiable on a domain ${\mathcal T}\subset{{\mathbb R}}^{m}.$
The Frobenius conditions for the total solvability \[4, p. 44\] of system (2.39) are (2.2) and\
$
\partial_{t_\zeta}f_j(t)-B_{\zeta}f_j(t)=
\partial_{t_j}f_\zeta(t)-B_{j}f_\zeta(t)$ for all $t\in {\mathcal T},
\quad
j,\zeta=1,\ldots, m.
\hfill (2.40)
$\
. [*Let $\nu$ be a real common eigenvector of the matrices $A_{j}$ corresponding to the eigenvalues $\lambda^{j},\ j=1,\ldots, m,$ respectively. Then the completely solvable system of equations in total differentials [(2.39)]{} has the first integral\
$
\displaystyle
F\colon (t,x)\to
\nu x\!\cdot\!\varphi(t)-
\int\sum\limits_{j=1}^m \nu f_j(t)\!\cdot\! \varphi(t)\,dt_j$ for all $(t,x)\in \widetilde{\mathcal T}\times{{\mathbb R}}^n,
$\
where $\widetilde{\mathcal T}$ is a simply connected domain from ${\mathcal T},$ the function\
$
\displaystyle
\varphi\colon t\to
\exp\biggl({}-\sum\limits_{j=1}^m \lambda^j\,t_j\biggr)$ for all $t\in {\mathcal T}.
\hfill
$\
*]{} . Let us introduce the 1-form\
$
\displaystyle
\omega\colon t\to
\sum\limits_{j=1}^m \nu f_j(t)\!\cdot\! \varphi(t)\,dt_j$ for all $t\in {\mathcal T},
\quad {\mathcal T}\subset {\mathbb R}^m.
\hfill
$\
The external differential has the form\
$
\displaystyle
d\,\omega(t)=\varphi(t)\sum\limits_{j=1}^m \sum\limits_{\zeta=1}^m
\bigl(\partial_{t_\zeta} \nu f_j(t) -
\lambda^{\zeta}\cdot\nu f_j(t)\bigr)\,dt_{\zeta}\wedge dt_j$ for all $t\in {\mathcal T}.
\hfill
$\
Using (2.40) and\
$
dt_j\wedge dt_j=0,\ \ \ dt_\zeta\wedge dt_j={}-dt_j\wedge dt_\zeta,
\quad
\zeta, j=1,\ldots, m,
\hfill
$\
we get the differential 1-form $\omega$ is closed on ${\mathcal T}.$ By the Poincaré theorem (see \[5, p. 14\]), it follows that $\omega$ is exact in a simply connected domain $\widetilde{\mathcal T}\subset {\mathcal T}.$
Taking into account Lemma 2.1 and (2.15), we obtain\
$
\displaystyle
{\frak P}_j F(t,x)=
{}-\lambda^j\nu x\cdot\varphi(t)
-\nu f_j(t)\varphi(t)+{\frak p}_j\nu x\cdot\varphi(t)
+\nu f_j(t)\varphi(t)=0
\hfill
$\
for all $(t,x)\in \widetilde{\mathcal T}\times{{\mathbb R}}^n,
\quad
j=1,\ldots, m,
\hfill
$\
where the linear differential operators\
$
\displaystyle
{\frak P}_j(t,x)=
\partial_{t_j}+ {\frak p}_j(x) + f_j(t)\,\partial_{x}$ for all $(t,x)\in {\mathcal T}\times {\mathbb R}^{n},
\ \ \
j=1,\ldots, m.
\hfill
$\
Therefore the function (2.41) is a first integral of the completely solvable system (2.39).
------------------------------------------------------------------------
[**Corollary 2.4**]{}. [ *Let $\nu={\stackrel{*}{\nu}}+\widetilde{\nu}\,i\
({\stackrel{*}{\nu}}={\rm Re}\,\nu,\ \widetilde{\nu}={\rm Im}\,\nu)$ be a common complex eigenvector of the matrices $A_j$ corresponding to the eigenvalues $\lambda^j={\stackrel{*}{\lambda}}{}^{j}+\widetilde{\lambda}{}^{j}\,i\
({\stackrel{*}{\lambda}}{}^{j}={\rm Re}\,\lambda^j,\ \widetilde{\lambda}{}^j={\rm Im}\,\lambda^j),$ $j\!=\!1,\ldots,\! m,\!$ respectively. Then the completely solvable system [(2.39)]{} has the first integrals\
$
\displaystyle
F_\tau\colon (t,x)\to \alpha_\tau(t, x)-
\int \sum\limits_{j=1}^m\alpha_\tau(t, f_j(t))\,dt_j$ for all $(t,x)\in \widetilde{\mathcal T}\times{{\mathbb R}}^n,
\ \ \ \tau=1,2,
\hfill
$\
where the scalar functions of the vector arguments\
$
\displaystyle
\alpha_1(t,x)=
\biggl({\stackrel{*}{\nu}}x\cdot
\cos\sum\limits_{j=1}^{m}\widetilde{\lambda}{}^j\;\!t_j +
\widetilde{\nu}x\cdot\sin\sum\limits_{j=1}^{m}\widetilde{\lambda}{}^j\;\!t_j\biggr)\!
\cdot\exp\biggl(-\sum\limits_{j=1}^{m}{\stackrel{*}{\lambda}}{}^j\;\!t_j\biggr)$ for all $(t,x)\in {\mathcal T}\times{{\mathbb R}}^n,
\hfill
$\
$
\displaystyle
\alpha_2(t,x)=
\biggl(\widetilde{\nu}x\cdot
\cos\sum\limits_{j=1}^{m}\widetilde{\lambda}{}^j\;\!t_j -
{\stackrel{*}{\nu}}x\cdot\sin\sum\limits_{j=1}^{m}\widetilde{\lambda}{}^j\;\!t_j\biggr)\!
\cdot\exp\biggl({}-\sum\limits_{j=1}^{m}{\stackrel{*}{\lambda}}{}^j\;\!t_j\biggr)$ for all $(t,x)\in {\mathcal T}\times{{\mathbb R}}^n.
\hfill
$\
*]{} For example, the completely solvable system of total differential equations\
$
dx_1 = x_1\,dt_1 + (x_2+1)\,dt_2,
\quad \
dx_2 = (x_2+1)\,dt_1 +({}- x_1+e^{t_1+at_2})\,dt_2,
\hfill
$\
$
dx_3 = (x_3+t_2-t_1)\,dt_1 +({}- x_3+t_1-t_2)\,dt_2
\hfill
$\
corresponding to the linear homogeneous system (2.8). This system has the first integrals\
$
F_1\colon (t,x)\to
(x_1\cos t_2-x_2\sin t_2-\sin t_2)e^{{}-t_1}+\dfrac{a\sin t_2-\cos t_2}{a^2+1}\ e^{at_2}$ (by Corollary 2.4),\
$
F_2\colon (t,x)\to
(x_1\sin t_2+x_2\cos t_2+\cos t_2)e^{{}-t_1}-\dfrac{a\cos t_2+\sin t_2}{a^2+1}\ e^{at_2}$ (by Corollary 2.4),\
$
F_3\colon (t,x)\to
(x_3+t_2-t_1-1)e^{t_2-t_1}$ for all $(t,x)\in {\mathbb R}^5$ (by Theorem 2.7),\
where $a$ is some real number.
[**Theorem 2.8**]{}.
[*Proof*]{}. The proof is by induction on $\theta.$
The case $\theta=0$ was considered in Theorem 2.7.
Suppose $\theta=1.$ Using Lemma 2.3 with condition (2.16), we obtain\
$
\displaystyle
{\frak P}_j F_1(t,x)= {\frak P}_j \biggl(\!\nu^{1l}x\cdot \varphi(t)-
\int\sum\limits_{\xi=1}^m \mu_{1l}^{\xi\zeta}\,dt_{\xi}\cdot F_{0}(t,x) -
\int\sum\limits_{\xi=1}^m
\bigl(\nu^{1l}f_{\xi}(t)\cdot \varphi(t)+\mu_{1l}^{\xi\zeta}\,C_{0}(t)\bigr)\,dt_{\xi}\!\biggr)=
\hfill
$\
$
\displaystyle
=\mu_{1l}^{j\zeta}\bigl(\nu^{0l}x\cdot \varphi(t)-C_{0}(t)-F_{0}(t,x)\bigr)=0$ for all $(t,x)\in \widetilde{\mathcal T}\times {\mathscr X},
\ \ \ j=1,\ldots, m.
\hfill
$\
Therefore $F_1\colon \widetilde{\mathcal T}\times {\mathscr X}\to {\mathbb R}$ is a first integral of the completely solvable system (2.39).
Suppose that the assertion of the theorem is valid for $\theta=\varepsilon-1,$ i.e., the scalar functions $F_{\theta}\colon \widetilde{\mathcal T}\times {\mathscr X}\to {\mathbb R},\ \theta=1,\ldots,\varepsilon-1$ are first integrals of the system (2.39). Then\
$
{\frak P}_j F_{\varepsilon}(t,x)=
{\frak P}_j \biggl( \nu^{\,\varepsilon l}x\cdot \varphi(t) -
{\displaystyle \sum\limits_{\tau=1}^{\varepsilon} }
K_{\tau-1}^{\varepsilon}(t)\cdot F_{\tau-1}(t,x)-C_{\varepsilon}(t)\biggr)=
\hfill
$\
$
={}-\mu_{0l}^{j\zeta}\,\nu^{\,\varepsilon l}x\cdot \varphi(t) +
{\displaystyle \sum\limits_{\tau=0}^{\varepsilon} }
\binom{\varepsilon}{\tau}\,\mu_{\tau l}^{j\zeta}\, \nu^{\,\varepsilon-\tau,l}x\cdot \varphi(t)
+ \nu^{\,\varepsilon l}f_{j}(t)\cdot \varphi(t)\ -
\hfill
$\
$
-\ {\displaystyle \sum\limits_{\tau=1}^{\varepsilon}}\biggl(\!\!
\binom{\varepsilon}{\tau-1}\,\mu_{\varepsilon-\tau+1,l}^{\, j\zeta}+
{\displaystyle \sum\limits_{\delta=1}^{\varepsilon-\tau}}\binom{\varepsilon}{\delta}\,
\mu_{\delta l}^{j\zeta}\, K_{\tau-1}^{\varepsilon-\delta}(t)\!\!\biggr) F_{\tau-1}(t,x) -
\biggl(\!\nu^{\,\varepsilon l}f_j(t)\cdot \varphi(t) +
{\displaystyle \sum\limits_{\tau=1}^{\varepsilon} }
\binom{\varepsilon}{\tau}\,\mu_{\tau l}^{j\zeta}\, C_{\varepsilon-\tau}(t)\!\!\biggr)\!=
\hfill
$\
$
={\displaystyle \sum\limits_{\tau=1}^{\varepsilon} }
\binom{\varepsilon}{\tau}\,\mu_{\tau l}^{j\zeta}
\biggl(\!
\Bigl( \nu^{\,\varepsilon-\tau, l}x\cdot \varphi(t) -
{\displaystyle \sum\limits_{\delta=1}^{\varepsilon-\tau} }
K_{\delta-1}^{\varepsilon-\tau}(t)\cdot F_{\delta-1}(t,x) -
C_{\varepsilon-\tau}(t)\Bigr)
-F_{\varepsilon-\tau}(t,x)\biggr)=0
\hfill
$\
for all $(t,x)\in \widetilde{\mathcal T}\times {\mathscr X},
\quad
j=1,\ldots, m.
\hfill
$\
Therefore $F_{\varepsilon}\colon \widetilde{\mathcal T}\times {\mathscr X}\to {\mathbb R}$ is a first integral of the linear system (2.39).
Thus the functions (2.42) are first integrals of the completely solvable system (2.39).
------------------------------------------------------------------------
[**Remark 2.3**]{}. In complex case from Theorem 2.8, we get the following real-valued first integrals of the completely solvable system of equations in total differentials (2.39):\
$
F_{\theta}^1\colon (t,x)\to {\rm Re}\,F_{\theta}(t,x),
\ \,
F_{\theta}^2\colon (t,x)\to {\rm Im}\,F_{\theta}(t,x)$ for all $(t,x)\in \widetilde{\mathcal T}\times {\mathscr X},
\ \theta=0,\ldots, s_{l}-1.
\hfill
$\
Moreover, we have\
$
F_{\theta}^1\colon (t,x)\to \alpha_{{}_{\scriptstyle \theta}}(t,x)-
{\displaystyle \sum\limits_{\tau=1}^{\theta}}
\Bigl({\rm Re}\,K_{\tau-1}^{\theta}(t)\cdot F_{\tau-1}^1(t,x) -
{\rm Im}\,K_{\tau-1}^{\theta}(t)\cdot F_{\tau-1}^2(t,x)\Bigr)
-{\rm Re}\,C_{\theta}(t),
\hfill
$\
$
F_{\theta}^2\colon (t,x)\to \beta_{{}_{\scriptstyle \theta}}(t,x)-
{\displaystyle \sum\limits_{\tau=1}^{\theta}}
\Bigl({\rm Re}\,K_{\tau-1}^{\theta}(t)\cdot F_{\tau-1}^2(t,x) +
{\rm Im}\,K_{\tau-1}^{\theta}(t)\cdot F_{\tau-1}^1(t,x)\Bigr)
-{\rm Im}\,C_{\theta}(t)
\hfill
$\
for all $(t,x)\in \widetilde{\mathcal T}\times {\mathscr X},
\quad
\theta=0,\ldots, s_{l}-1,
\hfill
$\
where the functions\
$
{\rm Re}\, K_{\tau-1}^{\theta}\colon t\to
{\displaystyle \int}
{\displaystyle \sum\limits_{j=1}^m}
\biggl(\binom{\theta}{\tau-1}\,{\stackrel{*}{\mu}}{}_{\theta-\tau+1,l}^{\,j\zeta} +
{\displaystyle \sum\limits_{\delta=1}^{\theta-\tau}}\binom{\theta}{\delta}\,
\bigl(\,{\stackrel{*}{\mu}}{}_{\delta l}^{\,j\zeta}\, {\rm Re}\,K_{\tau-1}^{\theta-\delta}(t) -
\widetilde{\mu}{}_{\delta l}^{\,j\zeta}\, {\rm Im}\,K_{\tau-1}^{\theta-\delta}(t)\bigr)\biggr)\,dt_j\,,
\hfill
$\
$
{\rm Im}\, K_{\tau-1}^{\theta}\colon t\to
{\displaystyle \int}
{\displaystyle \sum\limits_{j=1}^m}
\biggl(\binom{\theta}{\tau-1}\,\widetilde{\mu}{}_{\theta-\tau+1, l}^{\,j\zeta} +
{\displaystyle \sum\limits_{\delta=1}^{\theta-\tau}}\binom{\theta}{\delta}\,
\bigl({\stackrel{*}{\mu}}{}_{\delta l}^{\,j\zeta}\, {\rm Im}\,K_{\tau-1}^{\theta-\delta}(t) +
\widetilde{\mu}{}_{\delta l}^{\,j\zeta}\, {\rm Re}\,K_{\tau-1}^{\theta-\delta}(t)\bigl)\biggr)\,dt_j
\hfill
$\
for all $t\in \widetilde{\mathcal T},
\quad
\tau=1,\ldots,\theta,
\ \ \ \theta=1,\ldots, s_{l}-1,
\hfill
$\
$
{\rm Re}\,C_{\theta}\colon t\to
{\displaystyle \int\sum\limits_{j=1}^m}
\biggl(\alpha_{{}_{\scriptstyle \theta}}(t, f_j(t)) +
{\displaystyle \sum\limits_{\tau=1}^{\theta}}\binom{\theta}{\tau}\,
\bigl(\,{\stackrel{*}{\mu}}{}_{\tau l}^{\,j\zeta}\, {\rm Re}\,C_{\theta-\tau}(t) -
\widetilde{\mu}_{\tau l}^{\,j\zeta}\, {\rm Im}\,C_{\theta-\tau}(t)\bigr)\biggr)\,dt_j\,,
\hfill
$\
$
{\rm Im}\,C_{\theta}\colon t\to
{\displaystyle \int\sum\limits_{j=1}^m}
\biggl(\beta_{{}_{\scriptstyle \theta}}(t, f_j(t)) +
{\displaystyle \sum\limits_{\tau=1}^{\theta}}\binom{\theta}{\tau}\,
\bigl(\,{\stackrel{*}{\mu}}{}_{\tau l}^{\,j\zeta}\, {\rm Im}\,C_{\theta-\tau}(t) +
\widetilde{\mu}{}_{\tau l}^{\,j\zeta}\, {\rm Re}\,C_{\theta-\tau}(t)\bigr)\biggr)\,dt_j
\hfill
$\
for all $t\in \widetilde{\mathcal T},
\quad
\theta=0,\ldots, s_{l}-1,
\ \ \ \widetilde{\mathcal T}\subset {\mathcal T},
\hfill
$\
the real numbers ${\stackrel{*}{\mu}}{}_{0l}^{\,j\zeta}={\rm Re}\,\lambda^j_l,\ \
\widetilde{\mu}{}_{0l}^{\,j\zeta}={\rm Im}\,\lambda^j_l,\ \
{\stackrel{*}{\mu}}{}_{\theta l}^{\,j\zeta}={\rm Re}\,\mu_{\theta l}^{\,j\zeta},\ \
\widetilde{\mu}{}_{\theta l}^{\,j\zeta}={\rm Im}\,\mu_{\theta l}^{\,j\zeta},\ j=1,\ldots, m,$ the vectors ${\stackrel{*}{\nu}}{}^{\,\theta l}={\rm Re}\,\nu^{\,\theta l},\
\widetilde{\nu}{}^{\,\theta l}={\rm Im}\,\nu^{\,\theta l},\ \theta=0,\ldots, s_{l}-1,$ and\
$
\displaystyle
\alpha_{{}_{\scriptstyle \theta}}(t,x) =
\biggl( {\stackrel{*}{\nu}}{}^{\,\theta l}x\cdot
\cos\sum\limits_{j=1}^{m}\widetilde{\mu}{}_{0 l}^{\,j\zeta}\;\!t_j +
\widetilde{\nu}{}^{\,\theta l}x\cdot\sin\sum\limits_{j=1}^{m}\widetilde{\mu}{}_{0l}^{\,j\zeta}\;\!t_j\biggr)\cdot
\exp\biggl({}-\sum\limits_{j=1}^{m}{\stackrel{*}{\mu}}{}_{0l}^{\,j\zeta}\;\!t_j\biggr),
\hfill
$\
$
\displaystyle
\beta_{{}_{\scriptstyle \theta}}(t,x)=
\biggl(\widetilde{\nu}{}^{\,\theta l}x\cdot
\cos\sum\limits_{j=1}^{m}\widetilde{\mu}{}_{0 l}^{\,j\zeta}(t)\;\!t_j -
{\stackrel{*}{\nu}}{}^{\,\theta l}x\cdot\sin\sum\limits_{j=1}^{m}\widetilde{\mu}{}_{0l}^{\,j\zeta}\;\!t_j\biggr)\cdot
\exp\biggl({}-\sum\limits_{j=1}^{m}{\stackrel{*}{\mu}}{}_{0l}^{\,j\zeta}\;\!t_j\biggr)
\hfill
$\
for all $(t,x)\in \widetilde{\mathcal T}\times {\mathbb R}^n,
\quad \theta=0,\ldots, s_{l}-1.
\hfill
$\
. Suppose the system (2.39) satisfies (2.2) and (2.40). Then 1-forms\
$
{\stackrel{*}{\omega}}{}_{\tau-1}^{\theta}\colon t\to
{\displaystyle \sum\limits_{j=1}^m}
\biggl(\binom{\theta}{\tau-1}\,\mu_{\theta-\tau+1,l}^{\,j\zeta}+
{\displaystyle \sum\limits_{\delta=1}^{\theta-\tau}}
\binom{\theta}{\delta}\,\mu_{\delta l}^{j\zeta}\cdot K_{\tau-1}^{\theta-\delta}(t)\biggr)\,dt_j,
\ \ \
\tau=1,\ldots, \theta,
\ \ \theta=1,\ldots, s_{l}-1,
\hfill
$\
$
\widetilde{\omega}_{\theta}\colon t\to
{\displaystyle \sum\limits_{j=1}^m}
\biggl(\nu^{\,\theta l}f_j(t)\cdot \varphi(t)+
{\displaystyle \sum\limits_{\tau=1}^{\theta}}
\binom{\theta}{\tau}\,\mu_{\tau l}^{j\zeta}\cdot C_{\theta-\tau}(t)\biggr)\,dt_j$ for all $t\in \widetilde{\mathcal T},
\quad
\theta=0,\ldots, s_{l}-1
\hfill
$\
are exact in a simply connected domain $\widetilde{\mathcal T}\subset {\mathcal T}.$
\
**3. First integrals of linear systems of ordinary differential equations**
\
\
Let us consider a linear autonomous homogeneous system of ordinary differential equations\
$
\dfrac{dx}{dt} = Ax,
$ (3.1)\
where $x = \mbox{colon}(x_{1},\ldots,x_{n})\in{\mathbb R}^n,$ the $A = \bigl\|a_{ij}\bigr\|$ is a real $n\times n$ matrix.
Let B be the transpose of the matrix $A.$
Using methods of Section 2 , we obtain the following statements.
[**Theorem 3.1**]{}. [*Let $\nu\in{\mathbb C}^n$ be an eigenvector of the matrix $B.$ Then the linear function $p\colon x\to \nu x$ for all $x\in {\mathbb R}^n$ is a partial integral of the system*]{} (3.1).\
[**Theorem 3.2**]{}. [ *Let $\nu^{1},\ \nu^{2}$ be real eigenvectors of the matrix $B$ corresponding to the eigenvalues $\lambda_{1}, \ \lambda_{2} \ (\lambda_{1}\ne\lambda_{2}),$ respectively. Then the system [(3.1)]{} has the first integral\
$
F\colon x\to \bigl|\nu^{1} x\bigr|^{h_1}\,
\bigl|\nu^{2}x{\bigr|}^{h_2}$ for all $x\in {\mathscr X},
\quad {\mathscr X}\subset {\rm D}(F),
\hfill
$\
where $h_1,\ h_2$ is a real solution to the equation $\lambda_1h_1+\lambda_2h_2=0$ with $|h_1|+|h_2|\ne 0.$* ]{}
[**Corollary 3.1**]{}. [*If $\nu$ is a real eigenvector of the matrix $B$ corresponding to the eigenvalue $\lambda=0,$ then the linear function $
F\colon x\to \nu x
$ for all $x\in {\mathbb R}^n
$ is an autonomous first integral of the system of ordinary differential equations [(3.1)]{}.* ]{}
[**Corollary 3.2**]{}. [*Let $\lambda\ne 0$ be an eigenvalue of the matrix $B$ corresponding to two real linearly independent eigenvectors $\nu^{1}, \ \nu^{2}.$ Then the system [(3.1)]{} has the autonomous first integral $
F\colon x\to \dfrac{\nu^1 x}{\nu^2 x}$ for all $
x\in {\mathscr X},
$ where a domain ${\mathscr X}\subset \bigl\{x\colon \nu^2 x\ne 0\bigr\}.$* ]{}
For example, the autonomous system of ordinary differential equations\
$
\begin{array}{ll}
\dfrac{dx_1}{dt}= x_{1} - 2x_{2} - x_{4},
\quad
&
\dfrac{dx_2}{dt} = {}- x_{1} + 4x_{2} - x_{3} + 2x_{4},
\\[4ex]
\dfrac{dx_3}{dt} = 2x_{2} + x_{3} + x_{4}, &
\dfrac{dx_4}{dt} = 2x_{1} - 4x_{2} + 2x_{3} - 2x_{4}
\end{array}
\hfill (3.2)
$\
has the eigenvectors $\nu^{1}\!=\!(1,-1,1,-1),
\nu^{2}\!=\!(2,2,1,1),
\nu^{3}\!=\!(1,0,1,0),\!$ $\nu^{4}=(0,2,0,1)$ corresponding to the eigenvalues $\lambda_1=0,\ \lambda_2=\lambda_3=1, \ \lambda_4=2,$ respectively. The functions\
$
F_{1}\colon x\to x_{1}-x_{2}+x_{3}-x_{4}$ for all $x\in {\mathbb R}^4
$ (by Corollary 3.1), (3.3)\
$
F_{23}\colon x\to
\dfrac{2x_{1}+2x_{2}+x_{3}+x_{4}}{x_{1}+x_{3}}
$ for all $x\in {\mathscr X}_1
$ (by Corollary 3.2), (3.4)\
$
F_{24}\colon x\to
\dfrac{(2x_{1}+2x_{2}+x_{3}+x_{4})^{2}}{2x_{2}+x_{4}}$ for all $x\in {\mathscr X}_2
$ (by Theorem 3.2), (3.5)\
where ${\mathscr X}_1\subset \{ x\colon x_{1}+x_{3}\ne 0\},\
{\mathscr X}_2\subset \{ x\colon 2x_{2}+x_{4}\ne 0\},$ are autonomous first integrals of the system (3.2). The set of functionally independent first integrals $F_1,\ F_{23},\ F_{24}$ is a general autonomous integral of the system of ordinary differential equations (3.2).
[**Theorem 3.3**]{}. [ *Let $\nu={\stackrel{*}{\nu}}+\widetilde{\nu}\,i\
({\stackrel{*}{\nu}}={\rm Re}\,\nu,\ \widetilde{\nu}={\rm Im}\,\nu)$ be an eigenvector of the matrix $B$ corresponding to the complex eigenvalue $\lambda={\stackrel{*}{\lambda}}+\widetilde{\lambda}\,i\
({\stackrel{*}{\lambda}}={\rm Re}\,\lambda,\ \widetilde{\lambda}={\rm Im}\,\lambda\not=0).$ Then the system of ordinary differential equations [(3.1)]{} has the autonomous first integral\
$
F\colon x\to
\bigl( ({\stackrel{*}{\nu}}x)^2
+(\widetilde{\nu}x)^2\bigr)\cdot
\exp\biggl(
{}-2 \
\dfrac{{\stackrel{*}{\lambda}}}{\widetilde{\lambda}} \
{\rm arctg}\,\dfrac{\widetilde{\nu}x}{{\stackrel{*}{\nu}}x}
\,\biggr)$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \bigl\{x\colon {\stackrel {*}{\nu}}x\ne 0\bigr\}.$* ]{}
[**Theorem 3.4**]{}. [ *Let $\nu^1\!=\!{\stackrel{*}{\nu}}{}^{\,1}+\widetilde{\nu}{}^{\,1}\,i\
({\stackrel{*}{\nu}}{}^{\,1}\!={\rm Re}\,\nu^1,\ \widetilde{\nu}{}^{\,1}\!={\rm Im}\,\nu^1)$ be an eigenvector of the matrix $B$ corresponding to the complex eigenvalue $\lambda_1={\stackrel{*}{\lambda}}_{1}+\widetilde{\lambda}_{1}\,i\
({\stackrel{*}{\lambda}}_{1}={\rm Re}\,\lambda_1,\ \widetilde{\lambda}_{1}={\rm Im}\,\lambda_1\not=0),$ $\nu^2$ be an real eigenvector of the matrix $B$ corresponding to the eigenvalue $\lambda_2\ne 0.$ Then the system of ordinary differential equations [(3.1)]{} has the autonomous first integral\
$
F\colon x\to
\nu^2x\cdot \exp\Biggl(
{}-\dfrac{\lambda_2}{\widetilde{\lambda}_1} \
{\rm arctg}\,\dfrac{\widetilde{\nu}{}^{\,1}x}
{{\stackrel{*}{\nu}}{}^{\,1}x}\,\Biggr)$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \bigl\{x\colon {\stackrel{*}{\nu}}{}^1x\ne 0\bigr\}.$* ]{}
The autonomous linear system of ordinary differential equations\
$
\dfrac{dx_1}{dt} = 2x_{1} + x_{2},
\quad
\dfrac{dx_2}{dt} = x_{1} + 3x_{2} - x_{3},
\quad
\dfrac{dx_3}{dt} = {}-x_{1} + 2x_{2} + 3x_{3}
$ (3.6)\
has the eigenvalues $\lambda_{1}=3+i,\ \lambda_{2}=2$ corresponding to two eigenvectors $\nu^{1}=(1,i,{}-1),$ $\nu^{2}=(3,{}-1,{}-1),$ respectively, and the first autonomous integrals\
$
F_{1}\colon x\to
\bigl( (x_{1}-x_{3})^2+x_{2}^2\,\bigr)
\exp\Bigl( {}-6\,{\rm arctg}\,\dfrac{x_2}{x_1-x_3}\,\Bigr)
$ for all $x\in {\mathscr X}$ (by Theorem 3.3), (3.7)\
$
F_{2}\colon x\to
(3x_1-x_2-x_3)\exp\Bigl( {}-2\,{\rm arctg}\,\dfrac{x_2}{x_1-x_3}\Bigr)
$ for all $x\in {\mathscr X}$ (by Theorem 3.4), (3.8)\
where a domain ${\mathscr X}\subset \{x\colon x_1-x_3\ne 0\}.$
[**Theorem 3.5**]{}. [*Let $\nu^1={\stackrel{*}{\nu}}{}^{\,1}+\widetilde{\nu}{}^{\,1}\,i$ and $\nu^2={\stackrel{*}{\nu}}{}^{\,2}+\widetilde{\nu}\,{}^{\,2}\,i$ be two eigenvectors of the matrix $B$ corresponding to the complex eigenvalues $\lambda_1=
{\stackrel{*}{\lambda}}_1 +\widetilde{\lambda}_1\,i$ and $\lambda_2={\stackrel{*}{\lambda}}_2 +
\widetilde{\lambda}_2\,i\ (\lambda_1\!\ne \overline{\lambda}_2),$ respectively. Then the system [(3.1)]{} has the autonomous first integral\
$
F\colon x\to\
{\stackrel{\sim}{\lambda}}_{1}\,
{\rm arctg}\,\dfrac{{\stackrel{\sim}{\nu}}{}^{\,2}x}
{{\stackrel{*}{\nu}}{}^{\,2}x}
\ -\ {\stackrel{\sim}{\lambda}}_{2}\,
{\rm arctg}\,\dfrac{{\stackrel{\sim}{\nu}}{}^{\,1}x}
{{\stackrel{*}{\nu}}{}^{\,1}x}$ for all $x\in {\mathscr X},
\hfill
$\
where the vectors ${\stackrel{*}{\nu}}{}^{\,\tau}\!={\rm Re}\,\nu^\tau,\ \widetilde{\nu}{}^{\,\tau}\!={\rm Im}\,\nu^\tau,$ the numbers ${\stackrel{*}{\lambda}}_{\tau}\!=\!{\rm Re}\,\lambda_\tau,\,
\widetilde{\lambda}_{\tau}\!=\!{\rm Im}\,\lambda_\tau\!\not=\!0,\, \tau\!=\!1,2,$ a domain ${\mathscr X}\subset \bigl\{x\colon {\stackrel{*}{\nu}}{}^{\,2}x\ne 0\wedge
{\stackrel{*}{\nu}}{}^{\,1}x\ne 0\bigr\}.$* ]{}
As an example, the linear autonomous system of ordinary differential equations\
$
\begin{array}{ll}
\dfrac{dx_1}{dt} = {}-3x_1 + x_2 + 4x_3+ 2x_4,
\quad
&
\dfrac{dx_2}{dt} = 8x_1 - 3x_2 -2x_3+ 6x_4,
\\[3ex]
\dfrac{dx_3}{dt} = {}-9x_1 + 3x_2 + 4x_3- 4x_4,
&
\dfrac{dx_4}{dt} = 6x_1 - 3x_2 - 4x_3+ 2x_4
\end{array}
\hfill (3.9)
$\
has the eigenvalues $\lambda_1\!=i,\, \lambda_2\!=2i$ corresponding to the eigenvectors $\!\nu^{1}\!\!=\!(1-i,-1+2i,2i,2),$ $\nu^{2}=(i,{}-1,i,1+2i),$ respectively. The functionally independent first integrals\
$
F_{1}\colon x\to
(x_{1}-x_{2}+2x_{4})^2+({}-x_{1}+2x_{2}+2x_{3})^2$ for all $x\in {\mathbb R}^4$ (by Theorem 3.3), (3.10)\
$
F_{2}\colon x\to ({}-x_2+x_4)^{2}+(x_1+x_3+2x_4)^2$ for all $x\in {\mathbb R}^4$ (by Theorem 3.3), (3.11)\
and (by Theorem 3.5)\
$
F_{3}\colon x\to\
{\rm arctg}\,\dfrac{x_{1}+x_{3}+2x_{4}}{{}-x_{2}+x_{4}}
-2\,{\rm arctg}\,\dfrac{{}-x_{1}+2x_{2}+2x_{3}}{x_{1}-x_{2}+2x_{4}}$ for all $x\in {\mathscr X}
\hfill (3.12)
$\
are a general autonomous integral on a domain ${\mathscr X}\subset \{x\colon x_1-x_2+2x_4\ne 0\wedge x_2-x_4\ne 0\}$ of the linear system of ordinary differential equations (3.9).\
. [*Let $\nu^{0}$ be an eigenvector of the matrix $B$ corresponding to the eigenvalue $\lambda$ with elementary divisor of multiplicity $m.$ A non-zero vector $\nu^{k}\in {\mathbb C}^n$ is called a ***generalized eigenvector of order*** [$k$]{} for $\lambda$ if and only if\
$
(B-\lambda E)\,\nu^{k}=k \cdot \nu^{k-1},
\quad
k=1,\ldots, m-1,
\hfill
$\
where $E$ is the $n\times n$ identity matrix.* ]{}
[**Theorem 3.6**]{}. [*Let $\lambda$ be an eigenvalue of the matrix $B$ with the elementary divisor of multiplicity $m\ (m{\geqslant}2)$ corresponding to the real eigenvector $\nu^{0}$ and to the real order [1]{} generalized eigenvector $\nu^{1}.$ Then the system [(3.1)]{} has the autonomous first integral\
$
F\colon x\to
\nu^{0}x\,
\exp\biggl( {}-\lambda\, \dfrac{\nu^{1}x}{\nu^{0}x}\,\biggr)
$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \{x\colon \nu^0x\ne 0\}.$* ]{}
[**Corollary 3.3**]{}. [ *Let $\lambda={\stackrel{*}{\lambda}}+\widetilde{\lambda}\,i\
({\stackrel{*}{\lambda}}={\rm Re}\,\lambda,\ \widetilde{\lambda}={\rm Im}\,\lambda\not=0)$ be a complex eigenvalue of the matrix $B$ with the elementary divisor of multiplicity $m\ (m{\geqslant}2)$ corresponding to the eigenvector $\nu^{0}={\stackrel{*}{\nu}}{}^{\,0}+\widetilde{\nu}{}^{\,0}\,i$ and to the order [1]{} generalized eigenvector $\nu^{1}={\stackrel{*}{\nu}}{}^{\,1}+\widetilde{\nu}{}^{\,1}\,i.$ Then the system [(3.1)]{} has the autonomous first integrals\
$
F_1\colon x\to\,
\Bigr(\bigl({\stackrel{*}{\nu}}\,{}^{0}x\bigr)^2+
\bigl({\stackrel{\sim}{\nu}}\,{}^{0}x\bigr)^2\,\Bigr)
\exp\biggl( {}-2\,\dfrac{{\stackrel{*}{\lambda}}\,\alpha(x)-
{\stackrel{\sim}{\lambda}}\,\beta(x)}
{\bigl({\stackrel{*}{\nu}}\,{}^{0}x\bigr)^2+
\bigl({\stackrel{\sim}{\nu}}\,{}^{0}x\bigr)^2}\,\biggr)
$ for all $x\in {\mathscr X}
\hfill
$\
and\
$
F_2\colon x\to\
{\rm arctg}\,
\dfrac{{\stackrel{\sim}{\nu}}\,{}^{0}x}{{\stackrel{*}{\nu}}{}^{0}x}
\ - \
\dfrac{{\stackrel{\sim}{\lambda}}\,\alpha(x)+
{\stackrel{*}{\lambda}}\,\beta(x)}
{\bigl({\stackrel{*}{\nu}}\,{}^{0}x\bigr)^2+
\bigl(\,{\stackrel{\sim}{\nu}}\,{}^{0}x\bigr)^2}
$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \bigl\{x\colon {\stackrel{*}{\nu}}{}^{\,0}x\ne 0\bigr\},$ the vectors ${\stackrel{*}{\nu}}{}^{\,\tau}\!={\rm Re}\,\nu^\tau,\ \widetilde{\nu}{}^{\,\tau}\!={\rm Im}\,\nu^\tau,\ \tau=0,1,$ the polynomials $
\alpha\colon x\to\,
{\stackrel{*}{\nu}}{}^{\,0}x\,{\stackrel{*}{\nu}}{}^{\,1}x +
{\stackrel{\sim}{\nu}}\,{}^{\,0}x\,{\stackrel{\sim}{\nu}}{}^{\,1}x,
\ \ \beta\colon x\to\,
{\stackrel{*}{\nu}}{}^{\,0}x\,{\stackrel{\sim}{\nu}}{}^{\,1}x -
{\stackrel{\sim}{\nu}}\,{}^{\,0}x\,{\stackrel{*}{\nu}}{}^{\,1} x$ for all $x\in {\mathbb R}^n.$* ]{}
[**Corollary 3.4**]{}. [ *Let $\lambda_1=0$ be an eigenvalue with the elementary divisor of multiplicity $m\ (m{\geqslant}2)$ of the matrix $B$ corresponding to the real eigenvector $\nu^{0}$ and the real order [1]{} generalized eigenvector $\nu^{1}.$ Let $\lambda_2$ be an eigenvalue of the matrix $B$ corresponding to the real eigenvector $\nu^{2}.$ Then the system [(3.1)]{} has the autonomous first integrals\
$
F\colon x\to
\nu^{2}x\,
\exp\biggl(-\,\lambda_{2}\ \dfrac{\nu^{1}x}{\nu^{0}x}\,\biggr)
$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \{x\colon \nu^0x\ne 0\}.$* ]{}
The linear autonomous system of ordinary differential equations\
$
\dfrac{dx_1}{dt} =4x_1 - 5x_2+ 2x_3,
\quad
\dfrac{dx_2}{dt} = 5x_1 - 7x_2+ 3x_3,
\quad
\dfrac{dx_3}{dt} = 6x_1 - 9x_2 + 4x_3
$ (3.13)\
has the eigenvalue $\lambda_1=0$ with the elementary divisor $\lambda^2$ of multiplicity $2$ corresponding to the eigenvector $\nu^{0}=(1,{}-2,1)$ and the order [1]{} generalized eigenvector $\nu^{1}=(0,{}-1,1),$ and the simple eigenvalue $\lambda_{3}=1$ with the elementary divisor $\lambda-1$ corresponding to the eigenvector $\nu^{3}=(3,{}-3,1).$ The functionally independent functions\
$
F_1\colon x\to x_1-2x_2+x_3$ for all $x\in {\mathbb R}^3
$ (by Theorem 3.6) (3.14)\
and (by Corollary 3.4)\
$
F_{2}\colon x\to
(3x_1-3x_2+x_3)\exp\dfrac{x_2-x_3}{x_1-2x_2+x_3}$ for all $x\in {\mathscr X},
\hfill (3.15)
$\
where ${\mathscr X}\subset\{x\colon x_1-2x_2+x_3\ne 0\},$ are autonomous first integrals of the system (3.13).
[**Corollary 3.5**]{}. [ *Let $\lambda_1=0$ be an eigenvalue with elementary divisor of multiplicity $m$ $(m{\geqslant}2)\!$ of the matrix $B$ corresponding to the real eigenvector $\nu^{0}$ and to the real order [1]{} generalized eigenvector $\nu^{1}.$ Let $\lambda_2={\stackrel{*}{\lambda}}_2+\widetilde{\lambda}_2\,i\
({\stackrel{*}{\lambda}}_2={\rm Re}\,\lambda_2,\ \widetilde{\lambda}_2={\rm Im}\,\lambda_2\not=0)$ be a complex eigenvalue of the matrix $B$ corresponding to the eigenvector $\nu^{2}={\stackrel{*}{\nu}}{}^{\,2}+\widetilde{\nu}{}^{\,2}\,i.$ Then the system of differential equations [(3.1)]{} has two autonomous first integrals\
$
F_1\colon x\to
\Bigl( \bigl(\,{\stackrel{*}{\nu}}{}^{\,2}x\bigr)^{2} +
\bigl(\,{\stackrel{\sim}{\nu}}{}^{\,2}x\bigr)^2\,\Bigr)
\exp\biggl({}-2\,
{\stackrel{*}{\lambda}}_{2} \ \dfrac{\nu^{1}x}{\nu^{0}x}\,\biggr)$ for all $x\in {\mathscr X}
\hfill
$\
and\
$
F_2\colon x\to \
{\rm arctg}\,
\dfrac{{\stackrel{\sim}{\nu}}{}^{\,2}x}{{\stackrel{*}{\nu}}{}^{\,2}x}
\ - \
{\stackrel{\sim}{\lambda}}_2 \ \dfrac{\nu^{1}x}{\nu^{0}x}$ for all $x\in {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \{x\colon \nu^0x\ne 0,\ {\stackrel{*}{\nu}}{}^{\,2}x\ne 0\}.$* ]{}
[**Theorem 3.7**]{}. [ *Let $\lambda$ be an eigenvalue with elementary divisor of multiplicity $m{\geqslant}2$ of the matrix $B$ corresponding to the real eigenvector $\nu^{0}$ and to the real generalized eigenvectors $\nu^{k},\ k=\overline{1,m-1}.$ Then the system [(3.1)]{} has the first integrals\
$
F_g\colon x\to \Psi_g(x)$ for all $x\in {\mathscr X},
\quad g=2,\ldots,m-1,
$\
where the functions $\Psi_{g}\colon {\mathscr X}\to {\mathbb R}$ is the solution to system\
$
\displaystyle
\nu^{k}x =
\sum\limits_{i=1}^{k}{\textstyle\binom{k-1}{i-1}}
\Psi_{i}(x)\nu^{k-i}x,
\ \ \ k=1,\ldots,m-1,
\quad
{\mathscr X}\subset \{x\colon \nu^0x\ne 0\}.
\hfill
$* ]{}\
. Let $\lambda={\stackrel{*}{\lambda}}+\widetilde{\lambda}\,i\
({\stackrel{*}{\lambda}}={\rm Re}\,\lambda,\ \widetilde{\lambda}={\rm Im}\,\lambda\not=0)$ be a complex eigenvalue with elementary divisor of multiplicity $m\ (m{\geqslant}2)$ of the matrix $B$ corresponding to the complex eigenvector $\nu^{0}={\stackrel{*}{\nu}}{}^{\,0}+\widetilde{\nu}{}^{\,0}\,i.$ Taking into account (3.16), we obtain the real-valued first integrals of the system of ordinary differential equations (3.1):\
$
F_{g,1}\colon x\to \mbox{Re}\,\Psi_g(x),
\ \ \
F_{g,2}\colon x\to \mbox{Im}\,\Psi_g(x)$ for all $x\in {\mathscr X}, \ \ \ g=2,\ldots, m-1,
\hfill
$\
where a domain ${\mathscr X}\subset
\bigl\{x\colon (\,{\stackrel{*}{\nu}}{}^{\,0}x)^2 +
(\,{\widetilde{\nu}}{}^{\,0}x)^2\ne 0\bigr\}.$
For example, the system of ordinary differential equations\
$
\dfrac{dx_1}{dt} = 4x_1 - x_2,
\quad \ \
\dfrac{dx_2}{dt} = 3x_1 + x_2 - x_3,
\quad \ \
\dfrac{dx_3}{dt} = x_1 + x_3
\hfill (3.17)
$\
has the eigenvalue $\lambda_{1}\!=\!2$ with the elementary divisor $(\lambda-2)^3\!$ of multiplicity 3 corresponding to the eigenvector $\!\nu^{0}\!=\!(1,-1,1)\!$ and the generalized eigenvectors $\!\nu^{1}\!=\!(1,0,-1),\, \nu^{2}\!=\!(0,0,2).$ First integrals of the system of differential equations (3.17) are the functions\
$
F_{1}\colon (x_1,x_2,x_3)\to
(x_1-x_2+x_3)\exp\Bigl({}-2\, \dfrac{x_1-x_3}{x_1-x_2+x_3}\,\Bigr)
$ (by Theorem 3.6) (3.18)\
and (by Theorem 3.7)\
$
F_{2}\colon (x_1,x_2,x_3)\to
\dfrac{(x_1-x_3)^2-2x_3(x_1-x_2+x_3)}{(x_1-x_2+x_3)^2}$ for all $(x_1,x_2,x_3)\in {\mathscr X},
$ (3.19)\
where a domain ${\mathscr X}\subset \{(x_1,x_2,x_3)\colon x_1-x_2+x_3\ne 0\}.$
The autonomous system of ordinary differential equations\
$
\begin{array}{ll}
\dfrac{dx_1}{dt} = x_1 - 2x_2+x_3-2x_6,
\ &
\dfrac{dx_2}{dt} = 3x_2-x_3-x_5+2x_6,
\\[2.5ex]
\dfrac{dx_3}{dt} = {}-x_1 +x_3+2x_4+2x_5,
\ &
\dfrac{dx_4}{dt} = {}-x_1+x_4+x_5+x_6,
\\[2.5ex]
\dfrac{dx_5}{dt} = x_1 +x_2+x_5,
\ &
\dfrac{dx_6}{dt} = x_1-x_2+x_3-x_4-x_6
\end{array}
\hfill (3.20)
$\
has the complex eigenvalue $\lambda_{1}=1+i$ with the elementary divisor $(\lambda-1-i)^3$ corresponding to the eigenvector $\nu^{0}=(1,1,0,0,i,0)$ and to the generalized eigenvectors $\nu^{1}=(0,1,0,i,i,1),$ $\nu^{2}=(0,1,i,0,i,0).$ A general autonomous integral of the system (3.20) is the functions\
$
F_{1}\colon x\to P(x)\,\exp({}-2\varphi (x))$ for all $x\in {\mathscr X}
$ (by Theorem 3.3), (3.21)\
$
F_{2}\colon x\to
P(x)\exp\biggl({}-2\,\dfrac{\alpha(x)-\beta(x)}{P(x)}\,\biggr)$ for all $x\in {\mathscr X}
$ (by Corollary 3.3), (3.22)\
$
F_{3}\colon x\to
\varphi(x)- \dfrac{\alpha(x)+\beta(x)}{P(x)}$ for all $x\in {\mathscr X}
$ (by Corollary 3.3), (3.23)\
$
F_{4}\colon x\to
\dfrac{\gamma(x)P(x)+\beta^2(x)-\alpha^2(x)}{P^2(x)}$ for all $x\in {\mathscr X}
$ (by Theorem 3.7), (3.24)\
and\
$
F_{5}\colon x\to
\dfrac{\delta(x)P(x)-2\alpha(x)\beta(x)}{P^2(x)}$ for all $x\in {\mathscr X}$ (by Theorem 3.7), (3.25)\
where\
$
P\colon x\to (x_1+x_2)^2 + x_5^2,
\quad
\alpha\colon x\to (x_1+x_2)(x_2+x_6) + x_5(x_4+x_5),
\hfill
$\
$
\beta\colon x\to (x_1+x_2)(x_4+x_5) - x_5(x_2+x_6), \
\quad
\gamma\colon x\to x_2(x_1+x_2) + x_5(x_3+x_5),
\hfill
$\
$
\delta\colon x\to (x_1+x_2)(x_3+x_5) - x_2x_5$ for all $x\in {\mathbb R}^6,
\hfill
$\
$
\varphi\colon x\to\ {\rm arctg}\,\dfrac{x_5}{x_1+x_2}$ for all $x\in {\mathscr X},
\quad {\mathscr X}\subset \{x\colon x_1+x_2\ne 0\}.
\hfill
$\
\
. [ *Let $\nu$ be a real eigenvector of the matrix $B$ corresponding to the eigenvalue $\lambda.$ Then the system [(3.1)]{} has the first integral\
$
F\colon (t,x)\to\ \nu x\,\exp({}-\lambda\,t)$ for all $(t,x)\in {\mathbb R}^{n+1}.
\hfill
$* ]{}\
For example, the four dimensional system (3.2) has the eigenvalue $\lambda_2=1$ corresponding to the eigenvector $\nu^2=(2,2,1,1)$ and the first integral\
$
F\colon (t,x)\to (2x_1+2x_2+x_3+x_4)\,e^{{}-t}$ for all $(t,x)\in {\mathbb R}^5
$ (by Theorem 3.8).\
The first integrals (3.3), (3,4), (3.5), and $F$ are a general integral on a domain ${\mathbb R}\times {\mathscr X}$ of the system (3.2), where ${\mathscr X}\subset \{x\colon x_1+x_3\ne 0\wedge 2x_2+x_4\ne 0\}\subset {\mathbb R}^4.$
Using the eigenvector $\nu^2=(3,{}-1,{}-1)$ corresponding to the eigenvalue $\lambda_2=2,$ we can build the first integral of the linear system (3.6):\
$
F\colon (t,x_1,x_2,x_3)\to (3x_1-x_2-x_3)\,e^{{}-2t}$ for all $(t,x_1,x_2,x_3)\in {\mathbb R}^4
$ (by Theorem 3.8).\
The functionally independent first integrals (3.7), (3,8), and $F$ are a general integral on a domain ${\mathbb R}\times {\mathscr X}$ of the system (3.6), where a domain ${\mathscr X}\subset\{(x_1,x_2,x_3)\colon x_1-x_3\ne 0\}.$
[**Corollary 3.6**]{}. [ *Let $\nu={\stackrel{*}{\nu}}+\widetilde{\nu}\,i\
({\stackrel{*}{\nu}}={\rm Re}\,\nu,\ \widetilde{\nu}={\rm Im}\,\nu)$ be an eigenvector of the matrix $B$ corresponding to the complex eigenvalue $\lambda={\stackrel{*}{\lambda}}+\widetilde{\lambda}\,i\
({\stackrel{*}{\lambda}}={\rm Re}\,\lambda,\ \widetilde{\lambda}={\rm Im}\,\lambda\not=0).\!$ Then the system of ordinary differential equations [(3.1)]{} has the first integrals\
$
F_1\colon (t,x)\to
\Bigl(\bigl(\,{\stackrel{*}{\nu}}x\bigr)^2
+ \bigl(\,{\stackrel{\sim}{\nu}}x\bigr)^2\,\Bigr)
\exp\bigl({}-2\,{\stackrel{*}{\lambda}}\,t\bigr)$ for all $(t,x)\in {\mathbb R}^{n+1}
\hfill
$\
and\
$
F_2\colon (t,x)\to\
{\rm arctg}\,\dfrac{{\widetilde{\nu}}x}{{\stackrel{*}{\nu}}x}\ -\
{\widetilde{\lambda}}\,t$ for all $(t,x)\in {\mathbb R}\times {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \bigl\{x\colon {\stackrel{*}{\nu}}x\ne 0\bigr\}.$* ]{}
As an example, the system of linear differential equations (3.9) has the eigenvalue $\lambda_1=i$ corresponding to the eigenvector $\nu^1=(1-i,{}-1+2i,2i,2)$ and the first integral\
$
F\colon (t,x)\to\
{\rm arctg}\,\dfrac{{}-x_1+2x_2+2x_3}{x_1-x_2+2x_4} \ -\ t$ for all $(t,x)\in {\mathbb R}\times {\mathscr X}_1
$ (by Corollary 3.6),\
where a domain ${\mathscr X}_1\subset \{x\colon x_1-x_2+2x_4\ne 0\}.$
The first integrals (3.10), (3.11), (3.12) and $F$ are a general integral on a domain ${\mathbb R}\times {\mathscr X}$ of the system (3.9), where a domain ${\mathscr X}\subset \{x\colon x_1-x_2+2x_4\ne 0\wedge x_4-x_2\ne 0\}\subset {\mathbb R}^4.$
[**Theorem 3.9**]{}. [*Let $\lambda$ be an eigenvalue of the matrix $B$ with the elementary divisor of multiplicity $m\ (m{\geqslant}2)$ corresponding to the real eigenvector $\nu^{0}$ and the real order [1]{} generalized eigenvector $\nu^{1}.$ Then the system [(3.1)]{} has the first integral\
$
F\colon (t,x)\to \dfrac{\nu^1x}{\nu^0x} - t$ for all $(t,x)\in {\mathbb R}\times {\mathscr X},
\hfill (3.26)
$\
where a domain ${\mathscr X}\subset \{x\colon \nu^0x\ne 0\}.$* ]{}
The system of ordinary differential equations (3.13) has the eigenvalue $\!\lambda_1\!=\!0\!$ corresponding to the eigenvector $\!\nu^0\!=\!(1,-2,1)\!$ and to the order [1]{} generalized eigenvector $\nu^1\!=\!(0,-1,1).$ From Theorem 3.9 it follows that the function\
$
F\colon (t,x_1,x_2,x_3)\to \dfrac{x_3-x_2}{x_1-2x_2+x_3}\ - \ t$ for all $(t,x_1,x_2,x_3)\in {\mathbb R}\times {\mathscr X}
\hfill
$\
is a first integral of the system (3.13), where a domain ${\mathscr X}\subset \{(x_1,x_2,x_3)\colon x_1-2x_2+x_3\ne 0\}.$
A general integral on a domain ${\mathbb R}\times {\mathscr X}$ of the system of ordinary differential equations (3.13) is the functionally independent first integrals (3.14), (3.15), and $F.$
Consider the system (3.17). Using the eigenvalue $\lambda_1=2$ corresponding to the eigenvector $\nu^0=(1,{}-1,1)$ and to the order [1]{} generalized eigenvector $\nu^1=(1,0,{}-1),$ we can build the first integral of the system of ordinary differential equations (3.17):\
$
F\colon (t,x_1,x_2,x_3)\to \dfrac{x_1-x_3}{x_1-x_2+x_3}\ - \ t$ for all $(t,x_1,x_2,x_3)\in {\mathbb R}\times {\mathscr X}
$ (by Theorem 3.9).\
The first integrals (3.18), (3.19), and $F$ are a general integral on a domain ${\mathbb R}\times {\mathscr X}$ of the system (3.17), where a domain ${\mathscr X}\subset \{(x_1,x_2,x_3)\colon x_1-x_2+x_3\ne 0\}\subset{\mathbb R}^3.$
[**Remark 3.2**]{}. Suppose $\!\lambda\!=\!{\stackrel{*}{\lambda}}+\widetilde{\lambda}\,i\,
({\stackrel{*}{\lambda}}\!=\!{\rm Re}\,\lambda,\, \widetilde{\lambda}\!=\!{\rm Im}\,\lambda\not\!=\!0)\!$ is a complex eigenvalue of the matrix $B$ corresponding to the eigenvector $\nu^{0}={\stackrel{*}{\nu}}{}^{\,0}+\widetilde{\nu}{}^{\,0}\,i$ and to the generalised eigenvector $\nu^{1}={\stackrel{*}{\nu}}{}^{\,1}+\widetilde{\nu}{}^{\,1}\,i.$ Using (3.26), we get the real-valued first integrals of the system (3.1):\
$\!
F_1\colon\! (t,x)\to
\dfrac{{\stackrel{*}{\nu}}\,{}^{0}x\,{\stackrel{*}{\nu}}{}^{1}x +
{\widetilde{\nu}}\,{}^{\,0}x\,{\widetilde{\nu}}{}^{\,1}x}
{\bigl(\,{\stackrel{*}{\nu}}\,{}^{0}x\bigr)^{2} +
\bigl(\,{\widetilde{\nu}}\,{}^{\,0}x\bigr)^{2}} \, - t,
\
F_2\colon\! (t,x)\to
\dfrac{{\stackrel{*}{\nu}}\,{}^{0}x\,{\widetilde{\nu}}{}^{\,1}x
-{\widetilde{\nu}}\,{}^{\,0}x\,{\stackrel{*}{\nu}}{}^{1}x}
{\bigl(\,{\stackrel{*}{\nu}}\,{}^{0}x\bigr)^{2} +
\bigl(\,{\widetilde{\nu}}\,{}^{\,0}x\bigr)^{2}}$ for all $(t,x)\!\in\! {\mathbb R}\!\times\!{\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset
\bigl\{x\colon (\,{\stackrel{*}{\nu}}{}^{\,0}x)^2 +
(\,{\widetilde{\nu}}{}^{\,0}x)^2\ne 0\bigr\}.$
Consider the system (3.20). Using the eigenvalue $\lambda_1=1+i$ corresponding to the eigenvector $\nu^{0}=(1,1,0,0,i,0)$ and to the order [1]{} generalized eigenvector $\nu^{1}=(0,1,0,i,i,1),$ we can build the first integral of the system of ordinary differential equations (3.20):\
$
F\colon (t,x)\to
\dfrac{(x_1+x_2)(x_2+x_6)+x_5(x_4+x_5)}{(x_1+x_2)^2+x_5^2}\ - \ t$ for all $(t,x)\in {\mathbb R}\times {\mathscr X},
\hfill
$\
where a domain ${\mathscr X}\subset \{x\colon x_1+x_2\ne 0\}\subset {\mathbb R}^6.$
The functionally independent first integrals (3.21), (3.22), (3.23), (3.24), (3.25), and $F$ are a general integral of the system of ordinary differential equations (3.20).\
\
We consider a system of $n$ first order constant-coefficient linear nonhomogeneous ordinary differential equations\
$
\dfrac{dx}{dt}=Ax+f(t),
$ (3.27)\
where $x = \mbox{colon}(x_{1},\ldots,x_{n})\in{\mathbb R}^n,$ $A = \bigl\|a_{ij}\bigr\|$ is a real $n\times n$ matrix, and the vector function $f\colon t\to {\rm colon}(f_1(t),\ldots,f_n(t))$ for all $t\in J$ is continuous on an interval $J\subset {\mathbb R}.$
Let B be the transpose of the matrix $A.$
[**Theorem 3.10**]{}. [*Suppose $\nu$ is a real eigenvector of the matrix $B$ corresponding to the eigenvalue $\lambda.$ Then a first integral of the system [(3.27)]{} is the function\
$
\displaystyle
F\colon (t,x)\to \nu x\cdot\exp({}-\lambda t)
-\int \nu f(t)\cdot\exp({}-\lambda t)\,dt$ for all $(t,x)\in \widetilde{J}\times {\mathbb R}^n,
\ \ \widetilde{J}\subset J.
\hfill
$\
*]{} . [ *Let $\nu={\stackrel{*}{\nu}}+\widetilde{\nu}\,i\
({\stackrel{*}{\nu}}={\rm Re}\,\nu,\ \widetilde{\nu}={\rm Im}\,\nu)$ be an eigenvector of the matrix $B$ corresponding to the complex eigenvalue $\lambda={\stackrel{*}{\lambda}}+\widetilde{\lambda}\,i\
({\stackrel{*}{\lambda}}={\rm Re}\,\lambda,\ \widetilde{\lambda}={\rm Im}\,\lambda\not=0).$ Then first integrals of the system of ordinary differential equations [(3.27)]{} are the functions\
$
\displaystyle
F_{\tau}\colon (t,x)\to \alpha_{\tau}(t, x)-
\int \alpha_{\tau}(t, f(t))\,dt$ for all $(t,x)\in \widetilde{J}\times {\mathbb R}^n,
\ \ \ \tau=1,2,\ \ \ \widetilde{J}\subset J,
\hfill
$\
where the functions $
\displaystyle
\alpha_1(t,x)=
\bigl(\,{\stackrel{*}{\nu}}x\cdot\cos\widetilde{\lambda}\,t +
\widetilde{\nu}x\cdot\sin\widetilde{\lambda}\,t \bigr)\cdot
\exp\bigl({}-{\stackrel{*}{\lambda}}\,t\bigr)$ for all $(t,x)\in J\times {\mathbb R}^n,
$ $
\displaystyle
\alpha_2(t,x)=
\bigl(\,\widetilde{\nu}x\cdot\cos\widetilde{\lambda}\,t -
{\stackrel{*}{\nu}}x\cdot\sin\widetilde{\lambda}\,t\bigr)\cdot
\exp\bigl({}-{\stackrel{*}{\lambda}}\,t\bigr)$ for all $(t,x)\in J\times {\mathbb R}^n.$\
*]{} . Under the conditions of Corollary 3.7, we have the function\
$
\displaystyle
F\colon (t,x)\to
\Bigr(\!({\stackrel{*}{\nu}}x)^2+(\widetilde{\nu}x)^2\Bigr)\!\exp\bigl(-2\,{\stackrel{*}{\lambda}}\,t\bigr) -
2\biggl(\!\!\alpha_1(t,x)\!\int\!\! \alpha_1(t,f(t))dt+\alpha_2(t,x)\!\int\!\! \alpha_2(t,f(t))dt\!\!\biggr) +
\hfill
$\
$
\displaystyle
+\ \biggl(\int \alpha_1(t,f(t))\,dt\biggr)^2+
\biggl(\int \alpha_2(t,f(t))\,dt\biggr)^2$ for all $(t,x)\in \widetilde{J}\times {\mathbb R}^n
\hfill
$\
is also a first integral of the system (3.27).
For example, the linear nonhomogeneous system of ordinary differential equations\
$
\dfrac{dx_1}{dt} = 2x_{1} + x_{2}+2e^{2t},
\quad
\dfrac{dx_2}{dt} = x_{1} + 3x_{2} - x_{3}+10,
\quad
\dfrac{dx_3}{dt} = {}-x_{1} + 2x_{2} + 3x_{3}+e^{3t}
\hfill
$\
has the eigenvalue $\lambda_{1}=2$ corresponding to the eigenvector $\nu^{1}=(3,{}-1,{}-1),$ the eigenvalue $\lambda_{2}=3+i$ corresponding to the eigenvector $\nu^{2}=(1,i,{}-1),$ and the first integrals\
$
F_{1}\colon (t,x_1,x_2,x_3)\to
(3x_1-x_2-x_3-5)e^{{}-2t}+e^{t}-6t
$ (by Theorem 3.10),\
$
F_{2}\colon (t,x_1,x_2,x_3)\to
\bigl( (x_{1}-x_{3}+1)\cos t+(x_{2}+3)\sin t\bigr) e^{{}-3t}+
(\cos t-\sin t)e^{{}-t}+\sin t,
\hfill
$\
$
F_{3}\colon (t,x_1,x_2,x_3)\to
\bigl( (x_{2}+3)\cos t+(x_{3}-x_{1}-1)\sin t\bigr) e^{{}-3t}-
(\cos t+\sin t)e^{{}-t}+\cos t
\hfill
$\
for all $(t,x_1,x_2,x_3)\in {\mathbb R}^4$ (by Corollary 3.7).\
. [ *Let $\lambda$ be an eigenvalue with the elementary divisor of multiplicity $m$ $(m{\geqslant}2)$ of the matrix $B$ corresponding to the real eigenvector $\nu^{0}$ and to the real generalized eigenvectors $\nu^{k},\ k=1,\ldots, m-1.$ Then the system [(3.27)]{} has the first integrals\
$
\displaystyle
F_{k+1}\colon (t,x)\to
(\nu^kx)\cdot\exp({}-\lambda t)-
\sum\limits_{i=0}^{k-1}
{\textstyle \binom{k}{i}}\, t^{k-i}\cdot F_{i+1}(t,x) \ -
\hfill
$\
$
\displaystyle
-\ \int\biggl(\bigl(\nu^kf(t)\bigr)\cdot \exp({}-\lambda t)+
k\cdot \int\biggl(\bigl(\nu^{k-1}f(t)\bigr)\cdot \exp({}-\lambda t) \ +
\hfill
$\
$
\displaystyle
+\, (k-1)\cdot\int\biggl(\bigl(\!\nu^{k-2}f(t)\!\bigr)\cdot \exp({}-\lambda t)+\ldots+
2\int\biggl(\bigl(\!\nu^{1}f(t)\!\bigr)\cdot \exp({}-\lambda t)\, +
\hfill
$\
$
\displaystyle
+\ \int\bigl(\nu^{0}f(t)\bigr)\cdot \exp({}-\lambda t)\,dt\biggr)\,dt\ldots\biggr)dt\biggr)dt\biggr)dt$ for all $(t,x)\in \widetilde{J}\times {\mathbb R}^n,
\ \ \ k=1,\ldots,m-1,
\hfill
$\
where the first integral [(]{}by Theorem [3.10)]{}\
$
\displaystyle
F_1\colon\! (t,x)\!\to\! (\nu^0 x)\cdot\exp({}-\lambda t)
-\int\!\! \bigl(\nu^0 f(t)\bigr)\cdot\exp({}-\lambda t)\,dt$ for all $(t,x)\in \widetilde{J}\times {\mathbb R}^n,
\ \ \widetilde{J}\subset J.
\hfill
$\
*]{} As an example, the system of ordinary differential equations\
$
\dfrac{dx_1}{dt} = 4x_1 - x_2+e^{3t},
\quad
\dfrac{dx_2}{dt} = 3x_1 + x_2 - x_3+8t,
\quad
\dfrac{dx_3}{dt} = x_1 + x_3+4
\hfill
$\
has the eigenvalue $\!\lambda_{1}\!=\!2\!$ with the elementary divisor $\!(\lambda-2)^3\!$ of multiplicity 3 corresponding to the eigenvector $\!\nu^{0}\!=\!(1,-1,1)\!$ and the generalized eigenvectors $\!\nu^{1}\!=\!(1,0,-1), \nu^{2}\!=\!(-2,2,0).$ First integrals of this system of differential equations are the functions\
$
F_{1}\colon (t, x)\to
(x_1-x_2+x_3-4t)e^{{}-2t}-e^{t}
$ for all $(t, x)\in {\mathbb R}^4$ (by Theorem 3.10),\
$
F_{2}\colon (t, x)\to
(x_1-x_3+2t-1) e^{{}-2t}-t\, F_1(t,x)-2e^{t}
$ for all $(t, x)\in {\mathbb R}^4$ (by Theorem 3.11),\
and (by Theorem 3.11)\
$
F_{3}\colon (t, x)\to
2(x_2-x_1+3t+2) e^{{}-2t}-t^2\,F_1(t,x)-2t\,F_2(t,x)-2e^{t}
$ for all $(t, x)\in {\mathbb R}^4.$\
. [ *Let $\lambda={\stackrel{*}{\lambda}}+\widetilde{\lambda}\,i\ (\widetilde{\lambda}\ne 0)$ be an eigenvalue with the elementary divisor of multiplicity $m\ (m{\geqslant}2)$ of the matrix $B$ corresponding to the eigenvector $\nu^0={\stackrel{*}{\nu}}\;\!{}^{0}+\widetilde{\nu}\;\!{}^{0}\,i$ and to the generalized eigenvectors $\nu^k={\stackrel{*}{\nu}}\;\!{}^{k}+\widetilde{\nu}\;\!{}^{k}\,i,\, k\!=\!1,\ldots,m\!-\!1.\!$ Then the system of ordinary differential equations [(3.27)]{} has the first integrals\
$
\displaystyle
F_{{}_{\scriptstyle \tau,k+1}}\colon (t,x)\to
\alpha_{{}_{\scriptstyle \tau k}}(t,x)-
\sum\limits_{\xi=0}^{k-1}
{\textstyle \binom{k}{\xi}}\, t^{k-\xi}\cdot F_{{}_{\scriptstyle \tau, \xi+1}}(t,x) \ -
\hfill
$\
$
\displaystyle
-\ \int\biggl(\alpha_{{}_{\scriptstyle \tau k}}(t, f(t)) +
k\cdot \int\biggl(\alpha_{{}_{\scriptstyle \tau,k-1}}(t, f(t)) +
(k-1)\cdot\int\biggl(\alpha_{{}_{\scriptstyle \tau,k-2}}(t, f(t))\ +
\hfill
$\
$
\displaystyle
+\ \ldots+
2\int\biggl(\alpha_{{}_{\scriptstyle \tau 1}}(t, f(t))
+\int \alpha_{{}_{\scriptstyle \tau 0}}(t, f(t))\,dt\biggr)\,dt\ldots\biggr)dt\biggr)dt\biggr)dt
\hfill
$\
for all $(t,x)\in \widetilde{J}\times {\mathbb R}^n,
\ \ \ k=1,\ldots, m-1,
\ \ \ \ \tau=1,2,
\quad
\widetilde{J}\subset J,
\hfill
$\
where the functions\
$
\displaystyle
\alpha_{{}_{\scriptstyle 1k}}(t,x)=
\bigl(\,{\stackrel{*}{\nu}}\;\!{}^kx\cdot\cos\widetilde{\lambda}\,t +
\widetilde{\nu}\;\!{}^kx\cdot\sin\widetilde{\lambda}\,t \bigr)\cdot
\exp\bigl({}-{\stackrel{*}{\lambda}}\,t\bigr)$ for all $(t,x)\in {\mathbb R}^{n+1},
\ k=0,\ldots, m-1,
\hfill
$\
$
\displaystyle
\alpha_{{}_{\scriptstyle 2k}}(t,x)=
\bigl(\,\widetilde{\nu}x\;\!{}^k\cdot\cos\widetilde{\lambda}\,t -
{\stackrel{*}{\nu}}\;\!{}^kx\cdot\sin\widetilde{\lambda}\,t\bigr)\cdot
\exp\bigl({}-{\stackrel{*}{\lambda}}\,t\bigr)$ for all $(t,x)\in {\mathbb R}^{n+1},
\ k=0,\ldots, m-1,
\hfill
$\
and the first integrals [(]{}by Corollary [3.7)]{}\
$
\displaystyle
F_{\tau 1}\colon (t,x)\to
\alpha_{{}_{\scriptstyle \tau 0}}(t, x)-\int \alpha_{{}_{\scriptstyle \tau 0}}(t, f(t))\,dt$ for all $(t,x)\in \widetilde{J}\times {\mathbb R}^n,
\quad \tau=1,2.
\hfill
$\
*]{}
[99]{}
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[**Estimation of Baryon Asymmetry from Dark Matter Decaying into IceCube Neutrinos**]{}
[ [**Tista Mukherjee$^{a}$**]{} [^1]]{}\
[*$^a$ Department of Physics, Presidency University,\
86/1, College Street, Kolkata- 700073, India*]{}\
,\
[*$^b$Astroparticle Physics and Cosmology Division,*]{}\
[*Saha Institute of Nuclear Physics, HBNI*]{}\
[*1/AF Bidhannagar, Kolkata 700064, India* ]{}\
[^2]\
[*$^{c}$Department of Physics, St. Xavier’s College,*]{}\
[*30, Mother Teresa Sarani, Kolkata - 700016, India*]{}\
0.5mm
[**Abstract**]{}
[The recent results of IceCube Neutrino Observatory include an excess of PeV neutrino events which appear to follow a broken power law different from the other lower energy neutrinos detected by IceCube. The possible astrophysical source of these neutrinos is still unknown. One possible source of such neutrinos could be the decay of non-thermal, long-living heavy mass Dark Matter, whose mass should be $> 10^{6} \rm {GeV}$ and could have produced at the very early Universe. They can undergo cascading decay via both hadronic and leptonic channels to finally produce such high energy neutrinos. This possibility has been explored in this work by studying the decay flux of these Dark Matter candidates. The mass and lifetime of such Dark Matter particles have been obtained by performing a $\chi^2$ fit with the PeV neutrino data of IceCube. We finally estimate the baryon asymmetry produced in the Universe due to such Dark Matter decay.]{}
Introduction
============
Recently IceCube has reported by analyzing six years of diffused flux data, a number of events detected at the energy range 60 TeV $\leq E_{\nu} \leq$ 10 PeV which do not follow the same power law that the other neutrinos detected by IceCube earlier seem to follow. In Ref. [@IC_PoS2017] the authors predicted an unbroken power law fit where $\gamma = 2.92^{+0.33}_{-0.29}$ for the high energy starting events (HESE) for a lifetime of 2078 days. However, by fitting the $\nu_{\mu}$ data separately for even higher energy track events in the energy range $10^5 < E_{\nu_{\mu}} < 10^7$ GeV, which are termed as ultra-high energy (UHE) events, seemingly large difference in the best fit value of the spectral index was noticed, which is softer than the HESE best fit $\gamma$ value. It suggests that there must be a break in the power law spectrum, hence their source candidate must be different as the spectral index depends on its source properties. In Ref. [@PeV-3] the authors analyzed the six years astrophysical diffused flux data only for up-going muons - the track events. It was pointed out that there is an excess of events beyond 100 TeV energy scale. They predicted an unbroken power law spectrum for these track-like events for which $\gamma = 2.13 \pm 0.30$, compatible with the previous assumption for high energy astrophysical neutrinos. But any possible source for them has not been confirmed yet [@AGN-nu; @cosmic-nu; @grb-nu; @SNe-nu; @ICnuExcess] as the expected neutrino energy distribution disagrees with the observed spectrum of the UHE excess events. These might be coming from some unknown galactic or extra-galactic sources or some other exotic astrophysical or cosmological phenomenon. In Refs. [@ICnuExcess; @esmaili2013; @esmaili2014; @esmaili2015; @esmaili2017; @esmaili2019; @kaz1; @kaz2] a possibility of heavy Dark Matter (DM) decay or self-annihilation for the production of such neutrinos has been indicated. It has also been emphasized particularly in Ref. [@ICnuExcess] that the possible self-annihilation or decay spectrum of PeV Dark Matter particle candidate would follow a power law which is given by $dN/dE \sim E^{-1.9}$ and is a softer spectrum compared to the HESE spectrum. Also, it clearly would not provide a good fit for the lower energy events which explains the background of the existing tension.
For these neutrinos to have originated from the decay of Dark Matter, the Dark Matter candidate should be super-heavy so that its decay process could produce such high-energy neutrinos. In literature, there are propositions about the existence of super-heavy (SHDM) Dark Matter which are considered to be non-thermal particle candidates as they were never in local thermodynamic equilibrium with the Universe’s plasma and also long-lived particles. However, they are metastable and might go through a rare decay under some favourable circumstances. These Dark Matter candidates are generally termed as “WIMPzilla"s. They could be produced during the inflationary epoch due to spontaneous symmetry breaking in the GUT scale via preheating or reheating [@chung1998SDM; @UHECR1998; @chungPRL; @chung1999reheating] or by classical gravitational effects [@chung2001gravSDM; @gondoloDMproduction; @bertone].
In this work, such super-heavy Dark Matter decay processes are considered in order to explain the UHE neutrinos of PeV range detected by IceCube. These heavy Dark Matter particles, as shown in Ref. [@parton] can decay through hadronic as well as leptonic cascades to finally produce known Standard Model particle-antiparticle pairs [@parton; @heavyDMdecay_PeVnu2018] among which all the three possible active flavours of neutrinos are also included. This may be mentioned in passing that the Dark Matter self-annihilation profile is mostly peaked at the centre of the galaxy due to galactic anisotropy constraints [@IND-ice]. But for the decay process, galactic anisotropy constraints are weaker [@IND-ice; @galacticPeV1; @galacticPeV2; @galacticPeV3]. Although the decay modes are quite model dependent, we adopt in this work the decay processes and the methodology given by Berezinsky et al. in Ref. [@parton] following Alterelli-Parisi formalism for QCD cascades. As described in Ref. [@parton], we in this work also adopted both the hadronic and leptonic channels of such a super-heavy Dark Matter decay to obtain the diffused muon neutrino flux for the decay processes from both galactic and extra-galactic origins. We first calculate the muon neutrino flux by considering only the hadronic channels of the super-heavy Dark Matter decay and fitted these results with the given PeV range harder spectrum obtained by IceCube from their detection of muon neutrinos in that region. This region is designated by a pink colour band in Fig. 2 from Ref. [@IC_PoS2017]. From this analysis the best fit value of Dark Matter mass and decay lifetime have been obtained. We then modify our analysis by including the decay processes through leptonic channel and by extending the data-set upto energy 50 PeV. This has modified our fitted values.
We also explore the baryon asymmetry in the Universe from such Dark Matter decay. Estimation of the baryon asymmetry from the Dark Matter decay time and mass are described in Ref. [@wimpzilla_decay]. In this work, as mentioned earlier, we consider a Dark Matter decay to have produced the PeV range neutrinos detected by IceCube and obtain the mass and the decay lifetime of such Dark Matter decay from the analysis of the IceCube Data. These are then used to calculate the baryon asymmetry of the Universe following the formalism given in Ref. [@wimpzilla_decay].
The paper is organised as follows. In section 2, the formalism for computing neutrino flux from Dark Matter decay is discussed. Also discussed in section 2, is the computation methodology of the baryon asymmetry for a decaying Dark Matter of a given mass and lifetime. In Section 3, we furnish the calculations and computational details. The results are discussed in section 4. Finally, in section 5, we conclude with a summary and discussion.
Formalism
=========
As mentioned before, heavy Dark Matter particles produced in the very early Universe can decay to highly energetic Standard Model particle-antiparticle pairs. It can also produce gamma rays. The flux of these decay products are termed as “fluxes at production". Further, from these particle-antiparticle pairs gamma ray, electron neutrino, muon neutrino, tau neutrino, positron, antideuteron, antiproton are produced. These are termed as secondary products. These secondary neutrinos suffer flavour oscillation while propagating towards Earth. So, the neutrino flux observed on Earth would be a mixed one. Recently in Refs. [@neronov2018; @heavyDMdecay_PeVnu2018], it has been mentioned that the UHE neutrino flux which had been observed at IceCube has a galactic contribution as well, though overall the neutrino flux is assumed to be dominantly extra-galactic. So, both galactic and extra-galactic flux for the decay processes must be included in the calculations and computations.
High energy neutrinos may also be produced from several other astrophysical sources through the processes involving highly energetic proton accelerating mechanisms. Usually, in this mechanism, the protons interact either with themselves (pp interactions) or with photons (p$\gamma$ interaction) or both to finally produce the UHE neutrinos. Such astrophysical sources may include extragalctic Supernova Remnants (SNR) [@SNR], Active Galactic Nuclei (AGN) [@AGN-nu; @AGN1; @AGN2; @AGN3], Gamma Ray Bursts (GRBs) [@grb-nu; @GRB] etc.
In this work, we also consider such astrophysical neutrino flux for our analysis and include the neutrino events detected by IceCube within the energy range $\sim 60 - 120$ TeV. Recently the authors of Refs. [@chianese; @bhupal] has given a formalism for the astrophysical neutrino flux. We in this work, follow the same formalism to compute the astrophysical neutrino flux for further analysis. Therefore, the total diffused flux for neutrinos consists of three components, namely galactic, extra-galactic and astrophysical flux as given below. $$\begin{aligned}
{\bigg( \frac{d\phi_{\nu}}{d\Omega dE_{\nu}} \bigg)}_{\rm th} &=& \frac{d\phi_{\nu}^{\rm G}}{d\Omega dE_{\nu}} + \frac{d\phi_{\nu}^{\rm EG}}{d\Omega dE_{\nu}} + \frac{d\phi_{\nu}^{\rm ast}}{d\Omega dE_{\nu}}.\end{aligned}$$ In the above, $\frac{d\phi_{\nu}^{G}}{d\Omega dE_{\nu}}$ and $\frac{d\phi_{\nu}^{EG}}{d\Omega dE_{\nu}}$ have contributions from both hadronic and leptonics channels of the Dark Matter decay cascade and $E_{\nu}$ denotes the neutrino energy and $\Omega$ is the solid angle. Thus, the Eqn. (1) is written as, $$\begin{aligned}
{\bigg( \frac{d\phi_{\nu}}{d\Omega dE_{\nu}} \bigg)}_{\rm th} &=& {\bigg( \frac{d\phi_{\nu}^{\rm G}}{d\Omega dE_{\nu}} \bigg)}_{\rm had} + {\bigg( \frac{d\phi_{\nu}^{\rm G}}{d\Omega dE_{\nu}} \bigg)}_{\rm lep} + \nonumber \\
&&{\bigg ( \frac{d\phi_{\nu}^{\rm EG}}{d\Omega dE_{\nu}} \bigg)}_{\rm had} + {\bigg ( \frac{d\phi_{\nu}^{\rm EG}}{d\Omega dE_{\nu}} \bigg)}_{\rm lep} + \nonumber \\
&& \frac{d\phi_{\nu}^{\rm ast}}{d\Omega dE_{\nu}}.\end{aligned}$$
For galactic Dark Matter decay, the secondary muon neutrinos are produced. The differential neutrino flux per solid angle is given by [@diff-flux], $$\begin{aligned}
\frac{d\phi_{\nu}^{G}}{d\Omega dE_{\nu}} &=& \frac{1}{4{\pi}{\alpha}M_X}\Gamma_{dec} \int_{los} dl \frac{dN}{dE} \rho[r(l,\theta)].\end{aligned}$$ Here, $M_X$ is the mass of the Dark Matter particle and $\alpha = 1$ for a Majorana-type particle. In order to obtain ultra-high energy PeV neutrinos, $M_X \geq 10^6$ GeV. The decay width is denoted as $\Gamma_{dec}$. The integral $\int_{los} \rho{[r(l, \theta)]}^2 dl$ is known as the line of sight integral, whereas $\rho(r)$ is the Dark Matter halo density profile which is a function of position from the centre of the halo distribution. Here we have used the Navarro-Frenk-White (NFW) profile as it provides nearly accurate result for a large range of Dark Matter mass and properly describes the cuspy nature of DM distribution which is widely accepted. The NFW profile is given by [@nfw1; @nfw2], $$\begin{aligned}
\rho_{\rm NFW}(r) &=& \rho_s \frac{r_s}{r}{ \bigg (1 + \frac {r_s}{r} \bigg )}^{-2}\,\, ,\end{aligned}$$ where $\rho_s$ = 0.259 $\rm{GeV/cm^{3}}$ and $r_s$ = 20 kpc.
For NFW profile, the position from the centre of the halo distribution is given by, $$\begin{aligned}
r &=& \sqrt{r^2_\odot\ + l^2 - 2 r_\odot\ l \cos\theta}\,\, ,\end{aligned}$$ where $l$ is the line of sight distance which is in this case, the distance between Earth and the Galactic Centre. The quantity $r_\odot\ $ has been considered to be $\sim 8.5$ kpc, distance between the observer located at solar system and the centre of the Dark Matter halo. The azimuthal angle, $\theta \sim \rm{0.5}$ between Earth and Galactic Centre. In Eqn. (3) the neutrino spectrum $\frac{dN}{dE}$ for channel $i$ is obtained from the simulation described in Ref. [@heavyDMdecay_PeVnu2018; @parton]. As we are considering super-heavy Dark Matter candidates, heavier hadronic and leptonic channels are going to contribute. For super-heavy Dark Matter particles of mass $M_X$, the lifetime $\tau$ is obtained from, $$\tau = \frac{1}{g^2 M_{X}} {\bigg (\frac{M_{pl}}{M_X} \bigg)}^2\,\, .$$ In the above, $M_{pl}$ denotes the Planck mass and $g$ is the coupling constant. Thus, evaluation of $\tau$, which is the inverse of $\Gamma_{dec}$, is model dependent. For heavy Dark Matter particle candidates, the value of $\tau$ should be at least $10^{17}$ seconds which is comparable to the age of the Universe. We can now compute the differential flux of muon neutrino termed as the primary differential muon neutrino flux corresponding to a single decay channel.
The differential neutrino flux for extra-galactic neutrinos in the present scenario is given as, $$\frac{d\phi_{\nu}^{EG}}{d\Omega dE_{\nu}} = \frac{1}{4{\pi}{\alpha}M_X}\Gamma_{dec} \int_{0}^{\infty} \frac{\rho_0 c/H_0}{\sqrt{\Omega_m (1 + z)^3 + (1 - \Omega_m)}} \frac{dN}{dE_z} dz\,\, ,$$ where $\rho_0$ is the present average cosmological Dark Matter density which has a value $1.15 \times 10^{-6}$ GeV/$cm^3$. The quantity $c/H_0 = 1.37 \times 10^{28}$ cm is known as the Hubble radius with $H_0$ as the Hubble cosntant. We consider the matter density $\Omega_m = 0.308$ [@heavyDMdecay_PeVnu2018]. In Eqn. (7) $\frac{dN}{dE_z}$ is the spectrum of the decay product which is evaluated as a function of the redshifted energy $E_z = E(z) = E(1 + z)$. With $$\begin{aligned}
\frac{dN}{dE_z} = \frac{dN}{dE} \frac{dE}{dE_z}\,\, .\end{aligned}$$ and $E = \frac{E(z)}{1 + z}$, we have, $$\begin{aligned}
\frac{dN}{dE_z} = \frac{dN}{dE} \frac{1}{(1 + z)}\,\, .\end{aligned}$$ The quantity $\frac{dN}{dE}$ in Eq. (3) and (7) are obtained by numerically solving the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations [@Gribov; @altarelli; @dokshitzer] related to the QCD cascade that is produced by the decaying superheavy dark matter. This spectrum has two parts namely hadronic and leptonic, originated due to the hadronic and the leptonic decay channels of the superheavy dark matter. The neutrino spectrum is given by [@heavyDMdecay_PeVnu2018] $$\displaystyle\frac {dN_{\nu}} {dx} = 2R \int_{xR}^{1} \displaystyle \frac
{dy} {y} D^{{\pi}^{\pm}} (y) + 2 \int_{x}^{1} \displaystyle \frac {dz} {z}
f_{\nu_i}
\left (\displaystyle\frac {y} {z} \right) D^{{\pi}^{\pm}}_i (z)\,\, ,
\label{form1}$$ where the parton fragmentation functions for pions are given as $D^{\pi}_i (x,s)$ ($\equiv [D_{q}^{\pi} (x,s) + D_{g}^{\pi} (x,s)]$, $i (=q(=u,d,s, ...),g)$ and $D^{\pi}_i (x,s)$, with $x (\equiv 2E/m_\chi)$ is a dimensionless quantity with $\sqrt{s}$ being the centre of mass energy. The functions $f_{\nu_i} (x)$ in the above equation are given in [@kelner] and are used in the present computation while $R = \displaystyle\frac {1} {1-r}$, where $r = (m_\mu/m_\pi)^2 \simeq 0.573$. For present analysis, the flavour ratio of neutrinos at the Earth is taken to be $\nu_e : \nu_{\mu} : \nu_{\tau} = 1:1:1$.
Furthermore, the consequence of late time decay of such heavy mass Dark Matter has significant implication on the aspect of matter-antimatter asymmetry as well because it can generate baryon and lepton number asymmetry which satifies Sakharov’s conditions for baryogenesis . To calculate the amount of asymmetry due to heavy Dark Matter decay, a Boltzmann-like equation needs to be solved to calculate the rate of baryon number production per decay. The equation is given as follows [@wimpzilla] $$\begin{aligned}
\frac{d(n_{b} - n_{\bar{b}})}{dt} + 3 \frac{\dot{a}(t)}{a(t)} (n_{b} - n_{\bar{b}}) = \frac{\epsilon n_X (t)}{\tau}\,\, ,\end{aligned}$$ where, $n_X (t)$ is the number density of the Dark Matter particles at the epoch of redshift $z$. $n_b$, $n_{\bar{b}}$ are the baryon and antibaryon number densities, $n_X$ is the number density of decaying DM (with decay time $\tau$) and $a$ is the scale factor. $\epsilon$ is the total baryon number violation per decay. Now from the standard cosmological scaling relations, the number density of any particle $\sim$ $a^{-3}$. With $a = \frac{1}{1 + z}$, $$\begin{aligned}
n_X &\propto& (z + 1)^3 \\
n_X (t) &=& n_X (t_0) \frac{(z + 1)^3}{(z_0 + 1)^3}\,\,\end{aligned}$$ ($t_0$, $z_0$ are the initial time and redshift respectively). Also, $$\begin{aligned}
n_{\gamma} &\propto& (z + 1)^3 \\
n_{\gamma} (t) &=& n_{\gamma} (t_0) \frac{(z + 1)^3}{(z_0 + 1)^3}\,\, .\end{aligned}$$ We define, $n_X (t_0)$ & $n_{\gamma} (t_0)$ is the number density of Dark Matter particles and photons respectively at time $t_0$ at redshift $z_0$ after Big Bang.
Solving this equation for time $t_0$ to $t$ which corresponds to the epochs of the Universe with redshifts $z_0$ to z within which the baryon-antibaryon difference $(n_b - n_{\bar{b}})$ evolves, we get, $$\begin{aligned}
\Delta (n_{b} - n_{\bar{b}}) &=& \frac{\epsilon n_X (t_0)}{3} \left [ 1 - \exp \left (-\frac{t - t_0}{\tau} \right ) \right ]\frac{{(1 + z)}^3}{{(1 + z_{0})}^3}\,\, .\end{aligned}$$ Defining the amount of baryon asymmetry $\Delta B$ as $\Delta B = \frac{(n_{b} - n_{\bar{b}})}{2 g_{*} n_{\gamma} (t)}$, it can be shown that (see Appendix), $$\begin{aligned}
\Delta B &=& \frac{\epsilon n_X(t_{0})}{3 \cdot 2 g_{*} n_{\gamma}(t_{0})} \left [ 1 - \exp \left (-\frac{t - t_{0}}{\tau} \right ) \right ]\,\, .\end{aligned}$$ In the comoving frame, we define $t_0 = t_{dec}$ to be the time when the Universe’s plasma decoupled from the CMB photons at redshift $z_{dec} \simeq 1100$. Thus, Eqn. (19) is rewritten as $$\begin{aligned}
\Delta B &=& \frac{\epsilon n_X(t_{dec})}{3 \cdot 2 g_{*} n_{\gamma}(t_{dec})} \left [ 1 - \exp \left (-\frac{t - t_{dec}}{\tau} \right ) \right ]\,\, .\end{aligned}$$ The number density of Dark Matter particles during the epoch of recombination, $n_X (t_{dec})$ is given by, $$\begin{aligned}
n_X (t_{dec}) &=& n_X (0) \frac{(1 + z_{dec})^3}{(1 + z(0))^3}\,\, , \\
n_X (t_{dec}) &=& \frac{\rho_{c}[\Omega_{m} - \Omega_{hot} - \Omega_{b} (1 + \frac{m_{e}}{m_{p}})]}{M_X} {(1 + z_{dec})}^3 \,\, .\end{aligned}$$ Here, $n_X (0)$ is the number density of Dark Matter particles at the present epoch at redshift $z(0) = 0$ and $\rho_{c}$ is the critical density of the Universe at present epoch given as, $$\begin{aligned}
\rho_{c} &=& \frac{3 {H_{0}}^2}{8 \pi G}\,\, .\end{aligned}$$ Here, $H_{0} = 67.27 \pm 0.60$ km $\rm sec^{-1} Mpc^{-1}$ is the Hubble’s constant [@planck18] and $\Omega_{hot}$ is given by, $$\begin{aligned}
\Omega_{hot} = \frac{{\pi}^2}{30} g_{*} {T_{CMB}}^4\,\, ,\end{aligned}$$ where $g_{*}$ is known as effective degrees of freedom and $T_{CMB} \sim 2.73$ K. $\tau$ is the lifetime of the Dark Matter particle. From Eqns. (13-22) with $n_{\gamma} (t_0) = 410 \rm cm^{-3}$, the CMB photon number density, $\Delta B$ can be computed for a given set of values of $M_X$ and $\tau$. As mentioned earlier, we have used the best fit values of $M_X$ and $\tau$ obtained from the present $chi^2$ fit of IceCube PeV neutrino data.
Calculations and Results
========================
We consider the IceCube neutrino data within in the energy region $\sim$ 60 TeV - $\sim 5 \times 10^7$ GeV for the present calculation. The data are obtained from Fig. 2 of Ref. [@IC_PoS2017] given by the IceCube Collaboration. The data having neutrino energy $E_\nu >$ 20 TeV are referred to as the HESE (high energy starting events) data by the IceCube Collaboration (IC). The HESE data have been fitted with the single power law $\sim E^{-\gamma}$ and $\gamma$ is obtained as $2.92^{+ 0.33}_{- 0.29}$ by the IC. However the region between $\sim 120\,\, {\rm TeV} - \sim 5\,\, {\rm PeV}$ exhibits the different power law and this is designated as the pink band in Fig. 2 of Ref. [@IC_PoS2017].
In the said figure the pink band contains three actual data points and a best fit line whereas the width of the pink band indicated the 1$\sigma$ uncertainty. In this work we consider from the pink band region, three actual data points and adopt 12 other points suitably chosen from the best fit line within the pink band with the widths of the pink band at a particular point to be the error corresponding to that chosen data point. The four data points beyond the energy range $\sim$ 5 $\times 10^6$ GeV in the same figure (from IceCube Collaboration) have only upper bounds and the nature of those four points differs in considerable extent from that of the pink band. We also consider these points in our analysis. As discussed earlier the UHE neutrino signals possibly from possible astrophysical sources are also included in the present analysis. This is represented by two data points of the same figure within the energy range $\sim$ 60 TeV to $\sim$ 120 TeV. In our analysis, we consider a spectrum of $\sim E^{-2.9}$ for the astrophysical part of the flux.
The $\chi^2$ is defined as $
\chi^2 = \sum_{i = 1}^{n} \left (\displaystyle\frac {E_i^2 \phi_i^{\rm th} -
E_i^2 \phi_i^{\rm Ex}} {({\rm err})_i} \right )^2 \,\, ,
$ where $\phi_i^a$ ($= \left (\frac{d\phi_{\nu}}{d\Omega dE_\nu} \right)_a, a \equiv $ th or Ex) is theoretical or experimental neutrino flux (with error (err)$_i$) for energy $E_i$, of the $i^{th}$ data point of $n$ number of total data points. The chosen data points are tabulated in Table 1. In Table 1 the data points marked ‘$\ast$’ are the actual data points given in Fig. 2 of Ref. [@IC_PoS2017] while the other points are chosen from within the pink band. Also note that for the last four data points (corresponding to the energy range $\sim$ 5 $\times 10^6$ GeV - $\sim$ 5 $\times 10^7$ GeV) no errors are given since only the upper limits of these four points are given in Fig. 2 of Ref. [@IC_PoS2017]. However for the present $\chi^2$ fit, the errors for these four points are chosen to be the same as the value of those points only.
--------------------------- ---------------------------------------------------------------------------- ----------------------
Energy Neutrino Flux ($E_\nu^2 \displaystyle\frac {d\Phi_{\nu}}{d\Omega dE_\nu}$) [Error]{}
(in GeV) (in GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$)
6.13446$\times 10^4\, ^*$ 2.23637$\times 10^8$ 2.16107$\times 10^8$
1.27832$\times 10^5\, ^*$ 2.70154$\times 10^8$ 1.30356$\times 10^8$
2.69271$\times 10^5\, ^*$ 7.66476$\times 10^9$ 8.5082$\times 10^9$
1.19479$\times 10^6\, ^*$ 5.14335$\times 10^9$ 7.6982$\times 10^9$
2.51676$\times 10^6\, ^*$ 4.34808$\times 10^9$ 8.4481$\times 10^9$
3.54813$\times 10^6$ 5.25248$\times 10^9$ 4.1258$\times 10^9$
2.30409$\times 10^6$ 5.71267$\times 10^9$ 4.1600$\times 10^9$
1.52889$\times 10^6$ 6.21317$\times 10^9$ 3.9882$\times 10^9$
1.05925$\times 10^6$ 6.61712$\times 10^9$ 3.7349$\times 10^9$
7.18208$\times 10^5$ 7.04733$\times 10^9$ 3.9777$\times 10^9$
4.46684$\times 10^5$ 7.66476$\times 10^9$ 3.6478$\times 10^9$
2.86954$\times 10^5$ 8.16308$\times 10^9$ 4.1571$\times 10^9$
1.90409$\times 10^5$ 8.87827$\times 10^9$ 6.2069$\times 10^9$
1.43818$\times 10^5$ 9.65612$\times 10^9$ 6.8856$\times 10^9$
2.51189$\times 10^6$ 4.16928$\times 10^9$ 8.2726$\times 10^9$
1.19279$\times 10^6$ 5.03649$\times 10^9$ 7.5383$\times 10^9$
2.68960$\times 10^5$ 7.50551$\times 10^9$ 8.1583$\times 10^9$
5.30143$\times 10^6\, ^*$ 1.55414$\times 10^9$
1.10473$\times 10^7\, ^*$ 4.08265$\times 10^9$
2.32705$\times 10^7\, ^*$ 6.08407$\times 10^9$
4.90181$\times 10^7\, ^*$ 1.05021$\times 10^8$
--------------------------- ---------------------------------------------------------------------------- ----------------------
Table 1. The chosen data points for the $\chi^2$ fit. See text for details.\
![The neutrino flux with the best fit values of $m_x$ and $\tau$ by considering all points ($\sim 60$ TeV - $\sim 50$ PeV) and both hadronic and leptonic channels. See text for details.[]{data-label="Fig:1"}](Flux.eps){width="14cm" height="10cm"}
With these data points the $\chi^2$ fit is performed where the theoretical data points are computed following the formalism discussed earlier. The $\chi^2$ analysis includes the contributions from the astrophysical flux and the flux calculated from the decay of superheavy dark matter via hadronic as well as leptonic channels and the best fit values for the mass of the decaying DM and the decay lifetime are obtained. These best fit values are $M_X = 1.5461 \times 10^8$ GeV and $\tau = 2.2136 \times 10^{29}$ sec. The data points and the best fit curve for the neutrino flux are shown in Fig. 1. The pink band is also shown in the same figure.
It is to be mentioned that we have made several other fits adopting the data from different energy zones (Table 1). We find that the region of the pink band is best represented if we consider only the hadronic channel for the dark matter decaying to neutrinos [@pandey2019]. We have also observed from our analyse that the contribution of the leptonic channel plays a major role in interpreting the last four points of Table 1 while the astrophysical contribution to the flux for the energy range considered is limited to the energy range $\sim$ 60 TeV to $\sim$ 120 TeV (the first two data points in Table 1).
With the best fit values of $M_X$ and $\tau$ we now estimate $\Delta B$, the amount of baryon asymmetry generated out of the superheavy dark matter decay following the formalism given in the previous section (also in Appendix). Thus for $M_X = 1.5461 \times 10^8$ Gev and $\tau = 2.2136 \times 10^{29}$ the amount of baryon asymmetry is calculated as $ \Delta B = 1.44 \times 10^{-10}$.
Summary and Discussions
=======================
In the present work, we have considered the UHE neutrino data given by IceCube between the energy range $\sim 60$ TeV - $\sim 50$ PeV and proposed the possibility that these neutrinos could have been produced by the decay of super-heavy Dark Matter (SHDM). The SHDM can decay to neutrinos by following mainly two cascade channels namely hadronic and leptonic. It appears that while the neutrino events in the energy range $\sim 120$ TeV - $\sim 5$ PeV (pink band) can be best represented by hadronic decay channel of Dark Matter, the leptonic decay channel is responsible for the behaviour of the neutrino flux for the energy range $\sim 5$ - 50 PeV. We also include the astrophysical flux (with power law behaviour $\sim E^{-2.9}$) in our $\chi^2$ fit for the IC flux in the energy region considered $\sim 60 - 120$ TeV in this work. From the $\chi^2$ fit with the hadronic and leptonic decay channels for SHDM decay as well as the astrophysical flux, the best fit values for $M_X$, the mass of SHDM and $\tau$, the decay lifetime of SHDM are obtained.
The decay of SHDM may invoke the baryon asymmetry in the Universe. The measure of the baryon asymmetry in terms of SHDM mass and decay lifetime is obtained analytically and then the estimation for the same is computed for the best-fit values of SHDM mass and decay lifetime obtained from the present analysis. This may be mentioned that the parameter $\epsilon$ which denotes total baryon number violation per decay in Eqn. (11) (and in other Eqns. that follow) is important in the sense that it embeds the effect related to the possible CP violation which is responsible for the generation of baryon asymmetry. The value of baryon asymmetry thus computed is found to be in the same ball park as given by PLANCK [@planck18] observational results.
Thus we make an attempt in this work to explain the UHE neutrino events at IceCube to have originated from the decay of SHDM and predict the generation of baryon asymmetry from such a scenario.
[**Acknowledgments**]{}
One of the authors (M.P.) thanks the DST-INSPIRE fellowship (DST/INSPIRE/\
FELLOWSHIP/IF160004) grant by Department of Science and Technology (DST), Govt. of India. One of the authors (A.H.) wishes to acknowledge the support received from St.Xavier’s College, Kolkata Central Research Facility and thanks the University Grant Commission (UGC) of the Government of India, for providing financial support, in the form of UGC-CSIR NET-JRF.
[**Appendix**]{}
Solving the Boltzmann-like Equation
===================================
In order to calculate the amount of baryon asymmetry due to heavy Dark Matter decay, a Boltzmann-like equation needs to be solved to evaluate the rate of baryon number production per decay. This is given as follows [@wimpzilla] $$\begin{aligned}
\frac{d(n_{b} - n_{\bar{b}})}{dt} + 3 \frac{\dot{a}(t)}{a(t)} (n_{b} - n_{\bar{b}}) = \frac{\epsilon n_X (t)}{\tau}\,\, ,\end{aligned}$$ where, $n_X (t)$ is the number density of the Dark Matter particles at the epoch of redshift $z$. It is easily recognisable that the above equation is a first order homogeneous linear differential equation of the form, $$\begin{aligned}
\frac{dy}{dt} + P y = Q\,\, ,\end{aligned}$$ where, $$\begin{aligned}
y &=& (n_{b} - n_{\bar{b}})\,\, , \nonumber \\
P &=& 3 \frac{\dot{a}(t)}{a(t)}\,\, , \nonumber \\
Q &=& \frac{\epsilon n_X (t)}{\tau}\,\, .\end{aligned}$$ We solve the Eqn.(23) defining an integrating factor (I.F.) of the form, $$\begin{aligned}
{\rm I.F.} = e^{\int P dt}\,\, .\end{aligned}$$ The solution of the equation would be, $$\begin{aligned}
y \times {\rm I.F.} = Q \int {\rm I.F.} dt\end{aligned}$$ From Eqn. (25), $P = 3 \frac{\dot{a}(t)}{a(t)}$ and thus, $$\begin{aligned}
e^{\int P dt} &=& e^{\int 3 \frac{\dot{a}(t)}{a(t)} dt}\,\, \nonumber \\
&=& a^3\,\, .\end{aligned}$$ From Eqn. (27), the solution is now written as, $$\begin{aligned}
y a^3 = \frac{\epsilon n_X (t)}{\tau} \int_{t_0}^t a^3 dt\,\, ,\end{aligned}$$ where, $t_0$ and $t$ are the corresponding times elapsed since Big Bang for the redshifts $z_0$ and $z$ respectively. Here, $z_0$ is the earlier epoch which can be related to the epoch when the baryon asymmetry was generated and $z$ could be any later epoch of the Universe.
Since during the generation of baryon asymmetry, the Universe is expected to be matter-dominated, from the well-known cosmological scaling relations, we know that for a matter-dominated Universe, $$\begin{aligned}
a &\propto& t^{2/3}\,\, ; \nonumber \\
a^3 &=& a_0 t^2\,\, .\end{aligned}$$ Substituting for $a$ in Eqn. (29), $$\begin{aligned}
y a_0 t^2 &=& \frac{\epsilon n_X (t)}{\tau} a_0 \int_{t_0}^t t^2 dt\,\, ; \nonumber \\
y a_0 t^2 &=& \frac{\epsilon n_X (t)}{\tau} a_0 \frac{a_0}{3} (t^3 - t_{0}^3)\,\, ; \nonumber \\
y &=& \frac{\epsilon n_X (t)}{\tau} \frac{1}{3} (t-t_0) \frac{(t^2 + t t_0 + t_{0}^2)}{t^2}\,\, ; \nonumber \\
y &=& \frac{\epsilon n_X (t)}{3} \frac{t - t_0}{\tau} \bigg[1 + \frac{t_0}{t} + \frac{t_{0}^2}{t^2}\bigg] \,\, .\end{aligned}$$ By definition, $\frac{t_0}{t}$ and $\frac{t_{0}^2}{t^2} << 1 \simeq 0$. We can approximate, $$\begin{aligned}
\frac{t - t_0}{\tau} = 1 - \left [ 1 - \frac{t - t_0}{\tau} \right ] \simeq 1 - \exp \left ( -\frac{t - t_0}{\tau} \right )\end{aligned}$$ Therefore, Eqn. (31) takes the form, $$\begin{aligned}
y = \frac{\epsilon n_X (t)}{3} \bigg [ 1 - \exp \left ( -\frac{t - t_0}{\tau} \right ) \bigg]\end{aligned}$$ Using the relation defined in Eqns. (12-13), it will have a final form which is given below. $$\begin{aligned}
y &=& \frac{\epsilon n_X (t_0)}{3} \frac{(z + 1)^3}{(z_0 + 1)^3} \left [ 1 - \exp \left ( -\frac{t - t_0}{\tau} \right ) \right ]\,\, ; \nonumber \\
y = (n_{b} - n_{\bar{b}}) &=& \frac{\epsilon n_X (t_0)}{3} \frac{(z + 1)^3}{(z_0 + 1)^3} \left [ 1 - \exp \left ( -\frac{t - t_0}{\tau} \right ) \right ]\,\, .\end{aligned}$$
[^1]: email: [email protected]
[^2]: email: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
A very particular connection between the commutation relations of the elements of the generalized Pauli group of a $d$-dimensional qudit, $d$ being a product of distinct primes, and the structure of the projective line over the (modular) ring ${{\mathbb Z}}_{d}$ is established, where the integer exponents of the generating shift ($X$) and clock ($Z$) operators are associated with submodules of ${{\mathbb Z}}^{2}_{d}$. Under this correspondence, the set of operators commuting with a given one — a perp-set — represents a ${{\mathbb Z}}_{d}$-submodule of ${{\mathbb Z}}^{2}_{d}$. A crucial novel feature here is that the operators are also represented by [*non*]{}-admissible pairs of ${{\mathbb Z}}^{2}_{d}$. This additional degree of freedom makes it possible to view any perp-set as a [*set-theoretic*]{} union of the corresponding points of the associated projective line.\
[**PACS Numbers:**]{} 03.65.-a – 03.65.Fd – 02.10.Hh – 02.40.Dr
[**Keywords:**]{} Qudit – Generalized Pauli Group – Projective Ring Line – Commutation Algebra of Generalized Pauli Operators
author:
- |
Hans Havlicek$^{1}$ and Metod Saniga$^{2}$\
\
$^{1}$Institut f" ur Diskrete Mathematik und Geometrie\
Technische Universit" at Wien, Wiedner Hauptstrasse 8-10\
A-1040 Vienna, Austria\
([email protected])\
\
$^{2}$Astronomical Institute, Slovak Academy of Sciences\
SK-05960 Tatransk' a Lomnica, Slovak Republic\
([email protected])
title: Projective Ring Line of a Specific Qudit
---
Introduction
============
The study of the finite-dimensional Hilbert spaces and their associated generalized Pauli operators has been a forefront issue of the quantum information theory within the past few years. A substantial mathematical insight has been possible thanks to a number of novel graph-combinatorial and algebraic geometrical concepts employed, see, e.g., [@ara]–[@psk] and references therein. Among the latter, it is the concept of a projective line defined over a(n associative) ring with unity that acquired a distinguished footing [@psk]–[@qic]. In this approach, one simply identifies the points of a projective ring line with the generalized Pauli operators (or the maximum commuting sets of them) pertaining to a given Hilbert space and rephrases their commutation relations in terms of neighbour/distant relations between the points on the line in question. Given this identification, it was possible to “projective-ring-geometrize" any $N$-qubit Hilbert space [@psk]–[@adv], two-qutrits [@spie; @qic], as well as to get important hints about the smallest composite case, viz. a six-dimensional Hilbert space [@pbs]. A detailed examination of these particular cases led soon to a discovery of a more complex and unifying approach based on group-theoretical considerations [@koen1; @koen2]. Adopting and properly generalizing the strategy pursued in the last two mentioned papers, we shall demonstrate, on the example of a specific single qudit, that the concept of a projective ring line naturally emerges also in a context slightly different from that introduced and elaborated in [@psk]–[@pbs], with the finest traits of the structure of the projective line coming into play.
The Pauli group $G$ of a single qudit
=====================================
Let $d>1$ be an integer and ${{\mathbb Z}}_d:=\{0,1,\ldots,d-1\}$. Addition and multiplication of elements from ${{\mathbb Z}}_d$ will always be understood modulo $d$.
We consider the $d$-dimensional complex Hilbert space ${{\mathbb C}}^d$ and denote by $$\{\, |s{\rangle}: s\in{{\mathbb Z}}_d \}$$ a computational basis of ${{\mathbb C}}^d$. Furthermore, let $\omega$ be fixed a primitive $d$-th root of unity (e.g., $\omega=\exp(2\pi i/d)$).
Now $X$ and $Z$ are unitary “shift" and “clock" operators on ${{\mathbb C}}^d$ defined via $X|s{\rangle}=
|s+1{\rangle}$ and $Z|s{\rangle}= \omega^s |s{\rangle}$, respectively, for all $s\in{{\mathbb Z}}_d$. With respect to our computational basis the matrices of $X$ and $Z$ are $$\begin{pmatrix}
0 & 0 &\ldots &0 & 1\\
1 & 0 &\ldots &0 & 0\\
0 & 1 &\ldots &0 & 0\\
\vdots &\vdots &\ddots &\vdots &\vdots\\
0 & 0 &\ldots &1 & 0
\end{pmatrix}
\mbox{~~and~~}
\begin{pmatrix}
1 & 0 &0 &\ldots& 0\\
0 &\omega &0 &\ldots& 0\\
0 & 0 &\omega^2 &\ldots& 0\\
\vdots &\vdots &\vdots &\ddots &\vdots\\
0 & 0 & 0 &\ldots &\omega^{d-1}
\end{pmatrix},$$ respectively.
The subgroup of the unitary group $\operatorname{U}_d$ generated by $X$ and $Z$, known by physicists as the (generalized) *Pauli group*, will be written as $G$. The operators $X^0=: I, X^1,\ldots,X^{d-1}$ form a cyclic subgroup of $G$ with order $d$; the same properties hold for $Z^0,Z^1,\ldots,
Z^{d-1}$. Hence $$\label{eq:d-1}
X^{d-1}=X^{-1} \mbox{~~and~~} Z^{d-1}=Z^{-1}.$$ For all $s\in{{\mathbb Z}}_d$ we have $XZ|s{\rangle}= \omega^s|s+1{\rangle}$ and $ZX|s{\rangle}=
\omega^{s+1}|s+1{\rangle}$. This gives the basic relation $$\label{eq:xzzx}
\omega XZ = ZX.$$ By virtue of (\[eq:d-1\]) and (\[eq:xzzx\]), each element of $G$, usually referred to as a (generalized) *Pauli operator*, can be written in the *normal form* $$\label{eq:normalform}
\omega^a X^b Z^c \mbox{~~for some integers~~}a,b,c\in {{\mathbb Z}}_d.$$ It is easy to see that this representation in normal form is *unique*: From $\omega^a X^b Z^c = \omega^{a'} X^{b'} Z^{c'}$ follows $\omega^{a-a'}
X^{b-b'} Z^{c-c'}=I$. As $|0{\rangle}$ remains fixed under $Z^{c-c'}$ we obtain $\omega^{a-a'} X^{b-b'}|0{\rangle}=|0{\rangle}$. This shows $b-b'=0$ and $a-a'=0$. Thus $Z^{c-c'}=I$ which implies $c-c'=0$, as required. The uniqueness of the normal form (\[eq:normalform\]) will be crucial for our further exhibition.
We immediately may read off from (\[eq:xzzx\]) the following rule for multiplication in $G$, when the factors are given in normal form: $$(\omega^a X^b Z^c) (\omega^{a'} X^{b'} Z^{c'}) = \omega^{b'c + a+a'}
X^{b+b'} Z^{c+c'}.$$ Observe that the product is also in normal form. The term $b'c$ in the exponent of $\omega$ on the right hand side shows that $G$ is a non-commutative group. The uniqueness of the normal form implies also that $G$ is a group of order $|G|=d^3$.
The *commutator*[^1] of two operators $W$ and $W'$ is $$\label{eq:defcommutator}
[W,W'] := W W'W^{-1}{W'} ^{-1}.$$ If $W=\omega^a X^b Z^c$ and $W'=\omega^{a'} X^{b'}Z^{c'}$ are given in normal form then it is immediate from (\[eq:xzzx\]) that $$\label{eq:commutator}
[\omega^a X^b Z^c, \omega^{a'} X^{b'}Z^{c'}] = \omega^{cb'-c'b}I.$$ Recall that two operators commute if, and only if, their commutator (taken in any order) is equal to $I$.
We shall be concerned with two important normal subgroups of $G$:
The *centre* $Z(G)$ of $G$ is the set of all operators in $G$ which commute with every operator in $G$. An operator $\omega^a X^b Z^c$ given in normal form lies in $Z(G)$ precisely when (\[eq:commutator\]) holds for any choice of $a'$, $b'$, and $c'$. Setting $b':=0$, $c':=1$ we get $b=0$, whereas $b':=1$ and $c':=0$ gives then $c=0$. These necessary conditions are also sufficient, whence $$\label{eq:centre}
Z(G) = \{\omega^a I: a\in{{\mathbb Z}}_d \}.$$ Note that $Z(G)$ is yet another cyclic subgroup of $G$ with order $d$.
The *commutator subgroup* $[G,G]$ is the smallest subgroup of $G$ which contains all commutators $[W,W']$ with $W,W'\in G$. We follow the usual convention to denote the commutator subgroup of $G$ by $G'$. From $[Z,X]=\omega
I$ follows that all powers of $\omega I$ are elements of $G'$. On the other hand (\[eq:commutator\]) shows that there are no other commutators but the powers of $\omega I$. Altogether we obtain $$\label{eq:G'}
G' = Z(G) = \{\omega^a I: a\in{{\mathbb Z}}_d \}.$$ It is easy to see from (\[eq:commutator\]) that each element of $G'$ is indeed a commutator, a property which need not be true for the commutator subgroup of an arbitrary group.
The ring associated with $G$
============================
By expressing the elements of our group $G$ in normal form we saw already that several basic algebraic relations can be expressed solely in terms of the exponents of $\omega$, $X$ and $Z$. These exponents are always elements of the *ring* $({{\mathbb Z}}_d,+,\cdot)$ of integers modulo $d$. To be more precise, this ring is unital ($1b=b$ for all $b\in{{\mathbb Z}}_{d}$) and commutative ($bc=cb$ for all $b,c\in{{\mathbb Z}}_d$). An element $b\in{{\mathbb Z}}_d$ is a unit (an invertible element) if, and only if $b$ and $d$ are coprime. If $d$ is a prime then every non-zero element of ${{\mathbb Z}}_d$ is invertible and ${{\mathbb Z}}_d$ is a field, otherwise there are non-invertible elements — see, e.g., [@mcd; @rag] for more details.
We show now that the ring ${{\mathbb Z}}_d$ “lives”, up to isomorphism, also within our group $G$. Let us consider the bijective mapping $$\psi : {{\mathbb Z}}_d \to G' : a \mapsto \omega^a I.$$ This is an isomorphism of the additive group $({{\mathbb Z}}_d,+)$ onto the multiplicative group $(G',\cdot)$, since clearly $$\psi(a+a')=\omega^{a+a'}I = \psi(a)\cdot\psi(a')
\mbox{~~for all~~} a,a'\in{{\mathbb Z}}_d.$$ However, in ${{\mathbb Z}}_d$ we also have the binary operation of multiplication. We obtain its counterpart in $G'$ via $$\psi(aa')=\omega^{aa'}I = (\omega^a I)^{a'}=\psi(a)^{a'}
\mbox{~~for all~~} a,a'\in{{\mathbb Z}}_d.$$ Thus we *could* use the bijection $\psi$ to turn $G'$ into an isomorphic copy of the ring $({{\mathbb Z}}_d,+,\cdot)$ by defining “new” binary operations on $G$ in accordance with the two formulas from the above. However, we refrain from doing so in order to avoid misunderstandings. (The “new” addition would be the “old” multiplication.) It is nevertheless important to emphasise that such a construction is possible.
A symplectic module associated with $G$ and the commutation algebra of Pauli operators
======================================================================================
As $(G,\cdot)$ is a non-commutative group, it cannot be isomorphic to the additive group of any module. Recall that the factor group of $G$ by any normal subgroup is commutative if, and only if, this normal subgroup contains the commutator subgroup $G'$. This means that the “largest” commutative group we can obtain from $G$ by factorisation is the factor group $$\label{eq:factorgroup}
G/G'.$$ Taking into account our normal form (\[eq:normalform\]) and the description of $G'$ in (\[eq:G’\]), the group $G/G'$ comprises all cosets $$\label{normalform'}
G'X^bZ^c \mbox{~~where~~}b,c\in{{\mathbb Z}}_d.$$ Each element of $G/G'$ can be written in a *unique* way in this *normal form*. As a by-product of this uniqueness, we learn from (\[normalform’\]) that the factor group $G/G'$ has order $d^2$. Multiplication in $G/G'$ is governed by the formula $$\label{eq:mult'}
(G'X^bZ^c)(G'X^{b'}Z^{c'}) = G'X^{b+b'} Z^{c+c'}
\mbox{~~for all~~} b,c,b',c'\in{{\mathbb Z}}_d.$$
Let us consider the bijective mapping $$\phi : {{\mathbb Z}}_d^2 \to G/G' : (b,c) \mapsto G'X^bZ^c.$$ Note that the elements of ${{\mathbb Z}}_d^2$ are written as rows. Sometimes they will be called *vectors*. We now consider $({{\mathbb Z}}_d^2,+)$ as a commutative group with the addition ($+$) defined componentwise. Then (\[eq:mult’\]) establishes immediately that $\phi$ is an isomorphism of the additive group $({{\mathbb Z}}_d^2,+)$ onto the multiplicative group $(G/G',\cdot)$.
But ${{\mathbb Z}}_d^2$ is also a module over ${{\mathbb Z}}_d$ in the usual way. Thus we *could* use the bijections $\psi:{{\mathbb Z}}_d\to G'$ and $\phi:{{\mathbb Z}}_d^2\to G/G'$ to turn $G/G'$ into an isomorphic module over $G'$. Like before, it is worth noting that this is possible, but the actual construction will not be needed. Let us just present an example: Given $a,b,c\in {{\mathbb Z}}_d$ we have on the one hand $a(b,c)=(ab,ac)$. On the other hand the “product” of the “scalar” $\omega^a
I\in G'$ with the “vector” $G'X^b Z^c$ would equal the “vector” $G'X^{ab}Z^{ac}$.
Recall that our main goal is to describe whether or not two operators of $G$ commute. Since $G/G'$ is a commutative group, any information of this kind is eliminated by our passage from $G$ to the factor group $G/G'$. This is why in the following construction we use not only the group $G/G'$, but also the group $G$ and the commutator subgroup $G'$:
Let $G'X^bZ^c$ and $G'X^{b'}Z^{c'}$ be elements of $G/G'$ in normal form. We associate with them the commutator $$\label{eq:lift}
[X^bZ^c, X^{b'}Z^{c'}] = \omega^{cb'-c'b}I\in G'.$$ This assignment uses the group $G$. It is independent of the choice of representatives from the cosets $G'X^bZ^c$ and $G'X^{b'}Z^{c'}$, since $a$ and $a'$ do not appear on the right hand side of (\[eq:commutator\]).
By virtue of the bijections $\phi^{-1}:G/G'\to{{\mathbb Z}}_d^2$ and $\psi^{-1}:G'\to{{\mathbb Z}}_d$ we are now in a position to transfer this construction to our ${{\mathbb Z}}_d$-module ${{\mathbb Z}}_d^2$. This gives a mapping[^2] $$\label{eq:alternating}
[\cdot,\cdot] : {{\mathbb Z}}_d^2 \to{{\mathbb Z}}_d : \big((b,c),(b',c')\big) \mapsto cb'-c'b$$ which just describes the commutator of two elements of $G$ (given in normal form) in terms of our ${{\mathbb Z}}_d$-module. There are several ways to rewrite the mapping (\[eq:alternating\]), for example $$\label{eq:determinant}
\big[(b,c),(b',c')\big]
= (b,c)\begin{pmatrix} 0 &-1\\1&\hphantom{-}0\end{pmatrix}\begin{pmatrix}b'\\c'\end{pmatrix}
= \det \begin{pmatrix} b' &c'\\b\hphantom{'}&c\hphantom{'}\end{pmatrix}.$$ By this formula, the mapping $[\cdot,\cdot]$ is a *bilinear form* on ${{\mathbb Z}}_d^2$. Clearly, this form is alternating, i. e., $\big[(b,c),(b,c)\big]=0$ for all $(b,c)\in{{\mathbb Z}}_d^2$. As usual, we write $(b,c)\perp(b',c')$ if $\big[(b,c),(b',c')\big]=0$ and speak of *orthogonal* (or: *perpendicular*) vectors (with respect to $[\cdot,\cdot]$). As an alternating bilinear form is always skew symmetric, our orthogonality of vectors is a symmetric relation. As our form $[\cdot,\cdot]$ is *non-degenerate*, i.e., only the zero-vector is orthogonal to all other vectors of ${{\mathbb Z}}_d^2$, we have indeed a *symplectic module*.
Summing up, we see that the set of operators in $G$ which commute with a fixed operator $\omega^aX^bZ^c$ corresponds to the *perpendicular set* (shortly the *perp-set*) of $(b,c)$, viz.$$(b,c)^\perp := \big\{(u,v)\in{{\mathbb Z}}_d^2 : (b,c)\perp (u,v) \big\}.$$ The perp-set of $(b,c)$ is closed under addition and multiplication by ring elements. Also, it is non empty, since $$\label{eq:perpset}
{{\mathbb Z}}_d(b,c)\subset(b,c)^\perp.$$ So, $(b,c)^\perp$ is a ${{\mathbb Z}}_d$-submodule of ${{\mathbb Z}}_d^2$. We shall exhibit perp-sets in detail in the following sections.
The projective line over ${{\mathbb Z}}_d$ and the commutation algebra of Pauli operators
=========================================================================================
In order to say more about perp-sets in ${{\mathbb Z}}_d^2$ we shall use some basic facts about the projective line over the ring ${{\mathbb Z}}_d$. We do not need the theory of projective ring lines in its most general form here, since our ring ${{\mathbb Z}}_d$ is commutative and finite. This will allow to work with determinants and state some definitions in a simpler way. While we sketch here some basic notions and results, the reader is referred to [@bh]–[@blhr] for further details and proofs.
First, let us consider any vector $(b,c)\in{{\mathbb Z}}_d^2$. It generates the cyclic submodule $${{\mathbb Z}}_d(b,c) = \{(ub,uc):u\in{{\mathbb Z}}_d\}$$ Such a cyclic submodule is called *free*, if the mapping $u\mapsto(ub,uc)$ is injective. In this case the vector (or: pair) $(b,c)$ is called *admissible*. Any free cyclic submodule of ${{\mathbb Z}}_d^2$ has precisely $d$ vectors, including the zero-vector. However, not all vectors $\neq(0,0)$ of a free cyclic submodule need to be admissible. If $(b,c)$ is an admissible vector then $(ub,uc)$ is also admissible if, and only if, $u\in{{\mathbb Z}}_d$ is an invertible element. Thus, if $d$ is not a prime each free cyclic submodule of ${{\mathbb Z}}_d^2$ contains at least one non-admissible vector other than $(0,0)$.
For our ring ${{\mathbb Z}}_d$ there are several other ways of describing admissible vectors, as the following assertions are equivalent for any vector $(b,c)\in{{\mathbb Z}}_d^2$:
1. The vector $(b,c)$ is *unimodular*, i. e., there exist elements $u,v\in{{\mathbb Z}}_d$ with $$ub + vc = 1.$$
2. The vector $(b,c)$ is the first row of an invertible $2\times 2$ matrix with entries in ${{\mathbb Z}}_d$.
3. The vector $(b,c)$ is the first vector[^3] of a basis of ${{\mathbb Z}}_d^2$. (This means that there is such a vector $(b',c')\in{{\mathbb Z}}_d^2$ that the mapping $${{\mathbb Z}}_d^2\to{{\mathbb Z}}_d^2:(u,u')\mapsto u(b,c)+u'(b',c')$$ is a bijection.)
In a more geometric language, motivated by classical analytic projective geometry over the real or complex numbers, a free cyclic submodule of ${{\mathbb Z}}_d^2$ is called a *point*. The point set $$\label{eq:projectiveline}
{{\mathbb P}}_1({{\mathbb Z}}_d):=\{{{\mathbb Z}}_d(c,d) : (c,d) \mbox{~is admissible} \}$$ is the *projective line* over the ring ${{\mathbb Z}}_d$. According to this definition a point is a set of vectors. In “genuine” projective geometry over a ring the individual vectors contained in a point are of no particular interest. They are merely a useful tool for doing geometry in terms of coordinates. For us, however, the vectors within a point will be significant. This is of course in sharp contrast to Euclid’s point of view: *A point is that which has no part.*
Two points ${{\mathbb Z}}_{d}(b,c)$ and ${{\mathbb Z}}_{d}(b',c')$ of ${{\mathbb P}}_1({{\mathbb Z}}_{d})$ are called *distant* if $(b,c),(b',c')$ is a basis of ${{\mathbb Z}}_d^2$. Two distant points share only the zero vector $(0,0)$. Otherwise, the points are called *neighbouring*. Thus, two neighbouring points have always a non-zero vector in common.
We are now in a position to state a first, preliminary result about perp-sets.
\[thm:weakresult\] Let $(b,c)\in{{\mathbb Z}}_d^2$ be any vector and let ${{\mathbb Z}}_d(b',c')$ be any point of the projective line ${{\mathbb P}}_1({{\mathbb Z}}_d)$ which contains the vector $(b,c)$. Then the following assertions hold:
1. The point ${{\mathbb Z}}_d(b',c')$ is a subset of the perp-set $(b,c)^\perp$.
2. Under the additional assumption that ${{\mathbb Z}}_d(b,c)$ is also a point, we have $$\label{eq:admissible}
(b,c)^\perp={{\mathbb Z}}_d(b,c)={{\mathbb Z}}_d(b',c').$$
Ad (a): By (\[eq:perpset\]) and the assumption of the theorem, $(b,c)\in{{\mathbb Z}}_d(b',c')\subset(b',c')^\perp$. We infer from the symmetry of the relation $\perp$ that $(b',c')\in (b,c)^\perp$. Also, since $(b,c)^\perp$ is a submodule, we obtain that the entire point ${{\mathbb Z}}_d(b',c')$ is a subset of $(b,c)^\perp$.
Ad (b): As $(b,c)$ is a unimodular vector, there exists a pair $(\widetilde
c,-\widetilde b)\in{{\mathbb Z}}_{d}^{2}$ such that $b\widetilde c - c\widetilde b=1$. This means $$\det\begin{pmatrix}
b & c \\\widetilde b & \widetilde c
\end{pmatrix} = 1$$ which in turn tells us that $(b,c)$ and $(\tilde b, \tilde c)$ form a basis of ${{\mathbb Z}}_d^2$. Each vector $(u,v)\in{{\mathbb Z}}_d^2$ can be expressed in a unique way as a linear combination $$(u,v) = w(b,c)+\widetilde w(\widetilde b,\widetilde c)
\mbox{~~with~~}w,\widetilde w\in{{\mathbb Z}}_d.$$ By (\[eq:determinant\]), a necessary and sufficient condition for $(u,v)$ to lie in $(b,c)^\perp$ reads $$\det\begin{pmatrix}
wb+\widetilde w\widetilde b &wc+\widetilde w\widetilde c
\\
b & c
\end{pmatrix}
= \widetilde w (\widetilde b c - b\widetilde c) = - \widetilde w = 0.$$ Therefore $(b,c)^\perp = {{\mathbb Z}}_d(b,c)$. Finally, we infer from $(b,c)\in{{\mathbb Z}}_d(b',c')$ that the point ${{\mathbb Z}}_d(b,c)$ is a subset of the point ${{\mathbb Z}}_d(b',c')$. These points coincide, as both have precisely $d$ vectors.
Let us give an example, where $d=6$. We consider the vector $(2,0)$ which cannot be unimodular, because $2b'+0c'=1$ has no solution in ${{\mathbb Z}}_6$. There are only three distinct multiples of $(2,0)$, namely $(0,0)$, $(2,0)$, and $(4,0)$. This indicates once more that $(2,0)$ is not unimodular (or: admissible). We infer from $$(2,0) = 4 (5,0) = 4(2,3) = 4(5,3)$$ that there are (at least) three points containing $(2,0)$. The subsequent remarks are immediate from Theorem \[thm:betterresult\] which will be established below. However, their verification is also an easy exercise which can be carried out without any background knowledge: The projective line ${{\mathbb P}}_1({{\mathbb Z}}_6)$ has precisely twelve points. There are no other points containing $(2,0)$ than those mentioned before. The perp-set of $(2,0)$ coincides with the set-theoretic union of those three points, hence $$\begin{aligned}
(2,0)^\perp &=& {{\mathbb Z}}_6(5,0)\cup {{\mathbb Z}}_6(2,3)\cup {{\mathbb Z}}_6(5,3)\\
&=& \{(5,0),(4,0),(3,0),(2,0),(1,0),(0,0),\\
&& \hphantom{\{} (2,3),(4,0),(0,3),(2,0),(4,3),(0,0),\\
&& \hphantom{\{} (5,3),(4,0),(3,3),(2,0),(1,3),(0,0) \}.\end{aligned}$$ This is a set of $18-6=12$ vectors, because $(2,0)$, $(4,0)$ and $(0,0)$ are vectors which belong to all three points.
A particular case: $d$ is square-free
=====================================
While Theorem \[thm:weakresult\] describes the perp-set of any admissible vector, the result for non-admissible vectors is unsatisfactory. The aim of this section is to improve the results of Theorem \[thm:weakresult\] under the additional hypothesis that the number $d$ is square-free. Throughout this section we adopt the assumption that $$\label{eq:factors}
d = p_1p_2\cdots p_r,$$ where $p_1,p_2,\ldots,p_r$ are $r\geq 1$ distinct prime numbers. The ring ${{\mathbb Z}}_d$ is isomorphic to the outer direct product $$\label{eq:outer}
{{\mathbb Z}}_{p_1}\times{{\mathbb Z}}_{p_2}\times\cdots\times{{\mathbb Z}}_{p_r}$$ of $r$ finite *fields*. Let us recall how this isomorphism arises: We consider the ring elements $$\label{}
q_k:=p_1\cdots p_{k-1}p_{k+1}\cdots p_{r}, \mbox{~~where~~} k\in\{1,2,\ldots,r\}.$$ (For $r=1$ this product is empty, whence $q_1=1$.) The ring ${{\mathbb Z}}_d$ is the inner direct product of the principal ideals $$\label{}
J^{(k)}:={{\mathbb Z}}_d q_k \mbox{~~where~~} k\in\{1,2,\ldots,r\}.$$ Given any element $y\in{{\mathbb Z}}_d$ there exists a unique decomposition $$\label{}
y = y^{(1)}+y^{(2)}+\cdots+y^{(r)} \mbox{~~with~~} y^{(k)} \in J^{(k)}.$$ We refer to the elements $y^{(k)}$ as the *components* of $y$. In terms of this decomposition we can add and multiply elements of $x,y\in{{\mathbb Z}}_d$ componentwise, i. e. $$(x+y)^{(k)} =x^{(k)}+y^{(k)} \mbox{~~and~~} (x\cdot y)^{(k)} =x^{(k)}\cdot
y^{(k)}.$$ In particular, the unit element $1\in{{\mathbb Z}}_d$ has the decomposition $$\label{eq:representation1}
1 = 1^{(1)}+1^{(2)}+\cdots+1^{(r)}.$$ For each $k\in\{1,2,\ldots,r\}$ the ideal $J^{(k)}$ is a *field* isomorphic to ${{\mathbb Z}}_{p_k}$. There is only one isomorphism $J^{(k)}\to
{{\mathbb Z}}_{p_k}$; it takes the element $1^{(k)}$ to the unit element $1\in{{\mathbb Z}}_{p_k}$. Note that each element of $J^{(k)}$ can be written as $1^{(k)}+1^{(k)}+\cdots+1^{(k)}$ with a finite number of summands. Then its isomorphic image in ${{\mathbb Z}}_{p_k}$ is the sum $1+1+\cdots+1$ with the same number of summands. Below we shall always use the representation of ${{\mathbb Z}}_d$ as the inner direct product of the ideals $J^{(k)}$ rather than the isomorphic model given in (\[eq:outer\]).
As a first application we obtain the following characterisation: An element $y\in{{\mathbb Z}}_d$ is invertible if, and only if, all its components are non-zero. In this case the $k$-th component of the element $y^{-1}$ is the unique solution in $J^{(k)}$ of the equation $y^{(k)}x=1^{(k)}$ in the unknown $x$. Therefore the number of invertible elements in ${{\mathbb Z}}_d$ is $$\label{eq:invertible}
\prod_{k=1}^{r}(p_k-1).$$ A similar description holds for the points of ${{\mathbb P}}_1({{\mathbb Z}}_d)$: A pair $(b,c)$ is unimodular (or: admissible) if, and only if, there exist elements $u,v\in{{\mathbb Z}}_d$ with $$\label{eq:unimodular_k}
u^{(k)}b^{(k)}+v^{(k)}c^{(k)}=1^{(k)}\mbox{~~for all~~}
k\in\{1,2,\ldots,r\}.$$ Since each ideal $J^{(k)}$ is isomorphic to a field, the last equation is equivalent to $$\label{eq:nonzero_k}
(b^{(k)},c^{(k)})\neq (0,0)\mbox{~~for all~~}k\in\{1,2,\ldots,r\}.$$
We are now in a position to state our main result. Note that the set-theoretic union of points gives a set of vectors.
\[thm:betterresult\] Let the square-free integer $d > 1$ be given as in *(\[eq:factors\])*. Also, let $(b,c)\in{{\mathbb Z}}_d^2$. We denote by $K$ the set of those indices $k\in\{1,2,\ldots,r\}$ such that $(b^{(k)},c^{(k)})=(0,0)$. Then the following hold:
1. The vector $(b,c)$ is contained in precisely $$\label{eq:points}
\prod_{k\in K} (p_k+1)$$ points of ${{\mathbb P}}_1({{\mathbb Z}}_d)$.
2. The set-theoretic union of these points equals the perpendicular set of the vector $(b,c)$.
3. The perpendicular set of the vector $(b,c)$ satisfies $$\label{eq:perpvectors}
|(b,c)^\perp| = d\prod_{k\in K}p_k.$$
Ad (a): First, let us determine all admissible vectors $(b',c')\in{{\mathbb Z}}_{d}^{2}$ such that $(b,c)=u(b',c')$ for some $u\in{{\mathbb Z}}_d$. So $$\label{eq:offK}
(b^{(j)},c^{(j)})=u^{(j)}({b'}{}^{(j)},{c'}{}^{(j)})\neq(0,0)
\mbox{~~for all~~}j\in\{1,2,\ldots,r\}\setminus K$$ and $$\label{eq:inK}
(b^{(k)},c^{(k)})=u^{(k)}({b'}{}^{(k)},{c'}{}^{(k)})=(0,0)\neq ({b'}{}^{(k)},{c'}{}^{(k)})
\mbox{~~for all~~}k\in K.$$ We obtain $u^{(j)}\neq 0$ from (\[eq:offK\]), whence $({b'}{}^{(j)},{c'}{}^{(j)})$ is one of the $p_j-1$ distinct multiples of $(b^{(j)},c^{(j)})$ by a non-zero factor in $J^{(j)}$. Next, (\[eq:inK\]) implies $u^{(k)}=0$, whence $({b'}{}^{(k)},{c'}{}^{(k)})$ is one of the $p_k^2-1$ non-zero pairs with entries from $J^{(k)}$. These necessary conditions are also sufficient so that we obtain $$\label{eq:total}
\prod_{j\notin K} (p_j-1) \prod_{k\in K} (p_k+1)(p_k-1)$$ admissible vectors with the required property. Dividing by the number of invertible elements of ${{\mathbb Z}}_d$, as stated in (\[eq:invertible\]), gives the number of points containing the vector $(b,c)$.
Ad (b): By Theorem \[thm:weakresult\] (a), each point containing $(b,c)$ is a subset of $(b,c)^\perp$. So the same property holds for the union of all these points. The proof will be accomplished by showing that for any vector $(x,y)\in
(b,c)^\perp$ there is a point ${{\mathbb Z}}_d(b',c')\in{{\mathbb P}}_1({{\mathbb Z}}_d)$ which has $(x,y)$ and $(b,c)$ among its vectors. We define $b'$ and $c'$ in terms of their components as follows: For all $j\in\{1,2,\ldots,r\}\setminus K$ we let $({b'}{}^{(j)},{c'}{}^{(j)}) := ({b}^{(j)},{c}^{(j)})$. For the remaining indices $k\in K$ we define $$\label{}
({b'}{}^{(k)},{c'}{}^{(k)}) :=
\left\{\begin{array}{ll}
(x^{(k)},y^{(k)}) & \mbox{if~}(x^{(k)},y^{(k)}) \neq (0,0),\\
(1^{(k)},1^{(k)}) & \mbox{otherwise.}
\end{array}\right.$$ According to our definition and (\[eq:nonzero\_k\]) the submodule ${{\mathbb Z}}_d(b',c')$ is a point. Letting $u^{(j)}:=1^{(j)}$ for all $j\notin K$ and $u^{(k)}:=0$ for all $k\in K$ yields $(b,c) = u(b',c')$.
Finally, we establish the existence of an element $v\in{{\mathbb Z}}_d$ with $(x,y)=v(b',c')$. For this purpose the components of $v$ can be chosen as follows: If $j\notin K$ then $(x,y)\perp(b,c)$ together with (\[eq:determinant\]) gives $$\label{eq:det_j}
\det \begin{pmatrix} x^{(j)} &y^{(j)}\\b^{(j)}&c^{(j)}\end{pmatrix}=0.$$ As this is a determinant over the field $J^{(j)}$, and because the second row is non-zero, we can define $v^{(j)}\in J^{(j)}$ via $(x^{(j)},y^{(j)})=v^{(j)}(b^{(j)},c^{(j)})$. If $k\in K$ and $(x^{(k)},y^{(k)}) \neq (0,0)$ we set $v^{(k)}:=1^{(k)}$, otherwise we let $v^{(k)}:=0$.
Ad (c): By (b), it suffices to count the number of vectors $(b'',c'')$ which are a multiple of an admissible vector as described in part (a) of the present proof. For each $j\notin K$ the pair $(b''{}^{(j)},c''{}^{(j)})$ can be chosen as any of the $p_j$ multiples of $(b^{(j)},c^{(j)})$ by a factor in $J^{(j)}$, whereas for each $k\in K$ the pair $(b''{}^{(k)},c''{}^{(k)})$ can be chosen arbitrarily in $p_k^2$ ways. Hence there are $$\prod_{j\notin K}p_j\cdot\prod_{k\in K}p_k^2 = d\prod_{k\in K}p_k$$ such vectors.
As a by-product of Theorem \[thm:betterresult\] we may infer from (\[eq:points\]) that the projective line ${{\mathbb P}}_1({{\mathbb Z}}_d)$ has precisely $\prod_{k=1}^{r} (p_k+1)$ points. Also, returning to our initial problem, we obtain the following result.
With the settings and notations of Theorem *2* the number of operators in the generalized Pauli group $G$ which commute with the operator $\omega^aX^bZ^c\in G$ equals the value given by *(\[eq:perpvectors\])* multiplied by d.
Conclusion
==========
Given the generalized Pauli group associated with a $d$-dimensional qudit, $d$ a product of distinct primes, a general formula was derived for the number of generalized Pauli operators commuting with a given one. This formula is based on the properties of (sub)modules of the associated modular ring ${{\mathbb Z}}_d$ and finds its natural interpretation in the properties of the projective line defined over ${{\mathbb Z}}_d$. When compared with other works on the subject [@psk]–[@pbs], our approach makes also use of [*non*]{}-admissible pairs of elements of the ring in question, thereby giving the physical meaning to the full structure of the line; moreover, it seems to be readily generalizable to tackle the case where $d$ also contains powers of primes.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the Science and Technology Assistance Agency under the contract $\#$ APVT–51–012704, the VEGA grant agency projects $\#$ 2/6070/26 and $\#$ 7012 and by the $\langle$Action Austria–Slovakia$\rangle$ project $\#$ 58s2 “Finite Geometries Behind Hilbert Spaces."
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[^1]: We shall always be concerned with commutator of operators in the sense of group theory. It must not be confused with the commutator from ring theory which uses addition and multiplication of operators.
[^2]: Of course the symbol $[\cdot,\cdot]$ has two different meanings in (\[eq:defcommutator\]) and (\[eq:alternating\]).
[^3]: All bases of ${{\mathbb Z}}_d^2$ consist of two admissible vectors. In general, a module over a ring may have bases of different size.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using Mach’s principle, we will show that the observed diurnal and annual motion of the Earth can just as well be accounted as the diurnal rotation and annual revolution of the Universe around the fixed and centered Earth. This can be performed by postulating the existence of vector and scalar potentials caused by the simultaneous motion of the masses in the Universe, including the distant stars.'
address: 'University of Zagreb, Department of Physics, Bijenička cesta 32, Zagreb, Croatia'
author:
- Luka Popov
title: Dynamical description of Tychonian Universe
---
Introduction {#intro}
============
The modern day use of the word *relativity* in physics is usually connected with Galilean and special relativity, i.e. the equivalence of the systems performing the uniform rectlinear motion, so-called *inertial frames*. Nevertheless, the physicists and philosophers never ceased to debate the various topics under the heading of *Mach’s principle*, which essentially claims the equivalence of all co-moving frames, including non-intertial frames as well.
Historically, this issue was first brought out by Sir Isaac Newton in his famous rotating bucket argument. As Newton saw it, the bucket is rotating in the absolute space and that motion produces the centrifugal forces manifested by the concave shape of the surface of the water in the bucket. The motion of the water is therefore to be considered as “true and absolute”, clearly distinguished from the relative motion of the water with respect to the vessel [@principia].
Mach, on the other hand, called the concept of absolute space a “monstrous conception” [@mach1], and claimed that the centrifugal force in the bucket is the result only of the relative motion of the water with respect to the masses in the Universe. Mach argued that if one could rotate the whole Universe around the bucket, the centrifugal forces would be generated, and the concave-shaped surface of the water in the bucket would be identical as in the case of rotating bucket in the fixed Universe. Mach extended this principle to the once famous debate between geocentrists and heliocenstrists, claiming that both systems can equally be considered correct [@mach2].
His arguments, however, remained of mostly philosophical nature. Since he was convinced empiricist, he believed that science should be operating only with observable facts, and the only thing we can observe are relative motions. Therefore, every notion of absolute motion or a preferred inertial frame, whether inertial or non-inertial, is not a scientific one but rather a mathematical or philosophical preference.
As Hartman and Nissim-Sabat [@hartman] correctly point out, Mach never formulated the mathematical model or an alternative set of physical laws which can explain the motions of the stars, the planets, the Sun and the Moon in a Tychonian or Ptolemaic geocentric systems. For that reason, some physicist in the modern days have tried to “Machianize” the Newtonian mechanics in various ways [@hood; @barbourN], or even try to construct new theories of mechanics [@assis]. There have also been attempts to reconcile Mach’s principle with the General Theory of Relativity, some of which were profoundly analyzed in the paper by Raine [@raine].
In the recent paper [@popov1] we have used the concept of the so-called pseudo-force and derived the expression for the potential which is responsible for it. This potential can be considered as a real potential (as shown by Zylbersztajn [@zylbersztajn]), which can easily explain the annual motion of the Sun and planets in the Neo-Tychonian system. In the same manner, one can explain the annual motion of the stars and the observation of the stellar parallax [@popov2].
It is the aim of this paper to use the same approach to give the dynamical explanation of the diurnal motion of the celestial bodies as seen from the Earth, and thus give the mathematical justification for the validity of Mach’s arguments regarding the equivalence of the Copernican and geocentric systems.
The paper is organized as follows. In section \[vectorpot\] the vector potential is introduces in general terms. This formalism is then applied to analyze the motions of the celestial bodies as seen from the Earth in section \[celestmot\]. Finally, the conclusion of the analysis is given.
Vector potential formalism {#vectorpot}
==========================
Following Mach’s line of thought, one can say that the simultaneously rotating Universe generates some kind of gravito-magnetic vector potential, $\mathbf{A}$. By the analogy with the classical theory of fields [@classf] one can write down the Lagrangian which includes the vector potential, $$\label{Lgen}
L = \frac{1}{2} m \dot{\mathbf{r}}^2 + m\, \dot{\mathbf{r}} \!\cdot\! \mathbf{A} +
\frac{1}{2} m \mathbf{A}^2 - m U_{\mathrm{ext}}\,,$$ where $m$ is the mass of the particle under consideration, and $U_{\mathrm{ext}}$ is some external scalar potential imposed on the particle, for example the gravitational interaction.
We know, as an observed fact, that every body in the rotational frame of reference undergoes the equations of motion given by [@goldstein] $$\label{eom}
m \ddot{\mathbf{r}} = \mathbf{F}_{\mathrm{ext}} - 2 m (\bm\omega_{\mathrm{rel}} \times \dot{\mathbf{r}}) -
m \left[ \bm\omega_{\mathrm{rel}} \times (\bm\omega_{\mathrm{rel}} \times \mathbf{r}) \right] \,,$$ where $\bm\omega_{\mathrm{rel}}$ is relative angular velocity between the given frame of reference and the distant masses in the Universe, and $\mathbf{F}_{\mathrm{ext}} = -\nabla U_{\mathrm{ext}}$ some external force acting on a particle.
It can be easily demonstrated that one can derive Equation (\[eom\]) by applying the Euler-Lagrange equations on the following “observed” Langrangian $$\label{Lobs}
L_\mathrm{obs} = \frac{1}{2} m \dot{\mathbf{r}}^2 +
m\, \dot{\mathbf{r}} \cdot (\bm\omega_{\mathrm{rel}} \times \mathbf{r}) +
\frac{1}{2} m (\bm\omega_{\mathrm{rel}} \times \mathbf{r})^2 -
m U_{\mathrm{ext}}\,.$$ By comparison of the general Lagrangian (\[Lgen\]) and the “observed” Lagrangian (\[Lobs\]) one can write down the expression for the vector potential $\mathbf{A}$, $$\label{vecpot}
\mathbf{A} = \bm\omega_{\mathrm{rel}} \times \mathbf{r} \,.$$
It is important to notice that there is no notion of the absolute rotation in this formalism. The observer sitting on the edge of the Newton’s rotating bucket can only observe and measure the relative angular velocity between him or her and the distant stars $\bm\omega_{\mathrm{rel}}$, incapable of determine whether it is the bucket or the stars that is rotating.
Trajectories of the celestial bodies around the fixed Earth {#celestmot}
===========================================================
Diurnal motion
--------------
It is one thing to postulate that rotating masses in the Universe generate the vector potential given by (\[vecpot\]), but quite another to claim that this same potential can be used to explain and understand the very motion of these distant masses. We will now demonstrate that this is indeed the case.
The observer sitting on the surface of the Earth makes several observations. First, he or she notices that there is a preferred axes (say $z$) around which all Universe rotates with the period of approximately 24 h. Then, according to the formalism given in Section \[vectorpot\], he or she concludes that the Earth must be immersed in the vector potential given by $$\label{Evecpot}
\mathbf{A} = \Omega \; \hat{\mathbf{z}} \times \mathbf{r} \,,$$ where $\Omega \approx (2\pi / 24 \textrm{ h})$ is the observed angular velocity of the celestial bodies [^1].
One can now re-write the Lagrangian (\[Lgen\]) together with the Equation (\[Evecpot\]) and focus only on the contributions coming from the vector potential $\mathbf{A}$, $$\label{lang2}
L_\mathrm{rot} = \frac{1}{2} m \dot{\mathbf{r}}^2 +
m\, \Omega\, \dot{\mathbf{r}} \cdot (\hat\mathbf{z} \times \mathbf{r}) +
\frac{1}{2} m\, \Omega^2\, (\hat\mathbf{z} \times \mathbf{r})^2 \,.$$
The Euler-Lagrange equations for this Lagrangian, written for each component of the Cartesian coordinates, are given by $$\begin{aligned}
\label{eomsCart}
\ddot{x} & = & -2 \, \Omega \, \dot{y} + \Omega^2 \, x \nonumber \\
\ddot{y} & = & 2 \, \Omega \, \dot{x} + \Omega^2 \, y \\
\ddot{z} & = & 0 \nonumber \,.\end{aligned}$$ The solution of this system of differential equations reads $$\begin{aligned}
\label{sol1}
x(t) & = & r \, \cos \Omega t \nonumber \\
y(t) & = & r \, \sin \Omega t \\
z(t) & = & 0 \nonumber \,,\end{aligned}$$ where $r$ is the initial distance of the star from the $z$ axes. The observer can therefore conclude that the celestial bodies perform real circular orbits around the static Earth due to the existence of the vector potential $\mathbf{A}$ given by Equation (\[Evecpot\]). This conclusion is equivalent to the one that claims that the Earth rotates around the $z$ axes and the celestial bodies don’t.
Annual motion
-------------
The second thing the observer on the Earth notices is the periodical annual motion of the celestial bodies around the $z'$ axes which is inclined form the axes of diurnal rotation $z$ by the angle of approximately $23.5^\circ$. This motion can be explained if one assumes that the Earth is immersed in the so-called pseudo-potential $$\label{Ups}
U_{\mathrm{ps}} (\mathbf{r}) = \frac{G M_S}{r_{SE}^2} \hat{\mathbf{r}}_{SE} \cdot
\mathbf{r} \,.$$ Here $G$ stands for Newton’s constant, $M_S$ stands for the mass of the Sun and $\mathbf{r}_{SE}(t)$ describes the motion of the Sun as seen form the Earth. The Sun’s trajectory $\mathbf{r}_{SE}(t)$ is shown to be an ellipse in $x'$-$y'$ plane (defined by the $z'$ axes from the above). Using this potential alone one can reproduce the observed retrograde motion of the Mars or explain the effect of the stellar parallax as the real motion of the distant stars in the $x'$-$y'$ plane. All this was demonstrated in the previous communications [@popov1; @popov2].
Total account
-------------
One can finally conclude that all celestial bodies in the Universe perform the twofold motion around the Earth:
1. circular motion in the $x$-$y$ plane due to the vector potential $\mathbf{A}$ (\[Evecpot\]) with the period of approximately 24 hours and
2. elliptical orbital motion in the $x'$-$y'$ plane due to the scalar potential $U_{\mathrm{ps}}$ (\[Ups\]) with the period of approximately one year.
Using Equations (\[Lgen\]), (\[Evecpot\]) and (\[Ups\]) one can write down the complete classical Lagrangian of the geocentric Universe, $$\begin{aligned}
\label{Lcompl}
L & = & \frac{1}{2} m \dot{\mathbf{r}}^2 +
m\, \Omega\, \dot{\mathbf{r}} \cdot (\hat\mathbf{z} \times \mathbf{r}) +
\frac{1}{2} m\, \Omega^2\, (\hat\mathbf{z} \times \mathbf{r})^2 \nonumber \\
& & {}- m
\frac{G M_S}{r_{SE}^2} \hat{\mathbf{r}}_{SE} \cdot \mathbf{r} -
m U_{\mathrm{loc}} \,,\end{aligned}$$ where $U_{\mathrm{loc}}$ describes some local interaction, e.g. between the planet and its moon.
It is a matter of trivial exercise to show that these potentials can easily account for the popular “proofs” of Earth’s rotation like the Faucault’s pendulum or the existence of the geostationary orbits.
Conclusion
==========
We have presented the mathematical formalism which can justify Mach’s statement that both geocentric and Copernican modes of view are “equally actual” and “equally correct” [@mach2]. This is performed by introducing two potentials, (1) vector potential that accounts for the diurnal rotations and (2) scalar potential that accounts for the annual revolutions of the celestial bodies around the fixed Earth. These motions can be seen as real and self-sustained. If one could put the whole Universe in accelerated motion around the Earth, the potentials (\[Evecpot\]) and (\[Ups\]) would immediately be generated and would keep the Universe in that very same state of motion *ad infinitum*.
[This work is supported by the Croatian Government under contract number 119-0982930-1016.]{}
References {#references .unnumbered}
==========
[99]{}
Newton I 1960 *The Mathematical Principles of Natural Phylosophy* (Berkeley CA: University of California Press) pp 10-11
Mach E 1960 *The Science of Mechanics* 6th ed (LaSalle IL: Open Court) p xxviii
Ref [@mach1] pp 279 and 284
Hartman H I and Nissim-Sabat C 2003 *Am. J. Phys.* [**71**]{} 1163–68
Hood C G 1970 *Am. J. Phys.* [**38**]{} 438–442
Barbour J 1974 *Nature* [**249**]{} 328–329
Assis A K T 1999 *Relational Mechanics* (Montreal: Aperion)
Raine D J 1981 *Rep. Prog. Phys.* [**44**]{} 1151–95
Popov L 2013 383–391 (Corrigendum 2013 817) (*Preprint* arXiv:1301.6045)
Zylbersztajn A 1994 , 1–8
Popov L 2013 arXiv:1302.7129
Landau L D and Lifshiz E M 1980 *The Classical Theory of Fields* 4th ed (Oxford: Butterworth-Heinemann) p 49
Goldstein H 1980 *Classical Mechanics* 2nd ed (Reading MA: Addison-Wesley) p 178
Wikipedia 26 Apr 2013 Sidereal time\
<http://http://en.wikipedia.org/wiki/Sidereal_time>
[^1]: The period of the relative rotation between the Earth and the distant stars is called *sidreal day* and it equals 23 h 56’ 4.0916”. Common time on a typical clock measures a slightly longer cycle, accounting not only for the Sun’s diurnal rotation but also for the Sun’s annual revolution around the Earth (as seen from the geocentric perspective) of slightly less than 1 degree per day [@wiki].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the effects of an in-plane Dzyaloshinskii-Moriya interaction under an external magnetic field in the highly frustrated kagome antiferromagnet. We focus on the low-temperature phase diagram, which we obtain through extensive Monte-Carlo simulations. We show that, given the geometric frustration of the lattice, highly non trivial phases emerge. At low fields, lowering the temperature from a cooperative paramagnet phase, the kagome elementary plaquettes form non-coplanar arrangements with non-zero chirality, retaining a partial degeneracy. As the field increases, there is a transition from this “locally chiral phase” to an interpenetrated spiral phase with broken $\mathcal{Z}_{3}$ symmetry. Furthermore, we identify a quasi-skyrmion phase in a large portion of the magnetic phase diagram, which we characterize with a topological order parameter, the scalar chirality by triangular sublattice. This pseudo-skyrmion phase (pSkX) consists of a crystal arrangement of three interpenetrated non-Bravais lattices of skyrmion-like textures, but with a non-(fully)-polarized core. The edges of these pseudo-skyrmions remain polarized with the field, as the cores are progressively canted. Results show that this pseudo-skyrmion phase is stable up to the lowest simulated temperatures, and for a broad range of magnetic fields.'
author:
- 'M. E. Villalba'
- 'F. A. Gómez Albarracín'
- 'H. D. Rosales'
- 'D. C. Cabra'
title: 'Field-induced pseudo-skyrmion phase in the antiferromagnetic kagome lattice'
---
=3.4 in
Introduction
============
Magnetic skyrmions are topological vortex-like spin structures where the spins point in all directions wrapping a sphere [@SkyrmionFirst]. In particular, in the last years, the Skyrmion crystal (SkX) phases have triggered a huge interest because of their important role in the electronic transport in conection with technological application devices [@TechDev]. The most simple situation where such SkXs are stabilized correponds to the ferromagnetic systems in a magnetic field including Dzyaloshinskii-Moriya (DM) interactions [@bogda1; @bogda2; @bogda3; @bogda4; @bogda5; @bogda6; @bogda7; @bogda8]. Also, it has been shown in numerous works that the SkX’s can be induced by competing interactions in ferromagnetic and mixed ferro/antiferro-magnetic systems [@17; @DMSkx]. Finally, the presence of local anisotropies can stabilize different skyrmion-like crystal phases under a magnetic field, which lead to merons-like structures in metastable states[@12].
In this direction, the search for new systems with skyrmions phases in a wide range of magnetic field and temperature is an central issue in the field of topological magnetic materials. One ingredient that may play a central role in this topic is the magnetic frustration, which in many cases, is associated with exotic spin orders having non-collinear or non-coplanar spin structures.
Recently, the emergence of skyrmion textures has been actively explored in frustrated lattices [@naga; @1; @osorio1; @osorio2; @Yu2018; @Loss2019]. In fact, in a previous work (see Ref. \[\]), some of the authors (see also Ref. \[\]) have shown that in the antiferromagnetic triangular lattice the competition between nearest neighbor exchange couplings and an in-plane DM interaction gives rise to a low temperature stable topological phase for a range of magnetic fields. This phase is characterized by three interpenetrated skyrmion crystals, one by sublattice.
In this context, the highly frustrated kagome antiferromagnet provides an alternative arena for studying emergent phenomena in magnets of strong frustration. A crucial point of the antiferromagnetic kagome lattice is its high degeneracy. This feature, combined with the chiral anisotropy induced by the DM interaction could induce different types of topologically non trivial phases. In the last years, materials with in- and out-of-plane DM interactions with an antiferromagnetic kagome structure have been thoroughly studied, for example by P. Mendels et al (Herbermisthite)[@kagomeDM1], B. Canals et al (Fe- and Cr-based jarosites, etc.)[@kagomeDM2] (for a review see Ref.\[\]). Last but not least, the possibility to generate a DM interaction in ultrathin films with perpendicular magnetic anisotropy in multilayer structures leads to the emergence of interfacial non-collinear spin textures (skyrmions and chiral domain walls) induced by DM interactions in such magnetic films [@Fertetal; @kagomeDM4].
A key question that arises is what is the role of the magnetic frustration and hugh degeneracy in the formation of skyrmion spin textures. Motivated by this, we consider the inclusion of a specific DM interaction in the pure antiferromagnetic Heisenberg model on the kagome lattice and study the consequences of the combination of high degeneracy, thermal fluctuations and anisotropic interactions. In particular, we explore the possibility of skyrmion-like textures in the proposed model.
We show that, under the action of an external magnetic field, there are a number of different exotic low-temperature phases: at low non-zero magnetic field a phase without global order, reminiscent of the pure Heisenberg model in the kagome lattice, but formed by clusters with non-zero local chirality, is stabilized. Then increasing the magnetic field, it leads to a tree-sublattice order with broken sublattice symmetry. For a larger magnetic field, a three sublattice pseudo-skyrmion crystal (pSkX) structure is established with the particularity that the hidden pSkX magnetic order appear in a non-Bravais sublattice. The emergent pseudo-skyrmion unit structures do not fully wrap the sphere, but can be distinguished with the help of another topological parameter, the sublattice chirality. These pseudo-skyrmion structures have a remarkable feature: instead of being the result of overlapping skyrmions, with a fully polarized core and a radius smaller than the separation between them, as in [@12], here the rims are completely polarized in the direction of the external field while the cores are not. In fact, these cores get canted as the field increases.
The rest of the manuscript is organized as follows: in Section \[sec:model\] we present and discuss the Heisenberg model on the kagome lattice including the DM interaction and the Zeeman coupling to an external magnetic field. In Sec. \[sec:MC\] we study the proposed model through extensive Monte Carlo simulations and examine the different phases stabilized by the Zeeman coupling, focusing on the topological pSkX. Conclusions are presented in Sec. \[sec:conclusions\].
Model {#sec:model}
=====
We consider an antiferromagnetic Heisenberg model on the kagome lattice, with in-plane DM interaction, immersed in a magnetic field. The Hamiltonian is given by $$\label{eq:H}
H = J\sum_{\langle i,j\rangle}\mathbf{S}_{i} \cdot \mathbf{S}_{j} + \mathbf{D}_{ij}\cdot(\mathbf{S}_{i} \times \mathbf{S}_{j})- h \sum_{j} S_{j}^{z}$$ where the magnetic moments $\mathbf{S}_{i}$ are three-component classical unit vectors at site ${\mathbf r}_i$, $\langle i,j \rangle$ indicates the sum over nearest neighbor sites, and $J>0$ is the antiferromagnetic exchange coupling. The DM interaction is defined by $\bold{D}_{ij}= D\, \delta \hat{r}_{ij}$, where $\delta\hat{r}_{ij}=(\mathbf{r}_{i}-\mathbf{r}_{j})/|\mathbf{r}_{i}-\mathbf{r}_{j}|$ is a unitary vector pointing along the nearest-neighbor bonds as shown in Fig. \[fig:latt\]. This implies that this interaction is constrained to the kagome plane, perpendicular to the external magnetic field $\bold{h}=h\hat{z}$. This choice of DM interaction proves to be adequate to develop Skyrmion phases in both ferromagnetic and antiferromagnetic systems [@1]. In this work, without loss of generality, we fix $D/J=0.2$, a value of $D/J$ that induces magnetic structures with sizes compatible with the systems size of the simulations.
![\[fig:latt\] (Color online) kagome lattice. The labels $1,2,3$ indicate the three sublattices (dashed blue line indicates the unit cell). Small (green) arrows are Dzyaloshinskii-Moriya vectors $\bold{D}_{ij}$, $\bold{D}_{jk}$ and $\bold{D}_{ik}$ involved in the sites (labeled as) $i,j,k$.](fig_1.png){width="0.7\columnwidth"}
The case $D = 0$ has been widely studied in the last decades [@2; @3; @4; @5; @6; @7; @19; @21; @9]. It is well known that the antiferromagnetic Heisenberg model for classical spins in the kagome lattice presents a rich phenomenology due to its high degeneracy. In zero field, magnetic moments form a infinitely degenerate $120^{\circ}$ spin-structure. Over the last few years, much effort has been devoted to study the mechanisms or interactions that can lift this degeneracy and the consequent emergence of non-trivial phases. One of these well known effects is that due to the inclusion of thermal fluctuations, the system goes from a paramagnetic (at high temperaure) to a cooperative paramagnetic phase [@3; @7]; while at low temperature the order-by-disorder mechanism selects a submanifold of coplanar states. A magnetic field partially relieves this degeneracy, and state selection by thermal fluctuations is still at play. Thermal fluctuations stabilize two coplanar states at finite fields with different symmetries. At very low temperature, each type of coplanar state can be studied through multipolar order parameters [@7]. The inclusion of further neighbor exchange couplings can select and induce different magnetic orders [@8; @fundamentales; @14]. Furthermore, the addition of an out-of-plane DM interaction favors a $q=0$ non-coplanar state [@18]
Due to the competition between the antiferromagnetic exchange $J$ which favors the coplanar configurations, and the in-plane DM interaction which favors the helical phases, we expect that the combination of these to terms results in a rich variety of chiral configurations which will be presented in the next section.
Monte Carlo Simulations And Phase Diagram {#sec:MC}
=========================================
To explore the low temperature behavior of the model presented in the previous section, we resort to Monte Carlo simulations. We use a combination of the Metropolis algorithm and the overrelaxation method, doing microcanonical updates, and lowering the temperature in an annealing scheme. We performed our simulations in $3\times L^2$ site clusters, $L=36-60$, with periodic boundary conditions. $10^5-10^6$ Monte Carlo steps (MCS) were used for initial relaxation, and measurements were taken in twice as much MCS.
As a first approach to identify and characterize the different low temperature phases, we first inspect the standard quantities: namely, specific heat $C_{v}=\frac{\langle E^{2} \rangle - \langle E \rangle^{2}}{NT^{2}}$, magnetization $M=\frac{1}{N} \left \langle \sum_{i} S_{i}^{z} \right \rangle $, absolute value of the magnetization $|M|=\frac{1}{N} \left \langle \sum_{i} |S_{i}^{z}| \right \rangle $ and magnetic susceptibility $\chi_{M}=\left \langle \frac{dM}{dh} \right \rangle$.
In Fig. \[fig:magH\] we show typical curves of magnetization $M$, its absolute value $|M|$ and the susceptibility $\chi_{M}$ as a function of the magnetic field at $T/J=2\times 10^{-3}$. We can identify four features in these curves, indicated by vertical arrows in the figure: a bump in $|M|$ at $h_{c1}/J\sim 1.5 $, a peak in the susceptibility which matches a change in the behavior of $|M|$ at $h_{c2}/J\sim 2.1$ and a second peak in $\chi_{M}$ at $h_{c3}/J\sim 4.4$. The last feature at the critical field $h_{c4}\sim 5.7$, indicates the transition to the state completely polarized with the magnetic field. To obtain valuable information on the nature of each phase we compute the static spin structure factor $S_{\bf q}$ in the reciprocal lattice to identify the Bragg peaks that characterize the different spin-textures.
![\[fig:magH\] (Color online) Magnetization curves vs external field $h/J$ for $L=60$ lattice size, at $T/J=2\times10^{-2}$. Average Magnetization $\langle M \rangle$ (blue open circles), Magnetization modulus $|M|$ (green open triangles), susceptibility $\chi_M= dM/dh $ (yellow open squares). The black arrows indicate four features in these curves. The value of the fields where these feature emerge, the critical fields, $h_{c1},h_{c2},h_{c3}$, are indicated by dashed lines. The critical field $h_{c4}$ correspond to the saturation field where all the spins are polarized.](fig_2.png){width="1.0\columnwidth"}
{width="100.00000%"} {width="100.00000%"}
The components $S_{\bf q}^{\perp} $ and $S_{\bf q}^{\parallel}$, perpendicular and parallel to the external field respectively, are defined as:
$$\begin{aligned}
S_{\bf q}^{\perp}&=&\frac{1}{N} \langle |\sum_{i} S_{i}^{x} \ e^{i {\bf q}\cdot {\bf r}_i} |^2 + |\sum_{i} S_{i}^{y} \ e^{i {\bf q} \cdot {\bf r}_i} |^2\rangle \label{factesperp}\\
S_{\bf q}^{\parallel}&=& \frac{1}{N} \langle |\sum_{i} S_{i}^{z} \ e^{i{\bf q}\cdot {\bf r}_i} |^2\rangle\label{factespara}\end{aligned}$$
In Fig. \[fig:snaps\] we show representative snapshots (top) and the corresponding structure factors $S_{\bf q}^{\perp}$ (bottom) of the low temperature phases as a function of the external magnetic field. In the $S_{\bf q}^{\perp}$ plots, the first Brillouin zone (1BZ drawn with solid lines) and the extended Brillouin zone (EBZ drawn with dashed lines) are indicated. By inspection of Fig. \[fig:snaps\] we find:
- [For very low magnetic fields $h<h_{c1}$, the magnetic structure retains some of the degeneracy present for the case $D = 0$ and $h = 0$. From a typical snapshot, it can be seen that elementary triangles form out of plane structures. Six bright peaks emerge around every high-symmetry point $\bf{M_e}$ in the spin structure factor as is shown in Fig. \[fig:snaps\](D). Half of these points (18 in total) are inside the extended Brillouin zone (EBZ). ]{}
- [For slightly higher fields $h_{c1}<h<h_{c2}$, the Zeeman coupling induces a striped/spiral-like structure, with single-q peaks in the $\bf{M_e}$ region of the EBZ, Fig. \[fig:snaps\](E).]{}
- [In a broad region of intermediate magnetic fields $h_{c2}<h<h_{c3}$ a non trivial swirling structure emerges (see Fig. \[fig:snaps\](C)). Visually, it is reminiscent of the interpenetrated skyrmion phase AF-SkX found in the triangular antiferromagnetic lattice[@1]. In the structure factor, 12 peaks emerge in the EBZ, indicating a triple-q structure, which may be a hint of a hidden skyrmion-like texture.]{}
Now, we proceed to further explore and characterize in detail these low temperature phases.
Lower $h/J$ multi-q states
--------------------------
![(Color online) Top: Specific heat as a function of temperature for: low magnetic field at $h/J=0.5$ (yellow open squares), spiral at $h/J=1.6$ (green open triangles) and pSkX at $h/J=2.7 $ (blue open circles). Bottom: structure factor obtained from Monte Carlo simulations at temperature $T/J=8\times 10^{-2}$, $h/J=0.5$, for $D/J=0.2$.[]{data-label="fig:SqLowH"}](fig_5b.png){width="1\columnwidth"}
At low external field, $h<h_{c1}$, there is an interesting behavior of the system with temperature. At $T/J>0.03$, the system seems to be in a copperative paramagnet (CP) phase. This is illustrated in the structure factor, presented in the bottom panel of figure Fig. \[fig:SqLowH\] showing similar behaviour to that obtained for the pure kagome antiferromagnet in the CP phase [@2]. It is characterized by the presence of “pinch points” in the $\mathbf{M_e}$ points of the EBZ, which are the signature of a classical algebraic spin liquid [@10; @11]. The CP and low temperature phases are separated by a phase transition at $T/J\approx 0.03$, where the specific heat exhibits a peak (top panel of Fig. \[fig:SqLowH\]).
However, these low temperature phases at low magnetic field do not show a clear periodic magnetic structure. As we mentioned before, although no clear order is seen, it is evident that there are numerous unit triangles where the spins are arranged in a non-coplanar way, with different orientations. This is most clearly shown inspecting the nearest neighbor scalar chirality per plaquette $\chi_{ijk}$ defined as:
$$\begin{aligned}
\chi_{ijk}&=&\bold{S}_{i}\cdot( \bold{S}_{j} \times \bold{S}_{k})
\label{eq:local_chira}\end{aligned}$$
where labels $i,j,k$ indicate the positions ${\bf r}_i$, ${\bf r}_j$, ${\bf r}_k$ of each of the three spins of every elementary triangular plaquette of the triangular lattice. In order to analyze the local distribution of the nearest neighbor scalar chirality in the plaquettes, we plot a histogram of the local values of $\chi_{ijk}$ obtained from snapshots of a $L=60$ lattice, at $T/J=2 \times 10^{-3}$, for $h/J=0.4,0.8$, in Fig. \[fig:histo\]. For a perfect translational invariant chiral state, a strong peak at a given value of $\chi_{ijk}$ is expected. However, in this phase the values of $\chi_{ijk}$ are widely spread, confirming the non-coplanar nature of the low temperature phases at low fields.
![ (Color online) Histogram of the nearest neighbor chirality $\chi_{ijk}$ per triangular plaquette, for two snapshots at $T/J=2 \times 10^{-3}$, $L=60$, $h/J=0.8$ (red) and $h/J=0.4$ (blue). []{data-label="fig:histo"}](fig_6.png){width="0.98\columnwidth"}
Due to the lack of periodicity of the magnetic structure, it is not possible to make a direct connection between the real space configuration and spin structure factor in the reciprocal space (Fig. \[fig:snaps\]A and Fig. \[fig:snaps\]D represent a typical spin texture and the structure factor $S_{\bf q}^{\perp}$ respectively). To further study this phase, we introduce what we call the “spherical snapshot”: it shows the values of the spins in the sphere where each point represents the tip of the spin centered at the origin, and the three axis correspond to the three components of the spins. This representation is a very useful tool in order to identify features of the spin configuration since it allows to differenciate the sublattices in a same plot and to compare, qualitatively, the spin textures between similar or different phases. In the top panel of Fig. \[fig:sphesnaplowh\] we show the spherical snapshots for $h/J=0.8$. Each color indicates the spins of each of the three triangular sublattices of the kagome lattice (see Fig.\[fig:latt\]). Clearly, there is a symmetric distribution of the spin values in the three sublattices. This is consistent with the symmetric peak distribution in the structure factor. Interestingly, even though the inclusion of a small in-plane DM interaction induces the emergence of non-coplanar arrangements, it is not enough to completely lift the degeneracy at low magnetic fields, leading to this “locally chiral” phase.
![\[fig:sphesnaplowh\] (Color online) Spherical snapshot at $T/J=2\times 10^{-3}$, for “locally chiral” phase at $h/J=0.8$ (top panel) and the spiral phase at $h/J=1.8$ (bottom panel). Each color indicates a different triangular sublattice. The right column is the top view of the spherical snapshot. ](fig_7.png "fig:"){width="50.00000%"} ![\[fig:sphesnaplowh\] (Color online) Spherical snapshot at $T/J=2\times 10^{-3}$, for “locally chiral” phase at $h/J=0.8$ (top panel) and the spiral phase at $h/J=1.8$ (bottom panel). Each color indicates a different triangular sublattice. The right column is the top view of the spherical snapshot. ](fig_8.png "fig:"){width="50.00000%"}
Spiral phase - $h_{c1}<h<h_{c2}$
--------------------------------
As the magnetic field increases, an interesting behavior is found at low temperatures. For magnetic fields $h_{c1} < h < h_{c2}$, coming from the “locally chiral” phase described above, a spiral-like texture emerges, where there is a clear real-space splitting in the three triangular sublattices. This is shown in the spherical snapshot presented in the bottom panel of Fig. \[fig:sphesnaplowh\]. The arrangement is not symmetric: two of the sublattices are described by the same modulation with different wave vector orientation and the $S^{z}$ component takes all values of the unitary sphere. In the remaining sublattice (indicated by blue points), the $S^{z}$ components are restricted to positive values, with an additional modulation. Clearly, which sublattice is arranged in which way depends on the MC realization as can be observed from the structure factor presented in Fig. \[fig:snaps\](E), which corresponds to a particular MC realization. However, the symmetry would be restored when averaging on several realizations, as we have checked. This simple analysis based in the inspection of the spherical snapshot suggests that there is a sublattice symmetry-breaking induced by the magnetic field. To further explore this, and to detect the spontaneous sublattice symmetry breaking, we introduce a $\mathcal{Z}_{3}$ complex order parameter $\phi_{tot}$ defined as:
$$\begin{aligned}
\phi_{\triangle}&=&S_{1}^z + wS_{2}^z+ w^2S_{3}^z\nonumber \\
\phi_{tot}&=&\left|\frac{1}{L^2}\sum_{\triangle}\phi_{\triangle} \right|\end{aligned}$$
where $w=\exp({i\,2\pi/3})$ and $S^z_{\alpha}$ is the $z$ component of the spins in each of the three triangular sublattices, indicated with $\alpha=1,2,3$, shown in Fig. \[fig:latt\].
In Fig. \[fig:mz3\] we show this parameter $\phi_{tot}$ as a function of the external magnetic field at $T/J=2\times 10^{-3}$ for lattice size $L=60$. It can be seen that this parameter is non-zero only on this spiral-like phase, where the symmetry between sublattices is broken. This feature is stable with the system size as can be seen in the inset in Fig. \[fig:mz3\] ($\phi_{tot}$ as a function of temperature for $h/J=1.8$ and $L=48,54,60$).
![\[fig:mz3\] (Color online) Order parameter $\phi_{tot}$ as a function of the external magnetic field at $T/J=2\times 10^{-3}$, $L=60$. Nonzero values of $\phi$ indicates the sublattice magnetization is not the same for each sublattice. Inset: $\phi_{tot}$ vs $T/J$ at $h/J=1.8$ for $L=48$, $L=54$ and $L=60$.](fig_9.png){width="1\columnwidth"}
Antiferromagnetic pseudo-skyrmion crystal
-----------------------------------------
For $h_{c2}<h<h_{c3}$, at low temperatures a highly non trivial chiral phase emerges, associated with a spin texture formed by a particularly intricate three-sublattice splitting, that we call pseudo-skyrmion crystal (pSkX), very similar to the one found in the antiferromagnetic triangular lattice[@1]. In the triangular lattice case, the hidden skyrmion texture can be revealed by splitting the system in three interpenetrated sublattices: in each sublattice a topological skyrmion crystal SkX is stabilised.
In the kagome lattice case, the “pseudo-skyrmion” name is due to the fact the these structures are similar to skyrmions, but their center is not fully polarized. As an example a typical snapshot is shown in Fig. \[fig:snaps\] (C).
However, in the model considered here, albeit its similarities with the one proposed for the antiferromagnetic triangular lattice, the picture is not that simple. For the kagome lattice, there are also three interpenetrated sublattices of pseudo-crystals, but these are not the three triangular sublattices constructed from the three sites in the unit cell of kagome lattice. In Fig. \[fig:pseudosnap\] (top) we isolate one such pseudo-skyrmion from different sublattices at $h/J=2.6$. A pseudo-skyrmion is formed by an hexagon of spins joined by second nearest neighbor bonds at the center, and it radially increases along third nearest neighbor bonds. The third-nearest neighbors form the three triangular sublattices of the kagome lattice. Spins belonging to each type of sublattice in the pseudo-skyrmions are highlighted in the top panel of Fig. \[fig:pseudosnap\]. Then, one way to extract information about the hidden structure is through the total third-nearest neighbors scalar chirality per site (i.e. the sublattice chirality) defined as:
$$\label{eq:chinn3}
\chi_{n3}= \frac{1}{8\pi\,N}\sum_{\alpha=1}^3\left \langle \sum_{m=1}^{N/3} \chi_{mpq}^{(\alpha)} \right \rangle$$
where $\chi_{mpq}^{(\alpha)}$ is the local sublattice chirality, defined as Eq. (\[eq:local\_chira\]) but taking the three spins $m,p,q$ in elementary triangles in the triangular sublattices of the kagome lattice, $\alpha=1,2,3$. In Fig. \[fig:pseudosnap\] (bottom) we plot this parameter as a function of the magnetic field in for $T/J=2 \times 10^{-3}$ and $L=60$.
![(Color online) Top: Snapshot showing the pseudo-skyrmion structure for $D/J=0.2$, $h/J=2.6$, $T/J=2 \times 10^{-3}$. A pseudo-skyrmions is shown, erasing the neighbouring spins. Every coloured triangle indicates a diferent triangular sublattice of the kagome lattice. Bottom: Sublattice chirality density as function of $h/J$ for $L=60$ (green open triangles) at $T/J=2\times 10^{-3}$. Inset: sublatice chirality density as function of $T/J$ for $L=48, 54, 60$ at $h/J=3.5$.[]{data-label="fig:pseudosnap"}](fig_10_b.png){width="0.9\columnwidth"}
In the low field boundary of the pSkX phase ($h/J \sim 2.5$) the local sublattice chirality $\chi_{n3}$ takes a value close to the number of pseudo-skyrmions that can be constructed. For example, in a system with $N=8748$ sites, we found $36$ pseudo-skyrmions while the local (sublattice) chirality $\chi_{n3}\approx 31$. This implies that the topological charge of each skyrmion is not $1$, but $Q\approx 0.86$. Hence dubbing this phase as a “pseudo-skyrmion” phase. Another way to see this is that the texture associated with each “pseudo-skyrmion”, when projected onto the sphere, does not fully wrap the sphere. Specifically, we find that the border of the pseudo-skyrmions is completely polarized (parallel to the field), but the core is not antipolarized, i.e.: the $S^z$ component never reaches the value $S_z=-1$. To illustrate this clearly, we show a typical spherical snapshot obtained from simulations in Fig. \[fig:sphersnappseudo\], for three values of the magnetic field $h/J=2.6, 3.8, 4.8$, at $T/J= 2 \times 10^{-3}$ and $L=60$. As before the spins from each site of a given triangular sublattice are represented with different colors. Two significant features are present in this plot: the projections of the spins are divided in three “slices”, one for each triangular sublattice, and the lowest value of the projection along the field (found for the lowest magnetic field) is $S_z=-0.8$.
Skyrmion-like structures that do not fully cover the sphere, i.e. with $Q<1$, have already been e.g. in the anisotropic triangular lattice [@12] and in [@25], where there is an emergent intermediate phase between skyrmions and merons. In the model presented here, as the magnetic field increases, the “pseudo-skyrmion” cores are further canted, while the edges of the magnetic structures remain parallel to the field. The evolution of the spin textures as a function of the magnetic field in the pSkX phase is shown in Fig.\[fig:sphersnappseudo\].
![\[fig:sphersnappseudo\] (Color online) Spherical snapshots in the pSkX phase for $L=60$ , $h/J=2.6$ (top), $h/J=3.8$ (middle) and $h/J=4.8$ (bottom) with $T/J=2\times 10^{-3}$, $D/J=0.2$. Each color indicates a different triangular sublattice.](fig_11.pdf "fig:"){width="0.95\columnwidth"} ![\[fig:sphersnappseudo\] (Color online) Spherical snapshots in the pSkX phase for $L=60$ , $h/J=2.6$ (top), $h/J=3.8$ (middle) and $h/J=4.8$ (bottom) with $T/J=2\times 10^{-3}$, $D/J=0.2$. Each color indicates a different triangular sublattice.](fig_12.pdf "fig:"){width="0.95\columnwidth"} ![\[fig:sphersnappseudo\] (Color online) Spherical snapshots in the pSkX phase for $L=60$ , $h/J=2.6$ (top), $h/J=3.8$ (middle) and $h/J=4.8$ (bottom) with $T/J=2\times 10^{-3}$, $D/J=0.2$. Each color indicates a different triangular sublattice.](fig_13.pdf "fig:"){width="0.95\columnwidth"}
![\[fig:phasediag\] (Color online) Complete $T/J$ vs $h/J$ magnetic phase diagram obtained from Monte Carlo simulations. The solid black line was obtained analysing the peaks in the specific heat, the dashed white lines restrict the nonzero $\phi_{tot}$ area $h_{c1}<h<h_{c2}$, and the CP phase was obtained computing the structure factor $S_{\bf q}$.](fig_14_b.png){width="0.98\columnwidth"}
The radius of this “pseudo-skyrmions” changes slightly depending on the field and system size. However, this phase is clearly present for all system sizes studied, as shown in Fig. \[fig:pseudosnap\] (bottom), and it is delimited for a certain range of magnetic fields, here $ h_{c2} < h < h_{c3}$. The sharp “saw-tooth” behavior of this parameter in this phase (see Fig. \[fig:pseudosnap\] bottom) is due to the fact that the radius of the pseudo-skyrmion changes with the field. A sharp change implies that pseudo-skyrmions with a different radii are found at that field. These magnetic structures with different radii are very close in energy, which explains the competition between pseudo-skyrmion crystals of different (but similar) sizes at lower temperatures. Despite this competition, this pseudo-skyrmion crystal phase emerges and is stabilized at low temperature, and can be distinguished through $\chi_{n3}$, the scalar chirality calculated in the elementary triangles of each sublattice of the kagome lattice. No system size effects are noticed in this phase, the inset of Fig. \[fig:pseudosnap\] (bottom) shows $\chi_{n3}$ as a function of temperature for $h/J=3.5$ for three different system sizes ($L = 48, 54, 60$) where cleary the behavior is the same for all the cases.
In the high field region $h_{c3}<h<h_{c4}$ the spin moments are continuously further aligned with the magnetic field and the pSkX is destroyed. Because of this, $\chi_{n3}$ decreases from its maximum value in $h_{c3}$ to zero in $h_{c4}$. We thus dub this phase the “chiral polarized phase”.
With al this, combining the $\chi_{n3}$ parameter and the information from the previous subsections, we construct the temperature vs. magnetic field phase diagram, presented in Fig. \[fig:phasediag\]. There are four clear low temperature phases: at low magnetic fields, there is a locally chiral phase with no clear order. In this region, at higher temperatures the system behaves like a cooperative paramagnet. As the field increases, at low temperatures, coming from the locally chiral phase, we find an intermediate spiral phase. Here, the sublattice symmetry is broken: two out of three sublattices form a complete spiral, and the third one only has positive projections along the external field. The most remarkable feature of the phase diagram is an extended pSkX region which is stabilized in a broad range of magnetic fields at low enough temperatures. In this phase, pseudo-skyrmion structures are periodically arranged in three non trivial sublattices, which are in turn constructed with groups of third nearest neighbours. Therefore, we identify this phase with a topological order parameter, the third nearest neighbor chirality. As the field increases, the spins are further canted, the pseudo skyrmions are destroyed, and the chirality decreases with the field, in a chiral polarized phase.
Conclusions {#sec:conclusions}
===========
The frustration in the kagome antiferromagnet is known to give rise to a plethora of exotic phenomena. The competition of different types of interactions and external fields has been shown to both relieve the frustration and induce topological phases. In this work, we present a study of the low temperature phases in the classical kagome antiferromagnet with competing in-plane antisymmetric Dzyaloshinskii-Moriya interactions under a magnetic field using extensive Monte Carlo simulations. We find that, differently to what was found in previous studies in other less frustrated geometries, the particular geometry of the kagome lattice gives rise to highly non trivial magnetic orders. Firstly, for lower fields, although the system retains some degeneracy of the pure kagome antiferromagnet, small clusters with local chirality can be identified. Interestingly, at higher temperatures, inspection of the structure factor and the specific heat shows that the system is in a cooperative paramagnet phase, as the pure kagome antiferromagnet. As the field is increased, at lower temperatures, a three-sublattice spiral order is stabilized with broken sublattice symmetry: two triangular (third nearest neighbours) sublattices form a complete spiral and in the third one the spin projection along the field only takes positive values. This allows us to construct an $\mathcal{Z}_{3}$ order-parameter, $\phi_{tot}$, to identify the extension of this phase. Finally, we find that the external field stabilizes a pseudo-skyrmion crystal (pSkX) structure in a large portion of the magnetic phase diagram, up to the lowest simulated temperatures. This texture is characterized by a periodical arrangement of three interpenetrated non-trivial sublattices formed by skyrmion-like magnetic clusters. These clusters are not skyrmions, since, when projected on a sphere, the spins do not fully cover it. They have a clear polarized border and a non-fully polarized core. Moreover, due to the fact that these pseudo-skyrmions are constructed with groups of third nearest neighbors, this phase can be characterized by a topological parameter, the scalar chirality defined in each of the three triangular sublattices that constitutes the kagome lattice. For large enough fields, this parameter decreases rapidly to zero, as the pseudo-skyrmions are destroyed and the spins are further canted along the field. In conclusion, we have presented and studied with extensive Monte Carlo simulations a model that combines the high geometric frustration of the pure exchange model in the kagome lattice with antisymmetric Dzyaloshinskii-Moriya interactions, which are known to induce topologically non trivial structures when an external field is applied. We have found that these competing terms give rise to a rich magnetic phase diagram, where highly non trivial and topological phases are stabilized at low temperatures. We hope our study further contributes to the understanding of the connection between topology and frustration, where the kagome lattice is one of the most emblematic and relevant systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by CONICET (PIP 2015-813), ANPCyT (PICT 2012-1724), SECyT UNLP PI+D X792 and X788, PPID X039. H.D.R. acknowledges support from PICT 2016-4083. MV thanks Santiago Osorio for fruitful discussions.
[99]{} A. O. Leonov and M. Mostovoy, Nat. Commun. 6, 8275 (2015); S.-Z. Lin and S. Hayami, Phys. Rev. B 93, 064430 (2016); S. Hayami, S.-Z. Lin, Y. Kamiya, and C. D. Batista, Phys. Rev. B 94, 174420 (2016). J. H. Yu, W. H. Li, Z. P. Huang, J. J. Liang, J. Chen, D. Y. Chen, Z. P. Hou, M. H. Qin, arXiv:1810.12704 \[physics.app-ph\].
|
{
"pile_set_name": "ArXiv"
}
|
---
address: |
Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA\
E-mail: [email protected]
author:
- 'M. M. FOGLER'
title: QUANTUM HALL LIQUID CRYSTALS
---
=cmr8 1.5pt
Charge density waves in high Landau levels {#CDW}
==========================================
Historically, most of the research in the area of the quantum Hall effect has been focused on the case of very strong magnetic fields where all the electrons reside at the lowest Landau level (LL). [@FQHE] In contrast, phenomena described below occur in moderate and weak magnetic fields, i.e., at high LLs. Recent progress in the high LL problem can be summarized as follows. [@Fogler_xxx] The low-energy physics is thought to be dominated by the electrons residing in the single spin subband of the topmost ($N$th) LL, which has a filling fraction $\nu_N$ where $0 < \nu_N < 1$. All other electrons play the role of a dielectric medium, which renormalizes the interaction among these “active” electrons. This picture holds at arbitrary small magnetic fields provided there is no disorder and the temperature is zero, $T = 0$. This is because the broadening of the $N$th LL by electron-electron interactions is set by the quantity [@Aleiner_95; @Fogler_96] $E_{\rm
ex} \sim 0.1 e^2 / \kappa R_c$, where $R_c$ is the classical cyclotron radius and $\kappa$ is the bare dielectric constant. In a metallic 2D system with not too large $r_s$, $E_{\rm ex}$ is always smaller than the cyclotron gap and $N$th LL is well isolated from the other LLs. The cyclotron motion is the fastest motion in the problem, and so on the timescale at which the ground-state correlations are established, quasiparticles of $N$th LL behave as clouds of charge smeared along their respective cyclotron orbits. This prompts a quasiclassical analogy between the partially filled LL and a gas of interacting rings with radius $R_c$ and the areal density $(N + 1 / 2) \nu_N / \pi R_c^2$. At $\nu_N > 1 / N$ the rings overlap strongly in the real space.
Within a mean-field Hartree-Fock theory a partially filled LL undergoes a charge-density wave (CDW) transition. [@Fukuyama_79] At high LLs it occurs at a critical temperature [@Fogler_96] $T_c^{m f} \sim
0.25 E_{\rm ex}$. At $0.4 < \nu_N < 0.6$ the resultant CDW is a unidirectional, i.e., the [*stripe phase*]{}. At other $\nu_N$, the CDW has a symmetry of the triangular lattice and is called the [*bubble phase*]{}, see Fig. \[Fig\_stripes\_and\_bubbles\] (left). In both cases the CDW periodicity is set by the wavevector $q_* \approx 2.4 / R_c$. As $T$ decreases, the amplitude of the local filling factor modulation increases and eventually forces expulsion of regions with partial LL occupation. The system becomes divided into depletion regions where the local filling fraction is equal to $2 N$, and fully occupied areas where the local filling fraction is equal to $2 N + 1$. At these low temperatures the [*bona fide*]{} stripe and bubble domain shapes are evident, [@Fogler_96] see Fig. \[Fig\_stripes\_and\_bubbles\] (right).
![ Left: Mean-field phase diagram. Right: Guiding center density domain patterns at $T = 0$. Shaded and blank areas symbolize filled and empty regions, respectively. \[Fig\_stripes\_and\_bubbles\] ](fogler_pphmf_1.ps){width="3.25in"}
The mean-field theory is expected to be valid in the quasiclassical limit of large $N$. At moderate $N$ the CDW compete with Laughlin liquids and other fractional quantum Hall (FQH) states. A combination of analytic and numerical tools suggests that the FQH states lose to the CDW at $N \geq 2$.
The existence of the stripe phase as a physical reality was evidenced by a conspicuous magnetoresistance anisotropy observed near half-integral fractions of high LLs [@Lilly_99; @Du_99; @Shayegan_00] (see Ref. 13 for review). This anisotropy develops at $T < 0.1\,{\rm K}$ in high-mobility samples. The anisotropy is the largest at total filling factor $\nu =
9/2$ ($N = 2$, $\nu_N = 1 / 2$) and decreases with increasing LL index. At $T = 25\, {\rm mK}$ it remains discernible up to $\nu \sim 11
\frac12$ whereupon it is washed out, presumably, due to disorder and finite temperature. The anisotropy is natural once we assume that the stripe phase forms. The edges of the stripes can be visualized as metallic rivers, along which the transport is “easy.” The charge transfer among different edges, i.e., across the stripes, requires quantum tunneling and is “hard” because the stripes are effectively far away.
The existence of the [*bubble phases*]{} at high LLs is supported by the discovery of reentrant integral quantum Hall effect (IQHE) at $\nu
\approx 4.25$ and $\nu \approx 4.75$. The Hall resistance at such filling factors is quantized at the value of the nearest IQHE plateau, while the longitudinal resistance is isotropic and shows a deep minimum with an activated temperature dependence. The current-voltage ($I$-$V$) characteristics exhibit pronounced nonlinearity, switching, and hysteresis. These observations are consistent with the theoretical picture of a bubble lattice pinned by disorder.
Liquid crystal analogy for the stripe phase
===========================================
In the wake of the experiments, a considerable amount of work has been devoted to the stripe phase in recent years, Refs. 14–27. It led to the understanding that the “stripes” may appear in several distinct forms: an anisotropic crystal, a smectic, a nematic, and an isotropic liquid (Fig. \[Fig\_four\_phases\]). These phases succeed each other in the order listed as the magnitude of either quantum or thermal fluctuations increases. Consequently, the phase diagram of Fig. \[Fig\_stripes\_and\_bubbles\] needs modifications to incorporate some of those phases. The general structure of the revised phase diagram for the quantum ($T = 0$) case was discussed in the important paper of Fradkin and Kivelson. [@Fradkin_99] However, pinpointing the new phase boundaries in terms of the conventional parameters $r_s$, $\nu$, and $T$ requires further work. The most intriguing are the phases which bear the liquid crystal names: the smectic and the nematic. They are the main subject of this report. Let us start with the basic definitions of these phases.
The smectic is a liquid with the 1D periodicity, i.e., a state where the translational symmetry is spontaneously broken in one spatial direction. [@DeGennes_book] The rotational symmetry is of course broken as well. An example of such a state is the original Hartree-Fock stripe solution [@Fogler_96] although a stable quantum Hall smectic must have a certain amount of quantum fluctuations around the mean-field state. [@MacDonald_00; @Fertig_99] The necessary condition for the smectic order is the continuity of the stripes. If the stripes are allowed to rupture, the dislocations are created. They destroy the 1D positional order and convert the smectic into the nematic. [@Toner_81]
By definition, the nematic is an anisotropic liquid. [@DeGennes_book] There is no long-range positional order. As for the orientational order, it is long-range at $T = 0$ and quasi-long-range (power-law correlations) at finite $T$. The nematic is riddled with dynamic dislocations. Other types of topological defects, the disclinations, may also be present but remain bound in pairs, much like vortices in the 2D $X$-$Y$ model. Once they unbind, all the spatial symmetries are restored. The resultant state is an isotropic liquid with short-range stripe correlations. As the fluctuations due to temperature or quantum mechanics increase further, it gradually crosses over to the “uncorrelated liquid” where even the local stripe order is obliterated.
![ Sketches of possible stripe phases. \[Fig\_four\_phases\] ](fogler_pphmf_2.ps){width="3.1in"}
It is often the case that the low-frequency long-wavelength physics of the system is governed by an effective theory involving a relatively small number of dynamical variables. In the remaining sections we will discuss such type of theories for the quantum Hall liquid crystals.
Smectic state
=============
[*Effective theory*]{}.— The collective variables in the smectic are (i) the deviations $u(x, y)$ of the stripes from their equilibrium positions and (ii) long-wavelength density fluctuations $n$ about the average value $n_0$. The latter fluctuations may originate, e.g., from width fluctuations of the stripes. Let us assume that the stripes are aligned in the $\hat{\bf y}$-direction, then the symmetry considerations fix the effective Hamiltonian for $u$ and $n$ to be [@DeGennes_book; @Fogler_00] $$H = \frac{Y}{2} \Bigl[\partial_x u - \frac12 (\nabla u)^2\Bigr]^2
+ \frac{K}{2} (\partial_y^2 u)^2 + \frac12 n U n,
\label{H_smectic}$$ where $Y$ and $K$ are the phenomenological compression and the bending elastic moduli, and $U(r) = e^2 / \kappa r$ should be understood as the integral operator. The dynamics of the smectic is dominated by the Lorentz force and is governed by the Largangean $${\cal L} = p \partial_t u - H,\quad
\partial_y p = -m \omega_c (n + n_0 \partial_x u),
\label{L_smectic}$$ where $m$ is the electron mass and $\omega_c = e B / m c$ is the cyclotron frequency.
From Eqs. (\[H\_smectic\]) and (\[L\_smectic\]) we can derive the spectrum of collective modes, the [*magnetophonons*]{}. It is natural to start with the harmonic approximation where one replaces the first term in $H$ simply by $(Y / 2) (\partial_x u)^2$. Solving the equations of motion for $n$ and $u$ we obtain the magnetophonon dispersion relation: [@Fogler_00] $$\omega({\bf q}) = \frac{\omega_p(q)}{\omega_c} \frac{q_y}{q}
\left[\frac{Y q_x^2 + K q_y^4}{m n_0}\right]^{1/2}.
\label{omega_smectic}$$ Here $\omega_p(q) = [n_0 U(q) q^2 / m]^{1/2}$ is the plasma frequency and $\theta = \arctan (q_y / q_x)$ is the angle between the propagation direction and the $\hat{\bf x}$-axis. For Coulomb interactions $\omega_p(q) \propto \sqrt{q}$. Unless propagate nearly parallel to the stripes, $\omega({\bf q})$ is proportional to $\sin 2
\theta\, q^{3 / 2}$. One immediate consequence of this dispersion is that the largest velocity of propagation for the magnetophonons with a given $q$ is achieved when $\theta = 45^\circ$.
[*Thermal fluctuations and anharmonisms*]{}.— From Eq. (\[H\_smectic\]) we can readily calculate the mean-square fluctuations of the stripe positions at finite $T$, e.g., $$\langle [u(0, 0) - u(0, y)]^2 \rangle = \frac{k_B T}{2 \sqrt{Y K}} |y|.
\label{uu_smectic}$$ As one can see, at any finite temperature magnetophonon fluctuations are growing without a bound; hence, the positional order of a 2D smectic is totally destroyed [@DeGennes_book] at sufficiently large distances along the $\hat{\bf y}$-direction, $|y| \gg \Lambda \sqrt{Y K} / k_B T
\equiv \xi_y$ where $\Lambda = 2 \pi / q_*$ is the interstripe separation. Similarly, along the $\hat{\bf x}$-direction, the positional order is lost at lengthscales larger than $\xi_x = (Y / K)^{1/2}
\xi_y^2$.
Another type of excitations, which disorder the stripe positions are the aforementioned dislocations. The dislocations in a 2D smectic have a finite energy $E_D \sim K$. At $k_B T \ll E_D$ the density of thermally excited dislocations is of the order of $\exp(-E_D / k_B T)$ and the average distance between dislocations is $\xi_D \sim \Lambda \exp(2 k_B
T / E_D)$. At low temperatures $\xi_x, \xi_y \ll \xi_D$; therefore, the following interesting situation emerges (Fig. \[Fig\_spaghetti\]). On the lengthscales smaller than $\xi_y$ (or $\xi_x$, whichever appropriate) the system behaves like a usual smectic where Eqs. (\[H\_smectic\]–\[omega\_smectic\]) apply. On the lengthscales exceeding $\xi_D$ it behaves[^1] like a nematic. [@Toner_81] In between the system is a smectic but with very unusual properties. It is topologically ordered (no dislocations) but possesses enormous fluctuations. In these circumstances the harmonic elastic theory becomes inadequate and anharmonic terms must be treated carefully.
![ Portraits of the stripe phase on different lengthscales. \[Fig\_spaghetti\] ](fogler_pphmf_3.ps){width="2.8in"}
As shown by Golubović and Wang, [@Golubovic_92] the anharmonisms cause power-law dependence of the parameters of the effective theory on the wavevector ${\bf q}$: $$Y \sim Y_0 (\xi_y q_y)^{1/2},\quad K \sim K_0 (\xi_y q_y)^{-1/2},
\label{limit_A}$$ for $q_x \ll \xi_x^{-1} (q_y \xi_y)^{3/2}$, $q_y \ll \xi_y^{-1}$, and $$Y \sim Y_0 (\xi_x q_x)^{1/3},\quad K \sim K_0 (\xi_x q_x)^{-1/3},
\label{limit_B}$$ for $q_x \ll \xi_x^{-1}$ and $q_y \ll \xi_y^{-1} (q_x \xi_x)^{2/3}$. The lengthscale dependence of the parameters of the effective theory is a common feature of fluctuation-dominated phenomena. It should be mentioned that the lower critical dimension for the smectic order is $d
= 3$, [@DeGennes_book] so that the 2D smectic is [*below*]{} its lower critical dimension. This is the reason why the scaling behavior (\[limit\_A\]) and (\[limit\_B\]) does not persist indefinitely but eventually breaks down above the lengthscale $\xi_D$ where the crossover to the thermodynamic limit of the nematic behavior commences.
The scaling shows up not only in the static properties such as $Y$ and $K$ but also in the dynamics. The role of anharmonisms in the dynamics of conventional 3D smectics has been investigated by Mazenko [*et al.*]{} [@Mazenko_83] and also by Kats and Lebedev [@Kats_Lebedev]. For the quantum Hall stripes the analysis had to be done anew because here the dynamics is totally different. It is dominated by the Lorentz force rather than a viscous relaxation in the conventional smectics. This task was accomplished in Ref. 17. The calculation was based on the Martin-Siggia-Rose formalism combined with the $\epsilon$-expansion below $d = 3$ dimensions. One set of results concerns the spectrum of the magnetophonon modes, which becomes $$\omega({\bf q}) \sim \sin \theta \cos^{7/6}\theta\,
(\xi_x q)^{5/3} \frac{\omega_p(\xi_x^{-1})}{\omega_c \xi_x}
\sqrt{\frac{Y_0}{m n_0}}.
\label{omega_m_R}$$ Compared to the predictions of the harmonic theory, Eq. (\[omega\_smectic\]), the $q^{3/2}$-dispersion changes to $q^{5/3}$. Also, the maximum propagation velocity is achieved for the angle $\theta \approx 53^\circ$ instead of $\theta = 45^\circ$. These modifications, which take place at long wavelengths, are mainly due to the renormalization of $Y$ in the static limit and can be obtained by combining Eqs. (\[omega\_smectic\]) and (\[limit\_B\]). Less obvious dynamical effects peculiar to the quantum Hall smectics include a novel dynamical scaling of $Y$ and $K$ as a function of frequency and a specific $q$-dependence of the magnetophonon damping. [@Fogler_00]
The latter issue touches on an important point. Our effective theory defined by Eqs. (\[H\_smectic\]) and (\[L\_smectic\]) is based on the assumption that $u$ and $n$ are the only low-energy degrees of freedom. It is probably well justified at $T \rightarrow 0$ but becomes incorrect at higher temperatures. The point of view taken in Ref. 17 is that in the latter case thermally excited quasiparticles (“normal fluid”) should appear and that they should bring dissipation into the dynamics of the magnetophonons. Another intriguing possibility is for quasiparticles or other additional low-energy degrees of freedom to exist even at $T = 0$. Such more complicated smectic states are interesting subjects for future study.
Nematic
=======
As discussed above, at finite temperature and in the thermodynamic limit, the smectic phase is always unstable. The lowest degree of ordering is that of a nematic. [@DeGennes_book] An intriguing possibility [@Fradkin_99] is to have a nematic phase already at $T =
0$, due to quantum fluctuations. The collective degree of freedom associated with the nematic ordering is the angle $\phi({\bf r}, t)$ between the local normal to the stripes ${\bf N}$ and the $\hat{\bf
x}$-axis orientation. The effective Hamiltonian for ${\bf N}$ is dictated by symmetry to be $$H_N = \frac{K_1}{2} (\nabla {\bf N})^2
+ \frac{K_3}{2} |\nabla \times {\bf N}|^2.
\label{H_nematic}$$ The phenomenological coefficients $K_1$ and $K_3$ are termed the splay and the bend Frank constants. [@DeGennes_book] A particularly simple form is obtained if $K_1 = K_3$, in which case $H_N = (K_3 / 2) (\nabla
\phi)^2$ just like in the $X$-$Y$ model.
Another obvious degree of freedom in the nematic are the density fluctuations $n({\bf r}, t)$. A peculiar fact is that in the static limit $n$ is totally decoupled from ${\bf N}$, and so it does not enter Eq. (\[H\_nematic\]). However, since the nematic is less ordered than even a smectic, the question about extra low-energy degrees of freedom or additional quasiparticles is highly relevant. We believe that different types of quantum Hall nematics are possible in nature. In the simplest case scenario ${\bf N}$ and $n$ are the only low-energy degrees of freedom. This is presumably the case when the nematic order is a superstructure on top of a parent uniform state. A concrete example is described by a wavefunction proposed by Musaelian and Joynt: [@Musaelian_96] $$\Psi = \prod\limits_{j < k} (z_{j} - z_{k}) [(z_{j} - z_{k})^2 - a^2]
\times \exp \Bigl(-\sum_j |z_j|^2 / 4 l^2\Bigr).
\label{nematic_trial_wavefunction}$$ Here $z_j = x_j + i y_j$ is a complex coordinate of $j$th electron, $l =
\sqrt{\hbar c / e B}$ is the magnetic length, and $a$ is another complex parameter that determines the degree of orientational order and the direction of the stripes. The rotational invariance is broken if $|a|$ exceeds some critical value. This particular wavefunction corresponds to $\nu = \frac13$ but can be easily generalized to higher Landau levels with $\nu_N = \frac13$ filling. This type of state has been studied by Balents [@Balents_96] and recently by the present author. [@Fogler_01] It was essentially postulated that the effective Largangean takes the form $${\cal L} = \frac12 \gamma^{-1} (\partial_t {\bf N})^2 - H.
\label{L_nematic}$$ (As hinted above, the full expression contains also couplings between $\partial_t {\bf N}$ and mass currents but they become vanishingly small in the long-wavelength limit). The collective excitations are charge-neutral fluctuations of the director. They have a linear dispersion, $$\omega({\bf q}) = q \sqrt{{K_1}{\gamma} \cos^2 \theta +
{K_3}{\gamma} \sin^2 \theta},
\label{omega_nematic}$$ and resemble spinwaves in the $X$-$Y$ quantum rotor model.
The quantum nematic phase must be separated from the stable zero-temperature smectic by a quantum phase transition. Further insights into the properties of the quantum nematics can be gained by analyzing the nature of such a transition. By analogy to the classical smectic-nematic transition in two [@Toner_81] and three [@Toner_82] dimensions, we expect the quantum one to also be driven by the proliferation of dislocations. Pictorially, the difference between the smectic and nematic can be represented as follows. The dislocations are viewed as lines in the (2 + 1)D space. In the smectic phase, they form small closed loops (Fig. \[Fig\_worldlines\]a) that depict virtual pair creation-annihilation events; in the nematic phase arbitrarily long dislocation worldlines exist and may entangle (Fig. \[Fig\_worldlines\]b), similar to worldlines of particles in a Bose superfluid.
![ Worldlines of dislocations in (a) smectic (b) nematic. \[Fig\_worldlines\] ](fogler_pphmf_4.ps){width="2.2in"}
To incorporate the dislocations into the effective Lagrangean (\[L\_smectic\]), we use a well known duality transformation, see, for example, Ref. 34. By means of such a transformation the original degrees of freedom $u$ and $n$ are traded for new variables: the second-quantized dislocation field $\Phi$ and an auxillary $U(1)$ gauge field $a_\mu$, which mediates the interaction among the dislocations. The imaginary-time effective action for $\Phi$ and $a_\mu$ has the form $$\begin{aligned}
A &=& \int\limits_0^\beta d \tau \int d^2 {\bf r}
\biggl\{\frac{t_\mu}{2}|(-i \partial_\mu - \Lambda a_\mu
- e_D a_\mu^{\rm ext}) \Phi|^2\biggr.
\nonumber\\
\mbox{} &+& \biggl.V(\Phi) + H_a[a]\biggr\},
\label{A_dual}\\
H_a &=& \frac{\sigma_x^2}{2 Y}
+ \sigma_y (-2 K \partial_y^2)^{-1} \sigma_y
+ \frac{l^4}{2} \partial_y \sigma_\tau U \partial_y \sigma_\tau,
\label{H_a}\\
\sigma_\mu &=& \epsilon_{\mu\nu\lambda}
\partial_\nu a_\lambda \equiv [\partial \times a]_\mu.
\label{a}\end{aligned}$$ Phenomenological parameters introduced above are as follows. Parameter $t_\tau$ is of the order of $\hbar^2 / E_c$, where $E_c$ is the dislocation core energy estimated within the Hartree-Fock approximation in Refs. 4 and 21. Unfortunately, this estimate is not reliable in the quantum nematic state where the quantum fluctuations are large. Parameter $t_x$ of dimension of ${\rm energy} \times {\rm
(length)}^2$ is the hopping matrix element for dislocation motion in the $\hat{\bf x}$-direction, i.e., dislocation [*glide*]{}. Such a glide requires quantum tunneling and is exponentially small unless $\Lambda <
l$. Parameter $t_y$ describes the dislocation climb, which also originates from the dynamics on the microscopic length scales. Yet another phenomenological variable is the potential $V(\Phi) = m_\Phi
|\Phi|^2 + r_\Phi |\Phi|^4 + \ldots$, which accounts for a self-energy and a short-range interaction between the dislocations; the scales of $m_\Phi$ and $r_\Phi$ are set by $E_c$ and $E_c \Lambda^2$, respectively. Finally, $e_D$ is electric charge of the dislocation that couples to the external vector potential $a_\tau^{\rm ext} = a_x^{\rm
ext} = 0$, $a_y^{\rm ext} = B x$. This coupling is introduced only for the sake of generality. Since we study electron liquid crystal phases derived from incompressible liquids, we expect dislocations to be electrically neutral, i.e., $e_D = 0$.
To recover Eq. (\[omega\_nematic\]) we assume that the dislocations have condensed, $\langle \Phi \rangle = \Phi_0 \neq 0$. Solving for the collective mode spectrum of the action (\[A\_dual\]), we find $$\omega_1({\bf q}) =
\left(\frac{m_x}{m_\tau} q_x^2 + m_x K q_y^2\right)^{1/2},
\:\: m_\mu \equiv t_\mu \Lambda^2 |\Phi_0|^2,
\label{omega_nematic_1}$$ which is consistent with Eq. (\[omega\_nematic\]) if $K_1 = 1 /
m_\tau$, $K_3 = K$, and $\gamma = m_x$.
Remarkably, the magnetophonon mode of the parent smectic (\[omega\_smectic\]), does not totally disappear from the spectrum. Instead, it acquires a small gap $\sqrt{m_y Y}$ at $q = 0$. This gapped mode anti-crosses with the acoustic branch (\[omega\_nematic\_1\]) near the point $\omega_1^2({\bf q}) \sim
m_y Y$, and at larger $q$ becomes the lowest frequency collective mode with the dispersion relation $$\omega_2({\bf q}) = \left[\frac{q^2_x q^2_y}{m^2 \omega_c^2} Y U(q)
+ m_y Y\right]^{1/2}
\label{omega_nematic_2}$$ only slightly different from (\[omega\_smectic\]). At such $q$ the structure factor of the nematic has two sets of $\delta$-functional peaks, $$S(\omega, {\bf q}) = \frac{\pi \hbar q_y^2}{m \omega_c^2}
\left[\frac{K q_y^4}{m n_0} \delta(\omega^2 - \omega_1^2) +
\frac{Y q_x^2}{m n_0} \delta(\omega^2 - \omega_2^2)\right],$$ which split between themselves the spectral weight of the single collective mode of the smectic. The presence of the two modes can be explained by the existence of two order parameters: the aforementioned unit vector (more precisely, director) ${\bf N}$ normal to the local stripe orientation and the complex wavefunction $\Phi_0$ of the dislocation condensate. Classical 2D nematics have two (overdamped) modes virtually for the same reason. [@Toner_81]
Recently, Radzihovsky and Dorsey [@Radzihovsky_02] formulated a qualitatively different theory of the quantum Hall nematics, whose predictions disagree with our Eqs. (\[L\_nematic\]) and (\[omega\_nematic\]). At this point it is unclear whether these authors study a different kind of nematic or they actually contest the theoretical models proposed by Balents [@Balents_96] and the present author. [@Fogler_01] To resolve some of these issues it is imperative to bring the discussion from the level of effective theory to the level of quantitative calculations. One promising direction is to investigate some concrete trial wavefunctions of quantum nematics, e.g., Eq. (\[nematic\_trial\_wavefunction\]). Recently, the work in this direction was continued by Ciftja nad Wexler. [@Wexler_02] It is also desirable to find a functional form of the electron-electron interaction which gives rise to the nematic ground state. An educated guess [@Fogler_xxx] is that even a realistic Coulomb interaction may be sufficient provided $r_s \ll 1$ and $1 \ll N \ll r_s^{-2}$. However, the quest for quantum nematic may not be easy. Finite-size study by Rezayi [*et al.*]{} [@Rezayi_00] suggests that the transition from the smectic to an isotropic phase as a function of the interaction parameters (Haldane’s pseudopotentials) can also occur via a first-order transition, without the intermediate nematic phase.
Experimentally, the nematic can be distinguished from the smectic by, e.g., the microwave absorption technique: the nematic will show two dispersing collective modes while the smectic will produce a single one. To circumvent disorder pinning effects, such measurements should be done at high enough $q$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the MIT Pappalardo Fellowships Program in Physics. I would like to thank A. A. Koulakov, B. I. Shklovskii, and V. M. Vinokur for previous collaboration on the topics discussed.
References {#references .unnumbered}
==========
[29]{}
For review, see [*Quantum Hall Effect*]{} edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1990); [*Perspectives in Quantum Hall Effect*]{} edited by S. Das Sarma and A. Pinczuk (Wiley, New York, 1997).
For review, see M. M. Fogler, cond-mat/0111001.
I. L. Aleiner and L. I. Glazman, , 11 296 (1995).
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H. Fukuyama, P. M. Platzman, and P. W. Anderson, , 5211 (1979).
M. M. Fogler and A. A. Koulakov, , 9326 (1997).
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E. H. Rezayi, F. D. M. Haldane, and K. Yang, , 1219 (1999); [*ibid*]{} [**85**]{}, 5396 (2000).
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M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, , 394 (1999).
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M. Shayegan, H. C. Manoharan, S. J. Papadakis, and E. P. DePoortere, Physica E [**6**]{}, 40 (2000).
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A. H. MacDonald and M. P. A. Fisher, , 5724 (2000).
H. A. Fertig, , 3693 (1999).
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R. Côté and H. A. Fertig, , 1993 (2000).
H. Yi, H. A. Fertig, and R. Côté, , 4156 (2000).
M. M. Fogler, cond-mat/0107306.
C. Wexler and A. T. Dorsey, , 115 312 (2001).
D. G. Barci, E. Fradkin, S. A. Kivelson, and V. Oganesyan, cond-mat/0105448.
A. Lopatnikova, B. I. Halperin, S. H. Simon, and X.-G. Wen, cond-mat/0105079.
K. Musaelian and R. Joynt, J. Phys. Cond. Mat. [**8**]{}, L105 (1996).
L. Balents, Europhys. Lett. [**33**]{}, 291 (1996).
O. Ciftja and C. Wexler, cond-mat/0108119.
L. Radzihovsky and A. T. Dorsey, cond-mat/0110083.
P. G. de Gennes and J. Prost, [*The Physics of Liquid Crystals*]{} (Oxford University Press, New York, 1995).
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[^1]: In a more precise treatment, [@Golubovic_92] the lengthscales $\xi_{D x} \propto
\xi_D^{6/5}$ and $\xi_{D y} \propto \xi_D^{4/5}$ are introduced such that $\xi_{D x} \xi_{D y} = \xi_D^2$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'I review some recent work (done in collaboration with G. Veneziano) which clarifies the existence of a correspondence between self-gravitating fundamental string states and Schwarzschild black holes. The main result is a detailed calculation showing that self-gravity causes a typical string state of mass $M$ to shrink, as the string coupling $g^2$ increases, down to a compact string state whose mass, size, entropy and luminosity match (for the critical value $g_c^2 \sim (M \, \sqrt{\alpha''})^{-1}$) those of a Schwarzschild black hole. This confirms the idea that the entropy of black holes can be accounted for by counting string states, and suggests that the level spacing of the quantum states of Schwarzschild black holes is exponentially small, and very much blurred by radiative effects.'
author:
- Thibault Damour
title: Strings and black holes
---
Introduction
============
This contribution deals with the “explanation” of the quantum properties of black holes (and, in particular, the statistical meaning of black hole entropy) hopefully provided by string theory. Most of the stringy literature has concentrated (for reasons recalled below) on some special, supersymmetric extreme black holes (BPS black holes). \[These black holes carry special (Ramond-Ramond) charges and their microscopic structure seem to be describable in terms of Dirichlet-branes.\] By contrast, we consider here the simplest, Schwarzschild black holes (in any space dimension $d$). It will be argued that their “microscopic structure” involves only fundamental string states. However, the lack of supersymmetry means that it becomes essential to deal with self-gravity effects.
To start with, let us recall that thirty years ago the study of the spectrum of string theory revealed [@FVBM] a huge degeneracy of states growing as an exponential of the mass. A few years later Bekenstein [@Bekenstein] proposed that the entropy of a black hole should be proportional to the area of its horizon in Planck units, and Hawking [@Hawking] fixed the constant of proportionality after discovering that black holes do emit thermal radiation at a temperature $T_{\rm Haw}
\sim R_{\rm BH}^{-1}$.
When string and black hole entropies are compared one immediately notices a striking difference: string entropy[^1] is proportional to the first power of mass in any number of spatial dimensions $d$, while black hole entropy is proportional to a $d$-dependent power of the mass, always larger than $1$. In formulae: $$S_s \sim {\alpha' M \over \ell_s} \sim M / M_s ~~~~ , ~~~~~~
S_{\rm BH} \sim \frac{{\rm Area}}{G_N} \sim \frac{R_{\rm
BH}^{d-1}}{G_N} \sim \frac{(g^2 \, M /
M_s)^{\frac{d-1}{d-2}}}{g^2}\; ,
\label{entropies}$$ where, as usual, $\alpha'$ is the inverse of the classical string tension, $\ell_s \sim \sqrt{\alpha' \hbar}$ is the quantum length associated with it[^2], $M_s \sim \sqrt{\hbar
/ \alpha'}$ is the corresponding string mass scale, $R_{\rm BH}$ is the Schwarzschild radius associated with $M$: $$R_{\rm BH} \sim (G_N \, M)^{1/(d-2)} \; ,\label{eq1.1}$$ and we have used that, at least at sufficiently small coupling, the Newton constant and $\alpha'$ are related via the string coupling by $G_N \sim g^2 (\alpha')^{(d-1)/2}$ (more geometrically, $\ell_P^{d-1}\sim g^2 \ell_s^{d-1}$).
Given their different mass dependence, it is obvious that, for a given set of the fundamental constants $G_N, \alpha', g^2$, $S_s > S_{\rm BH}$ at sufficiently small $M$, while the opposite is true at sufficiently large $M$. Obviously, there has to be a critical value of $M$, $M_c$, at which $S_s = S_{\rm BH}$. This observation led Bowick et al. [@Bowick] to conjecture that large black holes end up their Hawking-evaporation process when $M = M_c$, and then transform into a higher-entropy string state without ever reaching the singular zero-mass limit. This reasoning is confirmed [@GVFC] by the observation that, in string theory, the fundamental string length $\ell_s$ should set a minimal value for the Schwarzschild radius of any black hole (and thus a maximal value for its Hawking temperature). It was also noticed [@Bowick], [@susskind], [@GVDivonne] that, precisely at $M= M_c$, $R_{\rm BH} = \ell_s$ and the Hawking temperature equals the Hagedorn temperature of string theory. For any $d$, the value of $M_c$ is given by: $$M_c \, \sim M_s g^{-2} \, . \label{eq1.2}$$
Susskind and collaborators , [@halyo] went a step further and proposed that the spectrum of black holes and the spectrum of single string states be “identical”, in the sense that there be a one to one correspondence between (uncharged) fundamental string states and (uncharged) black hole states. Such a “correspondence principle” has been generalized by Horowitz and Polchinski [@hp1] to a wide range of charged black hole states (in any dimension). Instead of keeping fixed the fundamental constants and letting $M$ evolve by evaporation, as considered above, one can (equivalently) describe the physics of this conjectured correspondence by following a narrow band of states, on both sides of and through, the string $\rightleftharpoons$ black hole transition, by keeping fixed the entropy[^3] $S = S_s = S_{\rm BH}$, while adiabatically[^4] varying the string coupling $g$, i.e. the ratio between $\ell_P$ and $\ell_s$. The correspondence principle then means that if one increases $g$ each (quantum) string state should turn into a (quantum) black hole state at sufficiently strong coupling, while, conversely, if $g$ is decreased, each black hole state should “decollapse” and transform into a string state at sufficiently weak coupling. For all the reasons mentioned above, it is very natural to expect that, when starting from a black hole state, the critical value of $g$ at which a black hole should turn into a string is given, in clear relation to (\[eq1.2\]), by $$g_c^2 \, M \sim M_s \, , \label{eq1.2'}$$ and is related to the common value of string and black-hole entropy via $$g_c^2 \sim \frac{1}{S_{\rm BH}} = \frac{1}{S_s}\; . \label{eq1.2''}$$ Note that $g_c^2 \ll 1$ for the very massive states ($M \gg
M_s$) that we consider. This justifies our use of the perturbative relation between $G_N$ and $\alpha'$.
In the case of extremal BPS, and nearly extremal, black holes the conjectured correspondence was dramatically confirmed through the work of Strominger and Vafa [@SV] and others [@others] leading to a statistical mechanics interpretation of black-hole entropy in terms of the number of microscopic states sharing the same macroscopic quantum numbers. However, little is known about whether and how the correspondence works for non-extremal, non BPS black holes, such as the simplest Schwarzschild black hole[^5]. By contrast to BPS states whose mass is protected by supersymmetry, we shall consider here the effect of varying $g$ on the mass and size of non-BPS string states.
Although it is remarkable that black-hole and string entropy coincide when $R_{\rm BH} = \ell_s$, this is still not quite sufficient to claim that, when starting from a string state, a string becomes a black hole at $g = g_c$. In fact, the process in which one starts from a string state in flat space and increases $g$ poses a serious puzzle [@susskind]. Indeed, the radius of a typical excited string state of mass $M$ is generally thought of being of order $$R_s^{\rm rw} \sim \ell_s (M / M_s)^{1/2} \, , \label{eq1.5}$$ as if a highly excited string state were a random walk made of $M/M_s = \alpha'M/\ell_s$ segments of length $\ell_s$ [@rw]. \[The number of steps in this random walk is, as is natural, the string entropy (\[entropies\]).\] The “random walk” radius (\[eq1.5\]) is much larger than the Schwarzschild radius for all couplings $g \le g_c$, or, equivalently, the ratio of self-gravitational binding energy to mass (in $d$ spatial dimensions) $$\frac{G_N \, M}{(R_s^{\rm rw})^{d-2}} \sim \left( \frac{R_{\rm BH}
(M)}{R_s^{\rm rw}} \right)^{d-2} \sim g^2 \left( \frac{M}{M_s}
\right)^{\frac{4-d}{2}} \label{eq1.6}$$ remains much smaller than one (when $d>2$, to which we restrict ourselves) up to, and including, the transition point. In view of (\[eq1.6\]) it does not seem natural to expect that a string state will “collapse” to a black hole when $g$ reaches the value (\[eq1.2’\]). One would expect a string state of mass $M$ to turn into a black hole only when its typical size is of order of $R_{\rm
BH} (M)$ (which is of order $\ell_s$ at the expected transition point (\[eq1.2’\])). According to Eq. (\[eq1.6\]), this seems to happen for a value of $g$ much larger than $g_c$.
Horowitz and Polchinski [@hp2] have addressed this puzzle by means of a “thermal scalar” formalism [@chi]. Their results suggest a resolution of the puzzle when $d=3$ (four-dimensional spacetime), but lead to a rather complicated behaviour when $d \geq
4$. Moreover, even in the simple $d=3$ case, the formal nature of the auxiliary “thermal scalar” renders unclear (at least to me) the physical interpretation of their analysis.
Here, I will review the results of a recent collaboration with G. Veneziano [@dv] whose aim was to clarify the string $\rightleftharpoons$ black hole transition by a direct study, in real spacetime, of the size and mass of a [*typical*]{} excited string, within the microcanonical ensemble of [*self-gravitating*]{} strings. Our results [@dv] lead to a rather simple picture of the transition, in any dimension. We find no hysteresis phenomenon in higher dimensions. The critical value for the transition is (\[eq1.2’\]), or (\[eq1.2”\]) in terms of the entropy $S$, for both directions of the string $\rightleftharpoons$ black hole transition. In three spatial dimensions, we find that the size (computed in real spacetime) of a [*typical self-gravitating*]{} string is given by the random walk value (\[eq1.5\]) when $g^2 \le g_0^2$, with $g_0^2 \sim (M/M_s)^{-3/2} \sim S^{-3/2}$, and by $$R_{\rm typ} \sim \frac{1}{g^2 \, M} \, , \label{eq1.8}$$ when $g_0^2 \le g^2 \le g_c^2$. Note that $R_{\rm typ}$ smoothly interpolates between $R_s^{\rm rw}$ and $\ell_s$. This result confirms the picture proposed by Ref. [@hp2] when $d=3$, but with the bonus that Eq. (\[eq1.8\]) refers to a radius which is estimated directly in physical space (see below), and which is the size of a typical member of the microcanonical ensemble of self-gravitating strings. In all higher dimensions[^6], we find that the size of a typical self-gravitating string remains fixed at the random walk value (\[eq1.5\]) when $g \le
g_c$. However, when $g$ gets close to a value of order $g_c$, the ensemble of self-gravitating strings becomes (smoothly in $d=4$, but suddenly in $d \geq 5$) dominated by very compact strings of size $\sim \ell_s$ (which are then expected to collapse with a slight further increase of $g$ because the dominant size is only slightly larger than the Schwarzschild radius at $g_c$).
Our results [@dv] confirm and clarify the main idea of a correspondence between string states and black hole states [@susskind], [@halyo], [@hp1], [@hp2], and suggest that the transition between these states is rather smooth, with no apparent hysteresis, and with continuity in entropy, mass, typical size, and luminosity. It is, however, beyond the technical grasp of our analysis to compute any precise number at the transition (such as the famous factor $1/4$ in the Bekenstein-Hawking entropy formula).
Size distribution of free string states
=======================================
For simplicity, we deal with open bosonic strings ($\ell_s \equiv \sqrt{2 \, \alpha'}$, $0 \leq \sigma \leq \pi$) $$X^{\mu} (\tau , \sigma) = X_{\rm cm}^{\mu} (\tau , \sigma) +
\widetilde{X}^{\mu} (\tau , \sigma) \, , \label{eq2.4}$$ $$X_{\rm cm}^{\mu} (\tau , \sigma) = x^{\mu} + 2 \, \alpha' \, p^{\mu} \,
\tau \, , \label{eq2.5}$$ $$\widetilde{X}^{\mu} (\tau , \sigma) = i \, \ell_s \sum_{n \not= 0} \
\frac{\alpha_n^{\mu}}{n} \ e^{-i n \tau} \, \cos \, n \, \sigma \, .
\label{eq2.6}$$ Here, we have explicitly separated the center of mass motion $X_{\rm
cm}^{\mu}$ (with $[x^{\mu} , p^{\nu}] = i \, \eta^{\mu \nu}$) from the oscillatory one $\widetilde{X}^{\mu}$ ($[\alpha_m^{\mu} ,
\alpha_n^{\nu}] = m \, \delta_{m+n}^0 \, \eta^{\mu \nu}$). The free spectrum is given by $\alpha' \, M^2 = N-1$ where $(\alpha \cdot
\beta\equiv \eta_{\mu \nu} \, \alpha^{\mu} \, \beta^{\nu} \equiv -
\alpha^0\, \beta^0 + \alpha^i \, \beta^i)$ $$N = \sum_{n=1}^{\infty} \ \alpha_{-n} \cdot \alpha_n =
\sum_{n=1}^{\infty} \ n \, N_n \, . \label{eq2.7}$$ Here $N_n \equiv a_n^{\dagger} \cdot a_n$ is the occupation number of the $n^{\rm th}$ oscillator ($\alpha_n^{\mu} = \sqrt{n} \
a_n^{\mu}$,$[a_n^{\mu} , a_m^{\nu \dagger}] = \eta^{\mu \nu} \,
\delta_{nm}$, with $n,m$ positive).
The decomposition (\[eq2.4\])–(\[eq2.6\]) holds in any conformal gauge ($(\partial_{\tau} \, X^{\mu} \pm \partial_{\sigma}
\, X^{\mu})^2 = 0$). One can further specify the choice of worldsheet coordinates by imposing $$n_{\mu} \, X^{\mu} (\tau , \sigma) = 2 \alpha' (n_{\mu} \, p^{\mu})
\, \tau \, , \label{eq2.8}$$ where $n^{\mu}$ is an arbitrary timelike or null vector ($n \cdot n
\leq 0$) [@scherk]. Eq. (\[eq2.8\]) means that the $n$-projected oscillators $n_{\mu} \, \alpha_m^{\mu}$ are set equal to zero. As we shall be interested in quasi-classical, very massive string states ($N \gg 1$) it should be possible to work in the “center of mass” gauge, where the vector $n^{\mu}$ used in Eq. (\[eq2.8\]) to define the $\tau$-slices of the world-sheet is taken to be the total momentum $p^{\mu}$ of the string. This gauge is the most intrinsic way to describe a string in the classical limit. Using this intrinsic gauge, one can covariantly define the proper rms size of a massive string state as $$R^2 \equiv \frac{1}{d} \ \langle (\widetilde{X}_{\perp}^{\mu} \,
(\tau, \sigma))^2 \rangle_{\sigma , \tau} \, , \label{eq2.9}$$ where $\widetilde{X}_{\perp}^{\mu} \equiv \widetilde{X}^{\mu} -
p^{\mu} (p \cdot \widetilde{X}) / (p \cdot p)$ denotes the projection of $\widetilde{X}^{\mu} \equiv X^{\mu} - X_{\rm cm}^{\mu}
(\tau)$ orthogonally to $p^{\mu}$, and where the angular brackets denote the (simple) average with respect to $\sigma$ and $\tau$.
In the center of mass gauge, $p_{\mu} \, \widetilde{X}^{\mu}$ vanishes by definition, and Eq. (\[eq2.9\]) yields simply $$R^2 = \frac{1}{d} \, \ell_s^2 \, {\cal R} \, , \label{eq2.10}$$ with (after discarding a logarithmically infinite, but state independent, contribution) $${\cal R} \equiv \sum_{n=1}^{\infty} \ \frac{\alpha_{-n} \cdot
\alpha_n}{n^2} = \sum_{n=1}^{\infty} \ \frac{a_n^{\dagger} \cdot
a_n}{n}= \sum_{n=1}^{\infty} \ \frac{N_n}{n} \, . \label{eq2.11}$$ We wish to estimate the distribution function in size of the ensemble of free string states of mass $M$, i.e. to count the number of string states, having some fixed values of $M$ and $R$ (or, equivalently, $N$ and ${\cal
R}$). An approximate estimate of this number (“degeneracy”) is [@dv] $${\cal D} \, (M,R) \sim \exp \, [ c \, (R) \, a_0 \, M ] \, ,
\label{eq2.30}$$ where $a_0 = 2 \, \pi \, ((d-1) \, \alpha' / 6)^{1/2}$ and $$c \, (R) = \left( 1 - \frac{c_1}{R^2} \right) \left( 1 - c_2 \,
\frac{R^2}{M^2} \right) \, , \label{eq2.31}$$ with the coefficients $c_1$ and $c_2$ being of order unity in string units. The coefficient $c \, (R)$ gives the fractional reduction in entropy brought by imposing a size constraint. Note that (as expected) this reduction is minimized when $c_1 \, R^{-2} \sim c_2 \, R^2 / M^2$, i.e. for $R \sim R_{\rm rw}
\sim \ell_s \, \sqrt{M/M_s}$.
Mass shift of string states due to self-gravity
===============================================
We also need to estimate the mass shift of string states (of mass $M$ and size $R$) due to the exchange of the various long-range fields which are universally coupled to the string: graviton, dilaton and axion. As we are interested in very massive string states, $M \gg M_s$, in extended configurations, $R \gg
\ell_s$, we expect that massless exchange dominates the (state-dependent contribution to the) mass shift. \[The exchange of spin 1 fields (for open strings) becomes negligible when $M \gg M_s$ because it does not increase with $M$.\]
The evaluation, in string theory, of (one loop) mass shifts for massive states is technically quite involved, and can only be tackled for the states which are near the leading Regge trajectory [@mshift]. \[Indeed, the vertex operators creating these states are the only ones to admit a manageable explicit oscillator representation.\] As we consider states which are very far from the leading Regge trajectory, there is no hope of computing exactly (at one loop) their mass shifts.
In Ref. [@dv] we could estimate the one-loop mass-shift by resorting to a semi-classical approximation. The starting point of this semi-classical approximation is the effective action of self-gravitating fundamental strings derived in Ref. [@BD98]. Using coherent-state methods [@AABO], [@scherk], [@GSW] and a generalization of Bloch’s theorem (see Eq. (3.13) of [@dv]) one finds $$\delta \, M \simeq - c_d \, G_N \,
\frac{M^2}{R^{d-2}}\, , \label{eq3.25}$$ with the (positive) numerical constant $$c_d = \left[ \frac{d-2}{2} \, (4\pi)^{\frac{d-2}{2}} \right]^{-1} \,
,\label{eq3.26}$$ equal to $1 / \sqrt{\pi}$ in $d=3$.
The result (\[eq3.25\]) was expected in order of magnitude, but it is important to check that it approximately comes out of a detailed calculation of the mass shift which incorporates both relativistic and quantum effects and which uses the precise definition (\[eq2.11\]) of the squared size.
Finally, let us mention that, by using the same tools Ref. [@dv] has computed the imaginary part of the mass shift $\delta \, M =
\delta \,M_{\rm real} - i \, \Gamma / 2$, i.e. the total decay rate $\Gamma$ in massless quanta, as well as the total power radiated $P$. In order of magnitude these quantities are $$\Gamma \sim g^2 \, M \ , \ P \sim g^2 \, M \, M_s \, .
\label{eq3.29}$$
Entropy of self-gravitating strings
===================================
Finally one combines the results of the previous sections, Eqs. (\[eq2.30\]) and (\[eq3.25\]), and heuristically extend them at the limit of their domain of validity. We consider a narrow band of string states that we follow when increasing adiabatically the string coupling $g$, starting from $g =
0$. Let $M_0$, $R_0$ denote the “bare” values (i.e. for $g \rightarrow
0$) of the mass and size of this band of states. Under the adiabatic variation of $g$, the mass and size, $M$, $R$, of this band of states will vary. However, the entropy$S(M,R)$ remains constant under this adiabatic process: $S(M,R) = S (M_0 , R_0)$. We consider states with sizes $\ell_s \ll R_0 \ll M_0$ for which the correction factor, $$c \, (R_0) \simeq (1 - c_1 \, R_0^{-2}) \, (1 - c_2 \, R_0^2 /
M_0^2) \,,
\label{eq4.1}$$ in the entropy $$S (M_0 , R_0) = c \, (R_0) \, a_0 \, M_0 \, , \label{eq4.2}$$ is near unity. \[We use Eq. (\[eq2.30\]) in the limit $g
\rightarrow 0$, for which it was derived.\] Because of this reduced sensitivity of $c \,(R_0)$ on a possible direct effect of $g$ on $R$ (i.e. $R(g) = R_0 + \delta_g \, R$), the main effect of self-gravity on the entropy (considered as a function of the actual values $M$, $R$ when $g \not= 0$) will come from replacing $M_0$ as a function of $M$ and $R$. The mass-shift result (\[eq3.25\]) gives $\delta
\, M = M - M_0$ to first order in $g^2$. To the same accuracy[^7], (\[eq3.25\]) gives $M_0$ as a function of $M$ and $R$: $$M_0 \simeq M + c_3 \, g^2 \, \frac{M^2}{R^{d-2}} = M \left( 1 + c_3
\,\frac{g^2 \, M}{R^{d-2}} \right) \, , \label{eq4.3}$$ where $c_3$ is a positive numerical constant.
Finally, combining Eqs. (\[eq4.1\])–(\[eq4.3\]) (and neglecting, as just said, a small effect linked to $\delta_g \, R
\not= 0$) leads to the following relation between the entropy, the mass and the size (all considered for self-gravitating states, with $g \not= 0$) $$S(M,R) \simeq a_0 \, M \left( 1 - \frac{1}{R^2} \right) \left( 1 -
\frac{R^2}{M^2} \right) \left( 1 + \frac{g^2 \, M}{R^{d-2}} \right)
\, . \label{eq4.4}$$ For notational simplicity, we henceforth set to unity (by using suitable redefinitions) the coefficients $c_1$, $c_2$ and $c_3$. The possibility of smoothly transforming self-gravitating string states into black hole states come from the peculiar radius dependence of the entropy $S(M,R)$. Eq. (\[eq4.3\]) exhibits two effects varying in opposite directions: (i) self-gravity favors small values of $R$ (because they correspond to larger values of $M_0$, i.e. of the “bare” entropy), and (ii) the constraint of being of some fixed size $R$ disfavors both small $(R \ll \sqrt{M})$ and large $(R \gg \sqrt{M})$ values of $R$. For given values of $M$ and $g$, the most numerous (and therefore most probable) string states will have a size $R_* (M;g)$ which maximizes the entropy $S(M,R)$. Said differently, the total degeneracy of the complete ensemble of self-gravitating string states with total energy $M$ (and no [*a priori*]{} size restriction) will be given by an integral (where $\Delta \, R$ is the rms fluctuation of $R$) $${\cal D} (M) \sim \int \frac{d R}{\Delta \, R} \, e^{S(M,R)} \sim
e^{S(M,R_*)} \label{eq4.5}$$ which will be dominated by the saddle point $R_*$ which maximizes the exponent.
The value of the most probable size $R_*$ is a function of $M$, $g$ and the space dimension $d$. We refer to Ref. [@dv] for a full treatment. Let us only indicate the results in the (actual) case where $d=3$. When maximizing the entropy $S(M,R)$ with respect to $R$ one finds that: (i) when $g^2 \ll M^{-3/2}$, the most probable size $R_* (M,g) \sim
\sqrt{M}$, (ii) when $g^2 \gg M^{-3/2}$, $$R_* (M,g) \simeq \frac{1 + \sqrt{1+3 \lambda^2}}{\lambda} \label{eq4.1new}$$ where $\lambda \equiv g^2 M$.
Eq. (\[eq4.1new\]) says that, when $g^2$ increases, and therefore when $\lambda$ increases (beyond $M^{-1/2}$) the typical size of a self-gravitating string decreases, and (formally) tends to a limiting size of order unity, $R_{\infty} = \sqrt 3$ (i.e. of order the string length scale $\ell_s = \sqrt{2 \alpha'}$) when $\lambda \gg 1$. However, the fractional self-gravity $G_N M / R_* \simeq \lambda / R_*$ (which measures the gravitational deformation away from flat space) becomes unity for $\lambda = \sqrt 5$ and formally increases without limit when $\lambda$ further increases. Therefore, we expect that for some value of $\lambda$ of order unity, the self-gravity of the compact string state already reached when $\lambda \sim 1$ (indeed, Eq. (\[eq4.1new\]) predicts $R_* \sim 1$ when $\lambda \sim 1$) will become so strong that it will (continuously) turn into a black hole state. Having argued that the dynamical threshold for the transition string $\rightarrow$ black hole is $\lambda \sim 1$, we now notice that, for such a value of $\lambda$ the entropy $S(M) = S (M,R_* (M))
\simeq a_0 \, M \left[ 1 + \frac{1}{4} \, (g^2 M)^2 \right]$ of the string state (of mass $M$) matches the Bekenstein-Hawking entropy $S_{\rm BH} (M)
\sim g^2 \, M^2 = \lambda M$ of the formed black hole. One further checks that the other global physical characteristics of the string state (radius $R_*$, luminosity $P$, Eq. (\[eq3.29\])) match those of a Schwarzschild black hole of the same mass ($R_{\rm BH} \sim GM \sim g^2 M \, , \,
P_{\rm Hawking} \sim R_{\rm BH}^{-2}$) when $\lambda \sim 1$.
Discussion
==========
Conceptually, the main new result of this paper concerns the most probable state of a very massive single[^8] self-gravitating string. By combining our estimates of the entropy reduction due to the size constraint, and of the mass shift we come up with the expression (\[eq4.4\]) for the logarithm of the number of self-gravitating string states of size $R$. Our analysis of the function $S(M,R)$ clarifies the correspondence [@susskind], [@halyo], [@hp1], [@hp2] between string states and black holes. In particular, our results confirm many of the results of [@hp2], but make them (in our opinion) physically clearer by dealing directly with the size distribution, in real space, of an ensemble of string states. When our results differ from those of [@hp2], they do so in a way which simplifies the physical picture and make even more compelling the existence of a correspondence between strings and black holes. The simple physical picture suggested[^9] by our results is the following: In any dimension, if we start with a massive string state and increase the string coupling $g$, a [*typical*]{} string state will, eventually, become more compact and will end up, when $\lambda_c =
g_c^2\, M \sim 1$, in a “condensed state” of size $R \sim 1$, and mass density $\rho \sim g_c^{-2}$. Note that the basic reason why small strings, $R \sim 1$, dominate in any dimension the entropy when $\lambda \sim 1$ is that they descend from string states with bare mass $M_0 \simeq M (1 + \lambda / R^{d-2}) \sim 2 M$ which are exponentially more numerous than less condensed string states corresponding to smaller bare masses.
The nature of the transition between the initial “dilute” state and the final “condensed” one depends on the value of the space dimension $d$. In $d=3$, the transition is gradual: when $\lambda < M^{-1/2}$ the size of a typical state is $R_*^{(d=3)}
\simeq M^{1/2}(1-M^{1/2} \, \lambda / 8)$, when $\lambda > M^{1/2}$ the typical size is $R_*^{(d=3)} \simeq (1 + (1+3 \,
\lambda^2)^{1/2}) / \lambda$. In $d=4$, the transition toward a condensed state is still continuous, but most of the size evolution takes place very near $\lambda = 1$: when $\lambda <1$, $R_*^{(d=4)} \simeq M^{1/2} (1-\lambda)^{1/4}$, and when $\lambda >
1$,$R_*^{(d=4)} \simeq (2\lambda / (\lambda - 1))^{1/2}$, with some smooth blending between the two evolutions around $\vert \lambda - 1
\vert \sim M^{-2/3}$. In $d \geq 5$, the transition is discontinuous (like a first order phase transition between, say, gas and liquid states). Barring the consideration of metastable (supercooled) states, on expects that when $\lambda = \lambda_2 \simeq \nu^{\nu} /
(\nu - 1)^{\nu - 1}$ (with $\nu = (d-2) /2$), the most probable size of a string state will jump from $R_{\rm rw}$ (when $\lambda <
\lambda_2$) to a size of order unity (when $\lambda > \lambda_2$).
One can think of the “condensed” state of (single) string matter, reached (in any $d$) when $\lambda \sim 1$, as an analog of a neutron star with respect to an ordinary star (or a white dwarf). It is very compact (because of self gravity) but it is stable (in some range for $g$) under gravitational collapse. However, if one further increases $g$ or $M$ (in fact, $\lambda = g^2 \, M$), the condensed string state is expected (when $\lambda$ reaches some $\lambda_3 > \lambda_2$, $\lambda_3 = {\cal O} (1)$) to collapse down to a black hole state (analogously to a neutron star collapsing to a black hole when its mass exceeds the Landau-Oppenheimer-Volkoff critical mass). Still in analogy with neutron stars, one notes that general relativistic strong gravitational field effects are crucial for determining the onset of gravitational collapse; indeed, under the “Newtonian” approximation (\[eq4.4\]), the condensed string state could continue to exist for arbitrary large values of $\lambda$.
It is interesting to note that the value of the mass density at the formation of the condensed string state is $\rho \sim g^{-2}$. This is reminiscent of the prediction by Atick and Witten [@AW88] of a first-order phase transition of a self-gravitating thermal gas of strings, near the Hagedorn temperature[^10], towards a dense state with energy density $\rho \sim g^{-2}$ (typical of a genus-zero contribution to the free energy). Ref. [@AW88] suggested that this transition is first-order because of the coupling to the dilaton. This suggestion agrees with our finding of a discontinuous transition to the single string condensed state in dimensions $\geq 5$ (Ref. [@AW88] work in higher dimensions, $d=25$ for the bosonic case). It would be interesting to deepen these links between self-gravitating single string states and multi-string states.
Let us come back to the consequences of the picture brought by the present work for the problem of the end point of the evaporation of a Schwarzschild black hole and the interpretation of black hole entropy. In that case one fixes the value of $g$ (assumed to be $\ll 1$) and considers a black hole which slowly looses its mass via Hawking radiation. When the mass gets as low as a value[^11] $M \sim
g^{-2}$, for which the radius of the black hole is of order one (in string units), one expects the black hole to transform (in all dimensions) into a typical string[^12] state corresponding to $\lambda = g^2 \, M \sim 1$, which is a dense state (still of radius $R \sim 1$). This string state will further decay and loose mass, predominantly via the emission of massless quanta, with a quasi thermal spectrum with temperature $T \sim T_{\rm Hagedorn} = a_0^{-1}$ which smoothly matches the previous black hole Hawking temperature. This mass loss will further decrease the product $\lambda = g^2 \, M$, and this decrease will, either gradually or suddenly, cause the initially compact string state to inflate to much larger sizes. For instance, if $d \geq 4$, the string state will quickly inflate to a size $R \sim \sqrt{M}$. Later, with continued mass loss, the string size will slowly shrink again toward $R \sim 1$ until a remaining string of mass $M \sim 1$ finally decays into stable massless quanta. In this picture, the black hole entropy acquires a somewhat clear statistical significance (as the degeneracy of a corresponding typical string state) [*only*]{} when $M$ and $g$ are related by $g^2 \, M \sim 1$. If we allow ourselves to vary (in a Gedanken experiment) the value of $g$ this gives a potential statistical significance to any black hole entropy value $S_{\rm BH}$ (by choosing $g^2 \sim S_{\rm BH}^{-1}$). We do not claim, however, to have a clear idea of the direct statistical meaning of $S_{\rm BH}$ when $g^2 \, S_{\rm BH} \gg 1$. Neither do we clearly understand the fate of the very large space (which could be excited in many ways) which resides inside very large classical black holes of radius $R_{\rm BH}\sim (g^2 \, S_{\rm BH})^{1/(d-1)}
\gg 1$. The fact that the interior of a black hole of given mass could be [*arbitrarily*]{} large[^13], and therefore arbitrarily complex, suggests that black hole physics is not exhausted by the idea (confirmed in the present paper) of a reversible transition between string-length-size black holes and string states.
On the string side, we also do not clearly understand how one could follow in detail (in the present non BPS framework) the “transformation” of a strongly self-gravitating string state into a black hole state.
Finally, let us note that we expect that self-gravity will lift nearly completely the degeneracy of string states. \[The degeneracy linked to the rotational symmetry, i.e. $2J + 1$ in $d=3$, is probably the only one to remain, and it is negligible compared to the string entropy.\] Therefore we expect that the separation $\delta
\, E$ between subsequent (string and black hole) energy levels will be exponentially small: $\delta \, E \sim \Delta \, M \, \exp
(-S(M))$, where $\Delta \, M$ is the canonical-ensemble fluctuation in $M$. Such a $\delta \, E$ is negligibly small compared to the radiative width $\Gamma \sim g^2 \, M$ of the levels. This seems to mean that the discreteness of the quantum levels of strongly self-gravitating strings and black holes is very much blurred, and difficult to see observationally.
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[^1]: As we shall discuss, the self-interaction of a string lifts the huge degeneracy of free string states. One then defines the entropy of a narrow band of string states, defined with some energy resolution $M_s \lsim \Delta \, E
\ll M$, as the logarithm of the number of states within the band $\Delta \, E$.
[^2]: Below, we shall use the precise definition $\ell_s \equiv \sqrt{2 \alpha' \hbar}$, but, in this section, we neglect factors of order unity.
[^3]: One uses here the fact that, during an adiabatic variation of $g$, the entropy of the black hole $S_{\rm BH} \sim ({\rm Area}) / G_N \sim R_{\rm
BH}^{d-1} / G_N$ stays constant. This result (known to hold in the Einstein conformal frame) applies also in string units because $S_{\rm BH}$ is dimensionless.
[^4]: The variation of $g$ can be seen, depending on one’s taste, either as a real, adiabatic change of $g$ due to a varying dilaton background, or as a mathematical way of following energy states.
[^5]: For simplicity, we shall consider in this work only Schwarzschild black holes, in any number $d \equiv D-1$ of non-compact spatial dimensions.
[^6]: With the proviso that the consistency of our analysis is open to doubt when $d\geq 8$.
[^7]: Actually, Eq. (\[eq4.3\]) is probably a more accurate version of the mass-shift formula because it exhibits the real mass $M$ (rather than the bare mass $M_0$) as the source of self-gravity.
[^8]: We consider states of a single string because, for large values of the mass, the single-string entropy approximates the total entropy up to subleading terms.
[^9]: Our conclusions are not rigourously established because they rely on assuming the validity of the result (\[eq4.4\]) beyond the domain ($R^{-2} \ll 1$, $g^2 \, M / R^{d-2}
\ll 1$) where it was derived. However, we find heuristically convincing to believe in the presence of a reduction factor of the type $1-R^{-2}$ down to sizes very near the string scale. Our heuristic dealing with self-gravity is less compelling because we do not have a clear signal of when strong gravitational field effects become essential.
[^10]: Note that, by definition, in our [*single*]{} string system, the formal temperature $T = (\partial S / \partial M)^{-1}$ is always near the Hagedorn temperature.
[^11]: Note that the mass at the black hole $\rightarrow$ string transition is larger than the Planck mass $M_P \sim
(G_N)^{-1/2} \sim g^{-1}$ by a factor $g^{-1} \gg 1$.
[^12]: A check on the single-string dominance of the transition black hole $\rightarrow$ string is to note that the single string entropy $\sim M / M_s$ is much larger than the entropy of a ball of radiation $S_{\rm rad}
\sim (RM)^{d/(d+1)}$ with size $R \sim R_{\rm BH} \sim \ell_s$ at the transition.
[^13]: E.g., in the Oppenheimer-Snyder model, one can join an arbitrarily large closed Friedmann dust universe, with hyperspherical opening angle $0 \leq \chi_0 \leq \pi$ arbitrarily near $\pi$, onto an exterior Schwarzschild spacetime of given mass $M$.
|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'In this paper, we show that the action of a characteristically simple, non-extremely amenable (non-strongly amenable, non-amenable) group on its universal minimal (minimal proximal, minimal strongly proximal) flow is effective. We present necessary and sufficient conditions, for the action of a topological group with trivial center on its universal minimal proximal flow, to be effective. A theorem of Furstenberg about the isomorphism of the universal minimal proximal flows of a discrete group and its subgroups of finite index ([@G76 Theorem II.4.4]) is strengthened. Finally, for a pair of groups $H < G$ the same method is applied in order to extend the action of $H$ on its universal minimal proximal flow to an action of its commensurator group $\mathrm{Comm}_G(H)$.'
address:
- 'Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China'
- 'Department of Mathematics, Tel Aviv University, Tel Aviv, Israel'
author:
- Xiongping Dai
- Eli Glasner
title: On universal minimal proximal flows of topological groups
---
Universal minimal proximal flow $\cdot$ Effective action $\cdot$ Strong/extreme amenability
37B05
Introduction {#sec1}
============
By a topological group we mean a group $G$ endowed with a T$_2$-topology such that the algebraic operation $(x,y)\mapsto xy^{-1}$ be continuous from $G\times G$ to $G$. Unless stated otherwise, a *flow* is a triple $(G,X,\pi)$, sometimes write $(G,X)$, where $X$ is a compact T$_2$-space, $G$ is a topological group with the neutral element $e$, and where the action map $\pi\colon G\times X\rightarrow X$ is such that
- $\pi\colon(t,x)\mapsto tx$ is jointly continuous; $\pi_{t}\circ\pi_s=\pi_{ts}\ \forall s,t\in T$; and $\pi_e=\textit{id}_X$ the identity map of $X$ to itself.
A flow $(G,X,\pi)$ is called *effective* if $tx=x$ for all $x\in X$ implies $t=e$; and we say $(G,X,\pi)$ is *free* if $t\in T$ with $t\not=e$ implies $tx\not=x$ for every $x\in X$ (see e.g. [@Aus; @GU]). Given any $t\in T$ with $t\not=e$, we will say that $(G,X,\pi)$ is *effective at $t$* if $\pi_t\not=\textit{id}_X$, and *free at $t$* if $tx\not=x$ for every $x\in X$.
By $\mathrm{Aut}(G)$ we will denote the group of topological automorphisms of $G$ and we will write $\mathrm{Aut}(G)t=\{a(t)\,|\,a\in\mathrm{Aut}(G)\}$. Let $\mathfrak{C}(G)$ be the center of the group $G$; that is, $$\begin{gathered}
\mathfrak{C}(G)=\{t\in G\,|\,tg=gt\ \ \forall g\in G\}.\end{gathered}$$ We will show that the groups $\mathrm{Aut}(G)$ and $\mathfrak{C}(G)$ are important for the dynamics of the universal flows with phase group $G$.
\[def1.1\]A topological group $G$ is said to be *(topologically) characteristically simple* if $\langle\mathrm{Aut}(G)t\rangle$ is dense in $G$ for all $t\in G$ with $t\not=e$, where $\langle\mathrm{Aut}(G)t\rangle$ is the subgroup generated algebraically by $\mathrm{Aut}(G)t$.
Of course every (topologically) simple group is (topologically) characteristically simple. But, for example, for the topological group of rational numbers $\mathbb{Q}$, equipped with the topology it inherits from $\mathbb{R}$, we have $\mathrm{Aut}(\mathbb{Q}) = \mathbb{Q}^* = \{t \in \mathbb{Q}\,|\, t \not=0\}$, so clearly this abelian group is characteristically simple. Similarly, the groups $\mathbb{R}^d, \ d \ge 2$ are obviously characteristically simple. In [@CN] the reader can find a classification of the locally compact, compactly generated, characteristically simple groups.
A flow $(G,X,\pi)$ is said to be *minimal* if and only if there is no proper invariant closed subset of $X$—that is, $\textrm{cls}_XGx=X$ for all $x\in X$. Since G. Birkhoff 1927 [@B], minimal flows play a central role in the theory of topological dynamical systems (cf. [@F67; @E69; @G76; @V77; @Aus] etc.). In this paper, we will be mainly concerned with the effectiveness of the universal minimal and the universal minimal (strongly) proximal flows of a general topological group $G$.
In Section \[sec3\] we show that the action of a characteristically simple, non-extremely amenable (non-strongly amenable, non-amenable) group on its universal minimal (minimal proximal, minimal strongly proximal) flow is effective.
In Section \[sec4\] we present necessary and sufficient conditions, for the action of a topological group with trivial center on its universal minimal proximal flow, to be effective.
In Section \[sec5\] we strengthen a theorem of Furstenberg about the isomorphism of the universal minimal (strongly) proximal flows of a discrete group $G$ and its subgroups of finite index. Finally, in Section \[Comm\], for a pair of groups $H < G$ the same method is applied in order to extend the action of $H$ on its universal minimal proximal flow to an action of its commensurator group $\mathrm{Comm}_G(H)$.
Statements of the main results
==============================
Universal minimal flows {#sec2.1}
-----------------------
Let $G$ be any topological group. Recall that a minimal flow $(G,X,\pi)$ is called a *universal minimal flow of $G$* if any minimal flow $(G,Y,\psi)$ is a factor of $(G,X,\pi)$; that is, there exists a (not necessarily unique) continuous map $h$ from $X$ onto $Y$ with $h(\pi(t,x))=\psi(t,h(x))$ for all $t\in G$ and $x\in X$, written as $(G,X,\pi)
\xrightarrow{h}(G,Y,\psi)$. See, e.g., [@E60; @V77; @Aus].
- *Given any topological group $G$, there exists a unique (up to isomorphism) universal minimal flow $(G,\mathrm{M}(G))$.* (See, e.g., [@E60 Corollary 1 and Theorem 2], [@Aus Theorem 8.1], and [@GL].) This theorem will be generalized to topological semigroups; see Theorem \[thm3.5\].
- *If $G$ is a locally compact group, then its universal minimal flow is free.* (See [@E60 Theorem 3] and [@Aus Theorem 8.3] for $G$ a discrete group; and [@V77 Theorem 2.2.1] for any locally compact group. See also [@P Section 3.3].)
However, for non-locally compact groups the question whether the universal minimal flow is free, is an interesting one and has drawn a lot of attention in the last decade; see e.g. [@P]. For example, when $G$ is *extremely amenable* (i.e., the universal minimal flow of $G$ is trivial with one-point phase space; cf., e.g., [@G98]), then the universal minimal flow of $G$ is of course not effective.
In the literature [@E60; @Aus; @V77], the freeness is usually proven by using the $\beta$-compactification of $G$. Using a different approach, established originally for proving the effectiveness of the universal minimal proximal flow in [@G76], we will show in $\S\ref{sec3.1}$ the following result.
\[thm2.1\]Let $G$ be a topological group which is not extremely amenable; and let $(G,\mathrm{M}(G))$ be its universal minimal flow. Then $(G,\mathrm{M}(G))$ is effective at every $t\in G$ with $\mathrm{cls}_G^{}\langle\mathrm{Aut}(G)t\rangle=G$.
In particular, if the canonical flow $\mathrm{Aut}(G)\times G\rightarrow G$ by $(a,t)\mapsto a(t)$ is transitive at $t\in G$, then either $G$ is extremely amenable or $(G,\mathrm{M}(G))$ is effective at $t$.
It is interesting to note that our effectiveness condition of the universal minimal flow of a topological group is in fact independent of the phase space.
\[cor2.2\]Let $G$ be a characteristically simple group which is not extremely amenable; then $G$ acts effectively on its universal minimal space $\mathrm{M}(G)$.
A [*semiflow*]{} is a triple $(T,X,\pi)$ where $X$ is a compact T$_2$-space, $T$ a topological semigroup with a neutral element $e$ (a monoid), and the action map $\pi\colon G\times X\rightarrow X$ satisfies the properties
- $\pi\colon(t,x)\mapsto tx$ is jointly continuous; $\pi_{t}\circ\pi_s=\pi_{ts}\ \forall s,t\in T$; and
- $\pi_e=\textit{id}_X$ the identity map of $X$ to itself.
Let $(T,X,\pi)$ and $(T,Y,\psi)$ be two semiflows; a continuous surjection $X\xrightarrow{h}Y$ is called a homomorphism from $(T,X,\pi)$ onto $(T,Y,\psi)$, written as $(T,X,\pi)\xrightarrow{h}(T,Y,\psi)$, if $h(\pi(t,x))=\psi(t,h(x))$ for all $t\in T$ and $x\in X$.
The idea of Theorem \[thm2.1\] is also useful for universal minimal semiflows of characteristically simple semigroups (with the obvious definition); see Theorem \[thm3.6\], Theorem \[thm3.7\] and Corollary \[cor3.8\] in $\S\ref{sec3.2}$.
Of course proving that a flow is free is a much stronger result than showing that it is effective; so in that respect our effectiveness results yield nothing new for locally compact groups. Nonetheless, we hope that for non-locally compact acting groups this approach will become useful.
Universal minimal proximal flows {#sec2.2}
--------------------------------
A flow $(G,X,\pi)$ is called *proximal* if for any $x,y\in X$ there is a net $\{t_n\}$ in $G$ such that $\lim t_nx=\lim t_ny$. Recall that a minimal proximal flow of a topological group $G$ is *universal* if it has every minimal proximal flow with the same phase group $G$ as a factor (cf. [@G76 $\S$II.4]).
- *For every topological group $G$, there exists a unique (up to isomorphism) universal minimal proximal flow for it.* We will denote this universal minimal proximal flow by $(G,\Pi(G))$. (See [@G76 Proposition II.4.2].)
We notice that a topological group $G$ is *strongly amenable* if and only if $\Pi(G)$ is a singleton (cf. [@G76 p. 22]). Abelian, and more generally nilpotent, groups are strongly amenable (cf. [@G76 Theorem II.3.4]).
Let $G$ be a topological group; we denote by $\mathrm{Homeo}(\Pi(G))$ the group of self homeomorphisms of $\Pi(G)$ in the sequel.
- *There is a homomorphism $a\mapsto\hat{a}$ of $\mathrm{Aut}(G)$ into $\mathrm{Homeo}(\Pi(G))$. The homeomorphism $\hat{a}$ satisfies the equation $\hat{a}(tx) = a(t)\hat{a}(x)$ for every $t \in G$ and $x \in \Pi(G)$. For each $t\in G$ it sends the inner-automorphism $\sigma_{\!t}\colon g\mapsto tgt^{-1}$ to the homeomorphism $\widehat{\sigma_{\!t}}\colon x\mapsto tx$ of $\Pi(G)$. The flow $(G, \Pi(G))$ is effective if and only if the map $t\mapsto\widehat{\sigma_{\!t}}$, from $G$ to $\mathrm{Homeo}(\Pi(G))$, is one-to-one.*
(This is due to Furstenberg; see [@G76 Proposition II.4.3].)
It is easy to check that if $(G,\Pi(G))$ is effective, then the universal minimal flow of $G$ is also effective. So we are now concerned with the effectiveness of the universal minimal proximal flow $(G,\Pi(G))$. Since a central element of $G$ must act as the identity on $\Pi(G)$ we have to assume that $G$ has a trivial center, $\mathfrak{C}(G) =\{e\}$. Also notice that $\mathfrak{C}(G)$ is the kernel of the homomorphism $t \mapsto \sigma_t$ from $G$ into $\mathrm{Aut}(G)$.
We can strengthen Furstenberg’s result (D) as follows:
\[thm2.3\] Let $G$ be a topological group with $\mathfrak{C}(G)=\{e\}$. Then the following statements are pairwise equivalent.
1. $(G,\Pi(G))$ is effective.
2. The map $t\mapsto\widehat{\sigma_{\!t}}$, from $G$ to $\mathrm{Homeo}(\Pi(G))$, is one-to-one.
3. $a\mapsto\hat{a}$ of $\mathrm{Aut}(G)$ to $\mathrm{Homeo}(\Pi(G))$ is one-to-one.
Hence if one of the above $(1), (2), (3)$ holds, then the universal minimal flow of $G$ is effective.
We will prove this theorem in $\S\ref{sec4}$ following the framework established in [@G76 $\S$II.4].
It follows from (D) ([@G76 Proposition II.4.3]) that the universal minimal proximal flow $(G,\Pi(G))$ can be extended to a flow $(\mathrm{Aut}(G),\Pi(G))$, with the discrete topology of $\mathrm{Aut}(G)$, as follows: $$\xi\colon\mathrm{Aut}(G)\times\Pi(G)\rightarrow\Pi(G);\quad (a,x)\mapsto\hat{a}(x).$$ However, usually in $(\mathrm{Aut}(G),\Pi(G))$, we can not take $\mathrm{Aut}(G)$ to be equipped with its natural compact-open topology.
A simple application of Theorem \[thm2.3\] is Corollary \[cor4.3\] which says that, for $G$ with trivial center, $(\mathrm{Aut}(G),\Pi(G))$ is effective if so is $(G,\Pi(G))$. In particular, in a way similar to the situation in Theorem \[thm2.1\], we may consider the effectiveness of the universal minimal proximal flow of a characteristically simple group.
\[thm2.4\]Let $G$ be a topological group which is not strongly amenable and let $(G,\Pi(G))$ be its universal minimal proximal flow; then the action of $G$ is effective at every $t\in G$ with $\mathrm{cls}_G^{}\langle\mathrm{Aut}(G)t\rangle=G$.
\[cor2.5\]Let $G$ be a characteristically simple group which is not strongly amenable; then $G$ acts effectively on $\Pi(G)$.
Recall that a topological group has the *Rohlin property* if it has a dense conjugacy class in $G \setminus \{e\}$, and the *strong Rohlin property* if it has a co-meager conjugacy class in $G \setminus \{e\}$ (cf. [@GW Definition 3.3]). To illustrate the subject, we give here several examples of Polish (non-locally compact) groups that have the Rohlin property:
- The group $\mathrm{Aut}(X,\mathscr{X},\mu)$ of measure-preserving automorphisms of a standard measure space $(X,\mathscr{X},\mu)$ equipped with the (Polish) weak topology.
- The group $U(H)$ of unitary operators on a separable infinite-dimensional Hilbert space $H$ equipped with the strong operator topology.
- The group of homeomorphisms of the Cantor set and the group of homeomorphisms of the Hilbert cube $[-1,1]^\mathbb{N}$, equipped with the topology of uniform convergence.
Examples of groups with the strong Rohlin property are:
1. The group $S_\infty(\mathbb{N})$ of all permutations of a countable set with the topology of pointwise convergence;
2. The group $H_+[0,1]$ of order preserving homeomorphisms of the unit interval;
3. The group $H(X)$ of homeomorphisms of a Cantor set.
We will say that a topological group has the *characteristic Rohlin property* if it has a dense ${\mathrm{Aut}(G)}$ orbit in $G \setminus \{e\}$ (i.e. there is some $g \in G$ with ${\mathrm{Aut}(G)} g$ dense in in $G \setminus \{e\}$), and the *characteristic strong Rohlin property* if it has a co-meager ${\mathrm{Aut}}(G)$ orbit in $G \setminus \{e\}$.
Clearly, the (strong) Rohlin property implies the characteristic (strong) Rohlin property. However, they are conceptually different. For instance, the abelian group $\mathbb{Q}$ has the strong characteristic Rohlin property.
By Theorem \[thm2.4\], we can easily obtain the following:
\[cor2.6\]Let $G$ be a topological group which is not strongly amenable. If $G$ has the characteristic (strong) Rohlin property, then there is a (co-meager) dense set $E$ of $G\setminus\{e\}$ such that $G$ acts effectively on $\Pi(G)$ at each $t\in E$.
See Corollary \[cor4.6\] for a related freeness criterion.
An isomorphism theorem of universal minimal proximal flows {#sec2.3}
----------------------------------------------------------
The following result strengthens a theorem of Furstenberg (see [@G76 Theorem II.4.4]).
\[thm2.7\]Let $G$ be a topological group and $S$ a closed subgroup of finite index in $G$. Then $(S,\Pi(S))$ can be extended to $(G,\Pi(S))$ so that the flows $(G,\Pi(G))$ and $(G,\Pi(S))$ are isomorphic. Also, $(G,\Pi(G))$ is free if and only if so is $(G,\Pi(S))$.
We will prove this theorem in $\S\ref{sec5}$ by making use of some arguments established in [@G76].
Universal minimal strongly proximal flows
-----------------------------------------
If a topological group $G$ acts on $Q$ which is a compact convex subset of a locally convex topological vector space and if each $t\in G$ acts as an affine transformation, then we say that $(G,Q)$ is an *affine flow*. See [@G76; @Aus].
Let $\mathcal{M}(X)$ be the set of all *quasi-regular* Borel probability measures on the compact T$_2$-space $X$, which is compact and convex under the weak-\* topology. Given any flow $(T,X,\pi)$, let $(T,\mathcal{M}(X),\pi_*)$ be the naturally induced affine flow on $\mathcal{M}(X)$. See [@G76 p. 31]. A flow $(T,X,\pi)$ is said to be *strongly proximal* if the induced affine flow $(T,\mathcal{M}(X),\pi_*)$ is proximal [@G75 p. 161] and [@G76 p. 31]. It is easy to see that a subflow of a strongly proximal flow is strongly proximal and every strongly proximal flow is also proximal. As observed in [@G76 p. 32] the diagonal-wise product of strongly proximal flows is strongly proximal. Here is a precise statement and a short proof:
- *Strong proximality is preserved under diagonal-wise product of any cardinality.*
Let $\{(G,X_i)\}_{i\in I}$ be a family of strongly proximal flows and set $X=\prod_{i\in I}X_i$ and $\Pr_i\colon x=(x_i)_{i\in I}\mapsto x_i$. Let $\mathfrak{m}\in\mathcal{M}(X)$ be any Borel probability measure. Then by [@G76 Lemma III.1.1], we may suppose that $\mathfrak{m}$ belongs to a minimal set $M$ of $(G,\mathcal{M}(X))$. Since $\Pr_i(M)$ is minimal for $(G,\mathcal{M}(X_i))$ and $(G,\mathcal{M}(X_i))$ is proximal, it follows that $\Pr_i(M)\subseteq X_i$ for all $i\in I$. This implies that $\Pr_i(\mathfrak{m})$ is a point mass for any $i\in I$ and thus $\mathfrak{m}$ itself is a point mass. This proves the statement.
Hence as in (C) before one proves that
- *Associated to any topological group $G$, there exists a unique, up to an isomorphism, universal minimal strongly proximal flow.* We will denote this flow by $(G,\Pi_s(G))$. See [@G76 p. 32].
In the same way, Theorem \[thm2.3\] and Theorem \[thm2.7\] can be restated for $\Pi_s$ instead of $\Pi$. Moreover, by using the characterizations of amenable group given in [@G76 Theorem III.3.1], we can restate Theorem \[thm2.4\], Corollary \[cor2.5\] and Corollary \[cor2.6\], respectively, as follows.
\[thm2.8\]Let $G$ be a topological group which is not amenable and let $(G,\Pi_s(G))$ be its universal minimal strongly proximal flow. Then $G$ acts effectively on $\Pi_s(G)$ at every $t\in G$ with $\mathrm{cls}_G^{}\langle\mathrm{Aut}(G)t\rangle=G$.
\[cor2.9\]Let $G$ be a characteristically simple group which is not amenable; then $G$ acts effectively on $\Pi_s(G)$.
\[cor2.10\]Let $G$ be a topological group which is not amenable. If $G$ has the characteristic (strong) Rohlin property, then there is a (co-meager) dense set $E$ of $G\setminus\{e\}$ such that $G$ acts effectively on $\Pi_s(G)$ at each $t\in E$.
In the following table we collect the corollaries \[cor2.2\], \[cor2.5\] and \[cor2.9\], all of which hold under the assumption that $G$ is characteristically simple.
$\mathrm{M}(G) \not = pt$ $G$ is not extremely amenable $G$ is effective on $\mathrm{M}(G)$
--------------------------- ------------------------------- ------------------------------------- -- -- --
$\Pi(G) \not = pt$ $G$ is not strongly amenable $G$ is effective on $\Pi(G)$
$\Pi_s(G) \not = pt$ $G$ is not amenable $G$ is effective on $\Pi_s(G)$
: Effective actions of a characteristically simple group
Universal irreducible affine flows {#sec2.5}
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We now consider the affine flow $(G,\mathcal{M}(\Pi_s(G)))$ induced on the compact convex set $\mathcal{M}(\Pi_s(G))$ of quasi-regular Borel probability measures on the universal minimal strongly proximal flow $(G,\Pi_s(G))$. Recall that an affine flow $(G,Q)$ is *irreducible* if it contains no proper non-empty closed convex invariant subset [@G76].
Recall that an irreducible affine flow $(G,Q)$ is called a *universal irreducible affine flow* of $G$ if for every irreducible affine flow $(T,Q^\prime)$ there exists an affine homomorphism $(G,Q)\xrightarrow{\phi}(G,Q^\prime)$.
- [*For any topological group $G$, $(G,\mathcal{M}(\Pi_s(G)))$ is the (unique) universal irreducible affine flow.*]{} It is strongly proximal and contains $\Pi_s(G)$ (identified with the collection of Dirac measures, or equivalently the closed set of its extremal points) as its unique minimal set See [@G76 Proposition III.2.4].
Denote by $\mathrm{AHomeo}(\mathcal{M}(\Pi_s(G)))$ the group of all affine homeomorphisms of $\mathcal{M}(\Pi_s(G))$ onto itself. In view of this result and because an action on $\Pi_s(G)$ determines the action on $\mathcal{M}(\Pi_s(G))$ we immediately deduce the following:
\[thm2.11\] There is a homomorphism $a\mapsto\hat{a}$ of $\mathrm{Aut}(G)$ into $\mathrm{AHomeo}(\mathcal{M}(\Pi_s(G)))$ which, for $t\in G$, sends the inner-automorphism $\sigma_{\!t}\colon g\mapsto tgt^{-1}$ to the affine homeomorphism $\widehat{\sigma_{\!t}}\colon x\mapsto tx$ of $\mathcal{M}(\Pi_s(G))$. The flow $(G, \mathcal{M}(\Pi_s(G)))$ is effective if and only if the homomorphism $$t\mapsto\widehat{\sigma_{\!t}},
\qquad
G\to \mathrm{AHomeo}(\mathcal{M}(\Pi_s(G))),$$ is one-to-one.
\[thm2.12\]Let $G$ be a topological group and $S$ a closed subgroup of finite index in $G$. Then $(S,\mathcal{M}(\Pi_s(S)))$ can be extended to $(G,\mathcal{M}(\Pi_s(S)))$ so that the flows $(G,\mathcal{M}(\Pi_s(G)))$ and $(G,\mathcal{M}(\Pi_s(S)))$ are isomorphic.
The effectiveness of some universal minimal flows {#sec3}
=================================================
This section is devoted to proving Theorem \[thm2.1\]. Our argument is also useful for the effectiveness of the universal minimal semiflow associated to characteristically simple semigroups such as the additive semigroup $\mathbb{R}_+$ with the usual Euclidean topology. See Theorem \[thm3.6\] below.
Group actions {#sec3.1}
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\[def-coal\]A flow $(G,X,\pi)$ is said to be *coalescent* if every endomorphism of $(G,X,\pi)$ is an automorphism (cf., e.g., [@Aus p. 115]). Similarly one defines the notion of coalescence for semiflows.
The following result is due to Ellis (see [@E69]).
\[lem3.2\]The universal minimal flow of any topological group is coalescent.
Let $(G,X,\pi)$ be the universal minimal flow of the topological group $G$ where $X$ is not one-point, namely, with $G$ not extremely amenable.
Write $\pi(t,x)=tx$ for all $t\in G$ and $x\in X$. Let $\textrm{Homeo}(X)$ be the group of homeomorphisms of $X$ onto itself. For each $a\in\mathrm{Aut}(G)$ define a flow $\alpha\colon G\times X\rightarrow X$ by $(t,x)\mapsto t\cdot_ax$, where $$t\cdot_ax=a(t)x\quad\forall t\in G\textrm{ and }x\in X.$$ Since $a(G)=G$, it follows that $(G,X,\alpha)$ is minimal and by the university of $(G,X,\pi)$ there is a homomorphism $(G,X,\pi)\xrightarrow{\hat{a}}(G,X,\alpha)$. Clearly, $\hat{a}(tx)=a(t)\hat{a}(x)$ for every $t\in G$ and $x\in X$.
Now for $a^{-1}$ in place of $a\in\mathrm{Aut}(G)$, define similarly a minimal flow $(G,X,\alpha^{-1})$ and to obtain another homomorphism $(G,X,\pi)\xrightarrow{\widehat{a^{-1}}}(G,X,\alpha^{-1})$ such that $$\widehat{a^{-1}}(tx)=a^{-1}(t)\widehat{a^{-1}}(x)$$ for every $t\in G$ and $x\in X$.
Now, the composition map $\hat{a}\circ\widehat{a^{-1}}$ satisfies $$\begin{gathered}
\hat{a}\circ\widehat{a^{-1}}(tx)=\hat{a}\left(a^{-1}(t)\widehat{a^{-1}}(x)\right)=a(a^{-1}(t))\hat{a}\left(\widehat{a^{-1}}(x)\right)=t\hat{a}\circ\widehat{a^{-1}}(x),\end{gathered}$$ so that $\hat{a}\circ\widehat{a^{-1}}$ is an endomorphism of $(G,X,\pi)$. By Lemma \[lem3.2\], it follows that $\hat{a}\in\textrm{Homeo}(X)$ for each $a\in\mathrm{Aut}(G)$.
Next, arguing by contradiction, assume that for some $t\in G$ with $\mathrm{cls}_G^{}\langle\mathrm{Aut}(G)t\rangle=G$, $tx=x$ for all $x\in X$. Then by the following commutative diagram: $$\begin{CD}
x@>>> tx\\
@V{\hat{a}}VV @VV{\hat{a}}V\\
\hat{a}(x)@>>>t\cdot_a\hat{a}(x)
\end{CD}\quad \forall x\in X\textrm{ and }a\in\mathrm{Aut}(G),$$ it follows that $$\hat{a}(tx) =a(t)\hat{a}(x)=\hat{a}(x)\quad \forall x\in X\textrm{ and }a\in\mathrm{Aut}(G).$$ Since $\hat{a}\in \textrm{Homeo}(X)$, it follows that $a(t)y =y$ for every $y \in X$. Then, for any $n\ge1$ $$a_1(t)\dotsm a_n(t)y=y\quad \forall y\in X\textrm{ and }a_1,\dotsc,a_n\in\mathrm{Aut}(G).$$ Thus by $\mathrm{cls}_G^{}\langle\mathrm{Aut}(G)t\rangle=G$, it follows that $Gy=y$ and by minimality $X=\{y\}$. This contradicts the hypothesis that $X$ is not a singleton. The proof of Theorem \[thm2.1\] is therefore completed.
\[thm-free\]Let $G$ be a topological group such that $\mathrm{Aut}(G)t$ is dense in $G\setminus\{e\}$ for all $t\in G$ with $t\not=e$. If $(G,\mathrm{M}(G))$ is free at some element $\tau\in G$, then $(G,\mathrm{M}(G))$ is free.
Let $t$ be an arbitrary element of $G$ with $t \not=e$. By contradiction, assume that $tx_0=x_0$ for some $x_0\in \mathrm{M}(G)$. Since $\mathrm{Aut}(G)t$ is dense in $G\setminus\{e\}$, we can choose a net $\{a_n\}$ in $\mathrm{Aut}(G)$ such that $a_n(t)\to\tau$ and $a_n(t)\hat{a}_n(x_0)=\hat{a}_n(x_0)$. Since $\mathrm{M}(G)$ is compact, then, passing to a subnet if necessary, we may assume $\hat{a}_n(x_0)\to y$ in $\mathrm{M}(G)$. Thus $\tau y=y$, which contradicts the assumption that $(G,\mathrm{M}(G))$ is free at $\tau$.
Universal minimal semiflows and effectiveness {#sec3.2}
---------------------------------------------
Given any topological semigroup $T$, there exists a unique (up to isomorphism) *universal minimal semiflow* of $T$, written as $(T,\textrm{M}(T))$, as in the group case, such that if $(T,X)$ is a minimal semiflow there is a homomorphism $(T,\textrm{M}(T))\xrightarrow{\phi}(T,X)$. For this we need the semigroup version of Lemma \[lem3.2\].
\[lem3.4\] Let $(T,X,\pi)$ be a semiflow with $T$ a topological semigroup. Then there is a cardinal number $\mathrm{a}$ and a minimal subset $M$ of $(T,X^\mathrm{a},\pi^\mathrm{a})$ such that $(T,M,\pi^\mathrm{a})$ is coalescent.
According to Zorn’s lemma, let $C$ be a maximal almost periodic set of $(T,X)$; that is, for any finite subset $\{x_1,\dotsc,x_n\}\subseteq C$, the point $(x_1,\dotsc,x_n)$ is an almost periodic point for the diagonal-wise product semiflow $(T,X^n)$; and no other almost periodic set of $(T,X)$ properly contains it (cf. [@Aus p. 67]).
Let $z\in X^C$ such that range $z=C$ and $z\colon C\rightarrow X$ is one to one (for example, $z_c=c$, for each $c\in C$). Let $M=\textrm{cls}_{X^C}{Tz}$, and let $z^\prime\in M$. Now $z^\prime$ is an almost periodic point for $(T,X^C)$, and so $C^\prime=\textrm{range } z^\prime$ is an almost periodic set of $(T,X)$. In fact, it is easy to verify that $C^\prime$ is also maximal, and $z^\prime\colon C\rightarrow C^\prime$ is one to one.
Now, let $\varphi$ be an endomorphism of the minimal semiflow $(T,M)$. Then $(z,\varphi(z))$ is an almost periodic point of $(T,M\times M)$, so range $z\cup\textrm{ range } \varphi(z)$ is an almost periodic set of $(T,X)$. But range $z$ and range $\varphi(z)$ are both maximal almost periodic sets of $(T,X)$, so $\textrm{range }z=\textrm{ range }\varphi(z)$.
If $\gamma$ is a permutation (bijection) of $C$, let $\gamma^*$ denote the induced automorphism of $(T,X^C)$ by $\gamma^*(z)_c=z_{\gamma(c)}$ for each $c\in C$. Define $\gamma$ by $z_{\gamma(c)}=\varphi(z)_c$, for $c\in C$. Since $\varphi(z)$, (regarded as a map of $C$ to $X$) is one to one and range $\varphi(z)=\textrm{ range }z$, $\gamma$ is a permutation of $C$ and $\gamma^*(z)=\varphi(z)$.
Since $\gamma^*=\varphi$ restricted to $Tz$ and $\textrm{cls}_{X^C}{Tz}=M$, then $\gamma^*=\varphi$ on $M$ and so $\varphi$ is an automorphism.
Now we can obtain the unique universal minimal semiflow of a topological semigroup following the framework of the proofs in [@E69], [@G76 Proposition II.4.2] and [@Aus Theorem 8.1].
\[thm3.5\] For any topological semigroup $T$, there exists a universal minimal semiflow $(T,\mathrm{M}(T),\pi)$, and any two universal minimal semiflows for $T$ are isomorphic.
Let $\mathscr{M}=\{(T,X_i,\pi_i)\,|\,i\in I\}$ be the collection of non-isomorphic minimal semiflows with the phase semigroup $T$. Define $$X={\prod}_{i\in I}X_i\quad \textrm{and}\quad \pi\colon(t,(x_i)_{i\in I})\mapsto(tx_i)_{i\in I}
\ \textrm{ from }T\times X\textrm{ to }X.$$ By Lemma \[lem3.4\] there is a cardinal number $\mathrm{a}$ and a minimal subset $M$ of $(T,X^\mathrm{a},\pi^\mathrm{a})$ such that $(T,M,\pi^\mathrm{a})$ is coalescent. Obviously $(T,M,\pi^\mathrm{a})$ is a universal minimal semiflow of $T$. Suppose $(T,Z,\pi_Z)$ is another universal minimal semiflow of $T$. Then there are $T$-homomorphisms $$(T,M,\pi^\mathrm{a})\xrightarrow{\phi}(T,Z,\pi_Z)
\xrightarrow{\psi}(T,M,\pi^\mathrm{a}).$$ Since $(T,M,\pi^\mathrm{a})$ is coalescent, $\psi\circ\phi$ and also $\phi,\psi$ are all isomorphisms.
Next we will be concerned with the effectiveness of the universal minimal semiflow of some topological semigroups including $\mathbb{R}_+=[0,+\infty)$ or $\mathbb{Q}_+ = \mathbb{Q} \cap [0, \infty)$ Of course for $\mathbb{R}_+$ this way of proving effectiveness is an overkill, as already the action of $\mathbb{R}_+$ on the $2$-torus via an irrationally oriented line is minimal and effective (hence free). However, for general acting non locally compact semigroup our next results may be of interest.
Let $T$ be a topological semigroup. By $\mathcal{E}nd\,(T)$ we denote the set of continuous self homomorphisms $a$ of $T$ such that $a(T)$ is dense in $T$. By $(T,\mathrm{M}(T))$ we denote the universal minimal semiflow of $T$. It is easy to check that:
- Given any $t\in\mathbb{R}_+$ with $t\not=0$, $\quad \mathcal{E}nd\,(\mathbb{R}_+)t=\{a(t)\,|\,a\in\mathcal{E}nd\, (\mathbb{R}_+)\}=(0,+\infty)$.
By a slight modification of the proof of Theorem \[thm2.1\] we can obtain the following:
\[thm3.6\]Let $T$ be a topological semigroup such that $\langle\mathcal{E}nd\,(T)t\rangle$ is dense in $T$ for all $t\in T$ with $t\not=e$. Then, either $T$ is extremely amenable or $(T,\mathrm{M}(T))$ is effective (i.e., $t\colon x\mapsto tx$ is not the identity map for any $t\in T$ with $t\not=e$).
It should be noted that even if $T$ is locally compact, Theorem \[thm3.6\] is already beyond the framework of Veech [@V77 Theorem 2.2.1] which is proven only for locally compact groups.
We say that a semiflow $(T,X,\pi)$ is *free at $t\in T$* if $tx\not=x\ \forall x\in X$. We now have a semigroup version of Theorem \[thm-free\]:
\[thm3.7\]Let $T$ be a topological semigroup such that $\mathcal{E}nd\,(T)t$ is dense in $T\setminus\{e\}$ for all $t\in T$ with $t\not=e$. If $(T,\mathrm{M}(T))$ is free at some element $\tau\in T$, then $(T,\mathrm{M}(T))$ is free.
\[cor3.8\]Let $T$ be an abelian semigroup such that $\mathcal{E}nd\,(T)t$ is dense in $T\setminus\{e\}$ for all $t\in T$ with $t\not=e$. Then $(T,\mathrm{M}(T))$ is free if $T$ is not extremely amenable.
If $(T,\mathrm{M}(T))$ is free at some $\tau\in T$, then by Theorem \[thm3.7\] it is free. Now let there be some $\tau\in T$ with $\tau\not=e$ such that $\tau x_0=x_0$ for some point $x_0\in X$. Then $\tau tx_0=tx_0$ for any $t\in T$. Since $Tx_0$ is dense in $X$ by minimality, hence $\tau x=x$ for all $x\in X$. However, this contradicts Theorem \[thm3.6\]. Thus $(T,\mathrm{M}(T))$ is free.
The effectiveness of some universal minimal proximal flows {#sec4}
==========================================================
This section will be devoted to proving Theorems \[thm2.3\] and \[thm2.4\]. Let $G$ be a topological group. First, we will need a lemma.
\[lem4.1\]The only endomorphism a minimal proximal $G$-flow admits is the identity automorphism.
In fact we can obtain a more general uniqueness result.
\[lem4.2\]If $(G,X,\pi_X)\xrightarrow{\theta}(G,Y,\pi_Y)$ is a homomorphism (not necessarily surjective) from a minimal proximal flow $(G,X,\pi_X)$ to another proximal flow $(G,Y,\pi_Y)$, then $\theta$ is the unique homomorphism admitted from $(G,X,\pi_X)$ to $(G,Y,\pi_Y)$.
Let $(G,X,\pi_X)\xrightarrow{\phi}(G,Y,\pi_Y)$ be a homomorphism. Then given any $x\in X$, set $y_1=\theta(x)$ and $y_2=\phi(x)$. By proximality there is a net $\{t_n\}$ in $G$ and some $y_\infty\in Y$ such that $$\lim_n t_n y_1 = \lim_n t_n y_2 = y_\infty.$$ We can assume that the limit, $x_\infty = \lim t_n x$ exists and then $\theta(x_\infty)=y_\infty=\phi(x_\infty)$. Since $(G,X)$ is minimal, we conclude that $\theta(x)=\phi(x)$. Thus $\theta=\phi$ on $X$.
Recall that $\sigma_{\!t}\colon s\mapsto tst^{-1}$ is the inner-automorphism of $G$ as in (D) in $\S\ref{sec2.2}$. Let $(G,\Pi(G),\pi)$ be the universal minimal proximal flow associated with $G$ and simply write $\pi(t,x)=tx$ for $t\in G$ and $x\in\Pi(G)$ as in (C) in $\S\ref{sec2.2}$. We also recall the construction of the homomorphism $a\mapsto\hat{a}$ of $\mathrm{Aut}(G)$ to $\mathrm{Homeo}(\Pi(G))$ introduced in [@G76 p. 23]. In fact, this is in essence the same as the construction described above for $\mathrm{M}(G)$ in the proof of Theorem \[thm2.1\]; the only difference is that here we use the fact that $\Pi(G)$ admits no non-trivial endomorphisms instead of the coalescence of $\mathrm{M}(G)$.
It should be mentioned that usually we cannot expect the continuity of the homomorphism $a\mapsto\hat{a}$, from $\mathrm{Aut}(G)$ to $\mathrm{Homeo}(\Pi(G))$, when the former is equipped with its natural compact-open or pointwise convergence topologies.
\(1) $\Leftrightarrow$ (2): First note that the map $t\mapsto\widehat{\sigma_{\!t}}$ from $G$ to $\mathrm{Homeo}(\Pi(G))$ is a homomorphism. Then the statement easily follows from $\widehat{\sigma_{\!t}}=t$.
\(1) $\Rightarrow$ (3): Let $a\in\mathrm{Aut}(G)$ be such that $\hat{a}$ is the identity map on $\Pi(G)$. Then by the equality $\hat{a}(tx)=a(t)\hat{a}(x)$, it follows that $tx=a(t)x$ for all $x\in X$ and $t\in G$. Since $(G,\Pi(G),\pi)$ is effective, then $a(t)=t$ for every $t\in G$ and so $a=\textit{id}_G$. This shows that the homomorphism $a\mapsto\hat{a}$ is one-to-one.
\(3) $\Rightarrow$ (2): Since $\mathfrak{C}(G)=\{e\}$, the map $t\mapsto\sigma_{\!t}$ from $G$ to $\mathrm{Aut}(G)$ is one-to-one. Thus by condition (3), it follows that $t\mapsto\widehat{\sigma_{\!t}}$ is one-to-one.
The proof of Theorem \[thm2.3\] is completed.
This proof is analogous to the proof of Theorem \[thm2.1\] and so we omit the details.
Let $(\mathrm{Aut}(G),\Pi(G))$ be the canonical extension of $(G,\Pi(G))$. Then we can easily obtain the following by Theorem \[thm2.3\].
\[cor4.3\]Let $G$ be a topological group with $\mathfrak{C}(G)=\{e\}$. Then $(G,\Pi(G))$ is effective if and only if so is $(\mathrm{Aut}(G),\Pi(G))$.
In many cases the group of inner-automorphisms $\mathrm{Inn}(G) = \{\sigma_{\!t}\,|\,t\in G\}$ is a proper subgroup of $\mathrm{Aut}(G)$ and so Corollary \[cor4.3\] seems to be interesting. The “if” part of Corollary \[cor4.3\] will be generalized in Theorem \[thm4.4\] below.
Next we consider the inheritance of freeness of the universal minimal flows associated to topological semigroups.
\[thm4.4\]Let $T$ be a topological semigroup and $H$ a subsemigroup of $T$. Then
1. If $(T,\mathrm{M}(T))$ is free, then $(H,\mathrm{M}(H))$ is also free.
2. If $(T,\Pi(T))$ is free, then $(H,\Pi(H))$ is also free.
3. If $(T,\Pi_s(T))$ is free, then $(H,\Pi_s(H))$ is also free.
Let $(T,\mathrm{M}(T))$ be free; then $(H,\mathrm{M}(T))$ is also free. Now let $X$ be an $H$-minimal subset of $\mathrm{M}(T)$ and then $(H,X)$ is obviously free. Then by the universality of $(H,\mathrm{M}(H))$, it follows that there is a homomorphism $(H,\mathrm{M}(H))\xrightarrow{\phi}(H,X)$. This implies that $(H,\mathrm{M}(H))$ is free. This proves (1). We can easily prove (2) and (3) similarly.
\[thm4.5\]Let $G$ be a topological group with $\mathrm{Aut}(G)t$ dense in $G\setminus\{e\}$ for some $t\in G$ with $t\not=e$. If $(G,\Pi(G))$ is free at some $\tau\in G$, then $(G,\Pi(G))$ is free at $t$.
The proof is almost verbatim the same as that of Theorem \[thm-free\] and thus we omit its details here.
The following is a simple consequence of Theorem \[thm4.5\], which is comparable with Corollary \[cor2.6\].
\[cor4.6\]Let $G$ be a topological group with the characteristic (strong) Rohlin property, such that $(G,\Pi(G))$ is free at some $\tau\in G$. Then there exists a (co-meager) dense subset $E$ of $G\setminus\{e\}$ such that $G$ acts freely on $\Pi(G)$ at each $t\in E$.
We note that in the same way the above Corollary \[cor4.3\], Theorem \[thm4.5\] and Corollary \[cor4.6\] can be restated for $\Pi_s(G)$ in place of $\Pi(G)$.
A generalization of Furstenberg’s isomorphism theorem {#sec5}
=====================================================
Based on the construction of the homomorphism $a\mapsto\hat{a}$ presented in $\S\ref{sec3}$, this section will be mainly devoted to proving Theorem \[thm2.7\]. For that, let $G$ be a topological group and let $S$ be a *closed* proper subgroup of *finite index* in $G$, unless otherwise specified.
First of all, we will need a useful lemma.
\[lem5.1\]Let $T$ be a topological group and $(T,X,\varphi)$ a minimal proximal flow.
1. If $T$ is a compact extension of its subgroup $L$, then $(L,X,\varphi{\upharpoonright_{\!L\times X}})$ is minimal and proximal.
2. If $L$ is a closed subgroup of finite index in $T$, then $(L,X,\varphi{\upharpoonright_{\!L\times X}})$ is minimal and proximal.
Here $T$ is called a *compact extension* of $L$ if $L$ is a closed normal subgroup of $T$ such that the quotient group $T/L$ is a compact group.
We also will need a technical lemma.
\[lem5.2\]There exists a normal closed subgroup $N$ of $G$ such that $N\subseteq S$ and that $N$ is of finite index in $G$.
Under the discrete topology of $G$, $S$ contains a normal subgroup, say $M$, of $G$ of finite index in $G$ (see [@HR p. 26] or [@G76 p. 24]). Now let $N=\textrm{cls}_GM$. Then it is easy to check that $N$ is a normal closed subgroup of $G$ satisfying the claim of the lemma.
From now on, let $N$ be as in Lemma \[lem5.2\]. Let $(G,\Pi(G)), (S,\Pi(S))$ and $(N,\Pi(N))$ be the universal minimal proximal flows of the topological groups $G, S$ and $N$, respectively, as in (C) in $\S\ref{sec2.2}$.
First we need to extend the natural action of $N$ on $\Pi(N)$ to an action of $G$ on $\Pi(N)$. For this we define an action of $G$ on $\Pi(N)$ by mapping $G$ into $\mathrm{Aut}(N)$ as follows: $$\zeta\colon G\times\Pi(N)\rightarrow\Pi(N);\quad (t,z)\mapsto\widehat{\sigma_{\!t}{\!\upharpoonright_{\!N}}}(z),$$ where $\widehat{{}}\,\colon\mathrm{Aut}(N)\rightarrow\mathrm{Homeo}(\Pi(N))$ is the canonical map introduced in $\S\ref{sec3}$, with $N$ in place of $G$.
\[lem5.3\]$(G,\Pi(N),\zeta)$ is a flow, and as such it is minimal and proximal.
We only need to verify $\widehat{\sigma_{\!t}{\!\upharpoonright_{\!N}}}(z)$ is jointly continuous with respect to $t\in G$ and $z\in\Pi(N)$. Since $N$ is closed and of finite index in $G$, there exists a finite set, say $\{s_1,\dotsc,s_k\}\subseteq G$, such that $G=s_1N\cup\dotsm\cup s_kN$ is a clopen partition and $s_iN\cap s_jN=\emptyset$ for $1\le i\not=j\le k$. Now suppose that the net $(t_i,z_i)\to(t,z)$ in $G\times\Pi(N)$. Then, passing to subnets (and relabeling) we can assume that for some fixed $j_0$ we have $t_i \in s_{j_0}N$ for every $i$. Thus for every $i$ there is some $r_i \in N$ so that $t_i = s_{j_0}r_i$ and $r_i \to r : = s_{j_0}^{-1}t$. Observe that for $r \in N$ and $z \in \Pi(N)$ we have $\widehat{\sigma_{\!r}\!\!\upharpoonright_{\!N}}(z)= rz$. Now, by the continuity of the $N$ action on $\Pi(N)$, $$\widehat{\sigma_{\!t_i}\!\!\!\upharpoonright_{\!N}}(z_i)
= \widehat{\sigma_{\!s_{j_0}r_i}\!\!\upharpoonright_{\!N}}(z_i)=
\widehat{\sigma_{\!s_{j_0}}\!\!\!\upharpoonright_{\!N}}
\left(\widehat{\sigma_{\!r_i}\!\!\upharpoonright_{\!N}}(z_i)\right)=
\widehat{\sigma_{\!s_{j_0}}\!\!\!\upharpoonright_{\!N}}
(r_i z_i)
\to \widehat{\sigma_{\!s_{j_0}}\!\!\!\upharpoonright_{\!N}}
(r z).
$$ But $$\widehat{\sigma_{\!s_{j_0}}\!\!\!\upharpoonright_{\!N}}(rz)=
\widehat{\sigma_{\!s_{j_0}}\!\!\!\upharpoonright_{\!N}}
\left(\widehat{\sigma_{\!r}\!\!\upharpoonright_{\!N}}(z)\right) =
\widehat{\sigma_{\!s_{j_0}r}\!\!\upharpoonright_{\!N}}(z) =
\widehat{\sigma_{\!t}\!\!\upharpoonright_{\!N}}(z).$$ Thus $\zeta\colon G\times\Pi(N)\rightarrow\Pi(N)$ is continuous.
One can relax here the assumption that $N$ has finite index in $G$. Assuming only that $N$ is a clopen normal subgroup will suffice. Indeed, under this assumption the space $G/ N$ is discrete and (denoting the quotient map $Q\colon G \to G/N$) we have, in the notation of the proof above, $Q(t_i) \to Q(t)$. Thus, eventually $Q(t_i) = Q(t)$ and we can assume that for every $i$, $t_i = s_{j_0}r_i$ for some fixes $s_{j_0} \in G$ and $r_i \in N$. Now proceed as before.
With the above preparations at hand, we are ready to complete our proof of Theorem \[thm2.7\].
By the universality of $(G,\Pi(G))$, there exists a $(G,\Pi(G))\xrightarrow{\phi}(G,\Pi(N))$, where $(G,\Pi(N))=(G,\Pi(N),\zeta)$ as in Lemma \[lem5.3\].
Next consider the flow $(N,\Pi(G))$ which is obtained by restricting the action of $G$ on $\Pi(G)$ to the action of its subgroup $N$. By Lemma \[lem5.1\] this flow is minimal and proximal; therefore there is a homomorphism $(N,\Pi(N))\xrightarrow{\psi}(N,\Pi(G))$. Now $(N,\Pi(N))\xrightarrow{\phi\circ\psi}(N,\Pi(N))$ is an endomorphism, hence it is the identity map by Lemma \[lem4.1\]. Therefore $(G,\Pi(G))\xrightarrow{\phi}(G,\Pi(N))$ is an isomorphism.
By Lemma \[lem5.1\] again, $(S,\Pi(G))$ is minimal and proximal, hence there exists a homomorphism $(S,\Pi(S))\xrightarrow{\theta}(S,\Pi(G))$. Similarly, $(N,\Pi(S))$ is minimal and proximal and there exists a homomorphism $(N,\Pi(N))\xrightarrow{\lambda}(N,\Pi(S))$. Thus the composition $$(N,\Pi(N))\xrightarrow{\lambda}(N,\Pi(S))\xrightarrow{\theta}(N,\Pi(G))
\xrightarrow{\phi}(N,\Pi(N))$$ is an endomorphism of $(N,\Pi(N))$, hence it is the identity map. Thus $(N,\Pi(N))\xrightarrow{\lambda}(N,\Pi(S))$ is an isomorphism. Using this isomorphism, together with the isomorphism $(G,\Pi(G))\xrightarrow{\phi}(G,\Pi(N))$, an action of $G$ on $\Pi(S)$ can be defined so that $(G,\Pi(S))$ and $(G,\Pi(G))$ are isomorphic.
It is well known that for locally compact groups every subgroup of an amenable group is amenable. How about strongly amenable groups?
\[cor5.5\]Let $G$ be a topological group and $S$ a closed subgroup of finite index in $G$. Then $G$ is strongly amenable if and only if so is $S$.
By Theorem \[thm2.7\], it follows that $\Pi(G)\cong\Pi(S)$. So $\Pi(G)$ is a singleton if and only if so is $\Pi(S)$. This proves Corollary \[cor5.5\].
\[q5.6\]Let $G$ be a locally compact group. If $G$ is a compact extension of $S$, is it true that $(G,\Pi(G))\cong(G,\Pi(S))$?
If the answer to Question \[q5.6\] is positive, then as in Corollary \[cor5.5\] we will conclude that $G$ is strongly amenable if and only if so is $S$.
Commensurators {#Comm}
==============
If $G$ is a group and $H<G$ a subgroup, we denote $H^g = gHg^{-1}$ and $H_g = H \cap H^g$. The [*commensurator of $H$ in $G$*]{} is defined by $${\boldsymbol{H}}=\textrm{Comm}_G(H) =\{g \in G\,|\, H_g \ {\text{ has finite index in both}} \ H\ {\text{and}}\ H^g\}.$$
As a corollary of [@G76 Theorem II.4.4] we obtain the following.
Assume that ${\boldsymbol{H}}$ is countable. Then, the canonical action of $H$ on $\Pi(H)$ can be extended to an action $({\boldsymbol{H}}, \Pi(H))$. That is, there is a homomorphism $g \mapsto \hat{g}$, from ${\boldsymbol{H}}$ to ${\mathrm{Homeo}}(\Pi(H))$, such that
- $\hat{g}$ satisfies the equation $\hat{g}(tx) = (gtg^{-1})\hat{g}(x)$ for every $t\in H\cap g^{-1}Hg$ and $x\in\Pi(H)$.
- for $h \in H$ and $x \in \Pi(H)$ we have $\hat{h}(x) = h(x)$.
- for every $g \in {\boldsymbol{H}}$ $$(H, \Pi(H)) \cong (H, \Pi(H_g)) \cong (H^g, \Pi(H_g)) \cong (H^g, \Pi(H^g)).$$ Analogous statements hold for $\Pi_s$.
Let $\{e= g_0, g_1, g_2, \dotsc, g_k, \dotsc\}$ be an enumeration of ${\boldsymbol{H}}$ and for each $k$ let $H_k = \bigcap_{i=0}^k g_i H^{g_i}$. Then, $H_k$ has finite index in $H$ and there is a normal subgroup $N $ of $H$, with $N < H_k$, such that $N$ is of finite index in $H$. The flow $(N, \Pi(H))$ is minimal, proximal and for each $0 \le i \le k$, the map $\sigma_{\!g_i}\!\!\upharpoonright\!\!N$ is an automorphism of $N$. Thus the corresponding map $\widehat{\sigma_{\!g_i}\!\!\upharpoonright\!\!N}
\colon\Pi(H) \to \Pi(H)$ is a homeomorphism and the map $\; \widehat{{}}\,\colon \langle g_0, g_1, g_2, \dotsc, g_k \rangle \to \mathrm{Homeo}(\Pi(H))$ is a group homomorphism. Note that when $N_1 < N_2 < H$ and $g \in {\boldsymbol{H}}$ normlizes both $N_1$ and $N_2$ we have, with $\widehat{g_j}$ the corresponding homeomorphisms induced by $\sigma_{\!g_j}\!\!\!\upharpoonright_{\!N_j}, \ j=1,2$, $$\widehat{{g_2}^{-1}} (\widehat{g_1}(tx)) =
\widehat{{g_2}^{-1}} (g t g^{-1} \widehat{g_1}(x))=
t\widehat{{g_2}^{-1}}(\widehat{g_1}(x)),$$ for every $t \in N_1$. Thus $\phi = \widehat{{g_2}^{-1}}\circ \widehat{g_1}$ is an automorphism of the minimal proximal flow $(N_1, \Pi(H))$, whence $\widehat{{g_2}^{-1}}\circ \widehat{g_1}=\textit{id}_{\Pi(H)}$, and $\widehat{{g_2}} = \widehat{g_1}$.
We now let $k\to\infty$ to conclude the proof.
With $G = \textrm{SL}(n, \mathbb{R})$ and $H=\textrm{SL}(n,\mathbb{Z})$ we have $\textrm{Comm}_G(H) = \textrm{SL}(n, \mathbb{Q})$ and a nice instance of this theorem is the result that the action $(\textrm{SL}(n,\mathbb{Z}), \Pi(\textrm{SL}(n,\mathbb{Z})))$ extends to an action $(\textrm{SL}(n, \mathbb{Q}), \Pi(\textrm{SL}(n,\mathbb{Z})))$. Again an analogous result is valid for $\Pi_s$ instead of $\Pi$.
Acknowledgments {#acknowledgments .unnumbered}
===============
X. Dai was partly supported by National Natural Science Foundation of China (Grant Nos. 11431012, 11271183) and PAPD of Jiangsu Higher Education Institutions; and E. Glasner was supported by a grant of the Israel Science Foundation (ISF 668/13).
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $X_{\mathbb{R}}$ be the zero locus in ${\mathbb{R}\mathrm{P}}^n$ of one or two independently and Weyl distributed random real quadratic forms. Denoting by $X_{\mathbb{C}}$ the complex part in ${\mathbb{C}\textrm{P}}^n$ of $X_{\mathbb{R}}$ and by $b(X_{\mathbb{R}})$ and $b(X_{\mathbb{C}})$ the sums of their Betti numbers, we prove that: $$\label{ex}\lim_{n\to\infty}\frac{\mathbb{E}b(X_{\mathbb{R}})}{n}=1.$$ In particular for one quadric hypersurface asymptotically Smith’s inequality $b(X_{\mathbb{R}})\leq b(X_{\mathbb{C}})$ is expected to be sharp. The methods we use combine Random Matrix Theory, Integral Geometry and spectral sequences.'
author:
- 'A. Lerario'
title: Random matrices and the average topology of the intersection of two quadrics
---
Introduction
============
Let us consider the real vector space $W_{n,d}$ of real homogeneous polynomials of degree $d$ and $n+1$ variables. Each $f\in W_{n,d}$ defines a complex algebraic set $X_{{\mathbb{C}}}$ in ${\mathbb{C}\textrm{P}}^n$ and for an open dense subset of $W_{n,d}$ all these algebraic sets have the same volume (induced from the Fubiny-Study one) and the same topology. The first statement directly follows from Wirtinger’s formula and the second essentially from the fact that the set of degenerate polynomials has real codimension two in the space of polynomials with complex coefficients. If we look at the zero locus $X_{\mathbb{R}}$ of $f$ in the real projective space ${\mathbb{R}\mathrm{P}}^n$, then the situation dramatically changes. The set of degenerate polynomials has now real codimension one and as we cross it the topology of $X_{\mathbb{R}}$ (and its volume) may change. It is however possible to compare the volume and the topology of the real and complex parts by mean of the following inequalities: $$\label{volsmith}\frac{\textrm{Vol}(X_{\mathbb{R}})}{\textrm{Vol}({\mathbb{R}\mathrm{P}}^{n-1})}\leq \frac{\textrm{Vol}(X_{\mathbb{C}})}{{\textrm{Vol}({\mathbb{C}\textrm{P}}^{n-1})}}\quad \textrm{and}\quad b(X_{\mathbb{R}})\leq b(X_{\mathbb{C}}).$$ The l.h.s. inequality directly follows from the integral geometry formula and the r.h.s. is the so called Smith’s inequality and involves the sum of the Betti numbers (in this paper all cohomology groups and related ranks are assumed to be with ${\mathbb{Z}}_2$ coefficients).\
This raises the question: if $f$ is picked up randomly, what do we expect the volume and the topology of its real zero locus to be? Clearly this question does not make sense for the complex part, since for any reasonable distribution of probability on $W_{n,d}$ the volume and the sum of the Betti numbers of $X_{\mathbb{C}}$ are constant functions outside of a zero probability set.\
To answer this question let us consider the example of a random polynomial $f$ in $W_{1,d}$. In this case both the volume and the sum of the Betti numbers of $X_{{\mathbb{R}}}$ equal the number of real (projective) roots of $f$. In the seminal paper [@Kac] Kac proved that if the coefficients of $f$ are distributed as standard independent gaussians with mean zero and variance one, then the expected number of real roots $E_d$ of $f$ satisfies: $$\lim_{d\to \infty} \frac{E_d}{\log d}=\frac{2}{\pi}.$$ In this paper we will assume $f=\sum f_\alpha x^\alpha$, where $x^\alpha=x_0^{\alpha_0}\cdots x_n^{\alpha_n}$ and the $f_\alpha$ are Gaussian independently distributed random variables with mean zero and variance $\frac{d! }{ \alpha_0!\cdots \alpha_n!}$. The resulting distribution of probability on $W_{n,1}$ is called the *Weyl* distribution (or the Kostlan distribution). For instance the expected number of zeroes $E_d$ of a random Weyl distributed polynomial in $W_{1,d}$ is given exactly by: $$E_d=\sqrt{d}.$$ The reader is referred to the paper [@EdKo] for a proof of both these limits and a survey of related results.\
More generally the expected volume of the random algebraic variety $X_{\mathbb{R}}$ defined by a set of polynomials $f_1, \ldots, f_k$ with each $f_i\in W_{n,d_i}$ Weyl and independently distributed is given by: $$\mathbb{E}[\textrm{Vol}(X_{\mathbb{R}})]=\sqrt{d_1\cdots d_k}\textrm{Vol}({\mathbb{R}\mathrm{P}}^{n-k}).$$ Indeed the previous formula was proved in a sequence of papers of Shub and Smale first [@ShSm] and BŸrgisser [@Bu] in this general form. In this last paper the formula follows from the more striking result on the computation of the expected curvature polynomial of $X_{\mathbb{R}}$ in ${\mathbb{R}\mathrm{P}}^n$. The same computation also gives a precise formula for the expected Euler characteristic of $X_{\mathbb{R}}$ (the hypersurface case was already done by Podkorytov in [@Po]). In the case of $n$ equations in ${\mathbb{R}\mathrm{P}}^n$ this expected volume gives the expected number of solutions of a random polynomial system.\
Concerning the sum of the Betti numbers of $X_{\mathbb{R}}$, very little is known. Even the case of the expected number of components of a random real, Weyl distributed curve of degree $d$ in ${\mathbb{R}\mathrm{P}}^2$ is not known. Gayet and Welschinger in [@GaWe1] proved that maximal curves, i.e. those with approximatively $d^2$ components, become exponentially rare in the degree. The same authors in [@GaWe2] proved that the expected total Betti number of a random Weyl distributed hypersurface of degree $d$ in ${\mathbb{R}\mathrm{P}}^n$ satisfies the following: $$\lim_{d\to \infty}\mathbb{E}\bigg[\frac{b(X_{\mathbb{R}})}{d^n}\bigg]=0.$$ In an unpublished letter [@Sarnak] Sarnak claims that in the case of a plane curve we have even $\lim_{d\to\infty}\mathbb{E}\big[\frac{b(X)}{d}\big]\leq c_1$, for a positive constant $c_1.$ Indeed in a different direction Nazarov and Sodin [@NaSo] proved that the expected number of connected components of a random[^1] spherical harmonic of degree $d$ is asymptotically $c_2d^2$, for some $c_2>0.$ Generalizing this result, the author together with E. Lundberg, was able to prove that the expectation of the number of connected components of a random[^2] hypersurface of degree $d$ in ${\mathbb{R}\mathrm{P}}^n$ is asymptotically of order $d^n$ (see [@LeLu]).\
In this paper we study the case the random algebraic set is the intersection of real quadrics in ${\mathbb{R}\mathrm{P}}^n$. In this case Barvinok’s bound (see [@Ba]) gives for the intersection $X_{\mathbb{R}}$ of $k$ quadrics in ${\mathbb{R}\mathrm{P}}^n$: $$b(X_{\mathbb{R}})\leq n^{O(k)}.$$ This bound suggests that the measure of the complexity of $X_{\mathbb{R}}$ is the number $k$ of quadrics we are intersecting. Motivated by this and Smith’s inequality (\[volsmith\]) we thus focus on a different asymptotics, namely we fix the number of equations, i.e. the codimension of $X_{\mathbb{R}}$, and we let the number of variables go to infinity. The case we study is somehow the simplest, i.e. the one when $X_{\mathbb{R}}$ is defined by one or two random Weyl independent quadratic equations, but offers some new perspectives. More specifically we prove that if $X_{\mathbb{R}}$ is the intersection of one or two independently and Weyl distributed quadrics then: $$\label{exp}\lim_{n\to\infty}\frac{ \mathbb{E} b(X_{\mathbb{R}})}{n}=1.$$ Thus as we increase the number of variables, Smith’s inequality for one quadric hypersurface is expected to be sharp.\
The key fact here is that given a quadratic form $q$ on ${\mathbb{R}}^n$ we can associate to it a symmetric matrix $Q$ of order $n$ (using a scalar product) and the form $q$ is Weyl distributed if and only if $Q$ is in the *Gaussian Orthogonal ensemble*. This simple observation allows to introduce the language of Random Matrix Theory into the problem. For the case of one quadric hypersurface it is then enough to study the expectation of the signature of $Q$, which characterizes the topology of the zero locus of $q$.\
For the case of the intersection of two quadric hypersurfaces, the idea for proving these limits is to relate the sum of the Betti numbers of $X_{\mathbb{R}}$ to that of its spectral variety, namely the intersection in the space of all quadratic forms of the linear system defining $X_{\mathbb{R}}$ with the set of singular quadrics. This is made rigorous by the introduction of a spectral sequence from [@AgLe] to compute the cohomology of the intersection of real quadrics. This kind of duality between the variables and the quadratic equations is the same that allows to prove Barvinok’s bound.\
In the case of the intersection of three random quadrics in ${\mathbb{R}\mathrm{P}}^n$, the spectral variety is a random curve, but its distribution of probability is fairly different from the Weyl or the standard one. This random curve is smooth with probability one and its topological complexity is essentially the topological complexity of $X_{\mathbb{R}}$ (see [@Le3]).\
The paper is organized as follows: in Section 2 we introduce some notation and review some notions from integral geometry and in Section 3 we discuss the technique from [@AgLe] to study the intersection of real quadrics, focusing on the case of one and two quadrics. In Section 4 we prove the limit (\[exp\]) for one quadric; this is obtained by a combination of a formula for the cohomology of one single quadric and Wigner’s semicircular law. In Section 5 we consider the case of two quadrics: here the result follows again from a formula for the cohomology derived from Section 3. This formula involves the number of singular quadrics in the linear system defining $X_{\mathbb{R}}$ and the maximum of the inertia index of the quadrics belonging to this linear system; both the expectation of these numbers are computed using the integral geometry formula. As a byproduct we compute the intrinsic volume in the Frobenius norm of the set $\Sigma$ of singular symmetric matrices of norm one; this computation is related to some limit of gap probabilities in the GOE and the theory of PainlevŽ equations. Finally in the Appendix we compute the expected value of the rank of the second differential of the spectral sequence from Section 3.
Acknowledgements {#acknowledgements .unnumbered}
================
The author is grateful to Sofia Cazzaniga, who implemented numerical simulations to verify the obtained results, and to Erik Lundberg for useful discussions.
Random quadratic forms and integral geometry
============================================
Let $q=\sum c_{ij}x_ix_j$ be a real quadratic form whose coefficients $c_{ij}$ are independent Gaussian random variables with mean zero and variance one for $i=j$ and two for $i\neq j$. The quadratic form $q$ is said to be a *Weyl distributed* random polynomial. This results in a distribution of probability on the space ${\mathcal{Q}}(n)$ of real quadratic forms in $n$ variables; this distribution of probability is invariant by the action (by change of variables) of the orthogonal group $O(n).$ If $q$ is a random quadratic form Weyl distributed as above and ${\mathbb{R}}^{m}$ is a linear subspace of ${\mathbb{R}}^{n}$, then the restriction $q|_{{\mathbb{R}}^{m}}$ is again a random quadratic form Weyl distributed (see [@Bu]). Equivalently, once a scalar product has been fixed, it is possible to associate to each quadratic form $q$ a symmetric matrix $Q$ by the equation: $$q(x)=\langle x, Qx\rangle,\quad \textrm{for all } x \in {\mathbb{R}}^{n}.$$ In this way a linear isomorphism between the space ${\mathcal{Q}}(n)$ of real homogeneous polynomials of degree two in $n$ variables and the space ${\textrm{Sym}(n,{\mathbb{R}})}$ of real symmetric matrices of order $n$ is set up; we denote by $N=\frac{1}{2}n(n+1)$ the dimension of this vector spaces. If $q$ is a random quadratic form Weyl distributed, the corresponding random matrix $Q$ is said to belong to the *Gaussian Orthogonal Ensemble*. The entries of $Q$ are independent Gaussian random variables with mean zero and variance one on the diagonal and one-half off diagonal. If we define the norm of a matrix $Q$ in ${\textrm{Sym}(n,{\mathbb{R}})}$ by $$\|Q\|^2=\textrm{tr}(Q^2),$$ then the induced probability distribution is uniform on the unit sphere $S^{N-1}$; this distribution is thus invariant by the action of the group $O(N)$ of orthogonal transformation of ${\textrm{Sym}(n,{\mathbb{R}})}$. The map $\rho:O(n)\to O(N)$ given by $$\rho(M)Q=MQM^{-1}$$ defines a homomorphism of group: in fact if $M$ is orthogonal, then $\textrm{tr}(MQM^{-1})=\textrm{tr}(Q);$ the induced action of $O(n)$ on ${\mathcal{Q}}(n)$ is by change of variables. Thus we see that on ${\textrm{Sym}(n,{\mathbb{R}})}$ there are two actions, one of $O(N)$ and one of $O(n)$, both by isometries.\
If $X$ is a compact riemannian manifold of dimension $d$ we denote its Riemannian density by $\omega_X$ and we define its normalized volume $p(X)$ to be the number $\frac{\textrm{Vol}(X)}{\textrm{Vol}{(S^d)}}$. An important tool to study the volume of submanifolds of the sphere, or the projective space, is the so called *integral geometry formula*. Let $A$ and $B$ be submanifolds, with or without boundaries, of the unit sphere $S^m$, of dimension respectively $a$ and $b$, with $a+b\geq m$. We endow the sphere $S^m$ with the standard Riemannian metric and $A$ and $B$ by the induced one. The integral geometry formula is: $$\frac{\int_{SO(m+1)} p(A\cap gB) dg}{\textrm{Vol}(SO(m+1))}= p(A)p(B).$$ where the integral is with respect to the Haar measure. A similar formula holds in the case $A$ and $B$ are submanifolds of the projective space ${\mathbb{R}\mathrm{P}}^m$; in this case the volumes are normalized by $\textrm{Vol}({\mathbb{R}\mathrm{P}}^m).$
The cohomology of the intersection of real quadrics
===================================================
We recall in this section a general construction to study the topology of the intersection of real quadrics. If we are given quadratic forms $q_{1},\ldots, q_{k}$ on ${\mathbb{R}}^{n}$, then we can consider their common zero locus $X_{\mathbb{R}}$ in ${\mathbb{R}\mathrm{P}}^{n-1}:$ $$X_{\mathbb{R}}=X_{{\mathbb{R}}}(q_{1},\ldots, q_{k})$$ To study the topology of $X_{\mathbb{R}}$ we consider the linear span $W$ of $\{q_1, \ldots, q_k\}$ in the vector space ${\mathcal{Q}}(n)$: $$W=\textrm{span}\{q_1, \ldots, q_k\}$$ The arrangement of $W$ with respect to the subset of degenerate quadratic forms (those with at least one dimensional kernel) determines the topology of the base locus in the following way. The simplest invariant we can associate to a quadratic form $q$ is its positive inertia index ${\mathrm{i}}^{+}(q)$, namely the maximal dimension of a subspace $V\subset {\mathbb{R}}^{n}$ such that $q|_{V}$ is positive definite. In a similar fashion we consider for $j\in {\mathbb{N}}$ the sets: $$\Omega^{j}=\{q\in W\backslash \{0\}\,|\, {\mathrm{i}}^{+}(q)\geq j\}.$$ In this way we get a filtration $\Omega^{n}\subseteq\Omega^{n-1}\subseteq\cdots\subseteq \Omega^{1}\subseteq\Omega^{0}$ of $W\backslash\{0\}$ by open sets. The following theorem is proved in [@AgLe].
\[AgLe\]There exists a cohomology spectral sequence of the first quadrant $(E_r, d_r)_{r\geq1}$ converging to $H^{n-1-*}(X_{\mathbb{R}})$ such that $$E_2^{i,j}=H^i(W, \Omega^{j+1}).$$
Notice that each set $\Omega^{j+1}$ deformation retracts to an open subset of the unit sphere in $W$; because of this deformation retraction in the sequel we will always think at each set $\Omega^{j+1}$ as a subset of the unit sphere. From the previous theorem we immediately derive the following inequality: $$b(X_{\mathbb{R}})\leq n+\sum_{j\geq 1} b(\Omega^j)$$ In the low codimension cases, i.e. for $k=1,2,3$, the previous formula can be sharpened as following. The spectral variety of the linear system $W$ is defined to be the intersection of the set $\Sigma$ of degenerate forms of norms one (i.e. of symmetric matrices with zero determinant and Frobenius norm one) with $W$: $$\Sigma_W=W \cap \Sigma.$$ Notice that by homogeneity of the determinant this definition does actually not depend on the norm on ${\mathcal{Q}}(n)$ (respectively $\textrm{Sym}(n,{\mathbb{R}}))$. For a generic choice of $q_1, \ldots, q_k$ the following properties are satisfied: the vector space $W$ has dimension $k$; the intersection of $W$ with the set of degenerate quadratic forms $\Sigma$ is transversal to every of its strata (the stratification is given by the dimension of the kernel). Thus, generically, for $k=1$, i.e. one quadric, the spectral variety is empty, for $k=2$ consists of a finite number of points and for $k=3$ it is a smooth curve (this follows from the fact that the codimension of the singular locus of $\Sigma$ is at least three). These are the only cases in which we may assume the spectral variety is generically smooth. If we define the number $$\mu_W=\max {\mathrm{i}}^+|_W,$$ then in the case $k=1$ generically we have $$\label{one}b(X_{\mathbb{R}})=2(n-\mu_W).$$ The reader is referred to [@AgLe], Example 2 for this formula, while for $k=2,3$ the following inequality holds: $b(X_{\mathbb{R}})\leq 3n-4\mu_W+\frac{1}{2}b(\Sigma_W)$.
The case of one quadric hypersurface
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In this section we study the expected total Betti number of the zero locus $X_{\mathbb{R}}$ of one single quadric Weyl distributed. We start by recalling some results from random matrix theory. Let $Q$ be a random matrix in the Gaussian Orthogonal Ensemble (recall that this is equivalent to the corresponding quadratic form $q$ being Weyl distributed). If $\lambda_1,\ldots, \lambda_n$ are the eigenvalues of $Q$, we define the *empirical spectral distribution* $$\tau_n=\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_i/\sqrt{n}}.$$ Strictly speaking $\tau_n$ is a random variable in the space of the probability distributions over ${\mathbb{R}}$; what is relevant for us is that once we have a continuous, compactly supported function $\psi$ we can define the random variable $X_n(\psi)=\int_{\mathbb{R}}\psi d\tau_n$. Wigner’s semicircular law concerns the limit of the expectation of such a random variable. Let $\tau_{sc}$ be the probability density on the real line $$\tau_{sc}=\frac{1}{2\pi}(4- |x|^2)_+^{1/2}dx,$$ and for every $\psi$ continuous and compactly supported let $X_{sc}(\psi)$ be the number $\int_{\mathbb{R}}\psi d\tau_{sc}$. The following theorem was proved by Wigner (see [@W] for the original work and [@TT] for a modern exposition).
\[sc\] For every interval $A\subset {\mathbb{R}}$ $$\lim_{n\to\infty}\mathbb{E}\int_Ad\tau_n=\int_Ad\tau_{sc}.$$ Moreover for every $\psi$ continuos and compactly supported and $F$ continuous bounded $$\lim_{n\to\infty}\mathbb{E}F(X_n(\psi))=F(X_{sc}(\psi)).$$
Using the previous theorem we can prove the following proposition.
Let $Q$ be a random symmetric matrix in the Gaussian Orthogonal Ensemble of dimension $n$. Let also $\mu_{n}=\max\{{\mathrm{i}}^+, {\mathrm{i}}^-\}$ and $\nu_n=\min\{{\mathrm{i}}^+,{\mathrm{i}}^-\}.$ Then: $$\lim_{n\to \infty}\frac{\mathbb{E}[\mu_n-\nu_n]}{n}=0.$$
First notice that $\mu_n(Q)=\mu_n(Q/\sqrt{n})$ and $\nu_n(Q)=\nu_n(Q/\sqrt{n})$, since the inertia index of a symmetric matrix is invariant by multiplication of a positive number. Thus we have the equality of random variables: $$\frac{\mu_n-\nu_n}{n}=\frac{|{\mathrm{i}}^+-{\mathrm{i}}^-|}{n}=\bigg|\int_{\mathbb{R}}Hd\tau_n\bigg|,$$ where $H(x)=\textrm{sign}(x)$; the first equality follows directly from the definition and the second comes from: $$\int_{\mathbb{R}}Hd\tau_n=\int_{(0,\infty)} d\tau_n-\int_{(-\infty,0)}d\tau_n=\frac{{\mathrm{i}}^+-{\mathrm{i}}^-}{n},$$ (notice that since the set of symmetric matrices with determinant zero is the complement of a full measure set, we can discard the term $\int_{\{0\}}d\tau_n$). For every ${\epsilon}>0$ let us now consider a continuous, compactly supported function $\psi_{\epsilon}$ satisfying: $\psi_{\epsilon}$ is odd, $|\psi_{\epsilon}|\leq 1$ and $\psi_{{\epsilon}}(x)=H(x)$ for $x\in A({\epsilon})=(-3, {\epsilon})\cup({\epsilon}, 3)$. The existence of such a function is obvious. Let also $F$ be any compactly supported function equal to $|x|$ for $|x|\leq 1$. We have now the following chain of inequalities of random variables:
$$\begin{aligned}
\bigg|\int_{\mathbb{R}}H d\tau_n\bigg| &\leq F(X_n(\psi_{\epsilon}))+\bigg|\int_{\mathbb{R}}H-\psi_{\epsilon}d\tau_n\bigg|
\\ &\leq F(X_n(\psi_{\epsilon}))+\int_{{\mathbb{R}}\backslash A({\epsilon})}d\tau_n.\end{aligned}$$
The first inequality comes from the fact that $|X_n(\psi_{\epsilon})|\leq 1$ and the definition of $F$; the second inequality is because $H-\psi_{\epsilon}$ is zero on $A({\epsilon})$ and $|H-\psi_{\epsilon}|\leq 1.$ Thus by the previous theorem \[sc\] we have: $$\lim_{n\to\infty}\mathbb{E}F(X_n(\psi_{\epsilon}))=F(X(\psi_{\epsilon}))=0,$$ since $\psi_{\epsilon}$ is odd, and $$\lim_{n\to \infty}\mathbb{E}\int_{{\mathbb{R}}\backslash A({\epsilon})} d\tau_n=\int_{{\mathbb{R}}\backslash A({\epsilon})} d\tau_{sc}\leq 2{\epsilon}.$$ Hence for every ${\epsilon}>0$ $$\lim_{n\to\infty}\frac{\mathbb{E}[\mu_n-\nu_n]}{n}\leq 2{\epsilon},$$ which together with $\mu_n-\nu_n\geq 0$ proves the proposition.
We derive the following theorem for the expected value of the total Betti number of a random quadratic hypersurface in ${\mathbb{R}\mathrm{P}}^{n-1}.$
\[one\] Let $q$ be a random, Weyl distributed, quadratic form on ${\mathbb{R}}^{n}$ and $X_{\mathbb{R}}$ be its zero locus in ${\mathbb{R}\mathrm{P}}^{n-1}.$ Then $$\lim_{n\to \infty}\frac{\mathbb{E}[b(X_{\mathbb{R}})]}{n}=1.$$
Since generically $\mu_{n}(q)=n-\nu_{n}(q)$, by theorem \[AgLe\] we have $b(X_{\mathbb{R}})=n+\nu_{n}(q)-\mu_{n}(q)$ (this is a restatement of formula \[one\]). Since if $q$ is Weyl distributed then the corresponding symmetric matrix is in the Gaussian Orthogonal Ensemble, the conclusion follows from the limit of the previous proposition.
If we notice that for a nonsingular real quadric in ${\mathbb{C}\textrm{P}}^n$ we have $b(X_{\mathbb{C}})=n+\frac{1}{2}(1+(-1)^{n+1})$, then the previous limit can be rewritten in a more fashionable way as: $$\lim_{n\to\infty}\mathbb{E}[b(X_{\mathbb{R}})/b(X_{\mathbb{C}})]=1$$ The fact that this limit had to be less or equal then one is the content of Smith’s inequalities (see the Appendix of [@Wilson]).
The case of the intersection of two quadrics
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In the case $X_{\mathbb{R}}$ is the intersection of two quadrics $q_1, q_2$ in ${\mathbb{R}\mathrm{P}}^{n-1}$, we can derive directly from Theorem \[AgLe\] the following. Recall the definition of $W$ as $\textrm{span}\{q_1, q_2\}\subset {\mathcal{Q}}(n)$, the number $\mu_{W}=\max {\mathrm{i}}^+|_{W}$ and the spectral variety $\Sigma_W=\Sigma\cap W$ (in this case it is a subvariety of $S^1$, i,e consists either of a finite number of points or is the whole $S^1$). The topology of the intersection of two quadrics was studied by the author in [@Le2]; in fact the following proposition follows directly from Theorem 8 of [@Le2], though we give a short proof here using Theorem 1.
For a generic pair $(q_1, q_2)$ we have $$b(X_{\mathbb{R}})= 3n-1-4\mu_W +(c_W+d_W) +\frac{1}{2}b(\Sigma_W).$$ where $c_W$ and $d_W$ belong to $\{0,1\}$.
In this case, summing the elements for the third (the last) page of the spectral sequence of theorem \[AgLe\] gives: $$b(X_{\mathbb{R}})=\textrm{rk}(E_3)=n-1-2(\mu_W -\min {\mathrm{i}}^+|_W) +c_W+d_W+\sum_{j=\min {\mathrm{i}}^+|_W}^{\mu_W-1}b_0(\Omega^{j+1}).$$ where we have called $c_W$ and $d_W$ respectively $\textrm{rk}(E_3^{0, \mu})$ and $\textrm{rk}(E_3^{2, \mu-1})$; a direct look at the second table of the spectral sequence gives $c_W,d_W\in \{0,1\}$. Now for a generic choice of $q_1, q_2$ we have $\min {\mathrm{i}}^+|_W=n+1-\mu_W$, $\Sigma_W$ consists of a finite number of points and the function ${\mathrm{i}}^+$ jumps exactly by $\pm 1$ when crosses $\Sigma_W.$ In particular each point of $\Sigma_W$ belongs exactly to one of the $\partial \Omega^{j+1}$, $\min {\mathrm{i}}^+|_W\leq j\leq \mu_W-1$. Thus Alexander-Pontryiagin duality gives: $$\sum_{j=\min {\mathrm{i}}^+|_W}^{\mu_{W}-1}b_0(\Omega^{j+1})=\frac{1}{2}b(\Sigma_W).$$ which yelds the desired formula.
In particular we see that $$\label{extwo}\mathbb{E}[b(X_{\mathbb{R}})]=3n-1-4\mathbb{E}[\mu_W]+\mathbb{E}[c_W+d_W]+\frac{1}{2}\mathbb{E}b(\Sigma_W).$$ We will compute each term of the previous sum; we start by introducing some auxiliary material. We recall now that the probability that the interval $(-{\epsilon}, {\epsilon})$ does not contain any of the eigenvalues of a $Q\in \textrm{GOE}(n)$ is called *gap probability*; we consider this probability as a function of $\epsilon$ and denote it by $f_n({\epsilon}).$ In the case $n$ is even, $f_n({\epsilon})$ can be evaluated using methods from integrable systems. Following [@FoWhi] we have:[^3] $$\label{f}f_n({\epsilon})=\tau_{\sigma_V}({\epsilon}^2),$$ where $\tau_{\sigma_V}$ is a function satisfying: $$\label{tau}\sigma_V(t)=t\frac{d}{dt}\log \tau_{\sigma_V}(t)\quad \textrm{and}\quad \lim_{t\to 0^+}\sigma_{V}(t)t^{-1/2}=-\frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})\Gamma(\frac{1}{2})\Gamma(\frac{3}{2})}=-c_n.$$ We denote by $\Gamma$ the Euler Gamma Function; $\sigma_{V}$ itself satisfies a second order differential equation (equation (2.9) of [@FoWhi]) but for our purposes is not necessary to write it down explicitly.
\[bound2\]For every $n\in 2\mathbb{N}$: $$\lim_{{\epsilon}\to 0^+}f_n'({\epsilon})=-2c_n.$$ Moreover as $n$ goes to infinity $c_n\sim \frac{\sqrt{2n}}{\pi}.$
For the first limit we argue as follows. Since $\sigma_V(t)=t\tau_{\sigma_V}'(t)/\tau_{\sigma_V}(t)$ we have $\tau_{\sigma_V}'(t)=\sigma_V(t)\tau_{\sigma_V}(t)/t$ and thus: $$\lim_{{\epsilon}\to 0^+}f_n'({\epsilon})=\lim_{{\epsilon}\to 0^+}2{\epsilon}\tau_{\sigma_V}'({\epsilon}^2)=\lim_{{\epsilon}\to 0^+}\frac{2\sigma_V({\epsilon}^2)\tau({\epsilon}^2)}{{\epsilon}}=-2c_n,$$ where for the last limit we have used the limit in equation (\[tau\]) and the fact that $\lim_{\epsilon}f_n({\epsilon})=\lim_{{\epsilon}}\tau_{\sigma_V}({\epsilon}^2)=1$ by definition.\
The second limit follows immediately using the known values of the Gamma function $\Gamma(1/2)=\sqrt{\pi}$, $\Gamma(3/2)=\sqrt{\pi}/2$ and the Stirling’s asymptotic.
We consider now the unit sphere $S^{N-1}$ in $\textrm{Sym}(n, {\mathbb{R}})$ (with respect to the Frobenius norm) and the set $\Sigma$ of singular matrices of norm one. We are interested in bounding the intrinsic volume of $\Sigma$. In order to do that we start by bounding the volume of an ${\epsilon}$-tube $\Sigma_{\epsilon}$ of $\Sigma$ in $S^{N-1}$ (the volume is computed with respect to the Riemannian structure induced on $S^{N-1}$ by the norm). Let: $$\sigma(Q)=\min_{\lambda \in s(Q)}|\lambda|;$$ notice that using this notation the gap probability $f_n({\epsilon})$ equals ${\mathbb{P}}\{\sigma(Q)\geq \epsilon\}.$
\[bound3\] $$\emph{\textrm{Vol}}(\Sigma_{\epsilon})\leq\emph{\textrm{Vol}}(S^{N-1})\cdot (1-\mathbb{P}\{\sigma(Q)\geq {\epsilon}\|Q\|\}).$$
We let $Z\subset \textrm{Mat}(n, {\mathbb{R}})$ be the set of degenerate matrices and consider the following chain of inequalities for a matrix $Q\in \textrm{Sym}(n, {\mathbb{R}})$: $$d_{\textrm{Mat}(n,{\mathbb{R}})}(Q, Z)\leq d_{\textrm{Sym}(n,{\mathbb{R}})}(Q, Z\cap \textrm{Sym}(n,{\mathbb{R}}))\leq d_{S^{N-1}}(Q, \Sigma).$$ In particular the set $\Sigma_\epsilon$ is contained in the set $\{d_{\textrm{Mat}(n,{\mathbb{R}})}(Q,Z)\leq \epsilon\}\cap S^{N-1},$ and by Eckart-Young theorem the last one equals $\{\sigma(Q)\leq {\epsilon}\|Q\|\}\cap S^{N-1};$ thus we get: $$\label{eq:bound1}
\textrm{Vol}(\Sigma_\epsilon)\leq\textrm{Vol}\{\sigma(Q)\leq {\epsilon}\|Q\|\}\cap S^{N-1}.$$ Since the probability distribution of the Gaussian Orthogonal Ensemble is uniform on the sphere $S^{N-1},$ then the conclusion follows.
The probability $\mathbb{P}\{\sigma(Q)\geq {\epsilon}\|Q\|\}$ *is not* the gap probability, because of the rescaling factor $\|Q\|$.
The following lemma gives an upper bound for the volume of $\Sigma$.
\[bound4\]If $n\in 2\mathbb{N}$: $$\emph{\textrm{Vol}}(\Sigma)\leq \emph{\textrm{Vol}}(S^{N-1})\cdot O(n c_n)$$
Let us start by defining the functions $g_n({\epsilon})={\mathbb{P}}\{\sigma(Q)\geq {\epsilon}\|Q\|\}$ and $\hat{\sigma}=\sigma|_{S^{N-1}}.$ Notice that in polar coordinates $(\theta, r)\in S^{N-1}\times (0, \infty)$ we have: $$\{\sigma(Q)\geq {\epsilon}\|Q\|\}=\{\hat{\sigma}(\theta)\geq {\epsilon}\}\quad \textrm{and}\quad \{\sigma(Q)\geq {\epsilon}\}=\{\hat{\sigma}(\theta)\geq {\epsilon}/r\}.$$ Both identities follow by: $\sigma(Q)=\|Q\|\sigma(Q/\|Q\|)=r\hat{\sigma}(\theta).$ In particular we can write: $$\begin{aligned}
\nonumber
f_n({\epsilon})&=\frac{1}{(2 \pi)^{N/2}}\int_{S^{N-1}}\int_{0}^{\infty}\chi_{\{\hat{\sigma}(\theta)\geq {\epsilon}/r\}}(\theta, r)r^{N-1}e^{-\frac{r^2}{2}}dr dS^{N-1}=\\
&\nonumber=\frac{1}{(2 \pi)^{N/2}}\int_{0}^{\infty}\underbrace{\left(\int_{S^{N-1}}\chi_{\{\hat{\sigma}(\theta)\geq {\epsilon}/r\}}(\theta, r)dS^{N-1}\right)}_{\textrm{Vol}(S^{N-1})g_n({\epsilon}/r)}r^{N-1}e^{-\frac{r^2}{2}}dr=\\
&=\frac{\textrm{Vol}(S^{N-1})}{(2 \pi)^{N/2}}\int_{0}^{\infty}g_n({\epsilon}/r)r^{N-1}e^{-\frac{r^2}{2}}dr\nonumber\end{aligned}$$ In particular from the last equation we get: $$\label{derivative}
f'_n({\epsilon})=\frac{\textrm{Vol}(S^{N-1})}{(2 \pi)^{N/2}}\int_{0}^{\infty}g'_n({\epsilon}/r)r^{N-2}e^{-\frac{r^2}{2}}dr.$$ The function $g_n$ is monotone decreasing (as we let ${\epsilon}$ increase the probability of $\{\sigma(Q)\geq {\epsilon}\|Q\|\}$ decreases); thus $g'_n\leq 0$ and by Fatou’s Lemma from (\[derivative\]) we get the following (notice reversed inequalities due to the sign of $g_n'$): $$\begin{aligned}
\nonumber\frac{\textrm{Vol}(S^{N-1})}{(2 \pi)^{N/2}}\int_{0}^{\infty}\lim_{{\epsilon}\to 0}g'_n({\epsilon})r^{N-2}e^{-\frac{r^2}{2}}dr&=\frac{\textrm{Vol}(S^{N-1})}{(2 \pi)^{N/2}}\int_{0}^{\infty}\lim_{{\epsilon}\to 0}g'_n({\epsilon}/r)r^{N-2}e^{-\frac{r^2}{2}}dr\\ \nonumber
&\geq\lim_{{\epsilon}\to 0}\frac{\textrm{Vol}(S^{N-1})}{(2 \pi)^{N/2}}\int_{0}^{\infty}g'_n({\epsilon}/r)r^{N-2}e^{-\frac{r^2}{2}}dr\\
&=\lim_{{\epsilon}\to 0}f_n'({\epsilon})=-2c_n
\end{aligned}$$ In particular we have obtained: $$\underbrace{\left(\frac{\textrm{Vol}(S^{N-1})}{(2 \pi)^{N/2}}\int_{0}^{\infty}r^{N-2}e^{-\frac{r^2}{2}}dr\right)}_{\frac{\Gamma(\frac{N-1}{2})}{\sqrt{\pi}\Gamma{(\frac{N}{2}})}} \cdot \lim_{{\epsilon}\to 0}-g'_n({\epsilon})\leq 2c_n$$ which can be rewritten as: $$\label{finaleq}\lim_{{\epsilon}\to 0}-g'_n({\epsilon})\leq 2c_n \frac{\sqrt{\pi}\Gamma(\frac{N}{2})}{\Gamma{(\frac{N-1}{2}})}=O(nc_n).$$ We finally turn to the definition of $\textrm{Vol}(\Sigma)=\lim_{{\epsilon}\to 0}\textrm{Vol}(\Sigma_{\epsilon})/2{\epsilon}$; plugging the result of Lemma \[bound3\] into this limit we get: $$\textrm{Vol}(\Sigma)\leq \textrm{Vol}(S^{N-1})\cdot \lim_{{\epsilon}\to 0}\frac{1-g_n({\epsilon})}{2{\epsilon}}= \textrm{Vol}(S^{N-1})\cdot \lim_{{\epsilon}\to 0}-\frac{g'_n({\epsilon})}{2}\leq \textrm{Vol}(S^{N-1})\cdot O(nc_n)$$ where in the second equality we have used De l’Hopital’s rule and in the last one we used (\[finaleq\]). This concludes the proof.
As a corollary we get the following Theorem.
\[sqrt\]If $n\in 2\mathbb{N}:$ $${\mathbb{E}}b(\Sigma_W)\leq O(\sqrt{n})$$
We start by noticing that by assumption for every $g\in SO(N)$ the random quadratic forms $q$ and $gq$ have the same distribution (here the action is not by change of variable, but directly on the space of the coefficients). Thus we have: $$\mathbb{E}b(\Sigma_W)=\frac{\int_{SO(N)}\mathbb{E}[b(\Sigma_{gW})]dg}{\textrm{Vol(SO(N))}}=\frac{\mathbb{E}\int_{SO(N)}b(\Sigma_{gW})dg}{\textrm{Vol}(SO(N))}=\frac{2\textrm{Vol}(\Sigma)}{\textrm{Vol}(S^{N-2})}.$$ The first equality is because for every $g\in SO(N)$ we have $\mathbb{E}b(\Sigma_{gW})=\mathbb{E}b(\Sigma_{W})$; the second is just linearity of expectation, and the third one is the integral geometry formula (there is no expected value because the integral is constant).\
Using now the bound given by Lemma \[bound4\] we get: $${\mathbb{E}}b(\Sigma_W)\leq \frac{\textrm{Vol}(S^{N-1})}{\textrm{Vol}(S^{N-2})}O(nc_n).$$ Recalling the formula for the volume of the sphere $\textrm{Vol}(S^{k-1})=\frac{2\pi^{k/2}}{\Gamma(k/2)},$ we see that $$\frac{\textrm{Vol}(S^{N-1})}{\textrm{Vol}(S^{N-2})}\sim\frac{\sqrt{2}}{n \pi}.$$ This, together with the asymptotic $c_n\sim \frac{\sqrt{2n}}{ \pi}$ from Lemma \[bound2\], concludes the proof.
\[p1\] $$\lim_{n\to \infty}\frac{\mathbb{E}[4\mu_{n}]}{n}=2.$$
We start by noticing that for an open dense set of $(Q_1, Q_2)$ the following inequalities hold: $${\mathrm{i}}^+(Q_1)\leq\mu_{W}\leq {\mathrm{i}}^+(Q_1)+b(\Sigma_W).$$ In fact for a generic pencil the index function “jumps” exactly by $\pm1$ when crosses $\Sigma_W$ and thus the maximum that can reach over $W$ is $\mu(Q_1)+b(\Sigma_W).$ Dividing by $n$ and taking expectations, Theorem \[sqrt\] gives the result for $n\in 2\mathbb{N}.$ To prove that the statement holds also for odd $n$ we notice that restricting a Weyl distributed random quadratic form $q$ to a subspace $V\subset {\mathbb{R}}^{n}$ gives again a Weyl distributed random quadratic form $q|_V$ on $V\simeq \mathbb{R}^{\textrm{dim}(V)}$; since ${\mathrm{i}}^+(q|_V)\leq {\mathrm{i}}^+(q)$ we have: $$\mathbb{E}[\mu_{n-1}]\leq \mathbb{E}[\mu_n]\leq \mathbb{E}[\mu_{n+1}].$$ This proves that the same limit holds for odd $n.$
As a corollary we prove the following theorem for the asymptotic of $\mathbb{E}[b(X_{\mathbb{R}})]$.
Let $X_{\mathbb{R}}\subset {\mathbb{R}\mathrm{P}}^n$ be the intersection of two random quadrics independent and Weyl distributed. Then $$\lim_{n\to \infty}\frac{\mathbb{E}[b(X_{\mathbb{R}})]}{n}=1, \quad \textrm{$n$ odd}.$$
The limit follows from formula (\[extwo\]), Theorem \[sqrt\] and the previous proposition, after noticing that $\mathbb{E}[c_W+d_W]\leq 2$.\
Notice in particular that since the total Betti number of the complete intersection of two quadrics in ${\mathbb{C}\textrm{P}}^n$ is $2n-2$, then in this case the expectation of Smith’s inequality is turned into an equality for large $n$ (up to a factor $\frac{1}{2}$).
Appendix: the expected second differential {#appendix-the-expected-second-differential .unnumbered}
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It is interesting now to compute also the expected value of the number $c_W+d_W$. By definition we have: $$c_W=\textrm{rk}(E_3^{0, \mu})\quad \textrm{and}\quad d_W=\textrm{rk}(E_3^{2, \mu-1})$$ where $(E_r, d_r)_{r\geq 0}$ is the spectral sequence of theorem \[AgLe\] and $\mu=\mu_W=\max {\mathrm{i}}^+|_W.$ We recall now from [@AgLe] the definition of the second differential of this spectral sequence. Consider the bundle $L_{\mu}\to \Omega^{\mu}$ whose fiber at the point $q\in \Omega^{\mu}$ is the positive eigenspace of $Q$ and whose vector bundle structure is given by its inclusion in $\Omega^{\mu}\times {\mathbb{R}}^{n+1}.$ We let $w_{1,\mu}\in H^{1}(\Omega^{\mu})$ be the first Stiefel-Whitney class of this bundle. From Theorem B of [@AgLe] it follows that for every $x\in E_2^{0,\mu}$ we have: $$d_2^{0, \mu}(x)=(x\smile w_{1, \mu})|_{(W, \Omega^j)}.$$ In particular, since $E_3^{0, \mu}=\ker d_2^{0,\mu}$ and $E_3^{2, \mu-1}=H^1(\Omega^\mu)/ \textrm{Im} d_2^{0, \mu}$ we immediately get: $$\label{whit}c_W+d_W=1+b_1(\Omega^\mu) -2 \omega_{1, \mu},$$ where $\omega_{1,\mu}$ is $\textrm{rk}(d_2^{0, \mu})$ (thus $\omega_{1, \mu}$ “is" the Stiefel-Whitney class $w_{1, \mu}$ thought as an element of $H^1(\Omega^\mu)\subset {\mathbb{Z}}_2$). Using this description we prove the following.
For two Weyl, independent random quadrics in ${\mathcal{Q}}(n)$ we have $$\mathbb{E}[c_W+d_W]=1+(-1)^{[\frac{n}{2}]}\mathbb{P}\bigg\{{\mathrm{i}}^+|_{W\backslash \{0\}}\equiv \bigg[ \frac{n}{2} \bigg]\bigg\}.$$
In the case $n$ is odd, for a generic pair of quadrics $(q_1, q_2)$ the group $H^1(\Omega^\mu)$ has to be zero: this is because any generic linear family of quadrics in an odd number of variables contains at least a line of degenerate quadrics and thus the index function cannot be constant on the nonzero elements of the family. Thus $w_{1,\mu}=0$ and equation \[whit\] gives the desired conclusion in this case.\
In the case $n$ is even we use the following fact (see Proposition 2 of [@Agrachev1]): for a generic pair of symmetric matrices $(Q_1, Q_2)$ there exists an invertible matrix $M$ such that both $M^TQ_1M$ and $M^TQ_2M$ have the same block-diagonal shape with blocks of dimensions one or two. In particular the index function for the family $x_1Q_1+x_2Q_2$ is the sum of the index functions for the families $x_1B_1^k+x_2B_2^k$ (because the number of positive eigenvalues of a symmetric matrix is invariant by congruence), where $M^TQ_iM=\textrm{diag}(B_i^1, \ldots, B_i^m)$. Let us focus on the term $b_1(\Omega^\mu)$ in equation (\[whit\]). Notice that $$\mathbb{E}[b_1(\Omega^\mu)]=\mathbb{P}\bigg\{{\mathrm{i}}^+|_{W\backslash \{0\}}\equiv \bigg[ \frac{n}{2} \bigg]\bigg\}.$$ This is because the only case in which $b_1(\Omega^\mu)$ is nonzero, for a generic pair, is when the index function is constant on the nonzero elements of $W$, and for a generic pair this constant has to be $\frac{n}{2}.$ On the other hand using the previous observation, we se that the only way for the index function to be constant on $W\backslash\{0\}$, for a generic pair, is when each block has dimension two and the index function for each block is constantly equal to one. It is a well-known result that the bundle of positive eigenspace for a two dimensional family of quadrics in two variables equals the Moebius bundle (see [@Le2]), hence for every block the corresponding Stiefel-Whitney class is nonzero. Thus it follows that for a generic pair $(Q_1, Q_2)$, if the index function is constant on $W\backslash\{0\}$, then it must be equal $\frac{n}{2}$ and by the Whitney product formula in this case: $$w_{1, \mu}\equiv\frac{n}{2}\textrm{ mod }2.$$ Thus $b_1(\Omega^\mu)$ equals $1$ with probability $p_1=\mathbb{P}\bigg\{{\mathrm{i}}^+|_{W\backslash \{0\}}\equiv \bigg[ \frac{n}{2} \bigg]\bigg\}$ and zero otherwise (both for the even and the odd case); when $n$ is even, $\omega_{1,\mu}$ equals $\frac{n}{2}$ modulo 2 i.e. $\frac{1}{2}(1+(-1)^{[\frac{n}{2}]+1})$ with probability $p_1$, and zero otherwise. Using equation (\[whit\]) and the definition of expectation we immediately get the conclusion.
In the case $W$ is two dimensional, by Theorem \[sqrt\] we have: $$\label{p11}\lim_{n\to \infty}\mathbb{P}\bigg\{{\mathrm{i}}^+|_{W\backslash \{0\}}\equiv \bigg[\frac{n}{2}\bigg]\bigg\}=0.$$ Hence $\mathbb{E}[c_W+d_W]\to 1;$ notice also that the previous statement also gives a probabilistic statement on the second differential of the spectral sequence of Theorem \[AgLe\]; in fact using the limit (\[p11\]) we immediately derive the following.
For the intersection of two independent, Weyl, random quadrics in ${\mathbb{R}\mathrm{P}}^n$ we have: $$\lim_{n\to \infty}{\mathbb{E}}[ \emph{\textrm{rk}}(d_2)]=0.$$
[15]{} A. A. Agrachev, R. V. Gamkrelidze: *Quadratic maps and smooth vector valued functions; Euler Characteristics ol level sets*, Itogi nauki. VINITI. Sovremennye problemy matematiki. Novejshie dostigeniya, 1989, v.35, 179–234
A. A. Agrachev, A. Lerario: *Systems of quadratic inequalities*, Proceedings of the London Mathematical Society, (2012) 105 (3). A. I. Barvinok: *On the Betti numbers of semialgebraic sets defined by few quadratic inequalities*, Discrete and Computational Geometry , 22:1-18 (1999). J. Bochnak, M. Coste, M-F. Roy: *Real Algebraic Geometry*, Springer-Verlag, 1998.
P. BŸrgisser, *Average Euler characteristic of random algebraic varieties*, C. R. Acad. Sci. Paris, Ser. I 345 (2007) 507Ð512. E. Edelman, A. Kostlan, *How many zeros of a random polynomial are real?*, Bulletin of the American Mathematical Society 32 (1995), 1-37. P. J. Forrester, N. S. White, *$\tau$-function evaluation of gap probabilities in orthogonal and symplectic matrix ensembles*, Nonlinearity 15, 2002, Pages 937-954 D. Gayet, J-Y. Welschinger, *Exponential rarefaction of real curves with many components*, Publ. Math. Inst. Hautes ƒtudes Sci. No. 113 (2011), 69Ð96. D. Gayet, J-Y. Welschinger, *What is the total Betti number of a random real hypersurface?*, arXiv:1107.2288v1 U. Helmke, *Critical Points of Matrix Least Squares Distance Functions*, Linear Algebra and its Applications, 215, 15 January 1995, Pages 1-19 M. Kac, *On the average number of real roots of a random algebraic equation*, Bull. Amer. Math. Soc. Volume 49, Number 4 (1943), 314-320. A. Lerario, *Convex Pencils of real quadratic forms*, Discrete and computational Geometry, Volume 48, Number 4 (2012), 1025-1047. A. Lerario, *The total Betti number of the intersection of three real quadrics*, Advances in Geometry, to appear. A. Lerario, E. Lundberg, *Statistics on Hilbert’s Sixteenth Problem*, arXiv:1212.3823. F. Nazarov, M. Sodin, *On the Number of Nodal Domains of Random Spherical Harmonics*, arXiv:0706.2409v1 S. S. Podkorytov, *The mean value of the Euler characteristic of an algebraic hypersurface*, Algebra i Analiz, 11(5):185Ð 193, 1999. English translation: St. Petersburg Math. J. 11(5) (2000), pp. 853Ð860. M. Reid: *The complete intersection of two or more quadrics*, 1972.
P. Sarnak, *Letter to B. Gross and J. Harris on ovals of random plane curves*, publications.ias.edu/sarnak/paper/510 (2011) M. Shub, S. Smale, *Complexity of Bezout’s teorem II: volumes and probabilities*, The collected papers of Stephen Smale, Volume 3 (pp 1402-1420) T. Tao, *Topics in Random Matrix Theory*, Graduate Studies in Mathematics 2012 E. Wigner *On the Distribution of the Roots of Certain Symmetric Matrices* Ann. of Math. 67, 325-328, 1958. G. Wilson, *Hilbert’s sixteenth problem*, Topology 17 (1978), 53-74.
[^1]: The distribution of probability here is such that the components of the spherical harmonic with respect to the $L^2_{S^2}$ orthonormal basis are i.i.n. distributed.
[^2]: Here the probability distribution is a real analogue of the Weyl one, suggested by P. Sarnak in [@Sarnak].
[^3]: Here we use the same notation as in [@FoWhi] to help the reader comparing with this reference. The subscript of $\sigma_V$ is due to the connection with the PainlevŽ fifth equation.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'A definition of quantum correlation is presented for an arbitrary bipartite quantum state based on the skew information. This definition not only inherits the good properties of skew information such as the contractivity and so on, but also is effective and almost analytically calculated for any bipartite quantum states. We also reveal the relation between our measure and quantum metrology. As applications, we give the exact expressions of quantum correlation for many states, which provides a direct support for our result.'
author:
- 'Chang-shui Yu$^1$'
- 'Shao-xiong Wu$^1$'
- Xiaoguang Wang$^2$
- 'X. X. Yi$^1$'
- 'He-shan Song$^1$'
title: Quantum correlation measure in arbitrary bipartite systems
---
*Introduction.*-When quantum correlation is mentioned, one might be immediately tempted to think of quantum entanglement which results from the merging of the superposition principle of states and the tensor structure of the composite quantum state space. This is not strange because as one of the most intriguing feature of quantum mechanics that distinguishes the quantum world from the classical one, quantum entanglement has been employed in most of the quantum information processing tasks (QIPTs) and has been paid close attention to in wide field \[1\]. However, quantum entanglement can not cover all the quantumness of correlations in a composite quantum system. It has been shown that some QIPTs without any quantum entanglement might still demonstrate quantum advantage if such QIPTs own quantum discord \[2-9\] which was first introduced as the discrepancy between the generalizations of two classically equivalent mutual information \[10,11\]. This could be one of the potential reasons why quantum discord has attracted so many interests in the past few years (see Ref. \[12-27\] and the references therein).
As an important branch of the researches on quantum correlation, the quantification of quantum correlation is a hot topic \[10,11,28-34\]. The information theoretic definition of quantum discord is found to be only analytically calculated for some special states \[10,11,15, 30,36\]. The geometric quantum discord can be calculated analytically for all $\left(
2\otimes d\right) $ -dimensional systems \[28,29\], but it confronts some contradictions \[37,38\]. For example, 1) It could be increased by the non-unitary evolution on the subsystem without measurements; 2) It could be reduced by an extra product state. In order to avoid these unexpected properties, some new measures based on the trace norm \[33,39-40\] are proposed at the cost of losing the computability including some variational attempts \[37,41,42\] only covering 2). It is fortunate that a recent progress \[34\] according to the skew information \[43\] has effectively covered both the aspects, which shed new light on the quantification of quantum correlation. However, much as the quantification of entanglement is still restricted to the pure states and low dimensional bipartite quantum states due to the complex optimization for a general high dimensional quantum systems \[1\], so far it has still been an open question how to provide a good quantification or even an effective (fast, steadily, reliable, with machine precision) algorithm for the quantum correlation in high dimensional systems.
In this Letter, we give a definition of quantum correlation pertaining to arbitrary bipartite quantum systems based on the skew information. This definition automatically inherits the good properties such as the contractivity, so it is guaranteed to be a good measure. It is found to have an interesting relation with quantum metrology . In addition, this definition looks like that in Ref. \[34\], but it is quite different from it even in pure states and the $\left (2\otimes d\right)$ -dimensional quantum states. In particular, our definition for any dimensional state can be converted to an existing and easy optimization question that can be fast, steadily, reliably and effectively solved with machine precision by the well developed technique. In this sense, we think our measure of quantum correlation is even almost analytic. As a demonstration, we give the exact expressions of quantum correlation for $\left
(2\otimes d\right)$-dimensional states, a type of positive partial transpose (PPT) states, the high-dimensional Werner states and the Isotropic states, by which, on the one hand, one will find the power of our definition of quantum correlation, on the other hand, one will find the effectiveness of the proposed numerical method.
*The definition of quantum correlation*.-To begin with, we would like to briefly introduce the skew information for a bipartite density matrix $%
\rho _{AB}$ and an observable $O$. It is defined by$$I\left( \rho _{AB},O\right) =-\frac{1}{2}Tr\left[ \sqrt{\rho _{AB}},O\right]
^{2}.$$$I\left( \rho _{AB},O\right) $ has been employed in many fields \[44-46\], and has many good properties \[34\]. For example, it vanishes if and only if $\rho
$ and $O$ commute and it is positive in other cases; It doesn’t increase under classical mixing; In particular, if we select an observable $%
O=K_{A}\otimes \mathbf{1}_{B}$ with $K_{A}$ some observable on subsystem $A$, $I\left( \rho _{AB},O\right) $ is contractive under completely positive and trace-preserving maps $\Phi $ on $B$, that is, $I\left( \left( \mathbf{1}%
_{A}\otimes \Phi \right) \rho _{AB},K_{A}\otimes \mathbf{1}_{B}\right) \leq
I\left( \rho _{AB},K_{A}\otimes \mathbf{1}_{B}\right) $.
In order to quantify the quantum correlation, Ref. \[34\] has required the non-degenerate traceless observable $K_{A}$ (full rank) operated on subsystem $A$. Here, we would like to restrict us to the *rank-1* local projectors. Let $\rho _{AB}$ be an $\left( m\otimes n\right) $-dimensional density matrix and suppose $$K_{k}=\left\vert k\right\rangle \left\langle k\right\vert \otimes \mathbf{1}%
_{n}$$ with $\left\vert k\right\rangle $ in arbitrary orthonormal set $S$ of subsystem $A$, we will be able to give our definition of quantum correlation as follows.
**Definition. 1**.-The quantum correlation $\mathcal{Q}$ of $\rho _{AB}$ is defined by the minimal skew information induced by an group of orthonormal projectors. This can be rephrased as $$\mathcal{Q}\left( \rho _{AB}\right) :=-\frac{1}{2}\min_{S}\sum_{k=0}^{m-1}Tr%
\left[ \sqrt{\rho _{AB}},K_{k}\right] ^{2},$$with $K_{k}$ defined by Eq. (2).
**Proof.** In order to show $\mathcal{Q}\left( \rho _{AB}\right) $ is a measure of quantum correlation, we have to prove that $\rho _{AB}$ is a classical-quantum state as $\rho _{AB}=\sum \tilde{\lambda}_{k}\left\vert
\tilde{k}\right\rangle \left\langle \tilde{k}\right\vert \otimes \varrho _{%
\tilde{k}}$ with $\left\vert \tilde{k}\right\rangle $ analogous to $%
\left\vert k\right\rangle $ in some set $\tilde{S}$ if and only if $\mathcal{%
Q}\left( \rho _{AB}\right) =0$.
Let $\rho _{AB}=\sum \tilde{\lambda}_{k}\left\vert \tilde{k}\right\rangle
\left\langle \tilde{k}\right\vert \otimes \varrho _{\tilde{k}}$, one can always find such a $\tilde{k}$ that $\left[ \rho _{AB},K_{\tilde{k}}\right]
=0$ holds for all $\tilde{k}$. On the contrary, given an arbitrary $\rho
_{AB}$, if $\left[ \rho _{AB},K_{k_{1}}\right] =0$ for some particular $%
k_{1} $, we can write $$\rho _{AB}=\lambda _{k1}\left\vert k_{1}\right\rangle \left\langle
k_{1}\right\vert \otimes \varrho _{1}+\rho _{k_{1\bot }},$$where $\rho _{k_{1\bot }}$ means that it can be completely expanded in the orthogonal space of $\left\vert k_{1}\right\rangle \left\langle
k_{1}\right\vert $ and $\lambda _{k1}\geq 0$. If $\rho _{AB}$ given in Eq. (4) continues to commuting with $K_{k_{2}}$ with $\left\langle k_{1}\right.
\left\vert k_{2}\right\rangle =0$, one can further write $\rho _{AB}$ as $$\rho _{AB}=\lambda _{k1}\left\vert k_{1}\right\rangle \left\langle
k_{1}\right\vert \otimes \varrho _{1}+\lambda _{k2}\left\vert
k_{2}\right\rangle \left\langle k_{2}\right\vert \otimes \varrho _{2}+\rho
_{k_{1\bot }\cap k_{2\bot }}.$$If we $\left[ \rho _{AB},K_{k_{j}}\right] =0$ holds for all $k_{j}$ such that $\sum_{j}\left\vert k_{j}\right\rangle \left\langle k_{j}\right\vert =%
\mathbf{1}_{m}$, one will draw the conclusion that $$\rho _{AB}=\sum_{j}\lambda _{k_{j}}\left\vert k_{j}\right\rangle
\left\langle k_{j}\right\vert \otimes \varrho _{j},$$which is obviously a classical-quantum state. The proof is completed.$%
\blacksquare$
Next, one will easily find that the quantum correlation measure $\mathcal{Q}%
\left( \rho _{AB}\right) $ satisfies all the good properties that a measure should meet.
\(i) $\mathcal{Q}\left( \rho _{AB}\right) $ *is invariant under local unitary operations.* It is apparent that $I\left( \rho _{AB},O\right) $ is invariant under local unitary operations, which is analogous to the proof in Ref. \[34\].
\(ii) $\mathcal{Q}\left( \rho _{AB}\right) $ *is contractive under competely positive and trace-preserving maps* $\Phi $* on* $B$.* *Since $I\left( \rho _{AB},K_{k}\right) $ given in Eq. (1) is constractive for observable $K_{k}$, the same property is inherited by $%
\sum_{k}I\left( \rho _{AB},K_{k}\right) $. For an optimal set $\left\{
K_{k}\right\} $, $\mathcal{Q}\left( \rho _{AB}\right) \geq $ $%
\sum_{k}I\left( \left( \mathbf{1}_{m}\otimes \Phi \right) \rho
_{AB},K_{k}\right) \geqslant \mathcal{Q}\left( \left( \mathbf{1}_{m}\otimes
\Phi \right) \rho _{AB}\right) .$
\(iii) $\mathcal{Q}\left( \rho _{AB}\right) $ *is reduced to entanglement for pure states.* Because $\mathcal{Q}\left( \rho _{AB}\right) $ is not changed by the local unitary operations, we can safely consider the pure state in the form of Schmidt decomposition which is given by $%
\left\vert \chi \right\rangle _{AB}=\sum_{i=0}^{r-1}\mu _{i}\left\vert
ii\right\rangle _{AB}$ with $\mu _{i}$ the Schmidt coefficients and $r=\min
\{m,n\}$. Substitute $\left\vert \chi \right\rangle _{AB}$ into Eq. (3), one will easily find that $$\begin{aligned}
&&\mathcal{Q}\left( \rho _{AB}\right) =1-\max_{S}\sum_{k=0}^{n-1}\left\vert
\sum_{i,j=0}^{r-1}\mu _{i}\mu _{j}\left\langle ii\right\vert \left(
\left\vert k\right\rangle \left\langle k\right\vert \otimes \mathbf{1}%
_{n}\right) \left\vert jj\right\rangle _{AB}\right\vert ^{2} \notag \\
&&=1-\max_{S}\sum_{k=0}^{n-1}\left\vert \left\langle k\right\vert
\sum_{i=0}^{r-1}\mu _{i}^{2}\left\vert i\right\rangle _{A}\left\langle
i\right\vert \left. k\right\rangle \right\vert ^{2} \notag \\
&&\geqslant 1-\sum_{k=0}^{r-1}\mu _{k}^{4}=1-Tr\varrho _{r}^{2},\end{aligned}$$where $\varrho _{r}$ is the reduced density matrix of $\left\vert \chi
\right\rangle _{AB}$ and the “=” in Eq. (7) can always be satisfied if the optimized set $S=\left\{ \left\vert i\right\rangle \right\} $.
*The almost analytic expression for* $Q\left( \rho _{AB}\right) .$- Based on the previous results, one can say that $\mathcal{Q}\left( \rho
_{AB}\right) $ is a good measure of quantum correlation. In the proceeding part, we will convert the complex optimization question presented in Eq. (3) into an existing and easy question, by which we will give the almost analytic expression of $\mathcal{Q}\left( \rho _{AB}\right) .$
**Theorem 1.**-Let $\left\{ \left\vert i\right\rangle \right\} $ and $%
\left\{ \left\vert j\right\rangle \right\} $ denote two sets of orthonormal bases of the subspace $B$ of the state $\rho _{AB}$, and let the Hermitian matrices $A_{ij}=\left( \mathbf{1}_{m}\otimes \left\langle i\right\vert
\right) \sqrt{\rho _{AB}}\left( \mathbf{1}_{m}\otimes \left\vert
j\right\rangle \right) $, the quantum correlation $\mathcal{Q}\left( \rho
_{AB}\right) $ of $\rho _{AB}$ defined in Eq. (3) can be explicitly given by $$\mathcal{Q}\left( \rho _{AB}\right)
=1-\sum\limits_{i,j=0}^{n-1}\sum_{k=0}^{m-1}\left\vert \lambda
_{k}^{ij}\right\vert ^{2},$$where $\lambda _{k}^{ij}$ means the $k$th joint eigenvalue of $A_{ij}$ which is defined by $\left( U_{o}A_{ij}U_{o}^{\dag }\right) _{kk}$ with $U_{o}$ the joint diagonalizer of all the $A_{ij}$.
**Proof.** From Eq. (3), one will directly arrive at$$\begin{aligned}
&&\mathcal{Q}\left( \rho _{AB}\right) =\min_{S}\sum_{k=0}^{m-1}\left[ Tr\rho
_{AB}K_{k}^{2}-Tr\sqrt{\rho _{AB}}K_{k}\sqrt{\rho _{AB}}K_{k}\right] \notag
\\
&&=1-\max_{S}\sum_{k=0}^{m-1}Tr\sqrt{\rho _{AB}}K_{k}\sqrt{\rho _{AB}}K_{k}.\end{aligned}$$Substitute $K_{k}=\left\vert k\right\rangle \left\langle k\right\vert
\otimes \mathbf{1}_{n}$ and any orthonormal bases of subsystem $B$ into Eq. (9), it follows that $$\begin{aligned}
\mathcal{Q}\left( \rho _{AB}\right)
&=&1-\max_{S}\sum\limits_{i,j=0}^{n-1}\sum_{k=0}^{m-1}Tr\sqrt{\rho _{AB}}%
\left( \left\vert k\right\rangle \left\langle k\right\vert \otimes
\left\vert i\right\rangle \left\langle i\right\vert \right) \notag \\
&&\times \sqrt{\rho _{AB}}\left( \left\vert k\right\rangle \left\langle
k\right\vert \otimes \left\vert j\right\rangle \left\langle j\right\vert
\right) \notag \\
&=&1-\max_{S}\sum\limits_{i,j=0}^{n-1}\sum_{k=0}^{m-1}\left\vert
\left\langle k\right\vert A_{ij}\left\vert k\right\rangle \right\vert ^{2}
\notag \\
&=&1-\max_{U}\sum\limits_{i,j=0}^{n-1}\sum_{k=0}^{m-1}\left\vert
UA_{ij}U^{\dag }\right\vert _{kk}^{2},\end{aligned}$$with $$A_{ij}=\left( \mathbf{1}_{m}\otimes \left\langle i\right\vert \right) \sqrt{%
\rho _{AB}}\left( \mathbf{1}_{m}\otimes \left\vert j\right\rangle \right) .$$Thus, our calculation of the quantum correlation is directly changed into the joint approximate diagonalization (JAD) of the series of matrices $%
A_{ij} $ \[48,49\]. Let $U_{o}$ be the joint diagonalizer of all the $A_{ij}$ such that the optimal value of Eq. (10) can be attained, and assume $\lambda
_{k}^{ij}=\left( U_{o}A_{ij}U_{o}^{\dag }\right) _{kk}$, one will easily find that the final expression of $\mathcal{Q}\left( \rho _{AB}\right) $ can be written as Eq. (8). It is obvious that if $[A_{ij},A_{kl}]=0$ holds for all $A$, the question can be exactly solved. Of course, that these $A_{ij}$ commute with each other is just a sufficient condition for the exact solution of Eq. (8), which will be seen from our latter examples. In addition, the number of the matrices that need to be JAD can be reduced further based on the Appendix. $\blacksquare$
Next we will briefly analyze why we say $\mathcal{Q}\left( \rho _{AB}\right)
$ given in Eq. (8) is effective and almost analytic. At first, we would like to claim that the effectivity of our result is completely attributed to the well developed technique on the JAD (throughout this Letter, we especially mean the Jacobi algorithm for JAD \[48,49\]), since we have succeeded in converting the original particular optimization into such an existing JAD question. In particular, we emphasize that by these well developed techniques, especially the Jacobi algorithm which is also used to diagonalize a single matrix, the JAD can be solved as steadily, reliably, fast and perfectly as the diagonalization of a single matrix \[49,50\]. From the point of practical applications of view, we would like to say that our result in Eq. (8) is almost analytic, which could be intuitive if $\lambda
_{k}^{ij}$ were the exact eigenvalues of some particular matrix. In the researches on the quantum correlation including quantum entanglement and quantum discord, it is usual to accept that the result is analytic, if it can be given by some eigenvalues of any given matrices. However, in practical operations, these eigenvalues are usually calculated (especially for large matrices) by computers with some default precision (or machine precision). The technique of JAD is completely the generalization of that for a single matrix. For example, the Jacobi algorithm has the complete same principle as that for a single matrix \[48,50\]. So the JAD is completely on the same level as the diagonalization of a single matrix, so is their precision, efficiency, and so on. Quantitively, the JAD of Eq. (8) needs at most $\frac{m(m-1)n^2}{2}$ Givens rotations for one ergodicity of the entries of the matrix in Jacobi algorithm \[49\], while the diagonalization of $\rho_{AB}$ needs $\frac{mn(mn-1)}{2}$ rotations for one ergodicity \[50\]. Thus from the practical applications, we can think our result is almost analytic in the sense of the “analytic diagonalization of a single matrix $%
\rho_{AB}$”.
*Relation with quantum metrology.-* Before proceeding, we will show that our quantum correlation in Definition 1 connects some quantum metrology scheme in an interesting way. Let $\left\{ \left\vert k\right\rangle
\right\} $ denote the group of optimal orthonormal set of projectors that achieves the exact value of $\mathcal{Q}\left( \rho _{AB}\right) $. Assume the state $\rho _{AB}$ is a probing state with subsystem $A$ undergoing a unitary transformation which endows some unknown phases $\varphi _{k}$ on $%
\rho _{AB}$ by $\rho _{\vec{\varphi}}=e^{-iH(\vec{\varphi})}\rho _{AB}e^{iH(%
\vec{\varphi})}$ with $H(\vec{\varphi})=\sum_{k}\varphi _{k}\left\vert
k\right\rangle \left\langle k\right\vert \otimes \mathbf{1}_{n}$. We aim to estimate these $\varphi _{k}$ one by one by $N$ runs of detection with high precision quantified by the uncertainty of the estimated phase $\varphi
_{k}^{est}$ $\left( \delta \varphi _{k}\right) ^{2}=\left\langle \frac{%
\varphi _{k}^{est}}{\partial \left\langle \varphi _{k}^{est}\right\rangle
/\partial \varphi _{k}}-\varphi _{k}\right\rangle $ \[51,52\]. This variance $%
\delta \varphi _{k}$, for an unbiased estimator, is bounded by the quantum Cramer-Rao bound $\left( \delta \varphi _{k}\right) ^{2}\geq \frac{1}{NF_{Qk}%
}$ that can be attained asymptotically by the projective measurements in the basis of the symmetric logarithmic derivative operator and the maximum likelihood estimation, where $F_{Qk}$ is the quantum Fisher information subject to the phase $\varphi _{k}$ \[51-54\]. What we would like to emphasize is that $F_{Qk}=-Tr\left[ \sqrt{\rho _{AB}},K_{k}\right] ^{2}$, so one can easily find that $\sum_{k}\frac{1}{N\left( \delta \varphi _{k}\right) ^{2}}%
\leq \sum_{k}F_{Qk}=2\mathcal{Q}\left( \rho _{AB}\right) $. That is, our quantum correlation measure characterizes the contributions of all the inverse variances of the estimated phases.
*The applications.-*From the following, one will find that $\mathcal{Q%
}\left( \rho _{AB}\right) $ for some states can be analytically solved, whilst these examples will demonstrate the effectiveness of the JAD method in the calculation for high-dimensional systems and illustrate the perfect consistency between the strictly analytic solutions and the almost analytic ones obtained by the JAD method.
{width="0.6\columnwidth"}
*(a) Qubit-qudit states.-*As a comparison with the previous jobs, we will first consider the quantum correlation* *of a $\left( 2\otimes
d\right) $-dimensional state. For such a state $\rho _{AB}$, Eq. (3) can be rewritten as $$\begin{aligned}
\mathcal{Q}\left( \rho _{AB}\right) &=&-\frac{1}{2}\min_{S}\sum_{k=0}^{1}Tr%
\left[ \sqrt{\rho _{AB}},K_{k}\right] ^{2} \notag \\
&=&-\min_{S}Tr\left[ \sqrt{\rho _{AB}},K_{0}\right] ^{2}.\end{aligned}$$Since any pure state can be expanded in the Bloch representation, one can always write $K_{0}$ as$$K_{0}=\frac{1}{2}\left( \mathbf{1}_{2}+\vec{n}\cdot \vec{\sigma}\right)
\otimes \mathbf{1}_{d}$$with $\sum n_{i}^{2}=1$. Substitute Eq. (13) into Eq. (12), one will arrive at $$\begin{aligned}
\mathcal{Q}\left( \rho _{AB}\right) &=&\frac{1}{2}-\frac{1}{2}\max_{\vec{n}%
}\sum_{ij}Trn_{i}T_{ij}n_{j} \notag \\
&=&\frac{1}{2}\left( 1-\upsilon _{\max }\right), \end{aligned}$$where $\upsilon _{\max }$ is the maximal eigenvalue of the matrix $T$ with $$T_{ij}=Tr\sqrt{\rho _{AB}}\left( \sigma _{i}\otimes \mathbf{1}_{n}\right)
\sqrt{\rho _{AB}}\left( \sigma _{j}\otimes \mathbf{1}_{n}\right) .$$ Eq. (14) happened to be the half of that in Ref. \[34\].
*(b) $\left( 3\otimes 3\right) $ -dimensional PPT states.*-Let’s consider such a PPT state given by \[55\] $$\rho _{PPT}=\frac{2}{7}\left\vert \Phi \right\rangle _{3}\left\langle \Phi
\right\vert +\frac{\alpha }{7}\rho _{+}+\frac{5-\alpha }{7}\rho _{-},\alpha
\in \lbrack 2,4],$$where $\left\vert \Phi \right\rangle _{m}=\frac{1}{\sqrt{m}}%
\sum_{k=0}^{m-1}\left\vert kk\right\rangle $ and $\rho _{+}=\frac{1}{3}%
\sum\limits_{k=0}^{2}\left\vert k,k\oplus 1\right\rangle \left\langle
k,k\oplus 1\right\vert $ and $\rho _{-}=\frac{1}{3}\sum\limits_{k=0}^{2}%
\left\vert k\oplus 1,k\right\rangle \left\langle k\oplus 1,k\right\vert $ with "$\oplus "$ the modulo-3 addition. Note that, only when $\alpha \in
(3,4]$, $\rho _{PPT}$ is entangled. If $\alpha \leq 3$, $\rho _{PPT}$ is separable. But if $4<\alpha \leq 5$, $\rho _{PPT}$ is not a PPT state, but a free entangled state. For integrity, we also consider this type free entangled states here. It is interesting that, $\mathcal{Q}\left( \rho
_{PPT}\right) $ can be analytically solved for $\alpha \in \lbrack 2,5]$, which is given by$$\mathcal{Q}\left( \rho _{PPT}\right) =\left\{
\begin{array}{cc}
\frac{21-\sqrt{6\left( 5-\alpha \right) }-\sqrt{6\alpha }-3\sqrt{\alpha
\left( 5-\alpha \right) }}{31.5}, & 2\leq \alpha \leq N_{T} \\
\frac{4}{21}, & N_{T}<\alpha \leq 5%
\end{array}%
\right. ,$$with $N_{T}=\frac{15+\sqrt{136\sqrt{94}-1307}}{6}=3.066885$. The numerical results based on our theorem 1 is plotted in Fig. 1, which shows the perfect consistency between our theorem 1 and the strict analytic expression. In particular, we can analytically find the sudden change point of quantum correlation near the critical point of the separable state and the bound entangled state.
{width="1\columnwidth"}
*(c) Werner states and Isotropic states in* $\left( m\otimes m\right)
$ dimension*.*-Besides the above examples, our quantum correlation measure $\mathcal{Q}\left( \cdot \right) $ for both the $\left( m\otimes
m\right) $ -dimensional Isotropic states and Werner states \[53\] can be analytically calculated. Thus they can serve as important examples that show the effectivity of $\mathcal{Q}\left( \cdot \right) $ for larger systems. The Werner state can be written as$$\rho _{W}=\frac{m-x}{m^{3}-m}\mathbf{1}_{m^{2}}+\frac{mx-1}{m^{3}-m}V,x\in
\lbrack -1,1],$$with $V=\sum_{kl}\left\vert kl\right\rangle \left\langle lk\right\vert $ the swap operator. This state has no quantum correlation if and only if $x=\frac{%
1}{m}$. Through a simple algebra, one can have the analytic expression of the quantum correlation as follows. $$\mathcal{Q}\left( \rho _{W}\right) =\frac{m-x-\sqrt{m^{2}-1}\sqrt{1-x^{2}}}{%
2(1+m)}.$$From Eq. (19), one will also find that $\mathcal{Q}\left( \rho _{W}\right)
=0 $ for $x=\frac{1}{m}$. Analogously, we also plot $\mathcal{Q}\left( \rho
_{W}\right) $ based on Eq. (19) and Eq. (8), respectively, in Fig. 2 (a) which shows the perfect consistency. The isotropic state can be given by$$\rho _{I}=\frac{1-x}{m^{2}-1}\mathbf{1}_{m^{2}}+\frac{m^{2}x-1}{m^{2}-1}%
\left\vert \Phi \right\rangle \left\langle \Phi \right\vert ,x\in \lbrack
0,1],$$with $\left\vert \Phi \right\rangle =\frac{1}{\sqrt{m}}\sum_{k=0}^{m-1}\left%
\vert kk\right\rangle $. Based on our definition, we can analytically obtain$$\mathcal{Q}\left( \rho _{I}\right) =\frac{1-2\sqrt{m^{2}-1}\sqrt{x(1-x)}%
+\left( m^{2}-2\right) x}{m(1+m)}.$$It is obvious that $x=\frac{1}{^{m^{2}}}$ will lead to $\mathcal{Q}\left(
\rho _{I}\right) =0$, which is consistent to Ref. \[15\]. As a comparison, we plot $\mathcal{Q}\left( \rho _{I}\right) $ given by Eq. (8) and Eq. (21), respectively, in Fig. 2 (b) which shows the perfect consistency again.
*Conclusions and Discussion.*-We have presented a new definition of quantum correlation for any bipartite quantum system with some good properties. In particular, this definition can lead to an effective and even almost analytic expression for any states. As applications, we have found that the quantum correlations of many quantum states can be strictly analytically solved, which also provides a direct support for the effectivity of our theorem.
Finally, we would like to emphasize that the JAD technique plays an important role in our job. Whether it can induce other contributions to the relevant researches such as quantum correlation of other forms, quantum entanglement measure etc. is worthy of our forthcoming efforts.
*Acknowledgements*-This work was supported by the National Natural Science Foundation of China, under Grants No.11375036 and No. 11175033, and by the Fundamental Research Funds of the Central Universities, under Grant No. DUT12LK42.
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*Appendix: Reduction of the number of the matrices in JAD.-*In fact, the number $N$ of the jointly diagonalized matrices given in theorem 1 can be reduced further, which will directly improve the efficiency. This can be seen from what follows. From Eq. (10), one can rewrite $\sum%
\limits_{i,j=0}^{n-1}\sum_{k=0}^{m-1}\left\vert UA_{ij}U^{\dag }\right\vert
_{kk}^{2}$ as$$\sum\limits_{i,j=0}^{n-1}\sum_{k=0}^{m-1}\left\vert UA_{ij}U^{\dag
}\right\vert _{kk}^{2}=\sum_{k=0}^{m-1}\left[ \left( U\otimes U\right)
S\left( U^{\dag }\otimes U^{\dag }\right) \right] _{kk}^{kk},$$where $S=\sum_{i,j=0}^{n-1}A_{ij}\otimes A_{ij}^{\dag }$. Based on the Kronecker Approximation technique \[56\], one can always express $S$ as $$S=\sum_{k=0}^{r-1}B_{k}\otimes B_{k}^{\dag },$$where $r$ is the rank of the matrix $\tilde{M}=V_{s}(SV_{s})^{T_{2}},$ the superscript $T_{2}$ denotes partial transposition on the second space \[57\], $%
V_{s}$ is the swap operator defined by $V_{s}\left\vert \varphi
\right\rangle \left\vert \psi \right\rangle =\left\vert \psi \right\rangle
\left\vert \varphi \right\rangle $ for any two states $\left\vert \psi
\right\rangle $ and $\left\vert \varphi \right\rangle $ with the same dimension. If the singular value decomposition of $\tilde{M}$ is given by $$\tilde{M}=U\Sigma V^{\dagger }=\sum_{i=0}^{r-1}\sigma _{i}u_{i}v_{i}^{\dag },$$where $u_{i}$ is the $i$th columns of the unitary matrix $U$; $\Sigma $ is a diagonal matrix with elements $\sigma _{i}$ in the decreasing order, one will easily find that $Vec(B_{k})=\sqrt{\sigma _{k}}u_{k}$. Thus the $%
\mathcal{Q}\left( \rho _{AB}\right) $ can be rewritten as$$\mathcal{Q}\left( \rho _{AB}\right)
=1-\max_{U}\sum_{j=0}^{r-1}\sum_{k=0}^{m-1}\left\vert UB_{j}U^{\dag
}\right\vert _{kk}^{2}$$with $\lambda _{k}^{t}$ the $k$th joint eigenvalue of $B_{t}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider a model for a two dimensional electron gas moving on a kagomé lattice and locally coupled to a chiral magnetic texture. We show that the transverse conductivity $\sigma_{xy}$ does not vanish even if spin-orbit coupling is not present and it may exhibit unusual behavior. Model parameters are the chirality, the number of conduction electrons and the amplitude of the local coupling. Upon varying these parameters, a topological transition characterized by change of the band Chern numbers occur. As a consequence, $\sigma_{xy}$ can be quantized, proportional to the chirality or have a non monotonic behavior upon varying these parameters.'
author:
- 'M. Taillefumier$^{1,2}$, B. Canals$^{1}$, C. Lacroix$^1$, V. K. Dugaev$^{1,2,3}$, and P. Bruno$^2$'
title: Anomalous Hall Effect due to the spin chirality in the Kagomé lattice
---
Introduction
============
In ferromagnetic systems, there are two contributions to the transverse resistivity $\rho _{xy\text{ }}$: one is due to the usual Lorentz force acting on the electrons when a magnetic field is applied, $R_{0}B$, the second one, $R_{s}M$, is proportional to the magnetization of the ferromagnet. This is called the anomalous Hall effect.
The origin of this anomalous Hall effect has long been controversial and both extrinsic (impurities) or intrinsic mechanisms were discussed. Karplus and Luttinger[@Karplus1954] proposed that this effect is a consequence of spin-orbit interaction in metallic ferromagnets. Then it was argued that impurities give the main contribution to the anomalous Hall effect and usually two mechanisms, both due to spin-orbit coupling, contribute: one is known as side-jump mechanism[@Berger1970; @Berger1972] and it predicts that the Hall resistivity $R_{s\text{ }}$ is proportional to the longitudinal resistivity $\rho $. The second one[@Smit1954; @Smit1958] is the skew-scattering mechanism, which gives a contribution proportional to $\rho ^{2}$.
In the recent years several groups measured anomalous Hall effect in various systems which cannot be attributed to usual mechanisms (skew-scattering or side-jump). A new intrinsic mechanism, related to non-collinear spin configuration, with a ferromagnetic component, was first proposed for manganites[@Matl1998; @Ye1999; @Chun2000]: in these systems deviation from collinearity is a consequence of the competition between double exchange, superexchange and spin-orbit interactions. Then it was proposed that a similar mechanism works in spin glass systems where spin configuration is highly non coplanar[@Tatara2002; @Kawamura2003]: in the weak coupling limit it was shown that the Hall resistivity is proportional to the uniform chirality parameter with a sign which depends on the details of the band structure. This chirality parameter was defined by Tatara and Kawamura[@Tatara2002] for three spins at sites i, j, k as a scalar quantity: $%
\chi =\langle \mathbf{S}_{i}.(\mathbf{S}_{j}\times
\mathbf{S}_{k})\rangle $, being non-zero if the spins are non-coplanar. Very recently several groups confirmed the existence of such a contribution in typical AuFe spin glasses[@Pureur2004; @Taniguchi2004].
In non-disordered systems, the same mechanism can work as soon as the ordered magnetic structure is non-coplanar. This is realized in several pyrochlore compounds such as Nd$_{2}$Mo$_{2}$O$_{7}$ for which Taguchi et al[@Taguchi2001] proposed that an anomalous contribution to the Hall effect is related to the umbrella structure of both Nd and Mo moments in the long range ordered phase. Detailed studies of this system and other pyrochlores (Sm$_{x}$Y$_{1-x}$Mo$_{2}$O$_{7}$, Nd$_{x}$Y$_{1-x}$Mo$_{2}$O$_{7}$) have been performed[@Taguchi2001; @Taguchi2003; @Yasui2003] leading to some controversy: using neutron diffraction experiments in magnetic field it is possible to calculate the T and H dependence of both Mo and Nd chiral order parameter and in Ref. it was concluded that the chiral mechanism alone cannot explain the T-dependence of the Hall coefficient in pyrochlores.
From the theoretical point of view, this mechanism is often called ”Berry phase contribution” because a non-vanishing spin chirality is associated with a non-vanishing Berry phase for conduction electrons coupled via local Hund’s exchange interaction to the spins. Ohgushi et al[@Ohgushi2000] calculated the Hall effect in a Kagomé lattice with a non-coplanar long range spin structure. They have shown that, in the adiabatic limit, when the electron conduction spins are aligned with localized spins at each site of the lattice (this corresponds to infinite Hund’s coupling), the Berry phase contribution to Hall conductivity is quantized for some values of the band filling. In this paper, we study a model which extrapolates between this strong coupling limit and the weak coupling case studied by Tatara and Kawamura[@Tatara2002]. We show also that the Berry phase contribution does not depend only on the chirality, but also on the strength of the local Hund’s coupling and on the band filling.
Model
=====
Hamiltonian
-----------
The aim of this work is to show how the transport properties of electrons are influenced by two different contributions. First, the electrons are restricted to move on a lattice and are therefore experiencing its geometry. This results in a peculiar band structure. Second, each electron has its spin locally coupled to a given distribution of magnetic moments on each site of the lattice.
Both effects are taken into account in the following Hamiltonian: $$\mathcal{H}=\sum_{\langle i,j\rangle ,\sigma }t_{ij}\left( c_{i\sigma
}^{\dagger }c_{j\sigma }+h.c.\right) -J\sum_{i}c_{i\alpha }^{\dagger }\left(
\mathbf{\sigma }_{\alpha \beta }\cdot \mathbf{S}_{i}\right) c_{i\beta }.
\label{eq:1}$$ The first term describes the electrons moving on the lattice: $t_{ij}$ is the hopping integral between two neighboring sites $i$ and $j$; $c_{i\sigma }^{\dagger }$ and $c_{i\sigma }^{{}}$ are the creation and annihilation operators of an electron with spin $\sigma $ on the site $i$. The second part of the Hamiltonian couples the electron spin to a local moment on each site.
The coupling constant to each local moment ${\bf S}_i$ is $J$, and these moments are treated below as classical variables. $\bsig
_{\alpha \beta }$ are the Pauli matrices. The underlying lattice is the Kagomé lattice, depicted in Fig. \[fig:1\]. It is a two dimensional tiling of corner sharing triangles, described as a triangular lattice of triangles throughout this article.
This Hamiltonian has already been discussed in the limit of $J\to
\infty $ by Ohgushi et al.[@Ohgushi2000] In this limit the two $\sigma =\uparrow ,\downarrow $ bands are infinitely splitted and the model describes a fully polarized electron gas subject to a modulation of a fictitious magnetic field, corresponding to the molecular field associated with the magnetic texture. The comparison with the present work will be done later on. In a general case of finite $J$, the calculations must be done numerically. It is worth noting that the sign of $J$ is unimportant in this classical treatment since changing $J$ to $-J$ is equivalent to an exchange of $\uparrow$ and $\downarrow$ spin states.
![(a) The Kagomé lattice is described as a triangular lattice of triangles. Its Bravais vectors are $a_1=(1,\, 0)$, $a_2=(-1/2 , \,
\protect\sqrt{3}/2)$, and $a_3=-(1/2 , \, \protect\sqrt{3}/2)$, connecting respectively, the points $A$ to $B$, $B$ to $C$, and $C$ to $A$. (b) The first Brillouin zone is an hexagon with the corners located at $k=\pm (2\protect\pi/3)\, a_1$, $k=\pm (2\protect\pi/3)\, a_2$ and $k=\pm (2\protect\pi/3)\, a_3$. (c) The umbrella structure on the triangular cell of the kagomé lattice.[]{data-label="fig:1"}](fig1){width="8cm"}
Parameters and motivation
-------------------------
We consider the spin texture as a ’parameter’ of the model. The local moments ${\bf S}_i$ are described classically, and the chosen magnetic phase is periodic, thus allowing to work in the reciprocal space. The choice of the magnetic arrangement is motivated by two reasons. First, we are interested in describing the anomalous transport properties of the type already addressed in previous works (see Refs. \[\]) in which it has been argued that the transverse conductivity may be related to the chirality of the magnetic phase. Second, an unusual mechanism of the AHE has recently been proposed, based on an unconventional chiral magnetic ordering due to geometrical frustration[@Taguchi2003; @Kageyama2001; @Katsufuji2000; @Yoshii2000] but with controversial interpretations. We therefore consider the present model as a simple model to address these problems and clearly identify each relevant contribution.
We consider the magnetic phase which has been obtained by studying a pure spin model with the anisotropic Dzyaloshinskii-Moriya interactions on the Kagomé lattice[@Elhajal2002]. It consists of an umbrella of three spins per unit cell of the Kagomé lattice (see Fig. \[fig:1\]). Each umbrella can be described by the spherical coordinates of the three spins $(\pi /6,\, \theta )$, $(5\pi /6, \, \theta )$ and $(-\pi /2,\, \theta )$. The angle $\theta $ ranges from $0$ (all the spins are perpendicular to plane, and the corresponding ordering is ferromagnetic) to $\pi $. It is worth noting that the usual three sublattice planar magnetic phase belonging to the ground state manifold of the kagomé antiferromagnet, and commonly denoted as the “$q = 0$ phase” corresponds to $\theta = \pi / 2$. In between, all phases are possible and chiral, where chirality is defined as the mixed product of three spins on a plaquette $$\chi _{ijk}={\bf S}_{i}\cdot ({\bf S}_{j}\times {\bf S}_{k})
=\frac{3\sqrt{3}}{2}\cos \theta \sin ^{2}\theta .
\label{eq:3}$$All these phases ($\theta \in [ 0, \pi ]$) are therefore translation-invariant but do not have time reversal symmetry.
We consider the local spins as classical spins with length M and we introduce in eq. \[eq:1\] an effective coupling constant $J_{0} = J M $ which allows to rewrite the hamiltonian as $$\mathcal{H} =
\sum_{\langle i,j\rangle ,\sigma }
t_{ij}\left( c_{i\sigma}^{\dagger }c_{j\sigma }+h.c.\right)
- J_{0} \sum_{i}c_{i\alpha }^{\dagger }
\left(\bsig_{\alpha \beta } \cdot
\mathbf{n}_{i}\right) c_{i\beta }$$ where $\mathbf{n}_{i}$ is a unit vector collinear to the local moment ${\bf S_i}$.
Setting $ t=| t_{ij} |$ as the energy unit, our free parameters are the angle $%
\theta $, parametrizing the chirality of the magnetic texture, and the value of $ J_{0}$ . The Fermi level, through the filling factor $p$ of the bands, can also be varied. Each of these variables gives a different contribution as will be discussed in the next sections.
Physical quantities
-------------------
Before calculating any physical observables, we have to diagonalize the Hamiltonian of Eq. \[eq:1\]. For this purpose, the Hamiltonian is rewritten in the reciprocal space as $$\mathcal{H}=\sum_{\mathbf{k}}\Psi _{\mathbf{k}}^{\dagger }h_{\mathbf{k}}\Psi
_{\mathbf{k}}\,+\,h.c.,$$ with $\Psi _{\mathbf{k}}=(c_{A{\bf k}\uparrow },c_{B{\bf k}\uparrow
},c_{C{\bf k}\uparrow },c_{A {\bf k}\downarrow },c_{B {\bf k}\downarrow
},c_{C{\bf k}\downarrow })$, $A,B$ and $C$ are the corners of the kagomé lattice unit cell (Fig. \[fig:1\]-(a)) and $$c_{A{\bf k}\sigma}=\sum_j c_{A,j,\sigma} \,e^{i {\bf k}\cdot
{\bf r}_{ij} },$$ is the Fourier transform of $c_{A,j,\sigma}$ and $r_{Aj}=r_A + R_j$ with $R_j$ the lattice vector and $r_A$ the position of the moment ${\bf S}_A$ in the unit cell. $h_{\mathbf{k}}$ is a $6\times 6$ matrix given by
$$h_{\bf k}=\left(
\begin{array}{cccccc}
-J_{0} \cos\theta &p^1_k &p^3_k &-J_{0} \sin\theta e^{i \frac{\pi}{6}} &0 &0 \\
p^1_k &-J_{0} \cos\theta &p^2_k &0 &-J_{0} \sin\theta e^ {i \frac{5\pi}{6}} &0 \\
p^3_k &p^2_k &-J_{0} \cos\theta &0 &0 &i J_{0} \sin\theta \\
-J_{0} \sin\theta e^{-i \frac{\pi}{6}} &0 &0 &J_{0} \cos\theta &p^1_k &p^3_k \\
0 &-J_{0} \sin\theta e^{-i\frac{5\pi}{6}} &0 &p^1_k &J_{0} \cos\theta &p^2_k \\
0 &0 &-i J_{0} \sin\theta &p^3_k &p^2_k &J_{0} \cos\theta\\
\end{array}\right),
\label{eq:6}$$
with $t_{ij}=t$ and $p_{\mathbf{k}}^{i}=2t\cos (\mathbf{k}\cdot
\mathbf{a}_{i})$.
To calculate the Anomalous Hall Effect, we use the expression of the off-diagonal conductivity using the following Kubo formula in the limit of disorder-free electron gas[@Onoda2002] $$\begin{aligned}
\label{eq:7}
\sigma_{xy}(\omega) &=&\frac{e^2\hbar}{S}\sum_{n\neq m}\sum_{ \mathbf{k}} \left(f_{n\mathbf{k}%
}-f_{m\mathbf{k}}\right)\nonumber\\
&\times&\frac{\langle n,\mathbf{k}|v_x|m,%
\mathbf{k}\rangle \langle n,\mathbf{k}|v_y|m,\mathbf{k}\rangle} {%
\left(\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}}\right)
\left(\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}%
}-\omega\right)},\end{aligned}$$ where $S$ is the surface of the unit cell, $\varepsilon_{n\mathbf{k}}$ is the eigenvalue of matrix (\[eq:6\]) corresponding to the energy of $n$th band for the wave vector $\mathbf{k}$, with the eigenvector $|n,\mathbf{k}\rangle$. Here $v_i=\frac1\hbar\frac{\partial h_{%
\mathbf{k}}}{\partial k_i}$ is the velocity of electrons ($i=x,y,z$). Taking the static limit of $\omega\rightarrow 0$, Eq. (\[eq:7\]) becomes $$\begin{aligned}
\label{eq:8}
\sigma_{xy}&=&\frac{e^2\hbar}{S} \sum_{n,\mathbf{k}}f_{n,\mathbf{k}} \sum_{m\neq n}\frac{%
(v_x)^{\mathrm{nm}}(v_y)^{\mathrm{nm}}-(v_x)^{\mathrm{mn}}(v_y)^{\mathrm{mn}}%
}{(\varepsilon_{n, \mathbf{k}}-\varepsilon_{m,\mathbf{k}})^2} \notag \\
&=& \frac{e^2}{\hbar S}\sum_{n,\mathbf{k}}f_{n,\mathbf{k}}\left[\mathbf{\nabla}_{\mathbf{k}%
}\times \mathbf{\mathcal{A}}_{n,\mathbf{k}}\right]_z,\end{aligned}$$ where $\mathbf{A}_{n\mathbf{k}} = -i \langle n \mathbf{k}|\nabla_{\mathbf{k}%
}| n \mathbf{k} \rangle$ is the geometric vector potential and $%
f_{n\mathbf{k}}$ is the Fermi-Dirac distribution function.
The expression (\[eq:8\]) is similar to that obtained by Thouless *et al.*[@Thouless1982] in the context of the Quantized Hall effect in two dimensions. In particular, when the band is completely filled, the circulation of $\mathbf{A}_{n\mathbf{k}}$ or the flux of Berry curvature defined as $\mathbf{\Omega}_{n\mathbf{k}}=\mathbf{\nabla}_{\mathbf{k}}\times
\mathbf{A}_{n\mathbf{k}}$ over the first Brillouin zone is equal to $2\pi i$ times an integer[@Thouless1982; @Thouless1983] called the Chern number $\nu_n$. Expression (\[eq:8\]) of the off-diagonal conductivity finally becomes $$\begin{aligned}
\sigma_{xy} = \frac{e^2}{\hbar S} \sum_{n\mathbf{k}} f_{n\mathbf{k}}\mathbf{\Omega}%
_{n\mathbf{k}} = \frac{e^2}{h} \sum_n\nu_n, \label{eq:9}\end{aligned}$$ where $\nu_n=\frac{1}{2\pi}\int f_{n\mathbf{k}} \Omega_{n\mathbf{k}}
d^2 {\bf k}$. As we shall see, this implies that the Hall conductivity is quantized when the Fermi level lies in a gap.
The Chern numbers have the following properties: (i) the sum over all bands is equal to $0$, (ii) if for some peculiar values of parameters of the model, the energy bands $m$ and $n$ cross each others at some point of the Brillouin zone, their respective Chern number obey the following conservation rule $$\label{eq:10}
(\nu_m+\nu_n)_b=(\nu_m+\nu_n)_a$$ where the indices $b,a$ refer to the values of the Chern number before and after the bands are touching each other.[@Avron1983] We will check each of these conservation rules in the next sections.
It should be noticed that the sum over $\mathbf{k}$ in Eq. (\[eq:9\]) runs over all occupied states. This may seem unusual as in metallic systems we expect that only the quasiparticles near the Fermi surface contribute to transport properties. However, it has been recently shown by Haldane[@Haldane2004] that Eq. (\[eq:9\]) can be reconciled with this point of view and reduced to a sum over the states near the Fermi energy.
Numerical results {#sec:numerical-results}
=================
This section is divided in two parts. In the first one, the band structure is calculated for different angles $\theta$ and coupling constants $J_0$. The evidences for two different regimes of couplings are given, illustrated by the variation of Chern numbers. In the second one, we focus on the transverse conductivity at zero temperature. It is shown that the chirality may be a relevant quantity for describing the transverse conductivity ($\sigma \propto \chi$) in some ranges of coupling only for a half-filled band.
Band structure and associated Chern numbers
-------------------------------------------
The energy spectrum of the Kagomé lattice in the absence of exchange interaction, is characterized by one flat band at $E=2t$ and two dispersive bands, which touch each other at the ${\bf
k}$-point $K$ of the Brillouin zone as shown in Fig. \[fig:2\]. This peculiarity has already been addressed in the context of spin models as well as for the metallic systems[@Mielke1991].
Once the coupling constant $J_{0}$ is nonzero, the energy spectrum splits in two parts due to the spin dependant potential. For very large values of $J_{0}$, the spectrum is divided into two groups of three bands, which is the case studied by Ohgushi [*et al.*]{}[@Ohgushi2000]. In between, for nonzero but not too large values of $J_{0}$, the spectrum can only be computed numerically, except for the special cases of $\theta = 0$ or $\pi$, or for general $\theta$ at high symetry points. For these intermediate values of $J_{0}$, the splitting of the spectrum depends qualitatively on two mechanisms.
First, the coupling $J_0$ gradually separates each group of three bands taking them degenerate for $J_{0}=0$ to fully separated for infinite $J_{0}$. Second, within each group of three bands, point like degeneracies are lifted (see Fig. \[fig:2\], points $\Gamma$ and $K$) when switching on $J_{0}$, and finally restored for $J_0 \to \infty$ (see Ref. ). The numerical calculation of the spectrum shows that it is either gapless for small values of $J_{0}$, or has gaps for higher values.
When gaps open, it always occurs at the $M$ point of the Brillouin zone. This allows to compute analytically the critical value of the coupling as a function of chirality parameterized by the angle $\theta$,
![ (a) Energy spectrum of (\[eq:5\]) calculated for $J_{0}=0$. Each band is twice degenerate due to the spin degeneracy. (b) Energy spectrum calculated for $J_{0}=t$ and $\protect\theta=\protect\pi/3$. (c) Energy spectrum calculated for $J_{0}=2 t$ and $\protect\theta=%
\protect\pi/3$. The critical value $J_c$ is equal to $4t/\protect\sqrt{7}%
\approx 1.51\ t$[]{data-label="fig:2"}](fig2){width="8cm"}
$$\label{eq:11}
J_c(\theta)=\pm\frac{2}{\sqrt{1+3\cos^2\theta}}.$$
Using Eq. (\[eq:11\]) we can distinguish between two different regimes (Fig. \[fig:2\]) depending on the value of $J_{0}$ as compare to $J_c(\theta)$. These regimes are characterized by a particular organization of the band structure as shown in Figs. \[fig:2\]-(b) for $J_{0}=t$ and \[fig:2\]-(c) for $J_{0}=2 t$ and different Chern numbers. For $J_{0}>J_c(\theta)$, the Chern numbers associated to each band are given by $-1,\, 0,\, 1,\, 1,\, 0,\, -1$ from the lower to the upper band. Those numbers were obtained by Ohgushi *et al.*[@Ohgushi2000] in the limit of $J_0\to \infty$. When $
J_{0}<J_c$, the Chern numbers associated to this configuration are $-1,\, 3,\, -2,\, -2,\, 3,\, -1$ given in the same order as in the previous case. The global sum rule $\sum_n \nu_n = 0$ is always verified as well as the local one around the critical coupling where two pairs of bands are crossing each other. It reads as $$\nu_2 + \nu_3 = 1 \quad\mathrm{{and}\quad \nu_4 + \nu_5 = 1 }$$ below and above the critical coupling $J_c$, respectively. The corresponding phase diagram is shown on figure \[fig:3\].
Off-diagonal conductivity at $T=0$
----------------------------------
The off-diagonal conductivity is computed using Eq. (\[eq:7\]) at $T=0$ for different values of the chirality $\theta$ and the amplitude of the exchange interaction $J_{0}$.
![Off-diagonal conductivity evaluated at $T=0$ as a function of $\varepsilon_f$ for $\protect\theta= \protect\pi/3$ and (a) $J_{0}=2\ t$ (b) $J_{0}=t$. In both cases, $\protect\sigma_{xy}$ is quantized when the Fermi level is lying in a gap. The values of the plateaus depend strongly on the topology of the energy spectrum.[]{data-label="fig:4"}](fig4){width="8cm"}
As shown in Fig. \[fig:4\] (the corresponding energy spectrum is presented in Figure \[fig:2\],(b) and \[fig:2\],(c)) at $T=0$, the Hall conductivity is quantized when the Fermi level is lying in the gap. The value of the plateaus depends explicitly on the Chern number of the filled bands. When the Fermi level is in a band, the sum (\[eq:9\]) can be reformulated in terms of a sum over the Fermi level as shown by Haldane[@Haldane2004] recently. One has to note that when the Fermi level and $\theta$ are fixed, any variation of $J_{0}$ around $J_c(\theta)$ may induce a jump in the Hall conductivity. For instance with a filling factor $p = 1/3$ (Fig. \[fig:5\]-a), the Fermi level always lies in a gap but the off-diagonal conductivity jumps from $-e^2/h$ for $J_{0}>J_c$ to $2e^2/h$ for $J_{0}<J_c$.
A similar jump can be obtained of $\theta$ is changed, while $J_0$ is constant: such a change in $\theta$ can for example be induced by application of an external magnetic field perpendicular to the kagomé plane. In this case, large change of Hall conductivity (even sign change) can be induced by magnetic field.
This is not the case for $p = 1/2$. For that filling factor, the Hall conductivity varies smoothly when $J_{0}$ passes through the point $J_{0}=3\,t/2$ which does not depend on $\theta$ (Fig. \[fig:5\]-b). When $J<3\,t/2$, the Fermi level crosses the bands three and four resulting to a nonzero Hall conductivity. When $J>3\,t/2$, the Fermi level is in the gap separating the band three and four and the off-diagonal conductivity is given by the sum off the Chern number of the first three bands which gives zero.
![Off-diagonal conductivity $\sigma_{xy}(J)$ evaluated at $T=0$ for a filling factor $p=1/3$ (a), $p=1/2$ (b), $p=1/4$ (c) and $p=1/5$ (d) and $\theta=\pi/3$ (continuous lines) and $\theta=\pi/4$ (dashed lines).[]{data-label="fig:5"}](fig5){width="8cm"}
Fig \[fig:5\] shows the variation of the Hall conductivity as a function of $J_{0}$ for filling factors 1/4 (Fig. \[fig:5\]-c) and 1/5 (Fig. \[fig:5\]-d): in both cases the Fermi level is not in a gap and Hall conductivity does not exhibit any jump.
From these comments, it is clear that the chirality is never a relevant parameter for the Hall conductivity when the Fermi level lies in a gap, because $\protect\sigma_{xy}$ is then quantized and obviously, not proportional to the chirality.
![$\protect\sigma_{xy}/\protect\sigma_0$ with $\protect\sigma_0=\max(|\protect\sigma_{xy}|)$ versus $\protect\theta$ angle, calculated for a filling factor $p = 1/2$ (a)and (b) and $p=1/4$ (c) and $p=1/5$ (d) and different values of $J_{0}$. The continuous line represents the chirality given by (\[eq:9\]). The Hall conductivity is represented by the dashed lines for different values of the parameter $J_{0}$. We remark that $\protect\sigma_{xy}$ changes sign when $J_{0}$ tends to $0.6 t $. That change is related to the position of the Fermi level within the band but not associated to change of the Chern numbers. It is remarkable that the conductivity is more or less proportional to the chirality in a large range of couplings, up to $J_{0} \approx t$ as well as for large $\protect\chi$ values. For $p=1/4$ and $p=1/5$, one can show that the off-diagonal conductivity is not proportional to the chirality even for small values of $J_0$.[]{data-label="fig:6"}](fig6)
Conversely, it is possible to choose a filling factor such that the Fermi level is not within a gap. In Fig. \[fig:6\], we present $\sigma_{xy}$ versus $\theta$ for the filling factor $p =
1/2$ (a) and (b) and different values of $J_{0}$. For this filling factor, the Fermi level is not in a gap if $J_{0}<3\,t/2$. Scaling the conductivity by its maximal value in each case shows that it is proportional to chirality in a large range of coupling $J_{0}$, but also for large values of the chirality. This is somewhat unexpected as the previous studies relating $\sigma_{xy}$ to $\chi$ were restricted to small values of $\chi$[@Tatara2002]. However, for different values of the band filling this is no longer true as shown on figure \[fig:6\] for $p=1/4$ (\[fig:6\]-c) and $p=1/5$ (\[fig:6\]-d): for these filling factors, $\sigma_{xy}$ is sensitive to the variation of the Chern numbers when $J_{0}$ varies. In fact we found that the Hall conductivity is proportional to chirality only for $p=1/2$ and small enough $J_{0}$.
Discussion
==========
The model studied in this paper cannot be directly applied to materials, in which the Anomalous Hall Effect attributed to the spin chirality has been observed.
However, some interesting features have been obtained in this model. This work shows that transverse conductivity may be proportional to chirality in a strong coupling regime, extending previous results obtained in the weak coupling case[@Tatara2002] but this occurs only for a half filled band. It is also shown that the dependence of $\sigma_{xy}$ is not only determined by the chirality but also depends strongly on the exchange coupling parameter and on the band filling factor. $\sigma_{xy}$ can present a large variety of behaviors: non-monotonic variation, sign change or plateaus can be observed. Thus, even if the underlying magnetic phase chirality is the main origin of this intrinsic anomalous Hall effect, it is far from being entirely determined by the chirality.
We suggest that in real systems, variation of coupling J can be induced by temperature: the temperature acts on the system in many different ways. It acts on the magnetic configuration of the localized spins $\mathbf{S}_i$ by changing the amplitude of the magnetization, and on the position of the Fermi level in the band through the Fermi-Dirac distribution.
We assume that the magnitude of each local moment follows the mean field relation $$M(T)= |{\bf S_i}|
= M_{0}\sqrt{1-\frac{T}{T_{c}}}. \label{eq:4}$$where $T_c$ is the critical temperature. This can be described by introducing in eq. 1 a temperature dependent coupling constant
$$J(T)=J_{0}\sqrt{1-\frac{T}{T_{c}}}. \label{eq:5}$$
where $J_{0} = J M_0$ is the zero temperature exchange constant and $J_{0}$ and $T_{c}$ are supposed to be independent.
![Off-diagonal conductivity $\sigma_{xy}(T)$ for a filling factor $p=1/2$ (a), $p=1/3$ (b), $p=1/4$ (c) and $p=1/5$ (d) and $\theta=\pi/3$ (continuous lines) and $\theta=\pi/4$ (dashed lines).[]{data-label="fig:7"}](fig7)
Consequently, decreasing the temperature from $T_{c}$ to $T=0$ is equivalent to increase the exchange coupling from $J=0$ to $J=J_{0}$. This increase of $J$ can be considered as done at zero temperature since in a large range of parameters, the additional effect of temperature (i.e. the $T$-dependance of the Fermi function in Eq. (8)), is significant only when the Fermi level is very close to a band edge.
Qualitatively, this means that when temperature is decreased, one moves vertically from the bottom to the top of the phase diagram in Fig. \[fig:3\]: if $J_{0}$ is large enough, the temperature decrease produces at some peculiar temperature $T_{\star }$ such that $J(T_{\star })=J_{c}(T=0)$, a change in conductivity due to the change of the Chern numbers. From this observation it follows that $\sigma
_{xy}$ can show abrupt changes with temperature. It is worth noting that during that process, in the frame of our model, the magnetic texture characterized by its $\theta $ angle has *not* moved. Of course, a change in $\theta $ can also be responsible for a sign change.
To be more specific, we consider several examples presented in Fig. \[fig:7\].
We start with the filling factor $p = 1/2$, the critical temperature $T_c = 0.1 t$ and $J(T=0)=2t$ (Fig. \[fig:7\]-a). With this choice of parameters, the Fermi level lies in a gap when $J_{0}>3\,t/2$, which means that at low temperatures (i.e., for large $J_{0}$), the system is insulating. When the temperature is increased, $\sigma_{xy}$ remains equal to zero untill $T$ reaches the temperature $T_{1}$ for which $J(T_{1})=3\,t/2$. Above this temperature, the gap closes, making the transverse conductivity increasing continuously. It reaches an extremum which is a function of the texture angle $\theta$. Just below $T_c$, $\sigma_{xy}$ changes sign because of a subtle balance between the states giving positive and negative contributions to the conductivity, namely, the states from bands 3 and 4 (negative contribution) and the states from band 2 (positive contribution). These contributions may be globally explained by the associated Chern numbers of these bands, at small $J_{0}$ : $\nu_3, \nu_4 = -2$ and $\nu_2 = 3$.
For the filling factor $p = 1/3$ (Fig. \[fig:7\]-b), the behavior is completely different. The transverse conductivity is quantized and finite at zero temperature as the Fermi level is located in a gap. When the temperature increases and reaches the value $T_{\star}$ defined by $J(T_\star)=J_c(\theta)$, for which gaps are finite but sufficiently small to authorize the interband processes, $\sigma_{xy}$ continuously increases. Upon increasing temperature, the conductivity increases and then changes its sign. This change of sign is not of the same origin as we considered for $p = 1/2$. In the first case, the Chern numbers are fixed but the temperature changes the balance between the weights of each band. In the second example, bands 3 and 4 get crossed thus changing their Chern numbers. The variation of Chern numbers produces the sign change in the transverse conductivity. Due to thermal fluctuations, the extremum of conductivity does not reach its quantized value of 2 (in units of $e^2/h$) but saturates at 1.8. The conductivity finally decreases down to zero when $T$ reaches $T_c$, as it should be.
It is also possible to choose the filling factors that place the Fermi level at zero temperature within a band. In these cases, there is no generic behavior, and the angle of the spin texture plays an important role. Some results are shown on figure \[fig:7\] for the different filling factors $p = 1/4$ (c) and $p = 1/5$ (d).
Thus, these results show that in the mean field approximation a large variety of behaviors are possible in our model. The next step would be to study a model closer to the experimental situation of pyrochlores, where the variation of magnetic structure with temperature and applied field has been studied by neutrons experiments[@Yasui2003], allowing to take into account the real cristallographic and magnetic structures of these systems.
This work is partly supported by Université Joseph Fourier (Grenoble), by FCT Grant POCTI/FIS/ 58746/2004 (Portugal) and by Polish State Committee for Scientific Research under Grants PBZ/KBN/ 044/P03/2001 and 2 P03B 053 25. V.D. thanks the Calouste Gulbenkian Foundation in Portugal for support.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Eigenfunctions of the Laplace-Beltrami operator on a hyperboloid are studied in the spirit of the treatment of the spherical harmonics by Stein and Weiss. As a special case, a simple self-contained proof of Laplace’s integral for a Legendre function is obtained.'
address:
- 'The Institute of Mathematical Sciences, CIT campus Taramani, Chennai 600113.'
- 'Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Padur PO, Siruseri 603103.'
author:
- Amritanshu Prasad
- 'M. K. Vemuri'
bibliography:
- 'refs.bib'
title: |
Eigenfunctions of the Laplace-Beltrami\
operator on hyperboloids
---
In [@MR0304972 Chapter IV, Section 2], Stein and Weiss described the spectral decomposition of the Laplace-Beltrami operator on the unit sphere. Their approach was to identify the eigenfunctions with homogeneous harmonic functions on Euclidean space.
In this article the eigenfunctions of the Laplace-Beltrami operator on a hyperboloid are identified with homogeneous harmonic functions (with respect to a Laplacian of type $(p,q)$) on an open cone. In the case treated by Stein and Weiss, Liouville’s theorem implies that the degree of homogeneity must be a non-negative integer, whereas here the degree of homogeneity can be any complex number. This identification is used to compute spherical functions for $O(1,q)$, and consequently Laplace’s integral formula for Legendre functions is obtained. Laplace’s integral formula can also be obtained by using the residue theorem [@WW §15.23]. Spherical functions for semisimple Lie groups in general are obtained using different methods (see, e.g., [@MR754767 Chapter IV]).
Let $n=p+q$. Let ${{\mathbf{R}}^{p,q}}$ denote the space of real $n$-dimensional vectors equipped with the indefinite scalar product of signature $(p,q)$: $$\mathbf{x}\cdot \mathbf{y} = {}^t{\mathbf{x}}Q {\mathbf{y}}$$ where $Q$ is the diagonal matrix with $p$ $1$’s followed by $q$ $(-1)$’s along the diagonal. Write $|{\mathbf{x}}|^2$ for ${\mathbf{x}}\cdot {\mathbf{x}}$. There should be no confusion with the usual positive definite dot product and norm as they are never used in this paper.
Let ${{\mathbf{R}}^{p,q}}_+$ denote the subset of ${{\mathbf{R}}^{p,q}}$ consisting of those vectors for which $|{\mathbf{x}}|^2> 0$. For ${\mathbf{x}}\in {{\mathbf{R}}^{p,q}}_+$, let $|{\mathbf{x}}|$ denote the positive square root of $|{\mathbf{x}}|^2$. Let $O(p,q)$ denote the group consisting of matrices such that ${}^tAQA=Q$. Denote by $O(p,q)_0$ the connected component of the identity element of $O(p,q)$. Let ${S^{p,q}}$ denote the connected component of $(1,0,\ldots,0)$ in the hyperboloid $$\{ {\mathbf{x}}\;:\: |{\mathbf{x}}|=1,\:x_1>0 \}.$$ Let $\rho$ be any complex number. Let $\mathcal{P}_\rho$ denote the space of all functions $f\in C^2({{\mathbf{R}}^{p,q}}_+)$ which are homogeneous of degree $\rho$, i.e., functions such that $$f(\lambda {\mathbf{x}})=\lambda^\rho f({\mathbf{x}}) \mbox{ for all } {\mathbf{x}}\in {{\mathbf{R}}^{p,q}}_+,\: \lambda>0.$$ Denote by $\Delta$ the differential operator $|\nabla|^2$ (using the indefinite dot product), where $\nabla$ is the gradient operator $$\nabla = \left( \frac{\partial}{\partial x_1},\ldots, \frac{\partial}{\partial x_n} \right).$$ Define $$\mathcal{H}_\rho = \{ f\in \mathcal{P}_\rho \;:\: \Delta f = 0 \}.$$ A function $u\in C^2({S^{p,q}})$ is called a *spherical harmonic[^1] of degree $\rho$* if $u$ is the restriction to ${S^{p,q}}$ of a function in $\mathcal{H}_\rho$. Let $H_\rho$ denote the space of spherical harmonics of degree $\rho$: $$H_\rho = \{ f|_{{S^{p,q}}} \;: \: f\in \mathcal{H}_\rho \}.$$ The *Laplace-Beltrami operator* ${\Delta_{{S^{p,q}}}}$ on ${S^{p,q}}$ is defined by $${\Delta_{{S^{p,q}}}}u = \Delta \tilde{u}|_{{S^{p,q}}},$$ where $\tilde{u}:{{\mathbf{R}}^{p,q}}_+\to {\mathbf{C}}$ is defined by $\tilde{u}({\mathbf{x}})=u({\mathbf{x}}/|{\mathbf{x}}|)$ (the *degree zero homogeneous extension* of $u$).
Let ${\mathbf{x}}^\#=Q{\mathbf{x}}$. The following is easily verified:
\[lemma:formulas\] Let ${\mathbf{x}}\in {{\mathbf{R}}^{p,q}}_+$. Then
1. $\nabla|{\mathbf{x}}|={\mathbf{x}}^\#/|{\mathbf{x}}|$.
2. $\nabla|{\mathbf{x}}|^\rho = \rho |{\mathbf{x}}|^{\rho -2} {\mathbf{x}}^\#$.
3. $|{\mathbf{x}}^\#|=|{\mathbf{x}}|$.
4. ${\mathbf{x}}^\#\cdot \nabla \tilde{u}({\mathbf{x}}) = 0$ for any $u\in C^1({S^{p,q}})$.
5. $\nabla \cdot {\mathbf{x}}^\# = n$.
\[lemma:eigenvalue\] If $u\in H_\rho$ then ${\Delta_{{S^{p,q}}}}u = -\rho(\rho+n-2)u$.
Since $u\in H_\rho$, $|{\mathbf{x}}|^\rho \tilde{u}({\mathbf{x}}) \in \mathcal{H}_\rho$. Therefore (using the formulas in Lemma \[lemma:formulas\]), $$\begin{aligned}
0 &=&\Delta(|{\mathbf{x}}|^\rho \tilde{u}({\mathbf{x}}))\\
&=&\nabla\cdot(\nabla (|{\mathbf{x}}|^\rho \tilde{u}({\mathbf{x}})))\\
&=&\nabla\cdot(\rho|{\mathbf{x}}|^{\rho-2}{\mathbf{x}}^\# \tilde{u}({\mathbf{x}})+|{\mathbf{x}}|^\rho \nabla \tilde{u}({\mathbf{x}}))\\
&=&(\nabla(\rho|{\mathbf{x}}|^{\rho-2}\tilde{u}({\mathbf{x}}))\cdot {\mathbf{x}}^\# + \rho|{\mathbf{x}}|^{\rho-2}\tilde{u}({\mathbf{x}})(\nabla \cdot {\mathbf{x}}^\#) + |{\mathbf{x}}|^\rho \Delta \tilde{u}({\mathbf{x}})\\
&=&\rho(\rho-2)|{\mathbf{x}}|^{\rho-4}\tilde{u}({\mathbf{x}})|{\mathbf{x}}^\#|^2 + \rho|{\mathbf{x}}|^{\rho-2} \nabla \tilde{u}({\mathbf{x}})\cdot {\mathbf{x}}^\# + n\rho|{\mathbf{x}}|^{\rho-2}\tilde{u}({\mathbf{x}})+\Delta \tilde{u}({\mathbf{x}})\\
&=&\rho(\rho-2)|{\mathbf{x}}|^{\rho-4}\tilde{u}({\mathbf{x}})|{\mathbf{x}}|^2 + n\rho|{\mathbf{x}}|^{\rho-2}\tilde{u}({\mathbf{x}})+\Delta \tilde{u}({\mathbf{x}}).
\end{aligned}$$ Setting $|{\mathbf{x}}|=1$ in the result of the above calculation yields $$0=\rho(\rho-2+n)\tilde{u}({\mathbf{x}})+\Delta \tilde{u}({\mathbf{x}}),$$ from which the lemma follows.
The following proposition gives a construction of spherical harmonics when $p=1$:
\[prop:harmonic\] Suppose $\mathbf{c}\in \mathbf{R}^{1,q}_+$ is an isotropic vector, (meaning that $|\mathbf{c}|^2=0$) such that $c_1>0$. Then $\mathbf{c}\cdot {\mathbf{x}}>0$ for all ${\mathbf{x}}\in {S^{1,q}}$. Let $f({\mathbf{x}})=(\mathbf{c}\cdot {\mathbf{x}})^\rho$. Then $f\in \mathcal{H}_\rho$.
The set of points where $\mathbf{c}\cdot {\mathbf{x}}=0$ form a hyperplane tangential to the cone $|\mathbf{c}|^2=0$. For fixed $\mathbf{x}$, the sign of $\mathbf{c}\cdot {\mathbf{x}}$ can change only when $\mathbf{c}$ crosses this hyperplane. However, the entire half-cone $$\{\mathbf{c}\;:\:|\mathbf{c}|^2=0, \:c_1>0\}$$ lies on one side of the hyperplane, because the cone is quadratic. Therefore, for each ${\mathbf{x}}\in S^{1,q}$, it suffices to verify that $\mathbf{c}\cdot {\mathbf{x}}>0$ for $\mathbf{c}=(1,1,0,\ldots,0)$. In this case, $\mathbf{c}\cdot {\mathbf{x}}=x_1-x_2$, which is positive since $x_1>0$ and $x_1^2-x_2^2-\cdots-x_n^2=1$, so that $x_1>|x_i|$ for each $i>1$.
If $g\in C^2({{\mathbf{R}}^{p,q}}_+)$ and $\phi\in C^2({\mathbf{R}})$, then $$\Delta (\phi\circ g)({\mathbf{x}}) = \phi''(g({\mathbf{x}}))|\nabla g({\mathbf{x}})|^2 + \phi'(g({\mathbf{x}}))\Delta g({\mathbf{x}}).$$ Let $g({\mathbf{x}})=\mathbf{c}\cdot {\mathbf{x}}$, then $\nabla g({\mathbf{x}})=\mathbf{c}$, so that $|\nabla g({\mathbf{x}})|^2=0$. Since $g$ is linear, $\Delta g({\mathbf{x}})=0$. Therefore $\Delta f({\mathbf{x}})=0$.
Let ${\mathbf{e}}= (1,0,\ldots,0)$. Then $K=\mathrm{Stab}_{O(1,q)_0}({\mathbf{e}})$ is isomorphic to $SO(q)$ and is a maximal compact subgroup of $O(1,q)_0$. The action of $O(1,q)_0$ on ${S^{1,q}}$ is transitive, and the $K$-invariant spherical harmonics on ${S^{1,q}}$ are precisely the $K$-invariant spherical functions for $O(1,q)_0$. It follows from Proposition \[prop:harmonic\] that
\[prop:invariant\_harmonic\] Let $\mathbf{c}$ be any isotropic vector in ${{\mathbf{R}}^{1,q}}$. Then $$\int_K (k\mathbf{c}\cdot {\mathbf{x}})^\rho dk$$ is a $K$-invariant spherical harmonic of degree $\rho$ on ${S^{1,q}}$.
Since $K$ acts transitively on the slices of ${S^{1,q}}$ by the hyperplanes on which the first coordinate $x_1$ is constant, the value of a $K$-invariant spherical harmonic is simply a function of $x_1$, which will be denoted by $P(x_1)$. A $K$-invariant spherical harmonic may be viewed as a solution to an ordinary differential equation in $x_1$:
\[theorem:ODE\] Suppose that $P_\rho(x_1)$ is the value of a $K$-invariant spherical harmonic which is homogeneous of degree $\rho$. Then $P_\rho$ is a solution to the differential equation $$\label{eq:ODE}
(1-x_1^2)P''_\rho(x_1)+(1-n)x_1P'(x_1)+\rho(\rho-2+n)P(x_1)=0.$$
For any $f\in C^2({S^{1,q}})$ we have $$\begin{aligned}
\nabla f(x_1/|{\mathbf{x}}|) & = & \nabla(x_1/|{\mathbf{x}}|)f'(x_1/|{\mathbf{x}}|)\\
&=& \frac{(\nabla x_1)|{\mathbf{x}}|-x_1\nabla|{\mathbf{x}}|}{|{\mathbf{x}}|^2}f'(x_1/|{\mathbf{x}}|)\\
&=& \frac{{\mathbf{e}}|{\mathbf{x}}|-(x_1/|{\mathbf{x}}|){\mathbf{x}}^\#}{|{\mathbf{x}}|^2}f'(x_1/|{\mathbf{x}}|)\\
&=& u\mathbf{v},
\end{aligned}$$ where $u=|{\mathbf{x}}|^{-3}f'(x_1/|{\mathbf{x}}|)$ and $\mathbf{v}={\mathbf{e}}|{\mathbf{x}}|^2-x_1{\mathbf{x}}^\#$. Since $\Delta=|\nabla|^2$, $$\label{eq:laplacian}
\Delta f(x_1/|{\mathbf{x}}|)=(\nabla u) \cdot \mathbf{v}+ u \nabla\cdot \mathbf{v}.$$ Now, $$\begin{aligned}
\nabla u & = & -3|{\mathbf{x}}|^{-5}{\mathbf{x}}^\#f'(x_1/|{\mathbf{x}}|)+|{\mathbf{x}}|^{-3}\nabla(x_1/|{\mathbf{x}}|)f''(x_1/|{\mathbf{x}}|)\\
&=& -3|{\mathbf{x}}|^{-5}{\mathbf{x}}^\#f'(x_1/|{\mathbf{x}}|)+|{\mathbf{x}}|^{-6}({\mathbf{e}}|{\mathbf{x}}|^2-x_1{\mathbf{x}}^\#)f''(x_1/|{\mathbf{x}}|).
\end{aligned}$$ and $$\nabla \cdot \mathbf{v} = {\mathbf{e}}\cdot \nabla |{\mathbf{x}}|^2-({\mathbf{e}}\cdot {\mathbf{x}}^\#+nx_1) = (1-n)x_1.$$ Suppose there exists a function $P$ such that $P(x_1)=f({\mathbf{x}})$ for each ${\mathbf{x}}$ such that $|{\mathbf{x}}|=1$. Substituting the above values of $\nabla u$ and $\nabla\cdot \mathbf{v}$ in (\[eq:laplacian\]) and then setting $|{\mathbf{x}}|=1$ we have, $$\begin{gathered}
{\Delta_{{S^{1,q}}}}f|_{{S^{1,q}}}({\mathbf{x}})=(-3{\mathbf{x}}^\#P'(x_1)+({\mathbf{e}}-x_1{\mathbf{x}}^\#)P''(x_1))\cdot({\mathbf{e}}-x_1{\mathbf{x}}^\#)\\
+ (1-n)x_1P'(x_1).
\end{gathered}$$ When $|{\mathbf{x}}|=1$, $|{\mathbf{e}}-x_1{\mathbf{x}}^\#|^2=(1-x_1^2)$ and $({\mathbf{e}}-x_1{\mathbf{x}}^\#)\cdot {\mathbf{x}}^\#=0$ so that the above equality simplifies to $${\Delta_{{S^{1,q}}}}f|_{{S^{1,q}}}({\mathbf{x}})= (1-x_1^2)P''(x_1)+(1-n)x_1P'(x_1).$$ Combining this with Lemma \[lemma:eigenvalue\] gives (\[eq:ODE\]).
\[cor:uniqueness\] For $n\geq 3$, there is (up to scaling) a unique $K$-invariant spherical function of degree $\rho$ given by $$\int_K(k\mathbf{c}\cdot {\mathbf{x}})^\rho dk,$$ where $\mathbf{c}$ is any non-zero isotropic vector in ${{\mathbf{R}}^{1,q}}$.
The ordinary differential equation (\[eq:ODE\]) is linear of degree 2 with a regular singular point at $x_1=1$. The indicial equation at this point is $$m(m+(n-1)/2-1)=0.$$ Therefore, it has (up to scaling) at most one solution defined on $[1,\infty)$. This solution is known by Proposition \[prop:invariant\_harmonic\].
The classical integral formula due to Laplace for Legendre functions is readily derived from the preceding analysis:
\[cor:Laplace\_integral\] Every solution of the ordinary differential equation $$(1-x^2)P''(x)-2xP'(x)+\rho(\rho+1)P(x)=0$$ that is defined on $[1,\infty)$ is a scalar multiple of $$P_\rho(x)=\frac{1}{2\pi}\int_0^{2\pi} (x+\sqrt{x^2-1}\,\cos\theta)^\rho d\theta.$$
Evaluate the formula from Corollary \[cor:uniqueness\] taking $q=2$, $\mathbf{c}=(1,0,-1)$ and $\mathbf{x}=(x,0,\sqrt{x^2-1})$.
[^1]: Perhaps a more apt name would be *hyperboloidal harmonic*.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper is devoted to stability analysis of continuous-time delay systems based on a set of Lyapunov-Krasovskii functionals. New multiple integral inequalities are derived that involve the famous Jensen’s and Wirtinger’s inequalities, as well as the recently presented Bessel-Legendre inequalities of A. Seuret and F. Gouaisbaut, (2015) [@seur14b], and the Wirtinger-based multiple-integral inequalities of M. Park et al. (2015) and T.H. Lee et al. (2015) [@park15; @leejfi15]. The present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of LMIs establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods. Numerical examples illustrate the efficiency of the method.'
address: |
Mathematical Institute, Budapest University of Technology and Economics, Budapest, Pf. 91, 1521, Hungary\
Corvinus University of Budapest, 8 Fővám tér, H-1093, Budapest, Hungary
author:
- 'É. Gyurkovics , T. Takács'
title: Multiple integral inequalities and stability analysis of time delay systems
---
Integral inequalities, stability analysis, continuous-time delay systems, hierarchy of LMIs
Introduction
============
Time delays are present in many physical, industrial and engineering systems. The delays may cause instability or poor performance of systems, therefore much attention has been devoted to obtain tractable stability criteria of systems with time delay during the past few decades (see e.g. the monographs [@Briat14]-[@WHSh], some recent papers [@seur14b], [@park15], [@kim16]-[@ze15] and the references therein). Several approaches have been elaborated and successfully applied for the stability analysis of time delay systems (see the references above for excellent overviews).
Lyapunov method is one of the most fruitful fields in the stability analysis of time delay systems. On the one hand, more and more involved Lyapunoov-Krasovskii functionals (LKF) have been introduced during the past decades. On the other hand, much effort has been devoted to derive more and more tight inequalities (Jensen’s inequlity and different forms of Wirtinger’s inequality [@seur14b]-[@frid2014], [@kim16]-[@sga15], [@gye15], [@zha15], etc.) for the estimation of quadratic single, double and multiple integral terms in the derivative of the LKF. Simultaneously, augmented state vectors are introduced in part as a consequence of the improved estimations, in part in an ad’hoc manner. The effectiveness of different methods is mainly compared using some numerical examples. Recently, the authors of [@seur14b], [@seur14] have introduced a very appealing idea of the hierarchy of LMI conditions offering a rigorous theoretical basis for comparison of stability LMI conditions. Based on Legendre polynomials, they proposed a generic set of single integral inequalities opening the way to the derivation of a set of stability conditions forming a hierarchy of LMIs. A further possibility for the derivation of improved stability conditions have been proposed by [@park15] and [@leejfi15] using multiple integral quadratic terms in the LKF, together with Wirtinger-based multiple integral inequalities. Naturally the question arises: how these two lines of investigations are related to each other, and how sufficient stability conditions can be derived unifying the approaches of using multiple integral quadratic terms in the LKF and refined estimations of these integral terms.
*The aim of the present work is to answer these questions. On the one hand, multiple integral inequalities based on orthogonal hypergeometric polynomials will be derived that extend the results of [@seur14b], [@seur14] to multiple integrals and improve the estimations of [@park15] and [@leejfi15]. On the other hand, a multi-parametric set of LMI conditions will be constructed, and it will be shown that a two parametric subset forms a bidirectional hierarchy of LMIs.*
Analogous results have been presented for discrete-time systems in [@gytkn].
The paper is organized as follows. In Section 2 it is shown, how the quadratic terms of the derivative of the LKF can be estimated by Bessel-type inequalities. It is also proven that these estimations relevantly improve a recently published result. A sufficient condition of asymptotic stability is presented in the form of an LMI in Section 3. The hierarchy of LMI conditions is established then in Section 4. Some benchmark numerical examples are shown in Section 5, the results of which are compared to earlier ones known from the literature. Finally, the conclusions will be drawn.
The notations applied in the paper are very standard, therefore we mention only a few of them. Symbol $A \otimes B$ denotes the Kronecker-product of matrices $A,B,$ while $\mathbf{S}_{n}$ and $\mathbf{S}_{n}^{+}$ are the set of symmetric and positive definite symmetric matrices of size $n\times n,$ respectively.
Multiple integral inequalities
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Preliminaries
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The paper deals with the stability analysis of the following continuous-time time delay system $$\begin{aligned}
\dot{x} (t)&=&Ax(t)+A_{d_1} x(t-\tau )+ A_{d_2}\int_{t-\tau}^{t} x(s) ds, \; t \geq 0, \label{x1} \\
x_0 (t) &=& \varphi (t), \; t \in [-\tau,0], \label{xx1}\end{aligned}$$ where $x(t) \in \mathbf{R}^{n_x}$ is the state, $A$, $A_{d_1}$and $A_{d_2}$ are given constant matrices of appropriate size, the time delay $\tau $ is a known positive integer and $x_0(.)$ is the initial function.
A.) *A Bessel-type inequality.* Let $\mathbf{E}$ be a Euclidean space with the scalar product $\langle . , . \rangle$, and let $\pi _i \in \mathbf{E}, \; (i=0,1, \ldots )$ form an orthogonal system. For any $f,g \in \mathbf{E}^{n},$ define $ \langle f, g\rangle =\sum _{i=1}^{n} \langle f_i, g_i\rangle . $ Let $W \in \mathbf{S}^{+}_{n}$. For any $f\in\mathbf{E}^{n},$ consider the functional $$\begin{aligned}
J_{W}(f) = \langle f,W f \rangle . \label{e4}\end{aligned}$$
\[lem:11\] If $\nu \geq 0$ is a given integer, then the following inequality holds $$\begin{aligned}
J_{W} (f)\geq \sum _{j=0}^{\nu}\frac{1}{\left\| \pi_{j} \right\|^2} w_{j}^{T} W w_j, \label{e6}\end{aligned}$$ where $w_j = \langle f, \pi_{j}\rangle ,$ and the scalar product is taken componentwise.
**Proof.** The proof is standard, therefore it is omitted. B.) *Orthogonal hypergeometric polynomials.* Suppose that $m\geq0$ is a given integer and consider the closed interval $[a,b].$ For functions $g_1,g_2 \in L_2 [a,b] $ define a scalar product by $$\begin{aligned}
\langle g_1,g_2\rangle _{m,[a,b]} = \int _{a}^{b} \left( \frac{s-a}{b-a} \right)^{m} g_1 (s) g_2 (s) ds. \label{e2}\end{aligned}$$ It is easy to see that $\langle g_1,g_2\rangle _{m,[a,b]}$ can equivalently be expressed as $$\begin{aligned}
\langle g_1,g_2\rangle _{m,[a,b]} &=& \frac{m!}{(b-a)^m} \int _{a}^{b} \int _{v_1}^{b} ... \int _{v_m}^{b} g_1 (s) g_2 (s)ds dv_m ... dv_1
, \mbox{ if } m>0. \label{e1}\end{aligned}$$ (If $m=0,$ then a single integral is considered.) Substitute $s \in [a,b]$ by $s=a+(b-a)x$, where $x \in [0,1],$ and set $G_i (x)=g_i (a+(b-a)x),$ $(i=1,2)$ on the right hand side of (\[e2\]), then we obtain that $$\begin{aligned}
\langle g_1,g_2\rangle _{m,[a,b]} = (b-a) \int _{a}^{b} x^{m} G_1 (x) G_2 (x) dx = (b-a) \langle G_1,G_2\rangle _{m,[0,1]}. \label{gy30}\end{aligned}$$ Thus it is sufficient to consider the orthogonal polynomials with respect to $\langle .,.\rangle _{m,[0,1]}.$
For any fixed non-negative integer $m$, let us denote by $P_{m,n}, \; (n=0,1, \ldots)$ the polynomials of degree $n$ orthogonal with respect to $\langle .,.\rangle _{m,[0,1]}.$ (For general theory see e.g. [@gau].) They can be given by the two parameters generalization of the Rodrigues-formula: $$\begin{aligned}
P_{m,0}(x) & \equiv& 1, \label{ee1}\\
P_{m,n}(x) &=& \frac{1}{n!} \frac{1}{x^m} \frac{d^n}{dx^n} \left( x^m (x^2-x)^n \right) , \hspace{0.5cm} n=1,2,... \label{e3}\end{aligned}$$ For $m=0,$ this is the usual Rodrigues formula for the shifted Legendre polynomials.
We note that that polynomials (\[ee1\])-(\[e3\]) satisfy certain hypergeometric-type differential equation (see e.g. [@Area03] and [@San97]). This is why they are frequently called “orthogonal hypergeometric polynomials”. By straightforward calculation, it can be shown that they have the properties $$\begin{aligned}
&(i) \;\;& \left\| P_{m,n} \right\| _{m,[0,1]}^{2} = \int _{0}^{1} x^m P_{m,n}^{2} (x) dx = \frac{1}{m+2n+1} , \label{e3b} \\
&(ii) \;&P_{m,n}(0)=(-1)^n \frac{m+n}{n} , \hspace{1cm} P_{m,n}(1)=1.\end{aligned}$$ The polynomials $$\begin{aligned}
p_{m,n} (t) = P_{m,n} \left( \frac{t-a}{b-a} \right) \label{e3a} $$ are orthogonal with the scalar product (\[e2\]), and $$\label{gy01}
\left\| p_{m,n} \right\| _{m,[a,b]}^{2}=\frac{b-a}{m+2n+1}, \hspace{0.5cm} p_{m,n}(a)=(-1)^n \frac{m+n}{n} ,
\hspace{0.5cm}p_{m,n}(b)=1.$$
Integral inequalities
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Let $W \in \mathbf{S}^{+}_{n},$ $[a,b] \subset \mathbf{R}$ with $b-a>0$ and $0 \leq m \in \mathbf{Z}$ be given. For any continuous $f: [a,b] \rightarrow \mathbf{R}^{n},$ consider the functional $$\begin{aligned}
J_{W,m,a,b} (f) = \frac{m!}{(b-a)^m} \int _{a}^{b} \int _{v_1}^{b} ... \int _{v_m}^{b} f^{T}(s)W f(s)ds dv_m ... dv_1, \label{e400}\end{aligned}$$ which can also be expressed as $$\begin{aligned}
J_{W,m,a,b} (f) = \int _{a}^{b} \left( \frac{s-a}{b-a} \right)^{m} f^T (s) W f(s) ds = \langle f, W f \rangle _{m,[a,b]}. \label{e5}\end{aligned}$$ Let $\nu _m\geq0$ be a given integer. One can apply now Lemma \[lem:11\] with $\mathbf{E}=L_{[a,b]}^2,$ the scalar product (\[e2\]), $\nu =\nu _m$ and $\pi _j=p_{m,j}, $ $(j=0,1,\ldots, \nu _m).$ Now, *our aim is to derive a lower estimation as a quadratic form with respect to variables independent of $m.$*
\[lem:21\] Let $M>0$ and $\nu _m\geq0$ be given integers satisfying the condition $m+\nu _m \leq M-1$. Let $J_{W,m,a,b} (f)$ be defined by (\[e5\]). Then the following inequality holds true: $$\begin{aligned}
J_{W,m,a,b} (f) \geq \frac{1}{b-a} \Phi_ M^{T} \left( \Xi_m \otimes I \right)^T \mathcal{W}_m \left( \Xi_m \otimes I \right) \Phi_ M, \label{e10}\end{aligned}$$ where $\mathcal{W}_m = \mbox{\emph{diag}} \left\{ (m+1), (m+3), \ldots, (m+2\nu_m +1) \right\}\otimes W, $ $\Phi_ M^T =
\begin{bmatrix}\phi_0^T,&\ldots ,&\phi_{M-1}^T\end{bmatrix} \; $ with $\phi _l = \int _{a}^{b} p_{0,l} (s)f(s)ds,$ and matrix $\Xi_m$ is given by (\[gy20\]) below.
**Proof.** Introduce the notation $w_{m,j}=\langle f,p_{m,j}\rangle _{m,[a,b]}$ $(j=0,1,\ldots,\nu _m, )$ needed to apply Lemma \[lem:11\]. Clearly, $$\begin{aligned}
w_{m,j} = \int _{a}^{b}
\left( \frac{s-a}{b-a}\right)^m P_{m,j} \left( \frac{s-a}{b-a} \right) f(s) ds . \label{e8}\end{aligned}$$ The degree of polynomials $q_{m,m+j}(x)=x^m P_{m,j} (x)$ appearing in (\[e8\]) is exactly $m+j,$ thus these polynomials can be expressed as $$\begin{aligned}
q_{m,m+j} (x) = \sum _{l=0}^{M-1} \xi _{j,l}^{m} P_{0,l} (x), \label{e9}\end{aligned}$$ where $\xi_{j,l}^{m}=0,$ if $m+j<l \leq M-1.$ Using the definition of $\phi _l$ and $\Phi_ M$ we obtain $$\begin{aligned}
w_{m,j} = \sum _{l=0}^{M-1} \xi _{j,l}^{m} \phi _l = \left(\underline{\xi}_{j}^{m}\otimes I\right) \Phi _M, \label{e900}\end{aligned}$$ where $\underline{\xi}_{j}^{m}=\left( \xi_{j,0}^{m}, \ldots, \xi_{j,M-1}^{m} \right).$ Introduce the notation $$\label{gy20}
\Xi_m =
\left[\left(\underline{\xi}_{0}^{m}\right)^T, \ldots,\left(\underline{\xi}_{\nu_m}^{m}\right)^T \right]^T \in \mathbf{R}^{(\nu_m +1)\times M}$$ Estimation (\[e10\]) can be obtained by direct substitution taking into account (\[gy30\]) and (\[e3b\]). $\Box$
\[rem:1\] We note that Lemma \[lem:21\] is closely connected with Theorem 2.2 of [@zha15]. Both results are based (explicitly or implicitly) on Lemma \[lem:11\], thus they are substantially equivalent. The estimation of Lemma \[lem:21\] may be more advantageous when it is applied for derivative of functions (see Lemma \[lem:31\] below) and for stability analysis of time delay systems. The advantage is twofold: on the one hand, the variables are expressed using a common set of orthogonal polynomials independent of $m,$ on the other hand, the dependence on the length of the interval is relatively simple, since the matrices $\Xi _m$ and $\mathcal{W}_m$ do not depend on $b-a.$
\[rem:2\] Paper [@zha15] gives a thorough and detailed discussion of the relation between their WOPs-based result and the Jensen’s and Wirtinger’s inequalities published in a wide range of previous literature, therefore we only compare Lemma \[lem:21\] to the recently published multiple integral inequality of Lemma 5 of [@leejfi15]. Using the notations of [@leejfi15], we can see that the relation of the investigated functionals can be given as $$\label{LP1}
G_l(f,a,b,W)=\frac{(b-a)^l}{l!}J_{W,m,a,b} (f).$$ To express the estimation of the present paper with the variables of [@leejfi15], we need a short computation to show that $w_{l,0}=\frac{l!}{(b-a)^l}g_l(f,a,b),$ and $w_{l,1}=-\frac{(l+1)!}{(b-a)^l}\Upsilon_l(f,a,b).$ Employing the proposed estimation with (\[LP1\]), we obtain that $$\begin{aligned}
G_l(f,a,b,W) &=& \geq \frac{(l+1)!}{(b-a)^{l+1} } g_l^T(f,a,b)W g_l(f,a,b) \nonumber\\
& & \hspace{0.5cm}+ (l+1)^2\frac{l!(l+3)}{(b-a)^{l+1}}\Upsilon_l^T(f,a,b)W\Upsilon_l(f,a,b).
\label{LP2}\end{aligned}$$ The first term of the lower bound of [@leejfi15] is the same as in (\[LP2\]), while the second term is smaller inasmuch as it has the coefficient $1$ in place of $(l+1)^2,$ thus the estimation of the present paper is tighter.
The considerations above indicate that the results of [@leejfi15] correspond to the choice of $\nu _l =1,$ but the authors of [@park15] do not derive any estimation that can be characterized with $\nu _l >1.$
Next, we shall derive a lower estimation also for the case, when the functional is applied to the derivative $f'(s)=\frac{d}{ds}f(s),$ i.e. consider $$\begin{aligned}
J_{W,m,a,b} (f^{\prime}) &=& \int _{a}^{b} \left( \frac{s-a}{b-a} \right)^{m} f^{\prime}(s)^T W f^{\prime}(s) ds = \langle f^{\prime}, W f^{\prime} \rangle _{m,[a,b]}. \label{gy40}\end{aligned}$$
\[lem:31\] Let $\ M \ $ and $\ \nu _m \ $ be given non-negative integers satisfying the condition $\ m+\nu _m
\leq \; $ $\max \left\{0,M-1\right\}$. Let $J_{W,m,a,b} (f')$ be defined by (\[gy40\]). Then the following inequality holds true: $$\begin{aligned}
J_{W,m,a,b} (f^{\prime}) \geq \frac{1}{b-a} \widetilde{\Phi}_ M ^{T} \left( \mathcal{Z}_m \otimes I \right)^T \mathcal{W}_m
\left( \mathcal{Z}_m \otimes I \right) \widetilde{\Phi}_ M, \label{e12}\end{aligned}$$ where $\mathcal{W}_m $ is the same as in Lemma \[lem:21\], $\widetilde{\Phi}_ M = \mbox{col} \left\{ f(b), \; f(a), \; \frac{1}{b-a}\phi_0, \ldots , \frac{1}{b-a}\phi_{M-1} \right\},$ if $ M>0$, $\widetilde{\Phi}_ 0 = \mbox{col} \left\{ f(b), \; f(a) \right\},$ and matrix $\mathcal{Z}_m $ is given by (\[gy50\]) below.
**Proof.** Set $\theta_{m,j} = \langle f^{\prime},p_{m,j}\rangle _{m,[a,b]},$ $(j=0,1,\ldots,\nu _m, )$ and apply Lemma \[lem:11\]. In order to obtain the estimation (\[e12\]), we have to perform a short computation, as follows. Consider the previously introduced polynomials $q_{m,m+j}$ again, then integrating by parts we obtain $$\begin{aligned}
\theta_{m,j} &=& \int _{a}^{b} \left( \frac{s-a}{b-a} \right)^m p_{m,j} (s) f^{\prime }(s) ds = \nonumber \\
&=& q_{m,m+j} (1)f(b) - q_{m,m+j} (0) f(0) - \frac{1}{b-a} \int _{a}^{b} q^{\prime }_{m,m+j} \left( \frac{s-a}{b-a} \right) f(s)ds.
\label{e11}\end{aligned}$$ Express now polynomials $q^{\prime }_{m,m+j} $ having degree $m+j-1$ by $P_{0,0},...,P_{0,m+j-1},P_{0,m+j},...,P_{0,M-1}:$ $$\begin{aligned}
q^{\prime }_{m,m+j} \left( \frac{s-a}{b-a} \right) = \sum _{l=0}^{M-1} \zeta_{j,l}^{m} P_{0,l} \left( \frac{s-a}{b-a} \right)
= \sum _{l=0}^{M-1} \zeta_{j,l}^{m} p_{0,l} (s), \label{gy100}\end{aligned}$$ where $\zeta_{j,l}^{m} =0,$ if $m+j \leq l \leq M-1,$ and $\nu_m +m \leq M.$ Thus $$\begin{aligned}
\theta _{m,j} = q_{m,m+j} (1)f(b)-q_{m,m+j} (0)f(a) - \frac{1}{b-a} \sum _{l=0}^{M-1} \zeta _{j,l}^{m} \phi _l.\end{aligned}$$ By straightforward calculation $$\begin{aligned}
q_{m,m+j} (1)=1, \; q_{m,m+j} (0)= \left\{ \begin{array}{ccc}
(-1)^j , & \mbox{if} & m=0, \\
0, & \mbox{if} & m > 0. \\
\end{array}
\right.
$$ Set now $$\begin{aligned}
\underline{\zeta} _{j}^{o} &=& \left( 1, \; (-1)^{j+1} , \; -\zeta_{j,0}^{0},\ldots , -\zeta_{j,M-1}^{0} \right),\hspace{0.5cm}
(m=0), \label{gy110} \\
\underline{\zeta} _{j}^{m} &=& \left( 1, \; 0, \; -\zeta_{j,0}^m,\ldots , -\zeta_{j,M-1}^m \right), \hspace{1.7cm}
(m>0) \label{gy120}\end{aligned}$$ for $0 \leq j \leq \nu _m,$ and $$\begin{aligned}
\mathcal{Z}_m &=& \left[ (\underline{\zeta} _{0}^{m})^T,..., (\underline{\zeta} _{\nu_m}^{m})^T \right]^T \in
\mathbf{R}^{(\nu_m +1)\times (M+2)}. \label{gy50}\end{aligned}$$ Estimation (\[e12\]) can be obtained by direct substitution taking into account (\[gy30\]) and (\[e3b\]). $\Box$
\[rem:3\] Paper [@zha15] gives the lower bound for functionals applied to derivatives of functions for several special cases together with comparisons with previously published estimations, therefore we refer the reader for discussions to [@zha15]. We only mention that the relation between Lemma \[lem:31\] and Lemma 6 of [@leejfi15] is analogous to the one pointed out in Remark \[rem:2\]. Moreover neither [@leejfi15] nor [@zha15] derive any estimation for functionals applied to derivative of functions relying to polynomials of degree higher than $1.$
Stability analysis of continuous delayed systems
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Consider equation (\[x1\]). Let $M> 0, \; m_1 \geq 0, \; m_2 \geq 1$ be given integers. Let $x_t (s)=x(t+s)$ be the solution of (\[x1\]), and let $\phi _j (t) $ and $\Phi_M (t)$ be defined for function $f=x_t$ as before with $\phi_j (t) = \int _{-\tau}^{0} p_{0,j} (s) x_t (s) ds,$ and $\Phi_M (t)= \mbox{col} \left\{ \phi_0 (t), ..., \phi_{M-1} (t) \right\}$, Set furthermore $$\begin{aligned}
\widetilde{x} (t) = \mbox{col} \left\{ x(t), \Phi _M(t) \right\} , \;
\widetilde{\Phi}_M (t) = \mbox{\emph{col}} \left\{ x(t), x(t- \tau ), \frac{1}{\tau} \Phi _M(t) \right\}.\end{aligned}$$ Consider the LKF candidate $$\begin{aligned}
V(x_t , \dot{x}_t ) = V_1 (x_t) + V_2 (x_t) + V_3 (\dot{x}_t), \label{e15}\end{aligned}$$
where $$\begin{aligned}
V_1 (x_t) &=& \widetilde{x} (t)^T P \widetilde{x} (t), \hspace{5cm} P\in \mathbf{S}_{n_x(M+1)}, \label{gy60} \\
V_2 (x_t) &=& \sum _{j=0}^{m_1} \int _{-\tau}^{0} \left( \frac{s+ \tau}{\tau } \right)^j x_t (s)^T Q_j x_t (s)ds ,
\hspace{0.8cm}Q_j \in \mathbf{S}_{n_x}^{+}, \hspace{2mm} j=0,...,m_1, \label{gy61} \\
V_3 (\dot{x}_t) &=&
\tau \sum _{j=1}^{m_2} \int _{-\tau}^{0} \left( \frac{s+ \tau}{\tau }\right)^j \dot{x}_t (s)^T R_j \dot{x}_t (s)ds ,
\hspace{0.5cm} R_j \in \mathbf{S}_{n_x}^{+}, \hspace{2mm} j=1,...,m_2.
\label{gy62}
$$ We note that $V_2$ and $V_3$ can also be written as multiple integrals (c.f. (\[e400\]), (\[e5\])).
\[Th:13\] Let $M> 0, \; m_1 \geq 0, \; m_2 \geq 1$ and $\nu_{1,j}\geq 0, \; (j=0, \ldots , m_1)$, $\nu_{2,j} \geq 0, \; (j=0, \ldots , m_2)$ be given integers satisfying the inequalities $m_1+\nu_{1,j}<M,$ $m_2+\nu_{2,j} \leq M,$ for all $j$. System (\[x1\]) is asymptotically stable, if there are matrices $P \in \mathbf{S}_{n_x(M+1)}$, $Q_j \in \mathbf{S}_{n_x}^{+}$, $j=0,...,m_1$ and $R_j \in \mathbf{S}_{n_x}^{+}$, $j=1,...,m_2$ such that the LMIs $$\begin{aligned}
\Psi_{M,m_1}^0(\tau) >0, \hspace{0.5cm}\Psi _{M}^{1}(\tau) + \Psi _{M,m_1}^{2} + \Psi _{M,m_2}^{3,1}(\tau) - \Psi _{M,m_2}^{3,2}(\tau) < 0 \label{e20}\end{aligned}$$ hold true, where
$$\begin{aligned}
\Psi_{M,m_1}^0 (\tau) &=& \tau P + \sum _{j=0}^{m_1} \mbox{diag}
\left\{ 0, \left( \Xi_j \otimes I \right)^T \mathcal{Q}_{j}^{(j)} \left( \Xi_j \otimes I \right)\right\}, \label{gy55} \\
\Psi _{M}^{1}(\tau) &=& \Gamma_{M}^{T} P \Lambda_M + \Lambda_{M}^{T} P \Gamma_M, \label{gy580} \\
\Psi _{M,m_1}^{2} &=& \mbox{diag} \left\{ \sum_{j=0}^{m_1} Q_j , \; -Q_0 , \; - \sum_{j=1}^{m_1} j
\left( \Xi _{j-1} \otimes I \right)^T \mathcal{Q}_{j-1}^{(j)} \left( \Xi _{j-1} \otimes I \right) \right\} ,\label{gy56} \\
\Psi _{M,m_2}^{3,1}(\tau) &=& \tau \mathcal{A}^T \sum _{j=1}^{m_2} R_j \mathcal{A} , \label{gy57}\\
\Psi _{M,m_2}^{3,2}(\tau) &=& \sum_{j=1}^{m_2} j
\left( \mathcal{Z}_{j-1} \otimes I \right)^T \mathcal{R}_{j-1}^{(j)} \left( \mathcal{Z}_{j-1} \otimes I \right),\label{gy58}\end{aligned}$$
matrices $\Xi_k$ and $\mathcal{Z}_{k}$ are given by (\[gy20\]) and (\[gy50\]) with $\nu_{1,k}$ and $\nu_{2,k}$, respectively, $$\begin{aligned}
\mathcal{Q}_{j}^{(k)} &=& \mbox{diag} \left\{ (j+1)Q_k, \ (j+3)Q_k, \ ...\ , \ (j+(2M-1))Q_k \right\}, \label{gy81}\\
\mathcal{A}&=& \left( A, \; A_{d_1} , \; \tau A_{d_2} , 0 ,..., 0 \right)\in \mathbf{R}^{n_x \times n_x(M+2)}, \label{gy82}\\
\Lambda _M &=& \begin{bmatrix}
\mathcal{A} \\
\widetilde{\mathcal{L}}_0 \otimes I \\
\end{bmatrix} , \;
\Gamma _M = \begin{bmatrix}
1 & 0 & 0\\
0 & 0 & \tau I_{M}\\
\end{bmatrix} \otimes I, \;
\label{gy83}\\
\widetilde{\mathcal{L}}_0 &=& \left[ (\underline{\zeta} _{0}^{0})^T,...,
(\underline{\zeta} _{M-1}^{0})^T \right]^T
\label{gy840} \\
\mathcal{R}_{j-1}^{(j)}&=&\mbox{diag} \left\{ j R_j, \ (j+2)R_j, \ ... \ , \ (j+2\nu_{j-1}) R_j \right\}, \label{gy84}\end{aligned}$$ where $\underline{\zeta} _{j}^{0}$ is given by (\[gy110\]), and $0$ denotes zero matrices of compatible size.
**Proof.** We shall prove first the existence of a $\mu _1>0$ such that $V(x_t,\dot{x}_t) \geq \mu _1 \left\| x(t) \right\|$. Consider the term of $V_2$ with $j=0$. Applying estimation (\[e10\]) with $\nu_0=M-1$ and $\Xi_0=I$ we obtain $$\begin{aligned}
\int _{-\tau}^{0} x_t (s)^T Q_0 x_t (s)ds &\geq&
\tau ^{-1} \widetilde{x} (t)^T \mbox{diag} \left\{ 0 , \mathcal{Q}_{0}^{(0)} \right\} \widetilde{x} (t). \label{e17}\end{aligned}$$ If $m_1 > 0$, apply (\[e10\]) to the terms of $V_2$ with $j>0$. We obtain $$\begin{aligned}
&&\int _{-\tau}^{0} \left(\frac{s+\tau}{\tau}\right)^j x_t (s)^T Q_j x_t (s)ds \geq -\tau ^{-1} \widetilde{x} (t)^T \mbox{diag} \left\{ 0_n , \left( \Xi _j \otimes I \right)^T \mathcal{Q}_{j}^{(j)}
\left( \Xi _j \otimes I \right) \right\} \widetilde{x} (t). \label{e18}\end{aligned}$$ It follows from (\[e17\]) and (\[e18\]) that $$\begin{aligned}
V_1 (x_t) + V_2 (t) \geq \widetilde{x} (t)^T \left( P+ \tau ^{-1} \sum _{j=0}^{m-1} \mbox{diag} \left\{ 0,
\left( \Xi _j \otimes I \right)^T \mathcal{Q}_{j}^{(j)} \left( \Xi _j \otimes I \right) \right\} \right) \widetilde{x} (t). \label{e19}\end{aligned}$$ Since $V_3 (\dot{x}_t) \geq 0,$ the existence of an appropriate $\mu _1$ follows from (\[e19\]).
We shall prove next the negativity of $\frac{d}{dt} V(x_t , \dot{x}_t).$ Introduce the notation $\overline{V}_i (t) = V_i (x_t)$, $i=1,2,$ and $\overline{V}_3 (t) = V_3 (\dot{x}_t)$, where $x_t$ is the solution of system (\[x1\]). The derivative of the first term of (\[e15\]) is $$\begin{aligned}
\dot{\overline{V}}_1 (t)= \dot{\widetilde{x}} (t)^T P \widetilde{x}(t) + \widetilde{x}^T (t)P \dot{\widetilde{x}}(t),\end{aligned}$$ where $\dot{\widetilde{x}}(t)= \mbox{col} \left\{ \dot{x}(t) , \frac{d}{dt} \phi_0 (t),...,\frac{d}{dt} \phi_{M-1} (t) \right\}$. The derivatives of $\phi_j$s can be obtained by integration by parts: $$\frac{d}{dt} \phi_j (t)= \int _{-\tau}^{0} p_{0,j} (s) \dot{x}_t (s) ds = p_{0,j}(0)x(t)-p_{0,j}(-\tau)x(t-\tau)- \frac{1}{\tau} \int _{-\tau}^{0} P_{0,j}^{\prime} \left( \frac{s+ \tau}{\tau} \right)
x_t (s)ds. \label{bet1}$$ In consistence with (\[gy01\]), (\[e900\]) and (\[gy20\]), it follows from from (\[bet1\]) that $$\begin{aligned}
\frac{d}{dt} \phi_j (t)=x(t)-(-1)^j x(t-\tau)- \sum _{l=0}^{M-1} \zeta _{j,l}^{0} \frac{1}{\tau} \phi _l(t) .\end{aligned}$$ Therefore, $\dot{\widetilde{x}} (t)= \Lambda_M \widetilde{\Phi}_M (t).$ On the other hand, $\widetilde{x} (t) = \Gamma _M \widetilde{\Phi}_M (t),$ thus we obtain $$\begin{aligned}
\dot{\overline{V}}_1 (t) = \widetilde{\Phi}_M (t)^T \Psi _{M}^{1}(\tau) \widetilde{\Phi}_M (t). \label{e21}\end{aligned}$$
The derivative of the first term of $V_2$ is $$\begin{aligned}
\frac{d}{dt} \int _{-\tau}^{0} x_t (s)^T Q_0 x_t (s)ds = x(t)^T Q_0 x(t)-x(t-\tau)^T Q_0 x(t-\tau), \label{e23}\end{aligned}$$ while the derivatives of the terms of $V_2$ corresponding to $j \geq 1$ can be obtained as $$\begin{aligned}
\frac{d}{dt} \int _{-\tau}^{0} \left( \frac{s+\tau}{\tau} \right)^j x_t (s)^T Q_j x_t (s)ds &=
x(t)^T Q_j x(t)-\frac{j}{\tau} \int_{t-\tau}^{t} \left(\frac{s-t+\tau}{\tau} \right)^{j-1} x(s)^T Q_j x(s)ds \nonumber \\
&= x(t)^T Q_j x(t)-\frac{j}{\tau} J_{Q_j , j-1, -\tau , 0} ( x_t ). \label{e24}\end{aligned}$$ Employing Lemma \[lem:21\], we obtain from (\[e24\]) that $$\begin{aligned}
J_{Q_j ,j-1, -\tau , 0} ( x_t ) \geq \tau ^{-1} \Phi _M (t) ^T
\left( \Xi _{j-1} \otimes I \right)^T \mathcal{Q}_{j-1}^{(j)} \left( \Xi _{j-1} \otimes I \right) \Phi _M (t). \label{e25}\end{aligned}$$ It follows from (\[e25\]) that $$\begin{aligned}
&\frac{d}{dt} \int _{-\tau}^{0} \left( \frac{s+\tau}{\tau} \right)^j x_t (s)^T Q_j x_t (s)ds \\ &\hspace{1.5cm} \geq x(t)^T Q_j x(t) - j \frac{1}{\tau} \Phi _M (t)^T
\left( \Xi _{j-1} \otimes I \right)^T \mathcal{Q}_{j-1}^{(j)} \left( \Xi _{j-1} \otimes I \right) \frac{1}{\tau} \Phi_M (t),\end{aligned}$$ which means that $$\begin{aligned}
\dot{\overline{V}}_2 (t) \leq \widetilde{\Phi}_M ^T \Psi _{M,m_1}^{2} \widetilde{\Phi}_M. \label{e26}\end{aligned}$$ Now compute the derivative of $\overline{V}_3(t)$. We obtain $$\begin{aligned}
\dot{\overline{V}}_3 (t)&=& \tau \sum _{j=1}^{m_2} \frac{d}{dt} \int_{-\tau}^{0}\left( \frac{s+\tau}{\tau} \right)^j
\dot{x}_t (s)^T R_j \dot{x}_t (s) ds= \nonumber \\
&=& \tau \sum_{j=1}^{m_2} \left\{
\dot{x} (t)^T R_j \dot{x} (t) - \frac{j}{\tau} J_{j-1} (R_j, \dot{x}_t, -\tau, 0)
\right\} . \label{e27x}\end{aligned}$$ Applying now Lemma \[lem:31\], it follows that $$\begin{aligned}
J_{j-1} (R_j, \dot{x}_t, -\tau, 0) \geq \tau^{-1} \widetilde{\Phi}_M (t)^T
\left( \mathcal{Z} _{j-1} \otimes I \right)^T \mathcal{R}_{j-1}^{(j)} \left( \mathcal{Z} _{j-1} \otimes I \right) \widetilde{\Phi}_M (t),
\label{e27}\end{aligned}$$ where $\mathcal{R}_{j-1}^{(j)}$ is given by (\[gy84\]). From (\[e27x\]) and (\[e27\]) we obtain $$\begin{aligned}
\dot{\overline{V}}_3 (t) \leq \dot{x}(t)^T \left( \tau \sum_{j=1}^{m_2} R_j \right) \dot{x}(t)-
\tau^{-1} \widetilde{\Phi}_M (t)^T \Psi _{M,m_2}^{3,2}(\tau) \widetilde{\Phi}_M (t). \label{e28}\end{aligned}$$ Since $\dot{x} (t)= \mathcal{A} \widetilde{\Phi}_M (t),$ (\[e28\]) implies that $$\begin{aligned}
\dot{\overline{V}}_3 (t) \leq \widetilde{\Phi}_M (t)^T \left(\Psi _{M}^{3,1}(\tau) -
\Psi _{M,m_2}^{3,2}(\tau) \right) \widetilde{\Phi}_M (t). \label{e29}\end{aligned}$$ The statement of the theorem follows from (\[e20\]), (\[e21\]), (\[e26\]) and (\[e29\]) using the standard Lyapunov-Krasovskii Theorem (see e.g. [@frid2014]). $\Box$
If $A_{d_2}=0$ is considered in (\[e1\]), Theorem \[Th:13\] with $M=N, \; m_1=0, \; m_2=1 $ gives back Theorem 5 of [@seur14b] (apart from a multiplier $\tau$ (i.e. $h$) in the derivative of $V_3.$)
*Delay range stability.* An analogous stability result can be proven, if $A_d =0$ and $\tau$ is supposed to be an *unknown constant*, but for which a lower and an upper bound is known, i.e. $\underline{\tau} \leq \tau \leq \overline{\tau}$ for some given $\underline{\tau}$ and $\overline{\tau}.$ One can see that $\Psi_{M,m_1}^{0}(\tau)$ is affine in $\tau ,$ and $\Psi_{M,m_1}^{0}(\tau)>0$ holds true for all $\tau \in [\underline{\tau},\overline{\tau}], $ provided that $\Psi_{M,m_1}^{0}(\overline{\tau})>0.$ Moreover $\Psi_{M,m_1}^{2}$ does not depend on $\tau ,$ while in this case, $\Psi_{M}^{1}(\tau)$ is affine in $\tau ,$ as well. One can modify the definition (\[gy62\]) of $V_3(\dot{x}_t)$ by taking the multiplier $\tau ^2$ in front of the summation instead of $\tau,$ then $\tau ^{-1}$ disappears from $\Psi_{M,m_2}^{3,2}(\tau).$ Apply Schur complements to (\[e20\]) with respect to the new $\Psi_{M}^{3,1}(\tau)$ and a congruence transformation, then we obtain $$\begin{aligned}
\overline{\Psi}(\tau) =
\begin{bmatrix}
\Psi _{M}^{1}(\tau) + \Psi _{M,m_1}^{2} - \Psi _{M,m_2}^{3,2} & \tau \mathcal{A}^{T} \sum_{j=1}^{m} R_j \\
* & -\sum_{j=1}^{m} R_j \\
\end{bmatrix} < 0 . \label{e31}\end{aligned}$$ The matrix valued function $\overline{\Psi}(\tau)$ is affine in $\tau$, which means that it is enough to require the fulfillment of the inequality (\[e31\]) at the endpoints, i.e. the LMIs $\Psi _{M,m_1}^{0}(\overline{\tau})>0,$ $\overline{\Psi}(\underline{\tau}) <0$ and $\overline{\Psi}(\overline{\tau}) <0$ have to hold true.
Hierarchy of the LMI stability conditions
=========================================
This section is devoted to the comparison of the stability conditions obtained in the previous section for different parameters. We observe that parameter $M$ determines the size of matries $P$ and $\widetilde{\mathcal{L}}_0$, the number of columns of $\Xi_j$ and $\mathcal{Z}_k$, while the number of rows of $\Xi_j$ and $\mathcal{Z}_k$ is $\nu_{1,j}$ and $\nu_{2,k}$. The number of matrices $Q_j$s and $R_k$s is $m_1$ and $m_2$. *The aim is to show that the LMI conditions can be arranged into a hierarchy table provided that the parameters are chosen to satisfy the following condition.* $$\begin{aligned}
\begin{array}{lll}
M\geq 1, & m_1 =m, & m_2 = m+1, \\
\nu_{1,j}=
M-j-1, & \nu_{2,j}=\nu_{1,j}+1, & j=0,1,...,m .
\end{array}
\label{H1}\end{aligned}$$ We shall refer to the LMI condition (\[e20\]) with parameters satisfying (\[H1\]) as $\mathcal{L}_{M,m}(\tau)$.
Let the pairs $(M,m)$ and $(\hat{M},\hat{m})$ be given. We will say that *$\mathcal{L}_{\hat{M},\hat{m}} $ outperforms $\mathcal{L}_{M,m} $,* if, for every $\tau$ for which $\mathcal{L}_{M,m} (\tau)$ has a feasible solution, $\mathcal{L}_{\hat{M},\hat{m}} (\tau)$ has a feasible solution, too. This is denoted by $\mathcal{L}_{M,m} \prec \mathcal{L}_{\hat{M},\hat{m}}.$
We will show that the parametric family of $\mathcal{L}_{M,m}$ is ordered according to both parameters.
\[Th:4\] Let the integer parameters satisfy (\[H1\]). Then $$\begin{aligned}
\mathcal{L}_{M,m} &\prec& \mathcal{L}_{M+1,m}, \label{H10}\\
\mathcal{L}_{M,m} &\prec& \mathcal{L}_{M,m+1}. \label{H11}\end{aligned}$$
**Proof.** *Part 1.* First we show (\[H10\]). Let matrices $P$, $Q_{0},...,Q_{m-1}$ and $R_{1},...,R_{m}$ denote a feasible solution of $\mathcal{L}_{M,m} (\tau)$ for some fixed $\tau .$ We seek the solution of $\mathcal{L}_{M+1,m} (\tau)$ in the form of $$\label{H20}
\hat{P}= \begin{bmatrix}
P & 0 \\
0 & \varepsilon I \\
\end{bmatrix} ,
\hspace{2mm} \hat{Q}_{i} = Q_{i} , \; (i=0,...,m-1), \hspace{2mm} \hat{ R}_{j} = R_{j}, \; (j=1,...,m )$$ for some positive constant $\varepsilon .$ In what follows, we shall denote matrices that belong to $(M+1,m)$ analogously by putting “hat” over them.
We show first that inequality $\Psi_{M,m}^{0} (\tau)>0 $ implies $ \Psi_{M+1,m}^{0} (\tau)$ independently of $\varepsilon>0$. In fact, matrix $\hat{\Xi}_{j}$ is obtained by adding a new row and a new column to $\Xi_{j}$, i.e $$\hat{\Xi}_{j} = \begin{bmatrix}
\Xi_{j} & 0 \\
\underline{\xi}_{M-j}^{j,1} & \xi_{M-j,M}^{j} \\
\end{bmatrix} ,$$ where $\underline{\xi}_{M-j}^{j}$ is partitioned as $\underline{\xi}_{M-j}^{j} = \left( \underline{\xi}_{M-j}^{j,1}, \xi_{M-j,M}^{j} \right).$ Thanks to the structure of the matrices, we obtain by standard algebra $$\begin{aligned}
&& \hspace{-0.8cm}
\Psi_{M+1,m}^{0}(\tau)=
\begin{bmatrix}
\Psi_{M,m}^{0} (\tau)& 0 \\
0 & \varepsilon I \\
\end{bmatrix}
+ \mbox{PSDTs,}
\label{e32}\end{aligned}$$ where PSDTs stands for positive semidefinite terms, therefore the statement follows.
Next we express $\Psi _{M+1}^{1}(\tau)$ by $\Psi _{M}^{1} (\tau).$ By the definition of $\Psi _{M+1}^{1}(\tau)$, we have $$\begin{aligned}
\Psi_{M+1}^{1}(\tau) &=& \Gamma_{M+1}^{T} \hat{P} \Lambda_{M+1} + \Lambda_{M+1}^{T} \hat{P} \Gamma_{M+1}, \hspace{1cm} \mbox{with} \\
\Gamma _{M+1} &=& \mbox{diag} \left\{ \Gamma_M , \tau I \right\} , \;
\Lambda_{M+1} =
\begin{bmatrix}
\Lambda_{M} & 0 \\
\underline{\zeta}_{M}^{0,1} \otimes I & \underline{\zeta}_{M}^{0,2} I \\
\end{bmatrix} , \\\end{aligned}$$ where $\underline{\zeta}_{M}^{0} $ is again partitioned as $\underline{\zeta}_{M}^{0} = \left(\underline{\zeta}_{M}^{0,1}, \underline{\zeta}_{M}^{0,2}\right)$ with $\underline{\zeta}_{M}^{0,2}= -\zeta_{M,M}^{0}.$ Using (\[H20\]), by standard algebra we obtain $$\Psi_{M+1}^{1}(\tau) =
\begin{bmatrix}
\Psi_{M}^{1}(\tau) & 0 \\
0 & 0 \\
\end{bmatrix}
+ \varepsilon \tau \left(
\begin{bmatrix}
0 & 0 \\
\underline{\zeta}_{M}^{0,1} \otimes I & \underline{\zeta}_{M}^{0,2} I \\
\end{bmatrix} +
\begin{bmatrix}
0 & \left( \underline{\zeta}_{M}^{0,1} \otimes I \right)^T \\
0 & \underline{\zeta}_{M}^{0,2} I \\
\end{bmatrix}
\right) . \label{e34}$$
Express now $\Psi _{M+1,m}^{2}$ by $\Psi _{M,m}^{2} .$ Since $$\hat{\Xi}_{j-1}=
\begin{bmatrix}
{\Xi}_{j-1} & 0 \\
\underline{\xi}_{M+1-j}^{j-1,1} & \xi _{M+1-j,M}^{j-1} \\
\end{bmatrix} , \;
\hat{\mathcal{Q}}_{j-1}^{(j)} = \mbox{diag} \left\{ {\mathcal{Q}}_{j-1}^{(j)} ,c_2(M,j)Q_j \right\} ,$$ where $c_2(M,j)=2M-j+2$, we obtain $$\begin{aligned}
\left( \hat{\Xi}_{j-1} \otimes I \right)^T \hat{\mathcal{Q}}_{j-1}^{(j)} \left( \hat{ \Xi} _{j-1} \otimes I \right)& =&
\begin{bmatrix}
\left( \Xi_{j-1} \otimes I \right)^T \mathcal{Q}_{j-1}^{(j)}
\left( \Xi_{j-1} \otimes I \right) & 0 \\
0 & 0 \\
\end{bmatrix} + \mbox{PSDTs,} $$ therefore $$\begin{aligned}
&&\Psi _{M+1,m}^2(\tau) \leq
\begin{bmatrix}
\Psi _{M,m}^2(\tau) & 0 \\
0 & 0 \\
\end{bmatrix} .
\label{e35}\end{aligned}$$ Express now $\Psi _{M+1,m}^{3,1}(\tau)$ by $\Psi _{M,m}^{3,1}(\tau) .$ Since $\hat{\mathcal{A}}= \left( \mathcal{A},0 \right),$ we obtain $$\Psi _{M+1,m}^{3,1}(\tau) =
\begin{bmatrix}
\Psi _{M,m}^{3,1}(\tau) & 0 \\
0 & 0 \\
\end{bmatrix} . \label{e36}$$ Express $\Psi _{M+1,m}^{3,2}(\tau)$ by $\Psi _{M,m}^{3,2}(\tau) .$ Since $$\hat{\mathcal{Z}}_{j-1}^{(j)}=
\begin{bmatrix}
\mathcal{Z}_{j-1}^{(j)} & 0 \\
\underline{\zeta}_{M+1-j}^{j-1,1} & \underline{\zeta}_{M+1-j}^{j-1,2} \end{bmatrix} , \;
\hat{\mathcal{R}}_{j-1}^{(j)} = \mbox{diag} \left\{ \mathcal{R}_{j-1}^{(j)} ,c_2(M,j) R_j \right\} ,$$ where $\underline{\zeta}_{M+1-j}^{j-1}= \left[ \underline{\zeta}_{M+1-j}^{j-1,1},\underline{\zeta}_{M+1-j}^{j-1,2}\right]$ and $\underline{\zeta}_{M+1-j}^{j-1,2}=-{\zeta}_{M+1-j,M}^{j-1}$, we obtain $$\begin{aligned}
\Psi _{M+1,m}^{3,2}(\tau) &=& \frac{1}{\tau}\sum_{j=1}^{m} j \left( \hat{\mathcal{Z}}_{j-1}^{(j)} \otimes I \right)^T \hat{\mathcal{R}}_{j-1}^{(j)} \left( \hat{\mathcal{Z}}_{j-1}^{(j)} \otimes I \right)= \nonumber \\
&\geq&\frac{1}{\tau} \sum_{j=1}^{m} j
\begin{bmatrix}
\left( \mathcal{Z}_{j-1}^{(j)} \otimes I \right)^T \mathcal{R}_{j-1}^{(j)}
\left( \mathcal{Z}_{j-1}^{(j)} \otimes I \right) & 0 \\
0 & 0 \\
\end{bmatrix} \nonumber\\&&+\frac{c_2(M,1)}{\tau}
\begin{bmatrix}
0 & \left( \underline{\zeta}_{M}^{0,1} \otimes I \right)^T \\
0 & -\zeta_{M,M}^{0} I \\
\end{bmatrix}
\begin{bmatrix}
0 & 0 \\
0&R_1 \\
\end{bmatrix}
\begin{bmatrix}
0 & \left( \underline{\zeta}_{M}^{0,1} \otimes I \right)^T \\
0 & -\zeta_{M,M}^{0} I \\
\end{bmatrix}^T , \label{e37}\end{aligned}$$ where we omitted several positive semidefinite terms on the right hand side of (\[e37\]).
Finally we show that $\overline{\Psi}_{M+1}(\tau)= \Psi_{M+1}^{1}(\tau)+ \Psi_{M+1,m}^{2}+ \Psi_{M+1}^{3,1}(\tau)
- \Psi_{M+1}^{3,2} (\tau)< 0.$ Applying (\[e34\])-(\[e37\]) we obtain $$\begin{aligned}
\overline{\Psi}_{M+1}(\tau) &\leq&
\begin{bmatrix}
I & \left( \underline{\zeta}_{M}^{0,1} \otimes I \right)^T \\
0 & -\zeta_{M,M}^{0} I \\
\end{bmatrix}
\begin{bmatrix}
\overline{\Psi}_{M}(\tau) & 0 \\
0 & -\frac{2M+1}{\tau}R_1 \\
\end{bmatrix}
\begin{bmatrix}
I & 0 \\
\left( \underline{\zeta}_{M}^{0,1} \otimes I \right) & -\zeta_{M,M}^{0} I \\
\end{bmatrix}
\nonumber \\
& +& \varepsilon \tau
\begin{bmatrix}
0 & \left( \underline{\zeta}_{M}^{0,1} \otimes I \right)^T \\
\underline{\zeta}_{M}^{0,1} \otimes I & 2 \underline{\zeta}_{M}^{0,2}I \\
\end{bmatrix}
\label{e38}\end{aligned}$$ Since $\mbox{diag} \left\{ \overline{\Psi}_{M}(\tau), -\frac{2M+1}{\tau}R_1 \right\} <0,$ and matrix $\begin{bmatrix}
I & 0 \\
\left( \underline{\zeta}_{M}^{0,1} \otimes I \right) & -\zeta_{M,M}^{0} I \\
\end{bmatrix}$ is non-singular, there exists a positive constant $\nu_1$ such that $$\begin{bmatrix}
I & \left( \underline{\zeta}_{M}^{0,1} \otimes I \right)^T \\
0 & -\zeta_{M,M}^{0} I \\
\end{bmatrix}
\begin{bmatrix}
\overline{\Psi}_{M}(\tau) & 0 \\
0 & -\frac{2M+1}{\tau}R_1 \\
\end{bmatrix}
\begin{bmatrix}
I & 0 \\
\left( \underline{\zeta}_{M}^{0,1} \otimes I \right) & -\zeta_{M,M}^{0} I \\
\end{bmatrix}
< -\nu_1 I,$$ and constant $\nu_2$ such that $$\tau
\begin{bmatrix}
0 & \left( \underline{\xi}_{M}^{0,1} \otimes I \right)^T \\
\underline{\xi}_{M}^{0,1} \otimes I & 2 \underline{\xi}_{M}^{0,2}I \\
\end{bmatrix} < \nu_2 I.$$ If $\varepsilon \nu_2 < \nu_1,$ then $\overline{\Psi}_{M+1}(\tau)<0$ holds true.
*Part 2*. Secondly we show that $\mathcal{L}_{M,m} \prec \mathcal{L}_{M,m+1}.$ First we show the positivity of $\Psi_{M,m+1}^{0}(\tau).$ Suppose that $\Psi_{M,m}^{0}(\tau) >0$ with the choice of $P$, $Q_0,...,Q_{m-1}.$ We seek matrix $Q_m $ as $Q_m = \varepsilon I$ with $\varepsilon>0.$ Then we obtain $$\begin{aligned}
\Psi_{M,m+1}^{0}(\tau)
&=& \Psi_{M,m}^{0}(\tau) +
\frac{\varepsilon}{\tau}
\mbox{diag} \left\{ 0,\left( \Xi_{m} \otimes I \right)^T \mathcal{D}_{M,m} \left( \Xi_{m} \otimes I \right) \right\}, \label{e40} \\
\mathcal{D}_{M,m}&=&\mbox{diag}\left\{ (m+1)I,(m+3)I,...,\left(2(M-m-1)+m+1\right)I \right\}.\end{aligned}$$ The first term of the right hand side of (\[e40\]) is positive, the second term is non-negative, therefore $\Psi_{M,m+1}^{0}(\tau)>0 $ for any $\varepsilon>0.$
Next we show that $\overline{\Psi}_{M,m+1}(\tau)<0$ has a feasible solution, provided that $\overline{\Psi}_{M,m}(\tau)<0$ has. Suppose that $\overline{\Psi}_{M,m} (\tau)<-\nu _3 I$ with $P$, $Q_0,...,Q_{m-1}$ and $R_0,...,R_{m-1}$. We seek matrices $Q_m = \varepsilon I$ and $R_{m+1}= \varepsilon I$, where $\varepsilon >0$ has to be chosen. Matrix $\Psi_M^1(\tau)$ is unchanged.
Denote $E_1=\begin{bmatrix}
I & 0 & \ldots 0 \\
\end{bmatrix} \in \mathbf{R}^{n_x\times n_x(M+2)},$ then $$\Psi _{M,m+1}^2 (\tau)= \Psi _{M,m}^2(\tau) + E_1^T Q_m E_1 -
(m+1)\left( \Xi_{m} \otimes I \right)^T \mathcal{Q}_m \left( \Xi_{m} \otimes I \right)
\label{e41}$$ holds true. On the one hand, $$\Psi _{M,m+1}^{3,1}(\tau) = \tau \mathcal{A}^T \sum _{j=1}^{m} R_j \mathcal{A}+\tau \mathcal{A}^T R_{m+1} \mathcal{A}=
\Psi _{M,m}^{3,1}(\tau)+\varepsilon \tau \mathcal{A}^T \mathcal{A}, \label{e42}$$ while $$\Psi _{M,m+1}^{3,2}(\tau) = \Psi _{M,m}^{3,2}(\tau) + \frac{m+1}{\tau}\varepsilon
\left( \mathcal{Z}_{m+1} \otimes I \right)^T \mathcal{D}_{M,m+1} \left( \mathcal{Z}_{m+1} \otimes I \right). \label{e43}$$ It follows from (\[e42\])-(\[e43\]) that $$\overline{\Psi}_{M,m+1} (\tau) = \overline{\Psi}_{M,m} (\tau) + \varepsilon \Omega_{M,m}(\tau)$$
with $$\begin{aligned}
\Omega_{M,m}(\tau)&=& E_1' E_1 -
(m+1)\left( \Xi_{m} \otimes I \right)^T \mathcal{D}_{M,m} \left( \Xi_{m} \otimes I \right) \\
&&+\tau \mathcal{A}^T \mathcal{A}-\frac{m+1}{\tau}
\left( \mathcal{Z}_{m} \otimes I \right)^T \mathcal{D}_{M,m+1} \left( \mathcal{Z}_{m} \otimes I \right)\end{aligned}$$ Then there exists a constant $\nu_4$ such that $\Omega_{M,m}(\tau) \leq \nu_4 I.$ If $\varepsilon >0$ is small enough to satisfy inequality $\varepsilon \nu_4 <\nu_3 ,$ then $\Psi_{M,m+1} (\tau)$ is negative. $\Box$
Numerical examples
==================
In this section, we apply the proposed method to three benchmark examples that have been extensively used in the literature to compare the results. The computations have been performed by using YALMIP [@yalmip] together with MATLAB.
Some remarks on the implementation
----------------------------------
Assume that the integer parameters are chosed according to (\[H1\]). In order to implement LMIs (\[e20\]) with (\[gy55\])-(\[gy84\]), matrices $\widetilde{\mathcal{L}}_0, $ $\Xi _i$ and $\mathcal{Z}_j$ are to be produced. This matrices can be computed employing the generalized Rodrigues formula (\[e3\]) as follows. Let $X_K =\left(1, x, \ldots, x^K \right)^T $ and $\Pi_{m,K}(x)=\left(P_{m,0}(x), P_{m,1}(x), \ldots, P_{m,K}(x) \right)^T .$ Then $\Pi_{m,K}(x)=G(m,K)X_K$ and $X_K= G(0,M-1)^{-1}\Pi_{0,K}(x),$ where $G(m,K) \in \mathbf{R}^{(K+1)\times (K+1)}$ with elements $G(m,K)_{1,1}=1,$ $$G(m,K)_{l+1,k+1}= (-1)^{l+k}\prod _{j=0}^{k-1}\frac{l-j}{k-j}\prod _{i=1}^{l}\frac{m+k+i}{i}, \;
\mbox{ if } \hspace{3mm} \begin{array}{l}
l=1,\ldots K-1, \\
K=0,\ldots,l,
\end{array}$$ and $G(m,K)_{l,k}=0,$ if $k>l.$ By taking into account (\[e9\]), (\[gy20\]) and (\[H1\]) we can see that $$\begin{aligned}
\Xi_m &=& G(m,-\nu _m)
\begin{bmatrix}
0_{\nu _m+1,m} & I_{\nu _m+1} \\
\end{bmatrix} G(0,M-1)^{-1}.\end{aligned}$$ Further, by taking into account (\[gy100\]), (\[gy110\]),(\[gy50\]) and (\[H1\]) we can see that $$\begin{aligned}
\widetilde{\mathcal{L}}_0 &=&
\left[
\begin{array}{ccc}
\underline{\ell}_{M-1}^{(1)} & \underline{\ell}_{M-1}^{(2)} & -L_0
\end{array}
\right]
\\
&& \hspace{0.1cm} L_0 =\begin{bmatrix} I_{M} \; 0\end{bmatrix} G(0,M) diag\left\{0,1,\ldots,M\right\}
\begin{bmatrix} 0 \\ I_{M-1}\end{bmatrix} G(0,M-1)^{-1}, \\
\mathcal{Z}_0 &=&
\left[
\begin{array}{ccc}
\underline{\ell}_{\nu_0}^{(1)} & \underline{\ell}_{\nu_0}^{(2)} & -Z_0
\end{array}
\right]
\\
&& \hspace{0.1cm} Z_0 = G(0,M-1) diag\left\{0,1,\ldots,M\right\}
\begin{bmatrix} 0 \\ I_{M-1}\end{bmatrix} G(0,M-1)^{-1}, $$ $$\begin{aligned}
\mathcal{Z}_m &=& \left[
\begin{array}{ccc}
\underline{\ell}_{\nu_m}^{(1)} & \underline{0}_{\nu_m} &-Z_m
\end{array}
\right] \\
&& \hspace{0.1cm} Z_m=
G(m,\nu _m+1)
\left[
\begin{array}{cc}
0_{\nu_m+1, m-1} & D_{m,\nu _m}
\end{array}
\right]
G(0,m+\nu_m)^{-1},\end{aligned}$$ where the vectors $\underline{\ell}_{k}^{(1)}, \underline{\ell}_{^k}^{(2)}, \underline{0}_{k} \in \mathbf{R}^{k+1}, $ are defined by $\underline{\ell}_{k}^{(1)}=(1, \, \ldots, \, 1)^T,$ $\underline{0}_{k}=(0,\ldots,0)^T,$ $\underline{\ell}_{^k}^{(2)}=(-1, 1, \ldots, \pm 1)^T$ and $D_{m,\nu _m}=\mbox{diag}\left\{m, \ldots, m+\nu _m+1 \right\}.$
Numerical experiments
---------------------
Consider system (\[x1\]) with coefficient matrices listed in Table 1.
\[Tab:1\]
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Example $A$ $A_{d_1}$ $A_{d_2}$ analytical bounds
--------- ------------------------------------------------------ -------------------------------------------------- --------------------------------------------------- ----------------------------------------
1 $\begin{bmatrix}-2 & 0 \\ 0 & -0.9 \end{bmatrix}$ $\begin{bmatrix}-1 & 0 \\-1 & -1 \end{bmatrix}$ $\begin{bmatrix} 0 & 0 \\0 & 0 \end{bmatrix}$ $\begin{array}{c}
\underline{\tau}=0 \\
\overline{\tau}\sim 6.17258
\end{array}$
2 $\begin{bmatrix}0.2 & 0 \\ 0.2 & 0.1 \end{bmatrix}$ $\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}$ $\begin{bmatrix} -1 & 0 \\-1 & -1 \end{bmatrix}$ $\begin{array}{c}
\underline{\tau}\sim0.2 \\
\overline{\tau}\sim 2.04
\end{array}$
3 $\begin{bmatrix}0 & 1 \\ -2 & 0.1 \end{bmatrix}$ $\begin{bmatrix}0 & 0 \\1 & 0 \end{bmatrix}$ $\begin{bmatrix} 0 & 0 \\0 & 0 \end{bmatrix}$ $\begin{array}{c}
\underline{\tau}\sim 0.1002 \\
\overline{\tau}\sim 1.7178
\end{array}$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: [Systems used as illustrative examples]{}
\
-2mm
*Example 1* is considered in numerous papers, among others, in [@seur14b; @kim16; @seur14], where extensive comparisons with previously reported results are given. The results obtained by Theorem \[Th:13\] are given in Table 2. The results obtained with $m=1,$ $M=1;2;3$ coincide with that of [@seur14b], which are the best previously reported results we are aware of. We note that the same bounds of $\tau$ are obtained for $m=2$ and $m=3$ with the same values of $M,$ however an improvement is resulted in in the case of $m=4, \ M=3$ compared to $m=3, \ M=4.$ NoDV denotes in all tables the number of decision variables.
\[Tab:2\]
$m$ $1$ $1$ $1$ $1$ 4
------------------- ----------- ----------- ----------- ----------- -----------
$M$ $1$ $2$ $3$ $4$ $3$
$\overline{\tau}$ $6.05932$ $6.16893$ $6.17250$ $6.17258$ $6.17258$
NoDV $16$ $32$ $48$ $61$ $72$
: [Delay bounds for Example 1 obtained by Theorem \[Th:13\]]{}
\
-2mm
*Example 2* is considered in many papers, among others, in [@park15; @S-GAut13; @sga15], where extensive comparisons with previously reported results are given. In [@S-GAut13], a delay bounding interval $[0.200, \ 1.877]$ was obtained with $16$ decision variables, while the authors of [@park15] derived the delay bounding interval $[0.2000, \ 1.9504]$ from Theorem 1 with $59$ decision variables. In [@sga15], the lower bound of the delay was found to be $0.2001$, while by Theorem \[Th:13\], we obtained the lower bound $0.20001.$ The upper bounds reported in [@sga15] and obtained by Theorem \[Th:13\] are given in Table 3. The values of $N$ in [@sga15] and $M$ in the present paper are related as $N=M+1.$ We note that the same upper bounds were obtained for different values of $m$ for a given value of $M.$
\[Tab:3\]
Method $ M$ $ 1$ $ 2$ $3$ $ 4$ $5$
------------------ ------------------- ----------- ----------------------- ---------------------- ---------------------- ----------------------
[@sga15] $\overline{\tau}$ $-$ $1.58 \hspace{3.3mm}$ $1.83\hspace{3.3mm}$ $1.95\hspace{3.3mm}$ $2.02\hspace{3.3mm}$
Theorem 1, $m=1$ $\overline{\tau}$ $1.9419 $ $2.0395$ $2.0412$ $2.0412$ $2.0412$
NoDV $16$ $32$ $48$ $61$ $84$
: [Delay upper bounds for Example 2 ]{}
\
-2mm
*Example 3* is is also widely used for comparing the effectiveness of different methods. Here we shall mention a recently published work in [@park15], where comparisons with previously reported results are given, as well (see also [@trinh15]). The results obtained by [@park15] and by Theorem \[Th:13\] are given in Table 4. We note that the same upper bounds were obtained for different values of $m$ for a given value of $M$ for this example, too.
\[Tab:4\]
Method $ $ $M$ $\underline{\tau}$ $\overline{\tau}$ NoDV
--------------------- ----- ----- ------------------------- ---------------------- ------
[@park15] Theorem 1 $ $ $ $ $ 0.1002\hspace{2mm}$ $1.5954\hspace{2mm}$ $59$
Theorem 1, $(m=1)$ $ $ $1$ $ 0.10055$ $1.5405\hspace{2mm}$ $16$
$ $ $2$ $0.10018 $ $1.7122\hspace{2mm}$ $32$
$ $ $3$ $0.10017$ $1.71799$ $48$
: [Delay bounds for Example 3]{}
\
-2mm
Discussion
----------
It can be seen that, in these examples, Theorem 1 yields better delay bounds than previously published methods except of Theorem 5 of [@seur14b] which is equivalent to Theorem 1, if it is applied with $m=1.$ It is worth noting that the better results are obtained with much smaller number of decision variables.
In these and in several other examples from the literature, on which we tested our approach, we observed that the improvement of the delay estimation is primarily due to the increase of the dimension of the extended state variable together with the improved lower bounds of Lemma \[lem:21\] and Lemma \[lem:31\]. This does not contradict to the reported improvements in the case of the application of triple, etc. integral terms in the LKF, since the applied lower estimations of the integrals of quadratic terms lead to the introduction of some extended state variables with increased dimension, as well. We emphasize that this remark is limited to the investigated examples, and the observed behavior may have the reason that the analytical bounds were rapidly reached up to 4-6 digits. Moreover, we note similarly to [@seur14b] that the formulated result does not establish any convergence to the analytical bounds.
Conclusion
==========
In this paper, new multiple integral inequalities are derived based on certain hypergeometric-type orthogonal polynomials. These inequalities are similar to that of [@zha15], and they comprise the famous Jensen’s and Wirtinger’s inequalities, as well as the recently presented Bessel-Legendre inequalities of [@seur14b] and the Wirtinger-based multiple-integral inequalities of [@leejfi15; @park15]. Applying the obtained inequalities, a set of sufficient LMI stability conditions for linear continuous-time delay systems are derived. It was proven that these LMI conditions could be arranged into a bidirectional hierarchy establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods. Numerical examples confirm that the proposed method enhances the tolerable delay bounds.
[99]{}
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|
{
"pile_set_name": "ArXiv"
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